Nature Physics  Letter
A laboratory model for deepseated jets on the gas giants
 Simon Cabanes^{1}^{, }
 Jonathan Aurnou^{1, 2}^{, }
 Benjamin Favier^{1}^{, }
 Michael Le Bars^{1}^{, }
 Journal name:
 Nature Physics
 Volume:
 13,
 Pages:
 387–390
 Year published:
 DOI:
 doi:10.1038/nphys4001
 Received
 Accepted
 Published online
The strong east–west jet flows on the gas giants, Jupiter^{1} and Saturn^{2}, have persisted for hundreds of years. Yet, experimental studies cannot reach the planetary regime and similarly strong and quasisteady jets have been reproduced in numerical models only under simplifying assumptions and limitations. Two models have been proposed: a shallow model where jets are confined to the weather layer and a deep model where the jets extend into the planetary molecular envelope. Here we show that turbulent laboratory flows naturally generate multiple, alternating jets in a rapidly rotating cylindrical container. The observed properties of gas giants’ jets are only now reproduced in a laboratory experiment emulating the deep model. Our findings demonstrate that longlived jets can persist at high latitudes even under conditions including viscous dissipation and friction and bear relevance to the shallow versus deep models debate in the context of the ongoing Juno mission^{3}.
Subject terms:
At a glance
Figures
Main
Azimuthally directed (that is, zonal, east–west) jet flows are one of the dominant characteristics in the surficial cloud features observed on the gas giants, Jupiter and Saturn. An essential question of planetary dynamics and structure is whether these jet motions exist only within the shallow troposphere or extend through the molecular envelope that exists above the deeper dynamo region^{3}. Determining the depth of these atmospheric jets is one of the prime directives of the NASA (National Aeronautics and Space Administration) Juno mission, which entered into lowaltitude Jovian orbit in August 2016^{4}. Despite the longlived scientific interest in these flows, dominant multiple jets have been problematic in fully threedimensional (3D) numerical models of convection. In particular, multiple banded flows are not found in the most recent, highresolution models that couple the molecular envelope to the deeper dynamo region. In these models, magnetic dissipation damps the higherlatitude deep jets out of existence^{5, 6, 7}. Similarly, dissipation has also proved overly important in laboratory experiments carried out to date. Laboratory approaches were analysed in the framework of the shallowlayer model and strong viscous damping by the container boundaries only allows for the formation of weak zonal jets with tenuous instantaneous signatures^{8, 9, 10, 11, 12, 13, 14}. Thus, it has yet to be demonstrated, as proposed for the gas giant planets^{15}, that deep zonally dominant jet flows can exist in the presence of boundary dissipation.
We have developed a new laboratory experimental device that is capable of generating strong zonal jets despite viscous friction on the boundaries (Fig. 1a). The working fluid is water, contained in a 1.37mhigh by 1mdiameter cylindrical tank. The depth of the fluid layer is h_{o} = 50 cm at rest, and the tank’s rotation rate is Ω = 7.85 rad s^{−1} (75 revolutions per minute). Once equilibrated at Ω, the water’s free surface takes the shape of a paraboloid, with the fluid layer depth ranging from h_{min} ≃ 20 cm on the axis of rotation to h_{max} ≃ 90 cm at the tank’s outer radius. This rotating surface shape is analogous to the largescale curvature of a deep spherical planetary fluid layer^{16, 17}. In addition, the rotation provides strong Coriolis forces, as exist in planetary settings. Once solid body rotation is reached, a submersible pump situated at the base of the tank is turned on, and smallscale turbulence is injected at the base of the fluid layer. The pump continuously circulates water through a lattice of 32 outlets (4 mm diameter) and 32 inlets (2 mm diameter) arranged on a flat base plate without any axisymmetric features (see the injection pattern in Supplementary Fig. 1a). Typical rootmeansquare (r.m.s.) fluctuating velocities are in the range u_{r.m.s.} ≃ 1–5 cm s^{−1}. This smallscale turbulence is analogous to the convective turbulence that exists in deep planetary interiors^{18, 19} and constitutes an appropriate source of energy to supply deep jets^{20}. Inspired by several preceding experimental attempts^{10, 11, 12, 13, 14, 21, 22} this laboratory device bridges the gap to the planetary ‘zonostrophic’ regime and suggests the possibility of deepseated jets on the gas planets.
