### A.3 Energy conservation in the anelastic approximation

The equation which describes the evolution of the kinetic energy density can be obtained by taking the
scalar product of Equation (40) with the velocity, , which yields
where
The thermal energy equation may be obtained from Equation (41) if we first note that

where the thermal energy per unit volume is defined as
In the convection zone, is assumed to be of order . Thus, Equations (37) and (38)
imply that the temperature gradient is equal to the adiabatic gradient to lowest order in :

Together with the linearized equation of state, Equation (43), Equation (51) then implies that the final
term in Equation (49) may be written as follows:
where the final equality follows from Equations (36), (38), and (39), to lowest order in . Equations (49)
and (52) can then be substituted into Equation (41) to yield
where is the enthalpy flux
and is the energy flux due to radiative diffusion
The magnetic energy equation follows directly from the induction equation (42):

where is the magnetic energy density
and is the Poynting flux
Combining Equations (47), (51), and (56) yields the total energy equation:

where is the kinetic energy flux
is the viscous energy flux
and is the work done against the background stratification:
If the reference state is adiabatic then everywhere. If it is not, then we may rewrite by
defining a reference state heating term, as follows:

Then
Equation (59) then becomes
where
Note that if there is no net mass flux, , through horizontal surfaces, as is the case in most
applications, then this term yields no net radial energy flux: .