Conservation Of Enstrophy Research Articles

Equivalence of nonequilibrium ensembles: Two-dimensional turbulence with a dual cascade.

We examine the conjecture of equivalence of nonequilibrium ensembles for turbulent flows in two dimensions in a dual-cascade setup. We construct a formally time-reversible Navier-Stokes equationin two dimensions by imposing global constraints of energy and enstrophy conservation. A comparative study of the statistical properties of its solutions with those obtained from the standard Navier-Stokes equationsclearly shows that a formally time-reversible system is able to reproduce the features of a two-dimensional turbulent flow. Statistical quantities based on one- and two-point measurements show an excellent agreement between the two systems for the inverse- and direct-cascade regions. Moreover, we find that the conjecture holds very well for two-dimensional turbulent flows with both conserved energy and enstrophy at finite Reynolds number.

Selective energy and enstrophy modification of two-dimensional decaying turbulence

In two-dimensional decaying homogeneous isotropic turbulence, kinetic energy and enstrophy are respectively transferred to larger and smaller scales. In such spatiotemporally complex dynamics, it is challenging to identify the important flow structures that govern this behaviour. We propose and employ numerically two flow-modification strategies that leverage the inviscid global conservation of energy and enstrophy to design external forcing inputs that change these quantities selectively and simultaneously, and drive the system towards steady-state or other late-stage behaviour. One strategy employs only local flow field information, while the other is global. We observe various flow structures excited by these inputs and compare them with recent literature. Energy modification is characterized by the excitation of smaller wavenumber structures in the flow than enstrophy modification.

Enstrophy from symmetry

We study symmetry principles associated with the approximately conserved enstrophy current, responsible for the inverse energy cascade in non relativistic 2+1 dimensional turbulence. We do so by identifying the accidental symmetry associated with enstrophy current conservation in a recently realized effective action principle for hydrodynamics. Our analysis deals with both relativistic and non relativistic effective actions and their associated symmetries.

Arbitrary Order Energy and Enstrophy Conserving Finite Element Methods for 2d Incompressible Fluid Dynamics and Drift-Reduced Magnetohydrodynamics

Arbitrary Order Energy and Enstrophy Conserving Finite Element Methods for 2d Incompressible Fluid Dynamics and Drift-Reduced Magnetohydrodynamics

Vortex Layer on the β-Plane in the Miles – Ribner Formulation. Pole on the Real Axis

Purpose. The problem of a non-zonal vortex layer on the β-plane in the Miles – Ribner formulation is considered. It is known that in the absence of the β-effect, the vortex layer has no neutral eigenmodes, and the available two ones (varicose and sinusoidal) are unstable. Initially, generalization of the problem to the β-plane concerned only the zonal case. The problem for a non-zonal vortex layer is examined for the first time in the paper. It is known that in the WKB approximation for the linear wave disturbances (regardless of whether a zonal or non-zonal background flow is considered), there is an adiabatic invariant in the form of the law of the enstrophy (vorticity) conservation. For the zonal vortex layer, the enstrophy conservation law also holds, and no vorticity exchange occurs between the waves and the flow in the zonal case. The non-zonal vortex layer has qualitatively different features; particularly, it does not retain enstrophy. Thus, as a result, there appears a new class of solutions which can be interpreted as pure radiation of the Rossby waves by a non-zonal flow. Generalizing the vortex layer problem on the β-plane to the non-zonal case constitutes the basic aim of the present study. Methods and Results. A new class of linear stationary wave solutions, namely the Rossby waves, is found. It is shown a non-zonal flow can be directed in one way, whereas the stationary wave disturbances can move in the opposite (contrary) direction. The coexistence of such solutions for the shear non-zonal flow and stationary wave disturbances takes place due to the influence of the external force and mathematically comes from a non-self-adjoining character of the linear operator for a non-zonal background flow. Conclusions. There exists a new class of solutions that can be interpreted as pure radiation of the Rossby waves by a non-zonal flow. There is no such solution for a zonal flow. It is just non-zoning that gives the effect of pure radiation and corresponds to the classical definition of radiation. This approach makes it possible to eliminate inconsistency in terminology, when instabilities are mistakenly called radiation, and radiation – pure radiation.

