Temporal evolution of magmatic processes in a subvolcanic plumbing system in a continental region: constraints from geochemistry of the late Neoproterozoic Wadi Dib alkaline ring complex in the Eastern Desert, Egypt

Eman SAAD; Kazuhito OZAWA; Takeshi KURITANI; Ali A. KHUDEIR

doi:10.2465/jmps.240110

Abstract

We examined trace element compositions and Sr, Nd, Pb isotope ratios of the Wadi Dib ring complex (WDRC), one of the alkaline ring complexes in the Eastern Desert of Egypt, to better understand temporal evolution of an alkaline magma in a small crustal magma body. The ring complex consists of plutonic, volcanic, and dike units and represents a continental subvolcanic plumbing system developed above a magma body. The whole-rock Rb-Sr isochron age was newly determined as 586.8 ± 10.1 Ma with initial ⁸⁷Sr/⁸⁶Sr = 0.70333 ± 0.00011 by critically evaluating isotopic heterogeneity at the time of solidification and excluding samples unsuitable for the age estimation, which have information on open magmatic processes involving isotopically distinct exotic materials. The consistent Rb-Sr isochron age and the systematic major and trace element variations show that trachybasalts of the dike unit represent a parental magma of the WDRC. We developed a new approach to quantify open magmatic processes governing temporal evolution of the magma body by combining geochemical data of multiple samples to extract model parameters. We adopted an assimilation and fractional crystallization (AFC) model to rocks of the plutonic units with SiO₂ > 63 wt%, whose radiogenic isotopic ratios requires involvement of exotic materials and successfully estimated the AFC parameters including major and trace element compositions and Nd isotope ratios of the exotic melts, which are geochemically similar to the country rocks. We adopted a boundary layer fractionation (BLF) model to rocks of the volcanic and plutonic units with SiO₂ < ∼ 63 wt% without evidence for assimilation and successfully estimated the BLF parameters including major and trace element compositions of a ‘precursor magma’ initially emplaced into the magma body. Diverse rock types of the WDRC were derived from the precursor magma by open-system fractional crystallization with or without assimilation. The temporal evolution of the WDRC constrained by the modeling results features: (1) BLF operated in the early stage of the complex evolution was driven by formation of a fractionated melt in the sidewall boundary layer, which was transported to the roof zone to form magmas erupted as the volcanic unit by mixing with the precursor magma and (2) AFC became extensive in the later stage, during which the exotic melt was derived from the bottom boundary layer by melting of the country rocks sunk from the carapace of the magma body by stoping in the earlier stage. The ratio of extent of assimilation and crystallization (Ma/Mc) increases from ~ 0 to 1-2 with time and the overall Ma/Mc integrated over the entire evolution of the WDRC is estimated to be ∼ 0.2, which indicates a critical role of heat generated in the earlier BLF stage to induce the later extensive assimilation.

INTRODUCTION

Magma plumbing system or magma storage system presumed to exist beneath volcanoes play an important role not only in controlling volcanic processes but also in creating diverse magma compositions through crystallization with or without assimilation of crustal materials (Cashman et al., 2017; Magee et al., 2018; Dufek et al., 2022). Magma chamber occupies the central part of such magma plumbing system in which the dominant mass of magmas with various melt-crystal-vapor ratios exists. It is an open system and maintained for a certain period by a thermal and material balance of supply from the depth, tapping to the surface, and in situ solidification and fractionation by heat loss. Igneous intrusions exposed on the surface have been regarded as part of ancient magma chambers existed in a shallow crustal level and used to extract information on crustal magma chamber processes (e.g., Marsh, 1989; Cawthorn, 1996; Marsh, 2000; Harper et al., 2004; Charlier et al., 2015).

There is, however, an essential difference between the magma chambers existing beneath volcanoes and igneous intrusions. Most intrusions represent part of cooled and solidified entity of an open magma plumbing system. The igneous intrusions underwent prolonged and complex cooling and solidification histories with involvement of magma supply, crustal assimilation, tapping, and ascent/emplacement before they were finally solidified. Moreover, the exposure is virtually a two-dimensional cross section of solidified mass having three-dimensional structures, from which a mass of magma once existed in the chamber could have been partially lost due to volcanism. For better understanding the magma chamber processes, therefore, it is imperative to know: (1) magma supply history, (2) tapping history, (3) assimilation history, (4) emplacement history, and (5) which section of a magma plumbing system the exposure represents. If we know (1)-(5), we may be able to extract accurate and useful information related to a magma plumbing system from igneous intrusions.

One of the useful igneous intrusions is sills, which have advantage in having one-dimensional structure and static nature. Many sills are supposed to have formed in a shallow rigid crust through a single intrusion pulse without tapping and assimilation, and the entire intrusion is accessible ideally by examining only one profile crossing the sills. Because of such simplicity, one-dimensional heat and mass transportation can be assumed to model solidification and fractionation of the sills (Shirley, 1987; Hoshide et al., 2006; Simura and Ozawa, 2006; Simura and Ozawa, 2011; Floess et al., 2019). However, magma chamber responsible for volcanism as defined above have essentially a three-dimensional morphology and structure, in contrast to the one-dimensional nature of sills, in which sidewalls do not play an important role. In this regard, one of the most promising igneous intrusions is ring complexes, which represent a nearly horizontal cross section of an ancient three-dimensional magma plumbing system.

This study applies this concept to one of the alkaline ring complexes in the Eastern Desert of Egypt, Wadi Dib ring complex (WDRC), which represent an association of igneous intrusions and volcanic materials (volcano-plutonic complex). The complex is shown to have formed by one time magma supply followed by cooling with some tapping (Saad et al., 2023) and is very useful to elucidate magma chamber processes by exploiting the volcano-plutonic nature. In this paper, we present geochemical data of the WDRC to clarify (1) parental magma composition, (2) fractionation processes, (3) assimilation processes, and (4) timing of events (2) and (3). We developed new modeling approaches utilizing volcanic and plutonic samples and their trace and major element compositions for (2) as well as isotope ratios for (3) to constrain respective model parameters. The (1)-(3) are combined with geology and petrography of the complex and constraints from the phase relation in the ‘Petrogeny’s Residua System’ (Bowen, 1937), and (4) was addressed to infer temporal evolution of the ring complex. Finally, general implications for role of sidewall boundary layer in boundary layer fractionation, role of bottom boundary layer in assimilation and fractional crystallization, and major element fractionation of critically undersaturated alkaline magmas are presented.

GEOLOGICAL BACKGROUND AND PREVIOUS STUDIES

About 20 alkaline ring complexes occur in the Eastern Desert and the Sinai Peninsula, which is northern part of the ring complex distribution area extending Egypt, Sudan, South Sudan, and Ethiopia (Vail, 1989). They occur in the Proterozoic rocks of gneisses, metasediments, volcanics, and granitoids (El Ramly and Hussein, 1985). The ages of the ring complexes range from latest Neoproterozoic (e.g., 554 Ma Wadi Dib) to Late Cretaceous (e.g., 89 Ma Abu Khruq) (Serencsits et al., 1979). The WDRC is one of the oldest ring complexes in the Eastern Desert of Egypt. The WDRC is located in the North Eastern Desert, the basement rocks of which consists dominantly of late Pan-African granitoids associated with the Dokhan Volcanics and the Hammamat volcano-sedimentary sequence (El Gaby et al., 1990). The WDRC was dated by Frisch (1982), who obtained a Rb-Sr whole-rock isochron age of 578 ± 16 Ma (2σ) with an initial ⁸⁷Sr/⁸⁶Sr ratio of 0.7048 ± 0.0010. Frisch and Abdel-Rahman (1999) studied geology, geochemistry, and mineralogy of the complex and inferred that the WDRC formed by fractional crystallization of a primary alkaline oceanic-island type basalt formed by partial melting of the upper mantle. These authors noted a role of crustal assimilation only for the late-stage extensively fractionated magmas. Recently, Saad et al. (2023) conducted further detailed study on geology, petrography, and whole-rock major element chemical compositions of the WDRC, the results of which are listed in Table 1 and are briefly summarized in the next section.

Table 1. Principal features of the Wadi Dib ring complex based on study of Saad et al. (2023) on its geology, petrography, and major element composition

Lithological division			Formation order	Exposure area (%)	Major rock types	Crystal morphology and size variation types	Modal abundance (%)			SiO₂ content (wt%)
Lithological division			Formation order	Exposure area (%)	Major rock types	Crystal morphology and size variation types	Olivine	Quartz	Cpx	SiO₂ content (wt%)
Dike unit			3	<1	Trachybasalt, basaltic trachyandesite, trachyte, and rhyolite	Aphyric, porphyriotic	As phenoc in trachyb	As phenoc in rhyolite	As mphenoc in trachyb	50-75
Plutonic unit	Granitic core		2-4	11	Syenogranite	Idiomorphic/micrographic, porphyritic/sub-porphyritic	0	20-23	0	70-72
	Inner ring 2*		2-3	7	Quartz syenite	Idiomorphic, sub-porphyroitic/porphyritic	0	5-11	0	∼ 66
	Inner ring 1		2-2	12	Quartz-bearing syenite	Idiomorphic, equisized/subporphyritic	0	1-5	0.1-2	∼ 64
	Outer ring (OR)	OR intrusive member	2-2 - 2-4	55 †	Alkali feldspar syenite, quartz alkali feldspar syenite, quartz syenite	Idiomorphic, porphyrotic/sub/ porphyritic/equisized	0-3 ‡	0-20 ‡	0-7	60-73
	Outer ring (OR)	Main lithology	2-1	55 †	Sodalite-bearing alkali feldspar syenite, sodalite-bearing syenite, syenite sensu stricto, monzonite	Hipidiomorphic, bimodal/equisized	0	0	0.1-0.6	60-63
Volcanic unit			1	1	Pyroclastic rocks of trachyte, trachyandesite	Aphyric, porphyritic	0	0	<0.1, phenoc, groundmass	55-62

*, Minor dikes belonging to the inner ring 2 intrusive member were identified; †, exposure % of the outer ring intrusive member was not accurately estimated but is less than a few % of the outer ring exposure area; ‡, olivine and quartz are mutually exclusive excepting fayalite and quartz in a quartz alkali feldspar syenite; trachyb, trachybasalt; phenoc, phenocryst; mphenoc, microphenocryst; Cpx, clinopyroxene.

SUMMARY OF GEOLOGY, PETROGRAPHY, AND MAJOR ELEMENT COMPOSITIONS

Geology and petrography

The WDRC occurs as circular mass, about 2 km in diameter (Fig. 1). It is in sharp contact with the country rocks consisting mostly of late Pan-African granodiorite-granite (Francis, 1972; Ghanem et al., 1973; Stern and Hedge, 1985), The contacts are nearly vertical or dip steeply away from the core of the complex. The WDRC consists dominantly of plutonic and volcanic units, which are intruded by volumetrically minor dike units (Table 1).

Figure 1. Simplified geological map of the WDRC, modified after Frisch and Abdel-Rahman (1999) and Saad et al. (2023). Sample localities are shown by solid circles with a number after the letter D. For localities D59 and D61, sampling locations are shown with capital letters. The inset map shows the location of the WDRC in the northern Eastern Desert of Egypt, in which patterned area is Neoproterozoic exposure. The Pan-African rocks is mostly granitoids with subordinate distribution of the Dokhan volcanics. See Table 1 for characteristics of each lithology.

The plutonic unit is predominant in the WDRC and shows multiple circular ring structure consisting of an outer ring, an inner ring 1, and an inner ring 2 from the margin to the center, which is occupied by a granitic core (Fig. 1; Table 1). The main lithology of the outer ring consists of sodalite-bearing alkali feldspar syenite, sodalite-bearing syenite, syenite sensu stricto, and monzonite. They are characterized by hypidiomorphic texture and the absence of quartz. Various porphyritic rocks with a fine-grained matrix <1 mm in grain size occur as dikes in the outer ring. They show a large lithological variation, such as alkali feldspar syenite, quartz alkali feldspar syenite, and quartz syenite. Such intrusive rocks in the outer ring are referred to as outer ring intrusive member (ORIM hereafter) to distinguish them from the main lithology of the outer ring (Table 1).

The inner ring 1 consists of quartz-bearing syenite (modal quartz < 5 vol%) and is characterized by idiomorphic and sub-porphyritic texture and occurrence of euhedral-subhedral clinopyroxene in direct contact with quartz (Table 1). The inner ring 2 consists of quartz syenite (modal quartz > 5%) and is characterized by sub-porphyritic texture with large euhedral-subhedral crystals of feldspar set in a fine-grained matrix (Table 1). The granitic core consists of syenogranite containing abundant quartz as high as 20-23 vol %. The syenogranites are either sub-porphyritic or porphyritic with micrographic texture developed in some of them (Table 1). The volcanic unit is distributed in the plutonic unit as an imperfect ring resting on or as isolated blocks within the inner ring 1 (Fig. 1; Saad et al., 2023). It comprises pyroclastic rocks of trachyte and trachyandesite. The dike unit consists of dikes of trachybasalt, basaltic trachyandesite, trachyte, and rhyolite. The trachybasalts and trachytes are mostly aphyric, whereas rhyolites are porphyritic (Table 1). The dikes steeply dipping and trending mostly NNW-SSE and rarely N-S crosscut all the other units (Frisch and Abdel-Rahman, 1999; Fig. 1).

It is inferred from the geological and petrographic information that the order of formation of the WDRC form the oldest to youngest is the volcanic unit, the plutonic unit, and the dike unit (Saad et al., 2023; Table 1). The solidification of the plutonic unit progressed from the margin to the center, started with the outer ring followed by the inner ring 1 and then inner ring 2 and ended with the granitic core.

Whole-rock major element compositions

Whole-rock oxides contents recalculated to be 100 wt% volatile free are listed in Supplementary Table S1 (Supplementary Tables S1-S8 are available online from https://doi.org/10.2465/jmps.240110). Whole-rock SiO₂ content of rocks from the WDRC ranges from ∼ 50 to ∼ 75 wt%. All the rocks, irrespective of units and members, yield common variation trends on Harker variation diagrams for major oxides (Saad et al., 2023; e.g., Fig. 2a). The variation trends are characterized by an inflexion at ∼ 62.5 wt% SiO₂, at which Na₂O, K₂O, and Al₂O₃ change from increase to decrease with increase in SiO₂ content, whereas other oxides continue to decrease with a slope change from steep to gentle (Saad et al., 2023). The SiO₂ content of the plutonic unit decreases from the outer ring, the oldest marginal ring, to the granitic core, the youngest central part (Table 1). The SiO₂ contents of the volcanic unit are consistent and mostly ∼ 62 wt%, whereas those of the dike unit varies widely from ∼ 50 to ∼ 75 wt%, covering all the range for the plutonic and volcanic units (Table 1). Saad et al. (2023) adopted a stepwise fractional crystallization model to explain the variation of major element compositions using the trachybasalts as an initial magma and successfully reproduced the entire variation with some exceptions.

Figure 2. Harker variation diagrams of selected trace element concentrations in rocks from the WDRC [(b)-(l)]. The TAS (total alkali - SiO₂) diagram is also shown in (a) after Saad et al. (2023). Data for Co and V are those obtained with XRF and the others with ICP-MS. All data are listed in Tables 2 and S1. Four types of variation patterns with increase of the whole-rock SiO₂ content are recognized with a few outliers (Table 3): (1) decrease for Co, V, Sr, and Er [(b)-(e)], (2) increase for Rb, Ta, and Th [(f)-(h)], increase followed by decrease for Zr (i), and increase followed by splitting into increasing and decreasing trends for La, Nb, and Nd [(j)-(l)]. Data reported by Frisch and Abdel-Rahman (1999) excepting Co and V are plotted for comparison with symbols according to their grouping, which is different from ours. Their Ta, Th, and Nb data are systematically lower than our data and are not used in discussion as explained in the text. ORIM, outer ring intrusive member.

GEOCHEMISTRY

Whole-rock trace element concentrations measured with ICP-MS and Sr, Nd, and Pb isotope ratios of selected rocks (14 samples) from the WDRC are listed in Table 2. Concentrations of trace elements measured with XRF for 21 samples are listed in Table S1. The analytical methods are described in Supplementary Document (ANALYTICAL METHODS; Supplementary Document is available online from https://doi.org/10.2465/jmps.240110).

Table 2. Whole-rock trace element contents and Sr, Nd, and Pb isotope ratios of rocks from the Wadi Dib ring complex

(a) Trace element contents (alkali fusion)
Unit/Lithology	Outer Ring		Outer Ring Intrusive Member			IR1	IR2	GC	Volcanic Unit			Dike Unit			CI chondrite (ppm)	Primitive mantle (ppm)
Rock type	Syenite	Monzo	Afs sye	Qz sye	QAfs sye	Qz-b sye	Qz sye	Syenognt	Trachyte	Trachyte	Trachya	Trachyte	Trachyb	Trachyb
Sample name	D14-C	D2	D58-B1	D14-A	D20-B	D3-A	D6.2	D5.2	D7.1	D18-F	D18-K	D16-F	D18-J	D56-A
Analyisis #	#7	#1	#14	#6	#12	#2	#4	#3	#5	#9	#11	#8	#10	#13
Rb §	102.9	103.7	133.6	226.3	256.0	224.5	214.5	267.2	229.0	152.7	122.5	145.4	89.6	130.3	2.3	0.6
Sr §	230.1	466.1	112.5	76.2	105.5	49.3	139.8	61.6	170.5	167.5	787.6	195.6	672.4	626.6	7.25	19.9
Y	37.3	42.1	33.7	97.1	64.5	64.7	37.2	52.2	73.5	58.4	27.0	59.9	34.0	38.2	1.57	4.3
Zr	310.9	532.9	269.5	645.6	711.3	656.0	482.6	326.6	983.9	670.4	499.5	1,046.0	272.2	301.4	3.82	10.5
Nb	118.5	133.1	101.3	250.4	152.1	192.1	90.4	123.3	202.2	164.4	115.0	245.4	66.6	67.3	0.24	0.658
Cs	0.63	0.95	0.53	0.84	1.15	3.15	1.48	1.87	1.15	0.58	3.82	0.69	1.34	3.09	0.19	0.021
Ba	1,219.0	2,149.0	780.1	194.8	281.4	178.4	435.9	179.6	554.5	720.7	985.3	262.8	791.8	253.2	2.41	6.6
La	69.2	72.2	59.2	228.4	156.7	165.3	109.3	98.0	135.3	114.0	75.5	167.0	57.5	45.1	0.237	0.648
Ce	135.7	138.4	121.6	372.6	275.5	271.9	175.1	159.6	245.1	204.2	132.6	306.0	108.7	93.8	0.613	1.675
Pr	15.5	15.7	14.8	37.8	29.8	26.7	17.1	15.8	25.8	21.8	13.8	30.8	12.5	11.8	0.0928	0.254
Nd	55.2	58.3	57.5	122.8	98.6	83.7	51.9	49.4	87.7	73.1	46.2	97.5	47.2	47.8	0.457	1.25
Sm	10.1	11.2	10.8	20.9	15.6	13.5	8.01	8.44	15.9	13.4	7.69	15.3	9.24	9.87	0.148	0.406
Eu	3.26	4.32	2.38	0.43	0.45	0.82	1.14	0.60	1.77	2.36	2.20	0.71	2.84	2.73	0.0563	0.154
Gd	8.81	10.1	9.22	18.1	12.8	11.7	6.47	7.56	14.2	11.8	6.19	12.0	8.31	9.03	0.199	0.544
Tb	1.27	1.49	1.27	2.89	1.90	1.73	0.97	1.22	2.15	1.78	0.87	1.79	1.15	1.31	0.0361	0.099
Dy	6.88	8.03	6.46	16.8	10.6	10.1	5.61	7.34	12.2	9.79	4.64	9.90	6.23	6.93	0.246	0.674
Ho	1.34	1.54	1.20	3.36	2.15	2.03	1.22	1.59	2.48	1.98	0.91	1.98	1.16	1.35	0.0546	0.149
Er	3.56	4.03	3.32	9.39	6.39	5.80	3.57	5.00	6.83	5.51	2.56	5.70	3.11	3.63	0.16	0.438
Tm	0.48	0.59	0.45	1.37	0.93	0.92	0.59	0.84	1.03	0.81	0.36	0.89	0.41	0.46	0.0247	0.068
Yb	2.97	3.58	3.31	8.58	6.20	6.23	4.10	5.82	6.62	5.23	2.36	5.96	2.52	2.85	0.161	0.441
Lu	0.47	0.53	0.58	1.14	0.95	1.00	0.62	0.85	0.96	0.78	0.37	0.88	0.36	0.44	0.0246	0.0675
Hf	7.36	11.1	6.24	18.1	16.9	15.3	11.1	10.0	20.2	13.9	8.69	20.1	6.04	6.67	0.103	0.283
Ta	7.00	6.48	5.23	13.5	8.94	11.0	8.02	13.0	12.2	8.89	6.30	13.5	3.72	3.77	0.0136	0.037
Pb	7.51	7.44	8.27	13.6	12.7	9.80	13.4	10.9	11.9	12.3	15.9	10.2	5.24	6.36	2.47	0.15
Th	9.56	11.3	7.01	76.4	60.7	38.2	49.5	72.1	33.1	22.4	14.9	45.3	7.64	6.82	0.029	0.0795
U	1.72	2.51	1.59	18.5	14.8	7.43	10.2	17.3	7.43	4.81	3.98	5.63	1.61	1.59	0.0074	0.0203

