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Beta-plane turbulence: experiments with altimetry

Results from a new series of experiments on barotropic and baroclinic turbulent flows generated in a rotating tank with topographic beta-effect are presented. The velocity fields are measured by the Altimetric Imaging Velocimetry. The turbulent flows observed in the experiments develop zonal jets which are latent in a stationary forced-dissipative regime of the flow but become prominent in the decaying flow. The two-dimensional energy spectra of the flows exhibit the development of anisotropy towards zonal motions. The experiments demonstrate dual turbulent cascade with energy and enstrophy ranges. The frequency-wavenumber spectra reveal the presence of Rossby waves at low wavenumbers which are excited by the turbulent motions. The experimental results are compared with available theory of beta-plane turbulence.

simulations by Fontane et al. 4 performed at high Reynolds number showed that the theoretical -5/3 scaling law can be steepened to -2 when long-lived vortex cores develop in the flow. An extension beyond two-dimensionality is provided by rotating shallow water (RSW) theory (e.g. Zeitlin 5 ) which takes into account vortex stretching, the effect of the finite radius of deformation and inertia-gravity waves. Recent laboratory experiments with RSW turbulence on the f-plane by Afanasyev and Craig 6 showed similarity to two-dimensional turbulence with the spectral slopes of approximately -5/3 and (somewhat steeper than) -3. These experimental results are in agreement with results of numerical simulations by Yuan and Hamilton 7 . An effect of asymmetry between anticyclonic and cyclonic vortices with the anticyclonic vortices prevailing was observed in the experiments in Ref. (6); this is one of the effects that make RSW turbulence different from its two-dimensional counterpart. In order to eliminate the -effect, a cylindrical tank with a paraboloidal bottom of the same shape as that of the rotating water surface was employed in Ref. (6). Note that in the present work we use a similar apparatus but with an inverted paraboloidal bottom to increase -effect rather than to eliminate it.
While two-dimensional and RSW turbulence are isotropic in the horizontal plane, the turbulence with the -effect is not. Even when the forcing is isotropic, an anisotropy develops in the form of zonal jets. This phenomenon is of great interest for applications to atmospheric circulations on gas giants (see e.g. review by Dowling 8 ) as well to recently discovered jets in the Earth's oceans (Maximenko et al. 9 , Maximenko et al. 10 and Sokolov and Rintoul 11 ). Note that while on Jupiter and Saturn jets are prominent features of the circulation, the oceanic jets are often "lost" in a much stronger eddy field and can be revealed only as a result of time averaging of the observed fields. In the spectral representation, the anisotropy of the flows manifests itself in the concentration of energy at the k x = 0 axis in the wavenumber plane. Theory predicts that the zonal spectrum of the flow defined as E Z = E(k x = 0, k y ) has a slope of -5 (Rhines 12 , Sukoriansky et al. 13 ). Here E(k x , k y ) is a two-dimensional energy spectrum which will be properly defined later in this article.
Baroclinic instability is a major underlying process which generates eddies in atmospheric and oceanic large-scale turbulent flows. At a mature stage of its development baroclinic instability becomes equilibrated such that the flow becomes barotropic to a significant degree. The barotropic component of the flow is coupled to the baroclinic one (e.g. Cushman-Roisin and Beckers 14 ). Barotropization process allows one to use barotropic models for the investigation of turbulent flows. Observations of -plane turbulence in numerical simulations of different authors suggested that barotropic flows tend to generate somewhat weaker jets than those in baroclinic flows. It seems that an energy source capable of sustaining persistently high energy level in the flow is an important factor of strong jet formation (e.g. Rhines 15 , Panetta 16 ).
In baroclinic flows (even when the dynamics is mainly barotropic) the energy source is provided by eddies slowly releasing their available potential energy.
