Investigation of Boussinesq dynamics using intermediate models based on wave–vortical interactions

Abstract

Nonlinear coupling among wave modes and vortical modes is investigated with the following question in mind: can we distinguish the wave-vortical interactions largely responsible for formation versus evolution of coherent, balanced structures? The two main case studies use initial conditions that project only onto the vortical-mode flow component of the rotating Boussinesq equations: (i) an initially balanced dipole and (ii) random initial data in the vortical modes. Both case studies compare quasi-geostrophic (QG) dynamics (involving only nonlinear interactions between vortical modes) to the dynamics of intermediate models allowing for two-way feedback between wave modes and vortical modes. For an initially balanced dipole with symmetry across thex-axis, the QG dipole will propagate along thex-axis while the trajectory of the Boussinesq dipole exhibits a cyclonic drift. Compared to a forced linear (FL) model with one-way forcing of wave modes by the vortical modes, the simplest intermediate model with two-way feedback involving vortical-vortical-wave interactions is able to capture the speed and trajectory of the dipole for roughly ten times longer at Rossby Ro and Froude Fr numbers Ro = Fr ≈ 0.1. Despite its success at tracking the dipole, the latter intermediate model does not accurately capture the details of the flow structure within the adjusted dipole. For decay from random initial conditions in the vortical modes, the full Boussinesq equations generate vortices that are smaller than QG vortices, indicating that wave-vortical interactions are fundamental for creating the correct balanced state. The intermediate model with QG and vortical-vortical-wave interactions actually prevents the formation of vortices. Taken together these case studies suggest that: vortical-vortical-wave interactions create waves and thereby influence the evolution of balanced structures; vortical-wave-wave interactions take energy out of the wave modes and contribute in an essential way to the formation of coherent balanced structures.

Quantifying resonant and near-resonant interactions in rotating turbulence

“…In a similar way, Moffatt (2014) analysed the effect of triad truncation on the velocity and vorticity field of the Euler equation. Finally, decimation models in which wave-wave-wave interactions and wave-vortex-wave interactions were differentiated have been used to study rotating stratified turbulence (Remmel et al 2010;Hernandez-Duenas et al 2014).…”

Turbulent exchanges between near-inertial waves and balanced flows

“…Similarly, we expand the balanced field in Fourier modes as In the above expansions we used cosine and sine series in the vertical direction since our domain consists of impenetrable rigid lids on top and bottom ends. In triply periodic domains one would use the full Fourier series in , as can be seen in several previous works such as Bartello (1995), Embid & Majda (1998), Smith & Waleffe (2002), Majda (2002), Deusebio, Vallgren & Lindborg (2013), Hernandez-Duenas, Smith & Stechmann (2014), Herbert et al (2016) and Waite (2017) for example.Substituting (A 1) in (2.5), (A 2) in (2.6) and simplifying gives us where and are the amplitudes of wave and balanced flow for each wavenumber. It is easy to see in the above expressions that the wave and balance vectors above are orthogonal, i.e.…”

A generalized wave-vortex decomposition for rotating Boussinesq flows with arbitrary stratification

“…This can be done diagnostically (as in § 6) or while directly integrating the equations of motion. By selectively turning off transfer mechanisms, one can also derive reduced-interaction models, such as in Hernandez-Dueñas et al (2014), but now also for vertically bounded flows with variable stratification.…”

“…domains (Remmel 2010;Remmel, Smith & Sukhatme 2010;Hernandez-Dueñas, Smith & Stechmann 2014). These models are reduced versions of the equations of motions that restrict interactions between certain modes.…”