Scaling of high-wavenumber energy spectra in the unit aspect-ratio rotating Boussinesq system

The Boussinesq approximation for hydrodynamic flow is the cornerstone for many applications in which a suitable model for scalar transport is needed. It is physically realistic in that it postulates that the fluctuations of the scalar are small compared to its background value and is computationally tenable for linear stratification profiles and simple boundary conditions. The Boussinesq approximation in the rotating frame forms the parent system for the derivation of further simplifications such as the shallow-water, hydrostatic and quasi-geostrophic (QG) models which are standard approximations in geophysical regimes which include rotation and stratification of the flow.

The Boussinesq system for a stably stratified flow in a rotating frame is given by:

where $\bm{u}$ is the velocity, $w$ is its vertical component, $p$ is the effective pressure and $\cal F$ is an external input or force. The total density is given by ${\cal\rho}_{T}(\bm{x})=\rho_{0}-bz+\rho(\bm{x}),$ such that $\quad|\rho|\ll|bz|\ll\rho_{0}$ where $\rho_{0}$ is the constant background, $b$ is constant and larger than zero for stable stratification in the vertical $z$-coordinate, $\rho$ is the density fluctuation. The density in normalized to $\displaystyle\theta=\rho({g}/{b\rho_{0}})^{1/2}$ which has the dimensions of velocity. The Coriolis parameter $f=2\Omega$ where $\Omega$ is the constant rotation rate about the $z$-axis, the Brunt-Väisälä frequency $\displaystyle N=({gb}/{\rho_{0}})^{1/2}$, $\nu=\mu/\rho_{0}$ is the kinematic viscosity and $\kappa$ is the mass diffusivity coefficient. We assume periodic or infinite boundary conditions with Prandtl number $Pr=\nu/\kappa\sim{\cal O}(1)$. The relevant non-dimensional parameters for this system are the Rossby number $Ro=f_{nl}/{f}$ and the Froude number $Fr=f_{nl}/{N}$, where $f_{nl}=(\epsilon_{f}k_{f}^{2})^{1/3}$ is the non-linear frequency given input rate of energy $\epsilon_{f}$ Smith & Waleffe (2002). Thus $Ro$ and $Fr$ are the ratios of rotation and stratification timescales respectively to the nonlinear timescale.

There are three overlapping ideas from turbulence theory which we will exploit in this analysis of the Boussinesq equations. First from Kolmogorov (1941) the notion of a universal range of scales governed by the dynamics of the flux of a conserved quantity. Second, the analysis of Kraichnan (1971) on the inertial ranges of two-dimensional flows based on the constraint imposed by the joint conservation of both energy and enstrophy. Kraichnan thus deduced the $k^{-3}$ scaling, up to logarithmic correction, of the energy in wavenumbers larger than the forcing wavenumber $k_{f}$ and $k^{-5/3}$ scaling for wavenumbers smaller than $k_{f}$. Finally Charney (1971) used Kraichnan’s ideas to show that similar constraints are placed on the quasi-geostrophic system due to the joint conservation of potential enstrophy and energy. Charney showed that in the quasi-geostrophic system, energy must flow upscale and potential enstrophy downscale, with predictable scaling exponents of the energy spectrum in the two ranges.

The general principle then, is that for a given (inviscidly) conserved quantity in a system there is a range of scales over which the flux of that quantity is a constant. The larger the system and the more separation there is between the source (forcing) and the sink (dissipation, boundaries), the more extended this universal ’inertial’ range of scales becomes. Once this is established, it is fairly standard procedure to deduce phenomenologically what the distribution of such quantities must be in spectral space.

In the inviscid and unforced (${\cal F}=\nu=\kappa=0$, ), Boussinesq system the potential vorticity $q$ is a local (Lagrangian) invariant, as is its square the potential enstrophy $Q$, while the total energy is a global invariant. That is,

At any point $\bm{x}$, $E=\frac{1}{2}|\bm{u}|^{2}$ is the kinetic energy, $P=\frac{1}{2}\theta^{2}$ is the potential energy of the density fluctuations. The absolute vorticity $\bm{\omega}_{a}=\bm{\omega}+f\hat{\bm{z}}$ and the relative (or local) vorticity $\bm{\omega}=\nabla\times\bm{u}$. PV may be written in terms of $\theta$ as

The constant part $fN$ does not participate in the dynamics and we will therefore neglect it from now on. In what follows we will assume that $\nu\rightarrow 0$ and $\kappa\rightarrow 0$ such that Prandtl number $Pr=\nu/\kappa=1$, and the force $\cal{F}$ is confined to the lowest modes. Thus we assume a conventional ‘inertial-range’ of turbulent scales wherein the transfer of conserved quantities dominates over both their dissipation and forcing. Note that the global potential vorticity is zero for unbounded flows. While the potential enstrophy is also Lagrangian invariant, we will use its global conservation properties in our analysis.

