2 Derivation of the anelastic equations

We start with a quick review of the derivation procedure of the anelastic equations, which will set the ground for the final simplified set of equations, assuming that the fluid satisfies the perfect gas equations of state. The full set of dynamical equations expressing the physical laws of conservation of momentum, mass and energy, supplied by the equations of state, is recalled here,

where $\unicode[STIX]{x1D60E}_{ij}^{s}=(\unicode[STIX]{x2202}_{j}u_{i}+\unicode[STIX]{x2202}_{i}u_{j})/2$ is the symmetric part of the velocity field gradient tensor, $\unicode[STIX]{x1D642}^{s}\boldsymbol{ : }\unicode[STIX]{x1D642}^{s}=\unicode[STIX]{x1D60E}_{ij}^{s}\unicode[STIX]{x1D60E}_{ij}^{s}$ and $Q$ denotes heat sources, e.g. due to radiation. Let us assume that the shear dynamical viscosity $\unicode[STIX]{x1D707}$ , the bulk viscosity $\unicode[STIX]{x1D709}$ and the thermal conduction $k$ are non-uniform for generality; however, the latter is a function of depth only. Furthermore, due to the applicability and vast usage of the anelastic liquid approximation for the study of planetary and stellar interiors where the gravity typically also significantly varies with depth, we will assume that the gravitational acceleration, and likewise the thermal conductivity coefficient, is a function of depth only. The specific heat ratio $\unicode[STIX]{x1D6FE}=c_{p}/c_{v}$ will be assumed to be constant and, since for a perfect gas $c_{p}-c_{v}=R=\text{const.}$ , the specific heats at constant volume, $c_{v}$ , and pressure, $c_{p}$ , are also assumed to be uniform. (Of course, in a real fluid/gas the specific heat ratio depends on the temperature since the number of degrees of freedom of fluid/gas particles increases with temperature; this dependence, however, will be neglected here.)

We take the $z$ -axis of the coordinate system so that the gravitational acceleration $\boldsymbol{g}=-g(z)\hat{\boldsymbol{e}}_{z}$ . For time-independent boundary conditions, we decompose the thermodynamic variables into the static (reference) state contributions, denoted by upper tildes, and fluctuations, denoted by primes,

The equations of hydrostatic equilibrium take the following forms:

which are ordinary and easy to solve once $Q$ is known (for the reference state solution when $Q=0$ and either $k=\text{const.}$ or $\unicode[STIX]{x1D705}=k/\tilde{\unicode[STIX]{x1D70C}}c_{p}=\text{const.}$ ; see, e.g., Mizerski & Tobias (Reference Mizerski and Tobias2011)). We now introduce the two fundamental assumptions of the anelastic approximation. First, guided by the observation that natural large-scale convective systems such as, e.g., planetary and stellar interiors or atmospheres in their long-time evolution develop states that are nearly adiabatic, and only a slight excess above the adiabatic gradient drives a very vigorous convective flow, we assume that

where $\langle \cdot \rangle =\iiint (\cdot )\,\text{d}V/V$ denotes a spatial average over the entire fluid volume $V$ , $L$ denotes the vertical span of the system and $\unicode[STIX]{x1D6E5}_{S}=\text{d}\tilde{T}/\text{d}z+g/c_{p}$ , i.e. it is assumed that the thermal gradient in the fluid, which undergoes convection, is only weakly superadiabatic. It should be noted that the general relation

obtained with the aid of the heat capacity definition $c_{p}=T(\unicode[STIX]{x2202}s/\unicode[STIX]{x2202}T)_{p}$ and the Maxwell relation $\unicode[STIX]{x1D70C}^{2}(\unicode[STIX]{x2202}s/\unicode[STIX]{x2202}p)_{T}=(\unicode[STIX]{x2202}\unicode[STIX]{x1D70C}/\unicode[STIX]{x2202}T)_{p}=-\unicode[STIX]{x1D70C}/T$ , allows us to write

thus,

and the departure from adiabaticity can also be expressed by the entropy gradient.

