Groundwater Contribution to River Water Temperature

Abstract

River water temperature serves as a critical indicator of numerous biological and chemical processes essential for ecosystem support and water quality maintenance. River temperatures are expected to rise due to the increasing impact of climate change, causing potential adverse consequences. Hence, a thorough understanding of the drivers influencing river temperature is imperative. Physically or process-based models are suitable for enriching our understanding of the mechanisms regulating river temperature. In this study, we collected articles on river water temperature and modeling and classified them according to their modeling type and energy components. We reviewed the physically based models to determine the relative proportions of various energy fluxes that affect the temperature of river water. The results indicated that despite its importance, groundwater flux has not been given as much consideration as the other fluxes, particularly for small rivers. We also reviewed the semi-distributed Soil and Water Assessment Tool (SWAT) model, which has been applied to the computation of stream temperature and found that some modifications made to the model primarily used the equilibrium temperature approach, whereas only a few studies considered the groundwater flux. Our findings highlight the need for further improvements in modeling techniques, with special emphasis on improving the representation of subsurface fluxes, particularly groundwater, for the better management of ecosystem preservation and water quality.

1. Introduction

River water temperatures have substantially risen in the recent past [1, 2, 3, 4, 5], increasing on average by at least 0.5℃ per decade worldwide as a result of global warming [6, 7, 8, 9, 10]. As global warming continues, stream temperature predictions indicate an increase of 1.5°C by 2030 and, in some regions of the world, up to 4℃ by 2050 [11,12]. It is estimated that over the last century, water temperature in lakes and rivers increased from 1 to 3°C [13]. Warming due to climate change is anticipated to differ globally, with annual water temperature increases projected to be around 0.03℃ in the UK, 0.001–0.08℃ in the USA, and 0.03–0.05℃ in China [14]. Changes in water temperature can affect the health of aquatic ecosystems [15]. For example, less dissolved oxygen in streams due to higher water temperatures can cause changes in biogeochemical processes, including respiration, nitrification, and denitrification [16, 17, 18, 19]. Increasing water temperatures can promote metabolic rates, leading to faster microbial nutrient cycling, altered reproductive success, and juvenile aquatic species development [20]; it can also accelerate the growth of algal blooms [21]. However, despite its influence on crucial water quality variables such as dissolved oxygen, nitrates, and toxic metals, water temperature has received less attention than other water quality properties, with research largely focusing on nutrients and other contaminants [22].

Given its importance in aquatic biochemistry and biodiversity, water temperature monitoring and estimation have received considerable attention [23]. Nevertheless, water temperature observations remain inadequate despite the establishment of numerous water discharge monitoring sites worldwide [12]. In addition, as climate change and anthropogenic activities continue to alter natural landscapes, not only observations but also the need for modeling approaches is increasing. According to various review articles [24, 25, 26, 27], water modeling approaches can be classified into two types: statistical modeling and physically based modeling.

Statistical (stochastic) models rely on the empirical relationships between water temperature and its predictors. Benyahya et al. [26] provided a detailed account of statistical water temperature models, which are simpler and require minimal data compared with deterministic models. Parametric models such as regression is useful for short-term predictions, whereas periodic models capture seasonal patterns. Nonparametric models can be used to analyze complex nonlinear relationships between water temperature and environmental factors. Notably, the use of statistical models to simulate or predict stream water temperature [26] is important for enhancing water resource management and aquatic habitat preservation measures. Various studies have shown that statistical temperature models can generate accurate stream temperature predictions [28, 29]. Their clear distinction from physically based models is their ability produce reliable stream temperature predictions with less input data [30, 31, 32]. Although numerous studies have attempted to infer the drivers of river temperature variables [32, 33, 34], meteorological factors, such as solar radiation and wind speed, and hydraulic factors, such as water depth, also play important roles in the regulation of aquatic water temperature [35]. However, these factors are generally not considered in statistical models, which also do not have the potential to consider the impact of changes in watershed hydrological conditions.

In contrast, physically based models that use physical equations to represent water temperature can capture changes in environmental conditions [27, 36]. They also provide reliable results for historical data reconstruction and future projections [12, 37, 38]. In cases where statistical methods might not succeed owing to scenarios outside their calibration range, physically based models are especially helpful in forecasting temperature responses to alterations in climate or land use scenarios [15, 39]. Although these models have found applications at the basin [37], regional [40, 41], continental [42], and global [12, 23] levels, their application is often limited to scale or small areas in a basin, as they require extensive data and detailed physiographic, hydrological, and meteorological inputs [43, 44].