Vikhrevoy sloy na b-ploskosti v formulirovke Maylsa – Ribnera. Polyus na deystvitel'noy osi

Purpose. The problem of a non-zonal vortex layer on the β-plane in the Miles – Ribner formulation is considered. It is known that in the absence of the β-effect, the vortex layer has no neutral eigenmodes, and the available two ones (varicose and sinusoidal) are unstable. Initially, generalization of the problem to the β-plane concerned only the zonal case. The problem for a non-zonal vortex layer is examined for the first time in the paper. It is known that in the WKB approximation for the linear wave disturbances (regardless of whether a zonal or non-zonal background flow is considered), there is an adiabatic invariant in the form of the law of the enstrophy flow (vorticity) conservation. For the zonal vortex layer, the enstrophy conservation law also holds, and no vorticity exchange occurs between the waves and the flow in the zonal case. The non-zonal vortex layer has qualitatively different features; particularly, it does not retain enstrophy. Thus, as a result, there appears a new class of solutions which can be interpreted as pure radiation of the Rossby waves by a non-zonal flow. Generalizing the vortex layer problem on the β-plane to the non-zonal case constitutes the basic aim of the present study. Methods and Results. A new class of linear stationary wave solutions, namely the Rossby waves, is found. It is shown a non-zonal flow can be directed in one way, whereas the stationary wave disturbances can move in the opposite (contrary) direction. The coexistence of such solutions for the shear non-zonal flow and stationary wave disturbances takes place due to the influence of the external force and mathematically comes from a non-self-adjoint character of the linear operator for a non-zonal background flow. Conclusion. There exists a new class of solutions that can be interpreted as pure radiation of the Rossby waves by a non-zonal flow. There is no such solution for a zonal flow. It is just non-zoning that gives the effect of pure radiation and corresponds to the classical definition of radiation. This approach makes it possible to eliminate inconsistency in terminology, when instabilities are mistakenly called radiation, and radiation – pure radiation.

Downscale transfer of quasigeostrophic energy catalyzed by near-inertial waves

Abstract

Phase Transition to Large Scale Coherent Structures in Two-Dimensional Active Matter Turbulence.

The collective motion of microswimmers in suspensions induce patterns of vortices on scales that are much larger than the characteristic size of a microswimmer, attaining a state called bacterial turbulence. Hydrodynamic turbulence acts on even larger scales and is dominated by inertial transport of energy. Using an established modification of the Navier-Stokes equation that accounts for the small-scale forcing of hydrodynamic flow by microswimmers, we study the properties of a dense suspension of microswimmers in two dimensions, where the conservation of enstrophy can drive an inverse cascade through which energy is accumulated on the largest scales. We find that the dynamical and statistical properties of the flow show a sharp transition to the formation of vortices at the largest length scale. The results show that 2D bacterial and hydrodynamic turbulence are separated by a subcritical phase transition.

Fast and slow resonant triads in the two-layer rotating shallow water equations

In this paper, we examine triad resonances in a rotating shallow water system when there are two free interfaces. This allows for an examination in a relatively simple model of the interplay between baroclinic and barotropic dynamics in a context where there is also a geostrophic mode. In contrast to the much-studied one-layer rotating shallow water system, we find that as well as the usual slow geostrophic mode, there are now two fast waves, a barotropic mode and a baroclinic mode. This feature permits triad resonances to occur between three fast waves, with a mixture of barotropic and baroclinic modes, an aspect that cannot occur in the one-layer system. There are now also two branches of the slow geostrophic mode, with a repeated branch of the dispersion relation. The consequences are explored in a derivation of the full set of triad interaction equations, using a multiscale asymptotic expansion based on a small-amplitude parameter. The derived nonlinear interaction coefficients are confirmed using energy and enstrophy conservation. These triad interaction equations are explored, with an emphasis on the parameter regime with small Rossby and Froude numbers.

