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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 (40View Equation) with the velocity, v, which yields
@Ek- v- @t = - \~/ .(vEk - v.D) - v. \~/ P - vrrg + c .(J× B) - P, (47)
where
1---2 Ek =_ 2 rv . (48)

The thermal energy equation may be obtained from Equation (41View Equation) 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
--- Et =_ rT S. (50)

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

-- dT- = - -g-. (51) dr CP
Together with the linearized equation of state, Equation (43View Equation), Equation (51View Equation) then implies that the final term in Equation (49View Equation) may be written as follows:
-- -- - rvrS dT- = rg--vrP - rvrg = P \~/ .v - rvrg, (52) dr gP
where the final equality follows from Equations (36View Equation), (38View Equation), and (39View Equation), to lowest order in e. Equations (49View Equation) and (52View Equation) can then be substituted into Equation (41View Equation) 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
EN (--- ) -- F =_ rT S + P v = rCP Tv, (54)
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 (42View Equation):

@E v 4pj ---m- = - \~/ .F PF - -.(J × B) - --2-J2, (56) @t c c
where Em is the magnetic energy density
B2- Em =_ 8p , (57)
and PF F is the Poynting flux
{ } F PF =_ jJ - -1- (v × B) × B. (58) c 4p

Combining Equations (47View Equation), (51View Equation), and (56View Equation) 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
F KE =_ Ekv, (60)
VD F is the viscous energy flux
VD F =_ - v.D, (61)
and B is the work done against the background stratification:
-- ---dS- B =_ - vrrT dr . (62)

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:

-- -- d-Q -d-S dr = T dr . (63)
Then
-- -- (-- -) B = - rv. \~/ Q = - \~/ . rv Q . (64)
Equation (59View Equation) then becomes
@-- ( KE EN RD PF VD BS) @t (Ek + Et + Em) = - \~/ . F + F + F + F + F + F , (65)
where
BS -- -- F =_ rv Q. (66)
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.
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