Anisotropic macroturbulence and diffusion associated with a westward zonal jet: From laboratory to planetary atmospheres and oceans

Turbulence with inverse energy cascade and its transport properties are investigated experimentally in a flow associated with a westward propagating jet. Turbulence and the jet were produced by an electromagnetic force in a rotating tank filled with an electrolytic saline solution. The parabolic free surface emulated the topographic $\beta$ effect which evoked the zonation. The spectral and transport flow characteristics were highly anisotropic. Turbulence is diagnosed by exploring the analogy between vertical and horizontal turbulent overturns in, respectively, stably stratified and quasigeostrophic flows which gives rise to a method of potential vorticity (PV) monotonizing. The anisotropization of transport properties of the flow is investigated using the finite scale Lyapunov exponent technique. After initial exponential particle separation, radial (meridional in geophysical and planetary applications) diffusion attains a short-ranged Richardson regime which transitions to the Taylor (scale-independent diffusivity) one. The azimuthal (zonal) diffusion exhibits a double-plateau structure which attains a superdiffusive regime on large scales. The transition to the Taylor regime for the radial diffusion takes place at a scale of turbulence anisotropization. The radial eddy diffusivity in both regimes as well as the transition scale are all determined by the rate of the inverse energy cascade, $\epsilon$, that can be diagnosed by the PV monotonizing. Conversely, $\epsilon$ can be deduced from the scale of the Richardson-Taylor regime transition in the radial eddy diffusivity which, thus, provides an additional tool of diagnosing anisotropic macroturbulence with inverse energy cascade.

Figure 1

Schematic representation of the experimental device. The magnets are placed in a ${90}^{\circ }$ arc of the radius $a=17$ cm (sector I).

Figure 2

Instantaneous [panels (a)–(c); experiments 29–31] and time-averaged [panels (d)–(f)] velocities and stream functions. The vertical line delineates the forced sector I located in the right halves of the semicircles.

Figure 3

Radial profiles of the instantaneous radial gradients of the potential vorticity, $\partial q/\partial r$, along the radial section at $\theta ={237}^{\circ }$ for $t=72$ s (left panel) and 95 s (right panel) for experiment 31. The five lines in each panel provide temporal representations. The dark line is the current instant, while the “older” lines (by only a frame or two) are shown in progressively lighter hue. This is done to ease the tracking of the movement of $\partial q/\partial r$.

Figure 4

Experimental zonal ($E_Z$, thin line) and residual ($E_R$, thick line) spectra and their rms fittings to Eqs. (5) and (6). The index $n$ denotes the $n\mathrm{th}$ zero of the Bessel function with an index $m, J_m\left({\alpha }_{mn}\right)=0$, as explained after Eq. (2). Left, middle, and right panels pertain to the experiments 29, 30, and 31, respectively.

Figure 5

Longitudinal structure functions for experiments 29 (lower line), 30 (middle line), and 31 (upper line). Normalized structure functions, $D_{LLL}/\epsilon \ell$, are shown in the inset with the order of the lines reversed and the horizontal line corresponding to $D_{LLL}/\epsilon \ell =0.5$; $\epsilon$ is obtained from Table 1.

Figure 6

Vertical diffusion of a tracer in the Antarctic Circumpolar Current. The ordinate is height relative to the center of the initial profile. The abscissa is the concentration in femtomolars (fM; $1\text{fM}=1\times {10}^{-15}$ moles/liter). Dashed curve: initial mean concentration measured shortly after release, multiplied by 0.005. Solid curve: final mean profile measured 1 yr later. The gray uncertainty envelope is based on variations of the shape of individual profiles. Dash-dotted curve shows final profile for an integration of the one-dimensional diffusion equation, starting with the initial profile, for a diapycnal diffusivity of $1.28\times {10}^{-5}{\mathrm{m}}^2{\mathrm{s}}^{-1}$. After Ledwell et al. [81]. ©American Meteorological Society. Used with permission.

Figure 7

Meridional spread of the debris following the collision of the Shoemaker-Levy 9 comet with Jupiter. The crosses represent longitude-mean perturbation optical depths at 230 nm derived from the changes in reflectivity after 18 May 1994: (a) 25 August 1994, (b) 4–5 March 1995, (c) 15 May 1996, and (d) 6 November 1997. Optical depths were integrated over longitude in each of the latitude bands between $-{60}^{\circ }$ and $+{30}^{\circ }$. Regions of negative mean optical depth are found where the average brightness of the bin was higher than it was prior to the arriving of impact debris. The solid lines are synthetic profiles calculated with the dynamical model that includes horizontal eddy diffusivity. Dash-dotted and dash-triple-dotted lines correspond to the results with the eddy diffusivity coefficient, respectively, reduced or increased by a factor of 5 compared to its nominal value. After Friedson et al. [27]; reprinted with the permission from Elsevier.

Figure 8

FSLE maps (top), samples of instantaneous (middle), and fully time- and sector-averaged (bottom row) profiles of the radial PV gradient. The left, middle, and right columns pertain to the experiments 29, 30, and 31, respectively. The Roman numbers mark different sectors; sector I contains the magnets. While the flow in this sector is barotropically unstable most of the time, the occurrence of the instability decreases away from the sector. The color bar numbers show powers of 10.

Figure 9

Radial (black squares) and zonal (open circles) FSLE computed with $\rho =2^{1/4}$. Straight lines represent the Richardson, $\lambda \left(\delta \right)\sim {\delta }^{-2/3}$ (solid), and Taylor, $\lambda \left(\delta \right)\sim {\delta }^{-2}$ (dotted), diffusion laws. Left, middle, and right panels pertain to the experiments 29, 30, and 31, respectively.