Energy–enstrophy conserving compatible finite element schemes for the rotating shallow water equations with slip boundary conditions

We describe an energy–enstrophy conserving discretisation for the rotating shallow water equations with slip boundary conditions. This relaxes the assumption of boundary-free domains (periodic solutions or the surface of a sphere, for example) in the energy–enstrophy conserving formulation of McRae and Cotter (2014). This discretisation requires extra prognostic vorticity variables on the boundary in addition to the prognostic velocity and layer depth variables. The energy–enstrophy conservation properties hold for any appropriate set of compatible finite element spaces defined on arbitrary meshes with arbitrary boundaries. We demonstrate the conservation properties of the scheme with numerical solutions on a rotating hemisphere.

Numerical simulation of the geometrical-optics reduction of CE2 and comparisons to quasilinear dynamics

Zonal flows have been observed to appear spontaneously from turbulence in a number of physical settings. A complete theory for their behavior is still lacking. Recently, a number of studies have investigated the dynamics of zonal flows using quasilinear (QL) theories and the statistical framework of a second-order cumulant expansion (CE2). A geometrical-optics (GO) reduction of CE2, derived under an assumption of separation of scales between the fluctuations and the zonal flow, is studied here numerically. The reduced model, CE2-GO, has a similar phase-space mathematical structure to the traditional wave-kinetic equation, but that wave-kinetic equation has been shown to fail to preserve enstrophy conservation and to exhibit an ultraviolet catastrophe. CE2-GO, in contrast, preserves nonlinear conservation of both energy and enstrophy. We show here how to retain these conservation properties in a pseudospectral simulation of CE2-GO. We then present nonlinear simulations of CE2-GO and compare with direct simulations of quasilinear (QL) dynamics. We find that CE2-GO retains some similarities to QL. The partitioning of energy that resides in the zonal flow is in good quantitative agreement between CE2-GO and QL. On the other hand, the length scale of the zonal flow does not follow the same qualitative trend in the two models. Overall, these simulations indicate that CE2-GO provides a simpler and more tractable statistical paradigm than CE2, but CE2-GO is missing important physics.

Impact of domain anisotropy on the inverse cascade in geostrophic turbulent convection

The effect of domain anisotropy on the inverse cascade occurring within the geostrophic turbulence regime of rapidly rotating Rayleigh–Bénard convection is investigated. In periodic domains with square cross-section in the horizontal, a domain-filling dipole state is present. For rectangular periodic domains, a Kolmogorov-like flow parallel to the short side and consisting of a periodic array of alternating unidirectional jets with embedded vortices is observed, together with an underlying weak meandering transverse jet. Similar transitions occurring in weakly dissipative two-dimensional flows driven by externally imposed small-amplitude noise and in classical hydrostatic geostrophic turbulence are a consequence of inviscid conservation of energy and potential enstrophy, and can be understood using statistical mechanics considerations. Rotating Rayleigh–Bénard convection represents an important three-dimensional system with only one inviscid invariant which nonetheless exhibits large-scale structures driven by intrinsically generated fluctuations.

Discrete conservation properties for shallow water flows using mixed mimetic spectral elements

A mixed mimetic spectral element method is applied to solve the rotating shallow water equations. The mixed method uses the recently developed spectral element histopolation functions, which exactly satisfy the fundamental theorem of calculus with respect to the standard Lagrange basis functions in one dimension. These are used to construct tensor product solution spaces which satisfy the generalized Stokes theorem, as well as the annihilation of the gradient operator by the curl and the curl by the divergence. This allows for the exact conservation of first order moments (mass, vorticity), as well as higher moments (energy, potential enstrophy), subject to the truncation error of the time stepping scheme. The continuity equation is solved in the strong form, such that mass conservation holds point wise, while the momentum equation is solved in the weak form such that vorticity is globally conserved. While mass, vorticity and energy conservation hold for any quadrature rule, potential enstrophy conservation is dependent on exact spatial integration. The method possesses a weak form statement of geostrophic balance due to the compatible nature of the solution spaces and arbitrarily high order spatial error convergence.