Given these considerations, a comprehensive overview of the current research on river water temperature modeling, particularly focusing on physically based models, will provide valuable insights into the future direction of studies on river water temperature. In particular, the ability to predict the impact of changes in environmental conditions such as climate change and land use change is essential for the selection of a physically based model.

Thus, firstly, we reviewed articles to assess existing physical process-based models used to understand river water temperature dynamics. Then, through investigating the treatment of energy budget components within the reviewed models, we attempted to identify the relative importance of groundwater flux. And we explored observational evidence of groundwater flux contribution and modeling approaches to incorporate groundwater flux into river water temperature simulations. Lastly, we further attempted to identify its current limitations, and highlighted the unresolved challenges that remained to achieve more accurate physically based river water temperature model.

2. Physical processes influencing river temperature

The energy exchange between river channels and the surrounding environment influences river channel temperatures [45, 46, 47]. Thus, the law of conservation of energy is usually applied to describe river channel temperature dynamics. The energy budget components relevant to river temperature are schematically shown in Fig. 1.

Figure 1: Energy budget components of river temperature

Consequently, the equation for the unit length of a river channel can be expressed as follows:

  

$$\frac{\partial c_w{\rho }_{w}AT}{\partial t}=\frac{c_w{\rho }_w{T}_w(Q_{\mathit{\text{in}}}-Q_{\mathit{\text{out}}})}{L}+\mathit{\text{Bϕ}}$$(1)

  

$$\varphi =H_{\mathit{\text{sw}}}+H_{\mathit{\text{lw}}}+H_e+H_s+H_b+H_a$$(2)

where, $t$ : time [s], $c_w$ : heat capacity of water [J/kg/K], ${\rho }_w$ : density of water [kg/m³], $A$ : cross-sectional area of river channel [m²], $L$ : channel length [m], $B$ : width of the reach [m], $T_w$ : river temperature [K], $Q_{\mathit{\text{in}}}$ : river water discharge flowing into the reach [m³/s], $Q_{\mathit{\text{out}}}$ : river water discharge flowing out from the reach [m³/s], and $\varphi$ : source term of energy per area [W/m²] which is separately given in Equation (2). In Equation (2), $H_{\mathit{\text{sw}}}$ : net shortwave radiation [W/m²], $H_{\mathit{\text{lw}}}$ : net longwave radiation [W/m²], $H_e$ : net latent heat exchange due to evaporation or condensation [W/m²], $H_s$ : net sensible heat due to convection and conduction [W/m²], $H_b$ : net heat exchange through riverbed by conduction [W/m²], and $H_a$ : heat inflow from surface runoff, lateral flow, groundwater flow and tributaries.

3. Article review

3.1 Identification and classification of models

Using Scopus as the article search engine, we first retrieved articles which included the phrases “river water temperature” and “model/modeling/modelling” in their title. The identification and classification procedures are illustrated in Fig. 2.. The PRE/0 operator was employed to ensure that the specified words appeared next to each other in the articles. The search yielded a total of 448 articles. These articles were initially screened to determine their relevance to modeling river water temperature. Many articles focused on water flow modeling and reservoir or pond water modeling and were excluded. Ultimately, 343 articles were selected for further analysis those address the specific topic of modeling river water temperature.

Figure 2: Article selection process

The identified articles were categorized into physically based models, statistical models, and a combination of both. While the category ‘physically based model’ includes any kind of models which directly formulate physical processes governing water temperature, the category ‘statistical model’ includes mathematical models which formulate relationships between water temperature and potential governing factors of water temperature. All articles that discuss various modeling techniques, such as principal component analysis, canonical correlation analysis, artificial neural networks, and machine learning, were included within this category. Lastly, the category ‘combination of both’ includes mixed use of both physically based models and statistical models.

Figure 3: Composition of model types

As shown in Fig. 3., the number of articles discussing river water temperature modeling over the years has shown an overall increasing trend, reflecting the growing interest and research activity in this area. Interestingly, before 2020, physically based models exhibited a more pronounced upward trend compared to statistical models. However, research on statistical modelling has become increasingly popular with the incorporation of advanced methods, such as machine learning and data-driven approach.

