Expert: Philip A. Stahl - 12/26/2006

Question
Hello. Can you tell me how exactly the magnetic Reynolds number is obtained and what its values is for solar conditions? Thanks!

Answer
Hello,

The magnetic Reynolds number is a (dimensionless) measure of the degree of coupling between the plasma flow in a region and the ambient magnetic field.

If you back to the plasma form of Ohm's law (recall from the previous answer):

J  = o (E  + v X B)

In solar physics cases, the formulation (toward obtaining magnetic Reynolds number) is a bit more complicated, and begins with re-casting the above form of Ohm’s law:

E  = J/ o  - (v X B)


we take the curl of both sides of the equation:

Curl E = 1/o [Curl J] - Curl (v X B)

But J = Curl B/u_o

(this is for coronal conditions so a near vacuum)


So:

Curl E = 1/o [Curl (Curl B)/ u_o] - Curl (v X B)

This leads to a result of the form:

@B/ @t =   D  + C

where 'D' is known as the diffusive term, and represents the resistive leakage of magnetic field lines across the conducting fluid (plasma). 'C' is called the "convective term" (more exactly C = Curl (v X B)) and represents the tendency of the magnetic field lines to be 'frozen' into the plasma.

The magnetic Reynolds number (R_m) is just the ratio of the two terms:

R_m =  C/ D  

with dimensional scaling according to:

R_m = L^-1 v B/ (D B L^-2)

where 'B' denotes magnetic field strength in Tesla (which cancels out) and v is velocity and D is the diffusion coefficent.

It can alternatively be expressed:

R_m =  L V(A)/ n

where L is a typical length scale for the environment, V(A) is the Alfven velocity and n is the magnetic diffusivity written as:

n ~   (5.2 x  10^ 7   ln C  T^-1.5 ) m^2 /s

where ln C is the so-called Coulomb logarithm

In conditions of the solar corona - where most flares originate- for a temperature T ~ 10^6 K, length scale L ~ 10^7 m, and typical fluid velocities v ~ 10^3 m/s we obtain:

R_m ~  10^10

In other words, the diffusive term is almost entirely negligible and we can regard the field as "frozen in" - which will mean the current density J is essentially parallel to the field lines B.

By comparison, for the Earth's magnetosphere - where the aurora occurs - one can work out R_m ~ 10^6.

For Earthly lab conditions, we have generally that:

R_m < <  1  (e.g. diffusion of field lines predominates over frozen in condition)

Hope this helps!