Dynamics of Convectively Driven Banded Jets in the Laboratory

a. Convective forcing

A key objective of this study was to investigate the properties of geostrophic turbulence maintained by a mechanism that was as “natural” and unconstrained as possible. In particular, it was considered important to apply forcing on a relatively small horizontal scale, but one that was not fixed in position relative to the rotating reference frame.

To achieve this, convective forcing was applied by gently spraying relatively dense, salty water onto the main fluid layer from a reservoir located above the center of the tank. A system of spray nozzles was set up along a radial arm, which could be rotated around the tank in either direction at a speed controlled independently of the rotation of the main tank. The spacing of the nozzles was carefully designed so that, as the arm was rotated, the flux of salty water deposited onto the free upper surface of the tank was as uniform as possible. In practice, tests indicated that the volume flux from this system was uniform across the entire radius of the tank to better than ±10%. The flow speed and direction of each jet was set so as to disturb the surface of the water as little as possible. For most runs carried out, the spray arm was rotated with a period of 30–50 s in either the prograde or retrograde direction while depositing approximately 5–7 L min⁻¹ of salty water.

By varying the density of the water spray a range of buoyancy fluxes F_B = g(Δρ/ρ)F_V (where g is the acceleration due to gravity, Δρ is the difference in density between the working fluid and spray, and F_V is the volume flux of the spray) could be obtained ranging from approximately 10⁻⁹ to 10⁻⁷ m² s⁻³. For most runs a buoyancy flux of around F_B = 4.5 × 10⁻⁸ m² s⁻³ was obtained. Together with a rotation rate of (1–2)π/40 rad s⁻¹ and a depth h of around 30–40 cm, this would be expected to lead to a field of convection with a “natural Rossby number,” Ro* (Fernando et al. 1991; Maxworthy and Narimousa 1994; Marshall and Schott 1999), defined by

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(where f = 2Ω) of around 1–3 × 10⁻³. Such a value would indicate a convective regime somewhat similar to that of deep convection in the terrestrial oceans (Marshall and Schott 1999), and imply that rotation should exert a strong influence on the form of convection.

b. Flow measurement

The horizontal velocity fields were obtained from plan view images from a charge-coupled device (CCD) camera by a correlation imaging velocimetry (CIV) method (Fincham and Spedding 1997; Fincham and Delerce 2000). Neutrally buoyant tracer particles of diameter 0.5 mm were suspended in the flow and illuminated by a beam from a 6-W argon ion laser located at the side of the tank and introduced through a glass panel in the outer wall. The beam was rapidly scanned in the horizontal to create a diverging light sheet across a chord through the annular channel, at a depth that could be selected by moving the laser on a computer-controlled carriage.

The illuminated tracer particles were observed from above at a height of approximately 4 m above the surface of the tank by two digital cameras, a low-resolution, wide-angle camera with a resolution of 768 × 484 and covering an area 4.5 m × 3.3 m, and a higher-resolution, narrow-angle camera with a resolution of 1024 × 1024 and covering an area 2.5 m × 2.5 m. This allowed the observation of the largest feasible area of the tank, given the limited height of the building in which the rotating facility was accommodated, though it represented a maximum of just 12% of the total surface area of the annular channel that was visible simultaneously. By aligning the long axis of the wide-angle camera with the radius of the tank, however, the optimum coverage of the zonal flow profile across the tank could be obtained. This did mean that a true instantaneous zonal average could not be measured directly, though use of time averaging would help to recover a more global sample of flow statistics.

Flow fields were obtained every 20 s from groups of images obtained by each camera over time intervals of up to 3 s. By applying CIV analysis to these bursts of images, velocities could be measured to a precision of better than ±0.1 mm s⁻¹ (relative to typical velocities of a few millimeters per second) on a 72 × 48 point (wide angle) or 98 × 98 point (narrow angle) rectangular grid. To diagnose the azimuthal anisotropy of the evolving flow, the velocity fields were subsequently decomposed into radial and zonal (i.e., azimuthal) components and mapped into (r, rθ) coordinates, where r is the radius from the center of the tank and θ the azimuthal angle (increasing counterclockwise from above; see Fig. 2a). For the purposes of presentation and analysis, we also make use of a simple local Cartesian system with x in the (clockwise) θ direction (so x = −rθ, usually measured from the radial center line of the camera image) and r in the outwards radial direction.

In addition to long series of horizontal flow fields, a few measurements were made of the flow in the azimuth–height [(rθ, z) or (−x, z)] plane near midradius by reorienting the laser light sheet into a vertical plane and viewing with the wide-angle camera through a second glass panel in the outer wall. This enabled the flow over a 0.5 m × 0.5 m area to be measured in order to examine the flow pattern forced directly by the overhead spray.

