2.1 Mass conservation

Mass conservation is expressed in Eulerian form by the continuity equation
∂ρ- ∂t = − ∇ ⋅ (ρu), (1 )
or in Lagrangian form
D ln ρ ∂ lnρ D lnV ------ ≡ ------+ u ⋅ ∇ ln ρ = − ∇ ⋅ (u ) ≡ −------, (2 ) Dt ∂t Dt
where D lnρ -Dt-- represents the rate of change of the logarithm of the mass density, ln ρ, following the fluid motion (with velocity u), and DlDntV- is the logarithmic rate of change of the specific volume V.

These two forms of the continuity equation may already, before one even considers the rest of the equations of hydrodynamics, be used to understand some very fundamental consequences that conservation of mass has for the structure of solar convection.

For example, the Lagrangian form (Equation 2View Equation) shows that a fluid parcel that ascends over a number of density scale heights has to expand correspondingly. The Eulerian form (Equation 1View Equation) shows, on the other hand, that if the local density does not change much with time, then a rapid decrease of mean density with height in an ascending flow, can be balanced by rapid expansion sideways.

A rapid vertical expansion could in principle also happen, but as we shall see in the next subsection, the equations of motion dictate that the main expansion/contraction of ascending/descending flows happens sideways.

The Eulerian form of the continuity equation is also a central player in the anelastic approximation (Gough, 1969), where one assumes, at least for the purpose of computing the pressure field P (r,t), that one may ignore ∂ρ ∂t in comparison to (separately) the contributions from vertical changes of mass flux and horizontal expansion. So, to lowest order one can then assume that

ρ ≈ ρ(z,t), (3 )
where z is height and t is time, so
∂-ln-ρ- ∂uz- ∂ux- ∂uy- uz ∂z + ∂z ≈ − ∂x − ∂y , (4 )
showing again that if ln ρ is changing rapidly with height then ascending/descending fluid must expand/contract rapidly.
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