(b) Rare earth element ratios and Eu anomay (alkali fusion) and isotope ratios (acid decomposition)
Unit/Lithology		Outer Ring		Outer Ring Intrusive Member			IR1	IR2	GC	Volcanic Unit			Dike Unit
Rock type		Syenite	Monzo	Afs sye	Qz sye	QAfs sye	Qz-b sye	Qz sye	Syenognt	Trachyte	Trachyte	Trachya	Trachyte	Trachyb	Trachyb
Sample name		D14-C	D2	D58-B1	D14-A	D20-B	D3-A	D6.2	D5.2	D7.1	D18-F	D18-K	D16-F	D18-J	D56-A
Analyisis #		#7	#1	#14	#6	#12	#2	#4	#3	#5	#9	#11	#8	#10	#13
Sm/Nd		0.183	0.192	0.187	0.170	0.158	0.161	0.154	0.171	0.182	0.183	0.166	0.157	0.196	0.207
Eu/Eu*		122.97	206.70	92.09	6.00	12.57	7.44	27.69	12.63	28.65	44.39	86.33	11.06	94.37	36.36
(La/Yb)_N		15.86	13.72	12.15	18.08	17.18	18.02	18.11	11.44	13.88	14.80	21.73	19.03	15.50	10.76
(La/Nd)_N		2.42	2.39	1.99	3.59	3.07	3.81	4.06	3.83	2.97	3.01	3.15	3.30	2.35	1.82
(Nd/Sm)_N		1.77	1.68	1.73	1.91	2.05	2.01	2.10	1.89	1.78	1.77	1.95	2.07	1.66	1.57	Standard material
(Dy/Yb)_N		1.52	1.47	1.28	1.28	1.12	1.06	0.90	0.83	1.20	1.22	1.29	1.09	1.62	1.59	JB-3 †	JB-3 ‡
(Lu/Yb)_N		1.03	0.97	1.15	0.87	1.00	1.05	0.99	0.95	0.94	0.97	1.02	0.96	0.94	1.01	JB-3 †	Av./RSD	n
⁸⁷Sr/⁸⁶Sr		0.714220	0.708798	0.731975	0.773885	0.762307	0.807833	0.739576	0.807179	0.734751	0.725918	0.706581	0.721519	0.706690	0.707957	0.703375	0.703384	32
⁸⁷Sr/⁸⁶Sr	2se	0.000012	0.000014	0.000008	0.000014	0.000035	0.000014	0.000014	0.000019	0.000015	0.000012	0.000009	0.000011	0.000008	0.000009	0.000012	0.0034	32
¹⁴³Nd/¹⁴⁴Nd		0.512427	0.512451	0.512457	0.512404	0.512397	0.512412	0.512414	0.512454	0.512426	0.512423	0.512488	0.512428	0.512453	0.512512	0.513054	0.513065	23
¹⁴³Nd/¹⁴⁴Nd	2se	0.000005	0.000006	0.000007	0.000005	0.000006	0.000006	0.000005	0.000005	0.000005	0.000004	0.000007	0.000004	0.000003	0.000006	0.000008	0.0022	23
²⁰⁶Pb/²⁰⁴Pb		20.0651	19.9416	19.3197	29.6727	25.4012	22.9268	23.5521	29.5737	22.0378	20.5231	19.3690	22.1944	19.8739	19.7819	18.2971	18.2966	21
²⁰⁶Pb/²⁰⁴Pb	2se	0.0004	0.0006	0.0003	0.0007	0.0007	0.0007	0.0005	0.0007	0.0006	0.0005	0.0004	0.0006	0.0004	0.0005	0.0005	0.0069	21
²⁰⁷Pb/²⁰⁴Pb		15.7356	15.7207	15.6859	16.2664	16.0172	15.8694	15.8971	16.2453	15.8460	15.7562	15.6428	15.8195	15.6810	15.6756	15.5396	15.5393	21
²⁰⁷Pb/²⁰⁴Pb	2se	0.0003	0.0005	0.0002	0.0005	0.0003	0.0006	0.0004	0.0004	0.0004	0.0005	0.0003	0.0004	0.0004	0.0005	0.0005	0.0086	21
²⁰⁸Pb/²⁰⁴Pb		40.6654	40.5152	39.6684	51.4392	46.8241	44.8756	45.0130	50.9337	43.4482	42.1044	39.2954	46.9511	40.3325	40.2009	38.2564	38.2551	21
²⁰⁸Pb/²⁰⁴Pb	2se	0.0008	0.0014	0.0007	0.0015	0.0011	0.0018	0.0012	0.0014	0.0014	0.0015	0.0008	0.0014	0.0011	0.0014	0.0013	0.0099	21

2se, 2 standard error; Eu/Eu*, (Eu)_N/(((Sm)_N + (Gd)_N)/2), where (Eu)_N is the CI chondrite-normalized Eu concentration; (La/Sm)_N, ratio of CI chondrite normalized La and Sm concentrations; 2s standard deviation; n, number of analyses; RSD, relative standard deviation; §, solutions prepared with acid decomposition method; †, results of JB-3 measurements during the analytical session for this study; ‡, mean isotope ratios of JB-3 analyses at the Hokkaido University.

IR1, inner ring 1; IR2, inner ring 2; GC, granitic core; Afs syenite, alkali feldspar syenite; Qz sye, quartz syenite; Qz-b sye, quartz-bearing syenite; Qz Afs sye, quartz alkali feldspar syenite; Syenogrnt, syenogranite; Trachyand, trachyandesite; Trachyb, trachybasalt.

The 2 se errors for sample isotope ratios represent the internal errors during the MC-ICP-MS analyses. The CI chondrite and primitive mantle trace element concentrations are after McDonough and Sun (1995).

The RSD of the isotopic ratios obtained by repeated analyses of JB-3 standard, which was used as the uncertainties (external errors) for sample isotopic ratios.

Whole-rock trace element concentrations

Selected trace element concentrations of rocks from the WDRC are plotted against SiO₂ content in Figure 2 and compared with those obtained by Frisch and Abdel-Rahman (1999). The two data sets are mostly consistent except for systematically lower abundances of Nb, Ta, and Th reported by Frisch and Abdel-Rahman (1999). Our Nb and Th data based on ICP-MS are consistent with two different data sets based on XRF analyses (Tables 2 and S1). Our Nb, Ta, and Th data show distinct correlations with Y, Yb, and Zr. Such plots of the data after Frisch and Abdel-Rahman (1999) do not show clear correlations. Therefore, our Nb, Ta, and Th data are evaluated to be more reliable than the literature data, which are not used in the following data description and discussion.

The trace element variations are grouped into four patterns with an increase of the whole-rock SiO₂ wt% (Table 3). They are: (1) decrease (Figs. 2b-2e), (2) increase (Figs. 2f-2h), (3) increase followed by decrease (Fig. 2i), and (4) increase followed by split into increasing and decreasing trends (Figs. 2j-2l). See Supplementary Document (‘Variation patterns on Harker variation diagrams’ in section TRACE ELEMENT VARIATION PATTERNS) for details of each variation pattern. The nonlinear variation patterns of groups 1, 2, and 3 are characterized by an inflexion at ∼ 62.5 wt% SiO₂. The ∼ 62.5 wt% SiO₂ is where a notable inflexion occurs in each of the Harker variation diagrams for major elements (e.g., Fig. 2a; Saad et al., 2023). The variation pattern of group 4 is characterized by splitting into two trends at ∼ 62.5 wt% SiO₂. Evident increase with further increase in SiO₂ from ∼ 62.5 wt% is exhibited by quartz syenite and quartz alkali feldspar syenite from the ORIM, whereas a weak variation is exhibited by quartz syenite from the inner ring 2 and syenogranite from the granitic core. The MREEs show variation patterns transitional between LREEs and HREEs. HREEs (Tm, Yb, and Lu), which are classified as group 2, show variation patterns transitional between typical elements belonging to groups 2 and 4. The data from outer ring (filled and open red circles in Fig. 2) tends to deviate from the main variation trends (e.g., Eu and Sr) irrespective of variation patterns.

Table 3. Grouping of trace elements showing peculiar variation patterns of concentrations with increase of SiO₂ contents in rocks from the Wadi Dib ring complex

Variation group	Elements	Number of elements	Example of variation patterns	Variation pattern with increasng SiO₂			Behavior if simple fractional crystallization is assumed †	Lithologies for high-SiO₂ segment with SiO₂ = 62.5-75 wt%
				Low-SiO₂ segment	At 62.5 wt%	High-SiO₂ segment
				50-62.5 wt%	At 62.5 wt%	62.5-75 wt%
1	Ti, P, Sc, V, and Co	5	Figure 2b, 2c	Sharp decrease	Remarkable inflexion	Slight decrease	Always cmpatible, change by ∼ 1 order of magnitude	All lithologies with SiO₂ > ∼ 62.5 wt% mostly from the inner ring 2, granitic core, and outer ring intrusive member of the outer ring
1	Zn, Sr, Eu, and Ba	4	Figure 2d, 2e	Continuous decrease without inflexion			Always compatible, change by a factor <5
2	Rb, Pb, Ta, and HREEs (Tm, Yb, and Lu)	6	Figure 2f, 2g	Continuous increase without inflexion			Always incompatible, change by a factor <4
2	Th and U	2	Figure 2h	Continuous increase	Weak inflexion	Continuous increase	Always incompatible, change by ∼ 1 order of magnitude
3	Zr, Ga, and K	3	Figure 2i	Increase	Remarkable inflexion	Decrease	Change from incompatible to compatible
4	Nb, Y, Hf, and LREEs - MREEs (La, Ce, Pr, Nd, Sm, Gd, Tb, and Dy)	11	Figure 2j-2l	Increase	Splitting into two trend, one show no and the other moderate inflexions	Increase	Always incompatible	Quartz alkali feldspar syenite, quartz syenite from the outer ring intrusive member
4		11	Figure 2j-2l	Increase		Decrease	Change from incompatible to compatible	Quartz syenite from the inner ring 2 and syenogranite from the granitic core

†, behavior (incompatible versus compatible and extent of variations) of trace elements if simple fractional crystallization is assumed to explain variation patterns. In order to specify actual mechanism of fractionation, quantitative examination of major element and isotope compositions is necessary in addition to trace element variations.

CI chondrite-normalized rare earth element patterns are shown in Figure 3 for each unit or lithology member. The REE abundances are lowest in trachybasalts (Fig. 3e) and highest in quartz syenite and quartz alkali feldspar syenite from the ORIM (Fig. 3b). The REE patterns show LREE enrichment to various extents, which ranges from 11 to 22 in CI chondrite normalized La-Yb ratio, (La/Yb)_N (Table 2). They have various Eu anomalies from absence [(Eu)_N/(((Sm)_N + (Gd)_N)/2) = Eu/Eu* = 1.0], weak positive anomaly, and moderate to strong negative anomaly. Syenites from the outer ring have smooth LREE-enriched pattern with inclined HREEs and very weak positive Eu anomaly (Eu/Eu* = 1.03-1.22; Fig. 3a; Table 2). Rocks from the ORIM shows diverse REE patterns. Quartz syenite and fayalite-bearing quartz alkali feldspar syenite from the ORIM are characterized by high REE concentrations, extremely enriched and steeply inclined LREEs, inclined MREEs-HREEs, and extremely strong Eu negative anomaly (Eu/Eu* = 0.07-0.09; Fig. 3b; Table 2). Porphyritic olivine-bearing alkali feldspar syenite from the ORIM is characterized by moderate enrichment of LREEs, weak negative Eu anomaly (Eu/Eu* = 0.71; Table 2), and convex-upward pattern of MREE-HREE (Fig. 3b). Quartz-bearing syenite from the inner ring 1, quartz syenite from the inner ring 2, and syenogranite from the granitic core have similar REE patterns (Fig. 3c), though the quartz-bearing syenite has higher REE concentrations than the other two. They are characterized by strong enrichment of LREE, distinct negative Eu anomaly (Eu/Eu* = 0.2-0.5; Table 2), and flat MREE-HREE. Trachytes from the volcanic unit show REE patterns intermediate between those of syenites from the outer ring and quartz-bearing syenite from the inner ring 1 and have distinct Eu negative anomaly (Fig. 3d; Eu/Eu* = 0.36-0.59; Table 2). Trachyandesite from the volcanic unit is low in REEs, has a smooth and LREE-enriched pattern with a weakly inclined HREE-MREE segment, and has slightly negative Eu anomaly (Eu/Eu* = 0.94; Fig. 3d; Table 2). Trachybasalts of the dike unit are low in REEs and show a gently inclined smooth LREE-enriched pattern with minor negative Eu anomaly and inclined HREEs (Eu/Eu* = 0.87-0.97; Fig. 3e; Table 2). Trachyte of the dike unit is characterized by a pattern similar to rocks from the inner rings and granitic core, and is characterized by strong enrichment of LREE, strong negative Eu anomaly (Eu/Eu* = 0.15; Fig. 3e; Table 2), and flat HREEs.

Figure 3. CI chondrite normalized rare earth element (REE) patterns of rocks from the WDRC. The REE concentrations listed in Table 2 were normalized with the CI chondrite compositions after McDonough and Sun (1995) listed in the second-to-the-last column in Table 2. Gray patterns in (a)-(d) are those of trachybasalt plotted in (e) for easy comparison of the concentration levels. OR, outer ring; Qz Sye, quartz syenite; Fa Qz Afs Sye, fayalite-bearing quartz alkali feldspar syenite; Ol Po Afs Sy, olivine-bearing porphyritic alkali feldspar syenite; Qz-b Sye, quartz-bearing syenite; Syenogrnt, syenogranite; Trchyt, trachyte; Trchand, trachyandesite; TrchyB, trachybasalt.

Primitive mantle (PM) normalized trace element patterns are shown in Figure 4 for each unit or lithology member. The trace element pattern show enrichment in highly incompatible trace elements and various anomalies, such as negative anomalies of Ba, Sr, P, Eu, and Ti. Syenites from the outer ring have an inclined pattern with several moderate negative anomalies of Ti, P, Sr, and Cs, and weak anomaly of U, and Th relative to Nb, Ta, and Rb (Fig. 4a). Quartz syenite and fayalite-bearing quartz alkali feldspar syenite from the ORIM are high in trace element abundances and their patterns are featured by strong negative anomalies of Ti, Eu, P, Sr, and Ba (Fig. 4b). They are high in Th and U relative to Nb and Ta. Porphyritic olivine-bearing alkali feldspar syenite from the ORIM has a pattern similar to that of the syenites from the outer ring excepting MREE-HREE. Trace element patterns of rocks from the inner rings and granitic core are similar in strong enrichment of highly incompatible elements and strong negative anomalies of Ti, P, Sr, Ba, and Cs, but quartz-bearing syenite from the inner ring 1 is distinct from the others in the higher abundance, lesser enrichment of highly incompatible elements, particularly U and Th, and more inclined pattern of less incompatible elements (Fig. 4c). Trachytes from the volcanic unit show a pattern intermediate between syenite from the outer ring and quartz-bearing syenite from the inner ring 1 and are characterized by moderate Ti, Eu, P, Sr, and weak Ba negative anomalies (Fig. 4d). Trachyandesite from the volcanic unit has minor negative anomalies of Ti, P, Sr, and Ba (Fig. 4d). Trachybasalts from the dike unit show a relatively smooth PM-normalized trace element pattern with the gentlest slope in the WDRC and minor negative anomalies of Pb, Sr, Th, and U (Fig. 4e). Trachyte from the dike unit has a trace element pattern intermediate between quartz-bearing syenite from the inner ring 1and quartz syenite from the granitic core (compare D16-F in Fig. 4e to D6.1 and D5.2 in Fig. 4c).

Figure 4. Primitive mantle normalized trace element patterns of rocks from the WDRC. The trace element concentrations listed in Table 2 were normalized with the primitive mantle composition after McDonough and Sun (1995) listed in the last column in Table 2. Gray patterns in (a)-(d) are those of trachybasalt plotted in (e) for easy comparison of the concentration levels. See caption of Figure 3 for abbreviations.

CI chondrite normalized Dy-Yb, Lu-Yb, Nd-Sm, and La-Nd ratios [abbreviated as (Dy/Yb)_N, (Lu/Yb)_N, (Nd/Sm)_N, (La/Sm)_N, and (La/Nd)_N, respectively] are calculated and listed in Table 2. These REE ratios provide quantitative measures of CI chondrite normalized REE patterns shown in Figure 3: slope of HREEs by (Lu/Yb)_N, slope of MREE HREE by (Dy/Yb)_N, slope of LREE-MREE by (Nd/Sm)_N, and slope of LREEs by (La/Nd)_N. A set of three ratios, (Dy/Yb)_N, (Nd/Sm)_N, and (La/Nd)_N, are comparable to shape coefficients of λ₁ and λ₂ proposed by O’Neill (2016) as two of the parameters capturing REE patterns of basalt to better than ±5%.

The (Dy/Yb)_N of rocks from the WDRC is mostly higher than the chondritic value and decreases with increase in the whole-rock SiO₂ content from 1.6 to subchondritic value of 0.8-0.9 with syenites from the outer ring and SiO₂-rich rocks from the ORIM shifting upwards from the linear trend defined by other rocks (Fig. 5a). The other three ratios are plotted against (Dy/Yb)_N in Figures 5b-5d. These plots are not a standard way of presentation of REE data for crustal igneous rocks but are useful to quantify CI-normalized REE patterns and for extraction of detailed information on the crustal magmatic processes. The (Lu/Yb)_N is chondritic and does not show any change with (Dy/Yb)_N (Fig. 5b). The (Nd/Sm)_N and (La/Nd)_N are higher than the chondritic values, negatively correlated with (Dy/Yb)_N (Figs. 5c-5d), and thus positively correlated with SiO₂ content. These relationships imply changes of chondrite normalized REE patterns with increase in SiO₂ content in two aspects: (1) decreasing slope of the MREE-HREE segment up to sloping even to the opposite direction and (2) continual steepening of the segments of LREEs and MREE-LREE.

Figure 5. Mutual relationships between CI chondrite normalized rare earth element ratios of rocks from the WDRC. CI normalized Dy-Yb ratios [(Dy/Yb)_N] are plotted against whole-rock SiO₂ wt% in (a) and those of Lu-Yb, Nd-Sm, and La-Nd [(Lu/Yb)_N, (Nd/Sm)_N, and (La/Nd)_N, respectively] are plotted against (Dy/Yb)_N in (b)-(d). Groups of the samples closer to each other in chemical compositions are encircled and numbered in the increasing order of SiO₂ wt% and decreasing order of (Dy/Yb)_N from 1 to 5, which are referred in the text. In (a), sample name and rock type are attached to each symbol. See Figure 3 for rock name abbreviations used in (a).

The tight trend in each diagram plotting two REE ratios (Figs. 5b-5d) has the trachybasalt data on one end of the trend, where (Dy/Yb)_N is highest and (Nd/Sm)_N and (La/Nd)_N are lowest (circles marked with 1 in Fig. 5). Among all the rocks of the WDRC, therefore, the trachybasalts have the steepest slope of MREE-HREE segment and the gentlest slopes of LREEs and MREE-LREE segments of the chondrite-normalized REE patterns. Syenites from the outer ring (circles marked with 2 in Fig. 5) and trachytes from the volcanic unit (circles marked with 3 in Fig. 5), which have the similar whole-rock major element contents but distinct trace element concentrations as explained above (Fig. 2), are clearly distinguished from each other: higher values of (Dy/Yb)_N, (La/Sm)_N, and (La/Nd)_N of the syenites than those of the trachytes at SiO₂ ∼ 62 wt%. The syenites are plotted closer to the trachybasalts (compare circles marked with 1 and 2 in Fig. 5), whereas the trachytes (circles marked with 3 in Fig. 5) are plotted in the middle of the entire variations. The similar relationships are observed between SiO₂-rich rocks of the ORIM (ellipses marked 4 in Fig. 5) and those of the inner ring 2 and granitic core (circles or ellipse marked 5 in Fig. 5) at SiO₂ 67-72 wt%. The rocks from the ORIM are plotted closer to the trachytes in the middle of the entire variation (compare circles or ellipses marked with 3 and 4 in Fig. 5), whereas the rocks from the inner ring 2 and granitic core (circles or ellipse marked 5 in Fig. 5) are plotted in the lowest-(Dy/Yb)_N end. These contrasts are clearly seen in Figure 5a.

Radiogenic isotope ratios

Isotope ratios of Sr, Nd, and Pb measured for 14 samples from the WDRC are listed in Table 2. The values of ⁸⁷Sr/⁸⁶Sr of the WDRC are extremely variable ranging from 0.706690 to 0.807833, which is wider than those reported by Frisch and Abdel-Rahman (1999) (0.7104-0.7805 excluding one weathered sample). The Pb isotope ratios of the WDRC are also variable ranging 19.7819-29.6727 for ²⁰⁶Pb/²⁰⁴Pb, 15.6756-16.2664 for ²⁰⁷Pb/²⁰⁴Pb, and 39.2954-51.4392 for ²⁰⁸Pb/²⁰⁴Pb. The lowest values are registered by trachybasalt from the dike unit or trachyte from the volcanic unit (Table 2). The highest values are registered by syenogranite from the granitic core or quartz syenite from the ORIM (Table 2). These radiogenic isotope ratios vary by 13, 40, 3.7, and 29% relative to the mean isotope ratios, respectively. They are higher than those of the estimated values of the bulk silicate earth (BSE): 0.7045, 17.3-18.7, 15.42-15.7, and 37.68-39.05, respectively (DePaolo, 1979; Salters and Stracke, 2004; Workman and Hart, 2005; Doucet et al., 2023). By contrast, the variation of ¹⁴³Nd/¹⁴⁴Nd of the WDRC is limited and ranges from 0.512397 to 0.512512, which vary only by 0.02% relative to the mean isotope ratio. The lowest value is registered by quartz alkali feldspar syenite from the ORIM, and the highest value by trachybasalt from the dike unit (Table 2). The Nd isotope ratios are lower than the present-day BSE value of 0.512638 (Workman and Hart, 2005).

The ⁸⁷Sr/⁸⁶Sr, ¹⁴³Nd/¹⁴⁴Nd, ²⁰⁶Pb/²⁰⁴Pb, ²⁰⁷Pb/²⁰⁴Pb, and ²⁰⁸Pb/²⁰⁴Pb of the outer ring range 0.708798-0.71422, 0.512427-0.512451, 19.9416-20.0651, 15.7207-15.7356, and 40.5152-40.6654, respectively (the first two columns in Table 2). Those of the ORIM range 0.731975-0.773885, 0.512397-0.512457, 19.3197-29.6727, 15.6859-16.2664, and 39.6684-51.4392, respectively (the third to fourth columns in Table 2). Those of the inner rings and granitic core range 0.739576-0.807833, 0.512412-0.512454, 22.9268-29.5737, 15.8694-16.2453, and 44.8756-50.9337, respectively (the sixth to eighth columns in Table 2). Those of the volcanic unit range 0.706581-0.734751, 0.512423-0.512488, 19.3690-22.0378, 15.6428-15.8460, and 39.2954-43.4482, respectively (the nineth to eleventh columns in Table 2). Those of the dike unit range 0.706690-0.721519, 0.512428-0.512512, 19.7819-22.1944, 15.6756-15.8195, and 40.2009-46.9511, respectively (the last three columns in Table 2). The ⁸⁷Sr/⁸⁶Sr, ²⁰⁶Pb/²⁰⁴Pb, ²⁰⁷Pb/²⁰⁴Pb, and ²⁰⁸Pb/²⁰⁴Pb of the inner rings and granitic core are higher than those of the outer ring and volcanic and dike units. There is no such tendency for ¹⁴³Nd/¹⁴⁴Nd. The ⁸⁷Sr/⁸⁶Sr, ²⁰⁶Pb/²⁰⁴Pb, ²⁰⁷Pb/²⁰⁴Pb, and ²⁰⁸Pb/²⁰⁴Pb of the ORIM showing wide variations, particularly for Pb isotope ratios, which cover the entire ranges for the WDRC.

