Meanders 133 



known to exert an influence upon the path of some surface currents, as is 

 shown, for example, in the case of the Antarctic Circumpolar Current, by 

 Sverdrup, Johnson, and Fleming (1942), 



STABLE MEANDERS IN A STREAM OF 

 UNIFORM POTENTIAL VORTICITY 



The following dynamical study treats the problem of stable meanders in a 

 Stream of uniform absolute vorticity. 



We shaU suppose, as before, that absolute vorticity is conserved in the 

 upper layer, and hence that 



„ 8v 



Hq h 



where h^ is the value of h{x) a,t x->oo. 



We shall also suppose that the Stream is not straight, but slightly curved, 

 with a radius of curvature ^. If ^ is much greater than the width of the 

 Stream, and if we assign a positive sign to ^ for cyclonic curvature, then 

 the dynamical equation of the upper layer is 



fv-] — = g' — , (10 



where g' =g{Aplp). Ehminate h by cross-differentiation; and in this way 

 we obtain 



v^ _l d^v 

 aMc a? 8x^ ' 



where c = ^J{g'hQ), and oc=flc. We regard the nonlinear term as small in 

 comparison with the others; hence, we express v in series form: 



Therefore, the functions Uq, i^^, Vg, ... are determined by the following 

 equations: 



Vq = (x-^vI, (13) 



Vi + c-'^vl = a-^vl, (14) 



V2 + c-'^2voVi = oc-^vl, (15) 



V3 + c~'^{2voV2 + vl)=a~^vl, (16) 



