Energy transfer in two-dimensional magnetohydrodynamic turbulence: formalism and numerical results

Spectral potential and kinetic energy densities and fluxes in horizontal wavenumber space are estimated in an eddy‐resolving model of the North Atlantic. In agreement to recent observational results near surface kinetic energy fluxes are negative over wide regions of the North Atlantic, indicative of an inverse energy cascade. This inverse kinetic energy cascade is found over a wide depth range but both the spectral kinetic energy density and the corresponding flux show no clear dependency on Rossby radius or Rhines scale. Potential energy fluxes tend to be positive and show a direct potential energy cascade towards a scale which is related to the Rossby radius in the subtropical North Atlantic.

Surface semigeostrophic (SSG) turbulence is examined in this study with emphasis on the effect of ageostrophy on energy cascades across the scales below the deformation radius. In our simulations, the strength of the ageostrophic component is controlled by the Rossby number , varying from 0.01 to 0.2. The flows are asymmetric with preference for cold cyclonic vortices and warm anticyclonic filaments. Strong vertical motions concentrate in small‐scale filaments and at the periphery of vortices where the lateral divergence becomes significant. A negative correlation between the divergence and the relative vorticity is identified using joint probability density functions. Slopes of the kinetic and potential energy spectra vary between −2.2 and −1.7. The features of the simulated flows including the asymmetry, strong vertical motion, and −2 spectral slope agree with the observations of the oceanic submesoscale flows. Analyses of spectral fluxes demonstrate an inverse kinetic energy cascade and a forward cascade of potential energy. As increases, the filaments become more numerous in the flows. They wrap around cyclones, weakening their interactions and subsequent mergers, thus suppressing the inverse cascade of kinetic energy. Ageostrophy promoting the forward potential energy cascade is important for the frontogenesis in the ocean. We characterize lateral dispersion in the SSG flows using the finite‐scale Lyapunov exponents (FSLEs). They are used to identify the Lagrangian coherent structures as well as to investigate the regimes of dispersion at different scales. The results show a smooth transition from hyper‐ballistic diffusion at small scales to normal diffusion at large scales.

An approximate energy-transfer function for isotropic turbulence is proposed on the basis of an analogy with radiative transfer in an inhomogeneous medium. An essential feature of the approximation is replacement of the actual triad interactions of the Fourier modes by interactions between pairs of modes. The interaction of each pair of modes satisfies detailed conservation of energy. Regardless of which mode has the higher wave number, the net transfer of energy is always from the more strongly excited to the more weakly excited mode of a pair. The transfer function gives an inertial range where the spectrum obeys the Kolmogorov law and where energy transfer is by local cascade. The spectrum in the far-dissipation range falls off sufficiently rapidly that all spatial derivatives of the velocity field exist in mean square.

Energy transfers between modes obtained from the proper orthogonal decomposition (POD) of a turbulent flow past a backward-facing step are analysed with the aim of providing guidelines for modelling unresolved modes in truncated POD–Galerkin models. It is observed that energy transfers are local in the POD basis, and that the Fourier-decomposition-based concepts of forward and backward energy cascades are also valid in the POD basis, the net effect being a forward energy cascade. General features of the eddy-viscosity representation of kinetic energy transfers are investigated through a priori tests. It is observed that the ideal eddy-viscosity model should exhibit a cusp behaviour near the cutoff mode.

