Global horizontal wavenumber kinetic energy spectra and spectral fluxes of rotational kinetic energy and enstrophy are computed for a range of vertical levels using a T799 ECMWF operational analysis. Above 250 hPa, the kinetic energy spectra exhibit a distinct break between steep and shallow spectral ranges, reminiscent of dual power-law spectra seen in aircraft data and high-resolution general circulation models. The break separates a large-scale ''balanced'' regime in which rotational flow strongly dominates divergent flow and a mesoscale ''unbalanced'' regime where divergent energy is comparable to or larger than rotational energy. Between 230 and 100 hPa, the spectral break shifts to larger scales (from n 5 60 to n 5 20, where n is spherical harmonic index) as the balanced component of the flow preferentially decays. The location of the break remains fairly stable throughout the stratosphere. The spectral break in the analysis occurs at somewhat larger scales than the break seen in aircraft data. Nonlinear spectral fluxes defined for the rotational component of the flow maximize between about 300 and 200 hPa. Large-scale turbulence thus centers on the extratropical tropopause region, within which there are two distinct mechanisms of upscale energy transfer: eddy-eddy interactions sourcing the transient energy peak in synoptic scales, and zonal mean-eddy interactions forcing the zonal flow. A well-defined downscale enstrophy flux is clearly evident at these altitudes. In the stratosphere, the transient energy peak moves to planetary scales and zonal mean-eddy interactions become dominant.

We propose a new similarity theory for the two-dimensional inverse energy cascade and the coherent vortex population it contains when forced at intermediate scales. Similarity arguments taking into account enstrophy conservation and a prescribed constant energy injection rate such that E ∼ t yield three length scales, l ω , l E , and l ψ , associated with the vorticity field, energy peak, and streamfunction, and predictions for their temporal evolutions, t 1/2 , t, and t 3/2 , respectively. We thus predict that vortex areas grow linearly in time, A ∼ l 2 ω ∼ t, while the spectral peak wavenumber k E ≡ 2πlWe construct a theoretical framework involving a three-part, time-evolving vortex number density distribution, n(A) ∼ t αi A −ri , i ∈ 1, 2, 3. Just above the forcing scale (i = 1) there is a forcing-equilibrated scaling range in which the number of vortices at fixed A is constant and vortex 'self-energy' v results in a small but significant correction to these temporal dependencies. High resolution numerical simulations verify the predicted vortex and spectral peak growth rates, as well as the theoretical picture of the three scaling ranges in the vortex population. Vortices steepen the energy spectrum E(k) past the classical k −5/3 scaling in the range k ∈ [k f , k v ], where k v is the wavenumber associated with the largest vortex, while at larger scales the slope approaches −5/3. Though vortices disrupt the classical scaling, their number density distribution and evolution reveal deeper and more complex scale invariance, and suggest an effective theory of the inverse cascade in terms of vortex interactions.

We study how the properties of forcing and dissipation affect the scaling behaviour of the vortex population in the two-dimensional turbulent inverse energy cascade. When the flow is forced at scales intermediate between the domain and dissipation scales, the growth rates of the largest vortex area and the spectral peak length scale are robust to all simulation parameters. For white-in-time forcing the number density distribution of vortex areas follows the scaling theory predictions of Burgess & Scott (J. Fluid Mech., vol. 811, 2017, pp. 742–756) and shows little sensitivity either to the forcing bandwidth or to the nature of the small-scale dissipation: both narrowband and broadband forcing generate nearly identical vortex populations, as do Laplacian diffusion and hyperdiffusion. The greatest differences arise in comparing simulations with correlated forcing to those with white-in-time forcing: in flows with correlated forcing the intermediate range in the vortex number density steepens significantly past the predicted scale-invariant $A^{-1}$ scaling. We also study the impact of the forcing Reynolds number $Re_{f}$, a measure of the relative importance of nonlinear terms and dissipation at the forcing scale, on vortex formation and the scaling of the number density. As $Re_{f}$ decreases, the flow changes from one dominated by intense circular vortices surrounded by filaments to a less structured flow in which vortex formation becomes progressively more suppressed and the filamentary nature of the surrounding vorticity field is lost. However, even at very small $Re_{f}$, and in the absence of intense coherent vortex formation, regions of anomalously high vorticity merge and grow in area as predicted by the scaling theory, generating a three-part number density similar to that found at higher $Re_{f}$. At late enough stages the aggregation process results in the formation of long-lived circular vortices, demonstrating a strong tendency to vortex formation, and via a route distinct from the axisymmetrization of forcing extrema seen at higher $Re_{f}$. Our results establish coherent vortices as a robust feature of the two-dimensional inverse energy cascade, and provide clues as to the dynamical mechanisms shaping their statistics.