In spite of recent increasing trend in statistical models, physically based models still have a powerful capability of future prediction founded by concrete understanding of energy transfer between water, atmosphere, and surrounding environment. Thus, in the subsequent section, we will focus on energy budget treatment in physically based models.

3.2 Treatment of energy budget components

We further examined the types of energy components in each model and summarized the formulation of each energy budget component. As it is difficult to quantify each energy budget components due to the varying geographical condition of watersheds, we aimed to show the relative importance of each heat budget component by analyzing the percentage of articles that consider specific energy budget components (Fig. 4.). As shown in Fig. 4., shortwave radiation was the primary heat source [39, 45, 48].

Figure 4: Percentage of articles where a specific energy budget component is considered

Latent heat flux is related to the energy transfer associated with the evaporation and condensation of water at the river surface [45, 48]. Due to the scarcity of direct measurements of latent heat flux, river temperature models rely on meteorological data to estimate it [49, 50]. Most models employ a common equation for latent heat fluxes:

  

$$H_e={\rho }_w{L}_{e}E$$(3)

where, ${\rho }_w$ : density of water [10³ kg/m³], $L_e$ : latent heat of vaporization [2.5×10⁶ J/kg], and $E$ : rate of evaporation [m/s]. Various models employ different equations to compute evaporation rates, ranging from simpler methods based on air temperature and daylight hours [51] to more complex physically based equations, such as Dalton’s equation [52]:

  

$$E=\frac{M_w}{R{\rho }_w}\frac{D_v}{T_a}\frac{(e_s\left(T_s\right)-e_a\left(T_a\right))}{{\left[\left({\rho }_a{[{\left(k{u}_{v,c}\right)}^{\varphi }+{\left(n{u}_{\mathit{\text{ref}}}\right)}^{\varphi }]}^{1/\varphi }L\right)/{\eta }_a\right]}^{q}L}$$(4)

where, $E$ : evaporation rate [m/s], $M_w$ : molecular mass of water [kg/mol], $R$ : ideal gas constant [J/mol/K], $D_v$ : diffusivity of water vapor in air [m²/s], $T_a$ : air temperature [K], $e_s\left(T_s\right)$ : saturation vapor pressure at the surface temperature $T_s$ [Pa], $e_a\left(T_a\right)$ : actual vapor pressure of the air at temperature $T_a$ [Pa], $k$ : dimensionless constant related to the vertical air movement, $u_{v,c}$ : characteristic speed of air movement in the vertical direction [m/s], $\varphi$ : dimensionless constant, $n$ : dimensionless constant related to the horizontal wind speed, $u_{\mathit{\text{ref}}}$ : horizontal wind speed measured at a reference height [m/s], $L$ : characteristic length of the evaporating surface [m], ${\eta }_a$ : dynamic viscosity of air [kg/m/s], and $q$ : dimensionless constant.

For longwave radiative fluxes, most models use the following equations:

  

$$H_{\mathit{\text{lw}}}=H_{\mathit{\text{lw}}\mathit{\text{atm}}}-H_{\mathit{\text{lw}}\mathit{\text{stream}}}$$(5)

  

$$H_{\mathit{\text{lw}}\mathit{\text{atm}}}=(1-R_L).{\epsilon }_{\mathit{\text{atm}}}.\sigma .T_a^4$$(6)

  

$$H_{\mathit{\text{lw}}\mathit{\text{stream}}}={\epsilon }_w.\sigma .T_w^4$$(7)

  

$$H_e={\rho }_w{L}_{e}E$$(8)

where, ${\epsilon }_{\mathit{\text{atm}}}$ : emissivity of the stream, ${\epsilon }_w$ : emissivity of atmosphere, $R_L$ : reflectance coefficient of the stream surface that is usually given as 1 – ${\epsilon }_w$ , $\sigma$ : Stefan-Boltzmann constant [5.670367 × 10⁻⁸ W $/m^2/K^4$ ], and $T_w$ : water temperature [K]. The calculation of ${\epsilon }_{\mathit{\text{atm}}}$ , which is a critical parameter in the equation, varies among river temperature models. Some models use empirically derived formulae based on air temperature or vapor pressure formulas [53, 54, 55, 56], whereas others employ the physically derived method of Brutsaert [57], which considers both air temperature and vapor pressure. In addition, some models offer multiple methods or composite approaches for characterizing ${\epsilon }_{\mathit{\text{atm}}}$ . Emissivity is significantly affected by cloud cover, leading some models to correct computed values using the Bolz approach [58], particularly in cloudy regions.

