Geoscience Reference
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The damping of long gravitational waves in a viscous liquid is known to be due to
energy dissipation within a thin bottom layer. The wave amplitude, here, decreases
exponentially with time,
A
exp
{− γ
t
}
.
The following formula has been obtained in the topic [Landau, Lifshits (1987)]
for the time decrement of amplitude damping for a wave propagating in a basin of
constant depth H :
= νω
8 H 2 1 / 2
γ
,
where
is the cyclic
frequency of the wave. If a long wave propagates only in the direction of axis 0 x ,
then its amplitude will also decrease exponentially with the distance,
ν
is the molecular kinematic viscosity of the liquid and
ω
exp
,
x
L 1
A
where
L 1 = g H
γ
= 8g H 3
νω
1 / 2
.
(5.18)
The physical meaning of quantity L 1 is the distance, over which the amplitude of
a long wave in a viscous liquid becomes e times smaller. We shall call this quantity
the viscous (linear) dissipation length.
Note that formula (5.18) is correct for any constant viscosity coefficient, which,
generally speaking, can be both molecular and turbulent. One must, however, bear
in mind that in the bottom boundary layer the turbulent viscosity, as a rule, depends
strongly on the vertical component, i.e. is not a constant value. Therefore, it would
not be quite correct to substitute into formula (5.18) any values of the turbulent
viscosity coefficient. On the other hand, in a real ocean exchange of momentum
does not proceed via molecular mechanisms, but by turbulence. Indeed, in spite
of the re lativ ely low flow velocity, characteristic of tsunami waves in open ocean,
u
A g / H
10 2 m / s, The Reynolds numbers turn out to be sufficiently large
for the development of turbulence.
Determination of the quantity L 1 from formula (5.18) actually gives an idea of
the minimum possible level of tsunami wave energy losses. Actually, owing to turbu-
lence these losses may turn out to be more significant. We shall estimate the damping
of tsunami waves on the basis of the known parameterization of frictional tension
exerted by the ocean bottom on the water flowing along it with a velocity v,
ρ
|
|
,
T B =
C B
v
v
(5.19)
where
is the density of water, C B is a dimensionless empirical coefficient, the value
of which is usually set to 0.0025 [Murty (1984)]. The minus sign in formula (5.19)
indicates that the water flow is hindered by a force directed against the flow velocity
vector. The absolute value of the force of friction is proportional to the square flow
ρ
 
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