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...

What Aspects Of The Coming Total Solar Eclipse Are Most Critical For Astrophysics?

Image showing totality in a previous  solar eclipse. The Sun is totally blocked out by the Moon - in the line of sight- and the solar corona is seen expanding from the solar limb


Graphic showing the path for Aug. 21 total eclipse - and the partial zones to either side, with the percentages of the Sun to be covered.

As we approach August 21 and the first total eclipse to appear in in the U.S. in38 years, many are trying to stay ahead of the fervid hype. Here are the only two things you need to know: 1) In most of the U.S. the eclipse will be seen as partial only and you need special glasses in order to observe the transition to maximum coverup, and 2) If you want to observe the total eclipse (see top image) you will have to travel to some place within the band of totality (see graphic).  For example, in our location (Colorado Springs) the Sun will be roughly 85 percent covered.

Here's the skinny on stats for this eclipse: Approximately 12 million Americans will be directly in the totality band shown in the graphic. Another 88 million will be within 200 miles of some place inside the totality band. Naturally then, those living in communities right in the band (e.g. Jackson Hole, WY, St. Joseph, MO) are expecting eclipse watchers to pour in and pump up their economies - buying t-shirts, special glasses and other paraphernalia.

While the basic image of totality will be the appeal for casual observers,  and even partial eclipse will occur for (and excite)  many others, this doesn't hold quite the same spell for solar physicists or astrophysicists. They look to good observations made during totality, namely to be able to record and analyze the solar corona.   This is the outermost region of the solar atmosphere, at millions of kelvins temperature and extending sometimes millions of kilometers into space.

The corona up to now has presented a mystery, especially in terms of its "inverse" increasing temperature profile considered in the context the general temperature profile is steadily decreasing from the Sun's interior to outside.   For example, the Sun's "surface" or photosphere has an effective temperature of 5,777 K .   The corona's temperature by contrast is at least 2 million degrees K.

How did we find this out?  The key breakthrough probably arrived in 1939 when astronomer Walter Grotrian found that a previously discovered spectral line (attributed to "coronium") was actually 13 times ionized iron. Since it takes enormous energy to ionize even one iron atom (meaning stripping one of its outer electrons away) this meant the iron had to be subject to enormously high temperatures.  Indeed, such ions can only exist in plasmas with temperatures between 1 million and 5 million K.

The first clue as the cause of the extraordinary hearing came with high altitude rocket flights in the 1970s, bearing x-ray telescopes. These showed  features down to a resolution of 1 arcsec or 730 km.  Careful observations showed a positive correlation between the x-ray brightness of active regions (i.e. containing complex sunspots) and plasma-filled magnetic loops, e.g.



















The above image was taken by the much more recent Atmospheric Imaging Assembly, built for the Solar Dynamics Observatory . But you easily get the idea noting the brightness (representative image taken at  211Å) and the magnetic loop structures.

Another milestone occurred when Eugene Parker predicted the hot corona must expand into space as a solar "wind". It could not remain stationary or in place with no expansion. The proof of this is left at the end for those interested.

The next question, of course, is where the energy comes from not only to heat the corona but accelerate the solar wind which is an extension of it.  We believe now that MHD or "magnetohydrodynamic" waves of some type provide an answer.   Field lines such as shown in the previous image are rooted at both ends in the Sun's convection zone where they are jostled by the churning of convective cells.  If sufficiently rapid these field line motions could generate magnetic waves capable of carrying energy upward into the loop.   It seems that Alfven waves e.g.

http://brane-space.blogspot.com/2011/04/getting-handle-on-alfven-waves-2.html

are the most effective in reaching coronal heights. (Alas, Alfven waves do not compress the ambient plasma so we still need to identify some other mechanism that can effectively transfer the wave energy to the plasma. A clue may lie with Landau damping, say associated with a beam (or 2-stream)  instability in the plasma. We have a dual Maxwellian profile.


In the region where the slope is positive (¶f(v)  / ¶ v > 0) there is a greater 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.