In natural settings, zonal jets develop from rapidly rotating turbulence in the presence of strong boundary curvature^{23}. In particular, the Coriolis force must dominate the fluid’s inertia (that is, low Rossby number Ro), which, in turn, must dominate the viscous dissipative effects (large Reynolds number Re and small Ekman number E) (see Supplementary Table 1). Such flows, because they are essentially twodimensional (2D), transfer energy upscale via a socalled inverse cascade^{24, 25} driven by Reynolds stresses^{14}. In addition, the spherical geometry of planetary fluid layers is ideal for the formation of zonal jet flows: it induces a zonal anisotropization of the inverse energy cascade following the socalled βeffect. Energy is consequently channelled into zonal jets with characteristic latitudinal width approximated by the Rhines scale , where U_{r.m.s.}^{T} is the total r.m.s. velocity used as the characteristic system scale flow velocity^{23, 25}. The spherical curvature of the fluid shell, expressed via β, acts to halt the 2D energy cascade, leading to the formation of Rhines scale zonal jets.
All these physical ingredients can be reproduced following two distinct scenarios, depending on the nature of the anisotropic β parameter and on the jets dynamical aspect ratio η = h/L_{Rh} on which jets develop (with h and L_{Rh} being the vertical and horizontal scales of jets). In the first scenario, jets are confined into a shallow troposphere where shallow turbulence is essentially 2D because of depth confinement (η 1). The zonal anisotropization of the inverse cascade is due to the variation of the Coriolis force with latitude θ (the planetary βeffect: β = 2Ω cosθ/r, with r being the planetary radius) and the relevant axis of rotation is the local normal to the planetary surface. In the second scenario, jets are deepseated and extend into a 3D convective deep layer where rotation dynamically constrains largescale motions to be invariant along the rotation axis (η ≳ 1). In such a flow, dynamical confinement leads to bidimensionality, the resulting inverse cascade is dependent on the rotation rate and 3D wave motions may play an important role in the dynamics^{22}. In deep spherical fluid layers the zonal anisotropy arises due to cylindrically radial variations in the depth of the fluid layer^{23} (the topographic βeffect: β = 2Ωh^{−1} dh/dr) and the relevant axis of rotation is the planetary spin axis. The distinction between shallow and deepseated jets has never been clear in previous laboratory devices with moderate aspect ratio (0.57 ≤ η ≤ 1). Midlatitude Jovian jet aspect ratios are η ≃ 5 and η ≃ 0.004 for the deep and shallow scenarios, respectively. In our laboratory device we emulate a gas planet deeplayer scenario with η = 5.15 (see Supplementary Table 1 for comparative nondimensional analysis).
Atmospheric zonal flows are detected in all four Solar System gas planets (see for example, Fig. 2a for Jupiter), and, recently, zonal jet structures have also been observed in Earth’s oceans^{26}. However, gas giant jets are known to be incredibly steady over time whereas oceanic flows are strongly fluctuating and zonal features are revealed only following careful timeaveraging of the flow fields. These differing regimes of jet flow behaviours can be characterized in terms of the zonostrophy index^{27} R_{β} (see Methods for details on its calculation). This index is the ratio of the Rhines scale and the transitional scale L_{β} ≃ (ϵ/β^{3})^{1/5} above which the βeffect affects the energy cascade, with ϵ being the inverse energy transfer rate through the system^{23}. Simplistically, R_{β} = L_{Rh}/L_{β} represents the strength of the zonal flows relative to the strength of the underlying mixing. If mixing effects are strong (large L_{β}), the zonal jets tend to be undulatory and unsteady in time, or fail to form altogether^{28}. As L_{β} is decreased, at a given value of L_{Rh}, quasisteady zonal jets are predicted to become the prevalent fluid motion in the system^{23, 28}. Such flows, argued to exist at R_{β} ≳ 2.5, are in the socalled zonostrophic regime^{27}. Deep Jovian midlatitude jets exceed this threshold^{29} with a typical value R_{β} ≃ 5, while Earth’s oceans are limited to R_{β} ≃ 1.5. In previous laboratory experiments, strong boundary dissipation and relatively low β led to large L_{β}, nonzonostrophic flows (see Supplementary Table 1 for comparative nondimensional analysis).
