Large Internal Solitary Waves in Shallow Waters
103
Here
K = u −
í µí»½
−
3h
(h
3 u x ) x , R = w −
í µí»½
+
3í µí¼
(í µí¼
3 w x ) x ,
í µí¼ = H − h − í µí¼, v =
Q(t) − hu − í µí¼ w
í µí¼
.
(26)
We apply Eqs. (25)–(26) for numerical calculations of nonstationary problems, concerning the propagation of large amplitude internal waves in the frame of models BM, ULM, BLM, SM. In all cases we consider wave generation inside of the
channel (the lock problem etc.), so we apply the “wall condition” at the left and
right boundaries of the calculation domain (Q ≡ 0). The variables h, í µí¼, K, R are
considered as dependent variables, describing the time evolution of the flow, and
the velocities u = u(t, x) and w = w(t, x) can be restored for given h, í µí¼, h x , í µí¼ x , K and
R from the linear system of ODE (26). This numerical scheme have been developed in [21] for open channel flows and applied in [9, 11] for internal wave calculations. The numerical scheme is formally a version of Godunov’s scheme, in
which the fluxes through the lateral boundaries of the meshes are calculated by a
Riemann solver. To find the appropriate time step for SM or USM, BSM, BM models we consider the equilibrium system without dispersion (í µí»½ ± = 0), which coincides with common two or three-layer shallow water equations, correspondingly.
In all calculations presented in the paper the boundary conditions are u = 0, w = 0
at the side walls of the simulating domain. As the initial conditions, we use the
exact solution (10) for SM or the step-wise functions h = h 0 (x), í µí¼ = í µí¼ 0 (x) together
with u = u 0 (x) ≡ 0, w = w 0 (x) ≡ 0, K = K 0 (x) ≡ 0, R = R 0 (x) ≡ 0. The topography
y = z(x) is also included, when the problem on the solitary wave shoaling is
considered.
Decay of Solitary Waves
In real stratified fluids solitary waves do not propagate at a constant velocity. Friction
and entrainment of an ambient fluid result in the wave deceleration. Experiments on
the symmetric solitary wave propagation along the pycnocline together with numerical calculations by SM have demonstrated the ability of the two-layer flow scheme
to simulate the dissipation processes in the flow [7, 9]. It was shown that the main
input in the energy dissipation of solitary waves was given by the interfacial friction
term in (4)
f
−
= −
c i (u − v)|u − v|
h
, ̄
f =
c i (u − v)|u − v|
H 1 − h
(27)
and the bottom friction could be neglected. The friction coefficient c i = 10 −2 was
chosen considerably larger then the usual value of the bottom friction coefficient c b ∼
4 × 10
−3 for turbulent flows, because the mechanisms of the bottom and interlayer
103
Here
K = u −
í µí»½
−
3h
(h
3 u x ) x , R = w −
í µí»½
+
3í µí¼
(í µí¼
3 w x ) x ,
í µí¼ = H − h − í µí¼, v =
Q(t) − hu − í µí¼ w
í µí¼
.
(26)
We apply Eqs. (25)–(26) for numerical calculations of nonstationary problems, concerning the propagation of large amplitude internal waves in the frame of models BM, ULM, BLM, SM. In all cases we consider wave generation inside of the
channel (the lock problem etc.), so we apply the “wall condition” at the left and
right boundaries of the calculation domain (Q ≡ 0). The variables h, í µí¼, K, R are
considered as dependent variables, describing the time evolution of the flow, and
the velocities u = u(t, x) and w = w(t, x) can be restored for given h, í µí¼, h x , í µí¼ x , K and
R from the linear system of ODE (26). This numerical scheme have been developed in [21] for open channel flows and applied in [9, 11] for internal wave calculations. The numerical scheme is formally a version of Godunov’s scheme, in
which the fluxes through the lateral boundaries of the meshes are calculated by a
Riemann solver. To find the appropriate time step for SM or USM, BSM, BM models we consider the equilibrium system without dispersion (í µí»½ ± = 0), which coincides with common two or three-layer shallow water equations, correspondingly.
In all calculations presented in the paper the boundary conditions are u = 0, w = 0
at the side walls of the simulating domain. As the initial conditions, we use the
exact solution (10) for SM or the step-wise functions h = h 0 (x), í µí¼ = í µí¼ 0 (x) together
with u = u 0 (x) ≡ 0, w = w 0 (x) ≡ 0, K = K 0 (x) ≡ 0, R = R 0 (x) ≡ 0. The topography
y = z(x) is also included, when the problem on the solitary wave shoaling is
considered.
Decay of Solitary Waves
In real stratified fluids solitary waves do not propagate at a constant velocity. Friction
and entrainment of an ambient fluid result in the wave deceleration. Experiments on
the symmetric solitary wave propagation along the pycnocline together with numerical calculations by SM have demonstrated the ability of the two-layer flow scheme
to simulate the dissipation processes in the flow [7, 9]. It was shown that the main
input in the energy dissipation of solitary waves was given by the interfacial friction
term in (4)
f
−
= −
c i (u − v)|u − v|
h
, ̄
f =
c i (u − v)|u − v|
H 1 − h
(27)
and the bottom friction could be neglected. The friction coefficient c i = 10 −2 was
chosen considerably larger then the usual value of the bottom friction coefficient c b ∼
4 × 10
−3 for turbulent flows, because the mechanisms of the bottom and interlayer