Ocean Currents in a Non-homogeneous Ocean All 



the vertical slope of the stream lines is so small that the current field can be regarded 

 as horizontal. Under stationary conditions the stream lines follow the stream function 

 i/rCxj') = Ci; the horizontal density distribution shall be given by p{x,}') = c^. The 

 angle between the two sets of curves may be y. If the stream lines are at an angle a to 

 the positive .Y-axis and correspondingly the isopycnals at an angle /3, then 



difj Idip Sp /dp 



tan a = — K-l^^ and tan j8 = — -- / 

 c.v/ dy dxj 



dy 



From this it follows that 



dip dp dip dp 



^ oj^^Z_^y^^ (xv.i) 



dijj dp dill dp 

 dx dx dy dy 



If the stream hnes are parallel to the density lines (y = 0), then consequently 



dj^d_P_djPd_p^^ (XV 2) 



dx dy dy dx 



Disregarding for the moment the effects of friction (turbulence), and if there are no 

 physical changes in the water masses due to external circumstances then, for stationary 

 conditions dujdt = dv/dt = 0, the equations of motion (XIII. 1) will also apply for a 

 non-homogeneous sea. Eliminating the pressure p and taking into account the con- 

 tinuity equation and introducing a stream function (equation X.35), equation (XV.2) 

 is obtained. In a non-homogeneous sea stationary conditions require that the stream 

 lines and the isopycnals (isosteres) are parallel. This result is self-evident since otherwise 

 these surfaces would be displaced and this would contradict the condition of a 

 stationary state. The same also applies to isothermal and isohaline surfaces. On the 

 other hand, the following equation can be derived from the equation of motion and 

 the hydrostatic equation (Ertel, 1933) 



/ dpu dpv\ d^p dp d^p dp 



^' Y' -dl ~ P""^) = ~ W^ dx-^ ^^z dy- 



By means of the hydrostatic equation 



dp 



-dz=^^P 



this equation can also be written in the form 



■' P dz \vj ^ \dx dy dy dx] 



If the total velocity V is at an angle x to the ^--axis so that u = V sin x and v = V cos x 

 then 



If the isobars and isopycnals are parallel in a horizontal plane, then the expression in 

 brackets, D, is zero. The mass field is therefore barotropic and dxjdz = 0, that is, the 