Turbulent flows on the -plane have been a subject of study in laboratory experiments by different authors. Zonal jets were observed in the experiment by Whitehead 17 where the flows were induced locally by a vertically oscillating disk. A formation of a zonal flow was also observed by Colin de Verdiere 18 who generated the flow by a periodic excitation of sources and sinks. Recent experiments by Afanasyev et al. 19 and Slavin and Afanasyev 20 showed that a pattern of zonal jets can be formed westward of the area where perturbations are located. The perturbations were created by buoyancy sources. It was shown that the jets formed via a -plume mechanism. A -plume is an almost zonal circulation consisting of two zonal jets flowing in opposite directions. The circulation is formed via radiation of Rossby waves by a localized perturbation (see e.g. Afanasyev et al. 19 , Stommel 21 , and Davey and Killworth 22 ). In the experiments in Ref. (19) eddies eventually filled a significant area of the domain. However, the jets were still prominent in this turbulent flow. While it is clear how a single localized perturbation can create jets to the west of its location, it is much less clear how eddies in a turbulent flow "cooperate" to sustain jets. Rather than using a forcing applied in a limited area of the domain, one can use a uniformly distributed forcing to create an eddy field in the entire domain. Afanasyev and Wells 23 used an electromagnetic (EM) forcing to generate a turbulent flow on the polar -plane. In EM experiments the Lorentz force is created by using a combination of permanent magnets located at the bottom of the tank and the horizontal electric current flowing through the layer of conducting fluid. Experiments in Ref. (23) showed the formation of zonal flows in the forced-dissipative -plane turbulence, but they were somewhat limited by the size of the apparatus. Further EM experiments in a setup similar to that used in Ref. (23) were performed recently by Espa et al. 24 . A different kind of distributed forcing was used by Read et al. 25 in their experiments on the Coriolis platform. The authors used convective forcing where the surface of the water in the tank was sprayed with saline water. More dense saline water formed small-scale convective plumes descending to the bottom thus creating smallscale turbulence. A system of zonal jets was observed in the experiments in Ref. (25). The slope of the turbulent energy spectrum was measured to be close to -5/3 in the energy range as predicted by theory. Since the forcing was at very small scale, the enstrophy range was not resolved in those experiments.
In what follows, we describe laboratory experiments on a rotating table with the (topographic) -effect. Two experiments were performed; in the first experiment the forcing was barotropic and generated by the EM method while in the second experiment the forcing was baroclinic and generated by heating the fluid at the bottom of the tank. The purpose of these experiments was to investigate the properties of the flows including their spectral characteristics both in wavenumber space and in frequency-wavenumber space. We used a relatively large tank and an intermediate/small-scale forcing in order to observe the dual cascade including the energy and enstrophy intervals. Our optical altimetry system allowed us to measure fields with high spatial and temporal resolution which was necessary for spectral analyses. The dynamics of Rossby waves in the forced-dissipative -plane turbulence is elucidated. Our experimental results show how idealized theory or numerical simulations hold in application to real flows.
In Sec. II of this paper we describe the setup of our apparatus as well as the optical altimetry technique used to measure the gradient of the surface elevation field, from which we obtain the velocity and vorticity fields. In Sec. III the results of the experiments and their analyses are reported. Concluding remarks are given in Sec. IV.

II. LABORATORY APPARATUS AND TECHNIQUE
The experiments were performed in a cylindrical tank of radius R t = 55 cm installed on a rotating table (Fig. 1). The tank was filled with water and rotated in an anticlockwise direction at a rate of  = 2.32 rad s −1 .
In the first experiment with an electromagnetic (EM) forcing an inner container in the form of a cylinder with a paraboloidal bottom was inserted into the tank and was concentric with The height h of the water surface under a solid-body rotation varies quadratically with the distance r from the axis of rotation where H 0 is the depth of the layer in the absence of rotation and g is the gravitational The Altimetric Imaging Velocimetry (AIV) system was used to observe perturbations of the surface and to measure two components of the gradient   = ( / , / ) xy      of the surface elevation  in the horizontal plane (x, y). The AIV technique was described in detail in Ref. 28. The   field was measured with a spatial resolution of approximately 2 vectors per millimeter such that a total size of the array was 1900  1900 and with a temporal resolution of 5 fields per second. The surface velocity of the flow can be determined from the measured gradient of surface elevation using shallow water and quasigeostrophic approximations which yields where U is the horizontal velocity vector, n is the vertical unit vector and

III. EXPERIMENTAL RESULTS AND ANALYSES
We performed two experiments with different forcing which will be referred as barotropic Finally the bottom row of Fig. 2 shows the decaying flow at the very end of the experiment when vortices no longer dominate but zonal currents still present. Note an interesting spiral pattern of jets in this flow. Most likely, the spiral manifests the Rossby waves with spiral crests which occur above the isolated underwater mountains (Rhines 29,30 ). In our case, the entire water layer with shallow water in the center and deep water at the wall of the tank, can be considered as that above a mountain. We will illustrate this effect using linear Rossby wave theory in Section C.  (Fig. 3 c) shows a spiral pattern similar to that in Bt experiment.
