Christiane Marliani

Adaptive Mesh Refinement for Singular Current Sheets in Incompressible Magnetohydrodynamic Flows

The formation of current sheets in ideal incompressible magnetohydrodynamic flows in two dimensions is studied numerically using the technique of adaptive mesh refinement. The growth of current density is in agreement with simple scaling assumptions. As expected, adaptive mesh refinement shows to be very efficient for studying singular structures compared to nonadaptive treatments.

Scaling of high-order structure functions in magnetohydrodynamic turbulence

A phenomenological model for the description of intermittency corrections in magnetohydrodynamic flows is presented. The strength of the model lies in its lack of adjustable parameters. A comparison to measurements in the solar wind is presented.

Geometry of singular structures in magnetohydrodynamic flows

The flattening of current sheets is investigated by means of numerical simulations of the ideal incompressible magnetohydrodynamic equations in two dimensions. The use of adaptive mesh refinement techniques allows one to resolve the more and more singular structures and to follow the exponential growth of current density. The numerical results are in good agreement with a scaling ansatz proposed by Sulem et al. [J. Plasma Phys. 33, 191 (1985)]. The geometry of the current sheets is characterized by the alignment properties of the deformation matrices.

Numerical and analytical estimates for the structure functions in two‐dimensional magnetohydrodynamic flows

In two‐dimensional magnetohydrodynamic turbulence, the Kraichnan–Iroshnikov dimensional analysis suggests a linear scaling law for the exponents ζp=p/4 of the structure functions for the Elsasser variables z±=u±B. Numerical simulations are presented and higher order structure functions are calculated using the extended self‐similarity hypothesis of Benzi et al. [Phys. Rev. E 48, R29 (1993)]. In addition, an estimate for the first structure function ζ1≥1/4 is derived using a geometric technique introduced by Constantin and Procaccia [Phys. Rev. E 47, 3307 (1993)] in the the context of the transport of a passive scalar in three‐dimensional Navier–Stokes turbulence.

Reconnection of coalescing magnetic islands

The dynamics of magnetic reconnection in the framework of two-dimensional incompressible magnetohydrodynamics is studied numerically. The case of a doubly periodic array of magnetic islands where coalescence of neighboring islands occurs due to self-consistent magnetic forces is investigated. To adequately resolve the current sheets which evolve in between two islands adaptive structured mesh methods are applied. At the onset of the reconnection process the kinetic energy rises and drops rapidly and afterward settles into an oscillatory phase. The time scale of the magnetic reconnection process is not affected by these fast events but consistent with the Sweet–Parker model of stationary reconnection. When the spatial extension of the system is enlarged it undergoes a sequence of merging processes and unstable equilibria towards a large-scale pattern of magnetic islands. During this process the frequency of the oscillations in the kinetic energy is found to scale with the island size.

Center manifold approach to controlling chaos

Abstract Existing methods of controlling chaos can be generalized using ideas of center manifold theory. This approach extends the existing linear theory into the nonlinear regime, thus enlarging the range in phase space where control is possible. At the same time, sensitivity of the stabilized system against noise is reduced. In addition, this procedure leads to nonlinear time delay feedback rules in a constructive way.

Analytical and Numerical Approaches to Structure Functions in Magnetohydrodynamic Turbulence

In magnetohydrodynamic turbulence, the classical theory by Kraichnan and Iroshnikov based on dimensional analysis gives a linear dependence of the exponents ζp = p/4 of the structure functions for the Elsasser variables z± = u ± B. This linear behavior contradicts observations of MHD turbulence in the solar wind, where anomalous scaling was found similar as in hydrodynamic turbulence. Since the experimentally observed scaling can not yet be derived by analytical theories, one is dependent also on numerical simulations. As an alternative to direct numerical simulations we present a stochastic approach that recently was introduced for two-dimensional hydrodynamic flows. Finally, we discuss the applicability of operator-product expansions on a direct cascade in strongly turbulent systems.