Interacting scales and triad enstrophy transfers in generalized two-dimensional turbulence

Watanabe, Takeshi; Iwayama, Takahiro

doi:10.1103/physreve.76.046303

Cited by 12 publications

(22 citation statements)

References 30 publications

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“…For α 2, the downscale cascade of generalized enstrophy, Z = θ 2 /2, where the overline represents a domain average, is known to be dominated by spectrally nonlocal interactions, with ultralocal transfers mediated by a distant infrared wavenumber (Watanabe & Iwayama 2007;Tran, Dritschel & Scott 2010). For α = 2, logarithmic corrections to the Kolmogorov spectrum in the enstrophy cascade were suggested early on (Kraichnan 1967(Kraichnan , 1971b, and for α > 2 the enstrophy spectrum shallows to k 334 B. H. Burgess and T. G.…”

Section: Introductionmentioning

confidence: 99%

Spectral non-locality, absolute equilibria and Kraichnan–Leith–Batchelor phenomenology in two-dimensional turbulent energy cascades

Burgess

Shepherd

2013

J. Fluid Mech.

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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.

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Section: Introductionmentioning

confidence: 99%

Spectral non-locality, absolute equilibria and Kraichnan–Leith–Batchelor phenomenology in two-dimensional turbulent energy cascades

Burgess

Shepherd

2013

J. Fluid Mech.

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“…5,10,11 In contrast, Burgess and Shepherd 18 (hereafter denoted as BS13) studied the energy spectrum E α (k) ≡ k −α Q α (k) in the energy cascading range and concluded that the strain rate for α < 4 there is locally dominated. Furthermore, they showed from an eddydamped quasi-normal Markovian (EDQNM) closure approximation equation for the energy spectrum that the KLB spectrum in the energy cascading range,…”

Section: Introductionmentioning

confidence: 89%

“…Here, we have used relation (11). T Q(<) α (k|k c ) represents the enstrophy transfer due to the interactions among modes with wavenumbers less than k c .…”

Section: Formulationmentioning

confidence: 99%

“…This system, which also includes three realizable members of 2D geophysical fluid systems, 5,6 has been actively investigated both theoretically and numerically over the past decade. [7][8][9][10][11][12][13][14][15][16][17][18][19]26 When α = 2, q is the familiar vorticity, and governing equation (1) reduces to the vorticity equation for a 2D incompressible barotropic fluid (2D NS system for D = ν∇ 2 q, where ν is the kinematic viscosity coefficient, and a 2D Euler system for D = 0). For α > 3, the velocity induced by a point vortex, which is a delta-functional distribution of q, increases with the distance from the point vortex, as shown by Iwayama and Watanabe.…”

Section: Introductionmentioning

confidence: 99%

“…In particular, the extent to which the so-called Kraichnan-Leith-Batchelor (KLB) phenomenology [20][21][22] works well in the energy and enstrophy cascading ranges of forced-dissipative α-turbulence has been actively discussed. 5,[7][8][9][10][11][12]14,15,18,26 KLB phenomenology relies on the spectral locality, i.e., the localness of triad interactions between wavenumbers. However, for α ≥ 2, the strain rate in the enstrophy cascading range is dominated by non-local triad interactions.…”

Section: Introductionmentioning

confidence: 99%

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Anomalous eddy viscosity for two-dimensional turbulence

2015

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We study eddy viscosity for generalized two-dimensional (2D) fluids. The governing equation for generalized 2D fluids is the advection equation of an active scalar q by an incompressible velocity. The relation between q and the stream function ψ is given by q = −(−∇ 2 ) α/2 ψ. Here, α is a real number. When the evolution equation for the generalized enstrophy spectrum Q α (k) is truncated at a wavenumber k c , the effect of the truncation of modes with larger wavenumbers than k c on the dynamics of the generalized enstrophy spectrum with smaller wavenumbers than k c is investigated. Here, we refer to the effect of the truncation on the dynamics of Q α (k) with k < k c as eddy viscosity. Our motivation is to examine whether the eddy viscosity can be represented by normal diffusion. Using an asymptotic analysis of an eddy-damped quasi-normal Markovian (EDQNM) closure approximation equation for the enstrophy spectrum, we show that even if the wavenumbers of interest k are sufficiently smaller than k c , the eddy viscosity is not asymptotically proportional to k 2 Q α (k), i.e., a normal diffusion, but to k 4−α Q α (k) for α > 0 and k 4 Q α (k) for α < 0, i.e., an anomalous diffusion. This indicates that the eddy viscosity as normal diffusion is asymptotically realized only for α = 2 (Navier-Stokes system). The proportionality constant, the eddy viscosity coefficient, is asymptotically negative. These results are confirmed by numerical calculations of the EDQNM closure approximation equation and direct numerical simulations of the governing equation for forced and dissipative generalized 2D fluids. The negative eddy viscosity coefficient is explained using Fjørtoft's theorem and a spreading hypothesis for the spectrum. C 2015 AIP Publishing LLC. [http://dx.

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Local and nonlocal dispersive turbulence

Sukhatme

Smith

2009

Physics of Fluids

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We consider the evolution of a family of two-dimensional ͑2D͒ dispersive turbulence models. The members of this family involve the nonlinear advection of a dynamically active scalar field, and as per convention, the locality of the streamfunction-scalar relation is denoted by ␣, with smaller ␣ implying increased locality ͑␣ = 1 gives traditional 2D dynamics͒. The dispersive nature arises via a linear term whose strength, after nondimensionalization, is characterized by a parameter ⑀. Setting 0 Ͻ ⑀ Յ 1, we investigate the interplay of advection and dispersion for differing degrees of locality. Specifically, we study the forward ͑inverse͒ transfer of enstrophy ͑energy͒ under large-scale ͑small-scale͒ random forcing along with the geometry of the scalar field. Straightforward arguments suggest that for small ␣ the scalar field should consist of progressively larger isotropic eddies, while for large ␣ the scalar field is expected to have a filamentary structure resulting from a stretch and fold mechanism, much like that of a small-scale passive field when advected by a large-scale smooth flow. Confirming this, we proceed to forced/dissipative dispersive numerical experiments under weakly nonlocal to local conditions ͑i.e., ␣ Յ 1͒. In all cases we see the establishment of well-defined spectral scaling regimes. For ⑀ ϳ 1, there is quantitative agreement between nondispersive estimates and observed slopes in the inverse energy transfer regime. On the other hand, forward enstrophy transfer regime always yields slopes that are significantly steeper than the corresponding nondispersive estimate. At present resolution, additional simulations show the scaling in the inverse regime to be sensitive to the strength of the dispersive term: specifically, as ⑀ decreases, quite expectedly the inertial-range shortens but we also observe that the slope of the power law decreases. On the other hand, for the same range of ⑀ values, the forward regime scaling is observed to be fairly universal.

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Interacting scales and triad enstrophy transfers in generalized two-dimensional turbulence

Cited by 12 publications

References 30 publications

Spectral non-locality, absolute equilibria and Kraichnan–Leith–Batchelor phenomenology in two-dimensional turbulent energy cascades

Spectral non-locality, absolute equilibria and Kraichnan–Leith–Batchelor phenomenology in two-dimensional turbulent energy cascades

Anomalous eddy viscosity for two-dimensional turbulence

Local and nonlocal dispersive turbulence

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