7.3 Kinetic Equation for Non-linear Waves in Sea with Ice Cakes
329
by a series of the variables ' TJ ( k) and ( k). Expanding this ratio relative to
¢ ( k) with an accuracy up to third-order terms the following formula can be
obtained:
( th(kH)) [
J {2)
(k)= 1 +Aq
x(k)+ D 1 X1'T/215o-1-2dk1,2
(7.48)
The following designations are used:
n<2) = n< 2 ) (k) = -~ [q (1 + Ak cth (kH) )] = -cQo
0
2n
1 + Aq
(7.49)
(7.50)
Substituting (7.48)-(7.50) into (7.44)-(7.46), the following is obtained:
K2 =(I E~ 2 )wxox0 dk + j E~ 2 )axox0 dk)
= ~ (I qxoxo dk + A 1 qxoxo dk) = ~ 1 qxoxo dk
2
(1 + Aq) 2
(1 + Aq) 2
2
(1 + Aq) '
(7.51)
where
K
I [(A {3)w E(3)w) (A{3)a E{3)a)] ~ dk
3 =
0,1 + 0,1 + 0,1 + 0,1 XOX1'T/2UO+l+2 0,1,2
(7.52)
K 4 = J [(B(4)w + c<4)w + E{4)w)
0,1,2,3
0,1,2,3
0,1
(B (4)a c(4)a E(4)a)]
~
dk
+ 0,1,2,3 + 0,1,2,3 + 0,1 XOX1'T/2'T/3U0+1+2+3 0,1,2,3 (7.53)
Designations similar to those used by Krasitskii (1974) are used here.
They are added by modifications taking ice into account (the upper indexes
wand a belong to water and ice, correspondingly).
It is important to note that the integrand in the right-hand side of (7.51)
provides the exact linear dispersion ratio w(k) of the form (7.3) in the first
term of the final Hamiltonian representation (7.38).