Enstrophy Cascade Research Articles

The Formation of Concentric Vorticity Structures in Typhoons

Abstract An important issue in the formation of concentric eyewalls in a tropical cyclone is the development of a symmetric structure from asymmetric convection. It is proposed herein, with the aid of a nondivergent barotropic model, that concentric vorticity structures result from the interaction between a small and strong inner vortex (the tropical cyclone core) and neighboring weak vortices (the vorticity induced by the moist convection outside the central vortex of a tropical cyclone). The results highlight the pivotal role of the vorticity strength of the inner core vortex in maintaining itself, and in stretching, organizing, and stabilizing the outer vorticity field. Specifically, the core vortex induces a differential rotation across the large and weak vortex to strain out the latter into a vorticity band surrounding the former. The straining out of a large, weak vortex into a concentric vorticity band can also result in the contraction of the outer tangential wind maximum. The stability of the outer band is related to the Fjørtoft sufficient condition for stability because the strong inner vortex can cause the wind at the inner edge to be stronger than the outer edge, which allows the vorticity band and therefore the concentric structure to be sustained. Moreover, the inner vortex must possess high vorticity not only to be maintained against any deformation field induced by the outer vortices but also to maintain a smaller enstrophy cascade and to resist the merger process into a monopole. The negative vorticity anomaly in the moat serves as a “shield” or a barrier to the farther inward mixing the outer vorticity field. The binary vortex experiments described in this paper suggest that the formation of a concentric vorticity structure requires 1) a very strong core vortex with a vorticity at least 6 times stronger than the neighboring vortices, 2) a large neighboring vorticity area that is larger than the core vortex, and 3) a separation distance between the neighboring vorticity field and the core vortex that is within 3 to 4 times the core vortex radius.

Rapid beat generation of large-scale zonal flows by drift waves: a nonlinear generic paradigm.

The generalized Charney-Hasegawa-Mima equation is unstable to a four wave modulational instability whereby a coherent, monochromatic drift wave can drive a band of modes and associated zonal flows unstable. Although initially the fastest growing modes dominate, a secondary nonlinear instability later drives the longest wavelength zonal flow and its associated sidebands at twice the growth rate of the fastest growing modulationally unstable modes. This results in a direct transfer from strongly unstable short wavelength modes to the weakly unstable long wavelength modes, which drains the short wavelength pump energy. A related but less efficient direct enstrophy cascade generates very short wavelength modes lying outside the band of modulationally unstable modes.

Dispersion in the enstrophy cascade of two-dimensional decaying grid turbulence.

The use of vertically flowing soap films allows one to obtain decaying two-dimensional turbulence with an enstrophy cascade range. Here we use this property to study the dispersion of a fine column of slightly heated liquid entering the turbulent flow. The width of this column increases with time in an exponential manner consistent with theoretical predictions for the average dispersion of particle pairs despite the multiparticle nature of the dispersion studied. Other features such as Gaussian mean profiles of the temperature across the channel width are also evidenced.

Robustness of the inverse cascade in two-dimensional turbulence.

We study quasisteady inverse cascades in unbounded and bounded two-dimensional turbulence driven by time-independent injection and dissipated by molecular viscosity. It is shown that an inverse cascade that carries only a fraction r of the energy input to the largest scales requires the enstrophy-range energy spectrum to be steeper than k(-5) (ruling out a direct cascade) unless 1-r<<1. A direct cascade requires the presence of an inverse cascade that carries virtually all energy to the largest scales (1-r<<1). These facts underlie the robustness of the Kolmogorov-Kraichnan k(-5/3) inverse cascade, which is readily observable in numerical simulations without an accompanying direct enstrophy cascade. We numerically demonstrate an instance where the k(-5/3) inverse-cascading range is realizable with 79% of the energy injection dissipated within the energy range and virtually all of the enstrophy dissipated in the vicinity of the forcing region. As equilibrium is approached, the respective logarithmic slopes -alpha and -beta of the ranges of wave numbers lower and higher than the forcing wave number satisfy alpha+beta approximately 8. These results are consistent with recent theoretical predictions.

