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The kinetic energy spectrum in the atmospheric mesoscale has a - 5/3 slope, which suggests an energy cascade. But the underlying dynamics of this cascade is still not fully understood. Is it driven by inertia–gravity waves, vortices or something else? To answer these questions, it is necessary to decompose the spectrum into contributions from waves and vortices. Linear decompositions are straightforward, but can lead to ambiguous results. A recent paper by Wang & Bühler ( J. Fluid Mech. , vol. 882, 2020, A16) addresses this problem by presenting a nonlinear decomposition of the energy spectrum into waves and vortices using the omega equation. They adapt this method for one-dimensional aircraft data and apply it to two datasets. In the lower stratosphere, the results show a mesoscale spectrum dominated by waves. The situation in the upper troposphere is different: here vortices are just as important, or possibly more than important, as waves, although the limitations of the one-dimensional data preclude a definitive answer.

The atmospheric kinetic energy spectrum and energy cascade are investigated in idealized simulations of radiative–convective equilibrium (RCE). WRF is employed to perform cloud-resolving simulations of an idealized radiative–convective equilibrium with and without aggregation with Δ x = 4 km. The horizontal kinetic energy (HKE) spectrum for the aggregated simulation in the upper troposphere is steeper than the nonaggregated case and closer to −5/3. The HKE spectra for the nonaggregated simulation in the upper troposphere and the lower stratosphere are much shallower than the −5/3 spectrum. In the upper troposphere, the divergent kinetic energy has a similar magnitude to the rotational kinetic energy in both the nonaggregated simulation and aggregated simulation. Energy is mainly gained from the buoyancy flux and mainly lost from the vertical energy flux for scales larger than 20 km. Downscale energy transfer is found in the upper troposphere. Numerical dissipation is the main source of energy loss at small scales. In the lower stratosphere, the divergent kinetic energy dominates the kinetic energy spectrum in both simulations. Energy is mainly gained from the vertical energy flux and is balanced by the loss from the buoyancy flux term, transfer term, and dissipation. An Eliassen–Palm flux analysis suggests that wave–mean-flow interaction may be responsible for the upscale energy transfer found in the lower stratosphere. The magnitudes of our kinetic energy spectra are similar to spectra calculated from aircraft data. Rotation is found to promote aggregation and steepen the energy spectrum.

Subgrid-scale (SGS) parameterizations in atmosphere and ocean models are often defined independently in the horizontal and vertical directions because the grid spacing is not the same in these directions (anisotropic grids). In this paper, we introduce a new anisotropic SGS model in large-eddy simulations (LES) of stratified turbulence based on horizontal filtering of the equations of motion. Unlike the common horizontal SGS parameterizations in atmosphere and ocean models, the vertical derivatives of the horizontal SGS fluxes are included in our anisotropic SGS scheme, and therefore the horizontal and vertical SGS dissipation mechanisms are not disconnected in the newly developed model. Our model is tested with two vertical grid spacings and various horizontal resolutions, where the horizontal grid spacing is comparatively larger than that in the vertical. Our anisotropic LES model can successfully reproduce the results of direct numerical simulations, while the computational cost is significantly reduced in the LES. We suggest the new anisotropic SGS model as an alternative to current SGS parameterizations in atmosphere and ocean models, in which the schemes for horizontal and vertical scales are often decoupled. The new SGS scheme may improve the dissipative performance of atmosphere and ocean models without adding any backscatter or other energizing terms at small horizontal scales.

Many high-resolution atmospheric models can reproduce the qualitative shape of the atmospheric kinetic energy spectrum, which has a power-law slope of −3 at large horizontal scales that shallows to approximately −5/3 in the mesoscale. This paper investigates the possible dependence of model energy spectra on the vertical grid resolution. Idealized simulations forced by relaxation to a baroclinically unstable jet are performed for a wide range of vertical grid spacings Δ z . Energy spectra are converged for Δ z 200 m but are very sensitive to resolution with 500 m ≤ Δ z ≤ 2 km. The nature of this sensitivity depends on the vertical mixing scheme. With no vertical mixing or with weak, stability-dependent mixing, the mesoscale spectra are artificially amplified by low resolution: they are shallower and extend to larger scales than in the converged simulations. By contrast, vertical hyperviscosity with fixed grid-scale damping rate has the opposite effect: underresolved spectra are spuriously steepened. High-resolution spectra are converged except for the stability-dependent mixing case, which are damped by excessive mixing due to enhanced shear over a wide range of horizontal scales. It is shown that converged spectra require resolution of all vertical scales associated with the resolved horizontal structures: these include quasigeostrophic scales for large-scale motions with small Rossby number and the buoyancy scale for small-scale motions at large Rossby number. It is speculated that some model energy spectra may be contaminated by low vertical resolution, and it is recommended that vertical-resolution sensitivity tests always be performed.

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.

In this paper large-eddy simulations (LES) of forced stratified turbulence using two common subgrid scale (SGS) models, the Kraichnan and Smagorinsky models, are studied. As found in previous studies using regular and hyper-viscosity, vorticity contours show elongated horizontal motions, which are layered in the vertical direction, along with intermittent Kelvin–Helmholtz (KH) instabilities. Increased stratification causes the layer thickness to collapse towards the dissipation scale, ultimately suppressing these instabilities. The vertical energy spectra are relatively flat out to a local maximum, which varies with the buoyancy frequency $\\def \\xmlpi #1{}\\def \\mathsfbi #1{\\boldsymbol {\\mathsf {#1}}}\\let \\le =\\leqslant \\let \\leq =\\leqslant \\let \\ge =\\geqslant \\let \\geq =\\geqslant \\def \\Pr {\\mathit {Pr}}\\def \\Fr {\\mathit {Fr}}\\def \\Rey {\\mathit {Re}}N$ . The horizontal energy spectra depend on the grid spacing $\\varDelta $ ; if the resolution is fine enough, the horizontal spectrum shows an approximately $-5/3$ slope along with a bump at the buoyancy wavenumber $k_b = N/u_{rms}$ , where $u_{rms}$ is the root-mean-square (r.m.s.) velocity. Our results show that there is a critical value of the grid spacing $\\varDelta $ , below which dynamics of stratified turbulence are well-captured in LES. This critical $\\varDelta $ depends on the buoyancy scale $L_b$ and varies with different SGS models: the Kraichnan model requires $\\varDelta < 0.47 L_b$ , while the Smagorinsky model requires $\\varDelta < 0.17 L_b$ . In other words, the Smagorinsky model is significantly more costly than the Kraichnan approach, as it requires three times the resolution to adequately capture stratified turbulence.