In our experimental device, we create for the first time in the laboratory, the appropriate conditions for generating gas planetlike deepseated zonal flows (that is, laboratory nondimensional parameters Ro = 7 × 10^{−3}, Re = 3 × 10^{4}, E = Ro/Re = 2.3 × 10^{−7}, mean aspect ratio η = 5.15 and large values of the mean zonostrophy index R_{β} ≃ 3.7 that exist in deep zonostrophic regime; see Methods and Supplementary Table 1). Experimental measurements of the surface velocities are acquired by recording the tracks of small floating particles from a top lid camera onboard the rotating frame and analysing their paths using a Lagrangian tracking method (see Methods). To validate our laboratory surface velocity measurements, we have carried out highresolution direct numerical simulations of the experiment. Using the same upper fluid surface shape as in the 75 r.p.m. experiments, but at slightly higher Ekman number, E = 10^{−6}, the simulated flow is forced at the base of the tank via an array of vortices that approximate the smallscale basal pumping in the experiment, starting from the fluid initially at rest in the rotating frame (see Methods and Supplementary Fig. 1b). Figure 1b shows an instantaneous sidecut rendering of the azimuthal velocity field after 1,000 simulated rotation times. The largescale surface zonal flows are deepseated, validating our laboratory surface velocimetry and confirming that the flow is dynamically quasi2D due to the dynamical effect of rotation (Ro 1). Further, it reveals that locally injected turbulence can drive largescale zonal jets, even in the presence of boundary dissipation (all boundaries except the upper paraboloid being noslip, see Supplementary Fig. 2c, d).
Our laboratory experiment produces six zonostrophic jets with strong, instantaneous signatures, similar to those on gas planets (Fig. 2). In Fig. 2b, we show experimental particle tracks acquired from 4,900 to 5,350 rotations. The tracks are coloured on the basis of their instantaneous zonal velocity, red (blue) being prograde (retrograde). The zonally elongated tracks show that the jets exist not only in time average, but instantaneously at all times (see also Supplementary Movie 1). This instantaneous signature is novel in a rapidly rotating turbulent jets experiment and has not yet been attained in numerical models of deep jet generation with boundary dissipation. Compiling all the velocity data in Fig. 2b yields the timeaveraged zonal velocity field map shown in Fig. 2c, revealing strong zonation. We calculate the surface average of the zonostrophy index in our experiment (that is, with an estimate of ϵ using the total horizontal kinetic energy, see equation (6) in Methods), to be R_{β} = 3.7, confirming the relevance of our experiment to the gas giants zonostrophic regime reached for R_{β} ≥ 2.5.
Figure 3a shows the time and azimuthalaveraged zonal velocity profile. The solid red line denotes the spatiotemporalaveraged jet flows as a function of the cylindrical radius, whereas the r.m.s. values of the velocity fluctuations are demarcated by the dashed blue line. Using kinetic energy spectrum analysis in our deep model we retrieve in Fig. 3b the theoretical^{25} anisotropic zonal flow spectrum E_{Z}(k) = C_{Z}β^{2}k^{−5} and the 2D residual (nonzonal) energy spectrum from velocity fluctuations E_{R}(k) = C_{R}ϵ^{2/3}k^{−5/3} (where k is the 2D horizontal wavenumber and C_{R} = 5, see Methods). Considering our radial evolution of β we obtain that the zonal constant C_{Z}, which best fits our zonal spectrum, ranges from 1.7 to 3.7. This constitutes the first experimental measurement that confirms the estimate from direct observations of Jupiter’s clouds that gives C_{Z} ≃ 2 (ref. 29). However, since both of these studies are based on a constant β approximation, one cannot discriminate between shallow and deep βeffects for Jupiter using this spectral analysis. Similarly to the Jupiter observations, we report that energy is maximum at jets scale (that is, at the Rhines scale k_{Rh} = 2π/L_{Rh}) where zonal kinetic energy strongly dominates over the residual fluctuating flow. Transition to a strongly anisotropic flow where energy becomes preferentially channelled into the zonal direction is well observed at the predicted transitional scale k_{β} = 1/L_{β}. We find that more than 50% of the total surface kinetic energy is concentrated in the zonal velocity component, and this value goes up to 70% in the sharpest prograde jet (see Methods for more details).
