SECT. 3] DYNAMICS OF OCEAN CURRENTS 389 



horizontal area. By introducing Ep for the potential energy, Igt]'^. and using 

 (228), we can transform (258) to 



+ 9HH:- + ^\=0. (259) 



8t \ dx dy 



We can interpret gHurj and gHv-q as the components of flux of total energy 

 associated with the motion. Taking the average of (258), we obtain 



du-n dvri 



Thus, the divergence of the mean energy flux is zero and the mean energy 

 density at a given jDoint is constant. 



The mean energy flux associated with zonal Rossby waves {v = Q, 8^, = So) is 

 proportional to 



ut] = igr]'- 



aj(Ao± So) — eof 



e+2^o2/ (261) 



mj = 0. (262) 



The energy flow is eastward for short Rossby waves (Ao+So) and westward 

 for long Rossby waves (Aq — So). For both waves, the energy flow is independent 

 of X. For the solutions in the range ct^ < o given in (257), we obtain 



UTj = i^grjO- 



ojAo— eof 



e±25o2;+2eo2/ (263) 



^= +lgr^02|-y 3^1(^2 _^2)]e±25o.T+2eo2/. (264) 



These flux components satisfy (260) provided we apply the assumptions made 

 for (244) and treat So and eo as constants. As ojAq— eo/is approximately equal 

 to — jS/2, the zonal energy flow is westward. The meridional flow is towards the 

 equator for + So and towards the poles for — So. For poleward flow of energy, 

 the amplitude of the motion decreases westwards. 



We can see how solutions for g'^ < arise by considering an ocean divided by 

 the y-axis into two regions of different depths. We can always choose the 

 angular velocity of the Rossby weaves in the deeper region such that ct^ < in 

 the shallower region. Solutions in the shallower region will then be of the form 

 given in (257). The Rossby waves will not propagate into the shallower region 

 and the energy from the waves will be transported meridionally on reaching 

 the shallower region. 



The flow of energy is zonal for each zonal Rossby wave separately. However, 

 there is meridional flow for a combination of the two waves at a given fre- 

 quency. If we represent the sum of the two waves by 77+ + 77-, where rj+ is the 