We study the degree to which Kraichnan–Leith–Batchelor (KLB) phenomenology describes two-dimensional energy cascades in $\alpha $ turbulence, governed by $\partial \theta / \partial t+ J(\psi , \theta )= \nu {\nabla }^{2} \theta + f$, where $\theta = {(- \Delta )}^{\alpha / 2} \psi $ is generalized vorticity, and $\hat {\psi } (\boldsymbol{k})= {k}^{- \alpha } \hat {\theta } (\boldsymbol{k})$ in Fourier space. These models differ in spectral non-locality, and include surface quasigeostrophic flow ($\alpha = 1$), regular two-dimensional flow ($\alpha = 2$) and rotating shallow flow ($\alpha = 3$), which is the isotropic limit of a mantle convection model. We re-examine arguments for dual inverse energy and direct enstrophy cascades, including Fjørtoft analysis, which we extend to general $\alpha $, and point out their limitations. Using an $\alpha $-dependent eddy-damped quasinormal Markovian (EDQNM) closure, we seek self-similar inertial range solutions and study their characteristics. Our present focus is not on coherent structures, which the EDQNM filters out, but on any self-similar and approximately Gaussian turbulent component that may exist in the flow and be described by KLB phenomenology. For this, the EDQNM is an appropriate tool. Non-local triads contribute increasingly to the energy flux as $\alpha $ increases. More importantly, the energy cascade is downscale in the self-similar inertial range for $2. 5\lt \alpha \lt 10$. At $\alpha = 2. 5$ and $\alpha = 10$, the KLB spectra correspond, respectively, to enstrophy and energy equipartition, and the triad energy transfers and flux vanish identically. Eddy turnover time and strain rate arguments suggest the inverse energy cascade should obey KLB phenomenology and be self-similar for $\alpha \lt 4$. However, downscale energy flux in the EDQNM self-similar inertial range for $\alpha \gt 2. 5$ leads us to predict that any inverse cascade for $\alpha \geq 2. 5$ will not exhibit KLB phenomenology, and specifically the KLB energy spectrum. Numerical simulations confirm this: the inverse cascade energy spectrum for $\alpha \geq 2. 5$ is significantly steeper than the KLB prediction, while for $\alpha \lt 2. 5$ we obtain the KLB spectrum.

We study the scaling properties and Kraichnan-Leith-Batchelor (KLB) theory of forced inverse cascades in generalized twodimensional (2D) fluids (α-turbulence models) simulated at resolution 8192 2 . We consider α = 1 (surface quasigeostrophic flow), α = 2 (2D vorticity dynamics) and α = 3. The forcing scale is well-resolved, a direct cascade is present and there is no large-scale dissipation. Coherent vortices spanning a range of sizes, most larger than the forcing scale, are present for both α = 1 and α = 2. The active scalar field for α = 3 contains comparatively few and small vortices. The energy spectral slopes in the inverse cascade are steeper than the KLB prediction −(7 − α)/3 in all three systems. Since we stop the simulations well before the cascades have reached the domain scale, vortex formation and spectral steepening are not due to condensation effects; nor are they caused by large-scale dissipation, which is absent. One-and two-point pdfs, hyperflatness factors and structure functions indicate that the inverse cascades are intermittent and non-Gaussian over much of the inertial range for α = 1 and α = 2, while the α = 3 inverse cascade is much closer to Gaussian and non-intermittent. For α = 3 the steep spectrum is close to that associated with enstrophy equipartition. Continuous wavelet analysis shows approximate KLB scaling E (k) ∝ k −2 (α = 1) and E (k) ∝ k −5/3 (α = 2) in the interstitial regions between the coherent vortices. Our results demonstrate that coherent vortex formation (α = 1 and α = 2) and non-realizability (α = 3) cause 2D inverse cascades to deviate from the KLB predictions, but that the flow between the vortices exhibits KLB scaling and nonintermittent statistics for α = 1 and α = 2.