Ensemble‐type Kalman filter algorithm conserving mass, total energy and enstrophy

For numerical discretization schemes, the violation of enstrophy conservation causes a systematic and unrealistic energy cascade towards high wave numbers. The same occurs in data assimilation schemes, where the total energy, enstrophy and divergence could be strongly affected. In this article, we construct an ensemble data assimilation algorithm that conserves mass, total energy and enstrophy. The algorithm uses B‐spline functions for localization and sequential quadratic programming to solve nonlinear constrained minimization problem. Idealized experiments are performed using a 2D shallow‐water model, with selected contraints derived from the nature run. It is found that all experiments exhibit comparable root‐mean‐square errors, with a slight advantage for those that include the conservation constraint on the globally integrated enstrophy. However, the kinetic energy and enstrophy spectra in experiments with the enstrophy constraint are considerably closer to the true spectra, in particular at the smallest resolvable scales. Therefore, imposing conservation of enstrophy within the data assimilation algorithm effectively avoids the spurious energy cascade of the rotational part and thereby successfully suppresses the noise generated by the data assimilation algorithm. The 14 day deterministic free forecast, starting from the initial condition enforced by both total energy and enstrophy constraints, produces the best prediction. The same holds for the ensemble free forecasts.

Vorticity budgets in premixed combusting turbulent flows at different Lewis numbers

A direct numerical simulations database of statistically planar turbulent premixed flames using a simple Arrhenius type irreversible chemistry for different values of global Lewis numbers, Le, (0.34, 0.60, 0.80, 1.00, 1.20) has been examined to analyze the effects of Le on vorticity transport within the flame. To meet this objective, a general enstrophy conservation equation has been considered, which distinctly describes contributions from vortex-stretching, destruction by volumetric dilatation rates, baroclinic and viscous force torques, viscous transport, and dissipation. The average statistical behavior of the various contributions conditioned upon the value of the reaction progress variable, c, has been analyzed in the preheat and reacting regions of the flame. The mean values of enstrophy monotonically decays with c from fresh reactants toward hot products for Le equal to 0.8, 1.0, and 1.2; vortex-stretching and viscous dissipation are the leading contributors, while the remaining contributions are slightly smaller although non-negligible. By contrast, the mean value of enstrophy decreases from the leading edge before increasing up to the trailing edge of the flame; in these cases, the mean value of baroclinic torque is significantly greater than the other contributions in most of the preheat and reacting regions; vortex-stretching, destruction by volumetric dilatation rates and viscous transport, and dissipation remain comparable over most of the flame. An explanation for the significant qualitative and quantitative differences in the enstrophy transport, taking place for Le between 0.6 and 0.8 for the given turbulence intensity, is sought in terms of the alignments of vorticity and the gradients of density, pressure, temperature, and reaction progress variable. The transport statistics of the enstrophies of the vorticity vector components tangential and normal to iso-scalar surfaces, c(x, t) = constant, provide further insight into the mechanisms of the differences in the enstrophy transport in response to the changes in the global Lewis number.

A Potential Enstrophy and Energy Conserving Scheme for the Shallow-Water Equations Extended to Generalized Curvilinear Coordinates

An energy and potential enstrophy conserving finite-difference scheme for the shallow-water equations is derived in generalized curvilinear coordinates. This is an extension of a scheme formulated by Arakawa and Lamb for orthogonal coordinate systems. The starting point for the present scheme is the shallow-water equations cast in generalized curvilinear coordinates, and tensor analysis is used to derive the invariant conservation properties. Preliminary tests on a flat plane with doubly periodic boundary conditions are presented. The scheme is shown to possess similar order-of-convergence error characteristics using a nonorthogonal coordinate compared to Cartesian coordinates for a nonlinear test of flow over an isolated mountain. A linear normal mode analysis shows that the discrete form of the Coriolis term provides stationary geostrophically balanced modes for the nonorthogonal coordinate and no unphysical computational modes are introduced. The scheme uses centered differences and averages, which are formally second-order accurate. An empirical test with a steady geostrophically balanced flow shows that the convergence rate of the truncation errors of the discrete operators is second order. The next step will be to adapt the scheme for use on the cubed sphere, which will involve modification at the lateral boundaries of the cube faces.

