with the theoretically calculated time of the wave travel time τ gives the following
value β = 0.878. Thus, these estimates allow us to conclude that the influence of the
density field inhomogeneities, which is taken into account in the above-described
method for asymptotic representation of the wave fields, is one of the factors
determining the scales of the space attenuation of IGW fields observed in field
measurements.
Fields of IGW in the Ocean of Variable Depth
We consider one of the problems of IGW propagating in the stratified ocean of
variable depth. In the framework of the linear theory, we study the non-viscous
incompressible inhomogeneous medium with unperturbed density ρ 0 ðzÞ, which is
bounded by surface z = 0 and ocean floor z = γ y (the z axis is directed upwards, γ is
the ocean floor slope). At point x = x 0 , y = y 0 , z = z 0 at the slope, there is a point
mass source of power Q depending on time as expð − iω tÞ. The system of hydrodynamic equations for small perturbations of density ρ
* , pressure p
* , and velocity
components ðu 1 , u 2 , wÞ is written as [2–6]
ρ 0
∂u 1
∂t = −
∂p
*
∂x , ρ 0
∂u 2
∂t = −
∂p
*
∂y , ρ 0
∂w
∂t = −
∂p
*
∂z + gρ
*
∂u 1
∂x +
∂u 2
∂y +
∂w
∂z = Q expð − iω tÞδðx − x 0 Þδðy − y 0 Þδðz − z 0 Þ,
∂ρ
*
∂t + w
∂ρ 0
∂z = 0,
ð2:1Þ
where g is the acceleration of gravity. As the boundary conditions we pose the
“rigid lid” condition at the ocean surface and zero mass flux at the ocean bottom
w = 0 at z = 0, w + u 2 γ = 0 at z = − γ y
ð2:2Þ
Under the assumption that the time-dependence of all solutions is harmonic
ðp
* , ρ
* , u 1 , u 2 , wÞ = expð − i ω tÞðp, ρ, U 1 , U 2 , WÞ, we obtain the following system of
equations with boundary conditions (2.2)
iωρ 0 U 1 =
∂p
∂x , iωρ 0 U 2 =
∂p
∂y , iωρ 0 W = −
c
2 ∂p
∂z ,
∂U 1
∂x +
∂U 2
∂y +
∂W
∂z = Q δðx − x 0 Þδðy − y 0 Þδðz − z 0 Þ, iωρ =
W∂ ρ 0
∂z ,
ð2:3Þ
where c
2 = ω
2
̸ ðN
2
− ω
2
Þ and N
2
ðzÞ = −
g
ρ 0
∂ρ 0
∂z is the Brunt-Väisälä frequency which
is assumed constant: NðzÞ = N = const. These assumptions can be used to study the
IGW fields in many regions of the World Ocean [12]. In the Boussinesq approximation, system (2.3) reduces to a single equation, for example, for pressure perturbations p with the corresponding boundary conditions
Internal Gravity Waves in Horizontally Inhomogeneous Ocean
117
value β = 0.878. Thus, these estimates allow us to conclude that the influence of the
density field inhomogeneities, which is taken into account in the above-described
method for asymptotic representation of the wave fields, is one of the factors
determining the scales of the space attenuation of IGW fields observed in field
measurements.
Fields of IGW in the Ocean of Variable Depth
We consider one of the problems of IGW propagating in the stratified ocean of
variable depth. In the framework of the linear theory, we study the non-viscous
incompressible inhomogeneous medium with unperturbed density ρ 0 ðzÞ, which is
bounded by surface z = 0 and ocean floor z = γ y (the z axis is directed upwards, γ is
the ocean floor slope). At point x = x 0 , y = y 0 , z = z 0 at the slope, there is a point
mass source of power Q depending on time as expð − iω tÞ. The system of hydrodynamic equations for small perturbations of density ρ
* , pressure p
* , and velocity
components ðu 1 , u 2 , wÞ is written as [2–6]
ρ 0
∂u 1
∂t = −
∂p
*
∂x , ρ 0
∂u 2
∂t = −
∂p
*
∂y , ρ 0
∂w
∂t = −
∂p
*
∂z + gρ
*
∂u 1
∂x +
∂u 2
∂y +
∂w
∂z = Q expð − iω tÞδðx − x 0 Þδðy − y 0 Þδðz − z 0 Þ,
∂ρ
*
∂t + w
∂ρ 0
∂z = 0,
ð2:1Þ
where g is the acceleration of gravity. As the boundary conditions we pose the
“rigid lid” condition at the ocean surface and zero mass flux at the ocean bottom
w = 0 at z = 0, w + u 2 γ = 0 at z = − γ y
ð2:2Þ
Under the assumption that the time-dependence of all solutions is harmonic
ðp
* , ρ
* , u 1 , u 2 , wÞ = expð − i ω tÞðp, ρ, U 1 , U 2 , WÞ, we obtain the following system of
equations with boundary conditions (2.2)
iωρ 0 U 1 =
∂p
∂x , iωρ 0 U 2 =
∂p
∂y , iωρ 0 W = −
c
2 ∂p
∂z ,
∂U 1
∂x +
∂U 2
∂y +
∂W
∂z = Q δðx − x 0 Þδðy − y 0 Þδðz − z 0 Þ, iωρ =
W∂ ρ 0
∂z ,
ð2:3Þ
where c
2 = ω
2
̸ ðN
2
− ω
2
Þ and N
2
ðzÞ = −
g
ρ 0
∂ρ 0
∂z is the Brunt-Väisälä frequency which
is assumed constant: NðzÞ = N = const. These assumptions can be used to study the
IGW fields in many regions of the World Ocean [12]. In the Boussinesq approximation, system (2.3) reduces to a single equation, for example, for pressure perturbations p with the corresponding boundary conditions
Internal Gravity Waves in Horizontally Inhomogeneous Ocean
117