are carried out for a separate wave mode. We use the eikonal equation
∂S ̸ ∂x
ð
Þ
2 + ∂S ̸ ∂y
ð
Þ
2 = K
2
ðx, yÞ to determine function Sðx, yÞ. In the plane case, the
initial conditions for eikonal S are posed on line L: x 0 ðαÞ, y 0 ðαÞ, Sðx, yÞj L = S 0 ðαÞ.
To solve the eikonal equation, we construct the rays, i.e., the characteristics of this
equation, which have the following form
dx
dσ
=
p
Kðx, yÞ
,
dp
dσ
=
∂Kðx, yÞ
∂x
,
dy
dσ
=
q
Kðx, yÞ
,
dq
dσ
=
∂Kðx, yÞ
∂y
,
ð1:5Þ
where p = ∂S ̸ ∂x, q = ∂S ̸ ∂y, dσ is the ray length element. The initial conditions of
p 0 and q 0 for solution (1.5) are determined by solving the following system
p 0
∂x 0
∂α
+ q 0
∂y 0
∂α
=
∂S 0
∂α
, p
2
0 + q
2
0 = K
2 x 0 ðαÞ, y 0 ðαÞ
ð
Þ
whose solution and the initial conditions x 0 ðαÞ, y 0 ðαÞ, p 0 ðαÞ, q 0 ðαÞ determine the
ray x = xðσ, αÞ, y = yðσ, αÞ. After the rays are constructed, eikonal S can be determined by integrating along the ray: S = S 0 ðαÞ +
R σ
0 K xðσ, αÞ, yðσ, αÞ
ð
Þ dσ. Eigenfunction W 0 ðz, x, yÞ is calculated up to multiplication by arbitrary function A 0 ðx, yÞ:
W 0 ðz, x, yÞ = A 0 ðx, yÞ f 0 ðz, x, yÞ, where f 0 ðz, x, yÞ is the solution of the basic vertical
spectral problem with normalization
R H
0 ðN
2
ðz, x, yÞ − ω
2
Þf
2
0 ðz, x, yÞdz = 1. Then,
after rather cumbersome analytic calculations, we obtain the conservation law along
the eikonal characteristics:
d
dσ ln
A
2
0 ðx, yÞIðx, yÞ
K 2 ðx, yÞ
= 0, where Iðx, yÞ is the geometric
divergence of the rays (characteristics). We note that the wave energy flux is
proportional to A
2
0 K
− 1 R, where R is the width of an elementary ray tube; therefore,
the quantity equal to the wave energy divided by the modulus of the wave vector is
preserved in this case.
The long-range IGW fields in the real ocean are, as a rule, non-harmonic wave
packets. Indeed, at a far distance form perturbation sources, the complete wave field
is a sum of separate wave modes whose asymptotics, depending on the stratification, depth, and other parameters of the ocean, can be expressed in terms of the Airy
function or the Fresnel integrals. Therefore, to study the problem of wave packet
evolution in a horizontally smoothly inhomogeneous and unsteady stratified medium, it is necessary to use another ansatz [2, 5, 7].
We introduce slow variables x
* = ε x, y
* = ε y, t
* = ε t (since z is not assumed
to be a slow variable, we omit the asterisk in the index), where ε = λ ̸ L ≪ 1 is a
small parameter characterizing the smoothness of the medium variations along the
horizontal line (λ is the characteristic wave length, and L is the scale of horizontal
inhomogeneity). Then system (1.2) for determining the velocity components
ðU 1 , U 2 , WÞ in these slow variables becomes
Internal Gravity Waves in Horizontally Inhomogeneous Ocean
113
∂S ̸ ∂x
ð
Þ
2 + ∂S ̸ ∂y
ð
Þ
2 = K
2
ðx, yÞ to determine function Sðx, yÞ. In the plane case, the
initial conditions for eikonal S are posed on line L: x 0 ðαÞ, y 0 ðαÞ, Sðx, yÞj L = S 0 ðαÞ.
To solve the eikonal equation, we construct the rays, i.e., the characteristics of this
equation, which have the following form
dx
dσ
=
p
Kðx, yÞ
,
dp
dσ
=
∂Kðx, yÞ
∂x
,
dy
dσ
=
q
Kðx, yÞ
,
dq
dσ
=
∂Kðx, yÞ
∂y
,
ð1:5Þ
where p = ∂S ̸ ∂x, q = ∂S ̸ ∂y, dσ is the ray length element. The initial conditions of
p 0 and q 0 for solution (1.5) are determined by solving the following system
p 0
∂x 0
∂α
+ q 0
∂y 0
∂α
=
∂S 0
∂α
, p
2
0 + q
2
0 = K
2 x 0 ðαÞ, y 0 ðαÞ
ð
Þ
whose solution and the initial conditions x 0 ðαÞ, y 0 ðαÞ, p 0 ðαÞ, q 0 ðαÞ determine the
ray x = xðσ, αÞ, y = yðσ, αÞ. After the rays are constructed, eikonal S can be determined by integrating along the ray: S = S 0 ðαÞ +
R σ
0 K xðσ, αÞ, yðσ, αÞ
ð
Þ dσ. Eigenfunction W 0 ðz, x, yÞ is calculated up to multiplication by arbitrary function A 0 ðx, yÞ:
W 0 ðz, x, yÞ = A 0 ðx, yÞ f 0 ðz, x, yÞ, where f 0 ðz, x, yÞ is the solution of the basic vertical
spectral problem with normalization
R H
0 ðN
2
ðz, x, yÞ − ω
2
Þf
2
0 ðz, x, yÞdz = 1. Then,
after rather cumbersome analytic calculations, we obtain the conservation law along
the eikonal characteristics:
d
dσ ln
A
2
0 ðx, yÞIðx, yÞ
K 2 ðx, yÞ
= 0, where Iðx, yÞ is the geometric
divergence of the rays (characteristics). We note that the wave energy flux is
proportional to A
2
0 K
− 1 R, where R is the width of an elementary ray tube; therefore,
the quantity equal to the wave energy divided by the modulus of the wave vector is
preserved in this case.
The long-range IGW fields in the real ocean are, as a rule, non-harmonic wave
packets. Indeed, at a far distance form perturbation sources, the complete wave field
is a sum of separate wave modes whose asymptotics, depending on the stratification, depth, and other parameters of the ocean, can be expressed in terms of the Airy
function or the Fresnel integrals. Therefore, to study the problem of wave packet
evolution in a horizontally smoothly inhomogeneous and unsteady stratified medium, it is necessary to use another ansatz [2, 5, 7].
We introduce slow variables x
* = ε x, y
* = ε y, t
* = ε t (since z is not assumed
to be a slow variable, we omit the asterisk in the index), where ε = λ ̸ L ≪ 1 is a
small parameter characterizing the smoothness of the medium variations along the
horizontal line (λ is the characteristic wave length, and L is the scale of horizontal
inhomogeneity). Then system (1.2) for determining the velocity components
ðU 1 , U 2 , WÞ in these slow variables becomes
Internal Gravity Waves in Horizontally Inhomogeneous Ocean
113