Looking At Some Plasma Instabilities Applicable to the Sun

There are many variants and types of plasma instabilities on the Sun which play both major and minor  roles in events ranging from X-class solar flares to coronal mass ejections and even filament ejection-displacement.  In this post I look at a few of these.

Conveniently, many of these instabilities are defined in terms of specific velocities. One of the more important of these is known as the electron drift velocity  v  _d  :

 v  _d  = I/ n e


where I is the current, n the number density (per cubic meter, for example) and e the electron charge. Often   v  _d  is defined instead based on current density (J) where: J = I/A (current in amps flowing through a magnetic flux tube or channel of some cross-section area A)

So: v  _d  = J / n e

In many other cases, where J, I are not so easily obtained, v  _d    will be the electron drift velocity in a combined magnetic and gravitational field, i.e.:

v  _d     = m _e  g _s  c / e B

where B is the magnetic induction, e the electronic charge (1.6 x 10^-19 C) ,  g _s = 273  ms -2  the gravitational acceleration on the Sun, c is the velocity of light (2.98 x 10 ⁸   m/s) and m _e the electron mass (9.1 x  10 ^-31 kg) .

In the case of the ion-acoustic instability, the threshold for its onset is when:

v  _d    > 43  v  _s

that is, the drift velocity is at least 43 times the ion sound speed, where:

 v  _s   = Ö  (kT/ m _i)

where k is the Boltzmann constant (1.38 x 10 ^-23  J/K), T the (Kelvin) temperature and m _i  the ion mass (e.g. m _i  ~ 1.7 x 10 ^-27 kg).

Sen and White in a 1972 paper dealing with the role of the Hall effect in flares, showed that the two-stream instability is incepted when the drift velocity:

  v  _d    >   v  _ith

where  v  _ith   is the ion-thermal velocity:

v  _ith   =   Ö  (2kT/ m _i)

(rms value = Ö  3(kT/ m _i)

In two-stream instability, when an electron flow is suddenly injected into a plasma – say for a coronal loop – the particles’ (Maxwellian) velocity distribution acquires a “bump” on its "tail" (higher velocity end of the distribution), consistent with two streams- an unperturbed one ( f  _ov) and perturbed one ( f  _eb ) applicable to the electron beam (See diagram below ).

In the region where the slope is positive (df  _v  /d v > 0) there is a greater number of faster i.e.  than slower particles so a greater amount of energy is transferred from particles to associated (e.g. Alfven) waves. Since  f  _eb contains more fast than slow particles a wave is excited, and there is inverse Landau damping such that plasma oscillations with v_ph (phase velocity) in the positive gradient region are unstable.

Resonant electrons (at v  _ph   >   w _e  / k) where w _e  is the electron plasma frequency, i.e.

w _e     =  [n_e e²/ m_e  ε_o] ^½ 

 are the first to be affected by the local wave-particle interactions and have distributions altered by the wave electric field, E1, such that the total energy balance:

E1 (TOT) = ½ E1 _w + ½  E1 _k

referencing the wave and kinetic (particle) contributions respectively.

Thus, for E1(TOT) = const. then as the electron velocity decreases, the particle kinetic energy decreases and the wave energy density increases.

In Landau damping the exact opposite occurs, so the gradient df(v)/d v decreases, and with it the wave amplitude, while the particle kinetic energy increases- i.e. wave energy lost is fed to the particles (electrons) which gain energy.

Note also that the "plasma beam instability" is just the finite temperature analog for the two stream instability. In other words, if one has a "two stream instability" one de facto also has the beam instability.

Lastly, for the Buneman instability, the relevant condition is:

v  _d   >  v  _eth

where  v  _eth    is the electron thermal velocity:

v  _eth    =   Ö  (2k T _e/ m _e)

where T _e   is the electron temperature and k the Boltzmann constant.

The rms (root mean square) value for   v  _eth    is   Ö 3 (k T _e/ m _e)

The objective for much better solar flare predictions is to be able to interpret which of these instabilities is near onset for a given solar plasma condition. This is not easy by any means, and probably requires much better optical resolution than available at present. However, solar physicists will "truck on" using all the resources they have to effect the best forecasts of which they are capable.

See also:

A Closer Analytical Look At Two Stream Instabilities...