Sensible heat flux pertains to the exchange of heat between the river and surrounding atmosphere, driven by temperature differentials [25]. Although the magnitude of sensible heat flux is generally lower than that of radiative or latent heat flux, it remains a significant factor in regulating river temperatures [45].

Riverbed heat flux represents the transfer of energy between the streambed and overlying water and is generally considered to be smaller in magnitude than surface heat fluxes [59]. Some river temperature models include calculations of the heat flux through the riverbed, whereas others consider it negligible. The riverbed heat flux is often calculated using a variation of Fourier’s Law [60], which considers the thermal gradient at the streambed–water interface and the bed’s thermal conductivity [61, 62].

Only a few models include routines for calculating fluid friction heat flux. In particular, the heat gain from fluid friction can be significant in steep high-roughness streams [46, 63, 64]. However, for most temperature modeling scenarios, fluid friction heat exchange is considered minor and can be neglected [65, 66].

In river systems, advective heat fluxes result from the inflows of tributaries or subsurface sources, which can introduce temperature gradients [67, 68]. Most models include routines for computing advective heat fluxes using the water budget equation [62]. Although some models incorporate groundwater inputs to include advective heat flux, most require a combination of surface and groundwater inputs in the form of bulk inflows at discrete intervals [27]. An explicit account of groundwater contributions was found in only 30 articles in this review (Fig. 4.). This indicates that the state-of-the-art model has a limited representation of the groundwater contribution. A schematic diagram of relative importance of energy budget components considered in physically based model is shown in Fig. 5..

Figure 5: Schematical diagram of relative importance of energy budget components considered in physically based model

When modeling large rivers, the impact of groundwater inflow on stream temperature is often neglected; nevertheless, in smaller streams with low-flow conditions, it might be important [69, 71, 72]. This point is further discussed in Section 4.

4. Groundwater contribution to river water temperature

4.1 Observational facts

As pointed out in Section 3, in rivers where the groundwater contribution is significant, heat fluxes by subsurface flow can represent a substantial fraction of the stream energy budget [46]. Groundwater discharge can have substantially different temperatures than those of the river, thereby significantly influencing the river’s overall temperature regime during periods of low flow. This may reduce the risk of thermal stress in aquatic species.

Groundwater has emerged as a critical factor, especially in small streams, where its contribution significantly affects seasonal variations and overall stream dynamics. Leach et al. [73] suggested that groundwater plays a more dominant role in both the summer and winter seasons in small streams compared with large rivers. Higashino and Stefan [74] showed that the temperature of the Oita River, where groundwater accounts for approximately 50% of the annual flow, was strongly influenced by the availability of groundwater storage. The Oita River maintains a cool temperature profile, with an annual average water temperature of 16℃, a peak of around 24℃, and a minimum of approximately 8℃. Its temperature changes are small, with day-to-day oscillations of around ±0.5℃ in the wet summer and ±1.5℃ in the dry winter season, which efficiently moderates the river’s thermal regime.

Some studies have shown that a substantial groundwater contribution leads to unexpectedly cool river temperatures in summer and warm temperatures in winter [75, 76]. Moatar and Gailhard [77] quantified the magnitude of the cooling effect of groundwater as 1.4℃ with a 10 m³/s groundwater inflow in the Loire River. Groundwater discharge-dominated streams are expected to be more resilient to climate warming than those with minimal groundwater influence [78, 79, 80]. Wawrzyniak et al. [81] found that when groundwater discharge represents 16% of the river discharge, it lowers the river temperature by 0.68 ± 0.13℃, contrasting with a minimal effect (0.11 ± 0.01°C) when the groundwater discharge contributes only 2% to the discharge.