The energy pathways in geostrophic turbulence are explored using a two-layer, flat-bottom, f-plane, quasigeostrophic model forced by an imposed, horizontally homogenous, baroclinically unstable mean flow and damped by bottom Ekman friction. A systematic presentation of the spectral energy fluxes, the mean flow forcing, and dissipation terms allows for a comprehensive understanding of the sources and sinks for baroclinic and barotropic energy as a function of length scale. The key new result is a robust inverse cascade of kinetic energy for both the baroclinic mode and the upper layer. This is consistent with recent observations of satellite altimeter data over the South Pacific Ocean. The well-known forward cascade of baroclinic potential and total energy was found to be very robust. Decomposing the spectral fluxes into contributions from different terms provided further insight. The inverse baroclinic kinetic energy cascade is driven mostly by an efficient interaction between the baroclinic velocity and the barotropic vorticity, the latter playing a crucial catalytic role. This cascade can be further enhanced by the baroclinic mode self-interaction, which is only present with nonuniform stratification (unequal layer depths). When model parameters are set such that modeled eddies compare favorably with observations, the inverse baroclinic kinetic energy cascade is actually much stronger than the well-known inverse cascade in the barotropic mode. The upper-layer kinetic energy cascade was found to dominate the lower-layer cascade over a wide range of parameters, suggesting that the surface cascade and time mean density stratification may be sufficient for estimating the depth-integrated cascade from ocean observations. This may find useful application in inferring the kinetic to gravitational potential energy conversion rate from satellite measurements.

We perform direct numerical simulations of quasi-static magnetohydrodynamic turbulence and compute various energy transfers including the ring-to-ring and conical energy transfers, and the energy fluxes of the perpendicular and parallel components of the velocity field. We show that the rings with higher polar angles transfer energy to ones with lower polar angles. For large interaction parameters, the dominant energy transfer takes place near the equator (polar angle θ≈π2). The energy transfers are local both in wavenumbers and angles. The energy flux of the perpendicular component is predominantly from higher to lower wavenumbers (inverse cascade of energy), while that of the parallel component is from lower to higher wavenumbers (forward cascade of energy). Our results are consistent with earlier results, which indicate quasi two-dimensionalization of quasi-static magnetohydrodynamic flows at high interaction parameters.

Using the local multiscale energy and vorticity analysis (MS-EVA) and based on the global high-resolution ocean reanalysis product GLORYS12V1 for 20 years, this study investigates the energy transfers and conversions of Kuroshio in the Luzon Strait and its adjacent regions through three scales, namely, the climatological scale, the seasonal scale, and the eddy scale. The results show that the inverse cascades of kinetic energy dominate the energy transfer east of Luzon (at both the eddy and seasonal scales). Kuroshio transfers the climatological kinetic energy to the eddy scale through a forward energy cascade in Luzon Strait and east of Taiwan. Because the topography of Luzon Strait and Kuroshio jointly block and limit the westward propagation of non-local eddies, the eddy energy in the South China Sea west of Luzon Strait tends to depend on local forward potential energy cascades. In these subregions, potential energy drives the accumulation of kinetic energy under the action of buoyancy conversion: interannual (seasonal) potential energy as the source of multiscale energy in the Luzon Strait (the east of Taiwan).

The pair separation model of Goto and Vassilicos [New J. Phys. 6, 65 (2004)] is revisited and placed on a sound mathematical foundation. A direct numerical simulation of two-dimensional homogeneous isotropic turbulence with an inverse energy cascade and a k−5/3 power law is used to investigate properties of pair separation in two-dimensional turbulence. A special focus lies on the time asymmetry observed between forward and backward separations. Application of the present model to these data suffers from finite inertial range effects and thus, conditional averaging on scales rather than on time has been employed to obtain values for the Richardson constants and their ratio. The Richardson constants for the forward and backward case are found to be (1.066±0.020) and (0.999±0.007), respectively. The ratio of Richardson constants for the backward and forward cases is therefore gb/gf=(0.92±0.03), and hence exhibits a qualitatively different behavior from pair separation in three-dimensional turbulence, where gb>gf [J. Berg et al., Phys. Rev. E 74, 016304 (2006)]. This indicates that previously proposed explanations for this time asymmetry based on the strain tensor eigenvalues are not sufficient to describe this phenomenon in two-dimensional turbulence. We suggest an alternative qualitative explanation based on the time asymmetry related to the inverse versus forward energy cascade. In two-dimensional turbulence, this asymmetry manifests itself in merging eddies due to the inverse cascade, leading to the observed ratio of Richardson constants.