Second, we assume that the fluctuations of the thermodynamic variables are much smaller than their static profiles and, since it is the departure from the adiabatic state that drives the convection, it is natural to assume that it is also a measure of the relative magnitude of fluctuations,

an assumption well justified by experimental and numerical evidence which, however, needs to be verified for consistency a posteriori. The latter assumption concerning the magnitude of the entropy fluctuation differs from those referring to the density, temperature and pressure fluctuations, since a departure from the constant entropy state is equivalent to a departure from the adiabatic state; thus, even at hydrostatic equilibrium, the entropy vertical variation constitutes only a weak correction to the mean (cf. (2.7)).

As a consequence, the equations of state (2.1d,e ) are now expanded about the hydrostatic equilibrium at every height,

(where, again, the Maxwell relation $\unicode[STIX]{x1D70C}^{2}(\unicode[STIX]{x2202}s/\unicode[STIX]{x2202}p)_{T}=(\unicode[STIX]{x2202}\unicode[STIX]{x1D70C}/\unicode[STIX]{x2202}T)_{p}=-\unicode[STIX]{x1D70C}/T$ was used), which results in Furthermore, the buoyancy force in the momentum balance (2.1a ), i.e. $-g\unicode[STIX]{x1D70C}^{\prime }\hat{\boldsymbol{e}}_{z}$ , drives the fluid motion, which implies that the flow acceleration is of order $g\unicode[STIX]{x1D70C}^{\prime }/\tilde{\unicode[STIX]{x1D70C}}\sim \unicode[STIX]{x1D716}g$ , and is thus much smaller than the acceleration of gravity. This, in turn, results in the following convective velocity and time scales:

and hence the viscosity scale, and likewise the thermal conductivity scale also have to be small for consistency,

(A well-known result of the assumptions leading to the anelastic approximation is that the Mach number, i.e. the ratio of the convective velocity scale to the mean speed of sound $\langle C\rangle$ , $Ma={\mathcal{U}}/\langle C\rangle =\mathit{O}(\unicode[STIX]{x1D716}^{1/2})\ll 1$ , is small, and hence at leading order the sound waves are infinitely fast.) It follows that by assumption

has to be satisfied in order for the anelastic approximation to be valid. In other words, the anelastic approximation can be applied to systems internally heated by radiation only if the heating is weak enough not to push the system too far away from the adiabatic state; that is, $\tilde{Q}/c_{p}\langle \tilde{\unicode[STIX]{x1D70C}}\rangle \langle \tilde{T}\rangle \sqrt{\langle g\rangle /L}\sim \unicode[STIX]{x1D716}^{1/2}$ , which allows the fundamental assumption (2.7) to be satisfied.

We are now ready to write down the dynamic equations for the fluctuations, i.e. with subtracted hydrostatic balance (2.3a,b ), which with the aid of (2.8a ), (2.11a,b ) and (2.12) take the following leading-order form:

where $Q^{\prime }=Q-\tilde{Q}$ and $Q^{\prime }/\tilde{Q}=\mathit{O}(\unicode[STIX]{x1D716})$ . Making use of (2.6), the energy equation can also be rewritten in the form

The following relation for the derivative of density $\unicode[STIX]{x1D70C}=\unicode[STIX]{x1D70C}(p,T)$ will turn out to be useful in the following analysis:

and from the definition of $\unicode[STIX]{x1D716}$ in (2.7) and the hydrostatic balance $\text{d}\tilde{p}/\text{d}z=-\tilde{\unicode[STIX]{x1D70C}}g$ , we obtain an estimate of the static vertical variation of density,

3 Simplification through a constant-thermal-diffusivity entropy formulation

The system of anelastic equations can be further reduced by making use of (2.14d,e ), which result from the equation of state, to eliminate the density and temperature fluctuations and express them by the entropy and pressure fluctuations,

where the thermodynamic identity $c_{p}-c_{v}=R$ was used in obtaining the above expression for $\unicode[STIX]{x1D70C}^{\prime }$ . Introduction of (3.1b ) into the energy balance (2.15) leads to

On the other hand, the pressure term together with the buoyancy force in the Navier–Stokes equation (2.14a ), divided by $\tilde{\unicode[STIX]{x1D70C}}$ , with the aid of (3.1a ), can be expressed in the following way:

Next, since the $z$ -derivative of the static density distribution is given by (2.16), and, again, making use of $c_{p}-c_{v}=R$ , one obtains

since $\unicode[STIX]{x1D6E5}_{S}=\mathit{O}(\unicode[STIX]{x1D716}\tilde{T}/L)$ and $p^{\prime }/\tilde{p}=\mathit{O}(\unicode[STIX]{x1D716})$ .