This set of invariants is analogous to those for 2-d turbulence where the vorticity is Lagrangian invariant and energy and enstrophy are globally conserved. It therefore seems reasonable to expect that an analysis similar to that of Kraichnan for 2D turbulence would yield similarly useful results. The raw Boussinesq system does not permit an exact relationship between energy and potential enstrophy as in the case of 2d turbulence. One notices that the potential enstrophy is, in general, a quartic quantity, while the energy is quadratic. However, when $f$ and $N$ are sufficiently large, PV linearizes in the dynamical variables and reduces in fourier space to:

where $\tilde{\cdot}$ denotes fourier coefficients, the total wavevector $\bm{k}$ = $\bm{k}_{h}+k_{z}\hat{\bm{z}}$, the horizontal wavevector component has length $k_{h}=({k_{x}^{2}+k_{y}^{2}})^{1/2}$, the vertical wavevector component has length $k_{z}$ and $\bm{u}_{h}$ is the horizontal velocity vector with magnitude $u_{h}=(u_{x}^{2}+u_{y}^{2})^{1/2}$. We assume that the vertical velocity $w=u_{z}\sim 0$ in the leading order, consistent with empirical observations, and use incompressibility to obtain the last equality of Eq. (5). We will next show how in the formal limits where PV is linear, asymptotically exact relationships can be extracted between potential enstrophy and components of the total energy.

First we briefly review the theoretical results that have been obtained in the limits in which PV linearizes. For strong rotation and stratification, Embid & Majda (1998) and Babin et al. (2000) showed that the limit as $Ro\sim Fr=\epsilon\rightarrow 0$ is QG after the inertia gravity waves are averaged out. The scalings predicted in Charney (1971) are recovered, namely $k^{-3}$ scaling of the energy for $k_{f}\ll k\ll k_{d}$, and $k^{-5/3}$ scaling for $k<k_{f}$ where $k_{f}$ is the forcing wavenumber and $k_{d}$ is the typical dissipation wavenumber. Embid & Majda (1998) also showed that the limiting dynamics for $Fr\rightarrow 0$ with finite $Ro$ includes the so- called vertically sheared horizontal flows (VSHF). In both these cases, one can obtain a set of closed reduced equations for the linear PV dynamics from which all other quantities of interest can be derived. There is as yet no equivalent rigorous theoretical result for the unit-aspect ratio strongly rotating $Ro\rightarrow 0$ case with finite $Fr$, since in that limit the scalar equation 2, which is independent of $Ro$, vanishes in the leading order. This is a key reason why a (closed) reduced QG-like set of equations for this limit in the unit aspect-ratio domain has not been derived. Julien et al. (2006) showed that if an additional small parameter is introduced, namely the aspect-ratio of the domain, then reduced equations are possible for the vanishing $Ro$ limit as well.

In recent theoretical work we predicted that the potential enstrophy associated with the Boussinesq rotating system exhibits universal statistical properties in the so-called ‘inertial range’ of scales (Kurien et al. (2006)) in physical space for the same limiting regimes of the Rossby and Froude numbers described above. Formally, keeping lowest-order terms in a perturbation expansion in powers of $\epsilon\rightarrow 0$, universal statistics are found when the potential vorticity is linear in the dynamical variables $\bm{u}$ and $\theta$. In those cases, the associated von Kármán-Howarth equation von Kármán & Howarth (1938) for the two-point correlation of $q$ yields a simple scaling law for the third-order velocity-potential-vorticity correlation

where $\bm{r}$ is a physical separation scale, subscript $L$ denotes the component along $\bm{r}$, and $\varepsilon_{Q}$ is the mean dissipation rate of potential enstrophy $Q$. The $2/3$-law (6) assumes that the two-point statistics of potential vorticity have a universal small-scale statistically isotropic component for statistically steady state flows. A related result was subsequently derived for the strict QG case (beginning with the evolution equation for $q_{QG}$) Lindborg (2007) relating the third-order moments of potential vorticity difference and horizontal velocity difference across scale $r$ to potential enstrophy flux, assuming axisymmetry of the small-scales, and restricted to components of the velocity that are in the horizontal plane (given that vertical velocity is strictly zero for pure QG). The numerical experiments to verify 6 are currently underway and will be presented in a separate work.