The ocean is a turbulent fluid with processes acting on a variety of spatio-temporal scales. The estimates of energy fluxes between length scales allows us to understand how the mean flow is maintained as well as how mesoscale eddies are formed and dissipated. Here, we quantify the kinetic energy budget in a suite of realistic global ocean models, with varying horizontal resolution and horizontal viscosity. We show that eddy-permitting ocean models have weaker kinetic energy cascades than eddy-resolving models due to discrepancies in the effect of wind forcing, horizontal viscosity, potential to kinetic energy conversion, and nonlinear interactions on the kinetic energy (KE) budget. However, the change in eddy kinetic energy between the eddy-permitting and the eddy-resolving model is not enough to noticeably change the scale where the inverse cascade arrests or the Rhines scale. In addition, we show that the mechanism by which baroclinic flows organise into barotropic flows is weaker at lower resolution, resulting in a more baroclinic flow. Hence, the horizontal resolution impacts the vertical structure of the simulated flow. Our results suggest that the effect of mesoscale eddies can be parameterised by enhancing the potential to kinetic energy conversion, i.e., the horizontal pressure gradients, or enhancing the inverse cascade of kinetic energy.

We examine the inverse cascade of kinetic energy to large scales in rotating stratified turbulence as occurs in the oceans and in the atmosphere, while varying the relative frequency of gravity to inertial waves, N/f. Using direct numerical simulations with grid resolutions up to 10243 points, we find that the transfer of energy from three-dimensional to two-dimensional modes is most efficient in the range 1/2 ⩽ N/f ⩽ 2, in which resonances disappear. In this range, the cascade is faster than in the purely rotating case, and thus the interplay between rotation and stratification helps creating large-scale structures. The ensuing inverse cascade follows a −5/3 spectral law with an approximately constant flux. This inverse cascade becomes negligible when stratification is dominant.

We give a Von Karman-type model for the inertial transfer to develop a generalized spectral law for the inverse energy cascade in a forced two-dimensional turbulence that is in agreement with the experimental observations (Sommeria, 1986) of the inverse energy cascade in a statistically steady two-dimensional turbulence. We also demonstrate that the transfer of turbulent kinetic energy at small wave numbers can be modeled as a stationary continuous spectral cascading process. The equipartition spectrum for two-dimensional turbulence is demonstrated to be viscous in character and to differ radically from the equipartition spectrum for three-dimensional turbulence which is inviscid in character.

Understanding the phenomenology captured in direct numerical simulation (DNS) of magnetohydrodynamic (MHD) turbulence rests upon models and assumptions concerning the scaling of field variables and dissipation. Here compressible MHD turbulence is simulated in two spatial dimensions by solving the isothermal equations of resistive MHD on a periodic square grid. In these simulations it is found that the energy spectrum decreases more slowly with k, and the viscous cutoff length is larger, than would be expected from the 1941 phenomenology of Kolmogorov (K41). Both these effects suggest that the cascade time is modified by the presence of Alfvén waves as in the phenomenology of Iroshnikov and Kraichnan (IK). Motivated by this, these scaling exponents are compared with those of the IK-based model of Politano and Pouquet [Phys. Rev. E 52, 636 (1995)], which is an extension of the model of She and Leveque [Phys. Rev. Lett. 72, 336 (1994)]. However, the scaling exponents from these simulations are not consistent with the model of Politano and Pouquet, so that neither IK nor K41 models would appear to describe the simulations. The spatial intermittency of turbulent activity in such simulations is central to the observed phenomenology and relates to the geometry of structures that dissipate most intensely via the scaling of the local rate of dissipation. The framework of She and Leveque implies a scaling relation that links the scaling of the local rate of dissipation to the scaling exponents of the pure Elsässer field variables (z±=v±B∕μoρ). This scaling relation is conditioned by the distinct phenomenology of K41 and IK. These distinct scaling relations are directly tested using these simulations and it is found that neither holds. This deviation suggests that additional measures of the character of the dissipation may be required to fully capture the turbulent scaling, for example, pointing towards a refinement of the phenomenological models. It may also explain why previous attempts to predict the scaling exponents of the pure Elsässer fields in two-dimensional magnetohydrodynamic turbulence by extending the theory of She and Leveque have proved unsuccessful.

