15. Solar MHD II. Solar MHD • Particle Motion in Electric Field • Magnetic Effects • Ohm’s Law • MHD Equations • Diffusion • Frozen Magnetic Flux Approximation • Magnetic Forces • MHD Waves • movie: https://www.youtube.com/watch?v=QArcTylNooQ MHD Equations Consider plasma in an electro-magnetic field. The Maxwell equations are (CGS units): MHD approximation 41 E Bj    neglect displacement current because MHD cct processes are slow compared to the speed of light B 0 

1 B E   ct

neglect separation of electric charges of E 4e  electrons and ions – plasma quasi-neutrality

1 j  ()EvB - Ohm’s law in a non-magnetized plasma c

(1)L  moving with velocity v; recall the electric field transformation in a moving 1 coordinate system: E EBv  c Equations of conservation of mass, momentum and energy

Then, we combine the Maxwell equations with the equations of conservation of mass, momentum and energy:

  ()0v t

dvv 1  ()vv P j B dt t c

dS TQL  dt where ρ is the mass density, P is the gas pressure, v is the velocity,  ScV log() P is the specific entropy, Q is an energy input, L is the energy loss rate (radiative losses).

Magnetic Field Diffusion

11 Applying curl to the Ohm’s equation: E  j ()vB  c 11 EjvB ()  c

4 and using the Maxwell equations: Bj  c 1 B  E  ct we get an equation for magnetic field strength B :

B 11 cc E  jvB  (),  or tc

B c2 1  ()vB     B  t 4 This is “the induction equation” - a central equation for solar MHD theories.

Magnetic Reynolds Number

B c2 1  ()vB     B  t 4

advection of magnetic field magnetic field diffusion

The first term in the right-hand side describes advection of magnetic field, the second term corresponds to magnetic field diffusion due to Joule dissipation.

The relative importance of these terms for a process of a characteristic scale L , velocity v is determined by the magnetic Reynolds number RM or ReM: vB L 4 Lv RM  2 c2 B c 42 L For typical coronal conditions: T  106 K,   1012 s 1 , L  108 cm, v  105 cm/s, 4 RM  10 1  Coefficient of magnetic diffusion

For uniform  the last term can be simplified: 22  ()()BBBB   

B 2 Then, if v  0 we get a diffusion equation:  DB t c2 where D  is a diffusion coefficient for magnetic field. 4 Exercises: 1. Estimate the characteristic scale of dissipation of magnetic field in solar flares. The duration of solar flares is 103 sec. ct2 L 105 cm 1km 4 2. This is smaller than the observed flare structure. What does that mean? 3. Estimate the decay time of sunspots ( L  109 cm, T  104 K,   109 s 1 ). 4 L2 t 107 sec  4 months c2 4. This is longer than the observed lifetime of sunspots. Why? Ideal MHD approximation The MHD equations without energy input and dissipation are called the `ideal MHD’ approximation: d  v 0 dt dv 1   P () BB  dt 4 dP

  0 dt  B  ()v B t The ideal MHD approximation is often used for modeling processes with high magnetic Reynolds number.

Linearized ideal MHD equations are used to describe MHD waves. Consider small perturbations of velocity, density, pressure and magnetic field: vb, , P , in uniform stationary plasma with constant magnetic field: velocity vB00 0, density  , magnetic field 0 .

Frozen Magnetic Flux Approximation

Consider a high-conductivity plasma, RM  1, or    (‘ideal plasma’). Then the equations for B magnetic field are:  ()vB   t  B 0 Consider a 1D case: B (0B 0), v  (00)v , - plasma motion across the magnetic field lines, with B  B and v depending only on x :  ()vB tx   From this and the mass equation:  ()v tx dB v we get  B 0 dt x dv    0 dt x dB B d  Then  0 dt dt B d   B or  0 or  const dt  d BB  For a general 3D case:   v dt   This equation shows that in ideal plasma magnetic field is coupled with plasma density. It follows the plasma motions. Magnetic Flux Conservation

Consider magnetic flux  through a plasma area S restricted by a closed curve  :  Bsd  S where s is a vector perpendicular to the area.

If we change the contour line of this plasma element then the total flux will change due to: 1) the change of the magnetic field strength (as follows from the MHD equations); 2) the change of the area of this element: ddd   dt dt dt d B where   ds dtS  t Now consider a small change of the area ds due to plasma motion with velocity v during time dt : ddtdsv  l where l is the change of the length of contour  . Then, the change of magnetic flux IS: dd BsBvl     ddtdtd () vB   l  where we used a vector-product relation. d d B Then,  vBl d   ddsvBl  dt  dtS  t  Magnetic Flux Conservation

d B  ddsvBl  dtS  t  Using Stokes’ theorem to replace the contour integral with the surface integral

vBldd () vB s  S we obtain for the total flux:

dB  ()vBd s 0 dtS  t The right-hand side is equal zero because it satisfies the equation for magnetic field in an ideal plasma.

d  0 Thus, dt .