However, the relative vorticity map (Fig. 3 d) reveals that the eddies are much more "alive" in this experiment compared to the barotropic one. Note also that a cyclonic circulation develops in the tank which causes the asymmetry between the strength of the eastward and westward jets.
This circulation is due to a gradual accumulation of warm water at the periphery of the tank. The circulation then occurs as suggested by the thermal wind balance with a radial gradient of density.
where bottom friction is parameterized by a linear term with linear drag coefficient  and  is the kinematic viscosity of fluid. The coefficient  can then be determined as Alternatively, a theoretical estimate  = () 1/2 /H can be obtained from the bottom Ekman layer theory. The coefficient is approximately equal to 0.05 s -1 in the Bt experiment. The coefficient was measured immediately after the forcing stops using the energy and enstrophy data shown in Fig. 4 (a, c). Note that the contribution of "bulk" viscosity constitutes only 11% of the total rate of change of energy such that most of the energy is dissipated at the bottom. Ekman dissipation is not uniform across the domain; the system loses energy faster in the shallow area in the center than in the deeper layer at the periphery. Theory gives  = 0.06 s -1 in the center and  = 0.01 s -1 at the wall of the tank in the Bt experiment.
The presence of an energy source (baroclinic eddies) in the flow during the decay period in the Bc experiment makes it impossible to measure  using mean energy and enstrophy data. In this experiment we have to rely on the theoretical estimate only. It gives more uniform distribution with radius than that in the Bt experiment with varying between 0.03 s -1 in the center and 0.01 s -1 at the wall. A bulk value (averaged over the area) is  = 0.02 s -1 .
An important control parameter in a forced-dissipative turbulence is the rate, , at which the forcing supplies energy to the system. In a stationary flow this energy is subsequently removed from the system by dissipation at the same rate. Assuming that immediately after the forcing stops the energy dissipation rate remains the same as it was during the forcing, , can be estimated as The value of  in the Bt experiment is 0.1 cm 2 /s 3 while in the Bc experiment  = 0.006 cm 2 /s 3 .   Two-dimensional energy spectrum is given by

B. Energy spectra in wavenumber space
where (k x , k y ) is the wavenumber and ( , ) xy kk u represents the discrete Fourier transform of the velocity vector. Figure 5 shows the evolution of the two-dimensional spectrum in the Bt experiment. The spectrum shortly after the forcing starts (Fig. 5 a) Fig. 6 b shows the spectrum after the forcing is switched off.
The spectrum is qualitatively similar to that in the Bt experiment (Fig. 5 c).
Solid white and black lines in Figs. 5 b and 6 b show anisotropic Rhines wavenumber, k R and a wavenumber k  correspondently. These wavenumbers are determined using a "synchronization" condition for turbulence and Rossby waves which requires the frequency of the turbulent motions to be equal to the frequency of the waves. The frequency of the Rossby waves is determined by their dispersion relation 22 ( , ) Turbulence is assumed to be isotropic with wavenumber k characterizing a vortex arrangement in the turbulent flow. Alternatively, the characteristic frequency of forced turbulence can be obtained using the energy rate  as a main control parameter. In the upscale energy cascade with spectral slope -5/3, the time required for a fluid parcel to move a distance 1/k is  -1/3 k -2/3 (see e.g. Vallis 32 ) which gives Equating turbulent frequencies (8) or (9) to the Rossby wave frequency (7) and solving for the wavenumber, we obtain the Rhines wavenumber and the k  wavenumber Here is the angle in the wavenumber plane such that cos = k x /k. In the derivations of the wavenumbers we assumed R d to be large and ignored it in the dispersion relation (7).
Physically k  can be interpreted as a wavenumber where turbulence starts "feeling" the -  (8) and (9). However, turbulent motions emit waves with the entire spectrum of frequencies. Rossby wave emission by a localized perturbation is described by the -plume theory (see e.g. Refs (21,22)). Afanasyev 36 employed the -plume approach to show the evolution of two-dimensional spectrum of turbulence towards the anisotropic state similar to that observed in in Figs. 5 b and 6 b. The emission of Rossby waves occurs due to a forcing in the RHS of a linearized quasi-geostrophic where  is the surface elevation. The forcing is isotropic with a known spectrum. Note that an isotropic small-scale turbulence with the developed (due to nonlinear triad interactions) -5/3 type energy cascade can be considered as forcing for this purpose. A solution for the Fourier transform of the surface elevation  can be easily obtained as where F(k x , k y ) is the forcing and  is the frequency of the Rossby waves given by the dispersion relation (7). Geostrophic velocity is obtained from the surface elevation as and the two-dimensional energy spectrum is then calculated using Eq. (6). Fig. 7 shows the evolution of the spectra generated by an isotropic Gaussian forcing.
