Enstrophy Transfer Research Articles

Energy and enstrophy transfer in numerical simulations of two-dimensional turbulence

Numerical simulations of statistically steady two-dimensional (2-D) turbulence are analyzed to determine the relative importance of the types of wave-vector triad interactions that transfer energy and enstrophy in the both the energy and enstrophy inertial ranges. In the enstrophy inertial range, it is found (in agreement with previous studies [J. Fluid Mech. 72, 305 (1975); Phys. Fluids A 2, 1529 (1990)]) that the important triads (i.e., those associated with the highest transfer rates) are typically very elongated. On the average, nearly all of the enstrophy transfer within these triads is directed from the intermediate to the largest wave-number mode (i.e., downscale transfer). Energy, too, is transferred downscale in this manner, but is also transferred upscale due to the interaction of the intermediate with the smallest wave-number mode of the triad, resulting in no net flux of energy in the enstrophy inertial range. Analysis of the geometry of the important triads indicates they are not of similar shapes at all scales, and that the enstrophy transferring triads generally consist of one wave vector near the scale of the energetic peak, no matter how large the other wave vectors are. In the energy inertial range, elongated triads are also important. As in the enstrophy inertial range, there is downscale transfer of energy and enstrophy due to the interaction of the intermediate with the largest wave-number mode. There is also upscale transfer of both energy and enstrophy due to a very nonlocal interaction involving the smallest wave-number modes. The result is a net upscale flux of energy and no net flux of enstrophy in the energy inertial range. Comparison of the transfer functions from the simulations with those calculated by an eddy-damped quasinormal closure show agreement in the gross functional forms, but display certain quantitative differences in integrated quantities such as total transfer into and flux past a given wave number.

The theory of turbulence in two dimensions

The methods of conformal field theory are used to obtain series of exact solutions of the fundamental equations of the theory of turbulence. The basic conjecture, proved to be self-consistent, is the conformal invariance of the inertial range. The resulting physical picture is different from the standard one, since the enstrophy transfer is catalyzed by the large-scale motions. The theory gives some unambiguous predictions for the corrections in the inertial range.

The dynamics of spectral transfer in a model of drift wave turbulence with two nonlinearities

The spectral transfer dynamics of two-dimensional (2-D) drift wave turbulence over a broad range incorporating long- and short-wavelength extremes is studied numerically in the context of dissipative trapped electron convective cell turbulence. The direction, locality, and isotropy of energy and enstrophy transfer in wave-number space are determined by examining energy and enstrophy transfer rates, the enstrophy generation rate, spectra, and the spectrum response to perturbative impulses. Energy transfer is characterized by two subranges, according to the dominant nonlinearity, and a dynamically complex crossover region dividing the subranges. In the long-wavelength E×B subrange, energy transfer is nonlocal and anisotropic, proceeding to shorter wavelengths with significant generation of enstrophy. In the short-wavelength polarization drift subrange, energy transfer is local and, in the absence of sources and sinks, is dominated by an inverse cascade, consistent with the near conservation of enstrophy on dynamical time scales. In the crossover region, there is isotropic nonlocal forward transfer, as well as cascading to long wavelength. A significant shift of the frequency spectrum peak in the crossover region is shown to result from the cross coupling of the two nonlinearities. The shift is in the electron diamagnetic direction and increases with increasing wave number, consistent with the behavior of the renormalized response function. The simulation model does not incorporate the effects of electron inertia, and therefore does not account for the feedback of the frequency shift on nonlinear mode stability. Nevertheless, the simulations provide numerical validation of many aspects of the accompanying analytical investigation [Liang et al., Phys. Fluids B 5, 1128 (1993)].

High-resolution calculations of reduced waveset two-dimensional turbulence

Motivated by the possible utility of studying turbulent astrophysical flows as dynamical systems, solutions to the incompressible two-dimensional Navier–Stokes equation in Fourier space are investigated employing a small subset of Fourier modes that span a range of nearly 3000:1 in size and that preserve the triadic nature of the nonlinear interactions. These solutions thus preserve the energy and enstrophy conservation properties of the full solutions. The statistical similarity of the reduced solutions to the full solutions is tested by comparing their respective energy spectra in the energy and enstrophy inertial ranges, as well as the energy and enstrophy cascade directions. The behavior of the reduced solutions is consistent with full solutions having similar physical conditions (rates of injection and dissipation of the energy and enstrophy), while providing a computational gain of about a factor ∼1000 for kmax∼104 in two dimensions (2D) and ∼106 at kmax∼103.5 in 3D. Also, the effect and importance of the forcing on the outcome of the calculations are discussed, namely the slope of the energy spectrum and the duration of the net energy and enstrophy transfer.