This is the frozen flux theorem: the total magnetic flux through a plasma element does not change under deformations of this element. This can be interpreted as magnetic field lines move with the plasma (“frozen into the plasma”). Magnetic Forces 1 j B The , c , in the momentum equation can be expressed in terms 4 of magnetic field using the Maxwell equation, B  j : c 1 fBB  4 1 aaaaa2 () Using the standard vector formula: 2 11 1 fBBBB BP2 ()    ()   84 M 4 2 The first term is a gradient of : PM=B /8; the second term describes magnetic tension force. The tension force is analogous to restoring force of rubber bands.

The magnetic forces can be written in a tensor form:  f  Tik xk  112 where TBBBik ik i k are Maxwell stresses. The magnetic forces are anisotropic.   42 MHD Waves Magnetic forces can produce additional restoring force to small perturbations in a magnetized plasma and cause oscillations. Consider a linearized system of the ideal MHD equations for perturbations , velocity v , and magnetic field, b :    v 0 t 0 v 1   P  bB  00t 4

 P0 2 Pc S  (cS is the adiabatic sound speed) 0 b ()vB  0 B0  t k wave vector Consider a periodic solution: vb eiitkr  for k (0 0k ) , B00(0BB sin  0 cos ) - wave propagation along z-axis;

 (the angle between k and B0 ). 222 If B0  0 we obtain the dispersion relation for ordinary sound waves:   ckS .

The phase speed: ukc / S. Alfven waves

If B0  0 then we have solutions (dispersion relations) of two types: 2 22B 2 1)   k cos  4

- Alfven waves; vBA 4 is the Alfven speed. These waves are incompressible   0 . Plasma moves perpendicular to magnetic field lines.

The phase speed uk  is: uvAlfven  A cos

Fast and slow MHD waves

11 2) ukcvvcvc222 () 22  4422  2cos2  22SA AS AS

where cS is the sound speed. The solution with "+" is called fast MHD wave, with "-" - slow MHD wave. For   0 (propagation along field lines) and:

 vcuA S fast  vu A slow  c S - strong magnetic field

 vcuA S fast  cu S slow  v A - weak magnetic field for   2 (propagation perpendicular to the field lines): 22  uvcufast A  S  slow 0 In the fast MHD waves the magnetic and pressure forces act together, in the slow MHD waves they act against each other.

Illustration of Alfven and magnetosonic (fast MHD) waves

(fast MHD waves)

• The Alfven waves are incompressible. They transfer vorticity along the field lines. Plasma oscillates across the initial field lines. • The fast MHD waves mostly travel across the magnetic field lines with a speed higher the speed of sound and the Alfven speed. • The slow MHD waves mostly travel along the field line with a speed slower than the sound speed. • In plasma of variable density (solar atmosphere) the waves can transform from one type to another. Animation of Alfven wave – magnetic field lines are frozen in plasma Hinode observations of Alfven waves in the solar corona Illustration of the wave propagation and energy transport from a localized pressure perturbation (T. Magara) Sound wave (B0=0)

Without magnetic field the solution represents an isotropic sound wave.

Pressure perturbation Sound wave (B0=0)

Without magnetic field the solution represents an isotropic sound wave.

Pressure perturbation Weak magnetic field (c >v ) S A In the case of weak field: ccSA , or  PB2  , or 4 P  1, or B2 /4 B0 PP M  the solution is a superposition of fast MHD waves (most pronounced across the field lines) and slow wave (mostly along the field lines) . This is the case of “high plasma beta”: PP8 Pressure perturbation  2 PBM Weak magnetic field (c >v ) S A In the case of weak field: cvSA , or  PB2  , or 4 P  1, or B2 /4 B0 PP M  the solution is a superposition of fast MHD waves (most pronounced across the field lines) and slow wave (mostly along the field lines) . This is the case of “high plasma beta”: PP8 Pressure perturbation  2 PBM Strong magnetic field (vA>cS)

In this case (“low plasma beta”) most of the energy is transported by the slow MHD waves along the magnetic B0 field lines.

Pressure perturbation Strong magnetic field (vA>cS)

In this case (“low plasma beta”) most of B0 the energy is transported by the slow MHD waves along the magnetic field lines.

Pressure perturbation Propagation of pure acoustic wave

Time-distance diagram Propagation of MHD waves: case vA>cS Strong magnetic field transports wave energy preferentially along the magnetic field. Numerical simulation of MHD wave propagation through a sunspot (K.Parchevsky)

Fast MHD waves travels across the magnetic field lines; slow MHD waves travels along the field lines into the deeper layers.