There have been key numerical studies of the theoretical limits described above of which we here review the most relevant to our present study. The limit of strong rotation and stratification including the case $f=N$ was examined by Bartello (1995) in a paper which studied geostrophic adjustment as a process in which wave-vortical-wave interactions acted as catalyst for highly efficient downscale transfer of energy downscale. Smith & Waleffe (2002) demonstrated the generation of slow large scales in strongly stratified turbulence with small scale forcing and the dominance of the $k_{h}=0$ (VSHF) wave modes in the low wavenumbers. Waite & Bartello (2004) and Lindborg (2006) explored the effect of vortical forcing in stratified turbulence without rotation, thus highlighting the difference that 2D vs. 3D forcing can make in the evolution of the spectra. Sukhatme & L.M. (2008) showed that fairly small departures from $f/N=1$ can have measurable effect on the scaling of the spectra for moderately low-resolution simulations. Waite & Bartello (2006a, b) and demonstrated the transition to QG from stratified turbulence as $Ro$ is decreased for fixed small $Fr$. Remmel et al. (2009) proposed a novel non-perturbative model to add non-QG effects to a flow and demonstrate the importance of resolving the small scales (waves) in order to obtain even gross large-scale features such as the energy growth accurately.

We will here study rotating Boussinesq turbulence both heuristically and using data from high-resolution simulations for $Ro$ and $Fr$ chosen such that the potential vorticity is nearly linear, and potential enstrophy is nearly quadratic. In our approach we retain the type of physically sound arguments used by Kolmogorov, Kraichnan and Charney, namely a focus on conserved quantities (in the inviscid, non-diffusive system), existence of inertial ranges where the downscale flux of a conserved quantity (in our case potential enstrophy) governs the dynamics and consequent spectral scaling of other related quantities (energy), and the notion of a statistically universal range of small scales.

We begin by assuming that velocity fluctuations and density fluctuations are about the same order. We can construct spectral relationships between the quadratic potential enstrophy $Q(k)=\displaystyle\frac{1}{2}|\tilde{q}(k)|^{2}$ and either the horizontal kinetic energy $E_{h}(k)=\displaystyle\frac{1}{2}|\tilde{u_{h}}(k)|^{2}$ or the potential energy $P(k)=\displaystyle\frac{1}{2}|\tilde{\theta}(k)|^{2}$ based on the parameter $\Gamma=\displaystyle\frac{fk_{z}}{Nk_{h}}$, which by definition resembles a local Burger number in spectra space. From Eq. 5, for $\Gamma\ll 1$,

Eq. (9) is interpreted to mean that for wavevectors with large $k_{h}$, that is the ‘wide’ wavevectors, the potential enstrophy $Q(k)$ dominates the downscale (horizontal scales smaller than $1/\kappa_{h}$) dynamics and consequently suppresses the transfer of horizontal kinetic energy into those modes. This is analogous to the relation in 2d turbulence

wherein the enstrophy $\Omega(k)$ suppresses the downscale transfer of kinetic energy $E(k)$ for sufficiently high wavenumber $\kappa$, and in turn forces the inverse cascade of energy.

In the opposite limit $\Gamma\gg 1$ one can show that,

which is interpreted to mean that potential enstrophy suppresses the transfer of potential energy $P(k)$ into the wavevectors with large $k_{z}$, that is the ‘tall’ wavemodes. In physical space this may be loosely interpreted to mean that potential energy transfer into the small vertical scales is vertical scales.

