Two-layer baroclinic turbulence with arbitrary layer depths

While heat transport by baroclinic turbulence in oceans and planetary atmospheres is well described by a two-layer model, the relative depth of the two layers varies greatly depending on the situation of interest, making it an important parameter governing the transport properties of the system. Focusing on the low-drag turbulent regime, we extend the vortex-gas scaling theory to address the case of arbitrary layer depths. To wit, we map the arbitrary-layer-depth system onto an equivalent equal-depth system with rescaled parameters, establishing the asymptotic validity of the mapping for weak bottom drag. This approach leads to quantitative predictions for the turbulent transport by two-layer baroclinic turbulence with arbitrary layer depths, without additional free parameters. We validate these predictions using an extended suite of numerical simulations with either linear or quadratic bottom drag.

Figure 1

Two layers of fluid with densities ${\rho }_1$ and ${\rho }_2$ sit on top of one another, in a frame rotating at a rate $f/2$ around the vertical axis. The base state consists of a vertically sheared zonal flow. The interface between the layers is tilted as a consequence of thermal wind balance.

Figure 2

Dimensionless diffusivity as a function of the dimensionless drag coefficient for four different values of the relative layer depths, using either linear drag (open symbols) or quadratic drag (filled symbols).

Figure 3

Rescaled diffusivity as a function of the rescaled drag coefficient for linear drag (open symbols) and quadratic drag (filled symbols). Same symbols as in Fig. 2. The low-drag data collapse onto the vortex-gas predictions Eqs. (9) and (10), plotted using $c_1=1.7128, c_2=0.7644$, and $c_3=0.3436$.

Figure 4

Diffusivity coefficient normalized by its equal-depth value for a fixed low value of the drag coefficient acting on the barotropic mode (linear drag, $K=5\times {10}^{-2}$; quadratic drag, $M=5\times {10}^{-4}$). The lines are the parameter-free predictions Eqs. (16) and (17).