where k f = 1 cm -1 is the forcing wavenumber. The ring-like Gaussian forcing approximately models the forcing applied by the magnets in our experiment. The dimensional control parameters used to calculate the spectrum shown in Fig. 7 are chosen to have the same values as in the Bt experiment. Fig. 7 shows that the energy spectrum develops from the initially isotropic spectrum towards a strongly anisotropic one where the energy is concentrated at the k x = 0 line similar to that in the experiment (in Figs. 5 b and 6 b). One-dimensional energy spectrum is an important characteristic of a turbulent flow which can be easily obtained from the two-dimensional spectrum such as that shown in Figs. 5 and 6. The one-dimensional (isotropic) spectrum is defined as where the two-dimensional spectrum ( , ) xy E k k is averaged over angle  in wavenumber domain. In anisotropic -plane turbulence, a zonal spectrum can be defined as: Theory predicts that if the certain criteria are satisfied the flow has a steep zonal spectrum (Rhines 12 , Sukoriansky et al. 13 ) while the non-zonal spectrum is expected to have a slope of -5/3 as predicted by Kolmogorov-Kraichnan theory for two-dimensional turbulence. Fig. 8 shows both the isotropic spectrum E(k) and zonal spectrum E Z (k y ) in the Bt experiment while Fig. 9 shows the spectra in the Bc experiment. In both experiments, in forceddissipative regime the isotropic spectrum shows a good agreement with the predictions of the two-dimensional turbulence theory. The enstrophy and energy ranges are separated by a forcing wavenumber. In the Bt experiment the isotropic forcing wavenumber is 2 f xf kk  1 cm -1 (arrow in Fig. 8). In the Bc experiment the forcing wavenumber corresponds to the size of the baroclinic eddies and is more difficult to pinpoint precisely but it is clearly smaller than the forcing wavenumber in the Bt experiment. In the enstrophy range (k > k f ) the slope of the energy spectra is close to -3 in both experiments. The slope in the energy range (k < k f ) is close to -5/3 in both experiments. Note, however, that the energy range in the Bt experiment is quite limited.
The RSW experiments by Afanasyev and Craig 6 showed that the finite radius of deformation can be a limiting factor for the cascade (see also Danilov  The spectra in Fig. 8 b and Fig. 9 b were measured after the forcing was switched off and the flows were decaying. In the Bt experiment the levels of energy decrease; the shape of the zonal spectrum remains almost the same while the isotropic total energy spectrum loses its -5/3 range and exhibits a uniform slope of approximately -3 from the low wavenumber cut-off at k  0.5 cm -1 towards larger wavenumbers. In contrast, in the Bc experiment the flow is still driven by the baroclinic energy source at relatively large wavenumbers even when the direct forcing is switched off. The -5/3 range remains and is even wider than that in Fig. 9 a. Fig. 10 shows the evolution of the energy spectra E(k) after the forcing is switched off in each experiment. The levels of energy drop and the peaks of spectra shift to lower wavenumbers. To characterize the shift of the spectral peak it is useful to consider the energy-weighted mean wavenumber k E defined as Wavenumber k E provides a measure of the energy containing scale. Evolution of k E just before and after the forcing was stopped is shown in Fig. 10

C. Frequency-wavenumber space and Rossby waves
In his pioneering paper on -plane turbulence Rhines 12 predicted that Rossby waves should dominate at low wavenumbers below the wavenumber k R . In what follows we shall provide experimental evidence of the Rossby waves in our experiments. It is not easy to identify waves in a turbulent flow. One way to do that is to use the fact that Rossby waves have a distinct dispersion relation (7). The dispersion relation shows in particular a well-known property, namely the westward phase propagation of the Rossby waves. Fig. 11 a, b shows the xcomponent of velocity, u, measured along a circle of radius r 0 = 2R/3 and r 0 = 2R t /3 at different times during the Bt and Bc experiments respectively. The Hovmöller plot in Fig. 11 shows The dispersion relation (7) can be used in a more direct way to identify the spectral signature of the Rossby waves. should be in between these two lines. According to Rhines 12 waves are excited most effectively at frequencies given by the turbulence "dispersion relation" Eq. (8). The straight white lines in Fig. 12 show Eq. (8); there is indeed some concentration of energy along these lines. Let us return now to the spiral pattern of jets that we pointed out earlier in Fig. 2 k and Fig. 3 c. The spiral patterns were observed in the decaying flows. The energy is dissipated faster in the center of the domain where water is shallow. Therefore, we can expect propagation of energy from the periphery where the turbulence is still active towards the center. Turbulent eddies excite Rossby waves which can be described by linear theory. Here, however, we need to use a polar domain rather than a regular -plane. Linearised quasi-geostrophic equation similar to Eq. (12) but in polar coordinates (r, ) can be written as   22 20 d Bc experiment (Fig. 3 c). Thus, this simple linear analysis allows one to see where the experiments are in the relevant wavenumber space.

