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Astron. Astrophys. 318, 963-969 (1997)

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2. Low-beta, 2-D equilibria: basic equations in spherical geometry

The steady, ideal MHD equations may be written in the non-dimensional form:

[EQUATION]

[EQUATION]

[EQUATION]

[EQUATION]

where all the quantities have been non-dimensionalized against typical coronal values and where the values of the three parameters

[EQUATION]

indicate the relative importance of the various terms in Eq. (4) (here [FORMULA] is a reference value of the sound speed and all the other symbols have their usual meaning).

The main assumption in our model is that the magnetic forces are dominant over all the others, namely pressure gradients, gravity and inertial forces. In the low solar corona this is a good approximation and the coronal plasma is thus regarded as low- [FORMULA]. Hence, in order to linearise the MHD equations with respect to the magnetic field, the following form for [FORMULA] is assumed:

[EQUATION]

where its zeroth order component [FORMULA] is necessarily force-free (from Eq. (4)).

Consider now a purely 2-D spherical coordinate system in which all the quantities lie in the r - [FORMULA] plane and do not depend upon the azimuthal coordinate [FORMULA] (the plume axis will coincide with the symmetry axis [FORMULA]). Using the formalism of the flux functions, Eqs. (1) to (3) give

[EQUATION]

[EQUATION]

[EQUATION]

where the magnetic flux function is [FORMULA] and [FORMULA] is a free function of [FORMULA] (note that the velocity and magnetic fields are parallel only at the zeroth order). In order to solve the equations a relation between pressure and density is needed. Here the isothermal case will be assumed, thus

[EQUATION]

where the temperature T is another free function of [FORMULA] (thus T is constant along the field lines). Making use of these assumptions, the component of Eq. (4) across [FORMULA] splits into the two transfield (or generalised Grad-Shafranov) equations

[EQUATION]

[EQUATION]

whereas the component along [FORMULA] yields the Bernoulli equation

[EQUATION]

and E is the third free function of [FORMULA]. For the mathematical demonstrations (in the general case) see Del Zanna & Chiuderi (1996).

The main result of the linearisation of the magnetic field is clearly the decoupling of the transfield and Bernoulli equations. This allows one to solve the problem in three distinct steps:

  1. Solve the transfield equation, Eq. (6), for the unperturbed field.
  2. Solve the Bernoulli equation, Eq. (8), for the density.
  3. Solve the transfield equation, Eq. (7), for the correction to the field.

Clearly, the corrections to the magnetic field must remain small and the condition for this is [FORMULA].

The same approach in solving the MHD equations through the magnetic field linearisation has been previously adopted by Surlantzis et al. (1994, 1996) in order to model stationary flows in coronal loops and arcades. As their investigation is only concerned with closed field structures in cartesian and cylindrical coordinates, our analysis may be also considered as an extension to the complementary cases not contemplated in that work.

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© European Southern Observatory (ESO) 1997

Online publication: July 3, 1998
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