Meanders 129 



A SIMPLE MEANDER THEORY FOR A VERY WIDE 

 CURRENT IN A STRATIFIED OCEAN 



Let us suppose that the lower layer is very deep, and hence that the hori- 

 zontal pressure gradients vanish in it at all times. In the undisturbed state 

 a steady current U flows in the cc-direction in the upper layer of thickness D. 

 Associated with this current is a cross-stream pressure gradient of the 

 following form : 



/— f- 



We now suppose that there are small perturbations u, v, in the velocity 

 components, and h, in the elevation of the free surface, and that these 

 quantities are independent of y, the cross -stream coordinate. The pertur- 

 bation equations may be A\Titten in the form 





S-S( 



p \ du 8D 



If the perturbations are all in the form e'etac-i'O ^e obtain the frequency 

 equation 



P£72(l_^)3_ 



P + gk^^D 



{1-P)+P = 0, (5) 



where p = clU = vjk U=p' + ip" , the real part p' of which is the ratio of the 

 velocity of propagation of the wave to the velocity of the current, and the 

 imaginary part p" of which gives the instabiHty of the wave motion. For 

 the particular range of parameters involved, no one of these terms is small 

 compared to the others. It is convenient to rewrite this equation in the 



^°™ y' + 2 = Py, (6) 



where 



and 



y 



{ n \ -1/3 



The roots of this equation are all real, provided P> 3, in which case there 

 are three types of stable wave present. If P < 3 there is a region of unstable 



9 SGS 



