^{1,a)}, Marek Stastna

^{1}and Michael L. Waite

^{1}

### Abstract

It is well-known that in certain parameter regimes, the steady flow of a density stratified fluid over topography can yield large amplitude internal waves. We discuss an embedded boundary method to solve the Dubreil-Jacotin-Long (DJL) equation for steady-state, supercritical flows over topography in an inviscid, stratified fluid. The DJL equation is equivalent to the full set of stratified steady Euler equations and thus the waves we compute are exact nonlinear solutions. The numerical method presented yields far better scaling with increase in grid size than other iterative methods that have been used to solve this equation, and this in turn allows for a more thorough exploration of parameter space. For waves under the Boussinesq approximation, we contrast the properties of trapped waves over hill-like and valley-like topography, finding that the symmetry of freely propagating solitary waves when the stratification is reflected across the middepth is not present for trapped waves. We extend the derivation of the DJL equation to the non-Boussinesq case and discuss the effect of the new, non-Boussinesq terms on the structure of the trapped waves, finding that the sharp transition between large and small amplitude waves observed under the Boussinesq approximation is much more gradual when the Boussinesq approximation is relaxed. Finally, we demonstrate the existence of asymmetric steady states over hill-like topography where the flow is subcritical upstream of the topography but transitions to supercritical somewhere over the hill. Waves in this new class of exact solutions are related to so-called downstream recovery jumps predicted on the basis of hydrostatic (shallow water) theories, but when breaking does not occur the recovery jump does not stop propagating downstream and an asymmetric state across the topography maximum is reached for long times.

This research was supported by the Natural Sciences and Engineering Research Council of Canada.

I. INTRODUCTION

II. METHODS

A. Equations

1. Boussinesq

2. Non-Boussinesq

B. Numerical methods

C. Description of experiments

III. RESULTS

A. Elevation vs. depression topography

B. Non-Boussinesq effects

C. Subcritical flows

IV. DISCUSSION

### Key Topics

- Topography
- 96.0
- Internal waves
- 30.0
- Fluid equations
- 26.0
- Stratified flows
- 22.0
- Boundary value problems
- 15.0

## Figures

A single density contour is drawn over the topography h(x). 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.

A single density contour is drawn over the topography h(x). 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.

A comparison of computation times from the embedded and mapped methods for several grid sizes (N grid ) and waves. Results for a small wave with U 0 = 1.25c j (solid line) and a large wave with U 0 = 1.11c j (dashed line). The embedded method is indicated with circles (o) and the mapped method with stars (*). The results are normalized by longest computation time. N grid = m × n where m and n are the number of grid points in x and z, respectively.

A comparison of computation times from the embedded and mapped methods for several grid sizes (N grid ) and waves. Results for a small wave with U 0 = 1.25c j (solid line) and a large wave with U 0 = 1.11c j (dashed line). The embedded method is indicated with circles (o) and the mapped method with stars (*). The results are normalized by longest computation time. N grid = m × n where m and n are the number of grid points in x and z, respectively.

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 z 0 = 0.75H and on the right (b) and (d), it is centered at z 0 = 0.25H. In these cases, U 0 = 1.06c j . The thick black line represents the bottom topography.

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 z 0 = 0.75H and on the right (b) and (d), it is centered at z 0 = 0.25H. In these cases, U 0 = 1.06c j . The thick black line represents the bottom topography.

Diagram of wave properties as a function of the background velocity U 0. (a) Maximum value of |η| (scaled by 1/H) for waves over hole topography with z 0 = 0.6H and hill topography with z 0 = 0.4H. (b) A measure of the wave width for the corresponding cases in panel (a). The wave width (scaled by 1/H) 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 c j = 1.02c lw .

Diagram of wave properties as a function of the background velocity U 0. (a) Maximum value of |η| (scaled by 1/H) for waves over hole topography with z 0 = 0.6H and hill topography with z 0 = 0.4H. (b) A measure of the wave width for the corresponding cases in panel (a). The wave width (scaled by 1/H) 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 c j = 1.02c lw .

(a) Extreme value of |η| (scaled by 1/H) with varying background speeds U 0 for comparison between non-Boussinesq (solid) and Boussinesq (dashed) results with Δρ = 0.05, z 0 = 0.75H and depression topography. (b) A measurement of wave width (scaled by 1/H) is also included. The background speed is scaled by the conjugate flow speed, c j , as calculated for the non-Boussinesq case. The wave width is measured as in Fig. 4 .

(a) Extreme value of |η| (scaled by 1/H) with varying background speeds U 0 for comparison between non-Boussinesq (solid) and Boussinesq (dashed) results with Δρ = 0.05, z 0 = 0.75H and depression topography. (b) A measurement of wave width (scaled by 1/H) is also included. The background speed is scaled by the conjugate flow speed, c j , as calculated for the non-Boussinesq case. The wave width is measured as in Fig. 4 .

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

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

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

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

Density fields after 5 (10) h on the left (right) column, where U 0 = [0.96, 0.91, 0.85, 0.80]c j from top to bottom.

Density fields after 5 (10) h on the left (right) column, where U 0 = [0.96, 0.91, 0.85, 0.80]c j from top to bottom.

Horizontal velocity fields with three white isopycnals superimposed for the double hill case. The background speed is U 0 = 0.96c j . The results are presented at non-dimensional times (a) t* = 745, (b) t* = 993.3, (c) t* = 1987, and (d) t* = 2483. The scaling time is taken to be t d = H/c j .

Horizontal velocity fields with three white isopycnals superimposed for the double hill case. The background speed is U 0 = 0.96c j . The results are presented at non-dimensional times (a) t* = 745, (b) t* = 993.3, (c) t* = 1987, and (d) t* = 2483. The scaling time is taken to be t d = H/c j .

(a) An example of an asymmetric solution of the DJL equation, with U 0 = 0.96c j . (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/H) at x/H = 20 (*) and at x/H = 0 (o) are shown.

(a) An example of an asymmetric solution of the DJL equation, with U 0 = 0.96c j . (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/H) at x/H = 20 (*) and at x/H = 0 (o) are shown.

Comparison of wave amplitude (left) and wave width (right) for two density stratifications with z 0 = 0.75H; Δρ = 0.05 in (a) and (b) and Δρ = 0.2 in (c) and (d). Three cases are shown: non-Boussinesq solutions with N NB (z) (solid), for the Boussinesq solutions with N NB (z) (dashed), and for the Boussinesq solutions with N B (z). The velocity is scaled by the non-Boussinesq conjugate flow speed in each case.

Comparison of wave amplitude (left) and wave width (right) for two density stratifications with z 0 = 0.75H; Δρ = 0.05 in (a) and (b) and Δρ = 0.2 in (c) and (d). Three cases are shown: non-Boussinesq solutions with N NB (z) (solid), for the Boussinesq solutions with N NB (z) (dashed), and for the Boussinesq solutions with N B (z). The velocity is scaled by the non-Boussinesq conjugate flow speed in each case.

## Tables

Local conjugate flow speeds using water column depth above the center of the topography H loc . Upstream of the topography the conjugate flow speed is c j = 1.17c lw .

Local conjugate flow speeds using water column depth above the center of the topography H loc . Upstream of the topography the conjugate flow speed is c j = 1.17c lw .

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