Stratified turbulence has a horizontally layered structure with quasi-two-dimensional vortices due to buoyancy forces that suppress vertical motion. The Prandtl number $\\textit {Pr}$ quantifies the relative strengths of viscosity and buoyancy diffusivity, which damp small-scale velocity and buoyancy fluctuations at different microscales. Direct numerical simulations (DNS) require high resolution to resolve the smallest flow features for large $\\textit {Pr}$. To reduce computational demand, $\\textit {Pr}$ is often set to 1. In this paper, we explore how varying $\\textit {Pr}$ affects stratified turbulence. DNS of homogeneous forced stratified turbulence with $0.7 \\le \\textit {Pr} \\le 8$ are performed for four stratification strengths and buoyancy Reynolds numbers $\\textit {Re}_b$ between 0.5 and 60. Energy spectra, buoyancy flux spectra, spectral energy flux and physical space fields are compared for scale-specific $\\textit {Pr}$-sensitivity. For $\\textit {Re}_b \\gtrsim 10$, $\\textit {Pr}$-dependence in the kinetic energy is mainly found at scales around and below the Kolmogorov scale. The potential energy and flux exhibit more prominent $\\textit {Pr}$-sensitivity. As $\\textit {Re}_b$ decreases, this $\\textit {Pr}$-dependence extends upscale. With increasing $\\textit {Pr}$, the spectra suggest eventual convergence to a limiting spectrum shape at large, finite $\\textit {Pr}$, at least at scales at and above the Ozmidov scale. The $\\textit {Pr}$-sensitivity of the spectra in the most strongly stratified $\\textit {Re}_b<1$ case differed from the rest, since large horizontal scales are affected by viscosity and diffusion. These findings suggest that $\\textit {Pr}=1$ DNS reasonably approximate $\\textit {Pr} > 1$ DNS with large $\\textit {Re}_b$, as long as the focus is on kinetic energy at scales much larger than the Kolmogorov scale, but otherwise stray from $\\textit {Pr} > 1$ spectra around and below the Kolmogorov scale, and even upscale when $\\textit {Re}_b \\lesssim 1$.

The dynamic Smagorinsky model for large eddy simulation (LES) of stratified turbulence is studied in this paper. A maximum grid spacing criterion of ${\\it\\Delta}/L_{b}<0.24$ is found in order to capture several of the key characteristics of stratified turbulence, where ${\\it\\Delta}$ is the filter scale and $L_{b}$ is the buoyancy scale. These results show that the dynamic Smagorinsky model needs a grid spacing approximately twice as large as the regular Smagorinsky model to reproduce similar results. This improvement on the regular Smagorinsky eddy viscosity approach increases the accuracy of results at small resolved scales while decreasing the computational costs because it allows larger ${\\it\\Delta}$ . In addition, the eddy dissipation spectra in LES of stratified turbulence present anisotropic features, taking energy out of large horizontal but small vertical scales. This trend is not seen in the non-stratified cases, where the subgrid-scale energy transfer is isotropic. Statistics of the dynamic Smagorinsky coefficient $c_{s}$ are investigated; its distribution is peaked around zero, and its standard deviations decrease slightly with increasing stratification. In line with previous findings for unstratified turbulence, regions of increased shear favour smaller $c_{s}$ values; in stratified turbulence, the spatial distribution of the shear, and hence $c_{s}$ , is dominated by a layerwise pancake structure. These results show that the dynamic Smagorinsky model presents a promising approach for LES when isotropic buoyancy-scale resolving grids are employed.

In this paper, the sensitivity of idealized squall-line simulations to horizontal resolution, subgrid turbulence mixing scheme, and subfilter-scale motion is discussed. Inconsistent results from numerical simulations of convective systems have suggested that there are issues with the behavior of the subgrid turbulent mixing parameterizations with increasing resolution that still need to be understood. WRF is used to perform large-eddy simulation of an idealized squall line with horizontal grid spacings of 4 km, 2 km, 1 km, 500 m, and 250 m. While 4 km grid spacing is able to produce the general structure of the squall line, higher-resolution simulations produce more detailed structures. Individual convective cell size decreases, the maximum cloud top height increases, and the subgrid turbulence kinetic energy (TKE) ratio decreases as resolution increases. As found in past studies, 4 km grid spacing is not recommended as it contains an unreasonable amount of subgrid TKE, is not sufficient to resolve the large energy-containing eddies, and may even suppress propagation of the squall line. While horizontal resolution of 1 km can produce a squall line, there are several discrepancies between the 1 km case and higher resolutions, including trailing banded structures and inhibited three-dimensionalization. These issues at 1 km resolution are investigated by examining the subfilter energy transfer for the highest-resolution simulation filtered to a horizontal scale of 1 km. The subfilter energy transfer rate at a scale of 1 km is dominated by the streamwise and shear components. While dissipation dominates the transfer, a significant amount of backscatter also exists, which is not represented by most subgrid models.

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.