More generally, for fixed $f/N$ there is a transition regime when $k_{h}/k_{z}\simeq f/N$ if the wavenumber regime is large enough. Thus, in an idealized infinite wavenumber flow, for fixed $f/N$ there will be some regime of wavevectors such that $k_{h}/k_{z}\gg f/N$ for which $\Gamma\ll 1$ and horizontal kinetic energy is suppressed in $k_{h}$ according to Eq. 16, and a regime of wavevectors such that $k_{h}/k_{z}\ll f/N$ for which $\Gamma\gg 1$ and potential energy is suppressed according to Eq. 17. The transition regime is defined by a cone in wavevector space with angle $\theta$ to the vertical such that $\tan\theta=f/N$. All wavenumbers inside the cone would behave according to $\Gamma\gg 1$ and all those outside would behave according to $\Gamma\ll 1$. In our simulations in finite sized boxes, we study three extreme cases: $f/N\ll 1$ where the cone is extremely narrow (small angle) so that most wavevectors lie outside the cone giving $\Gamma\ll 1$, $f/N\gg 1$ where the cone is very broad (large angle) so that most wavevectors lie inside the cone and $\Gamma\gg 1$ and finally $f/N=1$ where the cone-angle is 45$\deg$ to the vertical and both $\Gamma\ll 1$ and $\Gamma\gg 1$ regimes exist.

There are several noteworthy differences between the relations Eq. (9) and (13) derived above and the 2d relation Eq. (10). The latter relates two (inviscidly) conserved quantities in spectral space, while the former relates the spectrum of a conserved quantity ($Q(k)$) with that of one ($E_{h}(k)$) that is not conserved in general. The 2d turbulence constraint forces enstrophy downscale and energy upscale. In rotating Boussinesq, for $\Gamma\ll 1$, when the horizontal kinetic energy is suppressed in the large $k_{h}$, there is no corresponding constraint in $k_{z}$ for either the kinetic or potential energies; conversely for $\Gamma\gg 1$ which potential energy is suppressed in $k_{z}$, there is no corresponding constraint in $k_{h}$. Therefore there is always the possibility of a downscale transfer of energy in the unconstrained wavevector component. That is, a dual cascade of potential enstrophy and energy is possible downscale. In addition, Eq. (10) is isotropic in wavenumber in that it depends on the total spherical wavenumber while our new relations Eqs. (9) and (13) are highly anisotropic in wavenumber. As we will show from the data, the spectra indeed indicate that energy is transferred downscale in all cases albeit in a highly anisotropic manner.

We perform pseudo-spectral calculations of the Boussinesq equations with rotation on grids of $640^{3}$ points and $1024^{3}$ points in unit aspect-ratio domains. The computational domain has length $L=1$ to a side, with wavenumbers in integer multiples of $2\pi$. The time-stepping is 4th-order Runge-Kutta with resolution of the smallest wave frequencies with five timesteps per wave period. In the simulations at $640^{3}$, the diffusion of both momentum and density (scalar) is modeled by hyperviscosity of laplacian to the 8th-power in order to extend the inertial scaling ranges. The Reynolds number in these cases is not computed explicitly but the hyperviscosity coefficient is chosen to resolve the total energy in the largest shell Chasnov (1994); Smith & Waleffe (2002) thus ensuring adequately turbulent flow for a given resolution. The simulations at $1024^{3}$ were run with regular Navier-Stokes viscosity. The energy input rate $\epsilon_{f}=0.5$ and $1$ respectively for the smaller and larger resolutions, and $Ro$ and $Fr$ are varied by varying the rotation and stratification rates for fixed energy input rate which determines the nonlinear timescale. The forcing is incompressible and equipartitioned between the three velocity components and $\theta$. This is equivalent to forcing the two wavemodes and the one vortical modes equally, thus providing an optimally unbiased calculation. The forcing is also stochastic and the phases of each mode change sufficiently rapidly that the forced modes are statistically isotropic. The forcing spectrum is peaked at $k_{f}=4\pm 1$, for large scale forcing. The simulations are dealiased according to the isotropic two-thirds dealiasing rule. Lower resolution runs at $256^{3}$ and $512^{3}$ (see Kurien et al. (2008)) were also performed in earlier studies, but are not reported here.

