### 2.4 Energy transport

The equation describing the evolution of internal energy is, in conservative form,
where is the internal energy per unit volume, is the radiative heating or cooling, and is the viscous dissipation
or, denoting by the symmetric part of the strain tensor ,
we have

The term 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,

where is the energy per unit mass. For an ideal gas
where . In this case is proportional to temperature . Since, as per Equation 2, the PdV-term may be written , the adiabatic compression/expansion of an ideal gas is characterized by
Equations 22 and 24 are useful when considering the energy balance of the solar photosphere – cf. further discussion below.

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

where
is the radiation frequency and is the radiation intensity in direction .

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

where is the radiative source function (equal to the Planck function if (strong) Local Thermodynamic Equilibrium (LTE) may be assumed), and
is the increment of the optical depth over the geometric interval , given the monochromatic absorption coefficient . Combining Equations 2528 one also finds that
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 and factors, the effect is largest in a thin layer near optical depth unity; at larger depths the difference tends to zero exponentially with optical depth, while at smaller depths the factor diminishes the effect as well.

The discussion above applies frequency by frequency, and because of variations in the opacity 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 29. An approximate way to estimate that integral using a binning method was developed by Nordlund (1982) and applied in subsequent works by Stein, Nordlund and collaborators (e.g., Stein and Nordlund, 19891998Asplund et al., 2000bCarlsson et al., 2004). 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, 2004Vögler et al., 2005). 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 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

and
in order to use Equations 518 to evolve the momentum and energy density forward in time (Gustafsson et al., 1975Mihalas et al., 1988Däppen 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

where is the convective, enthalpy, flux
is the kinetic energy flux
and is the (generally small) viscous flux
is the radiative energy flux defined by Equation 26. For an ideal gas one may expect the fluctuations in pressure, , internal energy per unit volume , and the kinetic energy density 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 29.