Focusing on the first 1,000 rotations of another 75 r.p.m. experiment, Fig. 4 shows the evolving width of the jets over time, with merging events occurring typically within the first 500 to 600 rotations. After this, strongly energetic, longlived zonal jets reach a quasisteady state (see Supplementary Fig. 3a for complementary results). The steadiness of the multiple jets system is an important feature shared by gas planets as well as our laboratory and numerical models. Even in the more dissipative noslip numerical simulation (where E = 10^{−6}), dissipation does not prevent the formation of intense jets (see Supplementary Fig. 3b). Comparison between the experimental and numerical cases, including the simulation with stressfree boundary conditions (see Supplementary Fig. 3c), highlights that the boundary dissipation is a key ingredient in determining the equilibrated scale and strength of the deepseated jets, as well as the typical time for reaching a steadystate regime.
Our experiments demonstrate for the first time that robust, quasisteady zonostrophic, deep (η ≥ 1) jets are realizable in a relatively simple laboratoryscale device that brings together the physical ingredients of planetary zonal flow systems: dynamically constraining rapid rotation, highly turbulent flows, and large fluid layer depth variations. Zonal jet formation then occurs even in the presence of bottom boundary friction, in sharp contrast to the results of overly viscous 3D simulations in which the mid to highlatitude jets are damped out^{5, 30}. We further claim that dissipation is an essential ingredient for jet equilibration^{15}. On the basis of our findings, we hypothesize that deep planetary jets will also form in the presence of a magnetic dissipation region, as exists within the gas planets. Indeed, a magnetodissipative layer localized at the transition between conductive and nonconductive regions will not be likely to impact the ability for geophysical turbulence to generate deep zonostrophic zonal jets extending across the gas giants’ molecular envelopes. Ultimately, our results suggest that the forthcoming Juno and Cassini measurements will support the existence of deepseated zonal jets on the gas giants.
Methods
Experimental device.
Our device is based on several previous setups^{10, 11, 12, 13, 14, 21, 22}. It consists of a 1mdiameter by 1.37mhigh cylindrical tank of polythene that is attached to the rotating table by a rigid aluminium superstructure. A 32cmhigh bottom table, made of PVC, encloses our basal injection system. We drilled 64 holes of, respectively, 2 mm and 4 mm diameter connected to 32 injection and 32 suction tubes distributed following a nonaxisymmetric square grid (see Supplementary Fig. 1a). This smallscale turbulence injection system is used to mimic convective turbulence in deep planetary interiors, in the manner of previous experimental devices^{8, 22, 31, 32}. These tubes are connected to an immersed AQUASON506MAL pump giving an operating flow of 2 < Q < 5 m^{3}/h. Consequently, the injected flow at the bottom of the domain is mainly threedimensional (3D) and weakly affected by rotation (injection velocity and Rossby number 0.1 < Ro < 0.4). An acrylic top lid carries a GoPro HERO3+ camera that records the surface flow at 59 frames per second. A lighting apparatus, including a powerful lightemitting diode lamp and a diffuser, sits on top of the aluminium superstructure. The whole device is attached to an ABRT 7686 rotary table using robust highprecision axial roller bearings with an air bearing in the horizontal axis for smooth, precise rotary motion. The table is 1.4 m in diameter with maximum loading of 1,000 kg and a typical operating speed of 1–90 r.p.m.
Experimental data acquisition.
Assuming the flow to be quasigeostrophic (that is, nearly invariant along the rotation axis, see Fig. 1b), we perform freesurface visualization. Lagrangian surface velocities are measured using a particles tracking code^{33}. We follow the surface motions of 90 black floating particles taken to be smaller than the dynamical flow scales (3 mm diameter spherical particles) and to be at the same density as the carrier fluid (for better optical contrast, the water is tinted white via a titanium dioxide suspension).