Physical mechanism and model of turbulent cascades in a barotropic atmosphere

In a barotropic atmosphere, new Reynolds mean momentum equations including turbulent viscosity, dispersion, and instability are used not only to derive the KdV-Burgers-Kuramoto equation but also to analyze the physical mechanism of the cascades of energy and enstrophy. It shows that it is the effects of dispersion and instability that result in the inverse cascade. Then based on the conservation laws of the energy and enstrophy, a cascade model is put forward and the processes of the cascades are described.

Nonlinear transfer and spectral distribution of energy in α turbulence

Two-dimensional turbulence governed by the so-called α turbulence equations, which include the surface quasi-geostrophic equation ( α=1), the Navier–Stokes system ( α=2), and the governing equation for a shallow flow on a rotating domain driven by a uniform internal heating ( α=3), is studied here in both the unbounded and doubly periodic domains. This family of equations conserves two inviscid invariants (energy and enstrophy in the Navier–Stokes case), the dynamics of which are believed to undergo a dual cascade. It is shown that an inverse cascade can exist in the absence of a direct cascade and that the latter is possible only when the inverse transfer rate of the inverse-cascading quantity approaches its own injection rate. Constraints on the spectral exponents in the wavenumber ranges lower and higher than the injection range are derived. For Navier–Stokes turbulence with moderate Reynolds numbers, the realization of an inverse energy cascade in the complete absence of a direct enstrophy cascade is confirmed by numerical simulations.

Empirical Low-Order Models of Barotropic Flow

Empirical low-order models are constructed in the framework of a quasigeostrophic barotropic spectral model, truncated at T21. The spectral model is projected onto a linear subspace spanned by only a limited number of spatial structures called principal interaction patterns. The expansion coefficients of these modes are assumed to be governed by a forced, dissipative dynamical system with a quadratic nonlinearity that conserves turbulent kinetic energy and turbulent enstrophy. A simple empirical scheme for modeling the energy and enstrophy cascade assuming localization of the nonlinear interactions with respect to spatial scale is applied. The optimal low-order model, that is, the optimal basis functions and the optimal interaction coefficients, is determined by minimizing the mean-squared error between trajectories of the reduced model and trajectories of the T21 model. A model with 40 degrees of freedom succeeds in well capturing, in a long-term integration, the behavior of the T21 model monitored by the mean state, the variance pattern, autocorrelation functions, and probability distributions of the streamfunction. The model based on principal interaction patterns offers a substantial improvement on a model based on empirical orthogonal functions with the same number of degrees of freedom in capturing the autocorrelation function, the probability distribution, and the response of the system to a change in the forcing.

Physical mechanism of the two-dimensional enstrophy cascade.

In two-dimensional turbulence, irreversible forward transfer of enstrophy requires anticorrelation of the turbulent vorticity transport vector and the inertial-range vorticity gradient. We investigate the basic mechanism by numerical simulation of the forced Navier-Stokes equation. In particular, we obtain the probability distributions of the local enstrophy flux and of the alignment angle between vorticity gradient and transport vector. These are surprisingly symmetric and cannot be explained by a local eddy-viscosity approximation. The vorticity transport tends to be directed along streamlines of the flow and only weakly aligned down the fluctuating vorticity gradient. All these features are well explained by a local nonlinear model. The physical origin of the cascade lies in steepening of inertial-range vorticity gradients due to compression of vorticity level sets by the large-scale strain field.

Velocity statistics in two-dimensional granular turbulence.

We studied the macroscopic statistical properties on the freely evolving quasielastic hard disk (granular) system by performing a large-scale (up to a few million particles) event-driven molecular dynamics systematically and found it to be remarkably analogous to an enstrophy cascade process in the decaying two-dimensional fluid turbulence. There are four typical stages in the freely evolving inelastic hard disk system, which are homogeneous, shearing (vortex), clustering, and final state. In the shearing stage, the self-organized macroscopic coherent vortices become dominant. In the clustering stage, the energy spectra are close to the expectation of Kraichnan-Batchelor theory and the squared two-particle separation strictly obeys Richardson law.