Total energy and potential enstrophy conserving schemes for the shallow water equations using Hamiltonian methods – Part 1: Derivation and properties

Abstract. The shallow water equations provide a useful analogue of the fully compressible Euler equations since they have similar characteristics: conservation laws, inertia-gravity and Rossby waves, and a (quasi-) balanced state. In order to obtain realistic simulation results, it is desirable that numerical models have discrete analogues of these properties. Two prototypical examples of such schemes are the 1981 Arakawa and Lamb (AL81) C-grid total energy and potential enstrophy conserving scheme, and the 2007 Salmon (S07) Z-grid total energy and potential enstrophy conserving scheme. Unfortunately, the AL81 scheme is restricted to logically square, orthogonal grids, and the S07 scheme is restricted to uniform square grids. The current work extends the AL81 scheme to arbitrary non-orthogonal polygonal grids and the S07 scheme to arbitrary orthogonal spherical polygonal grids in a manner that allows for both total energy and potential enstrophy conservation, by combining Hamiltonian methods (work done by Salmon, Gassmann, Dubos, and others) and discrete exterior calculus (Thuburn, Cotter, Dubos, Ringler, Skamarock, Klemp, and others). Detailed results of the schemes applied to standard test cases are deferred to part 2 of this series of papers.

Scaling theory for vortices in the two-dimensional inverse energy cascade

We propose a new similarity theory for the two-dimensional inverse energy cascade and the coherent vortex population it contains when forced at intermediate scales. Similarity arguments taking into account enstrophy conservation and a prescribed constant energy injection rate such that $E\sim t$ yield three length scales, $l_{\unicode[STIX]{x1D714}}$, $l_{E}$ and $l_{\unicode[STIX]{x1D713}}$, associated with the vorticity field, energy peak and streamfunction, and predictions for their temporal evolutions, $t^{1/2}$, $t$ and $t^{3/2}$, respectively. We thus predict that vortex areas grow linearly in time, $A\sim l_{\unicode[STIX]{x1D714}}^{2}\sim t$, while the spectral peak wavenumber $k_{E}\equiv 2\unicode[STIX]{x03C0}l_{E}^{-1}\sim t^{-1}$. We construct a theoretical framework involving a three-part, time-evolving vortex number density distribution, $n(A)\sim t^{\unicode[STIX]{x1D6FC}_{i}}A^{-r_{i}},~i\in 1,2,3$. Just above the forcing scale ($i=1$) there is a forcing-equilibrated scaling range in which the number of vortices at fixed $A$ is constant and vortex ‘self-energy’ $E_{v}^{cm}=(2{\mathcal{D}})^{-1}\int \overline{\unicode[STIX]{x1D714}_{v}^{2}}A^{2}n(A)\,\text{d}A$ is conserved in $A$-space intervals $[\unicode[STIX]{x1D707}A_{0}(t),A_{0}(t)]$ comoving with the growth in vortex area, $A_{0}(t)\sim t$. In this range, $\unicode[STIX]{x1D6FC}_{1}=0$ and $n(A)\sim A^{-3}$. At intermediate scales ($i=2$) sufficiently far from the forcing and the largest vortex, there is a range with a scale-invariant vortex size distribution. We predict that in this range the vortex enstrophy $Z_{v}^{cm}=(2{\mathcal{D}})^{-1}\int \overline{\unicode[STIX]{x1D714}_{v}^{2}}An(A)\,\text{d}A$ is conserved and $n(A)\sim t^{-1}A^{-1}$. The final range ($i=3$), which extends over the largest vortex-containing scales, conserves $\unicode[STIX]{x1D70E}_{v}^{cm}=(2{\mathcal{D}})^{-1}\int \overline{\unicode[STIX]{x1D714}_{v}^{2}}n(A)\,\text{d}A$. If $\overline{\unicode[STIX]{x1D714}_{v}^{2}}$ is constant in time, this is equivalent to conservation of vortex number $N_{v}^{cm}=\int n(A)\,\text{d}A$. This regime represents a ‘front’ of sparse vortices, which are effectively point-like; in this range we predict $n(A)\sim t^{r_{3}-1}A^{-r_{3}}$. Allowing for time-varying $\overline{\unicode[STIX]{x1D714}_{v}^{2}}$ results in a small but significant correction to these temporal dependences. High-resolution numerical simulations verify the predicted vortex and spectral peak growth rates, as well as the theoretical picture of the three scaling ranges in the vortex population. Vortices steepen the energy spectrum $E(k)$ past the classical $k^{-5/3}$ scaling in the range $k\in [k_{f},k_{v}]$, where $k_{v}$ is the wavenumber associated with the largest vortex, while at larger scales the slope approaches $-5/3$. Though vortices disrupt the classical scaling, their number density distribution and evolution reveal deeper and more complex scale invariance, and suggest an effective theory of the inverse cascade in terms of vortex interactions.

