2.2.1 Continuity equation
In the limit of MHD, plasma can be described as the motion of a quasi-neutral fluid. The plasma mass density is a conserved quantity, and it flows in space in a continuous way (thus the name continuity equation). Mathematically the mass conservation is described by the continuity equation,
∂ρ
∂t +∇·(ρu)=0, (2.22)
whereρis the mass density anduis the plasma velocity. Eq. (2.22) states that an increase of density at some point requires a mass flow into the neighboring region and a density decrease requires a mass flow out of the surrounding region. Alternatively, Eq. (2.22) can be expressed using the Langragian derivative DtD = ∂t∂ +u·∇as,
Dρ
Dt +ρ∇·u=0. (2.23)
For an incompressible flow,∇·u=0 and, Dρ
Dt =0. (2.24)
2.2.2 Momentum equation
The fluid motion is subject to forces acting on it. Hence, Newton’s second law describes the motion of the fluid. The equation of motion can be expressed with the momentum equation,
ρDu Dt =X
i
fi (2.25)
where D/Dt is the Lagrangian derivative and fi are the forces per unit volume. More precisely, this forces are,
• the pressure gradient−∇p
• the Lorentz force j×B
• the viscous force 2ν∇◦(ρS)
• gravitational force -ρ∇Φ,
whereνis the viscosity andS is the strain tensor. Forces that arise from rotation, such as Coriolis and centrifugal force, are generally not taken into account in the coronal study, and thus they are ignored.
2.2.3 Equation of state
To fully describe a fluid it is required to establish an equation of state. For most astro-physical objects that consist of plasma at low pressure, the equation of state of an ideal gas,
p= kB
µmp
ρT (2.26)
can be considered as a good approximation. Hereµis the atomic weight,kBis the Boltz-mann constant andmp is the proton’s mass. For a fully ionised hydrogen plasma, like the one we find in the solar corona, we have the same amount of protons and electrons, thereforeµ= 0.5 and for the densities,
n= ne +np= 2ne, ρ= npmp+neme ' nemp. (2.27)
2.2.4 Energy equation
The last equation necessary to fully describe the plasma evolution is the energy equation, ρTDs
Dt = −L, (2.28)
where sis the entropy per unit mass andL are all the sources and sinks of energy. The previous equation shows that an increase of heat per unit volume as it flows in space is only due to the total effect of all the sinks and sources. In the absence of sinks and sources, entropy is a conserved quantity.
A more convenient way to express Eq. (2.28) is by using the rate of temperature changeT instead of entropy s. To do that, we have to introduce the change of the in-ternal energyde,
de =T ds− p
ρ2dρ. (2.29)
Now Eq. (2.28) can be expressed as a function of the internal energy, ρDe
Dt − p ρ
Dρ
Dt =−L. (2.30)
For an ideal gas the internal energy is given bye= cvT, withcv being the specific heat for a constant volume. Using now the continuity Eq. (2.23) we derive the final expression,
cvρDT
Dt + p∇·u=−L (2.31)
The sources and sinks are described in the following section.
2.2.4.1 Ohmic dissipation
The term that describes the conversion of magnetic energy to thermal energy is called Ohmic or Joule dissipation. The expression is given by,
LH =−ηµ0j2, (2.32)
whereη is the magnetic resistivity and for the solar corona it is on the order of 1 m2/s.
However, following the work of Bingert and Peter (2011) we have, for numerical reasons, to set a much larger value ofηin the numerical simulations (see Sect. 4.2.1). The Ohmic heating term has a significant contribution in the heating of the coronal plasma in our numerical models.
2.2.4.2 Viscous heating
The viscous force can be considered to act analogously as friction does in our everyday life, converting the kinetic energy of a body into heat. Similarly, for fluids, the conversion of kinetic energy into heat is given by the viscous heating term,
Lν = −2ρνS2, (2.33)
whereνis the kinematic viscosity and typical values in the solar corona, from transport theory, are on the order of 1010m2/s. TheS is the rate of strain tensor,
S = 1 2
∂ui
∂xj
+ ∂uj
∂xi
− 2
3δi j∇·u
!
. (2.34)
2.2.4.3 Spitzer heat conduction
The energy transport in the solar corona is described by the heat conduction. The heat flux vector along the magnetic fieldlines reads,
q=−K∇T, (2.35)
where K is the thermal conduction tensor. The minus sign indicates that the heat flux points downwards to parts with lower temperatures and is proportional to the temperature gradient. The steeper the gradients, the more efficient it is. The heat is transported by the electrons, and for this term to hold, the electrons mean free path should be much smaller than the temperature scale height, which is true for the corona.
For the fully ionised plasma of the solar corona, the heat conduction tensorK has the general form,
K =K⊥δi j+(Kk−K⊥) ˆbibˆj, (2.36) whereK⊥, Kkare the perpendicular and parallel coefficients and ˆbi, ˆbjare the unit vectors along the magnetic fieldlines. For the components i and j, the perpendicular component
of the heat conduction K⊥ in the solar corona is by many orders of magnitude weaker compared to the parallel componentKk. Thus, we setK⊥= 0 and the heat flux reads,
q= −Kkb(ˆ bˆ ·∇T). (2.37)
For fully ionised gas Spitzer (1962) gives, Kk =K0T5/2
W mK
, (2.38)
where typical coronal values yieldK0 =1.8×10−10W m−1K−7/2. To conclude, the Spitzer heat conduction term in the energy equation is the divergence of Eq. (2.37), thus,
Ls =∇·q= −K0∇·(T5/2b(ˆ bˆ ·∇T)) (2.39) The efficiency of the Spitzer heat conduction poses a serious numerical challenge mainly because it can increase the computational time tremendously. In Chap. 4 we are going to describe a numerical method in order to overcome this problem.
2.2.4.4 Radiative cooling
An equally important process that takes place also in the solar corona is radiative cooling.
The corona is considered to be optically thin. The intensity I(λ) of an optically thin spectra line produced by photons at a specific wavelengthλvia spontaneous emission is expressed as,
I(λ)= Z
AXG(T, λ,ne)nenHdz, (2.40) where,AX is the elemental abundance in the corona,G(T, λ,ne) is the contribution func-tion which can be calculated through the CHIANTI (Dere et al. 1997) atomic database andne,nHare the number densities of electron and hydrogen respectively. To further con-tinue the analysis of Eq. (2.40) we need the density profile along the line of sight, which gives rise to differential emission measure. The coronal abundances are calculated based on the work from Meyer (1985), Murphy (1985), Cook et al. (1989).
For the optically thin corona, the radiative losses are not coupled to the radiation field, and hence it can be derived by integrating Eq. (2.40) in a specific temperature range. The expression is given by,
LR =nenHQ(T) (2.41)
The function Q(T) is called the radiative loss function with units of Wm3. For the imple-mentation in the code see Bingert (2009).
To conclude, radiative cooling is most efficient for lower temperatures and higher density, thus it is significant in the lower, more dense parts of the corona and transition region.
Figure 2.1: Coronal loops in the solar atmosphere as observed by the Transition Region and Coronal Explorer (TRACE) instrument on November 6th 1999. The image shows the EUV emission of the hot coronal plasma (T ' 1 MK) confined inside the loop. The colors in this picture are inverted. Credit: NASA/LMSAL.