The stability of propagating plane inertial waves in rotating fluids

TL;DR

This paper develops a Floquet stability framework for finite-amplitude, plane inertial waves in rotating fluids, valid for arbitrary perturbation wavelengths, and validates it with direct numerical simulations. It shows that the most unstable perturbations are strongly frequency-dependent, with a characteristic maximum growth rate near $\sigma^{\max}\approx 0.3\,A'|\omega|$, and that the perturbation orientation shifts from highly anisotropic to more isotropic as frequency increases. Nonlinearly, the IW breakdown transfers energy either to geostrophic modes or to a forward cascade, with lower-frequency, larger-amplitude waves more efficiently pumping energy into geostrophic structures. Together, these results connect linear instability properties to nonlinear energy partitioning in rotating turbulence and inform the understanding of inertial wave beams in geophysical and astrophysical contexts.

Abstract

Inertial waves transport energy and momentum in rotating fluids and are a major contributor to mixing and tidal dissipation in Earth's oceans, gaseous planets, and stellar interiors. However, their stability and breakdown mechanisms are not fully understood. We examine the linear stability and nonlinear breakdown of finite-amplitude propagating plane inertial waves using Floquet theory and direct numerical simulations. The Floquet analysis generalizes previous studies as it is valid for arbitrary perturbation wavelengths and primary wave amplitudes. We find that the wavenumber orientation of the most unstable perturbations depends strongly on the wave frequency and weakly on the wave amplitude. The most unstable perturbations have wavelengths that are small relative to the primary wave wavelength for low wave amplitudes, but become comparable for large wave amplitudes. We then use direct numerical simulations to follow the nonlinear breakdown of the wave and examine how the wave energy is either dissipated in a forward cascade or accumulated into long-lived geostrophic modes. Simulations reveal that the conversion efficiency into geostrophic modes increases with increasing wave amplitude, as expected for pumping of geostrophic modes by nearly-resonant triadic interactions. We also find that the conversion efficiency increases with decreasing primary wave frequency, which may be due to the more efficient coupling of quasi-2D waves to geostrophic modes. These results on the stability and breakdown of single plane inertial waves provides additional foundation for understanding the role of inertial waves in rotating turbulence, transport properties of inertial wave beams, and inertial wave propagation in more complex environments such as those with magnetic fields or shear flows.

Figures (11)

• Figure 1: The maximum normalized growth rate $\sigma^{\max}/A'|\omega|$ of unstable perturbations predicted by Floquet theory as a function of the perturbation's horizontal wave vector components $\alpha,\beta$ for primary waves with frequencies $\omega/2\Omega=0.17,0.71,0.98$ (a-c), corresponding to angles $\theta=80^\circ,45^\circ, 10^\circ$. The Floquet parameter is fixed to $\gamma=0$ and all primary waves have the same normalized amplitude $A'=0.3$, helicity $s=1$, and $\mathrm{Ek}=10^{-6}$.
• Figure 2: The maximum normalized growth rate of unstable perturbations with Floquet parameters $\gamma=0,1/3,1/4$ (a-c) for the same primary wave with $A'=0.3$, $\omega/2\Omega=0.71$, $s=1$, and $\mathrm{Ek}=10^{-6}$. The panels focus on the origin where the effects of $\gamma$ are most pronounced since $\alpha\sim\beta\sim\gamma$. The $\gamma=0$ case in panel (a) is a zoomed-in region of panel (b) in Figure \ref{['fig:omega_scan']}.
• Figure 3: The maximum normalized growth rate of unstable perturbations for primary waves with amplitudes $A'=0.03,0.3,3.0$ (a-c), all with the same frequency $\omega=\Omega$ (angle $\theta=60^\circ$), helicity $s=1$, and $\mathrm{Ek}=10^{-6}$. The Floquet parameter is $\gamma=0$.
• Figure 4: The maximum normalized growth rate $\sigma^{\max}/A'|\omega|$ versus the primary wave frequency $\omega/2\Omega$ for a range of amplitudes $0.1\leq A'\leq 3.0$. For each $\omega$ and $A'$, the $\sigma^{\max}$ is determined by finding the fastest growth rate among perturbations with wave vector components in the range $-16\leq\alpha\leq16$, $0\leq\beta\leq16$, and $\gamma=0$ using the numerical Floquet solver. The magnitude of $\sigma^{\max}/A'|\omega|$ is nearly independent of $\omega$ and is weakly dependent on $A'$. At low $A'\ll1$, the growth rate based on Floquet theory agrees with the maximum growth rate of the PSI, $\sigma_{\rm PSI}/A'|\omega|=3\sqrt{3}/16$, marked by the dashed, gray line.
• Figure 5: The DNS in the comoving frame is cross-validated against the Floquet theory for a primary wave with amplitude $A'=0.3$ and frequency $\omega/2\Omega=0.71$ (angle $\theta=45^\circ$). Panels (a) and (b) show the Floquet prediction for $\sigma^{\max}$, where the fastest growing perturbation at each $\alpha,\beta$ is found for fixed $\gamma=0$ in panel (a) and for all possible $\gamma$ allowed on the discrete Fourier grid of the DNS in panel (b). Panel (c) shows the $\sigma^{\max}$ measured at each $\alpha,\beta$ in the DNS.