Coronal Mass Ejections and Collisionless Shocks (1)

The CME captured by the Solar Dynamics and Heliospheric Observatory in February, 2013.

Coronal mass ejections are powerful, energetic events on the Sun which can also have serious repercussions for terrestrial life, see e.g.

http://brane-space.blogspot.com/2012/06/cme-that-you-dont-want-to-see.html

How do coronal mass ejections occur? This is believed to happen via a sudden, mass transient ejection probably via a high Mach number shock wave. It has also been found that collisionless shocks form ahead of CMEs when the latter’s velocity exceeds the Alfven velocity of the ambient plasma in the corona. Recall this Alfven velocity is given by:

V_A     =  Ö (B_o / m_o   r _o)

To fix ideas, let us recall the corona is a collisionless plasma . (Recall in the limit 1/N << 1 and 1/N ® 0  we say a plasma is “collisionless” where:N  =   4p n _o  l³ _D  / 3   and n _o  is the number density with   l _D  the Debye length.   (See example problems from Jan, 23rd). Let a coronal plasma have number density n = 10 ¹⁶ /m³)  and a kinetic temperature T =10 ⁶ K.  The plasma parameter is defined:

L  =  n _o  l³ _D

Where the Debye length

l_D,s =[kT _s ε_o / 4p  Z _s² n _s  e² ] ^½

Here:  Z _s   = 1 so that:

l _D,s   =

[(1.38 x 10^-23  (10 ⁶ K) (8.85 x 10 ^-12 F/m)  / (4p)  (10 ¹⁶/m³)  e² ] ^½

l _D,s   =  1.9 x 10 ^{- 4}   m

L  =  7.3 x 10 ⁴   

And N  =   4p n _o  l³ _D  / 3 

N =  (4p)(7.3 x 10 ⁴   ) / 3  »   3.1 x 10 ⁵   

Therefore:   1/N   =   1/ (3.1 x 10 ⁵ )

 

1/N = 3.2 x 10 ^{- 6}   

 1/N ® 0  , hence the  plasma is “collisionless” so that this meets the condition.

Then we may use the collisionless Vlasov-Boltzmann equation for a plasma description, e.g.:

v ¶ f _x/ ¶ x  - q(s)/m(s) [¶ j/ ¶ x  · ¶ f _s/ ¶ v]  = 0

where (f _s) is any function of the constant of the motion.

This, along with its collisional form, has been called “the most important equation in plasma physics”.[1] The associated Poisson equation is:

Ñ E = 4p e (ò f_e d³ V - òf _i d³V) = 0

where the e, i –subscripts denote electrons and ions, respectively and f refers to the appropriate distribution for each.

In the classical theory of shock waves in collisionless plasmas, one would not expect dispersion to be a sufficient condition to incite a shock.  (A dispersion relation implies that a relationship exists between the plasma frequency w and the wave number k. )  What it can do is form a large amplitude fixed wave called a “soliton”. (‘Steepening’ + dispersion yields solitons, while steepening + dissipation yields shocks). These soliton solutions ensure no turbulence (due to shocks), thereby meeting the collisionless condition. To estimate M we use (e.g. Sakai and Ohsawa, 1987)[2]:

B _m =  (2M – 1) B_o

Or in terms of the ratio of the magnetic fields:

M =  2 [B _m  / B_o] + 1

To reach an M= 80 value, say which might conceivably apply for coronal shocks, one would need:

[B _m  / B_o]  »  39.5

I.e. the ratio of the maximum magnetic field intensity to the equilibrium value is at least 39.5.

 Compression is also critical to have an effective shock and we have the condition, based on the plasma number density N:

DN/ N   =  DB/ B_o  [1 - DB/ 2 B] ^{- 1}

We already saw for the equilibrium collisionless corona:

N »   3.1 x 10 ⁵   

It is instructional here to determine the trigger or break point for the transient, or CME. This should be reached when the massive influx of dissipating particles is so great that:

DN/ N    ® ¥

 

By inspection we see that if DB = 2B :

¥   =   2B/ B_o  [1 – 2B/ 2 B] ^{- 1}

 

To achieve this one would most likely need a strong two-stream instability requiring (cgs units):

M > 1 +  [8p N T_e / B² ]  ^1/3

This elicits the necessary magnitude for B given our values. A typical soliton –like plot will appear:

The relevant equation for a soliton embodying a dissipative term to evolve into a shock can be written:

(- v _o  + c _s   + v) ¶  v / ¶x’  -  m ¶ ² v / ¶x²’  +  a ¶ ³ v / ¶x³

Where the underlined term above is the dissipative term. In a collisionless plasma such as the corona under consideration, the dissipation can be attributed to Landau damping and this may be the most common form to excite a shock associated with CMEs. For all such collisionless shocks the dissipative length scale, L << l _mfp   , the mean free path.  

The preceding equation can be derived using an analog to Newton’s 2^nd law of motion, viz.

a v” = F(v)  = a d²v/ dt²   =    -dF / dv

Where:   F  =  (c _s   -  v _o) v ² / 2   +     v³/ 6

Is the Sagdeev Potential.

(To be continued - with problems)

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[1] D.R. Nicholson:1983, Introduction to Plasma Theory,  John Wiley, p. 43

[2]J.-I. Sakai and Y. Ohsawa:1987, Space Sci. Rev., 46, 113.