The vertical profile of density in the flow could be measured at intervals to an accuracy of approximately ±1 kg m⁻³ using a conductivity probe that was mounted on the operator gantry at midradius and could be driven downward into the flow from the surface to within 10 cm of the bottom of the tank on command via a motor-driven carriage.

a. Vertical measurements

As discussed above, the forcing of the flow was effected by spraying dense, salty water onto the free surface of the tank from a rotating arm in the frame of the rotating table. This initially produced a shallow layer of dense fluid on the top surface of the flow that rapidly became statically unstable (on a time scale of a few seconds), leading to the formation of discrete plumes of salty water that would then descend toward the bottom of the tank. This is illustrated in Fig. 3, which shows two dye visualizations of descending plumes in the azimuth–height [(rθ, z) or (−x, z)] plane of the tank close to midradius for a case with a flat bottom, a rotation period of 80 s and buoyancy flux of approximately 1.7 × 10⁻⁷ m² s⁻³.

Fluorescein dye was introduced at the surface of the tank as the spray arm moved overhead above the field of view. This was then entrained into the plumes, which could be seen descending and spreading horizontally as they moved toward the bottom of the tank. It is noteworthy from such images (e.g., in Fig. 3b) that, although the convective mixing is initially most vigorous in the uppermost 25% of the depth of the tank (Fig. 3a), individual plumes actually reach the bottom of the tank, albeit with some azimuthal tilting, indicative of deep, penetrative convection.

The velocity field associated with these plumes was obtained via CIV analysis of sequences of these images, using the dispersion of neutrally buoyant particles to trace the flow. A typical example of such a velocity field is shown in the (x, z) plane at midradius in Fig. 4.

The velocity field is evidently dominated by irregular descending jets of fluid associated with each plume, on a horizontal scale of around 10 cm, again clearly traversing the entire depth of the tank along a sloping trajectory. The flow pattern appears to evolve rapidly, on time scales less than the rotation period of 80 s, thus representing a significantly ageostrophic, three-dimensional convective flow on these scales.

Although this convective flow would suggest the presence of a statically unstable stratification, in practice the conductivity probe always measured either a neutral or weakly stable density gradient. The vertical density gradient was typically found to evolve during the course of an experiment lasting a few hours, beginning with essentially neutral stratification and then gradually developing a weakly stable gradient in the lower part of the tank with a maximum value of ∂ρ/∂z of around 1–2 kg m⁻⁴. The upper part of the tank remained approximately neutrally stratified with no clear evidence of unstable stratification, which was probably confined to a thin layer near the top surface, though this could not be measured reliably.

These measurements would suggest a crude estimate of an upper limit on N ² of ∼5 × 10⁻³ s⁻², where

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ρ is the fluid density (with ρ₀ a reference value), and g is the acceleration due to gravity. This would also imply a corresponding lower limit on the internal deformation wavenumber,

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where k_d ≥ 13 rad m⁻¹ for cases I and III and k_d ≥ 6.5 rad m⁻¹ for case II.

b. Horizontal flow patterns

For each main experiment, the fluid in the tank was brought into solid-body rotation with the tank over a period of at least half a day prior to the start of the experiment, without any form of forcing. For the majority of experiments carried out, horizontal velocity fields were derived at 20-s intervals, beginning shortly after the initiation of convective forcing, and continued for as long as possible. In at least two cases, the flow continued to be monitored some time after the convective forcing was removed, thereby providing information on the decay of the turbulence field, due mainly to Ekman friction. The horizontal flow was typically measured at a level close to middepth (approximately 25 cm below the top surface in each case), although in at least two runs (with a flat bottom) a short sequence of measurements were made in which the horizontal flow field was sampled at a series of five different levels spanning the full depth of the tank.

1) Flat bottom flows

The first case we consider is for a flat bottom, in which the fabric sheet was loosened sufficiently to lie at the bottom of the tank. In this case there is only a weak topographic gradient across the tank, due solely to the centrifugal distortion of the free upper surface. In practice at a rotation period τ_r = 40 s this led to a mean slope in the top surface of around 0.65° and a corresponding value for β [cf. Eq. (1)] of ∼0.0036 m⁻¹ s⁻¹. Although this case corresponds most closely to an f plane, therefore, there is a weak residual β effect that should not be ignored. However, the effect is comparatively weak, and leads to characteristic length scales based on β [such as the Rhines scale

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where U_rms is a typical horizontal velocity scale based on the root-mean-square velocity fluctuations] that are comparable with the size of the tank. For run I, for example, L_R ≃ 2 m, indicating a Rhines wavelength of around 4 m.

Figure 5 shows a typical example of an equilibrated horizontal flow, taken from run I using the wide-angle camera. In this case, the flow is apparently dominated by a single complex anticyclonic gyre, around 2 m across in radius though perhaps more extended in azimuth beyond the field of view. In addition, there is a general tendency for broadly retrograde drift in the azimuthal direction toward the outer radius and prograde drift in the inner part of the tank. This trend is clearly seen in the azimuthal mean azimuthal velocity profile in the left frame of Fig. 5. This overall pattern was seen to persist for long periods of the experiment, with the anticyclonic gyre remaining almost stationary, though evolving erratically in shape and intensity.

