Unidirectional Modes Induced by Nontraditional Coriolis Force in Stratified Fluids

Using topology, we unveil the existence of new unidirectional modes in compressible rotating stratified fluids. We relate their emergence to the breaking of time-reversal symmetry by rotation and vertical mirror symmetry by stratification and gravity. We stress the role of the Coriolis force’s nontraditional part, induced by a rotation field tangent to the surface. In contrast with horizontally trapped equatorial waves induced by the traditional component of the Coriolis force perpendicular to the surface, we find vertically trapped modes that propagate along interfaces between regions with distinct stratification properties. We show that such modes are generalized atmospheric Lamb waves whose direction of propagation can be selected by the nontraditional component of the Coriolis force.

Figure 1

Geometry of the planar problem on a rotating planet. The traditional component $f_t$ cancels at the equator, while the nontraditional term $f_{\mathrm{nt}}$ is maximal. Modes are trapped in the direction $z$ where $S=+f_{\mathrm{nt}}$ (or $S=-f_{\mathrm{nt}}$) and propagate along the zonal direction $x$.

Figure 2

Degeneracy points (in black) between the positive-frequency bands ($n=+1$ and $+2$) of the spectrum of Eq. (3), with $f_{\mathrm{nt}}>0$. The pairs of integers indicate the topological charges $({\mathcal{C}}_1,{\mathcal{C}}_2)$ of the two bands involved in a given degeneracy point. The insets on the left show their dispersion relations associated with the green, red, and blue planes in parameter space. We also show the degeneracy points (gray) between the negative-frequency bands (not shown on the left), with $({\mathcal{C}}_{-2},{\mathcal{C}}_{-1})$.

Figure 3

Numerical solutions of the eigenvalue problem [Eq. (1)] projected on the Fourier basis—such that $({\partial }_x,{\partial }_t)\to i(k_x,-\omega )$—with constant sound speed $c_s$, obtained with Dedalus [25]. (a) Spectrum for $f_{\mathrm{nt}}=0.1$: $S$ does not take the value $f_{\mathrm{nt}}$; no mode transits through the gap. $N_m=\min \{N(z)\}$. (b) $f_{\mathrm{nt}}=2$: $S$ takes twice the value of $f_{\mathrm{nt}}$; a mode is localized at each interface $S=f_{\mathrm{nt}}$ and transits through the gap around $k_x^{\star }$. The pressure perturbations’s amplitude $|\tilde{p}(z)|$ of the topological modes at $k_x=k_0$ are plotted with the background stratification $S$ in green. (c) Plot of the velocity field of the topological modes with wave number $k_0$, with pressure levels in background (positive values in solid lines and negative in dashed lines).