By Winter P. A.

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26) f + > « ' > + ! 21) of the form u' = uexp i (kx + ly — wt), v' = vexp i (kx + ly — ut),h' = hexpi (kx + ly — ut). 21) can then be rewritten as —iu> —/ gik \ lu f -iw gil \ I v ) = 0. 26) have the trivial solution u = v = h = 0. Non-trivial solutions are only possible if the determinant —iu) —f gik f —iu gil ikH UH —iu> 0. 29) which has roots w= 0 = ±y/gH(k2+l2) + f2. 27). The nature of the roots can be further studied by examining the associated eigenfunctions. 32) where a is a constant. This eigenfunction is both geostrophic, satisfying and non-divergent, satisfying l(Hu>) + ly(HV>) = 0.

49) Large-scale atmosphere flow 28 where u a = (ua,va) is the rotational part of (u,v) satisfying £-£-C. 50, dv^ _ _ dy We can write (ua,va) in terms of a stream-function ip, so that ua = —dtp/dy, va = dtp/dx. 47) has been approximated by + gV2h - V • / W = 0. 51). The resulting equations are called the nonlinear balance shallow water equations. Slightly different versions of these appear in the various references cited above. 25) is not changed by this approximation. 34) satisfied by linear Rossby waves.

Write (u,v,0) = uz. As before, we differentiate the second equation with respect to time and substitute from the first equation. 93) + remainder. 83). dw\ „ 2 ( V2Z (j-uz + — (-fuz-Vz(C + 2f) + 2—J(u,v)\ The linearisation of this equation is gd&\ • V^'j + new remainder. 89). 96) • V(C + 2/) + 2 ^ J ( u , t , ) ) , where once again it can be shown that it is consistent to have neglected the terms in 'new remainder'. 50). 97) and the first equation by dw d2tp dw d2tp — -\ — = 0. 99) are essentially the equations used in [McWilliams et al.

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2-3 graphs which have Vizings adjacency property by Winter P. A.


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