3.13 Exercise 7: Lee Waves
55
case studies consider variations of the barotropic forcing with the background sea
level varying between 0.1 and 1.5 cm over the length of the model domain.
Density diffusion is included in both scenarios with horizontal and vertical density diffusivities being set to small uniform values of K h = K z = 1 × 10
−4 m
2
/s.
The total simulation time is 60 min with data outputs at one-minute intervals. The
pressure accuracy for the S.O.R. iteration is set to = 1 × 10
−3 Pa. The time step
is set to Δt = 1 s, using the rigid-lid approximation. Note that neither momentum
diffusion nor boundary friction is yet included in the momentum equations.
3.13.2 Results: Continuous Density Stratification
For a weak initial stratification (N
2
= 0.5 × 10
−4 s
−2 ), the flow over the sill triggers
a standing internal lee wave of an enormous wave height of 40 m on a wavelength of
approximately 150 m (Fig. 3.28a). Near-bottom water is lifted across the sill where
it becomes subject to vigorous mixing via the breaking of internal waves. This instability mechanism is responsible for the localised generation of internal waves in the
ocean.
The situation is different for a stronger initial stratification (N
2
= 5 × 10
−4 s
−2 )
(Fig. 3.28b). In this case, the sill operates as a barrier for the flow with the consequence that flow below sill depth is almost absent. Interaction between flow at mid
depth and bathymetry creates lee waves of a reduced height of 10 m. This demonstrates that, for strong density stratification, bathymetric obstacles can operate as a
barrier for flows.
According to Eq. (3.55), the phase speed of internal waves in a fluid of continuous
density stratification depends on the wavelength of the disturbance. In this situation,
a hydraulic jump flow will always generate lee waves of a wavelength corresponding
Fig. 3.28 Exercise 7.
Scenario 1. Density
distribution (shading) and
flow field (arrows) after
40 min of simulation.
Panel (a) shows results
for an initial value of
N
2 = 0.5 × 10
−4 s
−2 , panel
(b) for N
2 = 5 × 10
−4 s
−2 .
Flow vectors are averaged
over 5 by 5 grid cells
55
case studies consider variations of the barotropic forcing with the background sea
level varying between 0.1 and 1.5 cm over the length of the model domain.
Density diffusion is included in both scenarios with horizontal and vertical density diffusivities being set to small uniform values of K h = K z = 1 × 10
−4 m
2
/s.
The total simulation time is 60 min with data outputs at one-minute intervals. The
pressure accuracy for the S.O.R. iteration is set to = 1 × 10
−3 Pa. The time step
is set to Δt = 1 s, using the rigid-lid approximation. Note that neither momentum
diffusion nor boundary friction is yet included in the momentum equations.
3.13.2 Results: Continuous Density Stratification
For a weak initial stratification (N
2
= 0.5 × 10
−4 s
−2 ), the flow over the sill triggers
a standing internal lee wave of an enormous wave height of 40 m on a wavelength of
approximately 150 m (Fig. 3.28a). Near-bottom water is lifted across the sill where
it becomes subject to vigorous mixing via the breaking of internal waves. This instability mechanism is responsible for the localised generation of internal waves in the
ocean.
The situation is different for a stronger initial stratification (N
2
= 5 × 10
−4 s
−2 )
(Fig. 3.28b). In this case, the sill operates as a barrier for the flow with the consequence that flow below sill depth is almost absent. Interaction between flow at mid
depth and bathymetry creates lee waves of a reduced height of 10 m. This demonstrates that, for strong density stratification, bathymetric obstacles can operate as a
barrier for flows.
According to Eq. (3.55), the phase speed of internal waves in a fluid of continuous
density stratification depends on the wavelength of the disturbance. In this situation,
a hydraulic jump flow will always generate lee waves of a wavelength corresponding
Fig. 3.28 Exercise 7.
Scenario 1. Density
distribution (shading) and
flow field (arrows) after
40 min of simulation.
Panel (a) shows results
for an initial value of
N
2 = 0.5 × 10
−4 s
−2 , panel
(b) for N
2 = 5 × 10
−4 s
−2 .
Flow vectors are averaged
over 5 by 5 grid cells