Green’s function for a generalized two-dimensional fluid

Iwayama, Takahiro; Watanabe, Takeshi

doi:10.1103/physreve.82.036307

Cited by 21 publications

(27 citation statements)

References 19 publications

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“…As such, the horizontal structure of a 3-D PV distribution may be expected to qualitatively satisfy (4.7) with 1 < α < 2. Iwayama & Watanabe (2010) derive expressions for the corresponding Green's functions and also argue that α > 3 is unphysical, so here the attention is restricted to the range 0 < α < 3 for which the Green's function of Iwayama & Watanabe (2010) can be written as…”

Section: Illustration For a Family Of Inversion Operatorsmentioning

confidence: 99%

Rossby wave propagation on potential vorticity fronts with finite width

2016

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The horizontal gradient of potential vorticity (PV) across the tropopause typically declines with lead time in global numerical weather forecasts and tends towards a steady value dependent on model resolution. This paper examines how spreading the tropopause PV contrast over a broader frontal zone affects the propagation of Rossby waves. The approach taken is to analyse Rossby waves on a PV front of finite width in a simple single-layer model. The dispersion relation for linear Rossby waves on a PV front of infinitesimal width is well known; here, an approximate correction is derived for the case of a finite-width front, valid in the limit that the front is narrow compared to the zonal wavelength. Broadening the front causes a decrease in both the jet speed and the ability of waves to propagate upstream. The contribution of these changes to Rossby wave phase speeds cancel at leading order. At second order the decrease in jet speed dominates, meaning phase speeds are slower on broader PV fronts. This asymptotic phase speed result is shown to hold for a wide class of single-layer dynamics with a varying range of PV inversion operators. The phase speed dependence on frontal width is verified by numerical simulations and also shown to be robust at finite wave amplitude, and estimates are made for the error in Rossby wave propagation speeds due to the PV gradient error present in numerical weather forecast models.

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Section: Illustration For a Family Of Inversion Operatorsmentioning

confidence: 99%

Rossby wave propagation on potential vorticity fronts with finite width

2016

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“…The 'rotating shallow flow' (RSF) equation (Tran 2004) is given by α = 3, and is the isotropic limit of a mantle convection model, wherein a thin fluid on a rotating domain is driven by uniform internal heating and heated from below (Weinstein, Olson & Yuen 1989). For α > 3, the azimuthal velocity induced by a point vortex increases with distance (Iwayama & Watanabe 2010), which calls into question the physical relevance of these systems. However, all values of α, including non-integer values, give, at least in principle, well-defined systems, and the entire family is of interest when studying spectral non-locality, since this varies continuously with α.…”

Section: Introductionmentioning

confidence: 99%

“…This triad exchanges more (generalized) energy with large than with small scales and more (generalized) enstrophy with smaller than with larger scales for all values of α considered, in both cases the more so for larger α. We include results for α = 4 to show the continued trend of growing energy exchange with larger scales and enstrophy exchange with smaller scales as α increases, noting again that the physical relevance of α > 3 is questionable since the velocity induced by a point vortex in these models increases with distance (Iwayama & Watanabe 2010). On the right are the ratios for the non-local triad 0.2p, p and 1.1p (v = 0.2, w = 1.1).…”

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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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“…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%

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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Green’s function for a generalized two-dimensional fluid

Cited by 21 publications

References 19 publications

Rossby wave propagation on potential vorticity fronts with finite width

Rossby wave propagation on potential vorticity fronts with finite width

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

Anomalous eddy viscosity for two-dimensional turbulence

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