The quantification of the effects of groundwater on stream temperatures [82] and energy sources [83] has increased because of the importance of groundwater to stream biota, and the literature on the identification and quantification of groundwater and surface water interactions is extensive [82, 84, 85, 86, 87]; however, the studies on the downstream impacts of groundwater inputs on rivers remain limited [75, 76, 88, 89, 90]. Moreover, these studies have mainly focused on hyporheic exchanges [88, 91] or small streams [75, 76]. In contrast, the thermal effects of groundwater inputs for larger rivers [77, 90] are not well known, and Loheide and Gorelick [75] emphasized that further research is required at coarse spatial scales.

4.2 Modeling groundwater contribution

4.2.1 Air temperature-based groundwater estimation

The simplest way to include the groundwater contribution to the river heat budget is by directly providing the groundwater temperature. Because direct measurements of groundwater temperature are difficult, it has been assumed to be equal to the mean annual air temperature plus a potential thermal offset of up to 3℃ [74, 92, 93]. Generally, groundwater obtained from depths of 6.1 to 61 meter maintain a uniform temperature, typically 1.5℃ to 3℃ above the mean annual air temperature [94]. The annual groundwater temperature ranges are small at depths of 18.3 meter but increase significantly when the depth to water is 3.1 meter or less. In addition, groundwater temperatures have a time lag from air temperatures by 1–2 months at shallow depths and 3–4 months at depths of 18.3 meter. Furthermore, daily temperature fluctuations are rare in groundwater reservoirs, primarily occurring in shallow aquifers where the depth to the water is 3.1 meter or less [94].

When the discharge to the stream is dominated by shallow subsurface flow, especially during rainfall and snowmelt events, lateral heat advection to the stream can be characterized by relatively dynamic temperatures that differ substantially from those of deeper groundwater [47]. Suzuki et al. [95] developed a model by incorporating an atmospheric and land surface process model that considered snow, a runoff model, and a water temperature estimation model to better understand the impact of climate change on stream temperatures in cold snowy regions. Several studies have used physically based models to predict future water temperatures in cold snowy regions [96, 97, 98, 99].

4.2.2 Direct consideration of groundwater temperature

Though direct measurements of groundwater temperature are difficult, there is an attempt to directly consider groundwater contribution to river water temperature in reach scale. Kurylyk et al. [100] showed that isolating the thermal impacts of advective heat flux induced by groundwater discharge is challenging when working at a reach scale because the accompanying groundwater mass flux also contributes to the longitudinal stream dynamics. Equation (9) provides the least ambiguous approach as the advective flux is proportional to the difference between the groundwater temperature and thermal datum.

  

$${\rho }_{w}C\Delta x\frac{\partial \overline{A(T-T_{\mathit{\text{dat}}})}}{\partial t}=J_i-J_{i+1}+\Delta x\overline{w{Q}_v}+\Delta \mathit{\text{xρC}}\overline{f_g(T_g-T_{\mathit{\text{dat}}})}$$(9)

where, ${\rho }_w$ : density of water [kg/m³], $C$ : specific heat of water [J/kg/℃], $T$ : water temperature [℃], $T_{\mathit{\text{dat}}}$ : reference temperature for computing the thermal energy of the water [°C] [101, 102], $J$ : longitudinal heat flux associated with advection and dispersion [W], $w$ : surface width of the channel [m], $\Delta x$ : stream segment length [m], $Q_v$ : net vertical energy exchange associated with radiation, sensible, and latent heat across the water surface and the bed heat conduction and friction [W/m²], $f_g$ : rate of groundwater discharge per unit length of channel [m²/s], $T_g$ : groundwater temperature [℃], and subscript $i$ and $i+1$ indicates values located at $x_i$ and $x_i+\Delta x$ respectively. The reference temperature is typically set, either explicitly or implicitly, to 0℃, although this is arbitrary from a thermodynamic perspective. In their calculations, they assigned a constant groundwater temperature based on measurements or reasonable assumptions.