DISCUSSION AND CONCLUSIONS
In this work, we have shown experimental evidence that zonal jets occur both in forceddissipative and in decaying turbulent flows on the topographic -plane. The jets are latent in the forced-dissipative case if the forcing is strong and eddies dominate; the jets in our barotropic (Bt) experiment are similar in this respect to oceanic jets (also called "striations") which can only be revealed by filtering and time-averaging of the velocity fields 9,10 . The jets are stronger, relative to the eddies, in the baroclinic (Bc) experiment where the forcing was weaker. The jets become prominent in the decaying regime in both experiments, when the flow is no longer driven to isotropy by strong forcing.
Numerical simulations of -plane turbulence by Galperin et al. 33 , Galperin et al. 34 , Sukoriansky et al. 13 and Scott and Dritschel 31 showed that the ratio R  = k  /k R controls the regime of the flow. When R  is large enough (R  > 2), the flow is in a so-called zonostrophic regime when the jets are a dominant feature of the flow. The jets on gas giants satisfy the criterion of zonostrophy. When R  is less than approximately 1.5, the flow is in a viscous regime. A regime between the viscous and zonostrophic regimes (1.5 < R  < 2) is transitional. The jets in the Earth's oceans seem to be in the transitional regime and so are the jets in our Bt experiments where R   1.7.
The measurements of the energy spectrum in the wavenumber space demonstrated that the spectrum evolves from initially almost isotropic one to an anisotropic spectrum of a typical dumbbell form which was previously observed in numerical simulations (Vallis and Maltrud 35 ).
The evolution of the experimental spectrum is in a qualitative agreement with that predicted by a -plume theory 36 . The energy cascades towards k x = 0 and nonzero k y close to the forcing wavenumber.
Investigations of the energy spectra in the frequency-wavenumber domain revealed an asymmetry of the spectra with respect to the sign of the zonal wavenumber k x . This asymmetry can be explained by the westward propagation of the Rossby waves. Some concentration of waves energy was observed along the line  = U rms k as predicted by Rhines 12 .
The asymmetric frequency-wavenumber energy spectra, which are qualitatively similar to those observed in our experiments, were also observed in the middle latitude ocean by Wunsch 37 .
In the ocean, the spectra also indicated a predominantly westward propagation of perturbations.
A significant energy was distributed along the so-called `non-dispersive' line given by  = R d 2 k.
This line was tangent to the first-mode baroclinic Rossby wave dispersion curve. Note that in our experiments the Rossby waves were barotropic, so the "non-dispersion" line represented a barotropic Rossby wave speed for the longest spatial scale (the maximum wave speed allowed by local water depth). However, in our case, shorter waves with wavenumbers close to the forcing wavenumber were more prominent than long non-dispersive waves.
In this work we do not distinguish between jets and almost zero-frequency Rossby waves with nearly zonal wavecrests. In Fig. 2 j-l and Fig. 3 c the (slightly spiralling) jets can be directly associated with Rossby waves emitted by eddies in the decaying flows. Moreover, we suggest that -plume mechanism or emission of low-frequency waves by perturbations is the underlying mechanism in jets formation by -plane turbulence. This notion is illustrated by Fig. 7 showing the evolution of the two-dimensional spectrum towards zonal motions solely by linear wave emission. Further sustaining the jets by turbulence will perhaps involve nonlinear mechanisms where a self-organization of the jet-eddies system is important and results in fluxes of momentum from eddies to jets.