Dispersion of passive tracers in the direct enstrophy cascade: Experimental observations

The paper reports a statistical analysis of the separation of pairs in the enstrophy cascade, in a two-dimensional flow. The flow is produced experimentally, using electromagnetic forcing. Two regimes of separation are found. At early times (i.e., within two integral times) the separation process is exponential, following Lin’s law [J. T. Lin, J. Atmos. Sci. 29, 394 (1972)]. In this range of time, the probability density functions (PDFs) of separations are self-similar in time, developing stretched exponential tails, and one may define a Lagrangian correlation time. Above two integral times, an algebraic regime takes place, with exponential tails for the PDFs of the separations, and self-similar Lagrangian correlation functions. The present work thus confirms the existence of two regimes, by analyzing a number of statistical quantities including the Lagrangian PDF and the temporal Lagrangian correlation functions which so far, were left undetermined for the Batchelor regime.

Relative dispersion at the surface of the Gulf of Mexico

We examine the relative motion of pairs and triplets of surface drifters in the Gulf of Mexico. The mean square pair separations grow exponentially in time from the smallest resolved scale (1 km) to 40‐ 50 km, with an e-folding time of 2‐ 3 days. Thereafter, the dispersion exhibits a power law dependence on time with an exponent of between 2 and 3 (depending on the measure used) up to scales of several hundred kilometers. The straining is for the most part isotropic, with only weak regional variations. But there are suggestions of anisotropy in the western basin, probably due to boundary current advection. The pair velocities are correlated during the early phase and a portion of the late phase. The relative displacement distributions during the early phase are, after an initial adjustment, nonGaussian and approximately constant, suggestive of local straining. The triplet results likewise suggest two growth phases. During the early phase, the mean area and the longest triangle leg grow exponentially in time, the latter with a rate consistent with the two-particle results. Most triangles are drawn out during this time. During the late period, the triangles grow and their aspect ratios systematically decrease, suggesting an evolution to an equilateral shape. Although surface divergences should affect these statistics, they nevertheless strongly resemble those found with two-dimensional turbulent e ows. If so, we would infer an enstrophy cascade at scales below the deformation radius (40‐ 50 km) which is probably spectrally local. The latter implies that growth in particle separations comes from e ow features the same size as the separations. It is also possible there is an inverse energy cascade to scales larger than the deformation radius, driven possibly by baroclinic instability. However, the late period statistics may also ree ect dispersion by a large scale shear.

Energy and enstrophy transfer in decaying two-dimensional turbulence.

We present experimental data on the direct enstrophy cascade in decaying two-dimensional turbulence. Velocity and vorticity fields are obtained using particle tracking velocimetry. From those fields we directly compute the enstrophy and energy flux by using a filtering technique inspired by large-eddy simulations. This allows considerable insight into the physical processes of turbulence when compared with structure-function or spectral analysis. The direct cascade of enstrophy is weakly forward, with almost as much backscatter as down-scale enstrophy transfer, whereas the inverse energy cascade is strongly upscale with a modest amount of backscatter.

Atmospheric Energetics in the Wavelet Domain. Part II: Time-Averaged Observed Atmospheric Blocking

Wavelet energetics (WE) is a useful generalization of traditional wavenumber energetics, for analyzing atmospheric dynamics. WE is doubly indexed by wavenumber band j and location k. The interpretation is that 2 to the jth power is proportional to zonal wavenumber bandwidth and bandcenter, and k is proportional to longitude. Here, all 44 Atlantic and 16 Pacific atmospheric blocking events that are observed in 53 winters of the NCEP–NCAR reanalysis, and last no less than 6 days, are analyzed. Temporal average and variance maps suggest that persistent blocking structures are associated with smaller-scale eddy activity concentrated on either side of the block. Wavelet energetics, partitioned by blocking state and sector, are averaged over 30°–80°N and 10–100 kPa. For j above two, WE indicates that kinetic energy (KE) and enstrophy increase upstream and decrease downstream of both Atlantic and Pacific blocking. At smaller j the increases are at the block; this includes zero j for Pacific, but not Atlantic. As measured by new wavelet flux functions, at j above one, on average there are localized upscale KE and enstrophy cascades upstream, and localized downscale cascades downstream of the blocks. This is not significantly determined for KE in the Atlantic. Correlating WE, blocking relative to nonblocking, suggests a similarity of Pacific to Atlantic energetic patterns if the former are shifted over the latter; this holds for all enstrophy WE, and for KE stocks, but not for other KE wavelet energetics. The theoretical conservation of wavelet flux is numerically supported. Statistical significance is strongly suggested, if not rigorously established.

On the dual cascade in two-dimensional turbulence

We study the dual-cascade scenario for two-dimensional (2D) turbulence driven by a spectrally localized forcing applied over a finite wavenumber range [ k min, k max] (with k min>0) such that the respective energy and enstrophy injection rates ϵ and η satisfy k min 2 ϵ≤ η≤ k max 2 ϵ. The classical Kraichnan–Leith–Batchelor paradigm, based on the simultaneous conservation of energy and enstrophy and the scale-selectivity of the molecular viscosity, requires that the domain be unbounded in both directions. For 2D turbulence either in a doubly periodic domain or in an unbounded channel with a periodic boundary condition in the across-channel direction, a direct enstrophy cascade is not possible. In the usual case where the forcing wavenumber is no greater than the geometric mean of the integral and dissipation wavenumbers, constant spectral slopes must satisfy β>5 and α+ β≥8, where − α (− β) is the asymptotic slope of the range of wavenumbers lower (higher) than the forcing wavenumber. The influence of a large-scale dissipation on the realizability of a dual cascade is analysed. We discuss the consequences for numerical simulations attempting to mimic the classical unbounded picture in a bounded domain.

Large‐scale non‐turbulent dynamics in the atmosphere

AbstractClassical two‐dimensional turbulence theory is often used to understand large‐scale atmospheric flows. However, the equations governing classical two‐dimensional turbulence can only be derived from the governing equations by assuming that the horizontal scale is much smaller than the deformation radius, which is the scale on which baroclinic instability takes place. Typically, the large‐scale quasi‐two‐dimensional disturbances have the vertical depth‐scale of the troposphere and are of larger scale than the baroclinic waves which maintain them. It is therefore more appropriate to study the quasi‐two‐dimensional disturbances with a model appropriate to scales larger than the deformation radius. The semi‐geostrophic model is an accurate model in this regime. It is shown that it does not permit the enstrophy cascades associated with classical two‐dimensional turbulence. This agrees with other results in the literature suggesting that two‐dimensional flow on scales larger than the deformation radius is essentially non‐turbulent. The observed quasi‐permanently unsteady behaviour of the atmosphere thus represents the natural internal dynamics, and does not require explanation by anisotropic forcing. In the ocean, on the other hand, the external deformation radius, which governs the behaviour of two‐dimensional eddies, is much larger than the internal deformation radius, which determines the scale of baroclinic development. It may thus be appropriate to use two‐dimensional turbulence as a model for scales between the internal and external deformation radius. Copyright © 2002 Royal Meteorological Society.

Topology of two-dimensional turbulence.

Velocity differences in the direct enstrophy cascade of two-dimensional turbulence are correlated with the underlying flow topology. The statistics of the transverse and longitudinal velocity differences are found to be governed by different structures. The wings of the transverse distribution are dominated by strong vortex centers, whereas the tails of the longitudinal differences are dominated by saddles. Viewed in the framework of earlier theoretical work, this result suggests that the transfer of enstrophy to smaller scales is accomplished in regions of the flow dominated by saddles.

Numerical solution of the two-layer shallow water equations with bottom topography

We present a simple, robust numerical method for solving the two-layer shallow water equations with arbitrary bottom topography. Using the technique of operator splitting, we write the equations as a pair of hyperbolic systems with readily computed characteristics, and apply third-order-upwind differences to the resulting wave equations. To prevent the thickness of either layer from vanishing, we modify the dynamics, inserting an artie cial form of potential energy that becomes very large as the layer becomes very thin. Compared to high-order Riemann schemes with e ux or slope limiters, our method is formally more accurate, probably less dissipative, and certainly more efe cient. However, because we do not exactly conserve momentum and mass, bores move at the wrong speed unless we add explicit, momentum-conserving viscosity. Numerical solutions demonstrate the accuracy and stability of the method. Solutions corresponding to two-layer, wind-driven ocean e ow require no explicit viscosity or hyperviscosity of any kind; the implicit hyperdiffusion associated with third-order-upwind differencing effectively absorbs the enstrophy cascade to small scales.

Stationary spectrum of vorticity cascade in two-dimensional turbulence.

The logarithmic renormalization predicted by Kraichnan (1971) for the direct cascade of enstrophy in the inertial range of two-dimensional turbulence has been observed in a numerical simulation. A moderate resolution allows for a very long time integration that provides very good statistics. Deviations from Gaussianity in the vorticity probability distribution are observed.

Constraints on the spectral distribution of energy and enstrophy dissipation in forced two-dimensional turbulence

We study two-dimensional (2D) turbulence in a doubly periodic domain driven by a monoscale-like forcing and damped by various dissipation mechanisms of the form ν μ (− Δ) μ . By “monoscale-like” we mean that the forcing is applied over a finite range of wavenumbers k min≤ k≤ k max, and that the ratio of enstrophy injection η≥0 to energy injection ε≥0 is bounded by k min 2 ε≤ η≤ k max 2 ε. Such a forcing is frequently considered in theoretical and numerical studies of 2D turbulence. It is shown that for μ≥0 the asymptotic behaviour satisfies ∥ u∥ 1 2≤ k max 2∥ u∥ 2, where ∥ u∥ 2 and ∥ u∥ 1 2 are the energy and enstrophy, respectively. If the condition of monoscale-like forcing holds only in a time-mean sense, then the inequality holds in the time mean. It is also shown that for Navier–Stokes turbulence ( μ=1), the time-mean enstrophy dissipation rate is bounded from above by 2 ν 1 k max 2. These results place strong constraints on the spectral distribution of energy and enstrophy and of their dissipation, and thereby on the existence of energy and enstrophy cascades, in such systems. In particular, the classical dual cascade picture is shown to be invalid for forced 2D Navier–Stokes turbulence ( μ=1) when it is forced in this manner. Inclusion of Ekman drag ( μ=0) along with molecular viscosity permits a dual cascade, but is incompatible with the log-modified −3 power law for the energy spectrum in the enstrophy-cascading inertial range. In order to achieve the latter, it is necessary to invoke an inverse viscosity ( μ<0). These constraints on permissible power laws apply for any spectrally localized forcing, not just for monoscale-like forcing.

Statistical Estimates for the Navier–Stokes Equations and the Kraichnan Theory of 2-D Fully Developed Turbulence

A mathematical formulation of the Kraichnan theory for 2-D fully developed turbulence is given in terms of ensemble averages of solutions to the Navier–Stokes equations. A simple condition is given for the enstrophy cascade to hold for wavenumbers just beyond the highest wavenumber of the force up to a fixed fraction of the dissipation wavenumber, up to a logarithmic correction. This is followed by partial rigorous support for Kraichnan's eddy breakup mechanism. A rigorous estimate for the total energy is found to be consistent with Kraichnan's theory. Finally, it is shown that under our conditions for fully developed turbulence the fractal dimension of the attractor obeys a sharper upper bound than in the general case.