Coronal Mass Ejections and Collisionless Shocks (2) - With Problems

In the previous instalment I noted that the dissipation associated with collisionless plasma can be attributed to Landau damping and this may be the most common form to excite a shock associated with CMEs. But even more germane is the two stream instability which can be thought of as the inverse of Landau damping, where a greater number of particles that move slower than the wave phase velocity v_ph (as compared with those that move faster), leads to an energy transfer from the wave to the particles.  In the case of the two stream instability, when an electron stream is injected to the plasma, the particle's velocity distribution function has a "bump" on its tail:


In the region where the slope is positive (¶f(v)  / ¶ v > 0) there is a greaterI number of faster than slower particles so a greater amount of energy is transferred from particles to associated (e.g. Alfven) waves.  Since f _eb contains more fast than slow particles a wave is excited, and there is inverse Landau damping such that plasma oscillations with v_ph in the positive gradient region are unstable. We expect  v_ph  to occur where the gradient:(¶f(v)/ ¶ v) ê_{wr/ k}  = maximum. Instability can result provided: 

w_e  ²   >   k²v²  or  êkv ê  <   w_e  ^.

Resonant electrons (at v_ph  »  w_e/ k)   are the first to be affected by the local wave-particle interactions and have distributions altered by the wave electric field, E1, such that the total energy balance:

S E_tot = ½ E1 _w + E1 _k    =  ½[ ½ e_o | E1|²]  +  ½ r | ṽ|²         

For which the wave energy (E1 _w) is balanced against the kinetic (particle) energy (E1 _k ). Thus, for S E_tot = const. then as ṽ decreases, ½ e_o | E1 |²]   increases. In Landau damping the exact opposite occurs, so as the decrement associated with ¶f(v)/ ¶ v decreases, the wave amplitude (and  E1 _w) decreases, particle kinetic energy increases- i.e. energy is fed to the particles at expense of waves.

For reference, the longest known radio signatures for coronal shocks are the Type II radio bursts. These are narrow band (Dn/n ~ 0.1)  ) radio emissions excited at the local plasma frequency for which the drift rate :

dn/dt ~ ( - 0.2 MHz/s)

Where the negative sign indicates a drift toward lower frequencies. This is as the shock propagates outwards through the corona. Derived measurements disclose a radial velocity on the order of 1000 km/s. In 60 per cent of cases there is a clear harmonic structure with a frequency of occurrence of once every 100 hours. The Type II bursts are accompanied by weak polarization. The original excitation is usually by a fast mode MHD shock though the mechanisms previously considered, i.e. via Landau damping and two stream instability, must also be considered.

If we know the ambient free electron density associated with the event, it is possible to estimate the pure plasma  frequency (n)  using:

1)      n   =   9000  Ö N    or    2) n   =   9  Ö N   

      

In the above (1) is used to get MHz if the units for N are in c.g.s. units, and (2) for S.I. units. Bear in mind:w_e   =   2p n.

It is also useful to note that radio waves cannot propagate through a medium in which the plasma frequency  is greater than the radio frequency.

Problems:

1)For a hydrogen plasma, a lab generated shock of Mach number M= 80  with the temperature at T =   10 ⁵    K is found to have an ion gyroradius of 0.22 cm.  If    Ω _e  =  1.9 x 10 ⁷  /s  what would be the magnitude of the associated magnetic field (B)?

2)Find the associated plasma frequency and use it to obtain the Debye length and plasma parameter if n is a value typical of the solar corona  (10 ¹⁶ /m³). Find also the frequency n of the associated radio emission.

3)If the shock speed is u =  10 ⁷  cm/ sec, estimate the wave number k _w.  

4)Hence, or otherwise, use the equation below to obtain the ion velocity  v _o   if the  frequency w  = 10 ¹⁰ / s

(Note: m is the electron mass, and M =  m_i, the ion mass.)

5)Based on all the above information, estimate the maximum magnetic field B _m using the shock equation.

6) The frequency of a critical layer as a function of electron density is given by:

n   =  e/2p  Ö( N/ ε_o m _e )

Show that this can be reduced to the simplified form:

n   =   9  Ö N

(Hint: ε_o  =  8.85 x  10 ^-12  farad m^-1 )