Large-scale anisotropy in stably stratified rotating flows

We present results from direct numerical simulations of the Boussinesq equations in the presence of rotation and/or stratification, both in the vertical direction. The runs are forced isotropically and randomly at small scales and have spatial resolutions of up to ${1024}^3$ grid points and Reynolds numbers of $\approx 1000$. We first show that solutions with negative energy flux and inverse cascades develop in rotating turbulence, whether or not stratification is present. However, the purely stratified case is characterized instead by an early-time, highly anisotropic transfer to large scales with almost zero net isotropic energy flux. This is consistent with previous studies that observed the development of vertically sheared horizontal winds, although only at substantially later times. However, and unlike previous works, when sufficient scale separation is allowed between the forcing scale and the domain size, the kinetic energy displays a perpendicular (horizontal) spectrum with power-law behavior compatible with $\sim k_{\perp }^{-5/3}$, including in the absence of rotation. In this latter purely stratified case, such a spectrum is the result of a direct cascade of the energy contained in the large-scale horizontal wind, as is evidenced by a strong positive flux of energy in the parallel direction at all scales including the largest resolved scales.

Figure 1

Contour plots of the logarithm of the axisymmetric kinetic energy spectrum $e(k_{\perp },k_{\parallel })$ for a rotating and stratified flow with $N/f=2$ (above, run I) and for a purely stratified flow with $\text{Fr}=0.04$ (below, run II), averaged in time over $20<t/{\tau }_{NL}<30$. Note that the spherical shell of externally forced modes is clearly visible at $k\in [40,41]$. The two insets show details of the axisymmetric spectra at wave numbers smaller than or equal to the forcing wave numbers.

Figure 2

Isotropic kinetic energy spectrum $E_V\left(k\right)$ at $t=27{\tau }_{NL}$ in simulations with the same Reynolds number, with $n_p=512$ linear grid points and forced at $k_F=[22,23]$ (dashed) and with $n_p=1024$ and $k_F=[40,41]$ (solid). Two cases are shown: a purely stratified case with $\text{Fr}=0.04$ (light or red curves) and a rotating stratified case with $N/f=2$ (dark or blue curves). A Kolmogorov slope is indicated as a reference.

Figure 3

Evolution of the kinetic energy in the isotropic shell in Fourier space with $k=1$ as a function of time in log-log scale for run I with $n_p=1024$ (see Table 1). After an initial transient, the kinetic energy grows steadily with time, following a power law that is well approximated by $E_V(k=1,t)\sim t^{3.7}$. Accumulation of most of the energy in the $k=1$ shell does not take place in this simulation until much later times.

Figure 4

Perpendicular kinetic energy spectra $E_V\left(k_{\perp }\right)$ at different times for a simulation with $N/f=2$ [run III, panel (a)] and for a purely stratified flow [run IV, panel (b)], both with $\text{Fr}=0.04$. A Kolmogorov slope and a flat line are indicated as references. The insets in panels (a) and (b) show the time evolution of the kinetic energy and of the (normalized) kinetic enstrophy in the same runs (solid and dashed lines respectively). Note the saturation of the kinetic enstrophy (i.e., of the power at the small scales) in both cases and of kinetic energy only in the purely stratified case. Panel (c) shows a time average over $\approx 100{\tau }_{NL}$ of the the perpendicular kinetic energy spectrum for the purely stratified run of panel (b) (run IV).

Figure 5

Perpendicular kinetic energy spectra $E_V\left(k_{\perp }\right)$ at $t=55{\tau }_{NL}$ for two purely stratified flows (runs II and IV) using grids with $n_p=512$ (solid) and $n_p=1024$ (dashed) points, with the same Froude and Reynolds numbers. A Kolmogorov slope is given as a reference. The inset shows details of the spectra for small wave numbers at later times for several runs with $\text{Re}\approx 1000$ (runs IV to VII) without rotation and with different Froude numbers. The labels in the top right box correspond to the curves in this inset.

Figure 6

(a) Isotropic, (b) perpendicular, and (c) parallel total energy fluxes normalized by the energy input ${\epsilon }_V=\langle \mathbf{u}·\mathbf{F}\rangle$ in a run with $N/f=2$ (solid black line), a run with pure rotation (dash-dotted red line), and a run with pure stratification (dashed blue line).

Figure 7

Schematic representation of the total energy flux in Fourier space, defined in Eqs. (18, 19, 20) for (a) rotating (or rotating and stratified) flows and (b) the purely stratified case. The gray circle indicates the shell of forced modes, and the dashed lines indicate planes across which the fluxes ${\Pi }_T\left(k_{\perp }\right)$ (represented by horizontal arrows) and ${\Pi }_T\left(k_{\parallel }\right)$ (represented by vertical arrows) are computed. The diagonal arrow represents the flux in terms of isotropic wave number, ${\Pi }_T\left(k\right)$ which, in case (b), is negligible with two projections (on the $k_{\perp }$ and $k_{\parallel }$ directions) that are almost equal but of opposite sign. Note, however, that the length of the arrows is arbitrary and does not denote the strength of each flux.