326 FOFONOFF [sect. 3 



where p is density, t time, jp pressure, Qt components of rotation corresponding 

 to the components at the earth's surface, assumed to be functions of X2,y only, 

 and oij the components of stress due to molecular viscosity. The symbol ^nk 

 denotes the permutation tensor having the values 



e^yfc = -\-\\ii, j and k are in cyclic order, 

 = —\ a i, j and k are in anticyclic order, 

 = if any pair, or all three, indices have the same value, 



and §31 is unity for i = 3, otherwise zero. 



The stress tensor aij can be expressed in terms of the rate of deformation of 

 a fluid element by the motion through the relationship 



where /x is the molecular viscosity. 



Conservation of mass is achieved by requiring the velocity components and 

 density to satisfy the equation 



In addition, we have conservation equations for other properties 9 of the 

 general form 



dp(p ^ dpcpuj ^ _^^q (4) 



dt 8xj 8xj 



where Fj are components of flux of the property cp set up by internal forces or 

 pressures and q is the total internal source of the property (p. Equations (1) 

 and (3) are special cases of (4). Similar equations can be written for the con- 

 servation of kinetic and internal energies, dissolved salts in sea-water and so on. 



2. Separation into Steady and Time-Dependent Motion 



The complete set of equations given by (1) and (3) cannot be solVed exactly 

 except in extremely simple cases. In order to obtain useful results from the 

 equations, we must carry out a series of simplifications. We shall first separate 

 the equations into a mean flow and a time-dependent flow that represents the 

 departure at any instant of time of the flow from its mean. Because of the non- 

 linearity of the general equations, we cannot carry out the separation into two 

 sets of independent equations. Each set contains terms representing inter- 

 actions between the steady and time-dependent modes of motion. 



By adding (3) multiplied by Ui to (1), substituting (2) for a^j and neglecting 

 compressibility in the frictional terms, we can express (1) in the form : 



which is essentially identical with the general form (4). 