Energy spectra and coherent structures in forced two-dimensional and beta-plane turbulence

Results from a wide range of direct numerical simulations of forced-dissipative, differentially rotating two-dimensional turbulence are presented, in order to delineate the broad dependence of flow type on forcing parameters. For most parameter values the energy spectra of simulations forced at low wavenumbers are markedly steeper than the classical k−3 enstrophy inertial-range prediction, and although k−3 spectra can be produced under certain circumstances, the regime is not robust, and the Kolmogorov constant is not universal unless a slight generalization is made in the phenomenology. Long-lived, coherent vortices form in many cases, accompanied by steep energy spectra and a higher than Gaussian vorticity kurtosis. With the addition of differential rotation (the β-effect), a small number of fairly distinct flow regimes are observed. Coherent vortices weaken and finally disappear as the strength of the β-effect increases, concurrent with increased anisotropy and decreased kurtosis. Even in the absence of coherent vortices and with a Gaussian value of the kurtosis, the spectra remain relatively steep, although not usually as steep as for the non-rotating cases. If anisotropy is introduced at low wavenumbers, the anisotropy is transferred to all wavenumbers in the inertial range, where the dynamics are isotropic.For those simulations that are forced at relatively high wavenumbers, a well resolved and very robust k−5/3 energy inertial range is observed, and the Kolmogorov constant appears universal. The low-wavenumber extent of the reverse energy cascade is essentially limited by the β-effect, which produces an effective barrier in wavenumber space at which energy accumulates, and by frictional effects which must be introduced to achieve equilibrium. Anisotropy introduced at large scales remains largely confined to the low wavenumbers, rather than being cascaded to small scales. When there is forcing at both large and small scales (which is of relevance to the Earth's atmosphere), energy and enstrophy inertial ranges coexist, with an upscale energy transfer and downscale enstrophy transfer in the same wavenumber interval, without the need for any dissipation mechanism between forcing scales.

The kinematics of quasi-2-D freely decaying turbulence in a stratified fluid

We have studied experimentally the dynamics of freely decaying turbulence in a stratified fluid. At low Froude number this flow has a quasi-2-D behavior1 that modifies the mixing (Fr=0.004, Re=1000; based on the integral scales). We characterize the vorticity evolution in the range scales where the enstrophy cascade is taking place. The quasi-2-D period of this turbulence is studied by photographic analysis of the velocity field V, which is interpolated on a regular grid (128×128). We are able to decompose it into its internal wave and turbulent component, and to compute its derivative fields up to the vector η=curlHΩV (related to the enstrophy cascade; where ΩV is the vertical vorticity). An enstrophy transfer model is developed by generalizing the model of Weiss to slightly divergent flows: the tensor F=∇HVH−DivH VHI2 (where H denotes the horizontal component, and I2 is the identity tensor of order 2) enters the equation of evolution for η as dtη=ηF+ΩV curlH (DivHVH). (1) An elongation of η corresponds to an increase of the local vorticity gradient, which is the mechanism of the enstrophy cascade to small scales. We neglect the second term of (1) since it occurs at the measurement noise scale. Thus eigenvalues λi and eigenvectors of F characterize Eq. (1) and we present maps of these functions. The analysis of these eigenvalues of F gives us the ‘‘coherent’’ zones of the flow where λ is complex, i.e., where the local effect of rotation counteracts the local strain induced by the whole vorticity field. In the remaining zones the λi are generally of opposite sign. Under this condition, the histogram of the angle α between the elongating (V+) and the contracting (V−) eigenvectors shows broad peaks centered on ±π/2, Where the separating curve of ΩV=0 on an Ω map corresponds to α=±π/2. In these regions the strain provokes a deformation of the vorticity field, which corresponds to the enstrophy cascade phenomenon. Statistics on the angles β±=∠(η,V±) reveals a mean alignment of η with V+ (elongation) and a mean right angle with V−. Thus, the relations between F and η are clearly shown in our experimental case. The main effect of vertical compression or expansion at every point of the field, due for instance to a weak internal wave, is to modify the eigenvalues of F, thus inhibiting or favoring the local enstrophy cascade. In some cases, it is able to impose its sign onto the two eigenvalues: this is observed at the border of the coherent zone.

The dynamics of enstrophy transfer in two-dimensional hydrodynamics

In this paper the qualitative properties of an inviscid, incompressible, two-dimensional fluid are examined. Starting from the equations of motion we derive a series of equations that govern the behavior of the spatial gradients of the vorticity scalar. The growth of these gradients is related to the transfer of enstrophy (integral of squared vorticity) to the small scales of the fluid motion.

Study of Two-Dimensional Interchange Turbulence

An eddy viscosity model describing enstrophy transfer in two-dimensional turbulence is presented. It is similar to that of Canuto et al . [Phys. Fluids 30 (1987) 3391] and provides an equation for the energy spectral function F ( k ) as a function of the energy input rate per unit wavenumber, γ s ( k ). In the enstrophy-transfer inertial range, F ( k )∝ k -3 is predicted by the model. Our model is applied to the interchange turbulence of a plasma in shearless magnetic field. Numerical simulation demonstrates that the energy spectrum for high wavenumbers is well described by this model. The turbulent transport driven by the interchange turbulence is expressed in terms of the Nusselt number Nu, the Rayleigh number Ra and the Prantl number Pr in the same manner as that of thermal convection problem. When we use the linear growth rate for γ s ( k ), our theoretical model predicts that Nu∝(Ra·Pr) 1/2 for a constant background pressure gradient and Nu∝(Ra·Pr) 1/3 for a self-consistent background pressure profil...

Statistical properties of decaying geostrophic turbulence

High-resolution, high-Reynolds-number numerical solutions of fully three-dimensional, decaying, geostrophic turbulence are examined. The results include the demonstration of a substantial degree of similarity between geostrophic and two-dimensional turbulence: transfer of energy to larger scales; transfer of potential enstrophy to smaller scales; vanishing energy dissipation as the Reynolds number increases; the emergence and growth to dominance of isolated, coherent vortices; and a competition between the vortices and Rossby waves, with an associated horizontal anisotropy when the latter are dominant. Properties that are distinct to geostrophic turbulence include the following: approximate three-dimensional wavenumber isotropy, with significant departures on large scales due to boundedness of the domain and on smaller scales due to anisotropic spectrum transfer rates; insensitivity of solution properties to anisotropy or vertical inhomogeneity in the dissipation; persistence of vertical inhomogeneity; development of inhomogeneity due to solid vertical boundaries; and the processes of alignment, attachment, and vertical straining associated with the finite vertical extent of the coherent vortices.

Rossby waves and two-dimensional turbulence in a large-scale zonal jet

The theory of homogeneous barotropic beta-plane turbulence is here extended to include effects arising from spatial inhomogeneity in the form of a zonal shear flow. Attention is restricted to the geophysically important case of zonal flows that are barotropically stable and are of larger scale than the resulting transient eddy field.Because of the presumed scale separation, the disturbance enstrophy is approximately conserved in a fully nonlinear sense, and the (nonlinear) wave-mean-flow interaction may be characterized as a shear-induced spectral transfer of disturbance enstrophy along lines of constant zonal wavenumber k. In this transfer the disturbance energy is generally not conserved. The nonlinear interactions between different disturbance components are turbulent for scales smaller than the inverse of Rhines's cascade-arrest scale κβ≡ (β0/2urms)½ and in this regime their leading-order effect may be characterized as a tendency to spread the enstrophy (and energy) along contours of constant total wavenumber κ ≡ (k2 + l2)½. Insofar as this process of turbulent isotropization involves spectral transfer of disturbance enstrophy across lines of constant zonal wavenumber k, it can be readily distinguished from the shear-induced transfer which proceeds along them. However, an analysis in terms of total wavenumber K alone, which would be justified if the flow were homogeneous, would tend to mask the differences.The foregoing theoretical ideas are tested by performing direct numerical simulation experiments. It is found that the picture of classical beta-plane turbulence is altered, through the effect of the large-scale zonal flow, in the following ways: (i) while the turbulence is still confined to KKβ, the disturbance field penetrates to the largest scales of motion; (ii) the larger disturbance scales K < Kβ exhibit a tendency to meridional rather than zonal anisotropy, namely towards v2 > u2 rather than vice versa; (iii) the initial spectral transfer rate away from an isotropic intermediate-scale source is significantly enhanced by the shear-induced transfer associated with straining by the zonal flow. This last effect occurs even when the large-scale shear appears weak to the energy-containing eddies, in the sense that dU/dy [Lt ] κ for typical eddy length and velocity scales.

Inertial range of two-dimensional turbulence in a Lagrangian renormalized approximation

The inertial range structure of two-dimensional turbulence is studied on the basis of a deductive Lagrangian renormalized approximation. The approximation is consistent with the energy spectra E(k)=Cε2/3 k−5/3 in the energy-transfer range and E(k)=C′β2/3 k−3 ×[ln(k/k1)]−1/3 (k≫k1) in the enstrophy-transfer range, and yields the estimates C=5.88 and C′=1.44. Here E(k) is the usual energy spectrum, ε and β are, respectively, the rates of energy and enstrophy transfer per unit mass, and k1 is the bottom of the enstrophy-transfer range.

A Spectral View of Nonlinear Fluxes and Stationary-Transient Interaction in the Atmosphere

Nonlinear spectral transfers of kinetic energy and enstrophy, and stationary-transient interaction, are studied using global FGGE data for January 1979. It is found that the spectral transfers arise primarily from a combination, in roughly equal measure, of pure transient and mixed stationary-transient interactions. The pure transient interactions are associated with a transient eddy field which is approximately locally homogeneous and isotropic, and they appear to be consistently understood within the context of two-dimensional homogeneous turbulence. Theory based on spatial wale separation concepts suggests that the mixed interactions may be understood physically, to a first approximation, as a process of shear-induced spectral transfer of transient enstrophy along lines of constant zonal wavenumber. This essentially conservative enstrophy transfer generally involves highly nonlocal stationary-transient energy conversions. The observational analysis demonstrates that the shear-induced transient enstrophy transfer is mainly associated with intermediate-scale (zonal wavenumber m > 3) transients and is primarily to smaller (meridional) scales, so that the transient flow acts as a source of stationary energy. In quantitative terms, this transient-eddy rectification corresponds to a forcing timescale in the stationary energy budget which is of the same order of magnitude as most estimates of the damping timescale in simple stationary-wave models (5 to 15 days). Moreover, the nonlinear interactions involved are highly nonlocal and cover a wide range of transient scales of motion.

Intermittency and coherent structures in two-dimensional turbulence

The authors study the mechanisms of enstrophy transfer in 2D turbulence. By theoretical discussion of the fragmentation process at very small scales, the authors show that enstrophy transfer must be space filling, due to the regularity of the flow at any time. Detailed analysis of a numerical decaying experiment confirms this picture. Yet 'intermittency' is acting at large scale due to the presence of coherent structures, namely vortices, which inhibit local enstrophy transfer.

The Integrated Enstrophy Budget of the Winter Stratosphere Diagnosed from LIMS Data

Abstract The Northern Hemisphere, quasi-geostrophic, integrated enstrophy budget for the 1978/79 winter has been analyzed from 10-0.1 mb using LIMS data. The stratospheric integrated enstrophy builds up during early winter as a result of the diabatic forcing of the polar vortex. Starting in January, an irregular and generally irreversible transfer of enstrophy from the zonal mean reservoir to planetary waves begins to reduce the total. This transfer of enstrophy to the waves produces significant imbalances in the integrated enstrophy budget at 10 mb. The imbalances appear to result from the transfer of enstrophy to smaller scales not resolved by the LIMS instrument. We believe that these imbalances are signature of Rossby wave breaking, as the imbalance episodes correspond well to the appearance of Ertel Vorticity filaments recently shown by McIntyre and Palmer. In the mesosphere the total enstrophy shows little seasonal trend. Our analysis indicates that the mesophere may be region of continuous wave bre...

Comparison of direct numerical simulation of two-dimensional turbulence with two-point closure: the effects of intermittency

We compare results of high-resolution direct numerical simulation with equivalent two-point moment closure (the test-field model) for both randomly forced and spin-down problems. Our results indicate that moment closure is an adequate representation of observed spectra only if the random forcing is sufficiently strong to disrupt the dynamical tendency to form intermittent isolated vortices. For strong white-noise forcing near a lower-wavenumber cut-off, theory and simulation are in good agreement except in the dissipation range, with an enstrophy range less steep than the wavenumber to the minus fourth power. If the forcing is weak in amplitude, red noise, and at large wavenumbers, significant errors are made by the closure, particularly in the inverse-cascade range. For spin-down problems at large Reynolds numbers, the closure considerably overestimates enstrophy transfer to small scales, as well as energy transfer to large scales. We finally discuss the possibility that the closure errors are related to intermittency of various types. Intermittency can occur in either the inverse-cascade range (forced equilibrium) or the intermediate scales (spin-down), with isolated concentrations of vorticity forming the associated coherent structures, or it can occur in the dissipation range owing to the nonlinear amplification of variations in the cascade rate (Kraichnan 1967).

Nonequilibrium statistical mechanics of two-dimensional turbulence

A complete set of independent real parameters and its dynamic equation are worked out to describe the vorticity dynamics of two-dimensional turbulence. The corresponding Liouville equation is solved by a perturbation method upon the basis of a Langevin–Fokker–Plank Model. The dynamic damping coefficient η of the LFP model is treated as an optimum control parameter to minimize the error of the perturbation solution. Thereby two integral equations, the enstrophy equation and the η equation, are obtained for two unknown functions: the spectrum and the η. The equilibrium spectrum for the inviscid case is obtained as a stationary solution of the enstrophy equation. The nonlocalness of the enstrophy transfer makes the enstrophy equation divergent for a simple power-law spectrum. In order to avoid the divergence problem, a localization factor g is introduced to characterize the actual spectrum. Finally, the localized forms of the two integral equations are numerically solved, leading to the inertial-range spectrum, E(k)=1.82(ln g−1.23)−2/3χ2/3k−3 for g≥10, χ is the dissipation rate of the enstrophy.

Observation of Wave-Wave Interactions in the Stratosphere

Data from the LIMS instrument for January 1979 are used to provide further evidence for the often observed vacillation between the amplitudes of waves 1 and 2 in the stratosphere. The vacillation is shown to result primarily from nonlinear wave-wave interactions within the stratosphere. Two ways of interpreting nonlinearity are discussed. In the first, the basic state is defined to include large amplitude waves as well as the mean zonal wind. A forced wave propagates with respect to this asymmetric basic state, which can lead to changes in the conventional zonal wavenumber measured at one latitude. The other view of nonlinearity, interaction of wave with the zonal flow and with other wavenumbers are considered separately. Wave-wave interactions among waves 1, 2 and 3 are calculated. The derivation and computation of wave-wave interaction terms in the potential enstrophy balance are given. The observations indicate that enstrophy transfer among waves can be substantial even when the amplitude of one of the contributing waves is small. The computed enstrophy balance also demonstrates that wave-wave interactions can have a large effect on the interaction of waves with the mean flow.

Nonlinear energy and enstrophy transfers in a realistically stratified ocean

We discuss the nonlinear transfers possible in a quasigeostrophic fluid with a basic stratification taken from oceanic data. The energy and enstrophy conservation laws imply a cascade of energy to larger total scale (including both the horizontal scale, defined as wavelength/2π, and the deformation radius of the vertical mode). The triplet interactions among components with various horizontal scales and vertical structures, represented by the vertical mode numbers, are considered in detail for exchanges involving the barotropic and first three baroclinic modes. The initial transfer rates from one component into the other two are estimated and the most rapid transfers described as a function of the initial scale and mode number. These results suggest that barotropic motions will cascade to larger-scale barotropic motions, first baroclinic small-scale motions will transfer to first baroclinic larger scales, and first baroclinic large-scale motions will cascade to barotropic and first baroclinic motions at the deformation scale. Second and third mode motions prefer to transfer energy into small-scale (second or third mode deformation radius) first and third baroclinic mode motions.We also show the relationship of these triplet interactions to Rossby wave instabilities and resonant triads. For the latter motions, the weakness of the nonlinearity adds additional constraints which impty that the motions will tend to become zonal.

Two-dimensional turbulence on the surface of a sphere

Large-scale atmospheric flow shares certain attributes with two-dimensional turbulence. In this paper, we study the effect of spherical geometry on two-dimensional turbulence.Energy transfer is multi-component in spherical geometry in contrast to energy transfer among triads of wave vectors in Cartesian geometry. It follows that energy transfer is more local in spherical than in Cartesian geometry. Enstrophy transfer to higher wavenumbers in spherical geometry is less than enstrophy transfer to higher wavenumbers in Cartesian geometry. Since both energy and enstrophy are inviscid constants of motion, the back transfer of energy is also less in spherical than in Cartesian geometry. Therefore, with a finite viscosity, enstrophy decays more slowly in spherical geometry than in Cartesian geometry. Here these conjectures are tested numerically by spectral methods. The numerical results agree well with the conjectures.

Subgrid-Scale Modeling of Enstrophy Transfer in Two-Dimensional Turbulence

Abstract A method of parameterization of the energy and enstropy transfers involving subgrid scales is proposed in order to predict the time evolution of the energy spectrum at larger scales. The method is based on the strong non-localness of enstrophy transfer within the enstrophy inertial range of turbulence in two dimensions. It is applied first to a master equation derived from an eddy-damped quasi-normal Markovian model. Comparisons with reference calculations including all the scales up to the dissipation range show that the method simulates the enstrophy inertial range accurately up to the cutoff wavenumber. It is then generalized to direct simulation of the two-dimensional Navier-Stokes equation: the inertial range is again accurately simulated.