The measured raw radiogenic isotope ratios listed in Table 2 reflect isotope ratios at the time of closure of respective isotope systems and subsequent radiogenic ingrowth if secondary modifications are negligible. Evaluation of the first factor requires accurate age estimation, which will be carefully made below in discussion sections. The second effect can be evaluated by plotting isotope ratios against respective parent-daughter isotope ratios (Fig. 6). The plots for Rb-Sr, U-Pb, and Th-Pb radiogenic isotope systems show tight positive correlations (Figs. 6a and 6c-6d) suggesting that radiogenic ingrowth played a dominant role in the overall variations of measured Sr and Pb isotope ratios. In these plots, there is a tendency that SiO₂-rich samples are more radiogenic. The Rb-Sr plot shows the tightest correlation, and our data are consistent with the isochron plot of Frisch and Abdel-Rahman (1999) (open symbols in Fig. 6a). The ¹⁴³Nd/¹⁴⁴Nd plotted against ¹⁴⁷Sm/¹⁴⁴Nd also shows a positive correlation and a tendency that SiO₂-rich samples are less radiogenic (Fig. 6b), but the correlation is not as good as the other four plots (Figs. 6a and 6c-6e). The Sm-Nd plot is sensitive to modification due to open processes because of the limited variation of Sm/Nd by smaller than 25% (Fig. 6b), which will be fully examined below in discussion section. By contrast, the Rb/Sr varies by two orders of magnitude (Fig. 6a) and the U/Pb and Th/Pb ratios vary by one order of magnitude (Figs. 6c-6e), which obscures secondary modification even if it was the case. Although it is not obvious, one sample with the highest Rb/Sr in the Rb-Sr isochron plot is deviated from the linear trend (Fig. 6a), suggesting secondary modification as mentioned below in the first part of discussion section.

Figure 6. Rb-Sr (a), Sm-Nd (b), U-Pb [(c), (d)], Th-Pb (e) isochron plots of rocks from the WDRC. The abscissas were calculated from our data for Rb/Sr, Sm/Nd, U/Pb, and Th/Pb ratios and the respective natural abundance ratios for isotopes. The Rb-Sr data reported by Frisch and Abdel-Rahman (1999) are plotted in (a). The dot-dashed and dashed lines in (a) are isochron for all samples with age 579.9 Ma and for selected 7 samples with age 586.8 Ma (Fig. 7), respectively. Dashed line in (b)-(e) are reference isochrons for 586.8 Ma. All isochron parameters are listed in Table 4. Error bars are ±2se (standard error), which are mostly smaller than the symbols.

DISCUSSION

Evaluation of low-temperature alteration on major, trace, and isotope compositions

The values of loss on ignition (L.O.I.) of samples from the volcanic and plutonic units and the ORIM of the WDRC are mostly lower than 1 wt% (Saad et al., 2023), indicating insignificant alteration due to formation of low-temperature hydrous minerals and/or carbonate. SiO₂-rich rocks rarely have higher L.O.I., though the values are lower than 2 wt%. We excluded samples with L.O.I. higher than 1.2 wt% for analyses of trace elements and isotope ratios.

Trachybasalts tend to have high L.O.I. ranging 2.6-5.4 wt%, which is due to static replacement of primary phases by secondary water-bearing minerals and carbonate, the assemblage of which suggests a greenschist facies metamorphism (Saad et al., 2023). However, their major and trace element contents including highly mobile elements are mostly identical or similar when recalculated for the sum to be 100 wt% volatile free (Fig. 2; Saad et al., 2023) even if the values of L.O.I. are different by a factor of two, suggesting that no significant modification of elemental concentrations took place during the metamorphism. This is also supported by the fact that the trachybasalts form consistent trends with other rocks from the WDRC in most of two element correlation plots. See Supplementary Document (‘Two trace element plots’ in section TRACE ELEMENT VARIATION PATTERNS; Supplementary Fig. S1; Supplementary Figs. S1-S17 are available online from https://doi.org/10.2465/jmps.240110) for details.

Quartz-bearing syenite from the inner ring 1 has values of ⁸⁷Sr/⁸⁶Sr and ⁸⁷Rb/⁸⁶Sr significantly deviated downward from the isochron (Fig. 6a), which is attributed to selective depletion of Sr during alteration as observed in some basaltic rocks of the Oman ophiolite (Kawahata et al., 2001). The Sr depletion is supported by deviation of the general trends in two element correlation diagrams, in which high field strength elements are plotted against Sr. There is no such deviation for Rb. The loss of Sr is suggested by local development of secondary Ca-free [<0.5% in An, An = 100 × Ca/(Ca + Na + K)] alkali feldspar perthite, whose typical An in rocks from the plutonic unit is higher than 3% and up to 10%.

Geochronology of the WDRC

The tight correlations shown in Figure 6 suggest that radiogenic ingrowth controls most of the variation of the isotope ratios, and thus each plot defines an isochron. This is most plausibly explained by fractional crystallization of a parental magma without extensive assimilation or contamination of isotopically distinct crustal materials. The dominant role of fractional crystallization is advocated by Frisch and Abdel-Rahman (1999) from the Rb-Sr isochron plot and trace element data and by Saad et al. (2023) from the major element data. The entire correlations cannot be explained by two components mixing because relationships between SiO₂ and trace elements are commonly nonlinear particularly for compatible elements (Figs. 2b-2e) or split into two trends (e.g., Figs. 2j-2l). Even if an open process could have involved in the formation of the WDRC, it must be limited to a group of some rock types and is not responsible for the entire correlation shown in Figure 6. Effects of open processes will be quantitatively evaluated below in the later part of discussion.

The ⁸⁷Sr/⁸⁶Sr and ⁸⁷Rb/⁸⁶Sr are most strongly correlated among the plots in Figure 6. The correlation for all samples excluding one outlier yields a whole-rock isochron age of 579.9 ± 6.0 (2se) Ma with initial ratio of 0.70324 ± 0.00011 (Table 4; dot dashed line in Fig. 6a), which is consistent with the age of 578 ± 16 (2se) Ma with initial ratio of 0.7048 ± 0.0010 obtained by Frisch and Abdel-Rahman (1999) if the errors are taken into consideration. We assessed the Rb-Sr isotope system for more accurate age estimation through careful evaluation of effects of open processes (e.g., crustal assimilation or contamination) during chemical evolution of the WDRC magmas, which will be discussed in detail below. We found that a half of the examined samples (7 samples) are principally unaffected by open processes. See Supplementary Document (‘Procedure of accurate geochronology for crustal intrusions: Approach combining evaluation of initial isotope ratios of samples’ in section ACCURATE GEOCHRONOLOGY AND EVALUATION OF INITIAL ISOTOPIC HETEROGENEITY) for details of the sample selection. Trachybasalts are included in the 7 samples indicating that a fractionation starting with the trachybasalts in a system closed to input of isotopically distinct exotic materials played an important role in development of the geochemical diversity of the WDRC. Figure 7 shows an isochron plot for samples whose fractionation is shown to have been unaffected by any material input of isotopically distinct materials, which yields 586.8 ± 10.1 (2se) Ma with initial ratio of 0.70333 ± 0.00011 (Figs. 7 and dashed line in 6a; Table 4). The revised age is again consistent with the age obtained by Frisch and Abdel-Rahman (1999) if the errors are taken into consideration. The revised initial ⁸⁷Rb/⁸⁶Sr is lower than that of Frisch and Abdel-Rahman (1999), which is important in that the literature initial ⁸⁷Rb/⁸⁶Sr is higher than that of the BSE at 578-587 Ma, whereas the revised initial ratio is lower than the BSE value.

Table 4. Geochronology of the Wadi Dib ring complex

Method		Isotope ratio	Age (Ma)	±2se	Initial ratio	±2se	MSWD	R²	Fractionation	Max/Min	Samples
Isochron	Rb-Sr	⁸⁷Sr/⁸⁶Sr	586.8	10.1 †	0.70333	0.00011 †	8.7	0.9998	Rb/Sr	18.2	7 *
	Rb-Sr	⁸⁷Sr/⁸⁶Sr	579.9	5.99	0.70324	0.000077	11	0.9978	Rb/Sr	34.2	14, all
	Sm-Nd	¹⁴³Nd/¹⁴⁴Nd	551	101	0.51204	0.000074	3.6	0.8493	Sm/Nd	1.32	7 *
	Sm-Nd	¹⁴³Nd/¹⁴⁴Nd	403.9	57.2	0.51216	0.00004	15	0.3500	Sm/Nd	1.36	14, all
	Pb-Pb	²⁰⁶Pb/²⁰⁷Pb	472.9	11.4	14.583	0.0055	890	0.9765	U/Pb	5.33	7 *
	Pb-Pb	²⁰⁶Pb/²⁰⁷Pb	486.5	6.22	14.570	0.0025	740	0.9923	U/Pb	7.28	14, all
	Th-Pb	²⁰⁸Pb/²⁰⁴Pb	559.1	16.5	38.041	0.068	100	0.9682	Th/Pb	5.10	7 *
	Th-Pb	²⁰⁸Pb/²⁰⁴Pb	584.9	9.78	37.847	0.040	65	0.7408	Th/Pb	7.24	14, all
Reference isochron	Sm-Nd	¹⁴³Nd/¹⁴⁴Nd	586.8	-	0.512011	0.000011	-	-	-	-	7 *
	U-Pb	²⁰⁶Pb/²⁰⁴Pb		-	18.194	0.217	-	-	-	-	7 *
	U-Pb	²⁰⁷Pb/²⁰⁴Pb		-	15.609	0.024	-	-	-	-	7 *
	Th-Pb	²⁰⁸Pb/²⁰⁴Pb		-	37.922	0.251	-	-	-	-	7 *

Ages were estimated with IsoplotR (Ludwig, 1988; Vermeesch, 2018). *: Seven samples, which are shown not to have been affected by AFC processes were used for dating. Used samples are: D2, D14-C, D18-F, D18-J, D20-B, D56-A, D58-B1. Max/Min, the ratio of the maximum and minimum values of parent-daughter element concentrations, representing degree of elemental fractionation; R², coefficient of determination, a measure of how well observed isotope ratios are replicated by the age model calculated from the estimated age. †: The augmented errors for Rb-Sr isochron age and the initial ⁸⁷Sr/⁸⁶Sr are ±13 and ±0.00033, respectively, which were calculated by assuming that the ‘geological scatter’ in excess of ‘analytical scatter’ is dominated by uncertainty of the initial isotope ratio. See Suppl. Doc. (‘Procedure of accurate geochronology for crustal intrusions: Approach combining evaluation of initial isotope ratios of samples.’ in section ACCURATE GEOCHRONOLOGY AND EVALUATION OF INITIAL ISOTOPIC HETEROGENEITY) for evaluation of ‘geological scatter’ and the method for calculation of the augmented errors.

Figure 7. Rb-Sr isochron plot of selected 7 rock samples from the WDRC. The isochron is defined by trachybasalts of the dike unit, syenite and monzonite of the outer ring (OR), trachyte of the volcanic unit, and alkali feldspar syenite and quartz alkali feldspar syenite of the outer ring intrusive member (Tables 2 and 4). The age and initial ratio were calculated using IsoplotR (Ludwig, 1988; Vermeesch, 2018). Error bars are ±2se (standard error), which are mostly smaller than the symbols.

We also calculated whole-rock isochron ages from the isochron plots presented in Figure 6 for Sm-Nd, U-Pb (Pb-Pb), and Th-Pb radiogenic isotope systems. The isochron ages range from ∼ 470 to 560 Ma for the selected 7 samples (Table 4). The ages obtained from Sm-Nd and Th-Pb systems are within the range of the Rb-Sr isochron age if the errors are taken into consideration, whereas the Pb-Pb method gives a younger age. The isochron ages are evaluated based on four parameters (Table 4): (1) degree of fractionation, which is quantified by the ratio of the maximum and minimum parent-daughter element ratios (Max/Min column in Table 4), (2) R², which represents coefficient of determination, a measure of how well observed isotope ratios are replicated by the age model calculated from the estimated age, (3) mean square of the weighted deviates (MSWD), and (4) ±2se errors for age and initial ratio. The fractionation of Sm-Nd is very limited, and the R² for the Sm-Nd isochron age gives lowest value. The parent-daughter fractionation for the Pb-Pb method is relatively large and the R² is close to 1.0, but its MSWD is extremely large. The parent-daughter fractionation and R² are notably smaller than unity and MSWD is large for the Th-Pb method (Table 4). The Rb-Sr isochron age is based on the largest parent-daughter fractionation and has smaller error, smaller MSWD, and R² closer to 1 than those of Sm-Nd, Pb-Pb, and Th-Pb irrespective of whether sample selection was made or not. Therefore, we conclude that the Rb-Sr whole-rock isochron age of 586.8 ± 10.1 Ma obtained from the selected 7 samples represents the most reliable age of a magmatism responsible for the fractionation of the WDRC closed to material input. We will adopt this age to evaluate radiogenic ingrowth of the four radiogenic isotope systems in the following discussion.

The consistent linear trends in isochron plots for Nd-Sm, U-Pb, and Th-Pb systems (Fig. 6) can be reproduced by 586.8 Ma reference isochrons constrained from the Rb-Sr isotope system by optimizing respective initial ratios or y-intercept based on the selected 7 samples (Table 4 bottom four rows; dashed lines in Figs. 6b-6e). The initial ratios are constrained within ±1% (2se) uncertainty and are regarded as isotope ratios of the parental magma of the WDRC and its source mantle.

Trachybasalts of the dike unit: a parental magma of the WDRC

The dike unit having a wide variety of compositions ranging from trachybasalt to rhyolite and cutting the volcanic and plutonic units represents the latest magma activity in the WDRC (Saad et al., 2023). Contrary to this contention, Frisch and Abdel-Rahman (1999) argued that general NNW-SSE trend of the dikes is parallel to the orientation of the Red Sea opening and that they are much younger and therefore genetically unrelated to the WDRC. This issue must be resolved, because trachybasalts of the dike unit could represent a parental magma of the WDRC (Saad et al., 2023). In this section, we will examine the major and trace element variations and geochronology of the WDRC presented above to argue for that the dike unit including trachybasalts is one of the essential members of the complex and that the trachybasalts represents a parental magma. Moreover, we compare major and trace element concentrations and isotope ratios of trachybasalts of the dike unit with those of young basaltic rocks formed during the opening of the Red Sea to confirm that the trachybasalts are distinct from the young basaltic rocks. See Supplementary Document (COMPARISON OF TRACHYBASALT WITH YOUNG BASALTS) for details of the comparison with young basalts.

Major element composition. The tight variation trends of whole-rock major oxide compositions of the WDRC on Harker variation diagrams, as summarized above (e.g., Fig. 2a), suggest a genetic link of all the rocks including rocks of the dike unit (Saad et al., 2023). The trachybasalts of the dike unit occupy the lowest SiO₂ end of the trends, from which the entire variations of major element compositions are reproduced by a stepwise fractional crystallization model (Saad et al., 2023). We thus contend that the dike unit including trachybasalts is one of the essential members of the WDRC and that the trachybasalt represents a parental magma of the WDRC, from which the lithological and the chemical diversities of the WDRC were produced through fractional crystallization. The enrichment of alkalis and depletion of CaO of the trachybasalts of the dike unit as compared with those of young basalts from the Red Sea Rift and the coastal plain and marginal region of the Red Sea show that the trachybasalts are distinct in major elements from younger rocks related to the Red Sea opening, which supports our contention. See Supplementary Document (‘Major elements’ in section COMPARISON OF TRACHYBASALT WITH YOUNG BASALTS; Supplementary Fig. S3) for details.

Geochronology. Unequivocal evidence for our contention comes from the Rb-Sr and Sm-Nd isochron plots. The Rb and Sr isotope ratios of examined samples from all the lithologies of the WDRC, including volcanic and dike units, are plot on the isochron yielding 586.8 ±10.1 (2se) Ma with initial ratio = 0.70333 ± 0.00011 as explained above (Figs. 6a and 7). Trachybasalts and trachyandesite of the dike unit are plotted on the lower segment of ⁸⁷Rb/⁸⁶Sr-⁸⁷Sr/⁸⁶Sr isochron defined by all the units of the WDRC including the plutonic and volcanic units. This clearly indicates that the formation age of the dike unit is indistinguishable from that of the plutonic and volcanic units. Even if only rocks from the dike unit are used, we obtain a two-points Rb-Sr isochron age of 577.8 ± 19.6 Ma, which is within the error range of the Rb-Sr isochron age of the WDRC based on 7 samples.

The values of ⁸⁷Sr/⁸⁶Sr of the trachybasalts from the dike unit are extremely higher and their ¹⁴³Nd/¹⁴⁴Nd values are much lower than those of young basalts from the Red Sea Rift and the coastal plain and marginal region of the Red Sea. The isotope ratios registered by the WDRC have never been reported from intra-plate basaltic volcanisms occurring in northeastern Africa and Arabia since late Mesozoic to Quaternary (Lucassen et al., 2008), which indicates the trachybasalts must be old enough to reconcile the inconsistency. See Supplementary Document (‘Geochronology’ in section COMPARISON OF TRACHYBASALT WITH YOUNG BASALTS; Supplementary Figs. S4a and S4b) for details. The Rb-Sr isochron plots (Fig. 7) and comparison with the young basalts from the Red Sea region demonstrate that the trachybasalts of the dike unit are as old as other lithologies of the WDRC.

Trace element compositions. The variation diagrams of trace element concentrations plotted against whole-rock SiO₂ contents show diverse variation patterns (Fig. 2), which are grouped into four (Table 3; See Supplementary Document, TRACE ELEMENT VARIATION PATTERNS). In spite of such diverse variation patterns, data of trachybasalts are plotted at the lowest SiO₂ end of the variations without exception. Moreover, plots of concentrations of two trace elements belonging to a particular variation pattern (e.g., Fig. S1) and those of two trace elemental ratios (e.g., Fig. 5) form consistent trends, in which the trachybasalts are plotted at the highest or lowest end (purple rectangles in Figs. 5 and S1). These behaviors of trace elements suggest that there is an intimate genetic link between trachybasalts of the dike unit and all the other rocks from the WDRC, which can be explained by fractional crystallization of a trachybasalt magma with negligible contribution of exotic materials (Saad et al., 2023). An involvement of a SiO₂-rich exotic material, which will be discussed below, is not responsible for the entire variations of trace and major elements of the WDRC. A SiO₂-poor magma mixed with a SiO₂-rich exotic material cannot be trachybasalt but should be richer in SiO₂ than trachyte. The notable inflexions or slope changes at SiO₂ = 62.5 wt% seen in Harker variation diagrams for major elements (Fig. 2a; Saad et al., 2023) cannot be explained by mixing of trachybasalt magma with an exotic material with SiO₂ content as high as or higher than syenogranite.

Trace element data of young basalts from the Red Sea Rift and the coastal plain and marginal region of the Red Sea are distinct from those of trachybasalts of the dike unit of the WDRC, which indicates the trachybasalts do not share source mantle and/or crustal open magmatic processes with the young basalts. See Supplementary Document (‘Trace elements’ in section COMPARISON OF TRACHYBASALT WITH YOUNG BASALTS; Figs. S4a, S4b, and S5) for details. The systematic trace element variations in the WDRC with the trachybasalt at the least-fractionated ends and the trace element characteristics of the trachybasalt distinct from those of the young basalts indicate (1) the trachybasalts have intimate genetic link with the other lithologies of the WDRC, (2) they have the least fractionated trace element composition, and (3) there is no genetic link with the young basalts from the Red Sea region.

All these lines of evidence from the major elements, geochronology, and trace elements demonstrate that the dike unit including trachybasalts formed during the final stage of the WDRC formation and is an important member of the complex. We contend that the trachybasalt represents a parental magma of the WDRC, from which the lithological and geochemical diversities were derived through fractionation (crystallization and crystal separation by a certain mechanism) with or without crustal assimilation or contamination. Mechanisms of the fractionation and assimilation are discussed below.

Fractional crystallization of trachybasalt to derive the geochemical diversity of the WDRC

Saad et al. (2023) applied stepwise fractional crystallization mass balance model to major element data of the WDRC, the results of which are listed in Table 5a (the first 4 columns) with estimated errors and statistics of the least-squares calculations. Complementary parent-daughter pairs were examined in this study (Table 5a, the last 3 columns). See Supplementary Document [‘Modeling stepwise fractional crystallization (Major elements)’ in section MODELING AND ERROR ESTIMATION PROCEDURES OF OPEN MAGMATIC PROCESSES] for modeling and error estimation procedures. Saad et al. (2023) concluded based on the geology, petrography, and variations of major elements of the WDRC that the rocks with SiO₂ > 62.5 wt% were derived from a trachyte magma by fractional crystallization. Because the trachybasalts of the dike unit are shown to be one of the essential members of the WDRC above, we contend regarding the major element compositions that most of the lithologies of the WDRC were derived from the trachybasalt by fractional crystallization.

