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∂ξ
+
M
+
N
= 0 ,
(5.35)
t
x
y
where M = UD , N = VD are components of the water release along the 0 x and 0 y
axes, respectively. Transition from system (5.29)-(5.31) to system (5.33)-(5.35)
is performed as follows. Equation (5.29) is multiplied by the quantity D , while
equation (5.31), in which the partial derivative
∂ξ
t is replaced by the equiv-
D
alent quantity
t , is multiplied by the velocity component U . Upon adding
up the obtained expressions and performing elementary transformations, we obtain
equation (5.33). Equation (5.34) is derived in a similar manner. Transition from for-
mula (5.31) to (5.35) is trivial and requires no comments.
Note that system (5.29)-(5.31) is not a rigorous consequence of the equations
of hydrodynamics. First of all, this is due to the expression for the force of bottom
friction having been obtained from an empirical dependence. Moreover, the stop-
ping of the flow is due to tangential tension, acting only on the lower boundary.
This circumstance hinders the rigorous derivation of non-linear equations for long
waves. However, linear equations (without advective terms) can be obtained in a rig-
orous manner by integration of the linearized Reynolds equations along the vertical
coordinate from the bottom up to the free water surface.
In calculating tsunami propagation along extended routes account must be taken
of the curvature of the Earth's surface. The form of the surface of our planet can
be considered spherical with a precision sufficient for our problem, therefore, it is
expedient to write the equations of the theory of long waves in spherical coordinates,
U
U
1
R cos
U
∂ψ
ϕ
U
∂ϕ
UV tan
ϕ
+
+ V cos
t
ϕ
R
C B U U 2 + V 2
D
g
R cos
∂ξ
∂ψ
+ fV ,
=
(5.36)
ϕ
U
+ U 2 tan
V
1
R cos
V
∂ψ
ϕ
V
∂ϕ
ϕ
+
+ V cos
t
ϕ
R
C B V U 2 + V 2
D
g
R ∂ξ
=
∂ϕ
fU ,
(5.37)
= 0 ,
∂ξ
1
R cos
( UD )
∂ψ
+
( VD cos
ϕ
)
+
(5.38)
t
ϕ
∂ϕ
where
is the latitude, U and V are the flow velocity compo-
nents, averaged over the depth, along the parallel (west-east) and along the meridian
(north-south), respectively, R
ψ
is the longitude,
ϕ
6,371 km is the mean radius of the Earth.
The set of equations of long-wave theory, written in a Cartesian or spherical
reference system, is usually resolved with initial conditions (initial elevation), rep-
resenting a free-surface displacement, equivalent to vertical residual deformations
of the ocean bottom, resulting from an earthquake. The initial field of flow velocities
is assumed to be zero.
Here, we shall no longer deal with a certain physical incorrectness of the tra-
ditional approach, related to the effects of water compressibility being neglected.
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