4.2 Energetics

Conservation of energy in the anelastic system is expressed as

$@ ( ) ---(Ek + Et + Em) = - \~/ . F KE + F EN + F RD + F PF + F VD + F BS , (2) @t$

where $Ek$ and $Em$ represent the kinetic and magnetic energy density respectively and $Et$ is the thermal energy. In the anelastic system, $Et$ incorporates both the internal energy density associated with the thermodynamic perturbations and the gravitational potential energy. It is proportional to the specific entropy perturbation, $---- Et = r TS$ , defined relative to a nearly adiabatic background stratification. The derivation of Equation (2

) is carried out in Appendix A.3 where complete expressions are given for all the energy and flux terms.

The terms $F KE$ and $F EN$ represent kinetic energy and enthalpy flux by convective motions. The latter of these, $EN F$ , dominates the energy flux throughout most of the convective envelope, transporting energy outward from the interior to the surface where it is then radiated into space. By contrast, the kinetic energy flux $F KE$ is much weaker and is directed inward as a result of the asymmetry between upflows and downflows which is characteristic of compressible convection (see Section 6.2).

In the deep interior, the energy flux is carried by radiative diffusion, $RD F$ , which falls off gradually above the base of the convection zone at $r ~ 0.7Ro .$ . The Poynting flux $PF F$ plays little role in the overall energy balance but can have a significant influence on dynamo processes, particularly if the magnetic boundary conditions permit leakage out of the domain (Brun et al., 2004 ). The viscous energy flux, $VD F$ , is generally negligible both in the Sun and in numerical models. Many numerical applications also include an additional diffusive heat flux which operates on the entropy gradient and which is intended to represent energy transport by unresolved convective motions (e.g., Miesch et al., 2000 ). This additional term is designed to carry flux outward near the upper boundary where the convective fluxes vanish and the radiative diffusion is small.

The final term in Equation (2 ), involving $F BS$ reflects the internal and gravitational potential energy associated with the background stratification. If the reference state is adiabatic, this term vanishes. Even if the reference entropy gradient is nonzero, the horizontal average of $BS F$ vanishes so it contributes nothing to the total radial energy flux (see Appendix A.3). However, this term together with the radiative heat flux, $F RD$ , provides the energy input which drives convective motions.

If the system is in thermal equilibrium, the fluxes must balance such that:

where $L o.$ is the solar luminosity and brackets indicate an average over the horizontal dimensions and time. The approach to equilibrium occurs on relatively long timescales because the energy flux through the convection zone is small relative to the internal energy of the plasma. An estimate for the relaxation timescale is $-- trad = MCZCV T /L o.$ , where $MCZ$ is the total mass in the convection zone: $-- 3 3 MCZ ~ r(4p/3)(R o. - rb)$ , with $rb ~ 0.7Ro .$ . This comes out to be $trad ~ 105 yr$ . By comparison, convective turnover timescales are thought to be of order a month.

If the anelastic equations are solved within a spherical shell Equation (2 ) implies that the total energy will be conserved if the net flux through the inner and outer boundaries vanishes. This will be the case if the boundary conditions are impenetrable and stress-free, if no net heat flux is applied, and if the magnetic field is required to be radial at the top and bottom of the shell. Other boundary conditions may lead to energy transport into or out of the domain.

Figure 5 summarizes the exchange of energy between the different reservoirs of the system. Energy is supplied from below via a radiative energy flux which ultimately originates from nuclear burning in the solar core. Convective motions tap this energy source through the buoyancy force which convert thermal energy to kinetic energy. This kinetic energy can then be converted into magnetic energy by the Lorentz force or back into thermal energy by pressure work on expanding or contracting fluid elements through the $P \~/ .v$ term in the mechanical and internal energy equations (see Appendix A.3). Kinetic and magnetic energy may also be converted into thermal energy by viscous and Ohmic heating. These heating terms are unidirectional, but the buoyancy force, Lorentz force, and compression can operate in both directions, either extracting or injecting kinetic energy. Because we have neglected the centrifugal force, the kinetic energy associated with the uniform component of the solar rotation cannot be tapped directly, although the differential rotation component can be (Section 4.3).