The Control of Surface Friction on the Scales of Baroclinic Eddies in a Homogeneous Quasigeostrophic Two-Layer Model

Abstract

In idealized models of the extratropical troposphere, both β and surface friction can control the equilibrated scales of baroclinic eddies by stopping the inverse cascade. A scaling theory on how surface friction alone sets these scales was proposed by Held in 1999 in the case of a quadratic drag law. However, the theory breaks down when friction is modeled by linear damping, and there are other reasons to suspect that it is oversimplified. An ideal system to test the theory is the homogeneous two-layer quasigeostrophic model in the β = 0 limit with quadratic damping. This study investigates some numerical simulations of the model to analyze two causes of the theory’s breakdown. They are 1) the asymmetry between two layers due to confinement of friction to the lower layer and 2) deviation from a spectrally local inverse energy cascade due to the spread of wavenumbers over which energy is input into the barotropic mode. The former is studied by comparing the simulations with drag appearing asymmetrically or symmetrically between the two layers. The latter is addressed with a heuristic modification of the theory. A regime where eddies equilibrate without an inverse cascade is also examined. A comparison is then made between quadratic and linear drag simulations. The connection to a competing theory based on the dynamics of equivalent barotropic vortices with thermal signatures is further discussed. Finally, we present an example of an inhomogeneous statistically steady state to argue that the diffusivity obtained from the homogeneous model has relevance to more realistic configurations.

The vortex gas scaling regime of baroclinic turbulence

“…The resulting scaling theory captures both the exponential dependence of these quantities on the inverse linear drag coefficient and their power law dependence on the quadratic drag coefficient. Our predictions are thus in quantitative agreement with the scaling laws diagnosed empirically by both Thompson and Young (7) and Chang and Held (8). Following Pavan and Held (9) and Chang and Held (8), we finally show how these scaling laws can be used as a quantitative turbulent closure to make analytical predictions in situations where the system is subject to inhomogeneous forcing at large scale.…”

“…The small-scale dissipation coefficient-a hyperviscosity in most studies-is also shown by Thompson and Young (7) to be irrelevant when low enough. The quantities D and thus depend only on the dimensional parameters U , the Rossby deformation radius λ, and the bottom drag coefficient denoted as κ in the case of linear friction (with dimension of an inverse time) and as µ in the case of quadratic drag (with dimension of an inverse length) (8,10). In dimensionless form, we thus seek the dependence of the dimensionless diffusivity D * = D/U λ and mixing length * = /λ on the dimensionless drag κ * = κλ/U or µ * = µλ.…”

A Quantitative Scaling Theory for Meridional Heat Transport in Planetary Atmospheres and Oceans

“…In the following, we thus augment the vortex gas scaling theory to capture the impact of planetary curvature on meridional heat transport, within the canonical 2LQG model on the E  plane. Following Pavan and Held (1996) and Chang and Held (2019), we then show that the resulting expression for the turbulent diffusivity provides a quantitative parameterization of meridional heat transport on a rotating planet. The 6 of 21 new theory further demonstrates that planetary atmospheres can exceed a marginally critical state, but the dependence of the criticality on the radiative forcing is very weak explaining why the baroclinic adjustment assumption gained traction.…”

“…where the PVs are still related to the streamfunctions through (3-4). Inhomogeneous models of this kind were initially introduced by Pavan and Held (1996) (see also Chang & Held, 2019) In Figure 7, we show snapshots of the temperature and barotropic kinetic energy in the equilibrated state of a numerical integration of Equations 20 and 21, for increasing values of E . For nonzero E , we notice again the emergence of zonal jets, together with a reduction in mixing length, noticeable through the smaller , the strength Q of the imposed heat flux and the drag coefficient.…”

Nonlinear generation of long waves and the reversal of eddy momentum fluxes in a two-layer quasi-geostrophic model

“…The halting wavenumber has been found to increase with β and r m in numerical simulations of two-layer QG turbulence (e.g., Thompson and Young (2007)). The best way of thinking about the halting scales in the two-layer context remains a topic of current research (Chang and Held, 2019;Gallet and Ferrari, 2020), but nevertheless the simplest barotropic halting scales appear to be useful here. which sets the width of the radiative-equilibrium jet.…”