2.4 Energy transport

The equation describing the evolution of internal energy is, in conservative form1,
∂E ∂t--= − ∇ ⋅ (Euu ) − P(∇ ⋅ u) + Qrad + Qvisc, (18 )
where E is the internal energy per unit volume, Qrad is the radiative heating or cooling, and Qvisc is the viscous dissipation
∑ ∂ui Qvisc = τij----, (19 ) ij ∂rj
or, denoting by s ij the symmetric part of the strain tensor ∂ui ∂rj,
( ) 1- ∂ui- ∂uj- sij = 2 ∂rj + ∂ri , (20 )
we have
∑ Qvisc = τijsij. (21 ) ij

The term P(∇ ⋅ u) is the “PdV”-work, which is responsible for adiabatic heating and cooling when gas is compressed or decompressed. This is more clearly brought out by the Lagrangian form of the energy equation,

De- = − P-∇ ⋅ u + Q + Q , (22 ) Dt ρ rad visc
where e = E ∕ρ is the energy per unit mass. For an ideal gas
P- = (Γ − 1)e, (23 ) ρ
where Γ = (∂ ln P ∕∂ ln ρ) S. In this case e is proportional to temperature T. Since, as per Equation 2View Equation, the PdV-term may be written P-D-lnρ- − ρ Dt, the adiabatic compression/expansion of an ideal gas is characterized by
Γ −1 e ∼ T ∼ ρ . (24 )
Equations 22View Equation and 24View Equation are useful when considering the energy balance of the solar photosphere – cf. further discussion below.

2.4.1 Radiative energy transfer

The radiative heating or cooling Qrad may be written as the negative divergence of the radiative flux,

Q = − ∇ ⋅ F , (25 ) rad rad
∫ ∫ Frad = ν Ω Iν(Ω, r,t) Ω dΩ dν, (26 )
ν is the radiation frequency and Iν is the radiation intensity in direction Ω.

The radiation intensity may be obtained along straight lines of sight by solving the radiative transfer equation

∂Iν ----= S ν − Iν, (27 ) ∂τν
where Sν is the radiative source function (equal to the Planck function B ν if (strong) Local Thermodynamic Equilibrium (LTE) may be assumed), and
dτ = ρκ ds (28 ) ν ν
is the increment of the optical depth τν over the geometric interval ds, given the monochromatic absorption coefficient κν. Combining Equations 25View Equation28View Equation one also finds that
∫ ∫ Q = ρ κ (I − S )d Ω dν, (29 ) rad ν Ω ν ν ν
which lends itself to a direct and intuitive interpretation; it shows that whenever the radiation intensity is lower than the local source function, there is cooling. This happens particularly in surface layers where the outgoing intensity (away from the surface) is similar to the source function, but the incoming intensity is much lower, due to the “dark sky” that is visible from a surface where the optical depth to infinity is less than unity.

Because of the interplay of the ρκ ν and Iν − Sν factors, the effect is largest in a thin layer near optical depth unity; at larger depths the difference Iν − Sν tends to zero exponentially with optical depth, while at smaller depths the ρκν factor diminishes the effect as well.

The discussion above applies frequency by frequency, and because of variations in the opacity κν with frequency, in particular across spectral lines, the net heating or cooling can only be found by evaluating the integral over frequency that occurs in Equation 29View Equation. An approximate way to estimate that integral using a binning method was developed by Nordlund (1982Jump To The Next Citation Point) and applied in subsequent works by Stein, Nordlund and collaborators (e.g., Stein and Nordlund, 1989Jump To The Next Citation Point1998Jump To The Next Citation PointAsplund et al., 2000bJump To The Next Citation PointCarlsson et al., 2004Jump To The Next Citation Point). The method has been tested and used by Ludwig and co-workers (Ludwig et al., 1994Steffen et al., 1995) and by Vögler and co-workers (Vögler et al., 2004Vögler, 2004Jump To The Next Citation PointVögler et al., 2005Jump To The Next Citation Point). Improvements are being developed by Trampedach and Asplund (2003).

2.4.2 Equation of state

Whenever the constituent atoms or molecules of a gas undergo ionization or dissociation, extra energy is required to heat or cool the gas; i.e., the internal energy e varies more with temperature than for an ideal gas. Near the surface of the Sun, both ionization and dissociation are important; ionization of hydrogen and helium influence the equation of state greatly just below the surface, and dissociation of hydrogen molecules is important in the cooler layers of the photosphere.

In general then, one needs to know the relations

P = P(ρ,e), (30 )
κν = κν(ρ,e) (31 )
in order to use Equations 5View Equation18View Equation to evolve the momentum and energy density forward in time (Gustafsson et al., 1975Jump To The Next Citation PointMihalas et al., 1988Jump To The Next Citation PointDäppen et al., 1988Mihalas et al., 1990Rogers, 1990Rogers et al., 1996Rogers and Iglesias, 1998Rogers and Nayfonov, 2002Trampedach, 2004b,aTrampedach et al., 2006). Since computing these relations from first principles in general is quite expensive the best approach for practical computational work is to tabulate and interpolate in these relations.

2.4.3 Convective and kinetic energy fluxes

By a few algebraic manipulations of the equations of motion and the energy equation one can show that

∂(E + Ekin) ------------= − ∇ ⋅ (Fconv + Fkin + Frad + Fvisc), (32 ) ∂t
where Fconv is the convective, enthalpy, flux
Fconv = (E + P )u, (33 )
Fkin is the kinetic energy flux
1 Fkin = -ρu2u, (34 ) 2
and Fvisc is the (generally small) viscous flux
Fvisc = τvisc ⋅ u; (35 )
Frad is the radiative energy flux defined by Equation 26View Equation. For an ideal gas one may expect the fluctuations in pressure, δP, internal energy per unit volume δE, and the kinetic energy density δ(1ρu2) 2 to be of similar magnitude, and thus the convective and kinetic energy fluxes to also be of similar magnitude (although not necessarily pointing in the same direction). However, as mentioned above, near the solar surface the internal energy becomes dominated by changes in the ionization energy, and the convective flux thus tends to dominate there.

At the solar surface there is a very rapid transition between primarily convective and primarily radiative energy transport. In the atmosphere it is more relevant to consider changes of the radiative flux, through Equation 29View Equation.

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