After camera calibration and correction of parabolic freesurface deformation, we obtain zonal and radial flow velocity components along Lagrangian tracks. We record the floaters’ motions over 5,000 rotations. A quasisteady state is achieved within 600 rotations (see Fig. 4 and Supplementary Fig. 3). We define a Cartesian grid of 142 × 142 square elements of 7 mm each. Temporal mean zonal velocity U_{φ} and azimuthalaveraged zonal velocity are obtained in each grid element from timeaveraging over 450 rotations once steady state is achieved (see Figs 2c and 3a). Figure 4 and Supplementary Fig. 3 represent azimuthal and timewindowed averages (the time average is computed over 50 rotations and sliding by steps of 3 turns) from the initial time at which we activate the pump. Velocity fluctuations reported in Fig. 3a are obtained by computing the r.m.s. velocity , where N is the total number of tracks in the ith grid elements and U_{φ}^{n, i} is the zonal velocity of the nth track. By using the same procedure for the radial component of the flow, we obtain the radial mean velocity and radial fluctuations u_{r.m.s.}^{r}. We report that the radial mean flow is nearly zero and radial fluctuations are of the same order as zonal fluctuations u_{r.m.s.}^{φ} ≃ u_{r.m.s.}^{r}. The total kinetic energy corresponds to the sum and E_{fluct} = (u_{r.m.s.}^{φ})^{2} + (u_{r.m.s.}^{r})^{2} is the energy of the fluctuating velocity. Within the jets, at the maximum of each zonal velocity peak, the kinetic energy of the fluctuations reaches 43% of the total kinetic energy (0.3 < E_{fluct}/E_{T} < 0.43), while the zonal kinetic energy exceeds 57% of the total (0.57 < E_{φ}/E_{T} < 0.7).
Numerical method.
The Navier–Stokes equations are solved in their weak variational form^{34} for an incompressible fluid in three dimensions with the spectral element code Nek5000 (http://nek5000.mcs.anl.gov). The spectral element method combines the geometric flexibility of finiteelement methods with the accuracy of spectral methods. It is particularly well adapted to our problem involving turbulent flows in complex geometries. Nek5000 has, for example, already been used in the context of rotating flows in ellipsoidal containers^{35}. The computational domain is decomposed into nonoverlapping hexahedral elements. Within each element, the velocity is decomposed onto Lagrange polynomial interpolants of order N, based on the tensorproduct of Gauss–Lobatto–Legendre quadrature points. For all the simulations discussed in this paper, numerical convergence was checked by fixing the number of elements and increasing the degree N of the polynomial decomposition. The temporal discretization is based on a semiimplicit formulation in which the nonlinear and rotation terms are treated explicitly and all remaining linear terms are treated implicitly. Note that our solution is dealiased following the 3/2 rule. The code is efficiently parallelized using MPI (Message Passing Interface) and we use up to 2,048 processors for the highest resolution considered in this study. The entire cylindrical domain is discretized using 53,760 elements and a spectral order of N = 11. The element distribution is denser close to the external and bottom boundaries. This, in addition to the Gauss–Lobatto–Legendre point distribution close to the element boundaries, ensures an appropriate resolution of the Ekman boundary layers with approximately six grid points to describe them in all cases.
The vertical extent of the cylindrical domain is stretched so that the upper boundary matches to the analytical paraboloid shape that depends on the rotation rate and on the gravitational acceleration. This surface is assumed to be stressfree and impenetrable. To mimic the experimental forcing, a smallscale steady volumic force is defined in Cartesian coordinates by
if 0 ≤ z ≤ 1/k_{f} and F = 0 elsewhere. The function S(r) is given by
A is the ratio of the force amplitude to the centrifugal acceleration and is related to the Rossby number (see Supplementary Fig. 1b). This forcing imposes a 3D smallscale turbulent flow at the base of the numerical domain, mimicking the turbulence in the experiments. The numerical simulations presented correspond to A = 0.002 and k_{f} = 12. Note that the typical jets observed in our simulations have a corresponding radial wavenumber k = 4 − 6 so that we reach a reasonable scale separation between the energy injection mechanism and the largescale zonal flow. We have checked that the results do not depend on k_{f} although it is difficult to modify it while keeping a large Reynolds number and a low Rossby number. Thus, we considered only 8 < k_{f} < 16.
Nondimensional analysis and zonostrophic regime.
