ALBERT

Description: We study the degree to which Kraichnan-Leith-Batchelor (KLB) phenomenology describes two-dimensional energy cascades in α turbulence, governed by ∑θ/ ∑t+ J(Ψ, θ)= ν∇2 θ+ f, where θ= (-Δ)α/2 Ψ is generalized vorticity, and Ψ(k)= k-αθ (k) in Fourier space. These models differ in spectral non-locality, and include surface quasigeostrophic flow ( α= 1), regular two-dimensional flow ( α= 2) and rotating shallow flow ( α= 3), which is the isotropic limit of a mantle convection model. We re-examine arguments for dual inverse energy and direct enstrophy cascades, including Fjortoft analysis, which we extend to general α, and point out their limitations. Using an α-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 α increases. More importantly, the energy cascade is downscale in the self-similar inertial range for 2. 5〈α〈10. At α= 2. 5 and α= 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 α〈4. However, downscale energy flux in the EDQNM self-similar inertial range for α〉2. 5 leads us to predict that any inverse cascade for α≥2. 5 will not exhibit KLB phenomenology, and specifically the KLB energy spectrum. Numerical simulations confirm this: the inverse cascade energy spectrum for α≥2. 5 is significantly steeper than the KLB prediction, while for α〈2. 5 we obtain the KLB spectrum. © Cambridge University Press 2013.

Description: 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, lωlE and lψ, associated with the vorticity field, energy peak and streamfunction, and predictions for their temporal evolutions,t1/2, t and t3/2, respectively. We thus predict that vortex areas grow linearly in time, A ∼ l2ω ∼ t, while the spectral peak wavenumber KE ≡ 2πl-1 E ∼ t-1 . We construct a theoretical framework involving a three-part, time-evolving vortex number density distribution, n(A) ∼ tαi 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' Ecm v = (2D)-1 ∫ω2 vA2n(A)dA is conserved in A-space intervals [μA0(t), A0(t)] comoving with the growth in vortex area,A0(t) ∼ t . In this range, α1 = 0 and n(A) ∼ A-3 At intermediate scales (i = 2 ) sufficiently far from the forcing and the largest vortex, there is a range with a scale-invariant vortex size distribution. We predict that in this range the vortex enstrophy Zcm v = (2D)-1 ∫ ω2 v(A)dA. is conserved and n(A) ∼ t-1A-1. The final range (i =3), which extends over the largest vortex-containing scales, conserves σcm v = (2D)-1 ∫ω2 vn(A)dA . If ω2 v is constant in time, this is equivalent to conservation of vortex number Ncm v =∫n(A)dA. This regime represents a 'front' of sparse vortices, which are effectively point-like; in this range we predict n(A) ∼ tr3-1A-r3. Allowing for time-varying ω2 v results in a small but significant correction to these temporal dependences. 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∈[kf,kv], where kv 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. © 2016 Cambridge University Press.

Description: 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 scaling. We also study the impact of the forcing Reynolds number , 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 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 , 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 . 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 . 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. © 2018 Cambridge University Press.

Description: We present a scaling theory that links the frequency of long frontal waves to the kinetic energy decay rate and inverse transfer of potential energy in freely evolving equivalent barotropic turbulence. The flow energy is predominantly potential, and the streamfunction makes the dominant contribution to potential vorticity (PV) over most of the domain, except near PV fronts of width, where is the Rossby deformation length. These fronts bound large vortices within which PV is well-mixed and arranged into a staircase structure. The jets collocated with the fronts support long-wave undulations, which facilitate collisions and mergers between the mixed regions, implicating the frontal dynamics in the growth of potential-energy-containing flow features. Assuming the mixed regions grow self-similarly in time and using the dispersion relation for long frontal waves (Nycander et al.Phys. Fluids A, vol. 5, 1993, pp. 1089-1091) we predict that the total frontal length and kinetic energy decay like, while the length scale of the staircase vortices grows like. High-resolution simulations confirm our predictions. © 2019 Cambridge University Press.

Description: We study the scaling properties and Kraichnan-Leith-Batchelor (KLB) theory of forced inverse cascades in generalized two-dimensional (2D) fluids ( α-turbulence models) simulated at resolution 81922. We consider α=1 (surface quasigeostrophic flow), α=2 (2D Euler flow) 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 p.d.f.s, 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)propto k-2(α=1) and E(k)propto 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 non-intermittent statistics for α=1 and α=2. © © 2015 Cambridge University Press.