Antecedentes

The behaviour of a continuous plasma is governed by Maxwell’s Equations, which for a plasma having a current density J, charge density pcharge» ^ magnetic field H with magnetic induction B and an electric field E with electrical displacement D are

V x H = J + — , (1.19) 9t Y . B = 0, (1.20) V x E = - ~ . (1.2 1 ) at and Y D = Pcharge* (1.2 2)

For a linear isotropic medium we have the following relationships connecting the electrical displacement with the electric field, and the magnetic induction with the magnetic field:

D =eE, B = |iH , (1.23)

where, for solar purposes, e and p are generally approximated by their vacuum values eo (« 8.854 X lQ-l2Fm‘^) and po (= 4tc x lQ-7 Hm-l), the permittivity and magnetic permeability of fi'ee space, respectively; eq and po are related to the speed of light c in a vacuum by

The relations given by Equation (1.23) may be used to eliminate the magnetic field H and the electric displacement D in Maxwell’s Equations.

Plasma moving with a non-relativistic velocity v in the presence of a magnetic field is subject to an electric field y_x B in addition to an electric field E which may act on the plasma at rest. Ohm’s Law states that

I = a(E + v_xB), (1.25)

so that the current density i is proportional to the total electric field E + y_x B. The constant of proportionality a is termed the electrical conductivity. It is measured in mho m“l.

It is convenient to eliminate the electric field E and the current density J between Equations (1.21) and (1.25), whilst ensuring that Equation (1.20) is obeyed. The result is

^ = V x(5L xa) + TlV2fi, (1.26)

ot

where Tj == (po)"^ is the magnetic diffusivity. This is the induction equation and may be used to determine the magnetic induction B (commonly, in the solar context, referred to as the magnetic field) when the velocity of the plasma is known. The induction equation indicates that changes in the magnetic field in time are the result of transport of the field with the plasma, together with diffusion of the field through the plasma.

In order of magnitude, the ratio of these two terms in the induction equation defines the magnetic Reynolds number (c.f. the viscous Reynolds number in fluid mechanics)

Rm = ^ . (1.27)

•n

where vq and Ig are typical plasma velocity and length scales. Except in regions of high current density, such as in filaments or sheets, most areas of the sun are such that R ^ » 1 (for example, R ^ = lO^-lO^ for typical coronal structures, since 1q is large) and

so the diffusion term in Equation (1.26) is negligible. Therefore the plasma acts as though it were a perfect conductor (a ®o). In this case the induction Equation (1.26) reduces to

^ = Vx(v_xB), (1.28)

at

indicating that the magnetic field is effectively “frozen” to the plasma. Ohm’s Law, given by Equation (1.25), in the perfectly conducting limit reduces to

E + v_xB = Ü, (1.29)

and the current density is determined from Ampere’s Law,

POI - V xB . (1.30)

1.5.2 Plasma Equations

The plasma motion is governed by the equations of mass continuity, motion and energy. The equation of mass continuity is

where

^ + p ( V . v ) = 0, (1.31)

(1.32,

is the convective or total derivative for time variations following the motion. Ignoring the effect of viscosity, the equation of motion for a plasma subject to the force of gravity, a pressure gradient V p, and a Lorentz force J_x B across the magnetic field may be written as

where g (= 274 ms‘^) is the solar gravitational acceleration and p the gas pressure. Considering only adiabatic perturbations gives the energy equation

§

where y is the ratio of specific heats. The gas pressure p is determined by the equation of state of the gas. We shall consider an ideal gas for which the equation of state is

^ p T . (1.35)

where kg (= 1.481 x 1 0 - ^ 3 Jdeg'l) is Boltzmann’s constant, m the mean particle mass

and T the temperature of the gas.

Then in general any disturbance in the solar atmosphere is subject to the three restoring forces of buoyancy, compressibility and magnetism. Motions are therefore expected to be anisotropic, reflecting the preferred directions from the inclusion of gravity and a magnetic field. We therefore expect wave motions, driven by these three restoring forces, that are distinct from a pure sound wave, characterised by the speed Cg defined by

Co — (1.36)

vPOy

where po and po are the equilibrium values of pressure and density of the gas and y the ratio of specific heat, and an Alfvén wave with speed v^ defined by

/' V/ 2

'0

W P07 (1.37)

where Bo is the equilibrium magnetic field strength. In fact, two magnetoacoustic modes arise from combinations of the two characteristic speeds Cg and v^:

C^ = c^ + v^ and cij? = (1.38) where Cf is referred to as the fast magnetoacoustic speed and Cj (< Cf) is the slow magnetoacoustic speed. The fast speed is both supersonic and super-Alfvénic, while the slow speed Cj is both subsonic and sub-Alfvénic. The speed Cy is also referred to as the cusp or tube speed (Roberts, 1981a).

The presence of gravity also means that there is a natural length-scale in the system; this is the density scale-height H, defined by

H = - (1.39)

dpo/dz

There is also a length-scale defined by pressure variations. This is the pressure scale height A, defined by

Associated with an imposed length-scale, we may introduce an imposed timescale, namely the time taken for a sound wave to travel the distance H and back again, i.e. 2H/cg. We can therefore construct a frequency defined by

This is the acoustic cut-off frequency of an isotheimal atmosphere.

In addition to the acoustic cut-off frequency we can construct on dimensional grounds, and through the use of gravity, the frequencies g/Cg and (g/H)l/2. In fact, the difference between the squares of both these terms arises, namely

We note that for an isothermal atmosphere in the absence of a magnetic field the density scale height and the pressure scale-heights are both constants and in fact equal (H = A = Hq, a constant). Then

0). yg

4Hf (1.43)

2 2

Thus with 7 = 5/3 we note that Cû^ exceeds cOg by 1/60 (g/Ho).

Chapter 2 : Magnetohydrodynamic Surface Waves in an