Part of the experimental challenge was to drive the whole apparatus, containing 400 l of water, at a spin rate of Ω = 75 r.p.m. for the purpose of reducing as low as possible the Ekman and Rossby numbers of a developed turbulent flow. The Rossby number is the ratio of inertial to Coriolis forces defined as Ro = U_{r.m.s.}^{T}/2Ωh_{o}, where we use U_{r.m.s.}^{T} the total r.m.s. velocity as the characteristic system scale velocity and h_{o} the fluid layer depth at rest. To simultaneously maintain a large Reynolds number, defined as Re = U_{r.m.s.}^{T}h_{o}/ν with ν being the kinematic viscosity, and a low Rossby number, one has to rapidly rotate the system for Coriolis to dominate over inertial forces. The ratio between Rossby and Reynolds numbers leads to the Ekman number E = ν/2Ωh_{o}^{2}. Our rapidly rotating deeplayer experiment strongly reduces this Ekman number to values that are almost unachievable for 3D direct numerical simulations. The Ekman number appears to be crucial to investigate jets saturation mechanisms. Another nondimensional parameter is the dynamical aspect ratio of the jets that we define as η = h_{o}/L_{Rh}, with L_{Rh} being the typical jets width. This parameter compares vertical (h_{o}) and horizontal (L_{Rh}) characteristic length scales, setting whether or not the jets develop in a ‘shallow’ or ‘deep’ domain. In the limit of η 0, the system tends to be 2D as the horizontal scale is much larger than the jet depth. Twodimensional flows prevent direct cascade of energy and kinetic energy is consequently transferred to largescale structures^{24}. Thus, in shallowlayer turbulence, the dynamic confines to a quasi2D system at a finite value of η 1, where the upscale cascade of energy is favoured. On the contrary, in deeplayer models, the vertical length scale can exceed the horizontal length scale, leading to 3D turbulent structures. Due to strong background rotation (that is, low Rossby number), however, largescale motions are nearly invariant in depth and become essentially 2D (ref. 23). The quasi2D components of the flow field tend to drive energy upscale via a socalled inverse cascade while 3D motions persist at smaller scales.
We report comparative values of the four nondimensional quantities estimated from recent experiments^{11, 12, 13, 14}, as well as for Jupiter and Earth’s oceans in Supplementary Table 1. This table shows that our experimental approach gets closer to Jupiter’s parameter range (low Ro, E and large Re) than any previous experiments. Moreover, we designed our experimental device to promote deepseated jets and exclude the possibility of a shallowlike scenario. Our setup is characterized by a large dynamical aspect ratio (η = 5.15), which clearly disqualifies the possibility of 2D turbulence due to depth confinement. Quasigeostrophic structures are then required to cascade energy upscale and, consequently, jet formation sensitively depends on the Rossby number. The geostrophic organization of the flow is well supported by our numerical simulation (Fig. 1). This distinction between shallow and deepseated jets is less clear in all other previous experiments with moderate aspect ratio (0.57 ≤ η ≤ 1) and in which the role of geostrophy is not well defined. The experimental device designed by Smith et al. ^{12} has a typical aspect ratio of η = 4.2 leading to deepseated jets. However, the associated zonostrophic index is much lower and largescale baroclinic vortices dominate the dynamics.
In the presence of background rotation, the free surface of a fluid takes the shape of a paraboloid. In our cylindrical domain, radial variation of the fluid depth is defined as
where r is the cylindrical radius and g is the gravitational acceleration. The centre of the tank, of minimum depth h_{min} = h_{o} − Ω^{2}R^{2}/4g, represents the planetary pole, while the edge of the domain corresponds to lower latitudes. Note that the paraboloidal surface shape is the firstorder Taylor expansion of the spherical curvature near the polar region. Under rapid rotation, the classical turbulent cascade, which usually dissipates kinetic energy from large to small scales, reverses direction to feed largescale coherent structures. Similarly to planetary spherical configuration, depth variation h(r) induces a zonal anisotropization of the reversed energy cascade (the socalled zonostrophic turbulence^{23}) following the topographic βeffect quantified by
where topographic paraboloidal β is zero at the pole and increases with radius.