But in Landau damping, with the slope negative (¶f(v)  / ¶ v < 0) the number of particles slower than the waves phase velocity exceeds the number of those that are faster. Thus, more particles gain energy from the wave than lose energy to it.

Whatever new data the eclipse generates, and there will be lots of it, we can be confident that the mysteries of the solar corona won't be solved even after it's all analyzed and multiple papers published. But I will keep readers abreast of any new findings say that appear in the next 8-12 months.

Those interested in actual scientific participation are invited to go to the Globe Observer site, where you can collect the app and then be prepared to make temperature measurements.

https://observer.globe.gov/

See also:

https://www2.mps.mpg.de/meetings/ccmag/dokumente/presentations/A.04.06.pdf

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Proof that the corona can't be stationary:

A static corona  superficially appears  to be quite reasonable but that’s why we need to test this is so. The first one to do this was Sydney Chapman. He began by first assuming the condition for hydrostatic equilibrium applied:

dp/ dr = - r {GM_s/ r²}

where G is the usual Newtonian gravitational constant, and r defines the plasma density for the corona, while M_s is the mass of the Sun, and r the distance from the solar center:

r = n(m_p)

with n the number density for protons

The coronal pressure (P) is given by:

P = 2 n T

Provided both protons and electrons are assumed to have the same temperature.

The thermal conductivity of the corona is dominated by electron thermal conductivity and takes the form:

k = k_o T ^5/2

for typical coronal conditions the value of k is about 20 times the value of copper at room temperature.

The coronal heat flux density is:

q = -k Ñ T

A static corona means heat inputs cancel heat outputs so that the divergence:

Ñ× q = 0

Assuming a spherical symmetry for the corona one can write:

1/r² [d/dr (r² k_o T ^5/2  dT/dr)] = 0

    Obviously the preceding assumptions mean there must be some distance where the coronal temperature becomes zero.

From the above equation one should be able to show:

d(T ^7/2) = 7/2 (F T_o ^5/2)/ 4 p k_o d(1/r) = C d(1/r)

where C is a constant.

The integral is:

T_o ^7/2 - T ^7/2 = C[ 1/R_o - 1/r]

Now, set the temperature at infinity (T) to zero and obtain:

C = R_o T_o ^7/2 

which fixes the total flux at:

F = 2/7   [4 p R_o k_o T_o ]

After another step, one finds:

T(r) = T_o (R_o / r) ^2/7

    This gives the temperature T at a distance from the Sun= r. This is based on using a defined value (say T_o = 2 x 10⁶ K) at a defined distance, say R_o = 7 x 10⁸ m.

    For example, at the Earth’s distance (r = 1.5 x 10¹¹ m) one would find: T = 4.3 x 10⁵ K

    This seems fine, until one examines the pressure.

    Analogous to the temperature formalism, we have, the pressure p(r) at some distance r defined by:

p(r) =

p(R_o) exp [7/5 GM_s m_p/ 2 T(R_o) R_o {( R_o / r)^5/7 – 1}]

    Now, if one allows r to approach infinity, e.g. r ®¥ an interesting thing occurs in the equation, as we can see. That is, the denominator of the first term in the end brackets becomes so large (R_o / ¥) that the first term vanishes.

Then we are left with the expression for the pressure:

p(¥) = p(R_o) [exp – 7k/5 * 1/ T(R_o) R_o]

where ‘k’ denotes a constant composed of all the constant quantities in the previous eqn. (G, M, m_p etc)

    Substituting the given values into the above, one finds p(R_o) multiplied by a factor

exp[0] = 1

    The reason is that the exponential of a very small and negative valued magnitude ® 0

Then:

p(¥)  »  p(R_o)

But this can’t be since the pressure of the coronal base would then be the same as the value at infinity!

    This led astrophysicists to conclude an unphysical result, and that the static coronal model couldn’t be accurate.

    If the static model were accurate, the pressure at infinity should be zero, p(¥)   = 0, not a small finite pressure that’s effectively equal to the coronal base pressure. This finding led to the further investigations that disclosed a solar “wind” had to flow outwards from the corona.

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 )