^{1}, Andrew B. G. Bush

^{1}and Bruce R. Sutherland

^{1,a)}

### Abstract

We examine the transmission of small and moderately large internal gravity wavepackets through uniformly and nonuniformly stratified fluids using fully nonlinear numerical simulations. The simulations of finite-amplitude waves in a uniformly stratified fluid show that the weakly nonlinear theory developed for horizontally periodic wavepackets extends well to the dynamics of wavepackets with horizontal extent comparable to the horizontal wavelength. In simulations of small-amplitude wavepackets in a nonuniformly stratified fluid the transmission coefficient is found to be comparable to that computed analytically for horizontally periodic waves that radiate continuously (nontransiently) on a reflection level. Simulations of finite-amplitude waves in a nonuniformly stratified fluid show little dependence of transmission coefficient on the wavepacket extent. However, for a wide range of incident wave frequencies the simulations exhibit a monotonic increase in the transmission coefficient as a function of the incident wave amplitude.

This work has been supported by the Natural Sciences and Engineering Research Council of Canada (NSERC), the Canadian Foundation for Climate and Atmospheric Science (CFCAS GR-615), and the Alberta Ingenuity Studentship program.

I. INTRODUCTION

II. THEORETICAL PRELIMINARIES

A. Small-amplitude wave transmission

B. Weakly nonlinear effects

III. NUMERICAL METHOD

A. Fully nonlinear simulation

B. Wave-induced mean-flow analysis

C. Transmission analysis

IV. RESULTS

A. Propagation in a uniform stratification

B. Tunnelling: Wavepacket evolution

C. Transmission: Amplitude effects

D. Transmission: Wavepacket extent effects

E. Transmission: Barrier extent effects

V. DISCUSSION AND CONCLUSIONS

### Key Topics

- Transmission coefficient
- 24.0
- Internal waves
- 20.0
- Flow instabilities
- 11.0
- Tunneling
- 9.0
- Wave interaction
- 7.0

## Figures

The time evolution of the wave-induced mean flow, given by Eq. (8), for fully nonlinear numerical simulations of a horizontally compact wavepacket in uniform background stratification. The vertical coordinate is set to translate at the vertical group speed: . Wavepackets are initialized with , and , , and in the left, center, and right panels, respectively. The initial amplitudes are prescribed by (a) and (b) .

The time evolution of the wave-induced mean flow, given by Eq. (8), for fully nonlinear numerical simulations of a horizontally compact wavepacket in uniform background stratification. The vertical coordinate is set to translate at the vertical group speed: . Wavepackets are initialized with , and , , and in the left, center, and right panels, respectively. The initial amplitudes are prescribed by (a) and (b) .

Results of fully nonlinear numerical simulations of a wavepacket given initially by Eq. (10) impinging on an -barrier of width , as depicted by the gray center at . Wavepackets are initialized with , , and (a) and (b) . Grayscale contours are of the normalized vertical displacement field, , at , , and in the left, center, and right panels, respectively. The background wind moves with speed so that small-amplitude wavepackets should remain approximately centered in the horizontal.

Results of fully nonlinear numerical simulations of a wavepacket given initially by Eq. (10) impinging on an -barrier of width , as depicted by the gray center at . Wavepackets are initialized with , , and (a) and (b) . Grayscale contours are of the normalized vertical displacement field, , at , , and in the left, center, and right panels, respectively. The background wind moves with speed so that small-amplitude wavepackets should remain approximately centered in the horizontal.

Horizontal slices of the normalized vertical displacement field, , at , for several simulations of a wavepacket impinging on an -barrier of width . Wavepackets are prescribed by Eq. (10) with . Profile (a) is taken at height for a wavepacket initialized with , , and . Profile (b) is the same as (a) except . Profile (c) is from a downward-propagating wavepacket at height with , , and .

Horizontal slices of the normalized vertical displacement field, , at , for several simulations of a wavepacket impinging on an -barrier of width . Wavepackets are prescribed by Eq. (10) with . Profile (a) is taken at height for a wavepacket initialized with , , and . Profile (b) is the same as (a) except . Profile (c) is from a downward-propagating wavepacket at height with , , and .

The wave-induced mean flow given by Eq. (8) at various times for fully nonlinear numerical simulations. The small- and large-amplitude simulations used in (a) and (b), respectively, are the same as those used in Fig. 2. The dashed lines in the right panels represent an idealized if the transmitted and reflected wavepackets retained Gaussian shapes.

The wave-induced mean flow given by Eq. (8) at various times for fully nonlinear numerical simulations. The small- and large-amplitude simulations used in (a) and (b), respectively, are the same as those used in Fig. 2. The dashed lines in the right panels represent an idealized if the transmitted and reflected wavepackets retained Gaussian shapes.

The wave-induced mean flow given by Eq. (8) at late times for three simulations with varying vertical wavenumbers. Wavepackets are initialized with (a) and , (b) and , and (c) and . In all simulations and . Profiles are taken at , , and in (a), (b), and (c), respectively. The dashed lines represent an idealized where the transmitted and reflected wavepackets are Gaussian.

The wave-induced mean flow given by Eq. (8) at late times for three simulations with varying vertical wavenumbers. Wavepackets are initialized with (a) and , (b) and , and (c) and . In all simulations and . Profiles are taken at , , and in (a), (b), and (c), respectively. The dashed lines represent an idealized where the transmitted and reflected wavepackets are Gaussian.

Wavepacket transmission values plotted against initial amplitude for horizontally localized wavepackets prescribed initially by Eq. (10) impinging on an -barrier of width . The computed transmission values are normalized by the linearly predicted transmission values for horizontal plane waves, , given by Eq. (2). Wavepackets are initialized with and (solid line), (dotted line), (dashed-dotted line), and (dashed line).

Wavepacket transmission values plotted against initial amplitude for horizontally localized wavepackets prescribed initially by Eq. (10) impinging on an -barrier of width . The computed transmission values are normalized by the linearly predicted transmission values for horizontal plane waves, , given by Eq. (2). Wavepackets are initialized with and (solid line), (dotted line), (dashed-dotted line), and (dashed line).

As in Fig. 6 but plotting relative transmission against the relative vertical wavenumber for horizontally localized wavepackets with four different initial vertical displacement amplitudes, as indicated.

As in Fig. 6 but plotting relative transmission against the relative vertical wavenumber for horizontally localized wavepackets with four different initial vertical displacement amplitudes, as indicated.

Wavepacket transmission values through an -barrier of width plotted against (a) the horizontal and (b) the vertical extent of the wavepackets for values of the vertical wavenumber as indicated. Computed transmission values are normalized by the linearly predicted transmission values for horizontal plane waves, , given by Eq. (2). Wavepackets are initially prescribed by Eq. (10) with (or ) and in (a) while in (b) .

Wavepacket transmission values through an -barrier of width plotted against (a) the horizontal and (b) the vertical extent of the wavepackets for values of the vertical wavenumber as indicated. Computed transmission values are normalized by the linearly predicted transmission values for horizontal plane waves, , given by Eq. (2). Wavepackets are initially prescribed by Eq. (10) with (or ) and in (a) while in (b) .

Localized wavepacket transmission values plotted against -barrier depths for various initial amplitudes. Wavepackets are initially prescribed by Eq. (10) with and . The dashed line represents the linearly predicted transmission values for horizontal plane waves, , given by Eq. (2).

Localized wavepacket transmission values plotted against -barrier depths for various initial amplitudes. Wavepackets are initially prescribed by Eq. (10) with and . The dashed line represents the linearly predicted transmission values for horizontal plane waves, , given by Eq. (2).

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