<p>Numerical studies of two-dimensional β-plane homogeneous magnetohydrodynamic turbulence are presented. The study of the fundamental properties of such turbulence allows understanding the evolution of various astrophysical objects from the Sun and stars to planetary systems, galaxies, and galaxy clusters. Energy spectra and cascade process in two-dimensional β-plane MHD are studied.</p><p>In this work the equations of two-dimensional magnetohydrodynamics with the Coriolis force in the β-plane approximation are used for the qualitative analysis and numerical simulation of processes in plasma astrophysics. The equations are solved on a square box of edge size 2π with periodic boundary conditions applying a the pseudospectral method using the 2/3 rule for dealiasing. The results of numerical simulation of two-dimensional β-plane MHD turbulence with a spatial resolution of 1024 × 1024 and 4096 × 4096 with different Rossby parameters β and different Reynolds numbers are presented.</p><p>It is found that only unsteady zonal flows with complex temporal dynamics are formed in two-dimensional β-plane magnetohydrodynamic turbulence. It is shown that flow nonstationarity is due to the appearance of isotropic magnetic islands caused by the Lorentz force in the system. The formation of Iroshnikov–Kraichnan spectrum is shown in the early stages of evolution of two-dimensional β-plane magnetohydrodynamic turbulence. The self-similarity of the decay of Iroshnikov–Kraichnan spectrum is studied. On long time scale violation of self-similarity of the decay and formation of Kolmogorov spectrum is discovered. The inverse cascade of kinetic energy, which is characteristic of the detected Kolmogorov spectrum, provides the formation of zonal flows.</p><p>This work was supported by the Russian Foundation for Basic Research (project no. 19-02-00016).</p>

In this paper we introduce a new method for computations of two-dimensional magnetohydrodynamic (MHD) turbulence at low magnetic Prandtl number $\mathit{Pm}= \nu / \eta $. When $\mathit{Pm}\ll 1$, the magnetic field dissipates at a scale much larger than the velocity field. The method we utilize is a novel hybrid contour–spectral method, the ‘combined Lagrangian advection method’, formally to integrate the equations with zero viscous dissipation. The method is compared with a standard pseudo-spectral method for decreasing $\mathit{Pm}$ for the problem of decaying two-dimensional MHD turbulence. The method is shown to agree well for a wide range of imposed magnetic field strengths. Examples of problems for which such a method may prove invaluable are also given.

The spectral kinetic-energy flux is an effective tool to analyze the kinetic-energy transfer across a range of length scales, also known as the kinetic-energy cascade. Three methods to calculate spectral energy fluxes have been widely used, hereafter the ΠA, ΠF, and ΠQ definitions. However, the relations among these three definitions have not been examined in detail. Moreover, the respective contribution of the normal strain and shear strain of the flow field to kinetic-energy cascade has not been estimated before. Here, we use the kinetic energy equations to rigorously compare these definitions. Then, we evaluate the spectral energy fluxes, as well as its decomposition into the normal-strain and shear-strain components for the North Pacific, using a dynamically consistent global eddying state estimate. We find that the data must be preprocessed first to obtain stable results from the ΠF and ΠQ definitions, but not for the ΠA definition. For the upper 500 m of the North Pacific, in the wavenumber ranges with inverse kinetic-energy cascade, both the normal and shear-strain flow components contribute significantly to the spectral energy fluxes. However, at high wavenumbers, the dominant contributor to forward kinetic-energy cascade is the normal-strain component. These results should help shed light on the underlying mechanism of inverse and forward energy cascades.