### 2.1 Mass conservation

Mass conservation is expressed in Eulerian form by the continuity equation
or in Lagrangian form
where represents the rate of change of the logarithm of the mass density, , following the fluid
motion (with velocity ), and is the logarithmic rate of change of the specific volume
.
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 2) shows that a fluid parcel that ascends over a
number of density scale heights has to expand correspondingly. The Eulerian form (Equation 1)
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
, that one may ignore in comparison to (separately) the contributions from vertical
changes of mass flux and horizontal expansion. So, to lowest order one can then assume that

where is height and is time, so
showing again that if is changing rapidly with height then ascending/descending fluid must
expand/contract rapidly.