where 17 is the Laplacian operator 17»\7 and J is the Jacobian vdth re- 

 spect to X and y . 



PERTURBATION ANALYSIS 



To subject the model to perturbation analysis, the x-axis and a constant 

 basic current U are aligned parallel to isopleths of g = f - Tq log h . 

 A velocity perturbation along the gradient of g is introduced in the form: 



V ~ Fe , (U) 



vrfiere i = "iZ-l , k is the wave nianber of the perturbation, and c is the 

 phase velocity of the perturbation. The total velocity is then: 



y ^Ui ^17 , (15) 



and the relative vorticity is given by: 

 Also note that 



§-ck'v , §-k\. 07) 



Substituting these perturbation relationships into the model, equation (12) 

 yields : 



g^ui^mhv^f = o, 



or 



which finally gives: 



/c'fc-A'U) f V ^C?, (18) 



where V = 3g/^y • _Equation (18) is similar to Rossby's long-wave speed 

 formula, except for A* and V ; V includes the effect of the sloping bot- 

 tom as well as the earth's rotation. Note here that the effect of bottcm 

 topography is to enhance (or diminish) the usual /& effect, and A*- effec- 

 tively increases the strength of the basic flow U . 



For the special case of stationary perturbation, the phase velocity c 

 is zero. Solving equation (18) for the stationary wavelength ^5= 2Tr/(^ 

 yields : f^ — • 



138 



