7.3 Kinetic Equation for Non-linear Waves in Sea with Ice Cakes
331
EC4)w _ -~ (-.!._) 2 (kk1- knq(k) + (kk1- k 2 )q(ki)
( 7 . 60 )
0 • 1 -
4 2n
(1+Aq(k))(1+Aq(k1))
.
There are similar expressions for the kernels B and C as for ice and for
water. The only difference is that the proper kernels E are taken for ice. So,
it is enough for ice to get only the last fourth-order term, namely:
(7.61)
Thus, the Hamiltonian is completely written in the Fourier representation
of canonical variables. Now it is possible to write it in the form of normal
variables.
Interaction coefficients and the kinetic equation. According to standard methods (Zakharov, 1968), the following expressions for the Fourier
components of canonical variables can be written with the help of normal
variables using the formulas (7.36). Substituting (7.36) in the Hamiltonian
terms into (7.43) and (7.51)-(7.53), under the condition that the quadratic
terms are presented in their standard form (7.38), the final expressions for
the interaction coefficients U and V can be obtained. Using the results of
Krasitskii & Kalmykov (1993), the coefficients of three-wave interactions can
be presented as:
U.w,a _ -L L M [A(3)w,a+ E(3)w,a]
0,1,2 -
0 1 2
0,1
0,1
--( /8
)1/2[A(3)w,a+E(3)w,a]
-
gw2
WOW1
0,1
0,1
(7.62)
and
u.(1) = """' [-uw,a - uw,a + uw,a ]
0,1,2
~
-0,1,2
-0,2,1
1,2,-0 '
(7.63)
w,a
u.C3) _ """' [U.w,a + u.w,a + uw,a ]
0,1,2 - ~
0,1,2
0,2,1
1,2,0
(7.64)
w,a
It should be noted that the summation in (7.63), (7.64) is taken over the
indexes w, a. The proper coefficients of four-wave interactions are as follows:
v;w,a
_
2 £ L M M [B(4)w,a + cC4)w,a + E(4)w,a]
0,1,2,3 - -
0 1 2 3
0,1,2,3
0,1,2,3
0,1
'
(7.65)
and they can be written for unsymmetrized coefficients in the final Hamiltonian as: