Coherent vortices and tracer cascades in two-dimensional turbulence

2007 ◽  
Vol 574 ◽  
pp. 429-448 ◽  
Author(s):  
ARMANDO BABIANO ◽  
ANTONELLO PROVENZALE
Keyword(s):  
Inertial Range ◽  
Two Dimensional ◽  
Passive Tracer ◽  
Simultaneous Presence ◽  
Inverse Cascade ◽  
Coherent Vortices ◽  
Small Scales ◽  
Elliptic Regions ◽  
Enstrophy Cascade

We study numerically the scale-to-scale transfers of enstrophy and passive-tracer variance in two-dimensional turbulence, and show that these transfers display significant differences in the inertial range of the enstrophy cascade. While passive-tracer variance always cascades towards small scales, enstrophy is characterized by the simultaneous presence of a direct cascade in hyperbolic regions and of an inverse cascade in elliptic regions. The inverse enstrophy cascade is particularly intense in clusters of small-scales elliptic patches and vorticity filaments in the turbulent background, and it is associated with gradient-decreasing processes. The inversion of the enstrophy cascade, already noticed by Ohkitani (Phys. Fluids A, vol. 3, 1991, p. 1598), appears to be the main difference between vorticity and passive-tracer dynamics in incompressible two-dimensional turbulence.

scholarly journals The effect of asymmetric large-scale dissipation on energy and potential enstrophy injection in two-layer quasi-geostrophic turbulence

2012 ◽  
Vol 694 ◽  
pp. 493-523 ◽  
Author(s):  
Eleftherios Gkioulekas
Keyword(s):  
Large Scale ◽  
Bottom Layer ◽  
Two Dimensional ◽  
Interlayer Interaction ◽  
Open Questions ◽  
Small Scales ◽  
Scale Range ◽  
Geostrophic Turbulence ◽  
Enstrophy Cascade ◽  
Potential Enstrophy

AbstractIn the Nastrom–Gage spectrum of atmospheric turbulence, we observe a${k}^{\ensuremath{-} 3} $energy spectrum that transitions into a${k}^{\ensuremath{-} 5/ 3} $spectrum, with increasing wavenumber$k$. The transition occurs near a transition wavenumber${k}_{t} $, located near the Rossby deformation wavenumber${k}_{R} $. The Tung–Orlando theory interprets this spectrum as a double downscale cascade of potential enstrophy and energy, from large scales to small scales, in which the downscale potential enstrophy cascade coexists with the downscale energy cascade over the same length scale range. We show that, in a temperature-forced two-layer quasi-geostrophic model, the rates with which potential enstrophy and energy are injected place the transition wavenumber${k}_{t} $near${k}_{R} $. We also show that, if the potential energy dominates the kinetic energy in the forcing range, then the Ekman term suppresses the upscale cascading potential enstrophy more than it suppresses the upscale cascading energy, a behaviour contrary to what occurs in two-dimensional turbulence. As a result, the ratio$\eta / \varepsilon $of injected potential enstrophy over injected energy, in the downscale direction, decreases, thereby tending to decrease the transition wavenumber${k}_{t} $further. Using a random Gaussian forcing model, we reach the same conclusion, under the modelling assumption that the asymmetric Ekman term predominantly suppresses the bottom layer forcing, thereby disregarding a possible entanglement between the Ekman term and the nonlinear interlayer interaction. Based on these results, we argue that the Tung–Orlando theory can account for the approximate coincidence between${k}_{t} $and${k}_{R} $. We also identify certain open questions that require further investigation via numerical simulations.

Strong MHD helical turbulence and the nonlinear dynamo effect

1976 ◽  
Vol 77 (2) ◽  
pp. 321-354 ◽  
Author(s):  
A. Pouquet ◽  
U. Frisch ◽  
J. Léorat
Keyword(s):  
Large Scale ◽  
Magnetic Energy ◽  
Three Dimensional ◽  
Power Laws ◽  
Inertial Range ◽  
Small Scale ◽  
Mhd Turbulence ◽  
Inverse Cascade ◽  
Small Scales ◽  
Dynamo Effect