Table 5. Results of mass balance calculations for major (a) and trace elements (b) by adopting stepwise fractional crystallization model to the Wadi Dib ring complex

(a)
Data type		Major elements
Source		Saad et al. (2023)				This study
Crystallization steps		TrBa → Trcht	Trcht → QbSy	QbSy → QzSy	QzSy → Sygrnt	TrBa → Syen	Trcht → QzSy*	QbSy → Sygrnt
All crystals (wt%)		57.7 ± 0.3	46.3 ± 16.5	43.8 ± 2.5	51.9 ± 2.6	61.5 ± 0.3	62.1 ± 1.4	69.8 ± 1.4
Abundances of fractioinated minerals (wt%)	Olivine	10.0 ± 0.79	-	-	-	9.7 ± 0.8	-	-
	Clinopyroxene	9.48 ± 0.55	2.0 ± 1.6	-	-	9.9 ± 0.6	-	−1.7 ± 0.4
	Amphibole	-	-	5.49 ± 0.31	5.73 ± 0.27	-	-	12.4 ± 0.7
	Anorthite	16.53 ± 0.28	6.1 ± 0.8	-	-	15.5 ± 0.3	7.6 ± 0.1	-
	Albite	11.57 ± 0.5	22.4 ± 9.6	22.72 ± 1.37	28.65 ± 1.35	16.1 ± 0.5	30.4 ± 0.7	36.2 ± 0.8
	Orthoclase	-	12.1 ± 5.9	13.6 ± 0.85	17.45 ± 0.86	-	18.3 ± 0.5	22.0 ± 0.5
	Magnetite ss	4.79 ± 0.49	3.4 ± 0.8	1.92 ± 0.09	-	5.2 ± 0.5	5.5 ± 0.1	0.82 ± 0.20
	Ilmenite ss	3.99 ± 0.62	-	-	-	3.9 ± 0.6	-	-
	Apatite	1.38 ± 0.16	0.21 ± 0.05	0.06 ± 0.01	0.11 ± 0.01	1.4 ± 0.2	0.26 ± 0.01	0.11 ± 0.01
Feldspar comp. (mol%)	X_An	57.4 ± 1.1	14.5 ± 4.4	-	-	48.5 ± 0.9	13.1 ± 0.2	-
	X_Ab	42.6 ± 1.1	56.6 ± 13.3	63.9 ± 2.0	63.5 ± 1.6	51.5 ± 0.9	55.5 ± 0.7	63.6 ± 0.7
	X_Or	-	28.9 ± 12.3	36.1 ± 2.0	36.5 ± 1.6	-	31.4 ± 0.7	36.4 ± 0.7
Statistical parameters	RSS	0.38	0.63	0.13	0.19	0.19	3.3	0.19
	χ²	10.4	13.5	66.9	127.2	5.14	22.7	171.9
	Prob Q	0.0056	0.0036	<0.0001	<0.0001	0.0764	<0.0001	<0.0001
	DGF	2	3	4	5	2	3	3
	Error type	from XRF	Trcht / Mean 3	from XRF	Grnt/ Mean 3	from XRF	Grnt/ Mean 3	Grnt/ Mean 3

(b)
Data type		Trace elements
Source		This study
Crystallization steps		TrBa → Trcht	Trcht → QbSy	QbSy → QzSy	QzSy → Sygrnt	TrBa → Syen	Trcht → QzSy*	QbSy → Sygrnt
Parent melt		TrBa, Mean §	Trcht, D18-F	QbSy, D3-A	QzSy, D6.2	TrBa, Mean §	Trcht, D18-F	QbSy, D3-A
Daughter melt		Trcht, D18-F	QbSy, D3-A	QzSy, D6.2	Grnt, D5.2	Syen, D14-C	QzSy, Mean §	Grnt, D5.2
Wt% crystallized ‡		57.0 ± 2.3	59.6 ± 3.1	−11.0 ± 0.8	28.1 ± 3.3	24.8 ± 3.2	53.2 ± 2.8	7.2 ± 2.6
Crystallization reaction stoichiometry (wt%)	Olivine	17.4	-	-	-	15.6	-	-
	Clinopyroxene	16.5	4.1	-	-	15.9	-	−2.3
	Amphibole	-	-	13.9	10.8	-	-	19.4
	Plagioclase †	48.9	58.3	57.7	54.2	50.9	60.0	50.3
	Orthoclase	0.0	24.8	34.5	33.0	-	28.8	30.5
	Magnetite ss	8.3	6.9	4.9	-	8.4	8.7	1.1
	Ilmenite ss	6.9	-	-	-	6.2	0.0	-
	Apatite ‡	1.86 ± 0.42	5.10 ± 0.68	−0.2 ± 3.5 #	1.81 ± 0.7	3.05 ± 1.07	2.14 ± 0.4	1.0 ¶
	Zircon ‡	-	1.12 ± 0.34	−10.9 ± 4.4 #	0.23 ± 0.11	-	0.52 ± 0.18	1.0 ¶
Reduced χ²		31	59	7.5	6.9	22	23	19

All errors are in ±1σ. Values of crystallization reaction stoichiometry except for apatite and zircon are constrained by the major element compositions, which is after Saad et al. (2023).

TrBa, trachybasalt; Trcht, trachyte; QbSy, quartz-bearing syenite; QzSy, quartz syenite; Sygrnt, syenogranite; QzSy*, quartz syenite (D14-A) and/or quartz alkali feldspar syenite (D20-B) from the outer ring intrusive member; Syen, syenite; ss, solid solution; X_An, 100Ca/(Ca + Na + K); X_Ab, 100Na/(Ca + Na + K); X_Or, 100K/(Ca + Na + K).

§, mean of D18-J and D56-A for trachybasalt and that of D20-B and D14-A for quartz syenite from the outer ring intrusive member; *, amounts of minerals that reproduce the whole-rock major oxide compositions; †, plagioclase stoichiometry = anorthite + albite, which are constrained with major elements; ‡, parameters constrained from trace element by least-squares method or assumed as indicated by ¶; ¶, not optimized but a value to maximize the crystallization fraction is given, which does not affect the fitting much; #, If apatite or/and zircon stoichiometry is constrained to be zero, the model does not reproduce the observation; RSS, residual sum of squares; χ², chi-squared; Prob Q, probability Q value (goodness-of-fit); DGF, degree of freedom; Error type, adopted error source for evaluation of χ²; Trcht/Mean 3, mean of 3 trachyte samples; from XRF, based on errors of XRF measurement; Grnt/Mean 3, mean of 3 syenogranite samples; reduced χ², reduced chi-squared for trace elements.

From the optimized assemblages and amounts of fractionated phases in the mass balance modeling of major elements as listed in Table 5a. There are three important points to be noted: (1) plagioclase, olivine, clinopyroxene, Fe-Ti oxides, and apatite were fractionated to derive trachyte from trachybasalt magma; (2) dominant alkali feldspar and plagioclase with lesser amounts of pyroxene, Fe-Ti oxide, and apatite to derive quartz-bearing syenite from trachyte magma; and (3) dominant alkali feldspar with lesser amount of amphibole to derive quartz syenite and syenogranite from the quartz-bearing syenite magma. If these modeling results are taken into consideration, the four groups of variation patterns of trace elements (Table 3; Fig. 2) are understood by incompatible/compatible behavior of trace elements during fractional crystallization. See Supplementary Document (‘Variation patterns on Harker diagrams’ in section TRACE ELEMENT VARIATION PATTERNS) for details of each group. The group 1 elements show always compatible behavior, and the group 2 elements always incompatible behavior. The group 3 elements change their behavior from incompatible up to ∼ 62.5 wt% SiO₂ to compatible above the SiO₂ content, which is attributable to a change of crystallization stoichiometry. The group 4 elements show incompatible behavior up to ∼ 62.5 wt% SiO₂ for all samples but they change either to continuously incompatible and weakly incompatible or even compatible above the SiO₂ content depending on samples. The continuous increase with further increase of SiO₂ is understood as consistent behavior as incompatible elements even above SiO₂ ∼ 62.5 wt%, though the decrease in various extent with further increase of SiO₂ may require other processes in addition to fractional crystallization.

These qualitative evaluations of behavior of trace elements indicate that the trace element diversities of the WDRC is mostly attributed to fractional crystallization starting with a trachybasalt parental magma whose composition is recorded in the dike unit. We apply mass balance model to trace element data to quantitatively examine how far fractional crystallization can explain the trace element abundances. If the model does not reproduce trace element abundances for some lithologies, we need to examine open processes for the lithologies, such as assimilation of crustal material with distinct geochemical characteristics, or elaborate fractionation mechanism, such as boundary layer fractionation.

We modeled trace element concentrations of the main lithologies covering the geochemical diversity of the WDRC by a least-squares method. See Supplementary Document [‘Modeling stepwise fractional crystallization (Trace elements)’ in section MODELING AND ERROR ESTIMATION PROCEDURES OF OPEN MAGMATIC PROCESSES] for modeling and error estimation procedures. A stepwise fractional crystallization model starting with the trachybasalt of the dike unit, which successfully reproduces the major element variations of the WDRC (Saad et al., 2023; Table 5a), was adopted. In the modeling, degree of crystallization and crystallization stoichiometries of apatite and zircon, which significantly affect trace element variations, are optimized by reproducing trace element concentrations of a chosen rock as a daughter melt from those of the other rock as a parent melt. The same parent-daughter pairs were examined as those adopted for the modeling based on major elements (Saad et al., 2023; Table 5). Batch crystallization is assumed for each fractional crystallization step and crystallization modes of minerals other than apatite and zircon are according to those constrained by major elements (Saad et al., 2023). Adoption of maximum (Rayleigh) fractional crystallization model does not alter the results much for incompatible trace elements so far as the extent of crystallization is less than ∼ 62.5 wt%. Used mineral-melt partition coefficients and their data sources are listed in Supplementary Table S2.

The modeling results are listed in Table 5b with estimated errors. The modeled trace element patterns normalized by the primitive mantle (PM) composition are plotted in Figure 8 with the observed patterns for daughter melts. The examined daughter melts are: (1) trachyte of the volcanic unit, (2) syenite of the outer ring, (3) quartz-bearing syenite of the inner ring 1, (4) quartz syenite of the inner ring 2, (5) syenogranite of the granitic core, and (6) quartz syenite from the ORIM. The parent melts are: trachybasalt from the dike unit for (1) and (2), trachyte of the volcanic unit for (3) and (6), quartz-bearing syenite of the inner ring 1 for (4), and quartz syenite of the inner ring 2 for (5). There is an overall consistency between the modeled and observed abundances and patterns (Fig. 8), though the reduced chi-squared values are mostly >20 (Table 5b). The optimized crystallization modes of apatite tend to be higher than those estimated from the major elements even if the errors are taken into consideration, which may be attributed to uncertainty in apatite-melt partition coefficient of phosphorus and other trace elements (Supplementary Table S2). The optimized crystallization model of zircon ranges from 0.2 ± 0.1 to 1.1 ± 0.3 except for derivation of the trachyte as a daughter melt from the trachybasalt parent, which does not require zircon crystallization (Table 5). The optimized degrees of crystallization are not necessarily consistent with those estimated from the major elements even if the errors are taken into consideration (compare ‘wt% crystallized’ for major and trace elements in Table 5) except for several cases.

Figure 8. Comparison of observed and modeled primitive mantle (PM) normalized trace element patterns of selected rocks from the WDRC. Shown by red and blue dots are PM normalized trace element patterns modeled with stepwise fractional crystallization starting from trachybasalt for selected rocks from the WDRC shown in colored solid symbols. The modeling results are listed in Table 5b. See Supplementary Document [‘Modeling stepwise fractional crystallization’ (Trace elements) in section MODELING AND ERROR ESTIMATION PROCEDURES OF OPEN MAGMATIC PROCESSES] for details of modeling procedure.

There are two cases in which modeling did not reproduce trace element patterns. The first case is derivation of quartz syenite of the inner ring 2 as a daughter melt (SiO₂ ∼ 66 wt%) from quartz-bearing syenite of the inner ring 1 as a parent melt (SiO₂ ∼ 63 wt%), the modeled pattern of which is extremely deviated from the observed values and is not shown in Figure 8. Moreover, the modeling results in negative values of crystallization extent and apatite and zircon reaction stoichiometries, implying an obvious modeling failure (the third column in Table 5b). This is because the most of trace element abundances of the daughter (quartz syenite) are lower than that of the parent (quartz-bearing syenite; cf. Figs. 3c and 4c), requiring derivation of the parent from the daughter by crystallization, which is clearly not the case from the major element compositions. The second case is derivation of syenogranite of the granitic core as a daughter melt (SiO₂ ∼72 wt%) from the quartz syenite parent melt, which results in poor fitting (Fig. 8d) and significantly low degree of crystallization than that constrained by major elements (the fourth column in Table 5b). Contrary to these cases of the SiO₂-rich rocks from the main lithology of the plutonic unit, quartz syenite and quartz alkali feldspar syenite from the ORIM are reasonably reproduced by crystallization from the trachyte (Fig. 8e; the sixth column in Table 5).

The disparity between the high SiO₂ rocks from the main lithology and those from the ORIM is also seen in variation behavior of trace elements. The variation patterns of trace elements from the WDRC are grouped into four as explained above (Table 3, Fig. 2). Trace elements of group 4 (e.g., LREEs-MREEs and Nb) exhibit variation patterns splitting into two trends with increase of SiO₂ higher than ∼ 62.5 wt%; one increases continuously and the other decreases to various extents (Figs. 2j-2l; Table 3), though their elements belonging to groups 1-3 behave similarly (Figs. 2b-2i; Table 3). The quartz syenite and quartz alkali feldspar syenite from the ORIM belongs to the former, whereas quartz syenite from the inner ring 2 and syenogranite from the granitic core belong to the latter. We thus conclude that other processes are necessary in addition to fractional crystallization to reproduce the disparity in group 4 trace element between rocks from the inner ring 2 and the granitic core and those from the ORIM. Frisch and Abdel-Rahman (1999) suggested minor crustal assimilation based on the high Y/Nb and Yb/Ta ratios of some SiO₂-rich rocks. We will explore such open processes below after examining Sr and Nd isotope ratios.

The modeled trace element patterns show that trachyte from the volcanic unit, syenite from the outer ring, quartz-bearing syenite from the inner ring 1, and quartz syenite and quartz alkali feldspar syenite from the ORIM are reasonably explained by stepwise fractional crystallization model (Figs. 8a-8c and 8e). However, the derivation of syenite of the outer ring as a daughter melt from the trachybasalt parent results in significantly low degree of crystallization (25 wt%; Table 3) than that constrained by the major elements (62 wt%). Moreover, a closer look at the trace element patterns in Figure 8 reveals some discrepancies between the observed and modeled patterns. This is more clearly seen in CI chondrite normalized ratios of REEs. See Supplementary Document, (‘Modeling simple fractional crystallization: CI chondrite normalized REE’ in section DETAILS OF RESULTS OF MODELING OPEN MAGMATIC PROCESSES; Supplementary Fig. S8; Supplementary Table S3) for details. Simple fractional crystallization with reaction stoichiometry constrained by the major element compositions cannot reproduce the REE ratios of trachyte and syenite from the trachybasalt and those of quartz-bearing syenite from the trachyte even if errors are taken into consideration. We thus conclude that the derivation of trachyte and syenite from trachybasalt and that of quartz-bearing syenite from trachyte require more elaborate fractionation mechanism. It will be shown below based on the isotope data that the fractionation does not require involvement of exotic material, and thus the chemical diversity should be developed from the trachybasalt parental magma via an elaborate fractionation mechanism without exotic material input.

Open magmatic processes (OMP) involving assimilation to derive geochemical diversity of the WDRC

Previous study on OMP involving assimilation. Frisch and Abdel-Rahman (1999) studied the WDRC and reported low Y/Nb and Yb/Ta ratios extending to the lower ends of the ocean island basalt field (e.g., Fig. S5a) and ⁸⁷Sr/⁸⁶Sr initial ratio (0.7048 ± 0.001) slightly higher than or comparable to the inferred value for the mantle at the time of the ring complex formation (0.70377 at 578 Ma for the bulk silicate earth; Workman and Hart, 2005) and argued for little or no crustal contamination. On the other hand, they reported high Y/Nb and Y/Ta ratios scattered towards the island arc basalt field in youngest quartz-rich lithologies and invoked heterogeneities in the magma chamber or crustal assimilation of island-arc type magmatic rocks, widespread in the Pan-African orogeny (e.g., Eliwa et al., 2006; Abdel Wahed et al., 2012; Obeid and Azer, 2015). Based on this, Frisch and Abdel-Rahman (1999) argued for assimilation of crustal materials during the late stages of WDRC evolution, though it was evaluated to be insignificant. Our data presented above qualitatively support the essence of their contention: assimilation in formation of the SiO₂-rich rocks to unknown extents. Frisch and Abdel-Rahman (1999) did not make quantitative evaluation of the role of crustal contamination or assimilation, which could have played an important role in thermal and material interaction of mantle-derived magmas with the continental crust.

Qualitative evaluation of OMP involving assimilation. The most unequivocal evidence for open magmatic process is obtained by examining isotope ratios of samples if exotic materials involved in the open process had isotope ratios distinct from those of the parental magma. For a present-day or recent magmatism, this can be made by using measured raw isotope ratios of samples as well as those of presumed exotic materials even for radiogenic isotopes. For examination of an ancient magmatism as old as ∼ 600 Ma, however, it is necessary to estimate radiogenic isotope ratios of samples registered at the time of closure of the isotope systems, which are called in this paper as ‘sample-specific initial isotope ratios’ or ‘sample initial ratios’ for short. See Supplementary Document, (‘Evaluation of isotopic heterogeneity at the time of closure of isotope systems’ in section ACCURATE GEOCHRONOLOGY AND EVALUATION OF INITIAL ISOTOPIC HETEROGENEITY; Supplementary Fig. S2) for details. These initial ratios must be distinguished from y-intercept of an isochron or reference isochron defined by the selected samples for age estimation, which is called ‘isochron (reference isochron) initial isotope ratio’ or ‘isochron initial ratio’ for short. In the following discussion, we use ‘sample initial ratio’ and ‘isochron initial ratio’ where meaning of ‘initial isotope ratio’ needs to be clarified. Estimation procedure of ‘sample initial ratio’ is explained in Supplementary Document, (‘Procedure of accurate geochronology for crustal intrusions: Approach combining evaluation of initial isotope ratios of samples’ in section ACCURATE GEOCHRONOLOGY AND EVALUATION OF INITIAL ISOTOPIC HETEROGENEITY), and the estimated values for the WDRC are listed in Table 6. The table bears information on isotopic heterogeneity of magmas solidified at the time of the ring complex formation (586.8 Ma).

Table 6. Estimated sample-specific initial ratios of Sr, Nd, and Pb isotopes for the Wadi Dib ring complex at 586.8 Ma

Isotope ratio	⁸⁷Sr/⁸⁶Sr		¹⁴³Nd/¹⁴⁴Nd		²⁰⁶Pb/²⁰⁴Pb		²⁰⁷Pb/²⁰⁴Pb		²⁰⁸Pb/²⁰⁴Pb
Sample	Initial ratio	2se	Initial ratio	2se	Initial ratio	2se	Initial ratio	2se	Initial ratio	2se
D2	0.70341	0.00020	0.511998	1.981E-05	18.126	0.0664	15.613	0.00460	37.904	0.0944
D3-A	0.69653	0.00401	0.512046	1.732E-05	18.536	0.1606	15.608	0.01071	37.862	0.2534
D5.2	0.70116	0.00382	0.512066	1.795E-05	19.188	0.3798	15.627	0.02518	37.723	0.4773
D6.2	0.70232	0.00134	0.512060	1.700E-05	18.892	0.1704	15.620	0.01136	38.115	0.2492
D7.1	0.70215	0.00117	0.512010	1.873E-05	18.203	0.1402	15.618	0.00938	37.784	0.2047
D14-A	0.70153	0.00260	0.512041	1.724E-05	18.613	0.4044	15.608	0.02680	36.205	0.5504
D14-C	0.70339	0.00039	0.512000	1.905E-05	18.600	0.0536	15.648	0.00380	37.911	0.0996
D16-F	0.70350	0.00065	0.512051	1.763E-05	18.339	0.1410	15.590	0.00943	37.723	0.3334
D18-F	0.70381	0.00080	0.512005	1.878E-05	18.155	0.0866	15.615	0.00589	38.305	0.1373
D18-J	0.70347	0.00012	0.511999	1.983E-05	17.534	0.0856	15.542	0.00582	37.034	0.1192
D18-K	0.70282	0.00014	0.512107	1.774E-05	17.947	0.0520	15.558	0.00370	37.592	0.0617
D20-B	0.70326	0.00213	0.512032	1.730E-05	18.052	0.2687	15.579	0.01784	37.085	0.3519
D56-A	0.70292	0.00018	0.512030	2.068E-05	18.177	0.0587	15.580	0.00411	38.013	0.0791
D58-B1	0.70316	0.00104	0.512010	1.964E-05	18.151	0.0428	15.616	0.00313	38.051	0.0586

Initial ratios are calculated for each sample at the Rb-Sr whole-rock isochron age of 586.8 Ma. ‘Augmented errors’ for the age (±13Ma) and isochron initial ⁸⁷Sr/⁸⁶Sr (±0.00033) were adopted to calculate errors of the listed initial isotope ratios. 2se, standard errors in 2s. See Supplementary Document, (‘Evaluation of isotopic heterogeneity at the time of closure of isotope systems’ in section ACCURATE GEOCHRONOLOGY AND EVALUATION OF INITIAL ISOTOPIC HETEROGENEITY; Fig. S2) for calculation procedures and error estimation.

Initial ratios of all the samples by correcting radiogenic ingrowth since 586.8 Ma (Table 6) are plotted with error bars of ±2se in Figure 9a against Sr contents and in Figure 9c against SiO₂ contents for ⁸⁷Sr/⁸⁶Sr and in Figure 9b against Nd contents and in Figure 9d against whole-rock SiO₂ contents for ¹⁴³Nd/¹⁴⁴Nd. The initial ratios of ¹⁴³Nd/¹⁴⁴Nd of all samples are plotted against those of ⁸⁷Sr/⁸⁶Sr and ²⁰⁶Pb/²⁰⁴Pb in Figures 9e and 9f, respectively. Initial ratios of Pb isotopes for all the samples (Table 6) were examined by plotting them against whole-rock SiO₂ and Pb contents as well as plotting their mutual values (Supplementary Document ‘Variations of sample initial ratios of Pb isotopes’ in section DETAILS OF GEOCHEMICAL VARIATIONS; Supplementary Fig. S6 and Table 6). The samples excluded in chronology have lower initial ⁸⁷Sr/⁸⁶Sr and higher ¹⁴³Nd/¹⁴⁴Nd and ²⁰⁶Pb/²⁰⁴Pb, indicating consistency between the different isotope systems in terms of deviation from the isochron or reference isochron initial ratios.

