32
1 General Problem Formulation
dk
tan( r.p) cos(/3)
1 {
[av
dt = -k
R
V sin(/3 -1)- kR sin(/3) ar.p cos(!- ,6)
.
a,] cos(/3) [av
a']}
+ V sm(/3 - 1) ar.p + cos( 'P) arJ cos(! - ,6) + V sin(/31) arJ
_ .!_! [ . (f.l) aH cos(/3) aH] .
R sm"' ar.p + cos(r.p) a{} '
(1.88)
d/3 = _ tan(r.p) cos(/3) [c + kV cos(!_ ,6)]- kcos(/3) [av cos(!_ ,6)
dt
R
g
R
ar.p
+ V sin(/3 - 1) a,] + !5_ sin(/3) [ av cos(! - j3) + V sin(/3 - 1) a,]
ar.p
R cos( arJ
1 [
aH sin(/3) aH]
+ Rf - cos(/3) ar.p + cos(rJ) arJ ;
(1.89)
dw
av
.
a1
-
= kcos(/3 - 1 ) - + kVsm(/3 -~)dt
at
at '
where v19 = v cos(!); v' P = v sin(!).
(1.90)
Thus, the problem of determining the spectral density of the wind wave action is reduced to solving the equation system (1.84), (1.86)-(1.90) under the
given initial (or boundary) conditions. It should be noted that the equation
system contains functions depending on the variable parameters such as the
depth field H(r.p,rJ), the current velocity field V = {V'P(r.p,{},t), V19 (r.p,rJ,t)},
and also the wind speed field U = {U'P( r.p, rJ, t), U19( r.p, rJ, t)}. The latter is
included in the source function G and determines the wind wave input.
In the general case the process of obtaining a numerical solution of the
problem (1.84)-(1.90) is a very complicated task demanding significant computer resources. A great variety of different physical factors, and their different spatial and temporal scales make the numerical implementation of the
problem very difficult.
1. 7 Spatial-Temporal Scale Considerations
in the Analysis of Problem Solution
Estimations of different components in the right-hand side of the equation
system (1.84), (1.86)-(1.90) show different spatial-temporal scales of the
physical mechanisms forming a wind wave field in the ocean. It should be
noted that the wave number and the change in the angle j3 is due to the current spatial and temporal variations and the non-uniformity of the depth in
shallow water. At the same time the angle j3 is changed without any current
or depth influence due to the ocean surface sphericity effect.