It was subsequently noted that the location of the inclined prograde jet on the inner flank of the gyre (at around r = 0.5 m and x = 0) coincided with some folds and irregularities in the fabric sheet that led to some residual topographic ridges 1–2 cm in height. It was possible, therefore, that some weak topographic effects were present to render the main flow pattern nearly stationary relative to the bottom of the tank. The weak differential azimuthal drift with radius is also in the same sense as might be expected from the residual effects of wind stress or drag, produced by differential motion between the moving surface of the tank and the stationary air in the laboratory, though it was difficult to estimate this effect theoretically. Nevertheless, the influence of these experimental imperfections needs to be borne in mind in interpreting these results.

c. Vertical flow structure

Although the basic buoyancy-driven forcing of the flows was essentially baroclinic in character, various theoretical and modeling studies suggest (Salmon 1980) that the evolution of any upscale nonlinear energy cascade will lead to the excitation of large-scale flow components of an essentially barotropic character. Although time did not permit an extensive study of this aspect of the flow evolution, a few near-simultaneous measurements were made of the horizontal flow field at several levels in the vertical for one case with a flat bottom.

The flow was allowed to develop for several hours while buoyancy forcing was maintained at a constant value of around F₀ ≃ 4 × 10⁻⁸ m² s⁻³. Figure 9 shows results from the near-simultaneous measurement of horizontal velocity at five vertical levels at a particular instant, spanning almost the full depth of the tank. Figure 9a shows shaded contours of the vertically averaged flow, while Fig. 9b shows a map of the standard deviation of KE about that vertical mean. Thus, Fig. 9a may be regarded as representing the KE of the barotropic component of the flow, while Fig. 9b represents a measure of the baroclinic velocity field.

From these maps, it is clear that the vertically averaged (barotropic) flow contains much more kinetic energy than the standard deviation (baroclinic) components, by a factor of 10 in peak amplitude (note the difference in contour scale between Figs. 9a and 9b). In addition, the vertically averaged flow appears to be dominated by relatively large-scale structures, while the baroclinic standard deviation field is dominated by small-scale patches of KE, with a typical diameter of around 10–20 cm. Such a partitioning of scale and magnitude between the barotropic and baroclinic components of the flow is quite striking, and appears to demonstrate strongly the trends suggested from simple model studies in forming a strongly barotropic large-scale flow from a baroclinic (convective) energy input.

The typical scale of the baroclinic features presumably represents the horizontal size of individual convective plumes in the flow, with reference to Figs. 3 and 4 above. It is of interest that this scale is comparable to the scale l_rot, defined by

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and suggested by Fernando et al. (1991) and Marshall and Schott (1999) as a likely horizontal scale for rotationally dominated convective plumes. Given the value of natural Rossby number Ro* ∼3 × 10⁻³ and h ≃ 30–40 cm, this would be expected to lead to l_rot ∼10–20 cm, much as observed here in Fig. 9b. Moreover, the observed velocity fluctuations in u_θ and w on the scale of the plumes is O(1 cm s⁻¹), again consistent with the scalings suggested by Fernando et al. (1991) and Marshall and Schott (1999) of |u, w| ∼ Ωl_rot ∼ 1–2 cm s⁻¹.

a. Reynolds stress accelerations

Similar issues must surely arise in the present experiments, though in this case we have already established that the large-scale flow (including, presumably, the main eddies resolvable in our measurements) is predominantly barotropic. Given detailed, accurate, and representative measurements of horizontal velocity fields, well sampled in time and covering an area of the flow sufficient to obtain meaningful statistics, it is feasible directly to derive not only the pattern of horizontal Reynolds stress at each time step for which velocities are measured, but also the divergence of this Reynolds stress and hence the implied eddy-induced acceleration of the azimuthal mean zonal flow. In cylindrical geometry (see, e.g., Pfeffer et al. 1974; Read 1985), the full azimuthally averaged zonal momentum equation reduces to

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[cf. Andrews et al. 1987, their Eq. (3.3.3)], where u, υ, and w are now the azimuthal, radial, and vertical velocity components, respectively. Here F^(x) represents residual forces, for example, due to molecular viscosity and/or wind stresses. Term [2] represents the combined Coriolis and centrifugal acceleration due to zonal mean meridional flow, while [3] and [4] represent advection of zonal momentum by . Terms [5] and [6] represent accelerations due to the divergence of the Reynolds stresses. If the present experiment develops zonal jets via the same barotropic nonlinear zonation mechanism as is believed to be responsible for the development of zonal jets in the numerical simulation studies discussed above, the zonal acceleration [1] should be dominated by the effects of term [5] representing the divergence of the horizontal Reynolds stress. Moreover, this balance should be realized independently of height z. Thus, if the zonal flow is indeed a direct result of barotropic eddy–zonal flow interactions, it should be possible to compare independent measurements of both the horizontal Reynolds stress divergence and actual changes in the azimuthal mean flow at any height in the flow.

In Fig. 10 the results of such a comparison are shown, for the case III with the most highly developed pattern of zonal jets featured above. In Fig. 10a, shaded contours of the instantaneous divergence of the measured Reynolds stress are shown, sampled twice per rotation period for a total interval of around 1 h. By this time the forcing had been applied continuously for several hours and the flow seemed to have reached a reasonably equilibrated level of kinetic energy. In Fig. 10b, measurements are shown of the acceleration of the azimuthal mean zonal flow ∂u/∂t, obtained by differencing the profile of azimuthal mean flow in successive time steps using centered differences. Both fields appear to be quite noisy, with many patches of both positive and negative accelerations apparent in each field, apparently fluctuating back and forth and mostly concentrated in the outer half of the annular channel. Even so, there are some similarities apparent in the spatial variations of both fields, suggesting that they could be correlated.

