Ocean Currents in a Non-homogeneous Ocean 479 



with those in deeper layers. This agreement in the course of these lines also extends 

 to the dynamic isobaths at any depth. It must therefore be concluded that the current 

 vectors are also tangential to all these sets of curves and that there is complete equahty 

 between all these hnes. This law allows deduction according to the Ekman theory of 

 the direction of the deep current outside the upper and lower frictional depth which 

 represents the layers in which the drift current and the bottom current are found. All 

 modem cartographic representations of the horizontal distribution of these factors at 

 different depths confirm the general validity of this law (see, for example, the ''Meteor''' 

 Reports, Vol. VI, Atlas). 



The basic prerequisites for the vahdity of this law are the same as in the rules derived 

 above for the relationships between the oceanographic factors and the current field 

 in any horizontal plane. These are satisfied for the deep currents except in those areas 

 where they are disturbed by discontinuity layers, or where due to mixing processes 

 there caimot be any stationary spatial density distribution. 



3. Horizontal Steady Currents in a Stratified Ocean 



The dependence of the vertical velocity distribution in a current on the stratification 

 of the water masses in the pressure field is already shown by the behaviour of two 

 adjacent water bodies. In steady state continuous changes in density require also a 

 definite mutual adjustment between the mass and pressure field. If the flow is directed 

 along the positive >'-axis, then for a steady frictionless motion 



Inserting the hydrostatic equation g = a(8pldz) (z counted positive downwards), 

 elimination of p leads to the relation 



8v 8 log a ? 2 log a 



p- = ^ — p^- - 7- ^^ • (XV.5) 



8z 8z f 8x 



This states that for a given vertical and horizontal mass distribution there will always 

 be a vertical velocity distribution given by (XV.5). Introducing the slope of the isobaric 

 surfaces tan ^ = — {flg)v and that of the isosteric surfaces tan y = — {8pl8x)l(8pl8z) 

 the equation takes the form 



dv 2 8 log a 



^ = -^(tan y - tan j8) -^ . (XV.6) 



8z J cz 



Since 8 log aj8z is always negative, the expression in parenthesis decides about 

 increase or decrease in the velocity with depth. In other words, this increase or decrease 

 in velocity depends on the difference in the slope of the two intersecting sets of surfaces 

 or lines in a dynamic section. Figure 136c (page 331) shows the two possible cases (r 

 is always positive); in that shown on the left-hand side the expression in brackets 

 is always positive, and therefore 8vl8z < 0, or there will be a decrease in velocity with 

 depth. In the case on the right-hand side 8vj8z > 0, and there will be an increase in 

 velocity with depth. When y = ^ then 8vjcz = which is the barotropic case with a 

 constant velocity at all depths. These results can be expressed by the following rule : 



