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This, for the Kelvin theorem, as discussed by Arnold,2'3 is A,,(AH) =fT(V8h n) IZ=,+E ds- T(V8. nh)I,=Z dsf= E a,[T(V8. nh) ] z= -ds =*E 6ZT(V8 n)|z=h Rds, (2) where we have chosen P =0 on the air-sea surface. Here n is the normal vector to the air-sea surface and ds the tangent vector. In the same way A E2H |[+ (VT)2 E__ | dx= 2 .f [( + (VT)2 | d L 2 (V)2J+iE2j =f ajz(gZ + (vT)2JI 2 = 2- |aZgz + t2 ) +T azq0 dx= 2 ||g+(2 ) |l x 2 gdx .(3) In formulas [see (i) of Ref. 1] A2H= ((V8)2+ d (2 q(1g ) j=(8q)2 dxdz + f- e d + 2 fEz(V8 n6) Iz= ,ds ; (4) a first necessary condition for A2H > 0 is d_ d _1 _ >0 (5) dq- 2 dq- (d/d;F)F(T) as discussed in Refs. 1-3. Dalrymple and Cox4 discuss a realistic case with F(T) - y2T. For similar waves, (VTP)2/2 is an increasing function of z: a= z[gz + (V;P)2/21 g. This is a second, realistic condition. But the last integral of Eq. (4) can be negative. Let us now discuss the simple case of
Physical Review A – American Physical Society (APS)
Published: Dec 1, 1984
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