Figure 11 shows the result of forming the correlation coefficient C(r, τ), computed as a function of radius and averaged over all time steps in the narrow-angle dataset for case III, between the two quantities plotted in Fig. 10. The definition of C is given by Eq. (8),

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where angle brackets denote the time average, the subscript t denotes time derivative, and div() = (1/r)∂r()/∂r, and allows for the possibility of a time lag in the response between the Reynolds stress divergence and the azimuthal mean zonal flow itself. The results show a clearly peaked response in correlation close to lag zero (τ = 0) and of magnitude up to 0.5, though with a tendency for the acceleration to respond slightly after the measured Reynolds stress event. There is a clear cutoff on the negative lag (time lead) side of the origin at all radii, with a more gradual decay in correlation with time lag in the positive lag direction. This is a clear signature that the horizontal Reynolds stress is strongly correlated with the measured zonal flow acceleration, and is entirely consistent with the hypothesis that horizontal Reynolds stress plays a major role in controlling and influencing subsequent changes in the behavior of the pattern of azimuthal mean flow. However, it is somewhat surprising that the zonal acceleration takes place following a short time delay. The explanation for this is not clear in the present work, though may indicate a finite time interval needed for disturbances detected at the midplane to propagate their influence throughout the water column.

The magnitude of the implied conversion of kinetic energy from eddies into the zonal flow, corresponding to −u/r∂()∂r integrated over the domain, is around 3.0 × 10⁻⁸ W kg⁻¹, which may be compared with the computed value for the rate of change of zonal kinetic energy ∂(u²/2)/∂t of 3.2 × 10⁻⁸ W kg⁻¹ for case III. Thus, the Reynolds stresses can account for more than 90% of the energy fluctuations in the zonal flow for this case. The missing remainder may represent the effects of measurement errors, but could also reflect other sources of zonal acceleration [e.g., due to vertical Reynolds stresses associated with baroclinic (convective) exchanges on small scales], and also the direct effects of wind stresses. The latter are almost certainly dominant for case I, for which Reynolds stresses are too weak to account for observed accelerations.

a. Spectral slopes and anisotropies

In the present series of experiments, the previous sections have indicated that the convectively driven turbulent flows result in the formation of jetlike, zonally elongated structures that appear to be the result of upscale nonlinear interactions, including strong and coherent interactions with the azimuthally symmetric component of the zonal flow. It is of interest, therefore, to examine whether the kinetic energy spectra reveal any similarities with those obtained in idealized numerical model studies.

Figure 12 shows kinetic energy spectra for each of the three main cases discussed above, illustrating the influence of increasing β on the equilibrated energy spectra. In the flat bottom experiment (Fig. 12a), the total and nonzonal spectra appear to follow closely a −5/3 slope over the full range of wavenumbers from around 5–50 rad m⁻¹, beyond which the spectrum steepens for k > 50. The zonal spectrum in fact follows a slope close to −8/3, broadly consistent with an isotropic energy spectrum, with no evidence of any steepening at the lowest wavenumbers accessible in these measurements. The amplitude of the total energy spectrum is approximately 40% below the expected KBK spectrum for C_K = 5 using the value of ϵ estimated from ϵ ≃ E_k/τ_E, though the reason for this is not clear.

Figure 12b shows the intermediate β case with the full sloping bottom but half the rotation rate Ω. In this case the nonzonal and total energy spectra have a somewhat steeper slope than −5/3 over the range 5 ≤ k ≤ 50 rad m⁻¹, though with an amplitude approaching that of the expected KBK spectrum around k ≤ 10 rad m⁻¹. In contrast to case I, however, the zonal spectrum exhibits a −8/3 slope only for k > 9 rad m⁻¹, becoming significantly steeper for lower values of k. This indicates that the kinetic energy distribution is starting to become somewhat anisotropic at low wavenumbers, at least qualitatively consistently with numerical studies on the β plane. It is of interest to note that the steepening of the zonal spectrum occurs around a wavenumber comparable to or somewhat greater than k_β (∼5.5 rad m⁻¹ for this flow). At the lowest wavenumbers, the nonzonal spectrum clearly flattens for k ≤ 5 rad m⁻¹. Spectra obtained from wide-angle images (not shown) indicate a flattening of the nonzonal spectrum around k ∼3 rad m⁻¹, indicating a value for k_fr of around 3 rad m⁻¹, which is comparable to the radial wavenumber of the zonal jets.

