We present numerical simulations of stably stratified, vortically forced turbulence at a wide range of Froude numbers. Large-scale vortical forcing was chosen to represent geophysical vortices which break down at small scales where Coriolis effects are weak. The resulting vortical energy spectra are much steeper in the horizontal direction and shallower in the vertical than typical observations in the atmosphere and ocean, as noted in previous studies. We interpret these spectra in terms of the vertical decoupling which emerges in the strongly stratified limit. We show that this decoupling breaks down at a vertical scale of $U/N$, where $N$ is the Brunt–Väisälä frequency and $U$ is a characteristic horizontal velocity, confirming previous scaling arguments. The transfer of vortical energy to wave energy is most efficient at this vertical scale; vertical spectra of wave energy are correspondingly peaked at small scales, as observed in past work. The equilibrium statistical mechanics of the inviscid unforced truncated problem qualitatively predicts the nature of the forced–dissipative solutions, and confirms the lack of an inverse cascade of vortical energy.

Numerical simulations of forced stratified turbulence are presented, and the dependence on horizontal resolution and grid aspect ratio is investigated. Simulations are designed to model the small-scale end of the atmospheric mesoscale and oceanic submesoscale, for which high horizontal resolution is usually not feasible in large-scale geophysical fluid simulations. Coarse horizontal resolution, which necessitates the use of thin grid aspect ratio, yields a downscale stratified turbulence energy cascade in agreement with previous results. We show that with increasing horizontal resolution, a transition emerges at the buoyancy scale 2pU=N, where U is the rms velocity and N is the Brunt-Väisälä frequency. Simulations with high horizontal resolution and isotropic grid spacing exhibit a spectral break at this scale, below which there is a net injection of kinetic energy by nonlinear interactions with the large-scale flow. We argue that these results are consistent with a direct transfer of energy to the buoyancy scale by Kelvin-Helmholtz instability of the large-scale vortices. These findings suggest the existence of a distinct subrange of stratified turbulence between the buoyancy and Ozmidov scales. This range must be at least partially resolved or parameterized to obtain robust simulations of larger-scale turbulence. V

We present numerical simulations of forced rotating stratified turbulence dominated by vortical motion (i.e. with potential vorticity). Strong stratification and various rotation rates are considered, corresponding to a small Froude number and a wide range of Rossby numbers $\hbox{\it Ro}$ spanning the regimes of stratified turbulence ($\hbox{\it Ro}\,{=}\,\infty$) to quasi-geostrophic turbulence ($\hbox{\it Ro}\,{\ll}\,1$). We examine how the energy spectra and characteristic vertical scale of the turbulence vary with Rossby number between these two regimes. The separate dependence on $N/f$, where $N$ is the Brunt–Väisälä frequency and $f$ is the Coriolis parameter, is found to be of secondary importance. As the macroscale $\hbox{\it Ro}$ decreases below 0.4 and the microscale $\hbox{\it Ro}$ (at our resolution) decreases below 3, the horizontal wavenumber energy spectrum steepens and the flat range in the vertical wavenumber spectrum increases in amplitude and decreases in length. At large Rossby numbers, the vertical scale $H$ is proportional to the stratified turbulence value $U/N$, where $U$ is the root mean square velocity. At small $\hbox{\it Ro}$, $H$ takes the quasi-geostrophic form $(f/N)L$, where $L$ is the horizontal scale of the flow. Implications of these findings for numerical atmosphere and ocean modelling are discussed.

The role of environmental moisture in the deepening of cumulus convection is investigated by means of cloud-resolving numerical experiments. Under idealized conditions of uniform SST and specified radiative cooling, the evolution of trade wind cumulus into congestus clouds, and ultimately deep convection, is simulated and analyzed. The results exhibit a tight coupling between environmental moisture and cloud depth, both of which increase over the course of the simulations. Moistening in the lower troposphere is shown to result from the detrainment of water vapor from congestus clouds, and the strength of this tendency is quantified. Moistening of the lower troposphere reduces the dilution of cloud buoyancy by dry-air entrainment, and the relationship between this effect and increasing cloud depth is examined. The authors confirm that the mixing of water vapor by subgrid-scale turbulence has a significant impact on cloud depth, while the mixing of sensible heat has a negligible effect. By contrast, the dependence of cloud depth on CAPE appears to be of secondary importance. However, the deepening trend observed in these simulations is not solely determined by the evolving mean vapor profile. While enhancing the horizontally averaged humidity does result in deeper clouds, this occurs only after an adjustment period of several hours, presumably because of the buildup of CAPE. The implications of these findings for large-scale simulations in which resolved mixing is reduced-for example, by coarser spatial resolution or 2D experiments-are also discussed.