We seek to verify our predictions for three particular “extreme” regimes in $Ro$, $Fr$ (see Table 1). The first is strongly stratified with moderate rotation, $f/N\ll 1$ (runs 1 and 4); the second regime has strongly rotating with moderate stratification, $f/N\gg 1$ (runs 2 and 5); and finally the regime which fixes $f=N$ large, (run 3) so that the dependence of potential vorticity on the ratio $\Gamma$ is reduced to a dependence on the cotangent $\tau=k_{z}/k_{h}$ alone, of the angle of the wavevector with respect to the vertical. Presumably, for sufficiently long inertial ranges any combination of $f$, $N$, $k_{h}$ and $k_{z}$ such that $\Gamma\gg$ or $\ll 1$, while maintaining near-linear PV, may be similarly investigated. As discussed in Bartello (1995), strict potential enstrophy conservation is not possible in a truncated inviscid spectral scheme due to the lack of conservation of Lagrangian invariants by individual triads. The best we can expect with this method in the viscous case is that truncation errors are swamped by viscosity thus giving conserved fluxes in the inertial range. We mention this fact because we will be using the notion of potential enstrophy conservation in this study. The spectral method itself is routinely used to compute flows very similar to those we study here and and we proceed while remaining aware of the caveats.

For each case in the Table, we will present the calculations to check specific predictions of the phenomenology proposed, namely the anisotropic suppression of horizontal kinetic energy and potential energy in the different $\Gamma$ regimes, and the scaling exponents predicted in these regimes by Eqs. (16) and (17). The spectra as a function of $(k_{h},k_{z})$ are defined as follows:

where $a$ is any of $\tilde{q}$, $\tilde{u}_{h}$ or $\tilde{\theta}$. We will present the horizontal kinetic energy and potential energy spectra (a) as a function of $k_{h}$ for various values of fixed $k_{z}$, (b) as a function of $k_{h}$, summed over $k_{z}$, (c) as a function of $k_{z}$ for various values of fixed $k_{h}$, and (d) as a function of $k_{z}$, summed over $k_{h}$. The (a)-(d) above correspond to subplots (a)-(d) in the sets of figures to follow. In each subplot, horizontal kinetic energy is plotted on top and potential energy on the bottom. We will also show the more conventional decomposition of the energy in contributions from the linear eigenmodes of the Boussinesq system. There are two inertial-gravity wave-eigenmodes with non-zero frequency and zero PV which contribute to the wave (or ageostrophic) energy spectrum. The remaining eigenmode has zero frequency and linear PV and contributes to the vortical (or geostrophic) energy spectrum. We will attempt to relate the mode-angle representation to the usual wave-vortical decomposition. Since this paper is largely an exploration of spectral scaling in the large wavenumbers, we will, for clarity’s sake, use the hyperviscous simulations data with their extended ranges to compute scaling exponents. The Navier-Stokes cases (runs 4,5) show no appreciable differences from their hyperviscous counterparts except for shortened scaling ranges, as is to be expected. Wherever appropriate, and when it does not result in a proliferation of figures, we will also present the data from the simulations with Navier-Stokes viscosity.

First we show in Fig. 1 that in each of the three cases, the global potential enstrophy (solid line) is captured by one half of the square of the corresponding linear PV, to within 3% or better agreement. potential enstrophy, after a period of spin-up and over-shoot, settles to a constant, indicating statistically steady state for this quantity, the mean rate of input of potential enstrophy balancing the mean rate of dissipation. We are thus able to stablish the time-frame over which the small scales, governed by potential enstrophy dynamics, have achieved statistically steady state. The small scale spectra of energy that we are interested in, achieve a statistically steady state. The total energy meanwhile (not shown) continues to grow since it can transfer upscale, where in our case, there is no sink to drain the energy. The calculation of all statistical quantities in what follows occurs over the time-frame where the mean potential enstrophy is constant, $3.5<t<6$, over about 25 frames.

To give a qualitative picture of the quantities of interest, we present volume rendering of snapshots of the fully developed flow for the horizontal kinetic energy and the potential energy in Fig. 9. It must be noted that it is not clear what correspondence the spectra have to the structures in the physical space visualizations of the flow. The spectra are related to the physical space correlations as fourier transform pairs. A particular distribution of a field in physical space does not indicate a unique spectral distribution in fourier. The visualizations presented here should be treated as qualitative impressions of the flow structures. Nevertheless, we will point out consistencies between what is observed in the spectra with what is observed in visualizing the corresponding physical space quantities.