To understand the turbulent generation of large-scale magnetic fields and to advance beyond purely kinematic approaches to the dynamo effect like that introduced by Steenbeck, Krause & Radler (1966)’ a new nonlinear theory is developed for three-dimensional, homogeneous, isotropic, incompressible MHD turbulence with helicity, i.e. not statistically invariant under plane reflexions. For this, techniques introduced for ordinary turbulence in recent years by Kraichnan (1971 a)’ Orszag (1970, 1976) and others are generalized to MHD; in particular we make use of the eddy-damped quasi-normal Markovian approximation. The resulting closed equations for the evolution of the kinetic and magnetic energy and helicity spectra are studied both theoretically and numerically in situations with high Reynolds number and unit magnetic Prandtl number.Interactions between widely separated scales are much more important than for non-magnetic turbulence. Large-scale magnetic energy brings to equipartition small-scale kinetic and magnetic excitation (energy or helicity) by the ‘Alfvén effect’; the small-scale ‘residual’ helicity, which is the difference between a purely kinetic and a purely magnetic helical term, induces growth of large-scale magnetic energy and helicity by the ‘helicity effect’. In the absence of helicity an inertial range occurs with a cascade of energy to small scales; to lowest order it is a −3/2 power law with equipartition of kinetic and magnetic energy spectra as in Kraichnan (1965) but there are −2 corrections (and possibly higher ones) leading to a slight excess of magnetic energy. When kinetic energy is continuously injected, an initial seed of magnetic field will grow to approximate equipartition, at least in the small scales. If in addition kinetic helicity is injected, an inverse cascade of magnetic helicity is obtained leading to the appearance of magnetic energy and helicity in ever-increasing scales (in fact, limited by the size of the system). This inverse cascade, predicted by Frischet al.(1975), results from a competition between the helicity and Alféh effects and yields an inertial range with approximately — 1 and — 2 power laws for magnetic energy and helicity. When kinetic helicity is injected at the scale linjand the rate$\tilde{\epsilon}^V$(per unit mass), the time of build-up of magnetic energy with scaleL[Gt ] linjis$t \approx L(|\tilde{\epsilon}^V|l^2_{\rm inj})^{-1/3}.$

scholarly journals Velocity statistics inside coherent vortices generated by the inverse cascade of 2-D turbulence

2016 ◽  
Vol 809 ◽  
Author(s):  
I. V. Kolokolov ◽  
V. V. Lebedev
Keyword(s):  
Velocity Profile ◽  
Structure Function ◽  
Correlation Functions ◽  
Two Dimensional ◽  
Strong Anisotropy ◽  
Velocity Fluctuations ◽  
Inverse Cascade ◽  
Velocity Statistics ◽  
Flow Fluctuations ◽  
Coherent Vortices

We analyse velocity fluctuations inside coherent vortices generated as a result of the inverse cascade in the two-dimensional (2-D) turbulence in a finite box. As we demonstrated in Kolokolov & Lebedev (Phys. Rev. E, vol. 93, 2016, 033104), the universal velocity profile, established in Laurie et al. (Phys. Rev. Lett., vol. 113, 2014, 254503), corresponds to the passive regime of the flow fluctuations. This property enables one to calculate correlation functions of the velocity fluctuations in the universal region. We present the results of the calculations that demonstrate a non-trivial scaling of the structure function. In addition the calculations reveal strong anisotropy of the structure function.

Vorticity and passive-scalar dynamics in two-dimensional turbulence

1987 ◽  
Vol 183 ◽  
pp. 379-397 ◽  
Author(s):  
Armando Babiano ◽  
Claude Basdevant ◽  
Bernard Legras ◽  
Robert Sadourny
Keyword(s):  
Similarity Theory ◽  
Physical Space ◽  
Passive Scalar ◽  
Inertial Range ◽  
Flow Dynamics ◽  
Two Dimensional ◽  
Closure Model ◽  
Coherent Vortices ◽  
Vorticity Transport ◽  
Kolmogorov Theory

The dynamics of vorticity in two-dimensional turbulence is studied by means of semi-direct numerical simulations, in parallel with passive-scalar dynamics. It is shown that a passive scalar forced and dissipated in the same conditions as vorticity, has a quite different behaviour. The passive scalar obeys the similarity theory à la Kolmogorov, while the enstrophy spectrum is much steeper, owing to a hierarchy of strong coherent vortices. The condensation of vorticity into such vortices depends critically both on the existence of an energy invariant (intimately related to the feedback of vorticity transport on velocity, absent in passive-scalar dynamics, and neglected in the Kolmogorov theory of the enstrophy inertial range); and on the localness of flow dynamics in physical space (again not considered by the Kolmogorov theory, and not accessible to closure model simulations). When space localness is artificially destroyed, the enstrophy spectrum again obeys a k−1 law like a passive scalar. In the wavenumber range accessible to our experiments, two-dimensional turbulence can be described as a hierarchy of strong coherent vortices superimposed on a weak vorticity continuum which behaves like a passive scalar.

Kraichnan–Leith–Batchelor similarity theory and two-dimensional inverse cascades

2015 ◽  
Vol 767 ◽  
pp. 467-496 ◽  
Author(s):  
B. H. Burgess ◽  
R. K. Scott ◽  
T. G. Shepherd
Keyword(s):  
Large Scale ◽  
Similarity Theory ◽  
Vortex Formation ◽  
Turbulence Models ◽  
Two Dimensional ◽  
Scaling Properties ◽  
Inverse Cascade ◽  
Coherent Vortices ◽  
Coherent Vortex ◽  
Non Gaussian

