Instability and breaking of internal waves in a horizontal shear layer

TL;DR

This study addresses how internal gravity waves interact with a horizontal shear layer and how such interactions lead to wave breaking and turbulence. It combines WKBJ-based ray-tracing with fully nonlinear direct numerical simulations to predict and test local wave states and instability mechanisms. A dimensionless parameter $F$ is introduced to quantify the relative importance of wave advection of momentum versus refraction-driven steepening, linking local energy partition to the tendency for convective versus shear-driven instability through metrics like $Ri_g$ and $s$. The DNS results confirm qualitative agreement with the theory, showing that energy dissipation from wave breaking can far exceed the initial wave energy and that momentum and energy exchanges depend sensitively on the breaking pathway, with distinct mixing efficiencies across regimes. These findings have implications for understanding energy cascades and mixing in stratified geophysical flows where horizontal shear interacts with IGWs.

Abstract

The behaviour of internal waves propagating in a background shear flow is studied in the case where the direction of shear is orthogonal to gravity. Ray-tracing theory is used to predict properties of the wave state at locations where instability occurs. It is found that local wave energy growth may result from two distinct mechanisms: an increase in wave steepness due to refraction by the shear, or an increase in streamwise velocity perturbations due to wave advection of the background flow. Based on the initial conditions, a dimensionless parameter $F$ is constructed to predict the relative importance of these two mechanisms in facilitating wave-breaking according to the local ratio of perturbation kinetic to potential energy. When $F$ is small but wave amplitudes are locally large, energy remains equipartitioned and subsequent instabilities are expected to develop due to a combination of shear and convection. On the other hand, as $F$ increases, kinetic energy dominates and wave advection of momentum may instead cause breaking to become increasingly driven by enhanced vertical shear. To test these predictions, fully nonlinear direct numerical simulations are conducted, spanning a range of wave-breaking dynamics. Good qualitative agreement with the theory is found despite substantial departures from the underlying assumptions. Wave breaking leads to significant turbulent dissipation, which in some cases greatly exceeds the initial wave energy. Momentum and energy transfers between the wave, background flow and turbulence are found to be sensitive to the dynamics of breaking, as are the mixing properties.

Figures (18)

• Figure 1: Schematic showing a representative initial condition for the problem under consideration.
• Figure 2: Plots of the wave steepness $s$ derived from the local wave action $\mathcal{A}$ (dark blue, left axes) and the parameter $F$ defined in equation \ref{['eq:Dparameter']} (dark red, right axes) along rays in the $y$ direction for parameter values corresponding to selected simulations from table \ref{['tab:simulation-results']}. $(a)$ F0.1s2.28(T) and $(b)$ F1.0s5.0(T) (wave trapping), $(c)$ F6.0s0.75_kx0 (no refraction) and $(d)$ F6.0s0.75 (wave reflection). Lines become dashed in panel $c)$ where the perturbation velocity exceeds $\Delta u$ and dotted in panel $(d)$ where $k_y<0.1$. Vertical dashed lines indicate a trapping level in $(a), (b)$ and a turning level in $(d)$. Horizontal dot-dashed lines represent the value of $F$ corresponding to a minimum $Ri_g$ smaller than $1/4$ (assuming $s=0.75)$.
• Figure 3: Snapshots in a vertical plane of the central region $-15<y<15$ taken at successive time points during the flow evolution for simulations $(a)$-$(d)$ F0.1s0.76(T); $(e)$-$(h)$ F0.1s2.28(T); $(i)$-$(l)$ F1.0s5.0(T). Colours illustrate the streamwise vorticity field $\zeta_x$. Triangle markers along the horizontal $y$-axis are located at $y = \pm \Delta u$, indicating the approximate width of the shear layer.
• Figure 4: $(a)$, $(d)$ and $(g)$ show vertical slices of the perturbation streamwise velocity field for simulations F0.1s0.76(T), F0.1s2.28(T) and F1.0s5.0(T) taken at times corresponding to the panels in the top row of figure \ref{['fig:trapped_panels']}. Solid black lines are evenly spaced isopycnals (surfaces of constant buoyancy $b=z+\theta$). Mean vertical profiles of the local buoyancy frequency $N^2 = 1 +\partial \theta/\partial z$ and vertical shear $S$ averaged in the plane indicated by the dashed magenta lines are shown in the panels to the right. Hatched regions show vertical locations where $N^2<0$.
• Figure 5: 3D renderings of the $b = 6.5$ isopycnal from simulation F0.1s2.28(T) at successive times during the flow evolution. Colours indicate the magnitude of the vertical vorticity field $\zeta_z$.