The role of moist processes in the development of the mesoscale kinetic energy spectrum is investigated with numerical simulations of idealized moist baroclinic waves. Dry baroclinic waves yield upper-tropospheric kinetic energy spectra that resemble a −3 power law. Decomposition into horizontally rotational and divergent kinetic energy shows that the divergent energy has a much shallower spectrum, but its amplitude is too small to yield a characteristic kink in the total spectrum, which is dominated by the rotational part. The inclusion of moist processes energizes the mesoscale. In the upper troposphere, the effect is mainly in the divergent part of the kinetic energy; the spectral slope remains shallow (around −) as in the dry case, but the amplitude increases with increasing humidity. The divergence field in physical space is consistent with inertia–gravity waves being generated in regions of latent heating and propagating throughout the baroclinic wave. Buoyancy flux spectra are used to diagnose the scale at which moist forcing—via buoyant production from latent heating—injects kinetic energy. There is significant input of kinetic energy in the mesoscale, with a peak at scales of around 800 km and a plateau at smaller scales. If the latent heating is artificially set to zero at some time, the enhanced divergent kinetic energy decays over several days toward the level obtained in the dry simulation. The effect of moist forcing of mesoscale kinetic energy presents a challenge for theories of the mesoscale spectrum based on the idealization of a turbulent inertial subrange.

We present numerical simulations of randomly forced internal gravity waves in a uniformly stratified Boussinesq fluid, and compare the resulting vertical wavenumber energy spectra with the saturation spectrum $E_z(k_z)\,{=}\,c\,N^2k_z^{-3}$ ($N$ is the Brunt–Väisälä frequency) observed in the atmosphere and ocean. Overall, we have been unsuccessful at reproducing the observed spectrum in our simulations. Our spectra are shallower than $k_z^{-3}$, although they steepen towards it with increasing stratification as long as wave breaking (in the form of static instability) is resolved. The spectral amplitude increases like $N^{1.1}$ rather than $N^2$. For a single stratification, our spectrum agrees well with the saturation spectrum with $c\,{=}\,0.1$, but only because it is spuriously steepened by insufficient resolution. We show that overturning occurs when the length scale $l_c\,{=}\,u_{rms}/N$ is larger than the dissipation scale, where $u_{rms}$ is the root mean square velocity. This scale must be at least three times larger than the dissipation scale for the energy spectrum to be independent of Reynolds number in our simulations. When this condition is not satisfied, the computed energy spectrum must be interpreted with caution. Finally, we show that for strong stratifications, the presence of vortical energy can have a dramatic effect on the spectrum of wave energy due to the efficiency of interactions between two waves and a vortical mode. Any explanation of the energy spectrum involving resonant interactions must take into account the effect of vortical motion.

The atmospheric mesoscale kinetic energy spectrum is investigated through numerical simulations of an idealized baroclinic wave life cycle, from linear instability to mature nonlinear evolution and with high horizontal and vertical resolution (Dx ' 10 km and Dz ' 60 m). The spontaneous excitation of inertia-gravity waves yields a shallowing of the mesoscale spectrum with respect to the large scales, in qualitative agreement with observations. However, this shallowing is restricted to the lower stratosphere and does not occur in the upper troposphere. At both levels, the mesoscale divergent kinetic energy spectrum-a proxy for the inertiagravity wave energy spectrum-resembles a 25/3 power law in the mature stage. Divergent kinetic energy dominates the lower stratospheric mesoscale spectrum, accounting for its shallowing. Rotational kinetic energy, by contrast, dominates the upper tropospheric spectrum and no shallowing of the full spectrum is observed. By analyzing the tendency equation for the kinetic energy spectrum, it is shown that the lower stratospheric spectrum is not governed solely by a downscale energy cascade; rather, it is influenced by the vertical pressure flux divergence associated with vertically propagating inertia-gravity waves.

The dynamics of a counter-rotating pair of columnar vortices aligned parallel to a stable density gradient are investigated. By means of numerical simulation, we extend the linear analyses and laboratory experiments of Billant & Chomaz (J. Fluid Mech. vol. 418, p. 167; vol. 419, pp. 29, 65 (2000)) to the fully nonlinear, large-Reynolds-number regime. A range of stratifications and vertical length scales is considered, with Frh < 0.2 and 0.1 < Frz < 10. Here Frh ≡ U/(NR) and Frz ≡ Ukz/N are the horizontal and vertical Froude numbers, U and R are the horizontal velocity and length scales of the vortices, N is the Brunt–Väisälä frequency, and 2π/kz is the vertical wavelength of a small initial perturbation. At early times with Frz < 1, linear predictions for the zigzag instability are reproduced. Short-wavelength perturbations with Frz > 1 are found to be unstable as well, with growth rates only slightly less than those of the zigzag instability but with very different structure. At later times, the large-Reynolds-number evolution diverges profoundly from the moderate-Reynolds-number laboratory experiments as the instabilities transition to turbulence. For the zigzag instability, this transition occurs when density perturbations generated by the vortex bending become gravitationally unstable. The resulting turbulence rapidly destroys the vortex pair. We derive the criterion η/R ≈ 0.2/Frz for the onset of gravitational instability, where η is the maximum horizontal displacement of the bent vortices, and refine it to account for a finite twisting disturbance. Our simulations agree for the fastest growing wavelengths 0.3 < Frz < 0.8. Short perturbations with Frz > 1 saturate at low amplitude, preserving the columnar structure of the vortices well after the generation of turbulence. Viscosity is shown to suppress the transition to turbulence for Reynolds number Re ≲ 80/Frh, yielding laminar dynamics and, under certain conditions, pancake vortices like those observed in the laboratory.