Figure 9. Relationship between whole-rock composition and initial values of Sr and Nd isotope ratios calculated for each sample (sample initial ratios) by correcting radiogenic ingrowth since the time of the WDRC formation at 586.8 Ma (Table 6). The initial ⁸⁷Sr/⁸⁶Sr and ¹⁴³Nd/¹⁴⁴Nd are plotted against Sr (a) and Nd (b) contents, respectively, and against whole-rock SiO₂ content [(c) and (d)]. The sample initial ratios for ¹⁴³Nd/¹⁴⁴Nd are plotted against those for ⁸⁷Sr/⁸⁶Sr in (e) and those for ²⁰⁶Pb/²⁰⁴Pb in (f). The initial ratios based on isochron for Sr (isochron initial ratio) and reference isochrons for Nd and Pb (reference isochron initial ratio) are shown with dashed lines. Thick vertical [and horizontal in (e) and (f)] bars on the dashed lines indicate ±2σ of the weighted mean of isochron initial ratios. Vertical [and horizontal in (e) and (f)] thick gray bars represent standard deviation (±2σ) of the initial ratios of the 7 samples selected for Rb-Sr isochron dating (Table 4). A mean value of initial ratios of the 7 samples for ²⁰⁶Pb/²⁰⁴Pb are shown with thin gray dashed line in (f). Error bars (±2σ) of the sample initial ratios are determined by propagating errors in isotope and elemental analyses and the augmented age error (Table 6). The augmented errors for the isochron initial ⁸⁷Sr/⁸⁶Sr (Table 4, footnote) are indicated with light gray bars behind the solid bars for the weighted mean of the sample initial ⁸⁷Sr/⁸⁶Sr.

The sample initial ratios are confined close to the isochron initial ratio for ⁸⁷Sr/⁸⁶Sr up to whole-rock SiO₂ ∼ 62.5 wt% and split into two groups for higher SiO₂ content (Fig. 9c). One group continues to be closer to the isochron initial ratio and the other decreases with increase in SiO₂. Because of the significant decrease of Sr contents with increase in SiO₂ (Fig. 2d), the sample initial ratios are almost constant at the values of isochron initial ratio for ⁸⁷Sr/⁸⁶Sr at high Sr contents and dramatically decreases at low Sr showing a large scatter (Fig. 9a). The sample initial ratios are confined close to the isochron initial ratio for ¹⁴³Nd/¹⁴⁴Nd up to whole-rock SiO₂ ∼ 62.5 wt% except for one outlier (trachyandesite from the volcanic unit, D18-K; See Supplementary Document (‘Trachyandesite with peculiar geochemical characteristics’ in section DETAILS OF GEOCHEMICAL VARIATIONS) for peculiarity of this sample. The sample initial ratios for ¹⁴³Nd/¹⁴⁴Nd tend to increase with increase of whole-rock SiO₂ content above 62.5 wt% (Fig. 9d). Because of the split of Nd contents with increase in SiO₂ above ∼ 62.5 wt% (Fig. 2l), the initial ratios also split into two trends in Figure 9b. One trend is characterized by increase of ¹⁴³Nd/¹⁴⁴Nd with keeping Nd content almost unchanged, and the other by a nearly constant ¹⁴³Nd/¹⁴⁴Nd with variable Nd contents showing a shift from the reference isochron initial ratio for samples with high Nd contents.

The initial ratios for ²⁰⁶Pb/²⁰⁴Pb and ¹⁴³Nd/¹⁴⁴Nd of each sample show a positive correlation and similar variation patterns when plotted against SiO₂ content (Figs. 9d, 9f, and S6d). There is a discrepancy in the initial ratios of ²⁰⁶Pb/²⁰⁴Pb between SiO₂-rich rocks from the main lithology (quartz syenite of the inner ring 1 and syenogranite of the granitic core) and those from the ORIM (Fig. S6).

The sample initial ratios of quartz syenite from the inner ring 2 and syenogranite from the granitic core for ⁸⁷Sr/⁸⁶Sr, ¹⁴³Nd/¹⁴⁴Nd, and ²⁰⁶Pb/²⁰⁴Pb show a large deviation from the respective isochron/reference isochron initial ratios, which is distinct in diagrams plotting sample initial ratios for ¹⁴³Nd/¹⁴⁴Nd against those for ⁸⁷Sr/⁸⁶Sr and ²⁰⁶Pb/²⁰⁴Pb (Figs. 9e and 9f) even if the errors are taken into consideration. The deviation may be explained only by open magmatic processes involving influx of a material with low ⁸⁷Sr/⁸⁶Sr, high ¹⁴³Nd/¹⁴⁴Nd, and high ²⁰⁶Pb/²⁰⁴Pb, which implies involvement of materials with more depleted geochemical characteristics than the WDRC at least in terms of Sr and Nd isotope systems. This is consistent with the behavior of trace elements belonging to group 4 (decrease in concentrations to various extent with increase of SiO₂; Figs. 2i-2l; Table 3) and the failure in fractional crystallization modeling for the two rocks (Fig. 8d; Table 5).

Quartz-bearing syenite from the inner ring 1 has a slightly higher sample initial ratios for ¹⁴³Nd/¹⁴⁴Nd (0.512046 ± 0.000017) (Fig. 9b) and ²⁰⁶Pb/²⁰⁴Pb (18.54 ± 0.16) (Fig. 9f; Table 6) than the reference initial values (0.512011 ± 0.000011 and 18.194 ± 0.217, respectively; Table 4). The difference is significant even if their uncertainties are taken into consideration, suggesting that an open magmatic process involving influx of a material with higher ¹⁴³Nd/¹⁴⁴Nd affected the quartz-bearing syenite magma. This tendency is also noticed for SiO₂-rich (>70 wt%) rocks from the ORIM (Figs. 9c and 9d), which have slightly higher initial ¹⁴³Nd/¹⁴⁴Nd than the reference isochron initial ratio, though the values are lower than that of the quartz-bearing syenite from the inner ring 1. This suggests effects of open magmatic processes even for the SiO₂-rich rocks from the ORIM, although the extent is minor.

Modeling procedures of OMP involving assimilation. Outline of modeling scheme of open magmatic processes is illustrated in Figure 10c. We quantify an open magmatic process to derive quartz syenite of the inner ring 2 and syenogranite of the granitic core from a magma that filled the magma body before the formation of the inner ring 2 (7 and 8 in Fig. 10c). We also quantify an open magmatic process to derive quartz-bearing syenite of the inner ring 1 from a magma that filled the magma body before the formation of the inner ring 1 (6 in Fig. 10c). For modeling the open magmatic processes for the two cases, we use the relationship between Nd contents and sample initial ratio for ¹⁴³Nd/¹⁴⁴Nd (Fig. 9b). The relationship between Sr contents and sample initial ratio for ⁸⁷Sr/⁸⁶Sr are not used because of the large errors of the estimated initial ratios (Figs. 9a and 9c), which is principally attributed to the large values of ⁸⁷Rb/⁸⁶Sr for SiO₂-rich rocks. The quartz-bearing syenite has a significantly low initial ⁸⁷Sr/⁸⁶Sr with large error (0.69653 ± 0.003), which is attributed to selective depletion of Sr during alteration as discussed above.

Figure 10. Magma evolution processes involved in the formation of chemical diversity in the WDRC. The parental magma of the WDRC, which is observed as trachybasalt of the dike unit, was derived from a primary magma formed in the mantle, which is not observed, through deep fractional crystallization in the lithospheric mantle or the bottom of the crust near the Moho [1 in (a)]. The trachybasalt parental magma of the WDRC underwent fractional crystallization in the crust to form a precursor magma [2 in (a)], from which trachyte of the volcanic unit and syenite and monzonite of the outer ring formed through BLF. In the BLF, a fractionated melt formed in the roof and sidewall boundary layers of a magma body from the precursor magma [3 in (b)] played an important role. The segregated fractionated melt transported through the boundary layer reacted with the mushy boundary layer inducing feldspar accumulation to form syenite of the outer ring [4 in (b)]. It mixed with the precursor melt upon eruption to formed trachyte of the volcanic unit [5 in (b)]. The trachyte magma underwent AFC to have solidified as the inner ring 1 [6 in (c)]. The quartz-bearing syenite magma underwent AFC involving an exotic melt derived from crustal materials to form SiO₂-rich rocks of the inner ring 2 and the granitic core [7 in (c)]. Contrary to this, quartz syenite and quartz alkali feldspar syenite of the outer ring intrusive member were derived from a magma dominantly by fractional crystallization with an earlier stage of minor assimilation [8 in (c)]. The fractionated melt was estimated in AFC modeling for the quartz-bearing sysnite (gray arrow marked with 3′).

Assimilation and fractional crystallization (AFC) model (Depaolo, 1981; O’Hara and Mathews, 1981; Kelemen, 1986; Aitcheson and Forrest, 1994; Spera and Bohrson, 2001; Bohrson and Spera, 2001; Ozawa, 2001) is adopted. In modeling AFC processes a two-step model consisting of batch crystallization followed by two-component mixing was adopted in this study. See Supplementary Document (‘Modeling assimilation and fractional crystallization’ in section MODELING AND ERROR ESTIMATION PROCEDURES OF OPEN MAGMATIC PROCESSES) for details of modeling method including the two-step approach and error estimation procedure.

For modeling quartz syenite from the inner ring 2 and syenogranite from the granitic core as daughter melts (products of AFC process), we assume a magma with the quartz-bearing syenite composition as assimilant [as defined by Thompson et al. (2002)], which occupied the WDRC magma body before the formation of the inner ring 2 (Saad et al., 2023). We need to determine composition of respective exotic materials involved in the AFC processes [assimilate used by Ghiorso and Kelemen (1987) and later defined by Thompson et al. (2002), though assimilant was used for exotic materials in several literatures such as Grove et al., 1988 and Spera and Bohrson (2001)] in addition to reaction parameters, such as crystallization degree, mass fraction of influxed material, and reaction stoichiometry, which poses an underdetermined problem. Thus, it is assumed that the same assimilate was involved in the formation of the two daughter melts through AFC to resolve this problem. Trace element contents and Nd isotope ratio of the common assimilate were optimize by reproducing the observed trace element and Nd isotope compositions combining the two samples as daughter melts (quartz syenite and syenogranite magmas).

For modeling quartz-bearing syenite from the inner ring 1 as a daughter melt, we assume a magma with the trachyte composition as assimilant, which occupied the WDRC magma body before the formation of the inner ring 1 (Saad et al., 2023). Because only one sample is available, a pattern of trace elements for the assimilate is assumed to be similar to that constrained from the quartz syenite and syenogranite as daughter melts, as explained above, and the level of the trace element abundance was optimized by least-squares method. See Supplementary Document (‘Modeling assimilation and fractional crystallization’ in section MODELING AND ERROR ESTIMATION PROCEDURES OF OPEN MAGMATIC PROCESSES) for details of modeling method.

As described in Supplementary Document (‘Two data sets for trace elements’ in section DETAILS OF RESULTS OF MODELING OPEN MAGMATIC PROCESSES), two data sets were used to modeling open magmatic processes: one includes only REEs and the other includes all trace elements measured with ICP-MS. The merits of usage of the two data sets in least-squares approaches are explained in the Supplementary Document. The following discussion is developed based on data sets limiting elements to REEs. Modeling results for AFC based on all trace elements are explained in Supplementary Document (‘Assimilation and fractional crystallization model based on all incompatible trace elements and major elements’ in section DETAILS OF RESULTS OF MODELING OPEN MAGMATIC PROCESSES; Supplementary Figs. S9-S11 and Table S4).

Modeling results of OMP involving assimilation. The results of AFC modeling for cases limiting elements to REEs are listed in Table 7, in which reaction parameters, reproduced daughter melt compositions, and optimized assimilate composition are tabulated with estimated errors. The optimized degree of crystallization based on REEs are 35 ± 5, 66.6 ± 9.4, and 57 ± 15% and the assimilate mass relative to the mass of daughter melt are 53.0 ± 1.7, 63.6 ± 2.5, and 16.9 ± 9.2% for the quartz syenite, syenogranite, and quartz-bearing syenite, respectively (Table 7). The most important reaction parameters optimized are mass of assimilate (exotic melt) influxed relative to mass of assimilant (quartz-bearing syenite or trachyte magma) crystallized. The ratio of the two values corresponds to ratio of mass assimilated against mass crystallized (Ma/Mc; Kelemen, 1986). The Ma/Mc attains 1-2 for the quartz syenite and syenogranite implying significant involvement of the exotic material, whereas the value is ∼ 0.15 for quartz-bearing syenite indicating minor assimilation. The differences are significant even if the large errors of Ma/Mc are taken into consideration (Table 7).

Table 7. Results and conditions for AFC modeling for the Wadi Dib ring complex by using rare earth elements

(a) Modeling results for quartz syenite from the inner ring 2 and syenogranit from the granitic core
Rare earth elements (ppm)	Observed daughter		Parent melt	Estimated Influxed exotic melt	Modeled fractionated melt		Modeled daughter		Reaction parameters			QzbSy-QzSy	QzbSy-Grnt
	Qz syenite	Syenogranite	Qz-b syenite		Qz syenite	Syenogranite	Qz syenite	Syenogranite				Qz syenite	Syenogranite
	D6.2	D5.2	D3-A		D6.2 **	D5.2	D6.2	D5.2				D6.2	D5.2
La	109.3	98.0	165.3	21.2 ± 2.7	197.8 ± 20.5	219.9 ± 10.2	104.3 ± 4.1	93.6 ± 4.6	Rare earth elements (Reduced χ² = 3.9)	Mass wt%	Crystallized §	35 ± 4.9 *	66.6 ± 9.4
Ce	175.1	159.6	271.9	55.2 ± 1.9	313.4 ± 32.1	347.1 ± 13.5	176.6 ± 5.3	161.6 ± 5.9		Mass wt%	Influxed ‡	53.0 ± 1.7	63.6 ± 2.5
Pr	17.1	15.8	26.7	6.01 ± 0.28	29.6 ± 3.0	33.7 ± 1.2	17.1 ± 0.4	16.1 ± 0.4		Mass ratio	Ma/Mc	2.09 ± 0.40	0.88 ± 0.38
Nd	51.9	49.4	83.7	19.5 ± 2.1	88.3 ± 8.9	101.4 ± 3.0	51.85 ± 0.00	49.4 ± 0.00		Reaction stoichiometry (wt%) #	Cpx	0.0	−2.76
Sm	8.01	8.44	13.5	4.21 ± 0.6	12.96 ± 1.30	15.77 ± 0.37	8.33 ± 0.28	8.42 ± 0.34			Amphibole	18.0 †	0.0
Eu	1.14	0.60	0.82	1.22 ± 0.11	0.38 ± 0.05	0.23 ± 0.02	0.83 ± 0.06	0.86 ± 0.07			Plagioclase	46.6	60.2
Gd	6.47	7.56	11.7	3.49 ± 0.67	10.81 ± 1.09	14.90 ± 0.57	6.93 ± 0.34	7.65 ± 0.39			Afs	27.9	36.5
Tb	0.97	1.22	1.73	0.52 ± 0.11	1.58 ± 0.16	2.32 ± 0.11	1.01 ± 0.05	1.18 ± 0.06			Magnetite_ss	3.94	1.36
Dy	5.61	7.34	10.1	2.79 ± 0.7	9.32 ± 0.94	14.59 ± 0.91	5.86 ± 0.35	7.09 ± 0.41			Ilmenite_ss	0.0	0.0
Ho	1.22	1.59	2.03	0.69 ± 0.12	1.88 ± 0.19	3.07 ± 0.22	1.25 ± 0.06	1.56 ± 0.08			Apatite	3.6 †	4.6 †
Er	3.57	5.00	5.80	2.22 ± 0.43	5.53 ± 0.55	9.59 ± 0.89	3.78 ± 0.24	4.90 ± 0.27			Zircon	0.0	0.0
Tm	0.59	0.84	0.92	0.37 ± 0.08	0.91 ± 0.09	1.60 ± 0.17	0.62 ± 0.04	0.82 ± 0.04	Estimation from major elements ¶
Yb	4.10	5.82	6.23	1.96 ± 0.6	6.44 ± 0.64	11.74 ± 1.5	4.06 ± 0.32	5.52 ± 0.28	Mass wt%		Crystallized §	35	66.6
Lu	0.62	0.85	1.00	0.22 ± 0.11	1.07 ± 0.11	2.07 ± 0.33	0.62 ± 0.06	0.90 ± 0.06	Mass wt%		Influxed ‡	28.0 ± 6.5	48.5 ± 5.6
¹⁴³Nd/¹⁴⁴Nd	0.512060	0.512066	0.512046	0.51212 (1)	-	-	0.512061 (1)	0.512065 (1)	Mass ratio		Ma/Mc	0.72 ± 0.49	0.47 ± 0.16

#, Values of reaction stoichiometry are mostly constrained by major elements [Saad et al. (2023) and Table 5]; *, Not optimized with error comparable to that for D5.2; **, Errors are calculated by propagating errors of the parent melt trace element concentrations and the error of crystallized fraction for D6.2 (See *); †, Not optimized and plausible values are assigned with tryal and error approach; §, Mass fractions relative to the parent melt mass; ‡, Mass fractions relative to the final assimilated melt mass; ¶, The mass crystallized was given to be consistent with that from REEs and the mass influxed were optimized by least-squares approach by assuming that the major element composition of the exotic melt was the same for melts crystallized as samples D6.2 and D5.2. Fractional crystallization path for major elements are estimated from D20-B and D14-C samples from the ORIM, for which effects of AFC are shown to be limited.

Ma/Mc, mass assimilated divided by mass crystallized (Kelemen, 1986); χ², chi-squared; Cpx, clinopyroxene; Afs, alkali feldspar; Qz, quartz; ss, solid solution; Qz-b, quartz-bearing; QzbSy-QzSy, derivation of quartz syenite from quartz-bearing syenite; QzSy-Grnt, derivation of syenogranite from quartz-bearing syenite.

(b) Modeling results for quartz-bearing syenite from the inner ring 1
Rare earth elements (ppm)	Obs daughter	Parent melt	Estimated Influxed exotic melt	Estimated fractionated melt ***	Mdl daughter	Reaction parameters			Trchyt-QzbSy
	Qz-b syenite	Trachyte			Qz-b syenite				Qz-b syenite
	D3-A	Mean *			D3-A				D3-A
La	165.3	124.7	104 ± 33	177 ± 21	164.5 ± 4.8	Rare earth elements (Reduced χ² = 0.3)	Mass wt%	Crystallized §	57 ± 15 **
Ce	271.9	224.7	147 ± 46	298 ± 33	272.9 ± 2.4		Mass wt%	Influxed ‡	16.9 ± 9.2
Pr	26.7	23.8	13.7 ± 4.3	29.5 ± 3.1	26.6 ± 0.06		Mass ratio	Ma/Mc	0.15 ± 0.11
Nd	83.7	80.4	41.7 ± 13.2	92.1 ± 9.4	83.6 ± 0.0		Reaction stoichiometry (wt%) #	Cpx	3.5
Sm	13.49	14.7	7.4 ± 2.3	14.5 ± 1.45	13.3 ± 0.17			Amphibole	16.2
Eu	0.82	2.06	0.54 ± 0.17	0.78 ± 0.10	0.74 ± 0.04			Plagioclase	49.9
Gd	11.69	13.0	5.62 ± 1.77	12.15 ± 1.22	11.05 ± 0.03			Afs	21.2
Tb	1.73	1.96	0.79 ± 0.25	1.79 ± 0.18	1.62 ± 0.00			Magnetite_ss	5.95
Dy	10.09	11.0	4.77 ± 1.50	10.25 ± 1.03	9.32 ± 0.03			Ilmenite_ss	0.0
Ho	2.03	2.23	1.00 ± 0.32	2.10 ± 0.21	1.91 ± 0.01			Apatite	3.24
Er	5.80	6.17	2.74 ± 0.86	6.07 ± 0.61	5.50 ± 0.00			Zircon	0.0
Tm	0.92	0.92	0.54 ± 0.17	0.94 ± 0.09	0.87 ± 0.02	Estimation from major elements ¶
Yb	6.23	5.93	2.98 ± 0.94	6.59 ± 0.67	5.98 ± 0.00	Mass wt%		Crystallized §	38
Lu	1.00	0.87	0.56 ± 0.18	1.02 ± 0.10	0.94 ± 0.02	Mass wt%		Influxed ‡	9.2
¹⁴³Nd/¹⁴⁴Nd	0.512046	0.512010	0.5124 (4)	-	0.512046 (17)	Mass ratio		Ma/Mc	0.15

*, mean of trachytes, D7.1 and D18-F; ¶, The mass crystallized and mass influxed were optimized by trial and error approach with constrained values of Ma/Mc to be consistent with those estimated from trace elements within the maximum values of possible degree of crystallization without material influx estimated from major elements, which are 46 wt% for the quartz-bearing syenite; **, Not optimized and plausible values are assigned with tryal and error approach with error comparable to that for D5.2; Mdl daughter, modeled daughter melt; ***, Errors are calculated by propagating errors of the parent melt trace element concentrations and the error of crystallized fraction for D6.2; Trchyt-QzbSy, derivation of quartz-bearing syenite from trachyte.

CI chondrite normalized REE patterns of reproduced daughter melts in the AFC modeling by using REE data are shown in Figures 11a-11c. The REE concentrations of the quartz syenite (Fig. 11a), syenogranite (Fig. 11b), and quartz-bearing syenite (Fig. 11c) are successfully reproduced by the modeled daughter melts (blue dots in Fig. 11). The optimized REE concentrations in the assimilates are higher than or in the higher range of those of 740-600 Ma Neoproterozoic granitoids from the North Eastern Desert (Eliwa et al., 2014) (Fig. 11d). The optimized assimilate has LREE-enriched CI chondrite normalized pattern with or without Eu anomaly, which is similar to some of the Neoproterozoic granitoids.

Figure 11. CI chondrite normalized REE patterns of melts reproduced by adopting AFC model to the WDRC with those of observed REE patterns. The modeling results [blue dots in (a)-(c)] are compared with quartz syenite of the inner ring 2 [green triangle; (a)], syenogranite of the granitic core [light blue diamond; (b)], and quartz-bearing syenite of the inner ring 1 [yellow square; (c)]. Only REEs were used for the modeling, the results of which are listed in Table 7. Shown in (d) are optimized chondrite normalized patterns of exotic melts supposed to have been influxed in the AFC process: joined open circles for modeling (a) and (b) and open squares for modeling (c). The patterns are compared with those of Neoproterozoic granitoids from the North Eastern Desert (Eliwa et al., 2014). Shown by red dots in (c) is a fractionated melt, which was derived from the trachyte magma by fractional crystallization and to which the exotic melt [open square in (d)] was added to derive quartz-bearing syenite shown as blue dots in (c). See Supplementary Document [‘Modeling assimilation and fractional crystallization’ (Trace elements) in section MODELING AND ERROR ESTIMATION PROCEDURES OF OPEN MAGMATIC PROCESSES; bottom panel in Fig. S7] for details of modeling procedure.

