Propagation and overturning of three-dimensional Boussinesq wave packets with rotation

The stability and overturning of fully three-dimensional internal gravity wave packets is examined for a rotating, uniformly stratified Boussinesq fluid that is stationary in the absence of waves. We derive through perturbation theory an integral expression for the mean flow induced by upward-propagating fully localized wave packets subject to Coriolis forces. This induced Bretherton flow manifests as a dipolelike recirculation about the wave packet in the horizontal plane. We perform numerical simulations of fully localized wave packets with the predicted Bretherton flow superimposed, for a range of initial amplitudes, wave-packet aspect ratios, and relative vertical wave numbers spanning the hydrostatic and nonhydrostatic regimes. Results are compared with predictions based on linear theory of wave breaking due to overturning, convection, self-acceleration, and shear instability. We find that nonhydrostatic wave packets tend to destabilize due to self-acceleration, eventually overturning although the initial amplitude is well below the overturning amplitude predicted by linear theory. Strongly hydrostatic waves, propagating almost entirely in the horizontal, are found not to attain amplitudes sufficient to become shear unstable, overturning instead due to localized steepening of isopycnals. Results are discussed in the broader context of previous studies of one- and two-dimensional wave packets overturning and recent observations of oceanic internal waves.

Figure 1

Slices of the initial nondimensionalized Bretherton flow, ${\tilde{u}}_{\text{BF}}=u_{\text{BF}}(x,y,z_0,0)/(N_0/k)$ through $z_0=0$, given by Eq. (11), for round (a), long (b), and wide (c) Gaussian wave packets with rotation set by $f_0=0.05N_0$. In each case, ${\sigma }_z=10k^{-1}$. The wave packets shown were initialized with a peak vertical displacement amplitude of $A_0=0.08k^{-1}$ and a relative vertical wave number of $m/k=-11.4$. The black contours are isolines of nondimensionalized streamfunction, ${\tilde{\psi }}_{\text{BF}}={\psi }_{\text{BF}}(x,y,z_0,0)/(N_0/k^2)$. The heavy and light white ellipses show the first and second standard deviations, respectively, of the Gaussian wave packet.

Figure 2

Curves of marginal stability emphasizing waves in the (a) nonhydrostatic and (b) hydrostatic wave-number ranges. The respective $\mathrm{\Theta }$ axes correspond to waves spanning the frequency ranges $N_0\ge \omega \ge f_0$ and $8.77\gtrsim \omega /f_0\ge 1$. The dotted and solid black curves in (a) and (b) indicate the amplitudes above which a plane internal gravity wave is predicted to be, respectively, statically and convectively unstable, given by Eqs. (14) and (15), respectively. The blue curves in (a) and (b) indicate the amplitudes above which a wave packet is predicted to be unstable owing to self-acceleration, and are given by Eq. (18) for $R_y={\sigma }_y/{\sigma }_x=1$ (solid), $R_y=1/4$ (dashed) and $R_y=4$ (dotted). The solid and dashed red curves in (b) indicate the amplitudes above which a wave is predicted to be shear unstable using respectively our definition of the gradient Richardson number Eq. (19) and that of Fritts and Rastogi [19] and Achatz [20].

Figure 3

Cross sections of the streamwise velocity field, $\tilde{u}=u/(N_0/k)$, through the horizontal plane $z=c_{gz}t$ (top row) and of the vertical displacement field, $\tilde{\xi }=k\xi$, through the vertical plane $y=0$ (bottom row) of a round, moderately large amplitude nonhydrostatic internal gravity wave packet (NH1), initialized with $A_0=0.5k^{-1}$ and $m=-0.2k$. Images have been cropped to focus on the region of the domain containing the wave packet. The black curves in the bottom row represent isopycnal surfaces. The isopycnals overturned at $t=49/N_0$.

Figure 4

As in Fig. 3 but for a round strongly hydrostatic internal gravity wave packet (H1), initialized with $A_0=0.08k^{-1}$ and $m=-11.4k$.

Figure 5

Time series of $min\left\{{\tilde{N}}_T^2\right\}$ (black curves), and $min\left\{{\text{Ri}}_g\right\}-1/4$ (red curves), in which ${\tilde{N}}_T^2$ and ${\text{Ri}}_g$ are given respectively by Eqs. (28) and (19), for (a) strongly nonhydrostatic wave packets with $m=-0.2k$ ($\mathrm{\Theta }={11}^{\circ }$) with initial amplitudes $A_0=0.5k^{-1}$ (NH1; solid curve), $A_0=0.35k^{-1}$ (NH2; dashed curve), and $A_0=0.2k^{-1}$ (NH3; heavy dotted curve), and (b) strongly hydrostatic wave packets with $m=-11.4k$ ($\mathrm{\Theta }={85}^{\circ }$) and $A_0=0.08k^{-1}$ with $f_0=0.01N_0$ (H1; solid curves) and $f_0=0.05N_0$ (${\mathrm{H1}}^{+}$; heavy dotted curves). The dashed red curve is a time series of $min\left\{{\text{Ri}}_g\right\}-1/4$ from H1 but using ${\text{Ri}}_g=N_T^2/{\parallel ({\partial }_{z}u,{\partial }_{z}v)\parallel }^2$ as in Refs. [19, 20].

Figure 6

Comparison of stability regimes with the results of simulations of round wave packets using initial vertical displacement amplitudes $A_0/{\lambda }_x$ and $\mathrm{\Theta }={tan}^{-1}\left(\right|m/k\left|\right)$ with rotation set by $f_0=0.01N_0$. Each cross corresponds to the initial amplitude and wave number of a single simulation. Greyscale values represent overturning times, $t_{\text{OT}}$, interpolated using cubic polynomials [28]. White curves are isolines of constant overturning time corresponding to $N_0{t}_{\text{OT}}=20$, 40, 60, and 80.