The strongest β case in Fig. 12c shows some even clearer signs of anisotropic structure, though evidence for the classical KBK −5/3 slope is not very clear. However, the total and nonzonal spectra become tangent to the KBK form around k = 10 rad m⁻¹, and in corresponding spectra obtained using the wide-angle camera (not shown) there is clearer evidence for a segment of the spectrum with a slope of −5/3 for ∼6 ≤ k ≤ 12 rad m⁻¹. The nonzonal and total energy spectra appear to rise to a peak at around k = 6 rad m⁻¹, flattening or even decreasing as k decreases below ∼6 rad m⁻¹, again suggesting a value for k_fr of around 6 rad m⁻¹, which, as in case II, corresponds closely to the radial wavenumber of the zonal jets. At smaller scales (k > 50 rad m⁻¹) the spectrum steepens, perhaps approaching the −3 slope characteristic of an enstrophy-cascading range in two-dimensional turbulence [sometimes known as Kraichnan’s −3 law; Kraichnan (1967)]. At these scales, however, corresponding to a horizontal wavelength of <10 cm, the flow is almost certainly dominated by three-dimensional baroclinic effects, and perhaps even viscous effects at the smallest scales. The azimuthal mean spectrum is significantly steeper than −8/3 for 6 < k < 15 rad m⁻¹, indicating fairly strongly anisotropic structure biased toward accumulation of energy in zonal modes at low wavenumber. We plot in Fig. 12c a straight line segment with the slope −5 and magnitude consistent with Eq. (10) for comparison with the measured zonal spectrum. In this case both the slope and the magnitude of the zonal spectrum are actually comparable with the theoretical spectrum suggested, for example, from the model studies of Huang et al. (2001) and Sukoriansky et al. (2002). Such a trend is somewhat evident for case II, though not at all for case I.

The role of wind stress in adding a directly driven component of the zonal flow in each case here may partly underlie why the effects of anisotropic inertial energy cascades are difficult to identify in those cases where β effects are relatively weak. It was mentioned above that wind stresses may be partly responsible for the presence of predominantly retrograde azimuthal flow near the outer boundary of the experiment. It is only in the case with the strongest β effect that a possible influence of the anisotropic energy cascade becomes apparent in the zonal energy spectrum, even though some weak indication of zonation in the azimuthal flow is apparent in the intermediate β case II.

b. Spectral transfer rates

The evolution of the anisotropic spectra discussed above is determined by the nature of the nonlinear interactions between various wavenumber components in the flow. Thus far we have concentrated on examining some of these interactions in the physical domain, by examining directly the energetic exchanges between the azimuthal mean (zonally symmetric) flow and the nonaxisymmetric eddies. However, most theoretical treatments of energy cascades consider exchanges within the spectral domain, by examining the transfer of energy from one region of the spectrum to another in wavenumber space. Anisotropy is then revealed in the channeling of energy in particular directions in wavenumber space.

Chekhlov et al. (1996), for example, computed a two-dimensional enstrophy transfer function in Fourier (wavenumber) space from a sequence of direct numerical simulations of β-plane turbulence. Starting with the enstrophy evolution equation,

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where Ξ(k, t) is the vorticity correlation function, ν is the kinematic viscosity of the fluid, k is the horizontal wave vector, and k its magnitude. The spectral enstrophy transfer, T_Ξ(k, t), is then given by

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where ζ(k, t) is the relative vorticity as a function of wavenumber k at time t, p and q are wave-vector variables of integration, [R] denotes the real part, and angle brackets denote the ensemble average. To compute a spectral energy transfer map in k space, a cutoff wavenumber k_c was introduced such that the energy injection into mode k is taken into account only via triads involving other modes with |k| > k_c. The k-dependent net time-averaged enstrophy transfer into mode k, T_Ξ(k|k_c), was then computed for each mode k such that |k| < k_c (the so-called explicit modes), by integrating Eq. (12) over all other modes with wave vectors p and q for which |p| and |q| > k_c (the “implicit” modes), replacing the ensemble average with the time-mean spectral covariance. The corresponding time-averaged energy transfer function T_E(k|k_c) can then be computed from

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Chekhlov et al. (1996) and Galperin et al. (2006) computed the integral in Eq. (12) for their run, designated “DNS3,” and this clearly showed a strong tendency for energy to accumulate into particular wavenumbers along the k_y axis, consistent with the formation of strong zonal jets with meridional wavenumbers <k_c.

In the present experiments, we have computed the same spectral energy transfer function in two dimensions, using the measured vorticity fields at middepth discussed above. Figure 13 shows two examples of such functions, computed from sequences of vorticity fields derived from images from the narrow-angle camera. Figure 13a shows the resulting kinetic energy transfer function for the flat bottom (case I), which indicates a general trend for upscale energy transfer toward the low wavenumber region around |k| = 5–10 rad m⁻¹. There is also apparently a weak tendency for the accumulation of kinetic energy into two peaks at k around (∓3, ±3) rad m⁻¹. This is consistent with the form of the velocity fields within this field of view shown above, with persistent diagonal flows across the domain in association with a near-stationary domain-filling eddy with a diameter ≥2 m.

