Basic Electrodyamics (IV): Maxwell's Equations in Solar Fields

In this installment we see how to apply Maxwell's famous equations to the case of solar magnetic fields. This is excellent in showing a practical application, as well as an example outside of terrestrial labs and experiments.

Maxwell’s equations in differential form (much easier to write than in integral form) for free space are:

(1) Div D = rho

(2) Curl E = - @B/ @t

(3) Div B = 0

(4) Curl H = J + @D/ @t

Where E is the electric field intensity, rho the charge density, B is the magnetic induction, H is the magnetic field intensity, J is the current density and D is the displacement current.

One also has the constitutive relations:

a) D = e_o E

b) B = u_0 H

c)J = o E

where e_o is the permittivity of free space, u_o is the magnetic permeability for free space, and o is the conductivity.

You can find the curl defined here:

http://mathworld.wolfram.com/Curl.html


and the divergence (Div) is, e.g. for some arbitrary vector F (F = F_x + F_y + F_z)

Div F = @F_x/ @x + @F_y/ @y + @F_z/ @z

Where ‘@’ is taken to be the partial derivative symbol

In solar applications, the vectors D and H are seldom, if ever, used, and neither is e_o . For example, for equation (4) one is more apt to make the following changes:

H = B/ u

curl (B/ u) = J + @(e E)/ @t

which leads directly to:

curl B = u [J + @(e E)/ @t]

(Note that u, e denote non-free space - e.g. non-vacuum, values!)

Now - if for E, @/ @ t -> 0, e.g. negligible variation in time:

curl B = uJ

This is also known as Ampere’s Law.

The Ohm’s law form is generally modified from equation (c):

J = o (E + curl(v X B)]


To consider what happens and what further changes in the equations are need for flare analysis, we introduce the ‘beta’ – the ratio of kinetic gas to magnetic energy density:

beta = (0.5 rho v^2)/ (B^2/ 2 u) < < 1

(around or near sunspots)

where B is the magnetic induction, and rho is the plasma density, u the magnetic permeability

In such conditions, an extraordinarily large (and unobserved) pressure gradient (grad p) would be required to balance and field presumed not force-free:

grad p = J X B

Note: in Cartesian coordinates,

grad p = (@p/ @x, @p/ @y, @p/ @z)

where J is the current density and B the induction and J x B is the Lorentz force (produced by a current density J perpendicular to the magnetic field B)

Thus, a force-free assumption (consistent with beta < < 1) requires:

J x B = 0

that is, the current density is essentially parallel to the magnetic induction.

The standard form for the force-free eqn. is obtained by substituting the force-free parameter (assumed to be constant):

alpha = (u J)/ B

into Ampere’s law for B, with J = alpha (B)/ u

thus,

curl B = u [alpha (B)/ u]

The mathematical starting point for an evolving solar magnetic field might be then:

Curl (B) = alpha (B)

(Force-free field)

For the special case, alpha = 0, then u J = 0 and we obtain a “current free” configuration for which there is no residual energy to be extracted from the field, e.g. for flares. This is also called a “potential” field. For any force-free field for which alpha > 0, magnetic free energy is available for flares.

Further working, by taking the curl of both sides, yields:

curl curl B = curl alpha (B)

Which, via vector identity, is:

DIV DIV B – DIV^2 B = alpha (curl B)

Into the above, substitute curl (B) = alpha (B)

and DIV B = 0 (Maxwell divergence free eqn.), to obtain:

DIV ^2(B) = (alpha)^2 B

where 'DIV' (“divergence”) is another vector operator.

From this, we can obtain:


DIV ^2(B) - (alpha)^2 B = 0

which is one form of the Helmholtz equation, also expressed:

[(DIV^2 - (alpha)^2] B = 0


In cylindrical geometry, one has:

DIV^2 = 1/r [@/ @r ( r * @/ @r)]

(where @ denotes the partial differential symbol)

This leads to:

1/r [@/ @r ( r * @/ @r)] - (alpha)^2 B = 0

for which the (axially symmetric) Bessel function solution is:

B_z (r) = B_o J_o(a r)

B_t(r)) = B_o J_1(ar)

t = theta

J_o(a r) is a Bessel function of the first kind, order zero and J_1(ar) is a Bessel function of the first kind, order unity (see also earlier blogs, from 2009, on Bessel functions!)

For a cylindrical magnetic flux tube (such as a sunspot represents, e.g. viewed in cross-section) the “twist” is defined:

T(r) = (L * B_t(r))/ (r * B_z (r))

Where L denotes the length of the sunspot-flux tube dipole

The primary focus in analysis of the above type field, is how it is stressed (via magnetic shear) to the point of instability and solar flares.

This provides the "basics" for using Maxwell's equations, and adapting them for much solar flare work