Effect of Prandtl’s ratio on balance in geophysical turbulence

Abstract

The fluid dynamics of the atmosphere and oceans is to a large extent controlled by the slow evolution of a scalar field called ‘potential vorticity’ (PV), with relatively fast motions such as inertia-gravity waves playing only a minor role. This state of affairs is commonly referred to as ‘balance’. Potential vorticity is a special scalar field which is materially conserved in the absence of diabatic effects and dissipation, effects that are generally weak in the atmosphere and oceans. Moreover, in a balanced flow, PV induces the entire fluid motion and its thermodynamic structure (Hoskins et al., Q. J. R. Meteorol. Soc., vol. 111, 1985, pp. 877–946). While exact balance is generally not achievable, it is now well established that balance holds to a high degree of accuracy in rapidly rotating and strongly stratified flows. Such flows are characterised by both a small Rossby number, $\mathit{Ro}\equiv |{\it\zeta}|_{max}/f$, and a small Froude number, $\mathit{Fr}\equiv |{\bf\omega}_{h}|_{max}/N$, where ${\it\zeta}$ and ${\bf\omega}_{h}$ are the relative vertical and horizontal vorticity components, while $f$ and $N$ are the Coriolis and buoyancy frequencies. In fact, balance can even be a good approximation when $\mathit{Fr}\lesssim \mathit{Ro}\sim \mathit{O}(1)$. In this study, we examine how balance depends specifically on Prandtl’s ratio, $f/N$, in unforced freely evolving turbulence. We examine a wide variety of turbulent flows, at a mature and complex stage of their evolution, making use of the fully non-hydrostatic equations under the Boussinesq and incompressible approximations. We perform numerical simulations at exceptionally high resolution in order to carefully assess the degree to which balance holds, and to determine when it breaks down. For this purpose, it proves most useful to employ an invariant PV-based Rossby number ${\it\varepsilon}$, together with $f/N$. For a given ${\it\varepsilon}$, our key finding is that – for at least tens of characteristic vortex rotation periods – the flow is insensitive to $f/N$ for all values for which the flow remains statically stable (typically $f/N\lesssim 1$). Only the vertical velocity varies in proportion to $f/N$, in line with quasi-geostrophic (QG) scaling for which $\mathit{Fr}^{2}\ll \mathit{Ro}\ll 1$. We also find that as ${\it\varepsilon}$ increases towards unity, the maximum $f/N$ attainable decreases towards 0. No statically stable flows occur for ${\it\varepsilon}\gtrsim 1$. For all stable flows, balance is found to hold to a remarkably high degree: as measured by an energy norm, imbalance never exceeds more than a few per cent of the balance, even in flows where $\mathit{Ro}>1$. The vertical velocity $w$ remains a tiny fraction of the horizontal velocity $\boldsymbol{u}_{h}$, even when $w$ is dominantly balanced. Finally, typical vertical to horizontal scale ratios $H/L$ remain close to $f/N$, as found previously in QG turbulence for which $\mathit{Fr}\sim \mathit{Ro}\ll 1$.