For the sloping boundary, on the other hand, a clearer pattern is found to emerge. Figure 13b indicates a clear tendency for kinetic energy to accumulate from |k| > k_c into a roughly circular ring in wavenumber space with a radius of around 8–10 rad m⁻¹, but with clear maxima close to the k_y axis at k_y = ±8 rad m⁻¹. The maxima are roughly twice the strength of the isotropic ring in T_E(k|k_c), indicating a modest accumulation rate of kinetic energy into zonally elongated structures with a wavelength of a little less than 1 m. Such a wavelength is roughly consistent with the development of the zonal jets described above, though indicates a much weaker tendency for strong jet formation than in the much clearer example presented in the numerical model simulations of Chekhlov et al. (1996) and Galperin et al. (2006).

a. Rayleigh–Kuo stability

Figure 14 shows radial profiles of the time and azimuthal mean azimuthal velocity and corresponding potential vorticity, averaged over approximately 1 h of the late phase of the experiment. From these profiles, it is clear that there are some zonal jetlike structures that survive the time averaging, but the potential vorticity profile associated with this profile of u is evidently monotonic in radius. Such a result would suggest that the spatiotemporal mean azimuthal flow is consistent with a state of neutral stability with respect to the Rayleigh–Kuo criterion for barotropic stability.

On shorter time scales, however, the stability status of the azimuthal flow profile is less clear. Figures 15 and 16 show the equivalent profiles of u(r) and q(r) for an example of an instantaneous flow (Fig. 15) and a shorter-term time average (this time over just 300 s; Fig. 16). In the first case, the instantaneous profile of u shows a stronger and more complex array of zonal jets in association with a q profile that is clearly nonmonotonic. Indeed, ∂q/∂r clearly changes sign in Fig. 15b several times, indicating positive violations of the Rayleigh–Kuo stability criterion consistent with active barotropic instability. Sharp positive gradients in q occur in association with prograde peaks in u, while weak or reversed gradients in q are found associated with retrograde azimuthal jets. This is much as found in idealized models of active Rossby wave critical layers (see, e.g., Haynes 1989). Similar traits are also found in the PV structure of Jupiter’s atmosphere (Read et al. 2006) in association with the cloud-level zonal jets.

If the profiles are averaged over an intermediate time scale, however, the violations of the Rayleigh–Kuo stability criterion are less marked but still evident, with clear indications of a PV “staircase” with radius. Figures 16a,b show profiles averaged over an interval of just 300 s around the time of the instantaneous flow in Figs. 15a,b. This would seem to indicate a tendency for persistent barotropic instability on time scales of a few hundred seconds, beyond which the instability is able to eliminate the regions of reversed ∂q/∂r in the PV “hyperstaircase” found on short time scales. Such a process would constitute a form of “barotropic adjustment,” in which the time-averaged flow on long time scales approaches a state of near-neutral stability with regard to PV gradients. The mechanism for this is almost certainly a manifestation of a form of Rossby wave breaking, in which PV gradients become temporarily reversed as waves reach a large enough amplitude to overturn.

a. Zonostrophic regime

Galperin and coworkers (Sukoriansky et al. 2002; Galperin et al. 2006) have attempted to define such a regime, dominated by the nonlinear generation of zonal jets (termed the “zonostrophic” regime), based on extensive studies using idealized barotropic numerical model simulations, as one in which (a) the flow is forced on relatively small scales (compared with those of the physical domain); (b) an inertial range exists over a large enough range of scales such that the β effect can exert sufficient influence on the flow to allow anisotropic interactions to develop fully; and (c) large-scale “friction” (i.e., removal of energy at large scales) is sufficiently small to permit (b), yet large enough to prevent the accumulation of energy in the largest (domain filling) modes of the system.

The existence of such a “universal” regime has been disputed, for example, by Danilov and Gurarie (2002), though subsequently Galperin et al. (2006) and Sukoriansky et al. (2007) have claimed that the cases considered by Danilov and Gurarie (2002) did not lie in the appropriate region of parameter space. Galperin et al. (2006) proposed a series of chain inequalities as a means of formulating a set of criteria for identifying the range of parameters over which such a regime might occur. In the present context, these can be expressed in the form

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where k_ξ is a wavenumber (rad m⁻¹) characteristic of the scales of energy input/forcing, k_β is defined as

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(where ϵ is the rate of input of kinetic energy into the upscale cascade by the forcing, m² s⁻³) and represents the scale at which the Rossby wave period is comparable with the eddy overturning time scale, k_fr is the “frictional” wavenumber determined by the balance between upscale energy transfer and energy removal at large scales, and L represents the physical horizontal scale of the domain (in the present case the radius of the apparatus L = 6.5 m). Given sufficient time to evolve, flows that satisfy Eq. (15) are expected to develop self-similar energy spectra that adopt the universal forms given in Eq. (9) (over most angles ϕ) and Eq. (10) (close to ϕ = ±π/2) over discernible regions of their spectra. Under such circumstances, the zonal component of the energy spectrum [given by Eq. (10)] can be integrated over wavenumber and shown (e.g., Galperin et al. 2001, 2006) to lead to k_fr ≃ k_R to within a factor O(1), where

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is the Rhines wavenumber [cf. Eq. (5)]. Thus, in this regime k_R and k_fr are more or less interchangeable as estimates of the meridional scale favored by the flow for the emergence of banded zonal jets. Most recently, Sukoriansky et al. (2007) have further elucidated this relationship between k_fr and k_R, noting that forced-dissipative flows for which k_β ≥ 2k_R will exhibit most of the salient properties of the zonostrophic regime, while those for which k_β ≤ 1.5k_R will lie in the so-called frictionally dominated regime and will not exhibit significant zonostrophic inertial ranges.

