 Award ID(s):
 1658564
 Publication Date:
 NSFPAR ID:
 10215027
 Journal Name:
 Journal of Fluid Mechanics
 Volume:
 912
 ISSN:
 00221120
 Sponsoring Org:
 National Science Foundation
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Abstract Anticyclonic vortices focus and trap nearinertial waves so that nearinertial energy levels are elevated within the vortex core. Some aspects of this process, including the nonlinear modification of the vortex by the wave, are explained by the existence of trapped nearinertial eigenmodes. These vortex eigenmodes are easily excited by an initialwave with horizontal scale much larger than that of the vortex radius. We study this process using a waveaveraged model of nearinertial dynamics and compare its theoretical predictions with numerical solutions of the threedimensional Boussinesq equations. In the linear approximation, the model predicts the eigenmode frequencies and spatial structures, and a nearinertial wave energy signature that is characterized by an approximately timeperiodic, azimuthally invariant pattern. The waveaveraged model represents the nonlinear feedback of the waves on the vortex via a waveinduced contribution to the potential vorticity that is proportional to the Laplacian of the kinetic energy density of the waves. When this is taken into account, the modal frequency is predicted to increase linearly with the energy of the initial excitation. Both linear and nonlinear predictions agree convincingly with the Boussinesq results.

The YBJ equation (Young & Ben Jelloul, J. Marine Res. , vol. 55, 1997, pp. 735–766) provides a phaseaveraged description of the propagation of nearinertial waves (NIWs) through a geostrophic flow. YBJ is obtained via an asymptotic expansion based on the limit $\mathit{Bu}\rightarrow 0$ , where $\mathit{Bu}$ is the Burger number of the NIWs. Here we develop an improved version, the YBJ + equation. In common with an earlier improvement proposed by Thomas, Smith & Bühler ( J. Fluid Mech. , vol. 817, 2017, pp. 406–438), YBJ + has a dispersion relation that is secondorder accurate in $\mathit{Bu}$ . (YBJ is firstorder accurate.) Thus both improvements have the same formal justification. But the dispersion relation of YBJ + is a Padé approximant to the exact dispersion relation and with $\mathit{Bu}$ of order unity this is significantly more accurate than the powerseries approximation of Thomas et al. (2017). Moreover, in the limit of high horizontal wavenumber $k\rightarrow \infty$ , the wave frequency of YBJ + asymptotes to twice the inertial frequency $2f$ . This enables solution of YBJ + with explicit timestepping schemes using a time step determined by stable integration of oscillations with frequency $2f$ . Other phaseaveraged equations have dispersion relations with frequency increasing as $k^{2}$more »

In the presence of inertiagravity waves, the geostrophic and hydrostatic balance that characterises the slow dynamics of rapidly rotating, strongly stratified flows holds in a timeaveraged sense and applies to the Lagrangianmean velocity and buoyancy. We give an elementary derivation of this waveaveraged balance and illustrate its accuracy in numerical solutions of the threedimensional Boussinesq equations, using a simple configuration in which vertically planar nearinertial waves interact with a barotropic anticylonic vortex. We further use the conservation of the waveaveraged potential vorticity to predict the change in the barotropic vortex induced by the waves.

Abstract We formulate the twodimensional gravitycapillary water wave equations in a spatially quasiperiodic setting and present a numerical study of solutions of the initial value problem. We propose a Fourier pseudospectral discretization of the equations of motion in which onedimensional quasiperiodic functions are represented by twodimensional periodic functions on a torus. We adopt a conformal mapping formulation and employ a quasiperiodic version of the Hilbert transform to determine the normal velocity of the free surface. Two methods of timestepping the initial value problem are proposed, an explicit Runge–Kutta (ERK) method and an exponential timedifferencing (ETD) scheme. The ETD approach makes use of the smallscale decomposition to eliminate stiffness due to surface tension. We perform a convergence study to compare the accuracy and efficiency of the methods on a traveling wave test problem. We also present an example of a periodic wave profile containing vertical tangent lines that is set in motion with a quasiperiodic velocity potential. As time evolves, each wave peak evolves differently, and only some of them overturn. Beyond water waves, we argue that spatial quasiperiodicity is a natural setting to study the dynamics of linear and nonlinear waves, offering a third option to the usual modeling assumptionmore »

Stationary longitudinal vortical rolls emerge in katabatic and anabatic Prandtl slope flows at shallow slopes as a result of an instability when the imposed surface buoyancy flux relative to the background stratification is sufficiently large. Here, we identify the selfpairing of these longitudinal rolls as a unique flow structure. The topology of the counterrotating vortex pair bears a striking resemblance to speakerwires and their interaction with each other is a precursor to further destabilization and breakdown of the flow field into smaller structures. On its own, a speakerwire vortex retains its unique topology without any vortex reconnection or breakup. For a fixed slope angle $\alpha =3^{\circ }$ and at a constant Prandtl number, we analyse the saturated state of speakerwire vortices and perform a biglobal linear stability analysis based on their stationary state. We establish the existence of both fundamental and subharmonic secondary instabilities depending on the circulation and transverse wavelength of the base state of speakerwire vortices. The dominance of subharmonic modes relative to the fundamental mode helps to explain the relative stability of a single vortex pair compared to the vortex dynamics in the presence of two or an even number of pairs. These instability modes are essentialmore »