Two-dimensional Inverse Cascade Research Articles

Investigation of a Vorticity-preserving Scheme for the Euler Equations

Abstract We investigate the vorticity-preserving properties of the compressible, second-order residual-based scheme, “RBV2.” The scheme has been extensively tested on hydrodynamical problems, and has been shown to exhibit remarkably accurate results on the propagation of inviscid compressible vortices, airfoil-vortex interactions on a curvilinear mesh, vortex mergers in an astrophysical accretion disk, and the establishment of a two-dimensional inverse cascade in high-resolution turbulent simulations. Here, we demonstrate that RBV2 sustains the analytic solution for a one-dimensional shear flow. We assess the fidelity by which the algorithm maintains a skewed shear flow, and present convergence tests to quantify the magnitude of the expected numerical dispersion. We propose an adjustment to the dissipation in the algorithm that retains the vorticity-preserving qualities, and accurately incorporates external body forces, and demonstrate that it indefinitely maintains a steady-state hydrostatic equilibrium between a generic acceleration and a density gradient. We present a novel numerical assessment of vorticity preservation for discrete wavenumber, vortical modes of discrete wavenumber up to the Nyquist wavenumber. We apply this assessment to RBV2 in order to quantify the extent to which the scheme preserves vorticity for the full Euler equations. We find that RBV2 perfectly preserves vorticity for modes with symmetric wavenumbers, i.e., k x = k y , and that the error increases with asymmetry. We simulate the dynamical interaction of vortices in a protoplanetary disk to demonstrate the utility of the updated scheme for rendering astrophysical flows replete with vortices and turbulence. We conclude that RBV2 is a competitive treatment for evolving vorticity-dominated astrophysical flows, with minimal dissipation.

On the energy of elliptical vortices

Consider a two-dimensional axisymmetric vortex with circulation Γ. Suppose that this vortex is isovortically deformed into an elliptical vortex. We show that the reduction in energy is ΔE=−Γ2 ln[(q+q−1)/2]/(4π), where q2 is the ratio of the major to the minor axis of any particular elliptical vorticity contour. It is notable that ΔE is independent of the details of vorticity profile of the axisymmetric vortex and, in particular, independent of its average radius. The implications of this result for the two-dimensional inverse cascade are briefly discussed.

Statistics of two-particle dispersion in two-dimensional turbulence

We investigate Lagrangian relative dispersion in direct numerical simulation of two-dimensional inverse cascade turbulence. The analysis is performed by using both standard fixed time statistics and an exit time approach. The latter allows a more precise determination of the Richardson constant which is found to be g≃4 with a possible weak finite-size dependence. Our results show only small deviations with respect to the original Richardson’s description in terms of diffusion equation. These deviations are associated with the long-range correlated nature of the particles’ relative motion. The correlation, or persistence, parameter is measured by means of a Lagrangian “turning point” statistics.

Intermittency in the two-dimensional inverse cascade of energy: Experimental observations

An extensive experimental study of the two-dimensional inverse energy cascade is presented. The experiments are performed in electromagnetically driven flows, using thin, stably-stratified layers. Complete instantaneous velocity fields are measured using particle imaging velocimetry techniques. Depending on the bottom-wall friction, two different regimes are observed: when the friction is low, the energy transferred from the forcing scale towards large scales accumulates in the lowest accessible mode, leading to a mean rotation of the flow and to an energy spectrum displaying a sharp peak at the minimum wave-number. This condensation is accompanied by the emergence of a very strong vortex around which the rotation is organized. At higher frictions, the inverse energy cascade conjectured by Kraichnan [Phys. Fluids 10, 1417 (1967)] is observed and is found to be stationary, homogeneous and isotropic, with a Kolmogorov constant consistent with numerical estimates. This inverse cascade does not appear to be characterized by the presence of strong coherent vortices. The characteristic size of the latter is of the order of the injection scale. Their statistical properties tend to show that the cascade is rather driven by a clustering mechanism involving same sign vortices rather than a sequence of merging events producing larger and larger vortices. Intermittency effects are also investigated for the inverse cascade range. It is found that, within experimental errors, there is no intermittency in the inverse cascade range of two-dimensional turbulence and that the statistics of velocity increments, either longitudinal or transverse, are close to Gaussian. These results constitute the first experimental study of intermittency in two-dimensional turbulence as well as the first observation of normal scaling in a field of research which has been increasingly concerned with anomalous exponents.