This allows us to rewrite the system of dynamic equations under the anelastic approximation in the forms

which constitute a closed problem for two thermodynamic variables $s^{\prime }$ and $p^{\prime }$ and the velocity field $\boldsymbol{u}$ . The momentum balance (3.5a ) multiplied by $\tilde{\unicode[STIX]{x1D70C}}$ , the continuity equation (3.5b ) and the estimate of the static density vertical variation in (2.17) can be used to get a stationary equation for $p^{\prime }/\tilde{\unicode[STIX]{x1D70C}}$ at leading order in $\unicode[STIX]{x1D716}$ ,

where $\unicode[STIX]{x1D735}\boldsymbol{\cdot }[\unicode[STIX]{x1D735}\boldsymbol{\cdot }(2\unicode[STIX]{x1D707}\unicode[STIX]{x1D642}^{s}-\tilde{\unicode[STIX]{x1D70C}}\boldsymbol{u}\boldsymbol{u})]=\unicode[STIX]{x2202}_{i}\unicode[STIX]{x2202}_{j}(2\unicode[STIX]{x1D707}\unicode[STIX]{x1D60E}_{ij}^{s}-\tilde{\unicode[STIX]{x1D70C}}u_{i}u_{j})$ and, again, $\unicode[STIX]{x1D6FE}=c_{p}/c_{v}=\text{const.}$ is the specific heat ratio.

The Poisson-like stationary problem for the pressure perturbation is a manifestation of the fact that, under the anelastic approximation, all of the terms of the order $\mathit{O}(Ma^{2})$ are neglected and thus pressure spreads infinitely fast. It should be noted that up to now, no additional assumptions have been made except for the standard anelastic assumption of small system departure from the adiabatic state, and the above reduction of the number of equations results solely from simple algebraic manipulations of the full anelastic system of equations.

Further simplification can be achieved by elimination of the pressure term from the energy equation. By virtue of (2.17), one obtains

and then using (3.6) the energy equation can be rewritten in the form

where we have introduced the thermal diffusivity coefficient

and the term

gathers all terms nonlinear in the velocity field and all associated with viscous diffusion. Next, the two terms on the right-hand side of (3.8) involving entropy derivatives on the basis of the definition of the parameter $\unicode[STIX]{x1D716}$ in (2.7) requiring $\text{d}\tilde{T}/\text{d}z=-g/c_{p}+\mathit{O}(\unicode[STIX]{x1D716}g/c_{p})$ can be manipulated to give at leading order

and then the energy equation can be cast in the following form:

It is clear now that the assumption of uniform thermal diffusivity $\unicode[STIX]{x1D705}$ allows the energy equation to be simplified by removing the entire term proportional to the $z$ -derivative of $\unicode[STIX]{x1D705}$ , and the full system of anelastic equations reads

where ${\mathcal{J}}$ is given in (3.10) and the Poisson-type problem for pressure fluctuation in (3.6). This constitutes a closed system of equations, obtained by no additional assumptions other than $\unicode[STIX]{x1D705}=\text{const.}$ , expressed solely in terms of the velocity field and the pressure and entropy fluctuations.

We have started the derivation of the final simplified energy equation (3.13c ) from (2.14c ), which is equivalent to (2.15). Although these two are already simplified with respect to the full energy equation (2.1c ), this simplification obviously does not affect the sign of the total entropy production, $\unicode[STIX]{x2202}_{t}\iiint _{V}(\tilde{\unicode[STIX]{x1D70C}}s^{\prime })$ , which in accordance with the second law of thermodynamics is positive, if the boundaries are assumed to be adiabatically insulating. In the process of derivation of (3.13c ), we have neglected some terms of the order $\mathit{O}(\unicode[STIX]{x1D716}^{5/2})$ . However, careful considerations show that all of the neglected terms in the energy equation can be gathered to yield a correction of the form $\unicode[STIX]{x1D705}\unicode[STIX]{x2202}_{z}[(p^{\prime }+\tilde{\unicode[STIX]{x1D70C}}\tilde{T}s^{\prime })(\unicode[STIX]{x1D6E5}_{S}/\tilde{T})]$ . These terms, divided by $\tilde{T}$ and integrated over the entire fluid volume, provide only a small $\mathit{O}(\unicode[STIX]{x1D716}^{5/2})$ correction to the total entropy production, which is of the order $\mathit{O}(\unicode[STIX]{x1D716}^{3/2})$ , and hence the total entropy production in (3.13c ) is necessarily positive.

