3D nonlinear wave heating of coronal loops

The heating of solar coronal loops by the resonant absorption or phase-mixing of incident wave energy is investigated in the framework of 3D nonlinear magnetohydrodynamics (MHD) by means of numerical simulations.     

MHD waves in coronal flux tubes

The basic MHD waves of a coronal flux loop are investigated for the rectangular box model of a plasma with oblique magnetic field and line-tied at the ends. The waves found are completely different from those in a periodic box, representative for tokamaks. They consist of a mixture of Alfvén components with a ballooning factor, favouring minimal field line bending, and fast components without such a factor. Pure Alfvén modes are only found as singular limiting cases of cluster spectra of Alfvén-fast waves, where the fast components are localised in a photospheric boundary layer which is dictated by the requirements of line-tying. This justifies the assumption of continuous spectra in coronal loops, required for the mechanism of resonant Alfvén wave heating. The waves consist of large amplitude Alfvén components in the corona and fast components with a small but rapidly varying amplitude in the boundary layer, so that they appear to have the right signature for effective transfer of energy from the photosphere to the corona.     

Transonic Flows in Two-Fluid Plasmas

Transonically rotating toroidal plasmas occur at all scales in the plasma universe and, recently, also in laboratory tokamak plasmas. This offers great opportunities for new insights of the effects of transonic transitions on the background equilibrium flows, and on the waves and instabilities excited. Transfer of knowledge and computational methods on MHD and two-fluid waves and instabilities in magnetically confined laboratory fusion plasmas to space and astrophysical plasmas is seriously hampered though by two related difficulties:

1. in contrast to laboratory plasmas, astrophysical plasmas always have sizeable plasma flows so that they can never be described as a static equilibrium;
2. these flows are usually ‘transonic’, i.e., surpass one of the critical speeds related to the different flow regimes with quite different physical characteristics.

Based on previously obtained MHD results on the stationary states and instabilities of transonically rotating accretion disks about compact objects, the extension to two-fluid plasmas is initiated: A variational principle for the computation of two-fluid stationary states is constructed which involves seven fields determining the different physical variables, and six arbitrary stream functions that should be determined by spatially resolved astrophysical observations. It exhibits all the intricacies due to the electron and ion flow excursions from the magnetic flux surfaces. New hyperbolic flow regimes are found with quite different properties than the MHD ones.     

MHD Spectroscopy of Transonic Flows

In previous publications (Keppens et al.: 2002, Astrophys. J. 569, L121; Goedbloed et al.: 2004a, Phys. Plasmas 11, 28), we have demonstrated that stationary rotation of magnetized plasma about a compact central object permits an enormous number of different MHD instabilities, with the well-known magneto-rotational instability (Velikhov, E. P.: 1959, Soviet Phys.–JETP Lett. 36, 995; Chandrasekhar, S.: 1960, Proc. Natl. Acad. Sci. U.S.A. 46, 253; Balbus, S. A. and Hawley, J. F.: 1991, Astrophys. J. 376, 214) as just one of them. We here concentrate on the new instabilities found that are driven by transonic transitions of the poloidal flow. A particularly promising class of instabilities, from the point of view of MHD turbulence in accretion disks, is the class of trans-slow Alfv’en continuum modes, that occur when the poloidal flow exceeds a critical value of the slow magnetosonic speed. When this happens, virtually every magnetic/flow surface of the disk becomes unstable with respect to highly localized modes of the continuous spectrum. The mode structures rotate, in turn, about the rotating disk. These structures lock and become explosively unstable when the mass of the central object is increased beyond a certain critical value. Their growth rates then become huge, of the order of the Alfv’en transit time. These instabilities appear to have all requisite properties to facilitate accretion flows across magnetic surfaces and jet formation.     

Numerical Simulations of Stellar Winds

We discuss steady-state transonic outflows obtained by direct numerical solution of the hydrodynamic and magnetohydrodynamic equations. We make use of the Versatile Advection Code, a software package for solving systems of (hyperbolic) partial differential equations. We model thermally and magneto-centrifugally driven stellar outflows as generalizations of the well-known Parker and Weber-Davis wind solutions. To obtain steady-state solutions efficiently, we exploit fully implicit time stepping. Wind solutions containing both a 'wind' and a 'dead' zone are presented. We emphasize the boundary conditions imposed at the stellar surface. For axisymmetric wind solutions, we use the knowledge of the flux functions to verify the numerical solutions. This revised version was published online in June 2006 with corrections to the Cover Date.     

Stability and waves of transonic laboratory and space plasmas

The properties of magnetohydrodynamic waves and instabilities of laboratory and space plasmas are determined by the overall magnetic confinement geometry and by the detailed distributions of the density, pressure, magnetic field, and background velocity of the plasma. Consequently, measurement of the spectrum of MHD waves (MHD spectroscopy) gives direct information on the internal state of the plasma, provided a theoretical model is available to solve the forward as well as the inverse spectral problems. This terminology entails a program, viz. to improve the accuracy of our knowledge of plasmas, both in the laboratory and in space. Here, helioseismology (which could be considered as one of the forms of MHD spectroscopy) may serve as a luminous example. The required study of magnetohydrodynamic waves and instabilities of both laboratory and space plasmas has been conducted for many years starting from the assumption of static equilibrium. Recently, there is a outburst of interest for plasma states where this assumption is violated. In fusion research, this interest is due to the importance of neutral beam heating and pumped divertor action for the extraction of heat and exhaust needed in future tokamak reactors. Both result in rotation of the plasma with speeds that do not permit the assumption of static equilibrium anymore. In astrophysics, observations in the full range of electromagnetic radiation has revealed the primary importance of plasma flows in such diverse situations as coronal flux tubes, stellar winds, rotating accretion disks, and jets emitted from radio galaxies. These flows have speeds which substantially influence the background stationary equilibrium state, if such a state exists at all. Consequently, it is important to study both the stationary states of magnetized plasmas with flow and the waves and instabilities they exhibit. We will present new results along these lines, extending from the discovery of gaps in the continuous spectrum and low-frequency Alfvén waves driven by rotation to the nonlinear flow patterns that occur when the background speed traverses the full range from sub-slow to super-fast. This revised version was published online in August 2006 with corrections to the Cover Date.