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Original page found at: http://www.fluid.tue.nl/WDY/vort/
Vortices in Plasmas Everyday fluid motions are often two-dimensional in nature. Flows in the atmosphere and oceans for example, have horizontal scales of hundreds of kilometres, while these geophysical flows are only a few kilometres in thickness. The reduction in dimensionality is thus due to the smallness of the vertical scale. Other examples of two-dimensional fluid flow are thin soap films and the flows in galactic disks. A purely two-dimensional inviscid flow in a nonrotating system is described by the Euler equation. When due to rotation of the system the pressure gradients in the fluid are balanced by the Coriolis force the flow is called geostrophic. This may for example be the case for motions in planetary atmospheres. The dynamics of small deviations from the geostrophic balance are described by the quasi-geostrophic equation. There is a strong resemblance between motions in the above-mentioned nonconducting fluids and those in a plasma. This is remarkable because at first sight the physical properties of "the fourth state of matter" are very different from those of fluids. A plasma consists of charged particles, electrons and ions. The temperature is so high, that the thermal energy of the particles is much greater than the binding energy between electrons and ions. In this completely ionized medium the interactions between particles are the combined Coulomb and Lorentz forces between the electron and ion fluids. Electromagnetic fields hence deeply affect the collective motions in the plasma. Furthermore, because of the high temperature the plasma is nearly collisionless and the system can be considered to be ideal, i.e. dissipative effects like resistivity and viscosity are negligible. The plasmas we are interested in are in general permeated by a, mainly external, strong magnetic field. Because of this, the ratio of the kinetic pressure and the magnetic pressure is small, and therefore the magnetic field is barely compressed by the plasma motions. Nearly compressionless perturbations of this system are called drift-Alfvén phenomena. A strongly magnetized plasma can maintain large gradients across the magnetic field, while the fast parallel motion quickly smoothes out gradients along the field. Parallel length scales are thus much larger than perpendicular scales making a two-dimensional description of the plasma dynamics possible too. Thus the role of the externally applied magnetic field in the dynamics of a plasma is reminiscent to the role of background rotation in the geostrophic dynamics of planetary atmospheres. The analogy between plasmas and fluids becomes even more apparent when considering the mathematical description of the different physical systems. The quasi-two-dimensional drift-Alfvén dynamics of a magnetized plasma, the quasi-geostrophic motion of rotating fluids, and the planar motion in an Euler fluid are all mathematically equivalent. In the absence of dissipation their dynamics is described by a Lagrangian equation. One or more conserved quantities are advected by so many incompressible velocity fields. In a classical inviscid two-dimensional fluid this conserved quantity is the vorticity of the fluid, the curl of the velocity vector field. In the general case these conserved quantities are therefore called generalized vorticities. Each vorticity field is advected by its own velocity field and these velocities are self-consistently generated by the generalized vorticity fields. The drift-Alfvén motions in a plasma conserve three generalized vorticity fields. They are combinations of the magnetic and electric field perturbations, the current density, the fluid vorticity, and the particle density of the plasma. The phenomenon which strengthens the analogy between these very different physical systems further is the occurrence of coherent structures in both plasmas and fluids. These are long-lived rotating structures in the velocity pattern of the fluid, and for a plasma, combinations of flow, electric currents and magnetic fields. When such coherent structures can be considered as separated entities that move in the fluid, they are called vortices. For a large part the research on magnetized plasmas in laboratories is aimed at achieving thermonuclear fusion reactions in a controlled way to generate electricity. A fusion reactor converts the energy that is released in the fusion reaction between deuterium and tritium into electricity. A way to overcome the repulsive Coulomb force between the nuclei is to heat them to temperatures of several hundred million degrees. To reach such conditions the confining properties of strong magnetic fields are used to isolate the plasma from its surroundings. The understanding of transport is therefore an important subject of study. Like turbulent air flow can enhance the transport of heat in everyday settings (cooling of engines, or blowing air over a nice hot cup of coffee), turbulence can also be an important transport mechanism of heat and particles in a thermonuclear plasma. Because the dynamics of vortex structures lies at the basis of fluid turbulence, its cross-fertilization to plasma physics may provide a basis for understanding the complex current-vortex dynamics in magnetized plasmas. Space physics is the second important field where the interplay of plasma and magnetic fields is investigated. Because almost all matter in the universe is ionized and in motion, plasma physics is an important part of the research of astrophysical phenomena. Filamentary and vortical structures, often arising from or in combination with turbulence, are for example the field-aligned Birkeland currents in the magnetosphere of planets and filamented coronal loops in the solar atmosphere. Vortical structures are also observed in magnetized temperature filaments in dilute plasmas. Magnetized nonneutral plasmas also exhibit (quasi) two-dimensional fluid behaviour. An illustrative example is a pure electron plasma. The evolution of the electron density distribution in an electrostatic Penning-Malmberg trap is exactly equal to the evolution of vorticity in an inviscid two-dimensional fluid. The two systems can thus be compared directly with each other. Particularly interesting is the emergence of regular point vortex crystals from a turbulent initial state as illustrated in the figure. The latter illustrates experimental vorticity distributions for two turbulent evolutions (two top rows) illustrating the formation of a vortex crystal. The bottom row of images is a selection of observed vortex crystals.
Despite the (mathematical) analogy between vortex dynamics in an ordinary fluid and in a plasma there also exist strong differences in the behaviour of vortices in both systems. One of the most striking examples of this is the recently discovered quasi-periodic dynamics of certain vortex patches that are subjected to the drift-Alfvén dynamics. Contrary to what is common in "classical" fluids, the interaction dynamics of patches of generalized vorticity in a plasma show a preferred length scale for structures that are formed. On this length scale structures can be formed either by merging of smaller vortices or by splitting of larger vortices resulting in quasi-periodic behaviour. A typical example of such behaviour is nicely illustrated by a movie that shows the evolution of an elliptical patch of generalized vorticity as a function of time. |
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