Convection Currents in the Atmosphere. 113 



The principal object of the present investigation is to find 

 out to what extent this fundamental hypothesis is justified. 

 It is rendered especially necessary at present by the recog- 

 nition by G. I. Taylor of Eddy Viscosity, which causes the 

 effective, viscosity to be enormously greater than the true 

 viscosity, and thus makes the validity of the neglect of the 

 viscosity terms doubtful. 



The method adopted is to choose a special problem which 

 can be solved exactly save for the neglect of second order 

 terms. Approximation is not resorted to till the exact 

 solution has been obtained. In order to obtain a soluble 

 problem it has been found necessary to replace the actual 

 atmosphere by an incompressible fluid, homogeneous and of 

 great but finite depth, with a free surface at the top. It 

 seems to the author highly improbable that these hypotheses 

 are of such a character as to change the order of magnitude 

 of the neglected portions of the solution. 



The case of a surface composed of a series of infinite 

 parallel strips of land and sea is first solved, and subse- 

 quently that of a surface with circular symmetry. In both 

 cases the lrypothesis tested is found to be valid when the 

 land masses are large and the temperature variation has a 

 yearly period, and usually in the case of a daily variation. 

 In a small island, however, the hypothesis is quite erroneous, 

 so that in such a case the vertical motion cannot be neglected. 

 The phase-differences between the variations of temperature, 

 pressure, and wind-velocity in both cases take very simple 

 forms, which seem unlikely to be qualitatively changed when 

 a hypothesis corresponding better to the actual atmosphere 

 is adopted. The ideal cases considered are, so far as I am 

 aware, the first in which a complete solution of the convec- 

 tion currents in a fluid due to a periodic supply of heat has 

 been obtained, which may, perhaps, add a further interest 

 to the present investigation apart from the atmospheric 

 problem. 



I. The Variation of Temperature. 



Whether we are considering air, land, or water, the trans- 

 mission of heat by conduction and eddies together is governed 

 by the equation 



DV 



T^_£V 2 V=/, « 



where / is the amount of heat supplied per unit time per unit 

 heat capacity, 

 V is the potential temperature, 

 and k the pseudo-conductivity, which is equal to the real con- 

 ductivity per unit heat capacity in the case of a solid. 

 Phil. Mag. S. 6. Vol. 34. No. 200. Aug. 1917. I 



