skip to main content


Title: A generalized wave-vortex decomposition for rotating Boussinesq flows with arbitrary stratification
The energetically independent linear wave and geostrophic (vortex) solutions are shown to be a complete basis for velocity and density variables $(u,v,w,\rho )$ in a rotating non-hydrostatic Boussinesq fluid with arbitrary stratification and non-periodic vertical boundaries. This work extends the familiar wave-vortex decomposition for triply periodic domains with constant stratification. As a consequence of the decomposition, the fluid can be unambiguously separated into decoupled linear wave and geostrophic components at each instant in time, without the need for temporal filtering. The fluid can then be diagnosed for temporal changes in wave and geostrophic coefficients at each unique wavenumber and mode, including those that inevitably occur due to nonlinear interactions. We demonstrate that this methodology can be used to determine which physical interactions cause the transfer of energy between modes by projecting the nonlinear equations of motion onto the wave-vortex basis. In the particular example given, we show that an eddy in geostrophic balance superimposed with inertial oscillations at the surface transfers energy from the inertial oscillations to internal gravity wave modes. This approach can be applied more generally to determine which mechanisms are involved in energy transfers between wave and vortices, including their respective scales. Finally, we show that the nonlinear equations of motion expressed in a wave-vortex basis are computationally efficient for certain problems. In cases where stratification profiles vary strongly with depth, this approach may be an attractive alternative to traditional spectral models for rotating Boussinesq flow.  more » « less
Award ID(s):
1658564
NSF-PAR ID:
10215027
Author(s) / Creator(s):
; ;
Date Published:
Journal Name:
Journal of Fluid Mechanics
Volume:
912
ISSN:
0022-1120
Format(s):
Medium: X
Sponsoring Org:
National Science Foundation
More Like this
  1. null (Ed.)
    Abstract Anticyclonic vortices focus and trap near-inertial waves so that near-inertial 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 near-inertial 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 wave-averaged model of near-inertial dynamics and compare its theoretical predictions with numerical solutions of the three-dimensional Boussinesq equations. In the linear approximation, the model predicts the eigenmode frequencies and spatial structures, and a near-inertial wave energy signature that is characterized by an approximately time-periodic, azimuthally invariant pattern. The wave-averaged model represents the nonlinear feedback of the waves on the vortex via a wave-induced 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. 
    more » « less
  2. The YBJ equation (Young & Ben Jelloul, J. Marine Res. , vol. 55, 1997, pp. 735–766) provides a phase-averaged description of the propagation of near-inertial 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 second-order accurate in $\mathit{Bu}$ . (YBJ is first-order 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 power-series 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 time-stepping schemes using a time step determined by stable integration of oscillations with frequency $2f$ . Other phase-averaged equations have dispersion relations with frequency increasing as $k^{2}$ (YBJ) or $k^{4}$ (Thomas et al. 2017): in these cases stable integration with an explicit scheme becomes impractical with increasing horizontal resolution. The YBJ + equation is tested by comparing its numerical solutions with those of the Boussinesq and YBJ equations. In virtually all cases, YBJ + is more accurate than YBJ. The error, however, does not go rapidly to zero as the Burger number characterizing the initial condition is reduced: advection and refraction by geostrophic eddies reduces in the initial length scale of NIWs so that $\mathit{Bu}$ increases with time. This increase, if unchecked, would destroy the approximation. We show, however, that dispersion limits the damage by confining most of the wave energy to low  $\mathit{Bu}$ . In other words, advection and refraction by geostrophic flows does not result in a strong transfer of initially near-inertial energy out of the near-inertial frequency band. 
    more » « less
  3. null (Ed.)
    In the presence of inertia-gravity waves, the geostrophic and hydrostatic balance that characterises the slow dynamics of rapidly rotating, strongly stratified flows holds in a time-averaged sense and applies to the Lagrangian-mean velocity and buoyancy. We give an elementary derivation of this wave-averaged balance and illustrate its accuracy in numerical solutions of the three-dimensional Boussinesq equations, using a simple configuration in which vertically planar near-inertial waves interact with a barotropic anticylonic vortex. We further use the conservation of the wave-averaged potential vorticity to predict the change in the barotropic vortex induced by the waves. 
    more » « less
  4. Abstract

    Weak but persistent synoptic-scale ascent may play a role in the initiation or maintenance of nocturnal convection over the central United States. An analytical model is used to explore the nocturnal low-level jets (NLLJ) and ascent that develop in an idealized diurnally varying frictional (Ekman) boundary layer in a neutrally stratified barotropic environment when the flow aloft is a zonally propagating Rossby wave. Steady-periodic solutions are obtained of the linearized Reynolds-averaged Boussinesq-approximated equations of motion on a beta plane with an eddy viscosity that is specified to increase abruptly at sunrise and decrease abruptly at sunset. Rayleigh damping terms are used to parameterize momentum loss due to radiation of inertia–gravity waves. The model-predicted vertical velocity is (approximately) proportional to the wavenumber and wave amplitude. There are two main modes of ascent in midlatitudes, an afternoon mode and a nocturnal mode. The latter arises as a gentle but persistent surge induced by the decrease of turbulence at sunset, the same mechanism that triggers inertial oscillations in the Blackadar theory of NLLJs. If the Rayleigh damping terms are omitted, the boundary layer depth becomes infinite at three critical latitudes, and the vertical velocity becomes infinite far above the ground at two of those latitudes. With the damping terms retained, the solution is well behaved. Peak daytime ascent in the model occurs progressively later in the afternoon at more southern locations (in the Northern Hemisphere) until the first (most northern) critical latitude is reached; south of that latitude the nocturnal mode is dominant.

     
    more » « less
  5. The data is from a direct numerical simulation of rotating stratified turbulence on a 4096-cubed periodic grid using a pseudo-spectral parallel code, GHOST. The simulations are documented in Ref. 1. The relative strength of stratification vs. rotation is characterized by the ratio of the Brunt-Väisälä to inertial wave frequency, N/f = 4.95. The code solves the Boussinesq equations with a solid body rotation force acting as the only external forcing mechanism. Time integration uses fourth-order Runge-Kutta. The simulation is initialized with large-scale isotropic conditions on a coarser grid. As the simulation progresses resolution is increased, peaking with 4096-cubed at maximum dissipation. After the simulation has reached a statistical stationary state, 5 frames of data, which includes the 3 components of the velocity vector and the temperature fluctuations, are generated and written in files that can be accessed directly by the database (FileDB system). 
    more » « less