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Register a new userDynamics of Potential Vorticity Staircase Evolution and Step Mergers in a Reduced Model of Beta-Plane Turbulence
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A two-field model of potential vorticity (PV) staircase structure and dynamics relevant to both beta-plane and drift-wave plasma turbulence is studied numerically and analytically. The model evolves averaged PV whose flux is both driven by and regulates, a potential enstrophy field, $varepsilon$. The model employs a closure using a mixing length model. Its link to bistability, vital to staircase generation, is analyzed and verified by integrating the equations numerically. Long-time staircase evolution consistently manifests a pattern of meta-stable quasi-periodic configurations, lasting for hundreds of time units, yet interspersed with abrupt ($Delta tll1$) mergers of adjacent steps in the staircase. The mergers occur at the staircase lattice defects where the pattern has not completely relaxed to a strictly periodic solution that can be obtained analytically. Other types of stationary solutions are solitons and kinks in the PV gradient and $varepsilon$ - profiles. The waiting time between mergers increases strongly as the number of steps in the staircase decreases. This is because of an exponential decrease in inter-step coupling strength with growing spacing. The long-time staircase dynamics is shown numerically be determined by local interaction with adjacent steps. Mergers reveal themselves through the explosive growth of the turbulent PV-flux which, however, abruptly drops to its global constant value once the merger is completed.
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Using a one-layer QG model, we study the effect of random monoscale topography on forced beta-plane turbulence. The forcing is a uniform steady wind stress that produces both a uniform large-scale zonal flow $U(t)$ and smaller-scale macroturbulence (both standing and transient eddies). The flow $U(t)$ is retarded by Ekman drag and by the domain-averaged topographic form stress produced by the eddies. The topographic form stress typically balances most of the applied wind stress, while the Ekman drag provides all of the energy dissipation required to balance the wind work. A collection of statistically equilibrated solutions delineates the main flow regimes and the dependence of the time-mean $U$ on the problem parameters and the statistical properties of the topography. If $beta$ is smaller than the topographic PV gradient then the flow consists of stagnant pools attached to pockets of closed geostrophic contours. The stagnant dead zones are bordered by jets and the flow through the domain is concentrated into a narrow channel of open geostrophic contours. If $beta$ is comparable to, or larger than, the topographic PV gradient then all geostrophic contours are open and the flow is uniformly distributed throughout the domain. In this case there is an eddy saturation regime in which $U$ is insensitive to changes in the wind stress. We show that eddy saturation requires strong transient eddies that act as PV diffusion. This PV diffusion does not alter the energy of the standing eddies, but it does increase the topographic form stress by enhancing the correlation between topographic slope and the standing-eddy pressure field. Last, using bounds based on the energy and enstrophy we show that as the wind stress increases the flow transitions from a regime in which form stress balances the wind stress to a regime in which the form stress is very small and large transport ensues.
Exact solutions of a magnetized plasma in a vorticity containing shear flow for constant temperature are presented. This is followed by the modification of these solutions by thermomagnetic currents in the presence of temperature gradients. It is shown that solutions which are unstable for a subsonic flow, are stable if the flow is supersonic. The results are applied to the problem of vorticity shear flow stabilization of a linear z-pinch discharge.
Eddy saturation describes the nonlinear mechanism in geophysical flows whereby, when average conditions are considered, direct forcing of the zonal flow increases the eddy kinetic energy, while the energy associated with the zonal flow does not increase. Here we present a minimal baroclinic model that exhibits complete eddy saturation. Starting from Phillips classical quasi-geostrophic two-level model on the beta channel of the mid-latitudes, we derive a reduced order model comprising of six ordinary differential equations including parameterised eddies. This model features two physically realisable steady state solutions, one a purely zonal flow and one where, additionally, finite eddy motions are present. As the baroclinic forcing in the form of diabatic heating is increased, the zonal solution loses stability and the eddy solution becomes attracting. After this bifurcation, the zonal components of the solution are independent of the baroclinic forcing, and the excess of heat in the low latitudes is efficiently transported northwards by finite eddies, in the spirit of baroclinic adjustment.
Zonal flows in rotating systems have been previously shown to be suppressed by the imposition of a background magnetic field aligned with the direction of rotation. Understanding the physics behind the suppression may be important in systems found in astrophysical fluid dynamics, such as stellar interiors. However, the mechanism of suppression has not yet been explained. In the idealized setting of a magnetized beta plane, we provide a theoretical explanation that shows how magnetic fluctuations directly counteract the growth of weak zonal flows. Two distinct calculations yield consistent conclusions. The first, which is simpler and more physically transparent, extends the Kelvin-Orr shearing wave to include magnetic fields and shows that weak, long-wavelength shear flow organizes magnetic fluctuations to absorb energy from the mean flow. The second calculation, based on the quasilinear, statistical CE2 framework, is valid for arbitrary wavelength zonal flow and predicts a self-consistent growth rate of the zonal flow. We find that a background magnetic field suppresses zonal flow if the bare Alfven frequency is comparable to or larger than the bare Rossby frequency. However, suppression can occur for even smaller magnetic fields if the resistivity is sufficiently small enough to allow sizable magnetic fluctuations. Our calculations reproduce the $eta/B_0^2 = text{const.}$ scaling that describes the boundary of zonation, as found in previous work, and we explicitly link this scaling to the amplitude of magnetic fluctuations.
In presence of an externally supported, mean magnetic field a turbulent, conducting medium, such as plasma, becomes anisotropic. This mean magnetic field, which is separate from the fluctuating, turbulent part of the magnetic field, has considerable effects on the dynamics of the system. In this paper, we examine the dissipation rates for decaying incompressible magnetohydrodynamic (MHD) turbulence with increasing Reynolds number, and in the presence of a mean magnetic field of varying strength. Proceeding numerically, we find that as the Reynolds number increases, the dissipation rate asymptotes to a finite value for each magnetic field strength, confirming the Karman-Howarth hypothesis as applied to MHD. The asymptotic value of the dimensionless dissipation rate is initially suppressed from the zero-mean-field value by the mean magnetic field but then approaches a constant value for higher values of the mean field strength. Additionally, for comparison, we perform a set of two-dimensional (2DMHD) and a set of reduced MHD (RMHD) simulations. We find that the RMHD results lie very close to the values corresponding to the high mean-field limit of the three-dimensional runs while the 2DMHD results admit distinct values far from both the zero mean field cases and the high mean field limit of the three-dimensional cases. These findings provide firm underpinnings for numerous applications in space and astrophysics wherein von Karman decay of turbulence is assumed.
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