Relationships between Nd contents and Nd isotope ratio of reproduced daughter melts in the AFC modeling by using REE data are shown in Figure 12. The relationships for the quartz syenite, syenogranite, and quartz-bearing syenite are well reproduced in the AFC modeling. The optimized Nd contents of assimilates are lower than any rocks from the WDRC (Table 7) and are comparable to those of Neoproterozoic granitoids (Eliwa et al., 2014) from the North Eastern Desert and the Dokhan volcanics from the Eastern Desert (Makovicky et al., 2016) and southern Sinai (Be’eri-Shlevin et al., 2011) (Figs. 11d and 12). The estimated Nd isotope ratio of the assimilate is 0.512121 (Table 4) for the quartz syenite and syenogranite pair and is slightly higher than age-corrected Nd isotope ratios of the Neoproterozoic granitoids from the North Eastern Desert and Dokhan volcanics from the southern Sinai. The estimated Nd isotope ratio of the assimilate for the quartz-bearing syenite (0.5124; Table 4) is much higher than Nd isotope ratio at 586.8 Ma of the observed country rocks including the Dokhan volcanics from the Eastern Desert (Makovicky et al., 2016) but is comparable to the depleted DMM (depleted MORB mantle: 0.512248) at 586.8 Ma (Workman and Hart, 2005) if large uncertainties originated from the Nd concentrations of the trachyte and Nd isotope ratio of the quartz-bearing syenite are taken into consideration (Table 7b).

Figure 12. Relationship between Nd contents and ¹⁴³Nd/¹⁴⁴Nd values reproduced by AFC modeling. The modeling results are compared with observed Nd contents and sample initial ratios for ¹⁴³Nd/¹⁴⁴Nd at 586.8 Ma of three rocks from the WDRC, which are taken from Figure 9b. The three rocks modeled as daughter melts are quartz-bearing syenite from the inner ring 1 (yellow square), quartz syenite from the inner ring 2 (green triangle), and syenogranite from the granitic core (light blue square). The modeling results are shown with open square for the quartz-bearing syenite, open triangles for the quartz syenite, and open diamond for the syenogranite. The modeling is based on REE data (See Table 7). The Nd isotope ratios at 586.8 Ma and Nd concentrations of parent melts (assimilants) are indicated by open circles: mean trachyte from the volcanic unit for the quartz-bearing syenite and quartz-bearing syenite for the other two rocks. The Nd isotope ratios and concentrations of influxed exotic melt (assimilate) optimized by modeling are shown by crosses, one of which for the quartz-bearing syenite is out of the plot range. Dashed curves are AFC path assuming a linear relationship between crystallization and assimilation, which implies an assumption of constant Ma/Mc (mass assimilated/mass crystallized; Kelemen, 1986). Data for Neoproterozoic granitoids are after Eliwa et al. (2014) (plus) and those of Dokhan volcanics are after Be’eri-Shlevin et al., (2011) (small star) and Makovicky, et al., (2016) (large star), three of which are larger than 0.51215 (Nd = 25.8-26.5, ¹⁴³Nd/¹⁴⁴Nd = 0.512151-0.512164).

CI chondrite normalized REE ratios of reproduced daughter melts in the AFC modeling by using REE data are shown in Figure 13. The REE ratios of the quartz syenite, syenogranite, and quartz-bearing syenite are well reproduced by the modeled daughter melts (Fig. 13; compare open triangles, open diamonds, and open square with light blue triangle, green diamond, and yellow square, respectively) if the estimated errors are taken into consideration, which cannot be reproduced by stepwise fractional crystallization model (cf. Fig. S8).

Figure 13. CI chondrite normalized REE ratios [(Lu/Yb)_N, (Nd/Sm)_N, and (La/Nd)_N] of melt compositions reproduced or optimized in application of BLF and AFC models (open symbols) to the data from the WDRC, plotted against (Dy/Yb)_N. The colored plot points for the WDRC are taken from Figures 5b-5d. The modeling results are based on REE data and are listed in Table 7. The shapes of open symbols are the same as those of respective observed rocks. Precursor magma (open circle with plus) is a magma initially present in a magma body beneath the WDRC derived by fractional crystallization of trachybasalt magma. Fractionated melt (open circle with cross) is a melt responsible for BLF process derived by fractional crystallization of the precursor magma. Error bars represent ±1σ standard errors calculated by Monte Carlo method. See the text and Figure S7 for details of BLF and AFC modeling procedures.

The results of AFC modeling using all trace elements including extents of crystallization and material influx and calculated Ma/Mc are essentially the same as those obtained by limiting elements to REEs as explained in Supplementary Document [‘Assimilation and fractional crystallization model based on all incompatible trace elements and major elements’ (Trace elements and Nd isotope ratio) in section DETAILS OF RESULTS OF MODELING OPEN MAGMATIC PROCESSES; Figs. S9-S11 and Table S4], if errors are taken into consideration. Major element compositions of daughter melts and exotic melts in the AFC modeling are also estimated, which is explained in Supplementary Document [‘Assimilation and fractional crystallization model based on all incompatible trace elements and major elements’ (Major elements) in section DETAILS OF RESULTS OF MODELING OPEN MAGMATIC PROCESSES; Supplementary Fig. S12 and Table S5]. The major element composition of the quartz syenite, syenogranite, and quartz-bearing syenite are well reproduced as daughter melts in the AFC modeling. The estimated major element compositions of exotic melts are rich in SiO₂ and are comparable to those of Neoproterozoic granitoids from the North Eastern Desert (Eliwa et al., 2014).

From the estimated characteristics of REEs, trace elements, Nd isotope ratios, and major elements of the assimilate, we argue that the exotic melt caused AFC of the inner rings and the granitic core was derived from the country rock of the WDRC, Neoproterozoic granitoids or Dokhan volcanics. The REE contents of the estimated exotic melt is comparable to or higher than those of Neoproterozoic granitoids (Eliwa et al., 2014) by up to a factor of 5 (Fig. 11d). This implies that the exotic melt could have undergone various degree of partial melting of the granitoids ranging from 20 to 100% if the bulk partition coefficients are assumed to range 0.1-0.01. See Supplementary Document (‘Geochemical characteristics of exotic material involving AFC’ in section DETAILS OF RESULTS OF MODELING OPEN MAGMATIC PROCESSES) for geochemical characteristics of the optimized exotic materials.

The AFC model adopted in this study consists of the first step of batch crystallization and the second step of two-component mixing. The results of the first step are regarded as fractionated melts that did not underwent any assimilation, which could have been formed in a magma body beneath the WDRC. The compositions of such fractionated melts based on REEs are listed with estimated errors in Table 7 and that related to the quartz-bearing syenite are plotted in Figure 11c with red dots. See Supplementary Document [‘Assimilation and fractional crystallization (AFC) model based on all incompatible trace elements and major elements’ (Estimation of fractionated melt compositions) in section DETAILS OF RESULTS OF MODELING OPEN MAGMATIC PROCESSES; Tables S4 and S5 and Figs. S9c, S12, and S13] for estimated fractionated melt compositions based on all trace elements and those of major elements. The concentrations of major and trace elements of the fractionated melts involved in the formation of the quartz syenite and syenogranite (daughter melts) are notably distinct from the daughter melts, whereas those involved in the formation of the quartz-bearing syenite are similar to the daughter melt. This is attributed to the difference in Ma/Mc; 1-2 for the quartz syenite and syenogranite but ∼ 0.15 for the quartz-bearing syenite.

The trace element concentrations of the fractionated melts estimated from the quartz syenite and synogranite are similar to those of quartz syenite and quartz alkali feldspar syenite from the ORIM (Figs. S13e and S13f). It is inferred that magmas crystallized as SiO₂-rich rocks of the ORIM underwent fractional crystallization after certain extents of AFC less extensive than that of the quartz-bearing syenite magma. The assimilation extents are evaluated to be 60-86% of that of the quartz-bearing syenite with the same Ma/Mc by using the initial Nd isotope ratios slightly lower than that of the quartz-bearing syenite (Fig. 12). The integrated Ma/Mc values for the SiO₂-rich rocks of the ORIM after additional fractional crystallization without assimilation are accordingly estimated to be ∼ 0.094. Such two-step formation of the SiO₂-rich rocks of the ORIM is supported by the fact that a single stage AFC with constant Ma/Mc and exotic melt composition in a plausible range cannot explain the observed relationship between Nd contents and sample initial ratios for ¹⁴³Nd/¹⁴⁴Nd. Another possibility of a two-step AFC process is the first stage of fractional crystallization followed by the second stage of assimilation, which is negated by considering phase relation in the ‘Petrogeny’s Residua System’ (Bowen, 1937) as discussed below.

It is notable that the rocks with SiO₂ > 62.5 wt% form tight trends on Harker variation diagrams for major elements even if they recorded various extents of assimilation ranging from dominantly fractional crystallization to extensive AFC with Ma/Mc ∼ 1-2. This implies that a liquid line of descent of SiO₂-rich rocks attributed principally to fractional crystallization almost coincides with a mixing trend in the WDRC. This is due to the similarity of major element compositions of melts derived from fractional crystallization to those of the exotic melts involved in AFC. Contrary to this, the trace element abundances belonging to variation group 4 and CI chondrite normalized REE ratios plotted against SiO₂ contents (Figs. 2j-2l and 5a) and the carefully estimated sample initial ratio for Nd isotopes (Fig. 9b) allow the distinction. The reason of the coincidence of liquid line of descent and mixing trend will be explained below with its implications.

It is concluded that the SiO₂-rich rocks from the inner ring 2 and the granitic core of the WDRC are solidified magma that derived from the quartz-bearing syenite magma through significant assimilation and fractional crystallization with Ma/Mc >1. The influxed material is a melt derived from the host Neoproterozoic granitoids by 20% partial melting to total melting. The Ma/Mc for quartz-bearing syenite from the inner ring 1 is ∼ 0.15, which is much smaller than Ma/Mc for quartz syenite from the inner ring 2 and syenogranite from the granitic core. Trachyte and syenite do not require any AFC process, whereas magmas crystallized as the quartz syenite and quartz alkali feldspar syenite of the ORIM, which formed after the formation of the outer ring (Table 1), recorded AFC with integrated Ma/Mc ∼ 0.094. These indicate that assimilation of the host country rocks was negligible in an earlier stage up to the period when the outer ring formed (Ma/Mc ∼ 0) but became more effective with time and attains very high in a later stage of the magma evolution (1-2 in Ma/Mc). Implications of the temporal and spatial variation of Ma/Mc are discussed below.

OMP involving boundary layer to derive geochemical diversity of the WDRC

Previous study on OMP involving boundary layer. Based on geology, petrography, and whole-rock major element compositions of the WDRC, Saad et al. (2023) invoked boundary layer fractionation (BLF) as plausible fractionation mechanism to form the chemical diversity of the WDRC, in which formation of a sidewall and roof boundary layer was followed by collapse of the roof boundary layer to segregate interstitial melt towards the top of the magma body. Saad et al. (2023) pointed out similarity of major element compositions of trachyte of the volcanic unit and syenite of the outer ring of the plutonic unit with low SiO₂ and Al₂O₃ and high Fe₂O₃ contents (low-SiO₂ syenite) as defined in Supplementary Document (‘Geochemical contrasts between trachyte of the volcanic unit and syenite of the outer ring’ in section DETAILS OF GEOCHEMICAL VARIATIONS) and their predominance in rocks solidified during the early stage of the WDRC evolution and argued that the boundary layer process started with a magma having the trachyte (and low-SiO₂ syenite) composition and proceeded without significant replenishment or crustal assimilation.

Qualitative evaluation of OMP involving boundary layer. As mentioned above, the trace element compositions of the examined trachytes and syenites are clearly distinct even if the major element compositions are similar though they are not exactly the same (Table S1). This distinction of the trachyte and the syenite in trace elements is the key to model the BLF processes. The critical differences of the trachyte and syenite are (1) the concentrations of most of trace elements at the similar SiO₂ contents are higher in the trachyte than in the syenite except for Sr, Eu, and Ba (Fig. 2), (2) the CI chondrite normalized REE pattern of trachyte is characterized by more steeply inclined LREEs and more gently inclined HREEs than that of syenite (Figs. 3a, 3d, and 5) and are more similar to those of quartz-bearing syenite from the inner ring 1 (Figs. 3c and 3d), (3) the PM normalized trace element pattern of trachyte is similar to that of quartz-bearing syenite (Figs. 4c and 4d), and (4) the syenite has a minor positive Eu anomaly whereas the trachyte has a negative Eu anomaly (Figs. 3a and 3d). The distinctions (1)-(3) suggest more advanced fractionation of trachyte than syenite and (4) suggests accumulation of feldspar for syenite because the major element compositions show that derivation of the syenite from the trachybasalt requires significant fractionation of plagioclase, which must result in a substantial negative Eu anomaly. We thus propose a chemical evolution scheme of BLF as illustrated in Figure 10b.

In order to explain the disparity of trace element compositions of trachyte and syenite, we assume a magma initially residing in a magma body beneath the WDRC formation site, which is called ‘precursor magma’ (Fig. 10b). The precursor magma had a chemical composition distinct from both of the trachyte and syenite magmas, which were thought to be the initial magma in Saad et al. (2023) by disregarding their minor differences in the major element compositions (the first two columns of Table 9a). It is presumed that distinction of trace element compositions of trachyte and syenite and minor differences of the major element compositions formed from the precursor magma through distinct fractionation mechanisms. The precursor magma was derived by fractional crystallization of the trachybasalt parental magma of the WDRC in a deep crustal level before its emplacement into the magma body beneath the formation depth of the WDRC (2 in Fig. 10a). From the facts that simple fractional crystallization of trachybasalt cannot explain the trace element ratios of trachyte and syenite as discussed above and that the trace element abundance and patterns of trachyte is more similar to that of quartz-bearing syenite (Figs. 3 and 4), it may be reasonable to assume that a fractionated melt with chemical composition similar to the quartz-bearing syenite formed in the sidewall boundary layer and was transported through the sidewall boundary layer reacting with crystal mush of syenite and that the fractionated melt was responsible for the trace element disparity of trachyte and syenite (4 and 5 in Fig. 10b). This process is a reactive melt transportation in the sidewall boundary layer advocated by Saad et al. (2023) based on the petrographic evidence, such as formation of small clinopyroxene grains along the rim of amphibole in syenites.

Modeling procedures of OMP involving boundary layer. Since the quartz-bearing syenite underwent AFC process with Ma/Mc ∼ 0.15, we cannot use its composition as the fractionated melt. However, we estimated a fractionated melt from the trachyte by batch crystallization of trachyte before mixing with an exotic material to derive the quartz-bearing syenite, the composition of which is similar to the quartz-bearing syenite (3′ in Fig. 10b; Table 7b for REEs). We use major and trace element compositions of this fractionated melt in BLF modeling. We do not know the trace and major element compositions of the precursor magma and thus we need to estimate them. See Supplementary Document (‘Modeling boundary layer fractionation’ in section Doc. MODELING AND ERROR ESTIMATION PROCEDURES OF OPEN MAGMATIC PROCESSES) for the BLF modeling and error estimation procedure. As in the AFC modeling two data sets (REEs and all incompatible trace elements), were examined. The following discussion is based on the data sets limiting elements to REEs. Modeling results for BLF based on all trace elements are explained in Supplementary Document [‘Boundary layer fractionation model based on all incompatible trace elements and major elements (Trace elements)’ in section DETAILS OF RESULTS OF MODELING OPEN MAGMATIC PROCESSES].

Modeling results of OMP involving boundary layer. The optimized model parameters in BLF modeling based on REEs are listed in Table 8 with estimated errors. The optimized crystallization degree to derive the precursor magma from the trachybasalt magma is 52.8 ± 6.1 wt%, which is smaller than that constrained by simple fractional crystallization based on major elements (57.7 ± 0.3 wt%; Table 5). The optimized feldspar addition to reproduce syenite composition is 39.0 ± 5.8 wt% (Table 8), which is ∼ 1/2 of the total feldspar in syenites of the outer ring (Saad et al., 2023) suggesting significant accumulation of feldspar in the boundary layer. The optimized fractionated melt additions to reproduce syenite and trachyte compositions are 10.4 ± 6.2 and 41.1 ± 11.3 wt%, respectively (Table 8), which shows higher contribution of the fractionated melt to the trachyte magma erupted on the surface than the syenite solidified in the sidewall boundary layer even if errors are taken into consideration. A small amount of amphibole crystallization is optimized (Table 8), which improves the fitting but is not so important.

Table 8. Results of modeling boundary layer fractional crystallization for the Wadi Dib ring complex based on rare earth elements by least-squares method

Rare earth elements (ppm)	Parent melt	Observed daughter		Fractionated melt estim. from D3-A	Precursor magma	Feldspar accumulated for syenite	Mdeled daughter		Reaction parameters (Reduced χ² = 0.64)		Fsp accum, frac mlt add †	Frac melt addition ††	Trachybasalt- precursor magma †††
	Trachybasalt	Syenite	Trachyte				Syenite	Trachyte			Fsp accum, frac mlt add †	Frac melt addition ††
	Mean *	Mean **	Mean ***				Mean **	Mean **			Syenite	Trachyte
La	51.3	70.7	114.0	177 ± 20	91.4 ± 8.9	59.8 ± 6.1	71.7 ± 5.0	125.5 ± 7.6	Mass wt%	Crystallized §	-	-	52.8 ± 6.1
Ce	101.2	137.1	204.2	298 ± 32	177 ± 17	114 ± 11	132 ± 8	225 ± 12		Melt add ‡	10.4 ± 6.2	41.1 ± 11.3	-
Pr	12.2	15.6	21.8	29.5 ± 3.1	20.9 ± 2.1	13.3 ± 1.4	15.0 ± 0.9	24.4 ± 1.3		Fsp Accum ‡	39.0 ± 5.8	-	-
Nd	47.5	56.8	73.1	92.1 ± 9.3	80.6 ± 8.6	50.8 ± 5.7	55.1 ± 4.0	85.2 ± 6.3	Mass ratio	Ma/Mc	0.17 ± 0.14	0.62 ± 0.33	-
Sm	9.6	10.66	13.36	14.5 ± 1.5	15.8 ± 2.3	9.91 ± 1.47	10.4 ± 1.3	15.26 ± 1.96	Reaction stoichiometry (wt%)	Olivine	-	-	15.8
Eu	2.78	3.79	2.36	0.78 ± 0.11	2.17 ± 007	4.59 ± 0.36	4.20 ± 0.25	1.61 ± 0.15		Cpx	-	-	16.1
Gd	8.7	9.43	11.75	12.1 ± 1.2	14.6 ± 3.0	9.16 ± 1.88	9.48 ± 1.75	13.57 ± 2.60		Amphibole	-	-	0.78 ± 1.6
Tb	1.23	1.38	1.78	1.79 ± 0.18	2.08 ± 0.49	1.33 ± 0.32	1.38 ± 0.30	1.96 ± 0.43		Plagioclase	-	-	51.5
Dy	6.6	7.46	9.79	10.2 ± 1.0	11.3 ± 2.8	7.09 ± 1.78	7.42 ± 1.67	10.87 ± 2.41		Afs	-	-	0.0
Ho	1.25	1.44	1.98	2.10 ± 0.21	2.18 ± 0.59	1.41 ± 0.38	1.48 ± 0.36	2.15 ± 0.50		Magnetite_ss	-	-	8.51
Er	3.37	3.80	5.51	6.07 ± 0.61	5.98 ± 1.63	3.77 ± 1.03	4.01 ± 0.96	6.02 ± 1.36		Ilmenite_ss	-	-	6.29
Tm	0.44	0.53	0.81	0.94 ± 0.09	0.79 ± 0.21	0.51 ± 0.14	0.56 ± 0.12	0.85 ± 0.17		Apatite	-	-	1.0
Yb	2.68	3.27	5.23	6.59 ± 0.66	4.9 ± 1.2	3.19 ± 0.77	3.54 ± 0.69	5.59 ± 0.92		Zircon	-	-	0.0
Lu	0.40	0.50	0.78	1.02 ± 0.10	0.74 ± 0.18	0.48 ± 0.11	0.53 ± 0.10	0.85 ± 0.13	Added Fsp ¶	X_An	41 ± 21	Pl mol%	29 ± 10

*, mean values of trachybasalts (D18-J and D56-A); **, mean values of syenite D14-C and monzonite D2; ***, mean values of trachytes (D7.1 and D18-F); §, mass fraction relative to the parent melt mass; ‡, mass fraction relative to the final accumulated and assimilated melt mass; ¶, anorthite content of plagioclase [X_An = 100Ca/(Ca + Na + K)] and fraction of plagioclase [Pl mol% = 100Pl/(Pl + Afs)] for added feldspars, which are optimized with the major element data (Table S6); Ma/Mc, mass assimilated divided by mass crystallized (Kelemen, 1986); Cpx, clinopyroxene; Afs, alkali feldspar; ss, solid solution; †, mass fractions of feldspar (Fsp Accum) with anorthite content of plagioclase (X_An) and plagioclase fraction in feldspar (Pl mol%), and fractionated melt addition (Melt add); ††, mas fraction of fractionated melt addition; †††, crystallization reaction stoichiometry and crystallized fraction to derive precursor magma from trachybasalt magma.

The reproduced CI chondrite normalized REE patterns of trachyte and syenite based on REE data (red dots in Fig. 14a and blue dots in Fig. 14b) are comparable with the observed respective patterns. The reproduced chondrite normalized REE ratios of trachyte and syenite are plotted in Figure 13 (open circle and open inverted triangle). As seen in Figures 13 and 14, the REE abundances, patterns, and ratios of the trachyte and syenite are well reproduced. The CI chondrite normalized REE pattern of the precursor magma is shown in Figure 14a (green dots) in comparison with trachyte. Its normalized REE ratios are shown in Figure 13 (circle with plus). The REE abundances of precursor magma are lower than those of trachyte for LREEs and HREEs, whereas their MREEs are comparable. The CI chondrite normalized REE pattern of the bulk composition of precursor magma with feldspar accumulation is shown in Figure 14b (green dots) in comparison with syenite. The REE abundances are lower than those of syenite for LREEs and comparable for MREEs and HREEs with weak positive Eu anomaly due to feldspar accumulation. Modeling results based on trace elements are essentially the same as those obtained from the REE data as explained in Supplementary Document [‘Boundary layer fractionation model based on all incompatible trace elements and major elements’ (Trace elements) in section DETAILS OF RESULTS OF MODELING OPEN MAGMATIC PROCESSES].