4.2.3 Fully distributed groundwater temperature modeling

Fundamentally, subsurface water temperature is governed by the advective – dispersive equation of heat. Qiu et al. [103] proposed an integrated, catchment scale hydrologic Process‐based Adaptive Watershed Simulator (PAWS) model [104, 105, 106, 107] that simulates key hydrologic processes, including surface and subsurface flow, channel flow, topography-induced overland flow, and soil water processes. As PAWS uses process-based descriptions of flow and transport, a general two-dimensional advective–dispersion equation [108] was used to model the groundwater temperature:

  

$$\frac{\partial {\rho }_b{c}_b{T}_g}{\partial t}=-{\rho }_w{c}_w.\left(T_{g}q\right)+k_e{\nabla }^2{T}_g+Q_{\mathit{\text{soil}}}+\Omega$$(10)

where, $T_g$ : the groundwater temperature [°C], ${\rho }_b$ and $c_b$ : density [kg/m³] and specific heat of aquifer‐water matrix [J/kg/K], respectively, ${\rho }_w$ and $c_w$ : density [kg/m³] and specific heat of the water [J/kg/K], respectively, $q$ : Darcy flux vector directly obtained from the PAWS groundwater flow module [m/s], $k_e$ : effective thermal conductivity [W/m/K], $Q_{\mathit{\text{soil}}}$ : soil heat flux exchange between soil and groundwater, and $\Omega$ : heat transfer due to advection and conduction through the stream bed [W/m³].

The same equation has been already implemented in various fully distributed hydrological model such as Hydrus [109], and GETFLOWS [110]. However, the spatial scale of all these models is usually limited to a relatively small area due to the parameter uncertainties and limited calculation resources.

4.2.4 Semi-distributed groundwater temperature modeling – case of SWAT

On the contrary to fully distributed model, semi-distributed model has an advantage in saving calculation time, which enables to extend calculation area to more large area such as continental scale basin. One of this type of model is the Soil and Water Assessment Tool (SWAT). The SWAT model, developed by the United States Department of Agriculture-Agricultural Research Service, is a hydro chemical model designed to predict how changes in land use and climate impact the water, energy, and nutrient cycles in watersheds [111]. This model divides watersheds into subbasins connected by a network of streams, which is further divided into hydrologic response units (HRU) based on unique combinations of land use and soil types in each sub-basin. It has been widely utilized to predict hydrological conditions and nutrient cycles in terrestrial and aquatic environments [112, 113, 114, 115, 116, 117, 118]. The default setting for the temperature calculation by SWAT is given by the following equation [119]

  

$$T_w=5.0+0.75T_a$$(11)

where, $T_w$ : water temperature [℃], and $T_a$ : air temperature [^oC]. If the air temperature is <20℃, the water will be warmer than the air. This pattern holds true for many large rivers but may not apply when the stream temperature is affected by factors such as snowmelt, surface runoff, and groundwater inflow, which can have a significant influence on the stream temperature.

4.2.5 Improvement in SWAT model for river temperature simulation

In our review, only 12 studies have attempted to simulate stream or river water temperature using the SWAT model. This might be attributed to the structure of default stream temperature module of SWAT. The hydrological runoff process and river water temperature are related, especially in watersheds with a significant groundwater contribution [Section 4.1]. But, in the default SWAT model, groundwater temperature is not considered.

The first attempt to explicitly consider groundwater temperature in the calculation of stream temperature was conducted by Ficklin et al. [94]. They developed a hydroclimatological stream temperature module within the SWAT model framework and implemented the combined effects of hydrological conditions and air temperature on the stream water temperature. The following equations describe the calculations for estimating the water temperature for each subbasin:

  

$$T_{w,\mathit{\text{local}}}=\frac{\left(T_{\mathit{\text{snow}}}{Q}_{\mathit{\text{snow}},\mathit{\text{sub}}}\right)+\left(T_{\mathit{\text{GW}}}{Q}_{\mathit{\text{GW}},\mathit{\text{sub}}}\right)+\lambda (Q_{\mathit{\text{surf}},\mathit{\text{sub}}}+Q_{\mathit{\text{lat}},\mathit{\text{sub}}})T_{\mathit{\text{air}},\mathit{\text{lag}}}}{Q_{\mathit{\text{sub}},\mathit{\text{yld}}}}$$(12)

  

$$T_{w,\mathit{\text{adj}}}=\frac{T_{w,\mathit{\text{upstream}}}{Q}_{\mathit{\text{upstream}}}+T_{w,\mathit{\text{local}}}{Q}_{\mathit{\text{sub}},\mathit{\text{yld}}}}{Q_{\mathit{\text{upstream}}}+Q_{\mathit{\text{sub}},\mathit{\text{yld}}}}$$(13)