When stratification dominates over rotation the horizontal kinetic energy exhibits flat horizontal structures at all values (Fig.9) which appear to have very little extent in the vertical direction. This is consistent with both Eq. 16 and the discussion and plots in section 3.1 which confirm complete independence from $k_{z}$. These structure are quite different from VSHF since the $k_{h}=0$ modes have a very small contribution in this regime. The potential energy for this flow (Fig. 9) appears to exhibit some tendency towards horizontally oriented blobs but retains more three-dimensionally coherent structures at the larger values, indicating that the potential energy distribution follows the distribution of horizontal kinetic energy to a degree, consistent with Fig. 2(b) but still retains some dependence on $k_{z}$ for small $k_{z}$ (large vertical scales) consistent with Fig. (2(d)).

Conversely, when rotation dominates over stratification, Fig. 9 shows that the potential energy exhibits structures that are strongly oriented in the vertical direction at all values with very little extent in the horizontal, consistent with Eq. (17). The horizontal kinetic energy also shows vertically coherent structures Fig. 9 but also shows high-valued larger blobs with more horizontal extent showing that $E_{h}$ also retains some horizontal scale dependence.

In the case of equal rotation and stratification (Figs. 9, 9), horizontal kinetic energy and potential energy look rather isotropically distributed with no preferred direction, in agreement with the spectra in Figs. 7(b) and 7(d) which show relatively shallow (ranging between -1 and -4/3) scaling in both $k_{z}$ and $k_{h}$ on average.

To summarize, equal rotation and stratification does not show any preferential direction for the energy isosurfaces, consistent with nearly the same shallow scaling in $k_{h}$ and $k_{z}$ (Figs. 7(b), 7(d)). Stratification dominated flows align the energy isosurfaces in the horizontal direction while rotation dominated flow align the energy isosurfaces along the vertical. The strongly stratified data are more in agreement with our asymptotic predictions than the strongly rotating data; the latter observation could indicate that even stronger rotation is required to see the asymptotic behavior.

The small scale, high wavenumber spectra of rotating Boussinesq have never before been studied at such asymptotically low values of $Ro$ and $Fr$ using high-resolution simulations with unbiased isotropic forcing and dissipation. The rigorous theoretical results for the $Ro\sim{\cal O}(1),Fr\rightarrow 0$ by Embid & Majda (1998) and the corresponding simulations of Smith & Waleffe (2002) extract the $k_{h}=0$ wave-modes as the VSHF contribution to the large scales. In our study of the small scales in this limit, which we have checked have negligible contribution of the $k_{h}=0$ modes (see Fig. 3), the vortical modes display a constant spectrum for $k>k_{f}$, apparently dominated by the $k_{z}^{0}$ behavior of the horizontal kinetic energy (see Fig. 2(d)). This appears to be a consequence of one of the key new results of this work, namely that the horizontal kinetic energy becomes independent of (constant in) $k_{z}$ for any $k_{h}$ (see Eq. 16 and Fig. 2(c)) for $\Gamma\ll 1$. This uniform spectral distribution implies a very efficient downscale transfer, with energy contained predominantly by the vortical modes. The exact value of the steep scaling exponent in $k_{h}$ is not as important for this result as the fact that the spectrum becomes independent of (constant in) $k_{z}$ while decaying sufficiently quickly in $k_{h}$ such that the spectrum becomes independent of spherical wavenumber $k$ as well (Fig. 3).

The strongly rotating case with finite stratification (runs 2 and 5) were expected to achieve the limiting behavior of Eq. 17 with collapse of the spectral curves for all $k_{h}$ as a function of $k_{z}$. While we see the correct trend in this direction, the curves in Fig. 4 show that the collapse is not complete. We are thus lead to believe that $Ro$ will need to be driven even lower and the resolution increased to see the collapse. We can postulate that, for sufficiently small $Ro$, the potential energy curves for various $k_{h}$ will collapse to the same function of $k_{z}$, corresponding to Eq. 17. Then, potential energy as a function of $k$ would be flat, governed by the flat spectrum in $k_{h}$ and strongly decaying spectrum in $k_{z}$. Again the exact exponent of the decay law would not affect this general conclusion.

