2DYBJ-QG

A pseudospectral solver for the coupled YBJ-QG system in 2D

The Coupled YBJ-QG System

The coupled YBJ-QG system describes the evolution of a NIW field in the presence of a QG eddy field [1] as well as the back-reaction of the waves on the evolution of the QG field [2] [3]. The 2D version of this system, where we make a vertical plane wave assumption for the NIW field, was formulated in [4]. The model equations are:

$$\begin{align} \partial_t\phi + J(\psi,\phi)+\frac{i\zeta}{2}\phi-\frac{i\eta}{2}\nabla^2\phi &= -\nu_\phi\nabla^4\phi + F,\\ \partial_tq +J(\psi,q) &= -\nu_q\nabla^4q,\\ q &= \nabla^2\psi + \frac{1}{f_0}\left[\frac{1}{4}\nabla^2|\phi|^2 + \frac{i}{2}J(\phi^*,\phi)\right], \end{align}$$

where $\phi$ is the back-rotated, complexified NIW velocity, $\psi$ is the QG streamfunction, $\zeta=\nabla^2\psi$ is the QG vorticity, $\eta=N_0^2/f_0m^2$ is the dispersivity (with $N_0$ being a reference stratification and $m$ being the vertical wavenumber of the NIWs), $f_0$ is the reference Coriolis parameter, $\nu_\phi$ is the wave hyperviscosity, $F$ is a forcing term, $q$ is the QGPV and $\nu_q$ is the QG hyperviscosity.

We use dedalus to solve these equations on a periodic domain.

The test file contains code to run the two examples from [4].

Animation of Lamb-Chaplygin Dipole

rocha_anim.mp4

(cf. [4] Figure 1.)

References

[1] Young, W. R., Ben Jelloul, M. (1997). Propagation of near-inertial oscillations through a geostrophic flow Journal of Marine Research, 55(4), 735-766.

[2] Xie, J-H., and Vanneste, J. (2015). A generalised-Lagrangian-mean model of the interactions between near-inertial waves and mean flow Journal of Fluid Mechanics, 774, 143-169.

[3] Wagner, G. L., and W. R. Young. (2016). A three-component model for the coupled evolution of near-inertial waves, quasi-geostrophic flow and the near-inertial second harmonic Journal of Fluid Mechanics, 802, 806-837.

[4] Rocha, C., Wagner, G., & Young, W. (2018). Stimulated generation: Extraction of energy from balanced flow by near-inertial waves Journal of Fluid Mechanics, 847, 417-451.