  

$$T_w=T_{w,\mathit{\text{adj}}}+K[T_{\mathit{\text{air}}}-T_{w,\mathit{\text{adj}}}]\mathit{\text{TT}}\mathit{\text{if}}{T}_{\mathit{\text{air}}}>0$$(14)

  

$$T_w=T_{w,\mathit{\text{adj}}}+K[{(T}_{\mathit{\text{air}}}+\epsilon )-T_{w,\mathit{\text{adj}}}]\mathit{\text{TT}}\mathit{\text{if}}{T}_{\mathit{\text{air}}}\le 0$$(15)

where, $T_{w,\mathit{\text{local}}}$ : local water temperature in the specific subbasin river [℃], $T_{w,\mathit{\text{adj}}}$ : adjusted water temperature at the point where upstream river water is merged with discharged water from the specific subbasin [℃], $T_w$ : final stream temperature at the outlet of specific subbasin [℃], $T_{\mathit{\text{air}}}$ : air temperature [℃], $T_{\mathit{\text{snow}}}$ : melted snow temperature [℃], $Q_{\mathit{\text{snow}},\mathit{\text{sub}}}$ : discharge in the specific subbasin [m³/day], $T_{\mathit{\text{GW}}}$ : groundwater temperature [℃], $Q_{\mathit{\text{GW}},\mathit{\text{sub}}}$ : groundwater discharge in the specific subbasin [m³/day], $T_{\mathit{\text{air}},\mathit{\text{lag}}}$ : air temperature calculated with a user-specified time lag [℃], and $Q_{\mathit{\text{surf}},\mathit{\text{sub}}}$ , $Q_{\mathit{\text{lat}},\mathit{\text{sub}},}$ and $Q_{\mathit{\text{sub}},\mathit{\text{yld}}}$ : surface runoff, lateral runoff, and total discharge in the specific subbasin [m³/day], λ: calibration coefficient relating $T_{\mathit{\text{air}},\mathit{\text{lag}}}$ , $Q_{\mathit{\text{surf}},\mathit{\text{sub}}}$ and $Q_{\mathit{\text{lat}},\mathit{\text{sub}}}$ , $T_{w,\mathit{\text{upstream}}}$ : water temperature entering the subbasin [℃], $Q_{\mathit{\text{upstream}}}$ : upstream discharge entering the subbasin, $\mathit{\text{TT}}$ : travel time of the stream [hr], $K$ : heat transfer coefficient [1/d], $\epsilon$ : air temperature coefficient.

After, local water temperature in the specific subbasin river is calculated by Equation (12), Equation (13) adjusts the local water temperature by accounting for contributions from upstream. Finally, Equations (14) and (15) describe the interaction between the stream temperatures and surrounding air temperature as the stream flows through the sub-basin. This interaction helps to model how the stream temperature changes as it moves through the area. The developed module improved the stream temperature simulation performance over SWAT’s original approach. Furthermore, it has been utilized to evaluate the effects of climate change on stream temperatures in the Sierra Nevada region in California [5] and the Columbia River basin in North America [35], and Zeiger et al. [120] evaluated its effectiveness in comparison to both linear and nonlinear regression models in a mixed-use, urbanizing watershed in the central United.

Brennan [121] found SWAT-Ficklin et al. [94] model is more suited for local resource management due to its higher spatial resolution and offer a user-friendly interface on the ArcGIS platform, making them suitable for interacting with stakeholders and visualizing data and findings. Mustafa et al. [122] added radiative components into the Ficklin et al. [94] stream temperature that provided a new mechanism for evaluating the effects of different land uses on stream temperature within SWAT. In addition to this, the model was used to assess the impact of climate change on the stream temperature regimes in the Athabasca River Basin [123] and the impact of hydrological processes on the stream temperature in the Elbow River watershed in the cold region of Western Canada [124].

Qi et al. [125] attempted to directly connect soil temperature with stream temperatures, assuming that each runoff components such as surface flow, lateral flow, and base flow matched those of their sources. Surface runoff temperature matched air temperature unless there was snow and remained constant at 0.1℃, despite snowmelt or rain-on-snow incidences (equation 16). Lateral flow temperatures were assumed to be equivalent to the temperatures of each soil layer (equation 17). The base flow temperature was considered to be equal to the temperature at the damping depth, where the soil temperature matches the average yearly air temperature (equation 18).

