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, $v$ , which yields

$@Ek- v- @t = - \~/ .(vEk - v.D) - v. \~/ P - vrrg + c .(J× B) - P, (47)$

where

The thermal energy equation may be obtained from Equation (41 ) if we first note that

$--DS @E dT-- rT ----= --t-+ \~/ .(Etv) - rvrS---, (49) Dt @t dr$

where the thermal energy per unit volume is defined as

In the convection zone, $-- dS/dr$ is assumed to be of order $e$ . Thus, Equations (37 ) and (38 ) imply that the temperature gradient is equal to the adiabatic gradient to lowest order in $e$ :

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:

$-- -- - rvrS dT- = rg--vrP - rvrg = P \~/ .v - rvrg, (52) dr gP$

where the final equality follows from Equations (36

), (38

), and (39

), to lowest order in $e$ . Equations (49

) and (52

) can then be substituted into Equation (41

) to yield

$-- @Et- ( EN RD) ---- d-S 4pj- 2 @t = - \~/ . F + F - r Tvr dr - P \~/ .v + rvrg + P + c2 J , (53)$

where $EN F$ is the enthalpy flux

and $RD F$ is the energy flux due to radiative diffusion

$( dT-- ) F RD =_ -krrCP \ ~/ T + --- ^r . (55) dr$

The magnetic energy equation follows directly from the induction equation (42 ):

$@E v 4pj ---m- = - \~/ .F PF - -.(J × B) - --2-J2, (56) @t c c$

where $Em$ is the magnetic energy density

and $PF F$ is the Poynting flux

${ } F PF =_ jJ - -1- (v × B) × B. (58) c 4p$

Combining Equations (47 ), (51 ), and (56 ) yields the total energy equation:

$@-- ( KE EN RD PF VD) @t (Ek + Et + Em) = - \~/ . F + F + F + F + F + B, (59)$

where $KE F$ is the kinetic energy flux

$VD F$ is the viscous energy flux

and $B$ is the work done against the background stratification:

If the reference state is adiabatic then $B = 0$ everywhere. If it is not, then we may rewrite $B$ by defining a reference state heating term, $-- Q$ as follows:

Then

$-- -- (-- -) B = - rv. \~/ Q = - \~/ . rv Q . (64)$

Equation (59

) then becomes

$@-- ( KE EN RD PF VD BS) @t (Ek + Et + Em) = - \~/ . F + F + F + F + F + F , (65)$

where

Note that if there is no net mass flux, $< rvr >hf$ , through horizontal surfaces, as is the case in most applications, then this term yields no net radial energy flux: $< F BS >hf= < B >hf= 0$ .