The equally strongly rotating and stratified case (run 3) show how underlying anisotropic scalings of the spectra, depending on the cotangent of the mode-angle $\tau$, can be masked by summing over one or other of the wavevector components, as is commonly done. In Fig. 7(b) we show that the sum over all $k_{z}$ of the horizontal kinetic energy $E_{h}(k_{h})$ shallow with scaling $\sim k_{h}^{-4/3}$. Thus the average over all $k_{z}$ smears out the effect of suppression by potential enstrophy and results in a much shallower scaling that the $\tau$ dependent $k_{h}^{-5}$ scaling. The net effect is that energy fills the high-wavenumbers with almost the same shallow scaling in $k_{h}$ and $k_{z}$ (Fig. 7(b) and 7(d)). In Fig. 8 the wave-vortical mode decomposition of the spectra shows that the total downscale energy scales as $k^{-1}$ and is dominated by wave modes. This is a well-known result and the scaling is identical to that of the passive scalar since the wave-modes are passively advected in this limit. What we further establish from this analysis is that the shallow spectrum is set up in spite of the underlying steeper, anisotropic, $\tau$-dependent spectra in $k_{h}$ and $k_{z}$.

We have made an attempt at a physically consistent phenomenology that results in a steeper than -3 scaling of the spectra within this framework. The $-5$ scaling that arises out of the assumption that the ratio $\varepsilon_{Q}/\varepsilon_{d}$, where $\epsilon_{d}$ is the net flux of the energy downscale, plays a role in the downscale dynamics seems justified a posteriori. The analysis and verification of such an ansatz requires a detailed study of the fluxes and dissipation of both potential enstrophy and energy and is the subject of a separate work. As we have said before, the main conclusions of this paper hold irrespective of the specific scaling exponent for the decay of energy in the $k_{h}$ and $k_{z}$.

The prediction for steep decay in $k_{h}$ according to Eq. (16) arises from the constraint that potential enstrophy places on the horizontal kinetic energy in $k_{h}$ for $\Gamma\ll 1$. While there is no explicit constraint on the potential energy, the data analysis in this regime reveals that potential energy scales very similarly to the horizontal kinetic energy as a function of $k_{h}$, indicating that it too is suppressed in $k_{h}$. Thus the total energy, where the contribution from the vertical component of the velocity is measured to be very small, appears to be suppressed in the large $k_{h}$ modes. The observed shallow scaling in $k_{z}$ of both horizontal kinetic and potential energies, suggests that the total energy is allowed to transfer into the large $k_{z}$ modes. Similarly, for $\Gamma\gg 1$, the suppression of potential energy in $k_{z}$ is mirrored to a large degree by the horizontal kinetic energy. These observations indicate that the exchange between kinetic and potential energies is local in wavenumber in both $f/N\ll 1$ and $f/N\gg 1$ regimes.

Recent work explored the connection between hyperviscosity, Galerkin truncation and the energy bottleneck Frisch et al. (2008) in Navier-Stokes turbulence without rotation or stratification. In that paper it was proposed that high values of hyperviscosity result in a partial thermalization of the high wavenumbers which manifests as energy ‘bottleneck’ phenomenon which contaminates the inertial range. In the present studies of rotating and stratified turbulence, the only substantial difference between the hyperviscous and the comparable runs with regular NS viscosity is the consistently less extended scaling ranges of the latter runs, which is to be expected. Therefore we use the hyperviscous runs to extract scaling exponents while checking against the NS runs to make sure that bottleneck-type artifacts are not introduced. It appears that for rotating and stratified flow and for the value of hyperviscous power used, the bottleneck-type artifacts, if any are minimal, and do not affect the purposes of this study.

While we have discussed three extreme parameter regimes, this discussion encompasses all values of $\Gamma$ thus allowing us to analyze arbitrary $f/N$ in the limit of linear (or near linear) PV. In an idealized infinite domain where $k\rightarrow\infty$, for any fixed $f/N$ the $\Gamma\ll 1$ regime is achieved when $k_{h}/k_{z}\gg f/N$ while the $\Gamma\gg 1$ regime is achieved when $k_{h}/k_{z}\ll f/N$. The transition between regimes occurs for $\Gamma=1$ or when $1/\tau=k_{h}/k_{z}=f/N$. Thus we observe that this type of analysis could be very relevant to geophysical flow regimes where $f$ and $N$ values are not as extreme but where the scaling ranges are much longer.