On a final note, let us comment on the case when the heat capacities are non-uniform, but at leading order correspond to the reference state, denoted now by $\tilde{c}_{p}$ and $\tilde{c}_{v}$ , and are thus only $z$ -dependent. The importance of the specific heat variation was emphasized, e.g., by Almgren et al. (Reference Almgren, Bell, Rendleman and Zingale2006) and Alboussière & Ricard (Reference Alboussière and Ricard2013). A very similar manipulation to that leading to (3.13a–c ) leads to the appearance of an additional term on the right-hand side of the energy equation of the form $-\unicode[STIX]{x2202}_{z}[(\unicode[STIX]{x1D705}\text{d}_{z}\tilde{c}_{p}/\tilde{c}_{p})(p^{\prime }+\tilde{\unicode[STIX]{x1D70C}}\tilde{T}s^{\prime })]$ , with the rest of the terms in all of the equations unchanged. Thus, in such a case, the pressure fluctuation cannot be eliminated and appears explicitly in the energy equation. This certainly makes the formulation less attractive from the point of view of modelling. Nevertheless, the final set of equations is still expressed through only two thermodynamic variables (which holds true even under inclusion of the non-uniform $\unicode[STIX]{x1D705}$ term in (3.12)); thus, determination of the flow velocity does not require calculation of the temperature and density fluctuations.

4 Final remarks

The system of anelastic equations (3.13a–c ) expressed solely through the pressure and entropy fluctuations and the velocity field (together with a stationary Poisson-like problem for the pressure field) was derived under the sole assumption of uniform constant thermal diffusivity of the fluid. No assumptions about the introduction of turbulent transport coefficients (such as, e.g., that of Braginsky & Roberts Reference Braginsky and Roberts1995) were made and the standard molecular heat flux expression, resulting from Fourier’s law, i.e. $-k\unicode[STIX]{x1D735}T$ , was utilized. The only difference between the final form of the obtained equations and those derived by Braginsky & Roberts (Reference Braginsky and Roberts1995) with the turbulent heat flux $-k_{t}T\unicode[STIX]{x1D735}s$ replacing the molecular one is the term denoted here by ${\mathcal{J}}$ and given in (3.10), which in the energy equation gathers all nonlinearities in the velocity field and all terms associated with viscous diffusion. In the case of Braginsky & Roberts (Reference Braginsky and Roberts1995), this nonlinear term contains the viscous heating effect only. Here, the final form of the energy equation was obtained by elimination of the pressure fluctuation through utilization of the momentum balance and mass conservation law (and, of course, the fundamental assumption of small departure of the system from an adiabatic state). The thus-obtained stationary equation for the pressure fluctuation involves divergence of other terms in the Navier–Stokes equation, which then replace the pressure contribution to the energy balance. The resulting nonlinearity is of the same type as the viscous heating, i.e. at most quadratic in the velocity, and does not introduce any significant complexity. Furthermore, since all the need for an appeal to the concept of turbulent mixing has been eliminated here, the current formulation does not suffer from the issues associated with negative entropy production known to appear in the Braginsky & Roberts (Reference Braginsky and Roberts1995) and Lantz & Fan (Reference Lantz and Fan1999) formulation due to the introduction of transport via turbulent mixing and averaging, and the entropy production in (3.13c ) is always positive.

Therefore, the system of anelastic equations obtained constitutes a very useful approximation from the point of view of modelling stratified systems such as stellar or planetary interiors, since it does not involve any additional assumptions about the properties of turbulence or the introduction of the concept of turbulent diffusivities and it is in mathematically fully rigorous relation with the general system of anelastic equations at constant $\unicode[STIX]{x1D705}$ .