2.3 The Physics of Coronal Plasma
2.3.1 Magnetohydrodynamics
The field of magnetohydrodynamics (MHD) was strongly influenced by Hannes Alvén, who was rewarded with the Nobel Price in 1970. The theory describes the hydrodynamics of magnetised fluids (e.g. salt water, liquid metals, plasma) through the combination of elec-trodynamics and fluid dynamics. The dynamics of the coronal plasma can be physically described by the theory of magnetohydrodynamics.
MHD is based on the fundamental concept that a moving conductive fluid in a magnetic field induces currents, which in the presence of the magnetic field, generates a force that acts on the motion of the fluid. In return, the fluid motion alters the geometry and strength of the magnetic field itself. The main quantities which characterise such electrically con-ductive fluids are the bulk plasma velocityv, the current densityj, the massm, the mass density%, the plasma pressure p, the magnetic FieldB and the electric fieldE.
Lorentz Force
In magnetised plasma, the Lorentz Force Fl =qv×B is effective on a particle with a chargeq. The particles move on trajectories perpendicular to the magnetic field, with the Larmor radiusrL and cyclotron frequencyωc. They are defined as
rL= mv⊥
|q|B (2.1)
and
ωc= |q|m
B . (2.2)
Conditions for the Application of Magnetohydrodynamics
The theory of magnetohydrodynamics describes the large-scale, slow dynamics of plasmas and can be applied when:
1. The characteristic time T is much greater than the ion gyroperiod and mean free path time of the system:
1
ωc T. (2.3)
2. The characteristic length L is much greater than the ion gyroradius and the mean free path length of the system:
rLL. (2.4)
3. The plasma velocitiesv are not relativistic:
vc. (2.5)
In the following the theory of magnetohydrodynamics is derived from the combination of electrodynamics and fluid theory.
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Maxwell’s Equations
Maxwell’s equations form the foundation of electromagnetism:
∇ ×B =µ0j+ε0µ0∂E
Together with the Lorentz Force, they fully describe all classical phenomena of electrody-namics.
Equation of State
Regarding fluid dynamics, the plasma pressurepcan be determined from the mass density and the plasma temperatureT by the equation of state (e.g. the ideal gas law). For a pure hydrogen plasma this equation is given as
p= 2kB
mp%T, (2.10)
where mp is the proton mass and kB the Boltzmann’s constant. The equation of state of the system is then given by
d
whereγ is the polytropic index and defined as the ratio of the specific heatsCp/CV, where Cp is the heat capacity at constant pressure and Cv is the heat capacity at constant vol-ume. It is taken as 5/3 in the adiabatic case.
Mass Continuity Equation
The fundamental equation of fluid dynamics is the mass continuity equation,
∂ρ
∂t +∇(%v) = 0, (2.12)
stating that mass is neither created nor destroyed.
Momentum Equation
The motion of the plasma is described in hydrodynamics by the momentum equation, also known as Euler’s equation which is a special case of the Navier-stokes equation for non-viscous elastic fluids. To account for magnetohydrodynamics, it is expanded by a Lorenz force term to:
%∂v
∂t%(v· ∇)v =−∇p+j×B. (2.13)
Additional viscosity terms can be neglected, because any transport process perpendicular to the magnetic field is strongly inhibited by the gyration whereas advection dominates
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2.3 The Physics of Coronal Plasma
along field lines. The field equation forB can be derived from the first Maxwell equation (Eq.2.6). In the MHD-approximation the second term, called displacement current, can be neglected, yielding:
∇ ×B=µ0j. (2.14)
The vector identity
(a· ∇)a=∇(a2/2) + (∇ ×a)×a (2.15) can now be used to eliminate the current and replace it with the magnetic field.
j×B= 1 where the termB2/(2µ0)is generally defined as the magnetic pressurepM. The substitution of Eq.2.16 into Eq.2.13 will eliminate the current and form the momentum equation of Magnetohydrodynamics:
The induction equation relates the velocity of an electrically conductive fluid to the mag-netic field. In particular, this equation describes the phenomenon of a magmag-netic dynamo.
The substitution of the electric fieldE in Faraday’s law (Eq.2.8) yields:
∂B
∂t =−∇ ×E, (2.18)
and with the help of the generalised Ohm’s Law
j=σ(E+v×B), (2.19)
this leads to the induction equation of Magnetohydrodynamics:
∂B
∂t =∇ ×(v×B)− 1
σµ0∇ ×(∇ ×B). (2.20)
Here, the electrical conductivity σ is regarded as constant. The model describes how a magnetised plasma responds to fluid motion and vice versa. The ratio of the two terms reveals which one dominates over the other. In a similar fashion to the fluid equations, one can define the dimensionlessMagnetic Reynolds NumberRM:
RM = ∇ ×(v×B)
η∇2B . (2.21)
In order to arrive at Eq. (2.21), the vector identity from Eq. (2.15), the solenoid constraint
∇B = 0 and the definition of the magnetic diffusion η = (σµ0)−1 were used. For high Reynolds numbers realised via a perfect conductive fluid (RM → ∞, σ→ ∞), drag effects can be neglected, which will reduce the induction equation to the following form:
∂B
t =∇ ×(v×B). (2.22)
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Ideal Magnetohydrodynamic Equations
Thus, the closed set of Magnetohydrodynamic equations is:
∂%
All dissipative processes resulting from thermal conductivity, electrical resistivity or finite viscosity are not considered in this formulation.
2.3.2 Application of Magnetohydrodynamics to the Solar Corona
The magnetic fieldB constraints particles in the coronal plasma to perform spiral gyro-motions along magnetic field lines. If the kinetic energy of a particle exceeds the magnetic field energy, it can escape from its gyroorbit. This behaviour can be described by the plasma-β parameter which is defined as the ratio of thermal against magnetic pressure in a plasma
β = pth
pmag
= nkbT
B2/2µ0 (2.27)
with the temperature T, the particle number density n, the Boltzmann constantkb, the magnetic fluxB and the magnetic constant µ. Table2.1gives an overview of the physical properties from the photosphere to the outer corona. Forβ <1, the structure of a plasma is dominated by the magnetic field. Forβ >1, the magnetic field is frozen to the plasma and tied to its motion. As illustrated in Figure (2.18), magnetic forces are controlling the structure formation in the upper chromosphere and the lower corona, creating structures such as filaments, coronal loops corona and helmet streamers. In the outer corona, the plasma-β increases, which forces the magnetic field to follow the motion of the plasma.
This leads to the effect that the interplanetary magnetic field winds up to the form of a spiral (compare with Section3.2).
Table 2.1: Plasma parameters in the photosphere and the corona. (Aschwanden,2004) parameters photosphere cool corona hot corona outer corona electron densityne (cm−3) 2×1017 1×109 1×109 2×107 temperatureT (K) 5×103 1×106 3×106 1×106
pressurep (dyne cm−2) 1.4×105 0.3 0.9 0.02
magnetic fieldB(G) 500 10 10 0.1
plasma-β 14 0.07 0.2 7
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2.3 The Physics of Coronal Plasma
Fig. 2.18:Typical plasma-β range for the different layers of the solar atmosphere for a magnetic field of the strength 100-2500 G. (Gary,2001)
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