All-101 7 



sphere. Such a mapping is not conforrnal since angles betv/een 



lines are not preserved. The region under consideration must 



be well removed from the north pole, 



3, The vertical acceleration terms and the viscous terms 



are neglected in the equation of vertical motion so that, in 



effect, hydrostatic pressure is assumed, i.e., p = g pdz, where 



rv z 

 T] is the free surface height and p = at z = t]. The density 



p may, of course, be a function of the space coordinates. In 

 Appendix 3 it is shown that for the problem v/hich is independ- 

 ent of time, the hydrostatic pressure assumption is necessary 

 only in the depths where there is no motion if one desires a 

 solution for the components of the mass transport only. If it 

 is necessary to find the shape of the free surface, however, or 

 if the non-steady problem is considered, this assumption or some 

 analogous one must be made. 



h» As stated in the introduction, the thermodynamic effects 

 are accounted for only empirically by stipulating a density dis- 

 tribution. We assume p = p[z - T(x,y,t)] v/here the function p 

 of the variable (z - T) can be prescribed to fit observational 

 data. This functional form for p makes the curves of constant 

 density parallel, 



% The equations of motion are integrated over the verti- 

 cal coordinate, z. 



In order to perform this integration it is necessary 

 that we specify the density distribution since p appears in 

 some of the integrands. V/e consider two cases. 



(i) The surface z = T separates two layers of constant 