Figure 14. CI chondrite normalized REE patterns of reproduced magma compositions by adopting BLF model. The modeling results are compared with the observed CI chondrite normalized REE patterns of trachyte and syenite of the WDRC (Figs. 3d and 3a, respectively). The BLF modeling is based on REEs and the results are shown with blue, red, and green dots. The observed data of trachyte from the volcanic unit is shown with filled blue inverted triangle for the mean and open blue inverted triangle for each trachyte sample in (a). The observed data of syenite from the outer ring is shown with filled red circle for the mean and open red circle for each syenite sample in (b). Modeling results plotted in panel (a) in comparison with trachytes are: (1) optimized precursor magma in green and (2) that reproduced the trachyte [precursor magma + fractionated melt (red dots in Fig. 11c)] in red. Modeling results plotted in panel (b) are: (1) the bulk composition of the precursor magma with feldspar accumulation in green and (2) that reproduce the syenite [precursor magma + accumulated feldspar (Fsp) + fractionated melt] in blue. The results are listed in Table 8. See Supplementary Document (‘Modeling boundary layer fractionation’ in section MODELING AND ERROR ESTIMATION PROCEDURES OF OPEN MAGMATIC PROCESSES; bottom panel in Fig. S7) for details of modeling procedure.

The major element compositions of the precursor magma estimated from syenite and trachyte are consistent with each other (within 20% differences) if errors are taken into consideration. See Supplementary Document [‘Boundary layer fractionation model based on all incompatible trace elements and major elements’ (Major elements) in section DETAILS OF RESULTS OF MODELING OPEN MAGMATIC PROCESSES; Fig. S12] for detail. The mean values are listed in Table 9a with estimated errors. The mean SiO₂ content is ∼ 58.9 ± 1.3 wt% and are plotted on the variation trends of the WDRC shifted to the lower SiO₂ side than the trachyte and syenite compositions. Parameters optimized in mass balance calculations deriving major element composition of the precursor magmas from the trachybasalt magma are listed in Table 9b with estimated errors, from which reaction stoichiometries were calculated and listed in Table 9c. The optimized crystallization degree is 54.5 ± 0.3 wt%, smaller than the value for derivation of the trachyte from the trachybasalt by simple fractional crystallization. The calculated reaction stoichiometries for derivation of the precursor magma and trachyte from the trachybasalt are essentially the same.

Table 9. Estimated major element compositions of magmas initially resided in a chamber beneath the Wadi Dib ring complex (precursor magmas) and results of related mass balance calculation

(a) Major element compositions of rocks and a magma used in estimation of precursor magma
Source	Saad et al. (2023)			This study
Oxides (wt%)	Syenite *	Trachyte *	Qz-b syenite *	PrcM	FracM
SiO₂	61.3	61.1	64.5	58.9 ± 1.3	63.3 ± 0.8
TiO₂	0.46	0.54	0.27	0.72 ± 0.23	0.31 ± 0.17
Al₂O₃	19.1	17.8	16.4	18.1 ± 0.6	16.9 ± 0.2
Fe₂O₃	4.1	5.9	5.3	7.22 ± 0.83	5.38 ± 0.53
MnO	0.10	0.16	0.16	0.14 ± 0.06	0.18 ± 0.04
MgO	0.47	0.65	0.18	0.79 ± 0.30	0.38 ± 0.22
CaO	2.7	2.1	1.3	2.94 ± 0.95	0.79 ± 0.53
Na₂O	6.7	6.3	6.3	6.18 ± 0.71	6.92 ± 0.46
K₂O	5.0	5.2	5.5	4.86 ± 0.69	5.81 ± 0.45
P₂O₅	0.08	0.12	0.05	0.14 ± 0.04	0.07 ± 0.03
Total	100.0	100.0	100.0	100.0	100.1
	Parameters for the accumulated feldspars		X_An	41 ± 15
	Parameters for the accumulated feldspars		Pl mol %	29 ± 8

All errors are in ±1σ. Major element composition of precursor magma, PrcM, which is the mean value of estimates from trachyte by subtracting quartz-bearing syenite in the amount estimated from the trace element modeling and from syenite by adding feldspars with plagioclase composition (X_An) and its amount relative to alkali feldspar (Pl mol%, plagioclase mole%) optimized to keep consistency between the two estimates. The composition of alkali feldspar is the value commonly observed in syenite and quartz-bearing syenite of the WDRC (X_Or = 40).

*, Mean values of D14-C and D2 for syenite, those of D18-F and D7.1 for trachyte, and those of D3-A and D2 for quartz-bearing syenite (Qz-b syenite).

FracM, fractionated melt estimated by modeling quartz-bearing syenite by assimilation and fractional crystallization model as a residual melt after the first stage of batch crystallization. The values are taken from the last column of Table S5.

(b) Mass balance calculation results related to the precursor magma
Mass balance calculation pair		Crystallization of trachybasalt		Derivation of qz-bearing syenite		PrcM → FracM
Mass balance calculation pair		TrBa → Trcht †	TrBa → PrcM	Trcht → QbSy †	PrcM → QbSy	PrcM → FracM
All crystals (wt%)		57.7 ± 0.3	54.5 ± 0.3	46.3 ± 16.5	63.8 ± 14.7	61.2 ± 15.9
Abundance of fractioinated minerals (wt%)	Olivine	10.0 ± 0.79	10.23 ± 0.79	-	-	1.03 ± 0.18
	Clinopyroxene	9.48 ± 0.55	9.27 ± 0.55	2.0 ± 1.6	4.0 ± 1.5	2.92 ± 0.79
	Amphibole	-	-	-	-	-
	Anorthite	16.53 ± 0.28	14.66 ± 0.28	6.1 ± 0.8	9.86 ± 0.83	8.86 ± 0.25
	Albite	11.57 ± 0.5	11.27 ± 0.5	22.4 ± 9.6	28.5 ± 8.7	27.3 ± 9.0
	Orthoclase	-	-	12.1 ± 5.9	15.7 ± 5.2	15.41 ± 5.41
	Magnetite ss	4.79 ± 0.49	3.6 ± 0.49	3.4 ± 0.8	5.5 ± 0.8	5.38 ± 0.78
	Ilmenite ss	3.99 ± 0.62	4.15 ± 0.62	-	-	-
	Apatite	1.38 ± 0.16	1.3 ± 0.16	0.21 ± 0.05	0.28 ± 0.05	0.27 ± 0.04
Feldspar comp. (mol%)	X_An	57.4 ± 1.1	55.1 ± 1.2	14.5 ± 4.4	17.7 ± 3.6	15.1 ± 3.4
	X_Ab	42.6 ± 1.1	44.9 ± 1.2	56.6 ± 13.3	54.2 ± 9.1	56.3 ± 9.9
	X_Or	-	-	28.9 ± 12.3	28.2 ± 8.1	28.6 ± 8.9
Statistical parameters	RSS	0.38	0.7	0.63	0.8	0.1
	χ²	10.4	18.4	13.5	15.9	5.0
	Prob Q	0.0056	0.0001	0.0036	0.0012	0.0816
	DGF	2	2	3	3	2
	Error type	from XRF	from XRF	Trcht / Mean 3	Trcht / Mean 3	from XRF

(c) Estimated reaction stoichiometry for crystallization reaction calculated from (b)
Crystallization raction		Crystallization of trachybasalt		Derivation of qz-bearing syenite		PrcM → FracM
Crystallization raction		TrBa → Trcht †	TrBa → PrcM	Trcht → QbSy †	PrcM → QbSy	PrcM → FracM
Crystallization reaction stoichiometry (wt%)	Olivine	17.3 ± 1.2	18.8 ± 1.2	-	-	1.7 ± 0.4
	Clinopyroxene	16.4 ± 0.9	17.0 ± 0.9	4.3 ± 3.5	6.2 ± 2.4	4.8 ± 1.5
	Amphibole	-	-	-	-	-
	Anorthite	28.6 ± 0.8	26.9 ± 0.8	13.2 ± 3.6	15.4 ± 2.7	14.5 ± 2.5
	Albite	20.0 ± 0.8	20.7 ± 0.9	48.4 ± 12.6	44.6 ± 8.4	44.6 ± 9.1
	Orthoclase	-	-	26.2 ± 10.9	24.6 ± 7.0	25.2 ± 7.6
	Magnetite ss	8.3 ± 0.8	6.6 ± 0.8	7.3 ± 2.4	8.7 ± 1.8	8.8 ± 1.9
	Ilmenite ss	6.9 ± 1.0	7.6 ± 1.1	-	-	-
	Apatite	2.4 ± 0.3	2.4 ± 0.3	0.5 ± 0.2	0.4 ± 0.1	0.4 ± 0.1

Abbreviations of rock names: Trcht, trachyte; QbSy, quartz-bearing syenite; TrBa, trachybasalt; PrcM, initially resided magma in the WDRC magma chamber (precursor magma); FracM, fractionated melt estimated by modeling quartz-bearing syenite by assimilation and fractional crystallization model as a residual melt after the first stage of batch crystallization. Abbreviations of mineral compositions: ss, solid solution; X_An, 100Ca/(Ca + Na + K), X_Ab; 100Na/(Ca + Na + K), X_Or; 100K/(Ca + Na + K). Other abbreviations: ss, solid solution; comp., composition; RSS, residual sum of squares; χ², chi-squared; Prob Q, probability Q value (goodness-of-fit); DGF, degree of freedom; Error type, adopted error source for evaluation of χ², which was chosen to maximize errors; †, after Saad et al. (2023).

Advanced modeling of OMP involving boundary layer. A fractionated melt formed in a boundary layer of the magma body beneath the WDRC played an important role in derivation of trachyte of the volcanic unit and syenite of the outer ring through boundary layer fractionation (4 and 5 in Fig. 10b). The composition of fractionated melt was estimated from quartz-bearing syenite of the inner ring 1 by evaluating contribution of assimilation and fractional crystallization separately from the trace element concentrations and sample Nd isotope ratio (gray arrow marked with 3′ in Fig. 10). In order to substantiate the estimated fractionated melt formed in the boundary layer, it is necessary to show that the melt can be derived from the precursor magma, which initially filled the magma body beneath the WDRC (3 in Fig. 10b). We tested batch crystallization and maximum Rayleigh fractional crystallization models, but such simple models do not reproduce the trace element composition of the fractionated melt, particularly REE ratios. See Supplementary Document [‘Multi-stage BLF model: derivation of fractionated melt from precursor magma’ (Simple fractional crystallization model) in section DETAILS OF RESULTS OF MODELING OPEN MAGMATIC PROCESSES] for application results of the simple models.

A multi-stage model involving crystallization, crystal separation, and mixing is invoked to overcome the problems of the simple models. See Supplementary Document (‘Modeling multi-stage BLF: Derivation of fractionated melt from precursor magma’ in section MODELING AND ERROR ESTIMATION PROCEDURES OF OPEN MAGMATIC PROCESSES) for the model and details of its modeling and error estimation procedures. The model considers extensive open environment of the sidewall boundary layer of a magma body. Contrary to the single-stage fractional crystallization models, the multi-stage crystallization/crystal separation/mixing model reasonably reproduces CI chondrite normalized REE pattern, primitive mantle normalized trace element patterns, and chondrite normalized REE ratios of the fractionated melt from the precursor magma with several exception of highly incompatible trace elements (Cs and Ba). See Supplementary Document [‘Multi-stage BLF model: derivation of fractionated melt from precursor magma’ (Multi-stage fractionation model) in section DETAILS OF RESULTS OF MODELING OPEN MAGMATIC PROCESSES; Supplementary Figs. S15 and S16] for details of modeling results.

Open magmatic process in the earlier stage of the WDRC formation: BLF

During the earlier stage of the WDRC formation, assimilation is limited because Ma/Mc ∼ 0.0. The successful application of the BLF model, which nicely reproduce trace element compositions of trachytes of the volcanic unit and syenites of the outer ring (Figs. 13, 14, and S14), substantiates operation of BLF mechanism proposed by Saad et al. (2023) based on studies on geology, petrography, and whole-rock major element compositions. Saad et al. (2023) argued that the trachyte magma erupted without significant fractionation from its similarity of the major element compositions to that of syenite of the outer ring (Fig. 20a in Saad et al., 2023). The distinct trace element compositions of the trachytes and the syenites, overall higher concentrations of incompatible trace elements in the former and higher concentration of elements compatible to feldspars in the latter, require an enrichment of incompatible trace elements in the trachyte magma (more advanced fractionation) and accumulation of feldspars in the syenite.

The advanced fractionation of the trachyte magma before its eruption is attributed to BLF through mixing of the initially resided magma (precursor magma) with a fractionated melt formed in the sidewall boundary layer. The fractionated melt has a quartz-bearing syenite like trace and major element compositions and can be produced from the precursor magma by multiple stages of crystallization/crystal separation/mixing (Figs. S15 and S16), which is part of BLF. The light fractionated melt formed in a boundary layer transported though the mushy sidewall boundary layer and was segregated in the upper boundary layer mush, a part of which might have seeped out and ponded at the top of the main magma body (Fig. 15a). This fractionated melt might have been mixed with the precursor magma upon eruption triggered by collapse of the upper boundary layer with keeping the Neoproterozoic carapace intact (Fig. 15b). This is followed by a large-scale collapse of the granitoids carapace accompanied by caldera formation on the surface (Fig. 15c) as advocated by Saad et al. (2023).

Figure 15. Schematic illustration of processes involved in formation of the WDRC in the early stage [(a)-(c)] and that in the later stage [(d)-(f)]. The illustrations are modified after Saad et al. (2023). The panel (a) is a modified version of (b) of Saad et al. (2023). Panel (b) is a snapshot between (c) and (d) of Saad et al. (2023). Panel (c) is the same as (d) of Saad et al. (2023). Panels (d)-(f) are modified versions of (f)-(h) of Saad et al. (2023). Precursor magma represents a magma initially occupied the magma body formed beneath the level of WDRC formation. Panels (a)-(c) depict a behavior of fractionated melt formed in the sidewall boundary layer, which played an important role in formation of syenites of the outer ring and trachytes of the volcanic unit. Panels (d)-(f) depict a behavior of exotic melt formed in the bottom boundary layer by melting of sunken granitoids in the early stage [panel (c)], which played an important role in formation of quartz-bearing syenite of the inner ring 1, quartz syenite of the inner ring 2, and syenogranite of the granitic core. See the text for details.

The optimized amounts of fractionated melt addition are 4-10% for syenite of the outer ring and ∼ 40% for trachyte of the volcanic unit (Tables 8 and S6). The difference between the syenite and the trachyte is distinct even if the errors are taken into consideration. Significantly higher contribution of a fractionated melt to trachyte than to syenite implies efficient magma mixing before the eruption of trachyte magma. The optimized accumulation of feldspar as high as 21-39% for the syenite is significant even if the estimated errors are taken into consideration (Tables 8 and S6). Because the syenite represents solidified boundary layer mush, the mechanism of accumulation of feldspar must not involve crystal settling in a melt dominant environment. It is attributed to reactive melt transportation, precipitating feldspar by reaction of the mush with migrating melt, the lines of evidence for which are provided by petrography of the WDRC (Saad et al., 2023), such as oligoclase-alkali feldspar intergrowth and reaction rim consisting of clinopyroxene and feldspars bordering anhedral amphibole. The very high reaction stoichiometry of feldspars, as high as 84%, required to derive the fractionated melt from the precursor melt, which is constrained by major elements (Table 9c), is consistent with significant precipitation of feldspars. Very high extent of crystallization, as high as 81-93%, to derive the fractionated melt from the precursor magma by multi-stage fractionation (Supplementary Table S7), which is constrained by trace elements, suggests low melt fraction of the boundary layer. It is consistent with the optimized amounts of fractionated melt addition to the syenites (4-10%) if errors are taken into consideration. These values may correspond to migrating melt entrapped in the syenite, which are comparable to the volume fraction of interstitial fine-grained parts developed in syenite (Saad et al., 2023).

Open magmatic process in the late stage of the WDRC formation: AFC

The AFC process contended above took place extensively in the later stage of the WDRC formation by incorporating materials of the country rocks. The Ma/Mc increased with increase in whole-rock SiO₂ content and with time from ∼ 0.0 (SiO₂ < 62.5 wt%) for the volcanic unit and outer ring, ∼ 0.15 (SiO₂ ∼ 64 wt%) for inner ring 1 and 1-2 (SiO₂ 68-73 wt%) for inner ring 2 and granitic core (Table 7). The variation is valid even if the estimated errors in Ma/Mc are taken into consideration. The Ma/Mc also shows spatial variation: it increases from the complex margin to the center. The quartz syenite of the inner ring 2 and the syenogranite of the granitic core occurring in the center of the WDRC without direct contact with the country rocks, mostly Neoproterozoic granitoids with subordinate distribution of Dokhan volcanics (Fig. 1), experienced the most extensive AFC process. There are no enclaves of the country rocks in the inner ring 2 and the granitic core (Saad et al., 2023). In order to explain the spatial variation of Ma/Mc, a particular mechanism is necessary. The crustal material cannot have derived from the upper zone of the magma body because the WDRC was overlain directly by the volcanic pile subsided from the surface by caldera formation and because the initially present host material as carapace of the magma body had sunk by stoping mechanism (Saad et al., 2023). It is also difficult to derive materials from the country rock through the sidewalls because a mostly solidified crust consisting of the outer ring inhibited such material interaction. Therefore, the crustal material must have derived from the bottom of the magma body (Fig. 15d).

Because the earlier stoping of the crustal rock carapace sunk to the bottom, the material could have undergone melting due to heating by the magma body, where heat loss was inefficient as compared with the sidewall and roof of the magma body. The melt formed in the bottom boundary layer might have segregated due to the density contrast or by filter pressing induced by settling of collapsed roof materials on the bottom (Fig. 15e). The reason why AFC became extensive in the later stage is due to protracted melting of the country rocks and slow segregation through the boundary layer. The collapse of the upper boundary layer and sinking of the detached mush on the bottom are coupled to enhance mixing of fractionated melt with the main magma body as well as emplacement of the mixed magma into the space formed by the collapse of the roof boundary layer (Fig. 15f). The estimated Ma/Mc ∼ 1 implies that heat of crystallization was consumed by melting of the host rocks, which might have been heated near the solidus before initiation and enhancement of melting (Thompson et al., 2002).

The spatial and temporal changes of thermal environments of the sidewall and bottom boundary layers of the magma body beneath the WDRC are contrasting in that efficiency of heat loss decreases with depth, which is an important factor in controlling processes of the sidewall and bottom boundary layers. The initial stage when the precursor magma formed a magma body starts with heating the country rocks accompanying minor partial melting (Saad et al., 2023), which took place almost homogeneously around the magma body. However, as time went by, temperature decreases rapidly in a shallower level due to effective heat loss from the surface accompanying solidification followed by collapse of the roof zone. By contrast, the bottom boundary layer was continuously heated up aided by a slow heat loss to the country rocks increasing temperature and inducing substantial melting.

The mechanism of AFC clarified above for the WDRC is contrasting to that reported in the Abu Khurq ring complex (AKRC) in the Eastern Desert (Landoll et al., 1994). The AKRC is 7.4 km in size and much larger than the WDRC (2.2 km) (Saad et al., 2023). It has a nepheline syenite core and marginal lithology rich in quartz (quartz syenite), which is associated with gabbros. The marginal quartz-rich lithology was explained by AFC processes with up to 20% assimilation of crustal material (Landoll et al., 1994). The association of gabbros and the marginal distribution of quartz syenite are features contrasting to the WDRC, which has syenogranite core and no association of gabbros. Saad et al. (2023) attributed the contrast to the differences in magma temperature and style of convection in a hidden magma body. This study reveals that thermal environment of the host crust of a crustal magma body, which was initially similar but became diverse depending on (1) depth of the magma body, (2) frequency of recharge, and (3) strength of thermal convection, played an essential role in the style of AFC processes. In the AKRC, repeated recharge of high temperature magma and induced thermal convection efficiently heated up the country rocks enhancing their melting and suppressing development of thick sidewall boundary layers, which is not the case in the WDRC.

IMPLICATIONS

The formation model concerning the early and late stages of evolution of a magma body formed beneath the WDRC via open magmatic processes discussed above could have implications on processes in present-day active magma bodies situated in the shallow crust (magma storage system or magma plumbing system). This is substantiated by comparing the ancient magma body beneath the upper boundary layer recorded as the WDRC, with present-day magma storage systems observed with geophysical observations (Paulatto et al., 2022). The comparison shows that processes inferred to have taken place in the magma body beneath the WDRC are applicable to small and shallow magma bodies with high melt fraction (<∼ 5 km in size, <∼ 7 km in depth, and >60% melt fraction; Saad et al., 2023; Costa et al., 2009). See Supplementary Document (CRUSTAL MAGMA CHAMBER AND RELEVANCE OF THE WDRC; Supplementary Fig. S17) for details of the comparison. We will explore implications for processes involved in (1) role of sidewall boundary layer in BLF, (2) role of bottom boundary layer in AFC, and (3) major element fractionation of critically undersaturated alkaline magmas, which has relevance to (2).

BLF: role of sidewall boundary layer

A role of sidewall boundary layer in magmatic fractionation has been pointed out by Marsh (2007) and McBirney (1984) based on conceptual models, McBirney et al. (1985) based on analog experiments, and Spera et al. (1995) based on numerical simulation in a simple system with sidewall boundary layer. This study proposed detailed mechanisms of sidewall boundary layer fractionation based on both extrusive and intrusive rocks combining evidence from trace element data of the early formed rocks of the WDRC in addition to its geology and petrography (Saad et al., 2023).

The outer ring of the WDRC represents an upper boundary layer of a cylindrical magma body and the volcanic unit represents magmas erupted on the surface above the magma body in the earlier evolution stage of the WDRC (Saad et al., 2023; Figs. 15a and 15b). They are contemporaneous and provide direct access to information on both the uppermost sidewall boundary layer (outer ring) and magma resided at the top of the magma body (volcanic unit). The information suggests operation of BLF by coupling formation of fractionated melt in the sidewall boundary layer, its migration to upper boundary layer, and collapse of the upper boundary layer to induce extensive mixing followed by volcanic eruption.

