The Tropospheric Circulation 619 



(Kuroshio, Peru Current) and others. Rossby's theoretical investigations are put 

 forward mainly along two lines. The first deals besides the vertical also with the lateral 

 frictional effect which is of influence on the horizontal velocity profile in currents. 

 The second deals with currents of constant momentum (impulse) transport and in 

 particular applies the theory of free jets to ocean currents. Since the exchange co- 

 efficients of lateral mixing are of considerably greater magnitude than those for vertical 

 exchange (see Pt. I, p. 103 et seq.) Rossby considered it absolutly necessary to account 

 for frictional forces due to lateral mixing and put strong emphasis on these forces. 

 The usual equilibrium conditions in a geostrophic current for mass elements along a 

 vertical line primarily determine — as always^the vertical velocity distribution. By 

 introduction of the lateral shearing forces this condition will not be changed in any 

 great extent, but the lateral shear imposes a definite transverse velocity profile to which 

 little attention has been paid in the past. 



A linear current in the positive v-direction with a mean velocity v will be fixed by 

 the geostrophic equilibrium between the pressure gradient —(1/ p)(dp/8x) and the 

 Coriolis term —fv. As a result of the horizontal turbulence, however, the individual 

 mass elements will have a movement at right angles to the mean direction of the current 

 and the equations of motion (XIII. 1), will apply for its horizontal components u and v. 

 If the deviations of w and v from the mean velocities « = and I; are denoted by 

 u' and v' then: 



dv' ^ , du' 1 dp 



T,=-^" and ^=A.' w,th -^£-/S = 0. (XIX.6) 



From (X.39) the lateral shearing stress is 



T = - pTlT. (XIX.7) 



Introducing the Prandtl mixing length / of lateral mixing (p. 388) allows (XIX.7) to 

 be rewritten as 



= p/«' (/+ ^)- (XIX-8) 



For a uniform horizontal current the lateral shearing stress will not approximate to 

 zero except when 



Under stationary conditions the lateral mixing imposes a definite horizontal velocity 

 profile, and indeed there must be a velocity decrease towards the right-hand edge of 

 the current (Northern Hemisphere). This is quite large and in middle latitudes (43°) 

 amounts to 1 cm/sec in 100 m).* 



Since such large transverse variations in velocity are hard to observe it must be 

 presumed that the right-hand edge of the current always tends to accelerate the left- 

 hand side even when the right-hand side has a lower velocity. This effect ceases only 

 when the condition (XIX. 9) is satisfied. 



* Against this conclusion the objection has been raised by Priebsch (1943), that besides the 

 lateral turbulence across the gradient current, also that in the direction of the current should be 

 taken into account. If this is done, it is found that the effect ot the earth's rotation mentioned 

 above no longer exists. On the average the effects in the two directions balance exactly. 