The presence of this βeffect selects a transitional wavenumber at which the inverse cascade transitions to Rossbywavedominated dynamics:
where ϵ is the upscale energy transfer rate. Following Vallis and Maltrud^{23}, the anisotropy wavenumber k_{β} defines the scale at which the eddy turnover timescale τ_{ϵ} = ϵ^{−1/3}k^{−2/3}, with k being the eddy wavenumber, becomes comparable with the inverse of the Rossby wave frequency τ_{Ro} = k^{2}/βk_{φ}, with k_{φ} being the azimuthal wavenumber. k_{β} is an estimation of the smallest scale at which anisotropic Rossby wave propagation, due to boundary curvature together with rotation, affects the turbulent inverse cascade. The other main scale of βturbulence comes out of the typical advection timescale τ_{a} = (Uk)^{−1}, where U ≃ U_{r.m.s.}^{T} is the characteristic systemscale flow velocity that advects turbulent eddies of scale k, and from which we recover the Rhines scale^{25} .
In the context of planetary inverse cascades, dimensional analysis can be reduced to the single zonostrophy index defined as the ratio between transitional wavenumber and Rhines wavenumber, R_{β} = k_{β}/k_{Rh} (ref. 27). This index can be seen as an estimate of the inverse turbulent cascade efficiency to grow zonal structures on a characteristic timescale τ_{ϵ} compared with the turbulent structures turnover time τ_{a}. Large βeffect coupled with a strong Coriolis force leads to large index value of R_{β} > 2.5, corresponding to τ_{ϵ} < τ_{a}. This threshold distinguishes between the zonostrophic and partially zonostrophic regimes.
It is not easy to access the value of ϵ in laboratory experiments. It can be estimated as ϵ ≃ (U_{r.m.s.}^{T})^{2}/2τ_{E} based on the total horizontal kinetic energy and the Ekman spinup time^{14} . Using U_{r.m.s.}^{T} = 0.035 m s^{−1} and U_{r.m.s.}^{T} = 0.015 m s^{−1} for respectively our experiment and the numerical simulation, we obtain ϵ = 3.4 × 10^{−6} m^{2} s^{−3} and ϵ = 6.3 × 10^{−7} m^{2} s^{−3}. We end up with zonostrophic index values in the experiment 4.9 < R_{β} < 5.2, using experimental energy transfer rate and with values of 4.5 < R_{β} < 4.8 in the numerical simulation. This zonostrophic index is estimated in the range of β from 60 m^{−1} s^{−1} to 89 m^{−1} s^{−1} in the cylindrical radius range 0.2 < r/R < 0.9 in which jets form. One can rewrite the zonostrophy index in the form
An overall view of the zonostrophic regime is given by spectral analysis where the total energy spectrum E(k) can be decomposed into zonal and residual (or nonzonal) components E(k) = E_{Z}(k) + E_{R}(k) with the total wavenumber in the horizontal plane , where k_{r} and k_{φ} are, respectively, the radial and azimuthal wavenumbers. The zonal spectrum can also be written as the axisymmetric component of the total energy spectrum E_{Z}(k) = E_{Z}(k_{r}, k_{φ} = 0). Following Sukoriansky et al. ^{36}, in the zonostrophic regime, the zonal and residual spectra are
where numerical simulations^{37} give C_{Z} ≃ 0.5 and C_{R} ≃ 5 − 6. More recently, Galperin et al. ^{29} reevaluated the zonal constant to C_{Z} = 2 from spectral analysis of highresolution images of Jupiter’s clouds. Using expressions (7) one can estimate the wavenumber at which zonal and residual spectra intersect, recovering the transitional wavenumber mentioned above,
In their simulations, Galperin et al. ^{38} estimated that (C_{Z}/C_{R})^{3/10} ≃ 0.5, which might lead to a more accurate estimation of k_{β} than from dimensional analysis equation (5). By considering the recent estimate of C_{Z} = 2 based on Jupiter’s zonal spectrum, one can recalculate the prefactor (C_{Z}/C_{R})^{3/10} = 0.73 in equation (8). This notably reduces the value of the transitional wavenumber and consequently the corresponding zonostrophic index. Similarly, we use our range of β along the cylindrical radius and we report an estimate of the zonal constant that best fits our zonal spectrum. We obtain 1.7 < C_{Z} < 3.7 with our experimental data and 0.42 < C_{Z} < 0.9 with our numerical data using equation (7). Interestingly, the zonal constant ranges evaluated from our experiment and our numerical simulation include respectively the values computed from direct observations of Jupiter and previous numerical simulations reported^{29, 38}. It is important to have in mind that Galperin et al. ^{29} reduce the zonal constant to a single value while β also varies with Jupiter’s latitude. Using the new expression (8) for k_{β}, one can rewrite the expression of the zonostrophy index in the form^{29, 38}:
The zonostrophic index values in the main text are based on C_{Z} = 2, which is in good agreement with our zonal kinetic energy spectrum and our mean β value.