There are two important implications for BLF in small, shallow, and melt dominant magma bodies, which are based on trace element data from the WDRC in addition to Saad et al. (2023). Firstly, fractionated melt was produced via repeated fractional crystallization and its mixing with the precursor melt as the fractionated melt ascent through the boundary layer and accumulated to the upper boundary layer. Such process is realized if melt in the main magma body flows into the boundary layer at all depths to mix with the melt becoming more fractionated as it ascent if the sidewall is vertical or inclined outwards. Second, efficient mixing of the precursor magma and the fractionated melt transferred from the sidewall boundary layer to be ponded in and beneath the upper boundary layer took place before or during trachyte magma eruption triggered by upper boundary layer collapse. Such efficient boundary layer fractionation is attributable to cooperation of vertical or outward inclined sidewall boundary layer as an environment of fractionated melt formation and upper boundary layer as an environment of the fractionated melt accumulation and mixing with the main magma body by forced convection. BLFs expected to operate solely in upper or bottom boundary layers are inefficient (Huppert and Sparks, 1984; Marsh, 1996; Simura and Ozawa, 2011; Saad et al., 2023).

AFC: role of bottom boundary layer

Spera and Bohrson (2001) modeled AFC with consideration of energy balance, which assumes several magma subsystems and thermal and material interaction between them. Bohrson et al. (2014) further incorporated thermodynamic constraints into the energy constrained AFC model and built a magma chamber simulator for exploration of magma chamber evolution. These studies, however, did not consider temporal and spatial controls of thermal and material structure of the magma system (magma body and surroundings involved in the AFC).

The time and spatial controls recorded in the WDRC are represented by temporal and spatial change of Ma/Mc from ∼ 0 for the outer ring to 1-2 for the inner ring 2 and the granitic core through ∼ 0.15 for the inner ring 1 (Fig. 16). The Ma/Mc increase with time, from the margin to the center, and with increase in SiO₂ content. The relationships indicate, as discussed above, that assimilation was enhanced in the later stage by mixing of exotic melt formed in the bottom boundary layer and transferred to and mixed with the overlying main magma body. Expected thermal environment of the bottom boundary layer when AFC with Ma/Mc 1-2 took place is consistent with isothermal or isenthalpic conditions (Kelemen and Ghiorso, 1986; Reiners et al., 1995; KG1986 and R1995 in Fig. 16). The extensive assimilation in the later stage is controlled by protracted melting of the sunken roof country rocks due to prolonged heating by latent heat of crystallization. Expected thermal environment of formation of magma solidified as quartz-bearing syenite of the inner ring 1, in which AFC with Ma/Mc ∼ 0.15 took place, is consistent with assimilation and boundary layer fractional crystallization (Kuritani et al., 2005) (K2005 in Fig. 16), which is controlled by effective heat loss to the country rocks through the lower boundary layer and minor entrainment of an exotic melt. The limited assimilation in the earlier stage of the WDRC is due to minor supply of exotic melt from the host rocks because it did not reach high temperature to be able to supply enough exotic melt.

Figure 16. Temporal and spatial change of ratios of mass assimilated and mass crystallized (Ma/Mc) for each lithology group of the plutonic unit and their integrated ratio for the WDRC (overall Ma/Mc). The estimated values of Ma/Mc (Tables 6 and S4) are plotted against respective whole-rock SiO₂ contents, which shows systematic temporal and spatial change as shown at the top of the diagram. The Ma/Mc increases with increase in the SiO₂ from the outer ring (OR) to the inner ring 2 (IR2) and the granitic core (GC) through the inner ring 1 (IR1) as shown by the red allow in the diagram. For the overall Ma/Mc, the range of maximum and minimum estimates listed in Table S8 and the range of SiO₂ variation of the magma body existed beneath the WDRC is shown with obliquely ruled rectangle. The magma body started with emplacement of the precursor magma with SiO₂ ∼ 59 wt% and ended with solidification of granitic magma with SiO₂ ∼ 75 wt%. The uncertainty of the integrated Ma/Mc is represented with greenish gradation (Table S8). Ranges of Ma/Mc and SiO₂ estimated in the literatures are shown with colored rectangles: first-stage isoenthalpic case (Reiners et al., 1995; R1995, light pink rectangle with dashed line), isothermal case (Kelemen and Ghiorso, 1986; KG1986, light yellow thick rectangle), compilation of geochemical modelling for igneous intrusions (Thompson et al., 2002; T2002, light blue rectangle with thin dash-dotted line), and assimilation and boundary layer fractionation (Kuritani et al., 2005; K2005, light purple thin rectangle). Inferred thermal conditions are given on the right-hand side of the diagram. Error bars are for ±1σ. See the text for details.

We calculated ratios of integrated mass assimilated and that crystallized from the initial emplacement to the final solidification of the WDRC (overall Ma/Mc). See Supplementary Document (‘Calculation method of overall Ma/Mc’ in section MODELING AND ERROR ESTIMATION PROCEDURES OF OPEN MAGMATIC PROCESSES) for details of the calculation method. The results are listed in Supplementary Table S8 with estimated uncertainties, and the maximum and minimum estimates with the error ranges are plotted in Figure 16. The overall Ma/Mc for derivation of quartz syenite or syenogranite from the precursor magma ranges from 0.19 to 0.24, slightly higher than the Ma/Mc for quartz-bearing syenite from the outer ring 1 and much lower than those for quartz syenite from the inner ring 2 and syenogranite from the granitic core. Such low value of the overall Ma/Mc is attributed to crystallization during earlier stage of magma evolution in a magma body beneath the WDRC by BLF with limited assimilation. This early BLF stage formed the outer ring releasing heat to the surrounding host rocks, which heated the host to high temperatures and induced higher Ma/Mc for the later AFC stages producing quartz syenite and syenogranite magmas. The range of the overall Ma/Mc of 0.19-0.24 for the entire history of the WDRC corresponds to the lowest range of geochemically constrained Ma/Mc compiled by Thompson et al. (2002) (Fig. 16). The reported cases of Ma/Mc ∼ 0.2 are shallow intrusions that have crystallized in country rocks with low initial temperature, which is consistent with the shallow depth of intrusion of the WDRC (∼ 4 km; Saad et al., 2023). See Supplementary Document (‘Comparison of overall Ma/Mc with temporal change of Ma/Mc’ in section DETAILS OF RESULTS OF MODELING OPEN MAGMATIC PROCESSES; Supplementary Table S8) for role of the thermal evolution of the host rocks in temporal change of Ma/Mc.

There is an important implication for temporal and spatial controls of AFC processes in small, shallow, and melt dominant magma bodies (Fig. S17). The bottom boundary layer AFC in small magma bodies formed by a single intrusion becomes effective in the later stage by melting of accumulated country rocks through prolonged and sufficient heating. Because of the small size of shallow magma body and single intrusion without replenishment, thick sidewall boundary layer relative to the size of magma body inhibits inflow of exotic melt and the cooler upper boundary layer cannot supply molten exotic materials. The melt dominant nature and highly alkaline magma compositions of the WDRC magma body is preferable for efficient homogenization of the mixed magma. The WDRC recorded magma body evolution without replenishment, which strongly affect the temporal change of thermal environment on AFC. If there are replenishment, not only bottom boundary layer might be effective from the earlier stage of new magma input but also thinned sidewall boundary layer may become transparent to assimilation through the sidewall boundary layer as in the case of larger ring complex of the AKRC as discussed above.

Major element fractionation of critically undersaturated alkaline magmas

Trachytic alkaline magmas critically undersaturated with SiO₂ often show liquid line of descent towards rhyolite, which is saturated with SiO₂ (e.g., Upton, 1974; Fitton, 1987; Liégeois and Black, 1987; Woolley and Jones, 1987; Downes, 1987; Wilson et al., 1995; Edwards and Russell, 2000; LeMasurier et al., 2011). Such trachyte-rhyolite trend, which is characterized by decrease in total alkalis with significant increase of SiO₂ (e.g., Fig. 2a), coexists in many cases with trachyte-phonolite trend characterized by increase in total alkalis with minor decrease of SiO₂. A trend between critically undersaturated trachyte and phonolite without extension toward rhyolite is also reported (e.g., Jung et al., 2013; Ackerman et al., 2015), which was attributed to fractional crystallization dominantly of feldspars. The most controversial is the origin of the transition of magma composition from critically undersaturated trachyte to SiO₂ saturated rhyolite. This issue has been discussed based on the ‘Petrogeny’s Residua System’ (Bowen, 1937), which corresponds to the ternary nepheline (Nph)-kalsilite (Kls)-silica system at 1 bar (Schairer, 1950; Fig. 17) with albite-orthoclase (Ab-Or) join as thermal barrier or ‘critical plane of silica undersaturation’ (Yoder and Tilley, 1962). Liquids with feldspar as liquidus phase located near the Ab-Or join on the opposite side of the silica apex are ‘critically undersaturated’ and the crystallization of feldspar drives the residual liquid to crystallization of feldspathoid. By contrast, liquids near the Ab-Or join on the side of silica apex evolve toward saturation with SiO₂. The fate of a melt having composition close to the Ab-Or join is strongly sensitive to SiO₂ content of the initial magma (Morse, 1980). Any fractional crystallization mechanisms do not provide reasonable explanation for trachytic melt to go beyond the critical plane of silica undersaturation (Foland et al., 1993). Contamination or assimilation of SiO₂-rich crustal materials has been invoked to explain the transition from trachyte or even phonolite to rhyolite on the basis of contrasting magma evolution trends between the continental and oceanic sectors of the Cameroon line (Fitton, 1987) or contrasting variations of Sr and Nd isotope ratios between quartz syenites and nepheline syenites in alkaline igneous complexes with modeling in the Nph-Kls-silica system (Foland et al., 1993). However, it has not been clarified yet how fractional crystallization and assimilation are coupled in time and space to give rise to the peculiar variation trend. The results of this study presented above have important implications on this issue.

Figure 17. CIPW normative compositions of rocks from the WDRC (Table S1) and those of estimated precursor magma (circle with plus), fractionated melt (circle with cross, solid triangle, and solid diamond), and exotic melts (five- or six-pointed solid stars) that caused AFC (Tables 8a and S5), projected on the Nepheline-Kalsilite-Quartz plane. The precursor magma is the mean of two estimates (circle with plus), which is shown as black dots in Figure S12. The fractionated melts are those involved in the AFC models to reproduce rocks from the inner ring 1 (circle with cross), the inner ring 2 (solid triangle), and the granitic core (solid diamond). The first one is used in the BLF modeling and is indicated so in the diagram. The Anorthite norm correction according to Blundy and Cashman (2001) was not applied because of the low Ca contents in rocks from the WDRC. Plotted samples are those with the sum of projected components greater than 80%, which excludes 5 samples including trachybasalts and trachyandesite. Liquidus boundaries (dashed lines), nepheline-saturated minimum (m_Ne), and the saddle point (open square) at 1 bar are after Schairer (1950). Silica-saturated minimum (m_S) is after Morse (1980). Water-saturated liquidus boundaries between silica mineral and feldspar at 0.05-0.2 GPa shown with dot-dashed curves are after Tuttle and Bowen (1958). Red dot-dashed line represents water-saturated trajectory of liquidus boundary minimum (and eutectic) melt compositions at pressure from 0.05 (light blue dot) to 1 GPa (red dot) experimentally determined by Tuttle and Bowen (1958) and Luth et al. (1964).

The precursor magma, which filled the magma body beneath the WDRC in the earliest stage (Fig. 15a; Table 9a), has 4.7 wt% Nph norm and is projected below the Ab-Or join in the Nph-Kls-silica system (Fig. 17). It has the most SiO₂ undersaturated composition among the rocks of the WDRC, though the sum of Nph-Kls-silica components is ∼ 60 wt%. Trachytes from the volcanic unit, which formed from the precursor magma via BLF, have 0.7-4.3 wt% Nph norm and are plotted below the Ab-Or join (Fig. 17). Syenites from the outer ring, which formed by closed-system crystallization of the precursor magma (Saad et al., 2023) with feldspar accumulation and minor addition of the fractionated melt (Tables 8 and S6), have up to 4.2 wt% Nph norm and are also plotted below the Ab-Or join (Fig. 17). The syenites contain sodalite as one of the interstitial phases (Saad et al., 2023), which indicates closed system crystallization drives the residual melt away from the Ab-Or join towards m_Ne in Fig. 17.

Rocks from the inner rings and granitic core contains various amounts of modal quartz and are plotted above the Ab-Or join. Quartz-bearing syenites from the inner ring 1 have 0.6-2.2 norm Qz and are plotted above the Ab-Or join closest among rocks with modal quartz (Fig. 17). Rocks from the inner ring 2 and the granitic core have 10-25 wt% norm Qz plotted in the middle of Ab-Or join and m_S or close to m_S (Fig. 17). Such highly SiO₂-saturated rocks from the inner ring 2 and the granitic core can be explained by extensive AFC, because the estimated Ma/Mc is higher than ∼ 1 and crystallization degree ranging 40-70%. By contrast, SiO₂-rich rocks from the ORIM, which have high Qz norm as high as 20 wt%, comparable to those of the inner ring 2 and the granitic core, underwent minor assimilation <∼ 0.1 in Ma/Mc. Therefore, fractional crystallization must be the dominant process for their derivation, which is only possible if their parental melt went beyond the Ab-Or join from the Nph normative field to the Qz normative field in an earlier stage by assimilation before main fractional crystallization stage started. This is shown to be plausible as discussed above by invoking two-step formation of the SiO₂-rich rocks of the ORIM: first step of minor assimilation and second step of extensive fractional crystallization. Its reverse sequence: first step of fractional crystallization followed by minor assimilation, does not give any explanation to go beyond the Ab-Or join and is implausible.

The fractionated melt derived from the precursor magma with 4.7 wt% Nph norm through BLF has 0.4 wt% Nph and ∼ 1 wt% Na₂SiO₃ norms. It is critically undersaturated, and the BLF mechanism involving multiple stages of fractional crystallization and mixing with the precursor magma does not make fractionated melts go beyond the Ab-Or join. The trachyte magma derived from the precursor magma by addition of the fractionated melt is thus critically undersaturated (Fig. 17). Addition of SiO₂-rich exotic melts to the trachyte magma is, therefore, essential to derive the quartz-bearing syenite magma, from which all the rocks of the WDRC with SiO₂ > 62.5 wt% formed by fractional crystallization with or without further assimilation. It is confirmed in the Nph-Kls-silica system that AFC with Ma/Mc ∼ 0.15 and ∼ 50% crystallization degree, which is consistent with the values constrained with trace elements, can go over the Ab-Or join to reproduce the quartz-bearing syenite composition.

The trachyte-rhyolite inflexions on the Harker major element variation diagrams reported from world alkaline magmatic provinces are located mostly ∼ 62.5 ± 5 wt% SiO₂ (e.g., Upton, 1974; Downes, 1987; Fitton, 1987; Liégeois and Black, 1987; Upton and Emeleus, 1987; Woolley and Jones, 1987; Edwards and Russell, 2000). The variance in the inflexion points results in diffuse trachyte-rhyolite trends. Even if assimilation could have taken place at less evolved stages with lower SiO₂ contents, extensive fractional crystallization of mafic minerals to attain trachytic magma compositions or to reach the ‘Petrogeny’s Residua System’ is necessary to give rise to the inflexions by enabling dominant fractionation of feldspar. The variance in inflexion points over 10 wt% in SiO₂ content and the diffuse trachyte-rhyolite segments may be attributed to diversities in (1) extent of assimilation relative to crystallization, which is represented by Ma/Mc, (2) timing and temporal change of assimilation, (3) chemical composition of exotic melt causing assimilation, and (4) timing of change in crystallization reaction stoichiometry from mafic minerals dominant to feldspar predominant.

The first three are controlled by chemical and thermal state of the crustal material hosting magma bodies and the last one is principally controlled by parental magma compositions and efficiency of fractional crystallization. Concerning the first point, extensive fractionation of feldspar is required to deviate critically undersaturated or saturated melt from the Ab-Or join irrespective of fractionation trends because of similarity of feldspar and trachytic melt compositions. By contrast, assimilation of only a small amount of SiO₂ rich melt (small Ma/Mc) enables critically undersaturated melt efficiently go beyond the Ab-Or join. This may be one of the reasons why trachyte-rhyolite trends are more frequently observed than trachyte-phonolite trends. The sharp inflexions observed in the WDRC (Fig. 2a) are attributed to strong coupling of (1) and (2) due to requirement of a significant extent of crystallization to heat up the host rock attaining a sufficient degree of melting for assimilation. The major element compositions of estimated exotic melts for the WDRC are very close to m_S or water-saturated liquidus minima at 0.1-0.2 GPa (Fig. 17), which is responsible for similar major element variations irrespective of extensive AFC with Ma/Mc as high as ∼ 1 or dominantly fractional crystallization under minor AFC with Ma/Mc smaller than ∼ 0.1. This is due to that both melting and crystallization in the ‘Petrogeny’s Residua System’ are governed by the phase relation (Fig. 17). This is another reason for the sharp inflexions and the tight trachyte-rhyolite trends documented in the WDRC as well as world alkaline igneous provinces.

CONCLUSIONS

The main conclusions of the present study on the WDRC are as follows.

(1) New geochemical data from the WDRC including trace element contents and Nd, Sr, Pb isotope ratios are presented. We assessed the geochemical data to identify modification of the primary isotope signatures by assimilation of country rocks and obtained a reliable whole-rock Rb-Sr isochron age of 586.8 ± 10.1 Ma with initial ratio of 0.70333 ± 0.00011 from carefully selected 7 samples including trachybasalt of the dike unit. The excluded 7 samples have important information on open-system processes involving isotopically distinct exotic materials. The consistent geochemical variations of rocks from the WDRC including trachybasalt from the dike unit, such as the trace element correlations and the tight isochron plot for Rb-Sr isotope system as well as Sm-Nd, U-Pb, and Th-Pb isotope systems giving consistent 586.8 Ma reference isochrons, strongly support that the trachybasalt represents parental magma of the WDRC, from which the lithological and the geochemical diversities of the WDRC were derived.
(2) Careful inspection of all the data of major and trace element concentrations and isotope ratios from the WDRC revealed that the systematic geochemical diversities are attributable to two key open processes in a magma body beneath the WDRC, a subvolcanic plumbing system including frozen upper boundary layers of the magma body. One is BLF, which played an important role in fractional of precursor magma initially intruded into the magma body beneath the WDRC, particularly in the earlier stage of magma evolution. The other is AFC, which became effective in the later stage. We constructed BLF and AFC models involving multiple samples and full data set (major and trace element concentrations and isotope ratios), which were applied to the WDRC with least-squares methods to put constraints on model parameters with evaluation of their uncertainties.
(3) The following points are clarified by BLF modeling. (1) The precursor magma was mixed with substantial amount of fractionated melt (∼ 40% in mass) in the roof zone shortly before its eruption by collapse of the roof boundary layer; (2) The fractionated melt was generated in the mushy sidewall boundary layer and transported through the boundary layer to have been segregated and accumulated in the roof zone; (3) Reactive melt migration of the fractionated melt through the sidewall boundary layer mush gave rise to feldspar accumulation with entrapment of 10% interstitial melt, which was solidified as the outer ring. The cooperation of vertical or outward inclined sidewall boundary layer as an environment of fractionated melt formation and upper boundary layer as an environment of the fractionated melt accumulation and mixing with the main magma body by forced convection induced by roof boundary layer collapse is effective BLF mechanism in a small, shallow, and melt dominant magma body.
(4) The following points are clarified by AFC modeling. (1) The exotic melt involved in the AFC process has major and trace element concentrations and Nd isotope ratio similar to those of Neoproterozoic granitoids from the Eastern Desert; (2) Ma/Mc, the ratio of extents of crystallization and assimilation increases from ∼ 0 for the outer ring (in the earlier stage) to 1-2 for the inner ring 2 and the granitic core (in the later stage) through ∼ 0.15 for the inner ring 1 (in the middle stage); (3) The substantial assimilation in the later stage and in the central part of the complex suggests protracted formation of the exotic melt in the bottom boundary layer by melting of granitoids carapace sunk by stoping mechanism accompanying caldera formation on the surface during an early stage of the WDRC formation; (4) The overall Ma/Mc integrated over the entire evolution of the WDRC quantifies heat budget in and around the magma body beneath the WDRC, which is estimated to be ∼ 0.2 demonstrating a critical role of the outer ring formation in releasing heat to have heated up the host rock in the earlier BLF stage and induced extensive assimilation in the later AFC stage. The bottom boundary layer AFC documented in the WDRC is applicable to a melt-dominant small magma body formed by a single intrusion. Because of the small size of shallow magma body and single intrusion without replenishment, thick sidewall boundary layer relative to the size of magma body inhibits inflow of exotic melt and the cooler upper boundary layer cannot supply molten exotic materials.
(5) Liquid line of descent from critically undersaturated alkaline trachytic magma to rhyolite documented in this study of the WDRC has been reported from world alkaline provinces. This study shows based on (4) that various extents of assimilation coupled with fractional crystallization are responsible for the trachyte-rhyolite trend without making any assumptions. Extensive fractionation of feldspar is required to deviate critically undersaturated melt from the Ab-Or join, whereas assimilation of a small amount of SiO₂ rich melt (small Ma/Mc) enables critically undersaturated melt go beyond the Ab-Or join. The exotic melt compositions are very close to water-saturated liquidus minima, which is responsible for similar major element variations irrespective of extensive AFC with Ma/Mc as high as ∼ 1 or dominantly fractional crystallization under minor AFC with Ma/Mc smaller than ∼ 0.1. This is attributable to that both melting and crystallization in the ‘Petrogeny’s Residua System’ are governed by the phase relation.

ACKNOWLEDGMENTS

We are grateful to Prof. S.R. Wallis of the University of Tokyo for his fruitful discussion and useful comments, which stimulate us to advance this study. E.S. thank Dr. S. Hamid for his kind help in the field and Dr. Y. Sato for his kind help during stay at the University of Tokyo. We thank Dr. A. S. Silpa of Shimane University and an anonymous reviewer for critical comments, which significantly improved the manuscript. We thank Prof. M. Satish-Kumar for helpful advice and useful comments in his editorial handling. This study was supported by RONPAKU (Dissertation PhD) program of Japan Society for Promotion of Science (JSPS) ID No. R11840 carried out at the Department of Earth and Planetary Science of the University of Tokyo during FY 2018-2022. This work was also supported by JSPS KAKENHI Grant Numbers JP20H02003 and JP23K03544 to K.O.

SUPPLEMENTARY MATERIALS

Supplementary Document, Tables S1-S8, and Figs. S1-S17 are available online from https://doi.org/10.2465/jmps.240110.

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