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Trapped internal waves over topography: Non-Boussinesq effects, symmetry breaking and downstream recovery jumps
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Image of FIG. 1.
FIG. 1.

A single density contour is drawn over the topography (). The isopycnal displacement η is defined as the distance the density contour is displaced from its far upstream value. For waves of depression, η is negative and for waves of elevation it is positive.

Image of FIG. 2.
FIG. 2.

A comparison of computation times from the embedded and mapped methods for several grid sizes ( ) and waves. Results for a small wave with = 1.25 (solid line) and a large wave with = 1.11 (dashed line). The embedded method is indicated with circles (o) and the mapped method with stars (*). The results are normalized by longest computation time. = × where and are the number of grid points in and , respectively.

Image of FIG. 3.
FIG. 3.

Plots of the density field for trapped depression waves (top) and elevation waves (bottom). On the left (a) and (c), the pycnocline is centered at = 0.75 and on the right (b) and (d), it is centered at = 0.25. In these cases, = 1.06 . The thick black line represents the bottom topography.

Image of FIG. 4.
FIG. 4.

Diagram of wave properties as a function of the background velocity . (a) Maximum value of |η| (scaled by 1/) for waves over hole topography with = 0.6 and hill topography with = 0.4. (b) A measure of the wave width for the corresponding cases in panel (a). The wave width (scaled by 1/) is measured as twice the distance between the wave center and location where the wave induced surface velocities reach half of their extreme value. In both cases the conjugate flow speed is = 1.02 .

Image of FIG. 5.
FIG. 5.

(a) Extreme value of |η| (scaled by 1/) with varying background speeds for comparison between non-Boussinesq (solid) and Boussinesq (dashed) results with Δρ = 0.05, = 0.75 and depression topography. (b) A measurement of wave width (scaled by 1/) is also included. The background speed is scaled by the conjugate flow speed, , as calculated for the non-Boussinesq case. The wave width is measured as in Fig. 4 .

Image of FIG. 6.
FIG. 6.

An example of the spatial structure of waves from the non-Boussinesq (solid) and Boussinesq (dashed) solutions in Fig. 5 for = 1.23 . (a) Normalized wave-induced velocities at the surface. (b) Wave-induced velocities through = 0 scaled by the conjugate flow speed.

Image of FIG. 7.
FIG. 7.

Comparison of non-Boussinesq (solid) and Boussinesq (dashed) maximum |η| (scaled by 1/) for several background speeds and stratifications. (a) Δρ = 0.05, (b) Δρ = 0.1, (c) Δρ = 0.15, and (d) Δρ = 0.2.

Image of FIG. 8.
FIG. 8.

Density fields after 5 (10) h on the left (right) column, where = [0.96, 0.91, 0.85, 0.80] from top to bottom.

Image of FIG. 9.
FIG. 9.

Horizontal velocity fields with three white isopycnals superimposed for the double hill case. The background speed is = 0.96 . The results are presented at non-dimensional times (a) * = 745, (b) * = 993.3, (c) * = 1987, and (d) * = 2483. The scaling time is taken to be = / .

Image of FIG. 10.
FIG. 10.

(a) An example of an asymmetric solution of the DJL equation, with = 0.96 . (b) Wave characteristics for asymmetric DJL solutions over several background speeds. The absolute value of the displacement of the ρ = 1 contour (scaled by 1/) at / = 20 (*) and at / = 0 (o) are shown.

Image of FIG. 11.
FIG. 11.

Comparison of wave amplitude (left) and wave width (right) for two density stratifications with = 0.75; Δρ = 0.05 in (a) and (b) and Δρ = 0.2 in (c) and (d). Three cases are shown: non-Boussinesq solutions with () (solid), for the Boussinesq solutions with () (dashed), and for the Boussinesq solutions with (). The velocity is scaled by the non-Boussinesq conjugate flow speed in each case.


Generic image for table
Table I.

Local conjugate flow speeds using water column depth above the center of the topography . Upstream of the topography the conjugate flow speed is = 1.17 .


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752b84549af89a08dbdd7fdb8b9568b5 journal.articlezxybnytfddd
Scitation: Trapped internal waves over topography: Non-Boussinesq effects, symmetry breaking and downstream recovery jumps