Table 2 presents estimates of the main parameters invoked in Eq. (15), derived from the experimental parameters noted in Table 1 and the apparatus dimensions, k_ξ is estimated as 2π/l_rot, where l_rot is computed using Eq. (6), and k_β is obtained from Eq. (16). Note that the estimates of ϵ used here were obtained as ϵ ≃ U²_rms/2τ_E, where τ_E is the Ekman spindown time scale (=h/νΩ) (see Table 1). This typically results in a value for ϵ around 3%–15% of the input buoyancy flux F_B, which is not unreasonable since it is likely that only a small fraction of the energy input associated with F_B will actually be utilized in the nonlinear inverse energy cascade [cf. the estimates used by Galperin et al. (2006) for the oceans and gas giant planets]. Here k_fr represents the dominant wavenumber at which large-scale friction exerts a strong influence on the energy spectrum, and is estimated from the low wavenumber peak or break in the total or zonal wavenumber spectrum for each case.

From these estimates, it is apparent that, although each experiment satisfies the first inequality in Eq. (15), cases I and II do not rigorously satisfy either of the remaining two criteria for the zonostrophic regime. Case III, however, with the strongest β effect, comes the closest to fulfilling all of the criteria, with k_β ≃ 2k_R. For case II, our estimates of k_β and k_R are probably too close together to allow much of an inertial range to develop between them, placing it on the margins of the zonostropic and frictionally dominated regimes. As discussed further by Galperin et al. (2006), this section of the spectrum is critical for the emergence of the fully formed k⁻⁵ spectrum in the zonal flow, and the separation of k_β and k_fr or k_R is much greater for the outer gas giant planets than could be obtained in our experiments here. This may well help to account for why our banded zonal jets are much less dominant in the laboratory than are typical in the atmospheres of Jupiter and Saturn. Even so, the flows obtained in case III do show a number of features suggestive of the zonostrophic regime, indicating that we were able to enter at least the margins of this class of dynamical behavior. Galperin et al. (2006) note that the oceans also do not quite satisfy the criteria on k_β/k_fr set out in Eq. (15), though they fulfill them somewhat more clearly than in our experiment III.

Also included in Table 2 are estimates of wavenumber scales based on both the internal and external deformation radii. In Eq. (4) k_d is defined, though only a lower limit can be estimated because N is poorly defined. The external deformation scale is estimated from k_dext ∼ f /gh, though this wavenumber turns out to be much less than the smallest wavenumber that can be accommodated in the annular domain. These estimates suggest that the external deformation radius does not play a role in the dynamics of this experiment, but internal baroclinic disturbances could be significant. However, there is no evidence of scales of motion that manifest themselves as different from those represented by k_β, k_R, or k_ξ.

b. Scales and spectral transfers

Insofar as our experiment III does lie parametrically in the zonostrophic regime, therefore, the diagnostics computed above help to reveal the dynamical mechanisms at work in sustaining a zonally banded circulation. We have shown that, provided β is strong enough, a flow excited by stirring at small scales will evolve a large-scale barotropic zonation, accompanied by an ensemble of barotropic vortical structures, many of which propagate zonally, at least qualitatively, like Rossby waves. The formation of the zonation in azimuthal flow appears to be largely as a result of direct eddy–zonal flow interactions, with a clear tendency for kinetic energy to accumulate upscale into the form of zonally elongated structures or jets. Such a tendency is at least consistent with the hypothesis of Balk (2005) that such zonally elongated structures will be favored in flows constrained simultaneously to conserve the three dynamical invariants: energy, enstrophy and the geometric invariant noted by Balk (1991) for Rossby wave systems. It would be of interest in future work to investigate further the latter invariant in actual diagnostics of actual flows and numerical simulations, and to examine in greater depth the validity of this hypothesis.

The scale favored by the zonation process is a significant factor in determining the effect of zonostrophic anisotropy on flow structure. The Rhines wavenumber, k_R [as defined in Eq. (17)] has often been suggested as representing the likely scale of zonal jet formation. As originally noted by Hide (1966), such a scale represents a rough balance between relative and planetary vorticity advection, though this may not be the most useful balance to consider. Galperin et al. (2001) put forward an alternative interpretation, relating the favored jet scale, represented by k_fr, to a balance between upscale energy transfer and large-scale energy removal. Both of these are effectively internal parameters, dependent on flow-dependent quantities, so are not particularly useful as predictive quantities. But they appear to be broadly consistent with the observed scale of jets found in the present experiments. However, our experiments II and III indicate in this case that k_fr ≃ 1.5k_R, which is not quite as suggested by Galperin et al. (2001) and Galperin et al. (2006) for the zonostrophic regime, though is close to the form suggested by Sukoriansky et al. (2007). In both cases II and III the scale k_fr corresponds quite closely to the radial wavenumber of the observed zonal jets, indicating that the scale separating adjacent zonal jets is reasonably well estimated by ∼1.5k_R. The Rhines scale k_R also seems to be quite closely consistent with observed flows in the outer planets (e.g., Vasavada and Showman 2005) and the oceans (Richards et al. 2006), which satisfy Eq. (15) even more closely.