We compute the zonal spectra E_{Z} defined by equation (7) from numerical and experimental data using 1D Fourier transforms along the radius of the mean zonal velocity (that is, temporal and azimuthal average of the zonal velocity). Being limited to Lagrangian tracers to access surface velocity in the experiment, we compute the residual spectrum from numerical simulation alone, using a horizontal square grid at a fixed depth h = 10 cm. The residual spectrum E_{R} defined by equation (7) is obtained from 2D Fourier transform of the nonzonal flow component defined as velocity deviations from the mean flow.
Figure 3b shows good agreement of the zonal and residual spectra with the theoretical predictions of equation (7). Residual and zonal energy spectra intersect at the transitional wavenumber predicted in equation (8) and kinetic energy cascades upscale to accumulate in the zonal component at Rhines scale^{25}. Injection scales presented in Fig. 3b range from an upper wavenumber defined as the half lattice grid size in the experiment and a lower wavenumber defined by numerical forcing. Using our estimation of the inverse energy cascade rate ϵ = 6.3 × 10^{−7} m^{2} s^{−3} and C_{k} = 5, we obtain a good prediction of our kinetic energy residual spectrum, which confirms our zonostrophic turbulence approach. Ultimately, with (C_{Z}/C_{R})^{3/10} ≃ 0.73 and 60 m^{−1} s^{−1} < β < 89 m^{−1} s^{−1}, we compute the zonostrophy index from equation (9). We obtain 3.6 < R_{β} < 3.8 in our laboratory experiment and 3.3 < R_{β} < 3.5 in the numerical simulation, which confirms that we are well into the zonostrophic regime. Mean values of R_{β} = 3.7 and R_{β} = 3.4, using the mean value β = 73 m^{−1} s^{−1}, are reported in the main text. Values of R_{β} for planetary objects and other experimental attempts are detailed in Supplementary Table 1. Expression (9) has been used to compare all experiments and Jupiter’s observations, reducing the zonostrophic index of previous experiments by a factor of 0.73.
Data availability.
The data that support the plots within this paper and other findings of this study are available from the corresponding author on request.
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Acknowledgements
This work was supported by the French Agence Nationale pour la Recherche (Grant No. ANR13JS05000401) and by the AixMarseille University Foundation (A^{∗}MIDEX project No. ANR11IDEX000102). J.A.’s sabbatical in Marseille was supported by S. A. Kremen, by the invited professor programmes of Ecole Centrale Marseille and AixMarseille University, and by the Labex MEC (grant ANR11LABX0092). We also acknowledge support from IDRIS for computational time on Turing (Project No. 100508) and from the HPC resources of AixMarseille University (project Equip@Meso No. ANR10 EQPX2901 of the programme Investissements d’Avenir).
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Affiliations

CNRS, Aix Marseille University, Centrale Marseille, IRPHE, 13013 Marseille, France
 Simon Cabanes,
 Jonathan Aurnou,
 Benjamin Favier &
 Michael Le Bars

Department of Earth, Planetary, and Space Sciences, University of California, Los Angeles, California 90095, USA
 Jonathan Aurnou
Contributions
This work is based on an initial design concept by J.A. All authors contributed to the final design and the building of the experimental setup. S.C. ran the experiment and carried out data processing. B.F. performed the numerical simulations. All authors contributed to data analysis and to writing the letter, with a major contribution from S.C.
Competing financial interests
The authors declare no competing financial interests.
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Simon Cabanes
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