  

$$T_{\mathit{\text{sf}}}=T_a\left(\mathit{\text{nonsnow}}\right)\mathit{\text{or}}{T}_{\mathit{\text{sf}}}=0.1℃$$(16)

  

$$T_{\mathit{\text{lf}}\left(i\right)}=T_{\mathit{\text{soil}}\left(i\right)}$$(17)

  

$$T_{\mathit{\text{bf}}}=T_{\mathit{\text{bot}}}$$(18)

where, $T_{\mathit{\text{sf}}}$ , $T_{\mathit{\text{lf}}}$ , and $T_{\mathit{\text{bf}}}$ : surface runoff, lateral flow and base flow temperatures generated from the HRUs, $T_{\mathit{\text{soil}}}$ and $T_{\mathit{\text{bot}}}$ : soil layers and the damping depth temperatures, and $i$ : soil layer numbers as defined by the SWAT model.

In the calculation method proposed by Qi et al. [125], they utilized the soil temperature calculation module of the SWAT. Though the module works well in temperate and tropical environments, predictions of soil temperature by the SWAT do not match field measurements in cold regions with seasonal snow cover. To address this problem, Qi et al. [125] also employed a physically based soil temperature module to calculate the soil temperature in each soil layer and at the damping depth based on heat transfer theory [126].

4.2.6 Remained issues in river water temperature modeling

Proposed method by Qi et al. [125] considerably accelerates the river temperature modeling structure in the SWAT. They explicitly consider the effect of flow paths (surface flow, lateral flow, and base flow) on river temperature, which means groundwater contribution to river water temperature is also explicitly considered. Though model structure itself can already consider the effect of underground thermal regime on river temperature, there remained some important issues yet to be solved.

The most important but difficult task might be to establish an appropriate method to determine damping depth. Damping depth is the soil depth where the soil temperature matches the average yearly air temperature. In other words, how deep the contribution depth to runoff processes should be determined. Usually, this depth is assumed to be same as the soil depth in agricultural areas. However, recent findings in steep mountainous area in Japan suggested that not only the flow in soil layer but also the flow in bedrock zone has a significant contribution to river water discharge [127, 128]. Moreover, this phenomenon is not unique to Japan but can be thought to occur in a wide range of steep mountainous regions especially with large rainfall amount. Thus, we need to identify the lower boundary of the zone where the water flow occurs. This requires that comprehensive understanding of interaction between water and heat regime occurring in soil and bedrock area. Combined use of hydrological and chemical observation and modeling might be promising to solve this problem.

5. Conclusions

This review provides that although physically based models offer valuable insights into the various energy fluxes influencing river temperature, a gap in the consideration of groundwater flux is evident, especially in smaller rivers. Among the articles in which a specific energy budget component is considered in river temperature models, shortwave radiation emerges as the dominant heat source, followed by sensible and latent heat fluxes, long wave radiation, riverbed fluxes, groundwater inflow, and friction, with each component playing a crucial role in regulating river temperatures.

The assessment of several modeling techniques reveals that groundwater temperature is often estimated using the mean annual air temperature with a thermal offset. Fully distributed models simulate groundwater temperatures but are limited by computational demands, while semi-distributed models like SWAT offer broader scale simulations and can save calculation time but they often rely on simplified equations. The method proposed by Qi et al. [125] significantly improves river temperature modeling by explicitly considering surface, lateral, and base flow paths on river temperature, thus incorporating groundwater’s contribution.

However, a challenge remains in determining the damping depth, particularly in regions with complex hydrological contributions from both soil and bedrock zones. Identifying the lower boundary for water flow is critical, and a comprehensive understanding of the water and heat interactions in soil and bedrock is necessary.

Future research can focus on the validation and comparison of different river temperature models using actual observational data to evaluate their accuracy and performance under various hydrological and climatic conditions. To enhance the better understanding and representation of the impact of groundwater on river temperature modeling is required to improve predictions and management strategies for water resources.

Acknowledgements

The author is grateful for the support and encouragement received from colleagues and family members during the preparation of this manuscript. The author acknowledges Japan’s Ministry of Education, Culture, Sports, Science, and Technology (MEXT) for financial support to live and study in Japan.

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