AbstractWe study the scaling properties and Kraichnan–Leith–Batchelor (KLB) theory of forced inverse cascades in generalized two-dimensional (2D) fluids (${\it\alpha}$-turbulence models) simulated at resolution $8192^{2}$. We consider ${\it\alpha}=1$ (surface quasigeostrophic flow), ${\it\alpha}=2$ (2D Euler flow) and ${\it\alpha}=3$. 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 ${\it\alpha}=1$ and ${\it\alpha}=2$. The active scalar field for ${\it\alpha}=3$ contains comparatively few and small vortices. The energy spectral slopes in the inverse cascade are steeper than the KLB prediction $-(7-{\it\alpha})/3$ 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 ${\it\alpha}=1$ and ${\it\alpha}=2$, while the ${\it\alpha}=3$ inverse cascade is much closer to Gaussian and non-intermittent. For ${\it\alpha}=3$ the steep spectrum is close to that associated with enstrophy equipartition. Continuous wavelet analysis shows approximate KLB scaling $\mathscr{E}(k)\propto k^{-2}~({\it\alpha}=1)$ and $\mathscr{E}(k)\propto k^{-5/3}~({\it\alpha}=2)$ in the interstitial regions between the coherent vortices. Our results demonstrate that coherent vortex formation (${\it\alpha}=1$ and ${\it\alpha}=2$) and non-realizability (${\it\alpha}=3$) 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 ${\it\alpha}=1$ and ${\it\alpha}=2$.

scholarly journals Dimensional analysis of two-dimensional turbulence

2019 ◽  
Vol 33 (19) ◽  
pp. 1950218
Author(s):  
Leonardo Campanelli
Keyword(s):  
Energy Spectrum ◽  
Dimensional Analysis ◽  
Two Dimensional ◽  
Scaling Properties ◽  
Finite Systems ◽  
Inverse Cascade ◽  
Small Scales ◽  
Decaying Turbulence ◽  
Forced Turbulence

We study the scaling properties of two-dimensional turbulence using dimensional analysis. In particular, we consider the energy spectrum both at large and small scales and in the “inertial ranges” for the cases of freely decaying and forced turbulence. We also investigate the properties of an “energy condensate” at large scales in spatially finite systems. Finally, an analysis of a possible inverse cascade in freely decaying turbulence is presented.

Freely decaying two-dimensional turbulence

2010 ◽  
Vol 659 ◽  
pp. 351-364 ◽  
Author(s):  
S. FOX ◽  
P. A. DAVIDSON
Keyword(s):  
Strain Field ◽  
Dissipation Rate ◽  
Inertial Range ◽  
Quasi Equilibrium ◽  
Two Dimensional ◽  
Vortex Filaments ◽  
Common Assumption ◽  
Coherent Vortices ◽  
The Common ◽  
Usual Interpretation

High-resolution direct numerical simulations are used to investigate freely decaying two-dimensional turbulence. We focus on the interplay between coherent vortices and vortex filaments, the second of which give rise to an inertial range. We find that Batchelor's prediction for the inertial-range enstrophy spectrum Eω(k, t) ~ β2/3k−1, where β is the enstrophy dissipation rate, is reasonably well satisfied once the turbulence is fully developed, but that the assumptions which underpin the usual interpretation of his theory are not valid. For example, the lack of a quasi-equilibrium cascade means the enstrophy flux Πω(k) is highly non-uniform throughout the inertial range, thus the common assumption that β can act as a surrogate for Πω(k) becomes questionable. We present a variant of Batchelor's theory which accounts for the wavenumber-dependence of Πω; in particular we propose Eω(k, t) ~ Πω(k1)2/3k−1, where k1 is the wavenumber marking the start of the observed k−1 region of the enstrophy spectrum. This provides a better collapse of the data and, unlike Batchelor's original theory, can be justified on theoretical grounds. The basis for our proposal is the observation that the straining of the vortex filaments, which fuels the enstrophy flux through the inertial range, comes almost exclusively from the strain field of the coherent vortices, and this can be characterized by Πω(k1)1/3. Thus Eω(k) is a function of only k and Πω(k1) in the inertial range, and dimensional analysis then yields Eω ~ Πω(k1)2/3k−1. We also confirm the prediction by Davidson (Phys. Fluids, vol. 20, 2008, 025106) that in the inertial range Πω varies as Πω(k)/Πω(k1) = 1 − a−1 ln(k/k1), where a is a constant of order 1. This corresponds to ∂Eω/∂t ~ k−1. Surprisingly, the measured enstrophy fluxes imply that the dynamics of the inertial range as defined by the behaviour of Πω extend to wavenumbers much smaller than k1, but this is masked in Eω(k, t) by the presence of coherent vortices which also contribute to Eω in this region. In fact, we find that kEω(k, t) ≈ H(k) + A(t), or ∂Eω/∂t ~ k−1 in this extended low-k region, where H(k) is almost independent of time and represents the signature of the coherent vortices. In short, the inertial range defined by ∂Eω/∂t ~ k−1 or Πω(k) ~ ln(k) is much broader than the observed Eω ~ k−1 region.

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

2013 ◽  
Vol 725 ◽  
pp. 332-371 ◽  
Author(s):  
B. H. Burgess ◽  
T. G. Shepherd
Keyword(s):  
Energy Spectrum ◽  
Energy Flux ◽  
Turnover Time ◽  
Inertial Range ◽  
Energy Cascade ◽  
Two Dimensional ◽  
Inverse Cascade ◽  
Energy Cascades ◽  
Self Similar ◽  
Non Locality

AbstractWe 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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