Zonal-flow dynamics from a phase-space perspective

The wave kinetic equation (WKE) describing drift-wave (DW) turbulence is widely used in the studies of zonal flows (ZFs) emerging from DW turbulence. However, this formulation neglects the exchange of enstrophy between DWs and ZFs and also ignores effects beyond the geometrical-optics limit. We derive a modified theory that takes both of these effects into account, while still treating DW quanta (“driftons”) as particles in phase space. The drifton dynamics is described by an equation of the Wigner–Moyal type, which is commonly known in the phase-space formulation of quantum mechanics. In the geometrical-optics limit, this formulation features additional terms missing in the traditional WKE that ensure exact conservation of the total enstrophy of the system, in addition to the total energy, which is the only conserved invariant in previous theories based on the WKE. Numerical simulations are presented to illustrate the importance of these additional terms. The proposed formulation can be considered as a phase-space representation of the second-order cumulant expansion, or CE2.

Role of helicity in triad interactions in three-dimensional turbulence investigated by a new shell model.

Fully developed homogeneous isotropic turbulence in two dimensions is fundamentally different from that in three dimensions. In two dimensions, the simultaneous inviscid conservation of both kinetic energy and enstrophy within the inertial range of scales leads to a forward cascade of enstrophy and a reverse cascade of energy. In three dimensions, helicity, the integral of the scalar product of velocity and vorticity, is also an inviscid flow invariant along with the energy. Unlike the enstrophy, however, the helicity does not block the forward cascade of energy to small scales. Energy and helicity are conserved not only globally but also within each nonlinear triadic interaction between three plane waves in the spectral form of the Navier-Stokes equation (NSE). By decomposing each plane wave into two helical modes of opposite helicities, each triadic interaction is split into a set of eight helical triadic interactions between helical modes [F. Waleffe, Phys. Fluids A 4, 350 (1992)PFADEB0899-821310.1063/1.858309]. Recently it was found that a subset of these helical interactions, which render both signs of helicity separately conserved (enstrophy-like), leads to an inverse cascade of (part of) the energy [L. Biferale et al., Phys. Rev. Lett. 108, 164501 (2012)PRLTAO0031-900710.1103/PhysRevLett.108.164501]. Motivated by this finding we introduce a new shell model, obtained from the NSE expressed in the helical basis, allowing the eight helical interactions to be coupled as in the NSE and their relative contributions evaluated as a function of both the net helicity input and triad geometry. By numerically integrating the new model, we find that the intermittency of the energy cascade decreases with the net helicity input. Studying the partitioning of the energy cascade between the eight helical interactions, we find that the decrease in intermittency is related to a shift in the dominating helical interactions when helically forced, two of which exhibit a larger cascade intermittency than the other six interactions. Among the relatively local triad geometries considered here, the partitioning of the energy and helicity cascades between the eight helical interactions shows no sign of change with triad geometry.