In this paper we consider a sort of eddy diffusion which may be expressed by the following differential equation:- where τ is time, a, b and m are constants depending on wind velocity, and the vertical distribution of wind velocity and temperature. Considering that the diffusion does not occur through the earth surface, we obtain solutions for a instantaneous point source and a continuous point source. These solutions show good agreement with experimental results throughout the whole range (100-2000m.) of all experiments. Values of constants will be reported in another paper.
Indirect evidence as to the prevalence of the Bergeron-Findeisen process(1) in the formation of precipitation may be obtained as follows. If it is assumed, with Findeisen, that ice particles cannot be formed in the atmosphere by the spontaneous freezing of undercooled drops, and if, as is supposed by Bergeron and Findeisen, most of the heavy rain originates as ice crystales, a low salt and acid content would be expected with high rainfall intensities while the rain collected from light intensity falls of rain would be more likely to have a high acid and salt content. Fortunately there has been simaltaneous determinations of the salt and acid content which can be correlated with the intensity of the historical heavy rainfall during June 28-30, 1938 in Tôkyô(2). During this period, wind was so steady that there was no marked change in the direction and speed of the wind. In his report on the chemistry of rain, Kita published the following set of chlorine, sulphate and nitrite determinations with the corresponding rainfall intensities. It will be seen that a well-defined inverse relationship exists between the average amount of rain for one-hour intervals and the corresponding chlorine content. Similar but ill-defined tendencies between the average amount of rain for one-hour intervals and the corresponding ammonia or nitrite content are indicated in the table. It appears that the high chlorinity and acidity for the last period of this heavy rainfall could not have resulted from the cleaning of the impurities from the air by the first part of the rainfall and this is only to be explained on the basis of the inverse relationship concept-which is in accordance with the Bergeron-Findcisen theory.(1) In his paper on the chemistry of rain, Dr. Miyake(2) published the following statistical table of chlorine, ammonia and nitrite determinations with the corresponding amount of rainfall in order to show how these contents may vary with the rainfall intensities. As is indicated in the statistical table, a marked low salt and acid content is generally found with high amount of rainfall while the rain collected from light rainfalls has a relatively high acid and salt content. This is in accordance with the Bergeron-Findeisen theory, but the possibility of the cleaning of the impurities from the air by the first part of the rainfall can be partially used to explain the low chlorine and acid content of relatively heavy rainfall.
The original data were obtained from the weather maps analysed by the Central Meteorological Observatory, Tôkyô. First, the locations where 4014 cyclones originate were determined carefully and were tabulated during the 8-year period extending 1933-1940. The frequencies were written for each 2° square of latitude and longitude (see Fig. 1). The place of origin of storms seems to be influenced to some extent by the distribution of land and water. The principal areas of origin may be roughly summed up as four groups (see Fig. 2) including storms which originate (1) in the sea to northeast of Formosa, passing ovor Loochoo Islands and thence move to Japan proper or to the Pacific' (2) in the sea to east of Korea, passing over the Sea of Japan and Northern Japan and thence move to the Pacific or the Okhotsk sea, (3) in the sea to south of Sikoku Island, advancing along the southern coast of Japan proper and thence move to the Pacific, and (4) in the sea to south of Kwanto district advancing to the Pacific. Next, the cyclones which originate during the 9-year period (1932-1940) in Japanese area (including China Eastern Sea, Yellow Sea. Sea of Japan, Sea to East, Southeast and South of Japan) have been enumerated and their seasonal distribution is Winter 614, Spring 556, Summer 499, Autumn 529. The frequency of the depressions which appeare in our weather chart is rather high in Spring and Autumn, but the frequency of the cyclones which originate in Japanese area is highest in Winter and lowest in Summer. Next the total number of storms that traversed each square, two degrees longitude in width and two degrees latitude in length, during the 8-year period 1933-1940 were enumerated. Chart 3 and 4 were constructed according to this plan. The location of the composite tracks is determined on the ground that the axis of the region of greatese cyclone frequency indicates the mean position of the storm path, which lays usually over sea along the Japanese Archeperago and tends to passing through strait or channel. Lastly, the average movements of Japanese storms that originated during the 3-year period 1935-1937, inclusive, are shown graphically in Figures 5 and 6. Recause the computations were highly troublesome, the smoothed averages were obtained by taking the average of all observations in each 10° square for which data were available. From these figures, we find that in winter cyclones advance to the east with high speed, while in summer cyclones move to the northeast or east in the middle latitudes (>30°N) and advance to the west in the tropical regions (<20°N) with relatively low velocity.
The prediction of the weathers in coming season is important for various purposes, especially for agriculture. For the purpose of any prediction, the most conservative character is usefull for extrapolation just as equivalent potential temperature is used in the air mass analysis. Calculating the correlations between succesive monthly mean values of various meteorological elements, it is found that the air temperature is most conservative. Next, the secular change of the weather is shown for the monthly mean air temperature of Singking in January and for others and the posibility of the seasonal forecast is proved. Lastly, the prediction of the weather in winter and Bai-u season (one of the rainys eason in Japan) is investigated, but the results are not yet sufficient for the practical use and the future investigations are Roped.
Recently, Y. Takahasi has solved various complicated problems of heat conduction and diffusion by an excellent method of graphical integration. In the present paper, I extended his method to the integration of the wave equations and obtained the following formulae corresponding to Y. Takahasi's. If τ be sufficiently small and τ3 be negligible, the value of a function u (x, t) at the time t+τ, which satisfies ∂2u/∂t2=V2•∂2u/∂x2, may be expressed in the form The first term of the right-band side expresses the mean value of u(x, t) at the time t over the interval (x-√3Vτ, x+√3Vτ) of x, and the second term is the effect of inertia, which does not appear in the case of heat conduction. The inertia term may be transformed into the following form; The corresponding formulae for two or three dimensioual problems may be obtained, in the same way, as follows; I introduced, also, modified formulæ which are more convenient for numerical calculations without the help If graphs. If a function u (x1, x2, … xn, t) in a n-dimensional space satisfies a wave equation where δji=1 when j=i, and 0 when j≠i. If we assume k to be eqnal to √3 and replace the integrals which appear in the right-hand side approximately by Simpson's 1/3 rule of intogration usiug the values of u at xi-Viτ, xi, xi+ViΤ, the process of calculation will be simplified. A simple example is shown: A string is stretched with a constant tension. One of its ends is fixed and the other is forced to move in a straight line perpendicular to the string with a small amplitude signified by f(t). When f(t)=0 for t<0, and sin2πt/T for t_??_0, the vibration of string is calculated. The errors of the results obtained by the above method are found to be always less than 0.001. The cases for heterogeneous medium and telegraphist's sequations may be also treated in the same manner, but with a little additional trouble.