An important aspect of the near-equivalence of k_fr with k_R, asserted by Galperin et al. (2001), Galperin et al. (2006), and Sukoriansky et al. (2007), is the assumption that energy removal takes place in a relatively scale-independent manner. This is consistent, for example, with Ekman damping, such as would apply to our present experiments, and in the earth’s oceans. The principal mechanism for energy removal on large scales on the gas giant planets, however, is not clear at the present time. None of the gas giants is believed to possess a solid underlying surface, so the relevance of Ekman damping seems unlikely. But presumably there must be some means of removing barotropic kinetic energy from the flow on large scales that, given the apparent consistency of the observed jets with the Rhines wavenumber, must act comparably on all scales. This is an aspect of the energy flow within the atmospheric circulation of these planets that deserves closer attention in future work.

c. Convective forcing

The method of forcing used herein seems to work well as a “natural” means of injecting energy at small scales, though the use of salt-driven convection does pose significant practical difficulties in sustaining a uniform and steady input of energy over long periods of time. In future experiments, it would be useful to explore the possibility of driving convection via imposed thermal contrasts, although providing sufficient heating and cooling (on the order of 10–100 W m⁻² on the scale of the Coriolis facility) would be a major undertaking. The imposition of unstable stratification has clear parallels with what may be happening in the atmospheres of the gas giant planets; though in their case, the role of moist convective processes needs to be taken into account (Gierasch et al. 2000; Ingersoll et al. 2000). The latter probably concentrates intense convection into small and highly intermittent regions though would certainly operate on a small scale compared with that of the planet itself. But an overall upwelling energy input of around 5–6 W m⁻² from the interior heat source on Jupiter (e.g., Vasavada and Showman 2005), coupled with a depth for convection of at least 50 km, would suggest an average natural Rossby number Ro* of 0.1 or less, placing it in a qualitatively similar convective regime to the present experiments.

Similar types of deep, rotationally controlled convective forcing may also occur in the oceans (Marshall and Schott 1999), though only in very restricted regions such as the polar oceans and the Mediterranean. More commonly, it is baroclinic instability on horizontal scales of 50–100 km that provides the dominant source of “stirring” in the wider ocean basins, and may be responsible for the energy input for the putative small-scale zonal jets now being found in ocean models and observations (Galperin et al. 2004; Maximenko et al. 2005; Richards et al. 2006). This was the original concept for exciting geostrophic turbulence by Salmon (1980), and there is certainly some evidence for zonation in small-scale laboratory experiments on baroclinic instability using sloping endwalls (Mason 1975; Bastin and Read 1998), which would support the notion that the zonostrophic regime does not depend strongly on the mechanism of small-scale energy injection. But laboratory experiments to date on baroclinic instability with sloping endwalls have been carried out only on a relatively small scale, and almost certainly have not satisfied the criteria expressed in Eq. (15). It would be of significant interest to investigate this further on the scale of the Coriolis facility.

d. Potential vorticity

The potential vorticity structure of the equilibrated banded zonal flow shows some clear signatures of intense interactions with eddies. Although the net effect of such interactions is to accelerate the zonal flow and to modify its structure, especially in the vicinity of Rossby critical layers that develop spontaneously as the flow evolves, the resulting instantaneous zonal flow ends up in a state that violates many of the “classical” criteria for stability. Instantaneous velocity fields show evidence for smoothing out and even reversals of the radial PV gradient, indicative of local Rossby wave breaking. Longer time averages, however, are much closer to a state of neutral stability. This would seem to suggest that barotropic instability acts as an adjustment mechanism on long enough time scales in these experiments, such that instantaneous flows are indeed unstable but adjust the PV distribution over time scales of a few hundred τ_r toward marginal stability. This would appear to be in contrast with results from unforced numerical model simulations, which seem to produce jets that are much closer to neutral stability, even for instantaneous fields. The continuous forcing must therefore play a major role in perturbing the zonal flow away from this marginal state, by an amount that presumably depends on the strength of forcing.

It has long been noted that the westward zonal jets on the outer planets tend to violate the classical Rayleigh–Kuo/Charney–Stern instability criteria (Vasavada and Showman 2005; Read et al. 2006), though they may be closer to neutrality with respect to Arnold’s second stability theorem (Dowling 1993, 1995). This may again be a signature of the effect of the processes forcing the jets on Jupiter and Saturn, whose origins are still poorly understood in detail, though may lead to Rossby wave breaking as energy cascades upscale from the smaller-scale convective forcing. Further studies of the relationship between eddy forcing and stability criteria in experiments and models may be a useful source of further information on these mechanisms for driving zonal flows on the outer planets. Finally, we remark that the extent to which the small-scale jets found in ocean observations and models violate the classical instability criteria has not yet been investigated in detail, but could form an important means of discriminating between zonostrophic mechanisms for jet formation and alternative mechanisms (e.g., see Berloff 2005).