Extension of the isothermal anelastic approximation to a two-component fluid

Extension of the isothermal anelastic approximation to a two-component fluid Kiyoshi Maruyama Department of Earth and Ocean Sciences, National Defense Academy, Yokosuka, Kanagawa 239-8686, Japan November 20, 2025 Abstract This paper extends a variant of the anelastic approximation to a two-component fluid. The energetics and the applicability of this extended approximation are also discussed, together with the relation to the Boussinesq approximation extended to a two-component fluid. 1. Introduction Ogura & Phillips (1962) devised an approximation called the anelastic approximation in order to study the motion under gravity of a deep layer of ideal gas with an isentropic basic state. Maruyama (2021a) reconstructed the approximation in such a manner that it can be applied to any kind of fluid with an isentropic basic state. Maruyama (2021b) further introduced a variant of the anelastic approximation which deals with a fluid having an isothermal basic state. As a limiting case of this isothermal anelastic approximation, the well-known Boussinesq approximation can be obtained The objective of this paper is to extend the isothermal anelastic approximation to a two-component fluid: the energetics of the extended approximation and the conditions for its applicability are also fully clarified; its relation to the Boussinesq approximation extended to a two-component fluid (Maruyama 2019) is discussed as well. 2. Extended isothermal anelastic approximation We consider the motion of an inviscid fluid consisting of two components, Aand B, in a uniform gravitational field. The concentration of component Ais denoted by c; the mass of Ain a unit volume of the fluid is given by ρc, with ρbeing the density of the fluid. The fluid is contained in a fixed finite domain Ω, and, in this domain, the z-axis is taken vertically upwards. We denote by kthe unit vector in the positive z-direction. 1 2.1. Equation of motion We first write down the equation of motion for the fluid as follows: Du Dt =−∇p/ρ −gk,(2.1) where D/Dt stands for the material derivative, and udenotes the velocity of the fluid; pis the pressure of the fluid, and gthe acceleration due to gravity. Let φdenote the specific Gibbs free energy of the fluid: it satisfies the relation dφ =−sdT +vdp +µdc, (2.2) in which sand Tare the specific entropy and the temperature of the fluid, respectively; v= 1/ρ denotes the specific volume of the fluid, and µis the chemical potential of the fluid (see Landau & Lifshitz 1987, § 58). In the following, all thermodynamic quantities are regarded as known functions of φ,Tand c. Using (2.2), we can rewrite ∇p/ρ as follows: ∇p/ρ =∇φ+s∇T−µ∇c. (2.3) The substitution of (2.3) into the equation of motion (2.1) yields Du Dt =−∇φ−s∇T+µ∇c−gk.(2.4) Now, let us decompose φ,T, and cas follows: φ=φ0+φ′, T =T0+T′, c =c0+c′.(2.5) Here φ0,T0, and c0are defined by φ0=−gz +α1, T0=α2, c0=α3,(2.6) with α1,α2, and α3being constants. Then (2.4) takes the following form: Du Dt =−∇φ′−s∇T′+µ∇c′.(2.7) Under the decomposition (2.5), sand µcan also be decomposed as follows: s=s0+s′, µ =µ0+µ′,(2.8) where s0and µ0are given by s0=s(φ0, T0, c0), µ0=µ(φ0, T0, c0).(2.9) We introduce here the following assumptions: |s′/s0| ≪ 1,|µ′/µ0| ≪ 1.(2.10) 2 Then the equation of motion (2.7) may be approximated as Du Dt =−∇φ′−s0∇T′+µ0∇c′.(2.11) Since s0∇T′=∇(s0T′)−T′∇s0and µ0∇c′=∇(µ0c′)−c′∇µ0, we have Du Dt =−∇(φ′+s0T′−µ0c′) + T′∇s0−c′∇µ0.(2.12) This is the equation of motion under the extended isothermal anelastic approximation. The second term on the right-hand side of (2.12) represents the buoyancy force that arises from changes in temperature. Regarded as a function of φ,T, and c,ssatisfies (∂s/∂φ)T,c =−β, (∂s/∂T )φ,c =T/cp−βs, (∂s/∂c)φ,T =βµ −(∂µ/∂T)p,c, (2.13) where β=v−1(∂v/∂T)p,c is the thermal expansion coefficient, and cpthe specific heat at constant pressure. Hence, in view of (2.6) and (2.9), we can write ∇s0= (∂s/∂φ)T,c|(φ0,T0,c0)∇φ0=β0gk,(2.14) in which the following notation has been introduced: β0=β(φ0, T0, c0).(2.15) The second term on the right-hand side of (2.12) can therefore be rewritten as T′∇s0=β0T′gk.(2.16) The third term on the right-hand side of (2.12) represents the buoyancy force due to changes in concentration. The chemical potential µin this term satisfies the relations (∂µ/∂φ)T,c =ρ(∂µ/∂p)T,c =−βc, (∂µ/∂T )φ,c = (∂µ/∂T )p,c +ρs(∂µ/∂p)T,c, (∂µ/∂c)φ,T = (∂µ/∂c)T,p −ρµ(∂µ/∂p)T,c, (2.17) where βc=ρ−1(∂ρ/∂c)T,p. Thus ∇µ0can be rewritten as ∇µ0= (∂µ/∂φ)T,c|(h0,s0,c0)∇φ0=βc0gk,(2.18) where βc0=βc(φ0, T0, c0). As a result, we obtain the following expression: −c′∇µ0=−βc0c′gk.(2.19) 3 2.2. Equation of continuity Under the decomposition (2.5), the density ρof the fluid can be written as ρ=ρ0+ρ′,(2.20) where ρ0is defined by ρ0=ρ(φ0, T0, c0).(2.21) We introduce here the following assumption: |ρ′/ρ0| ≪ 1.(2.22) On this assumption, ρmay be approximated as follows: ρ=ρ0.(2.23) Substituting (2.23) into the equation of continuity ∂ρ/∂t +∇ · (ρu) = 0, we get ∇ · (ρ0u)=0.(2.24) This is the equation of continuity under the extended approximation. 2.3. Concentration equation In the absence of diffusion, the equation for the rate of change of the concentration c of component Ais given by (see Landau & Lifshitz 1987, § 58) ρDc Dt = 0.(2.25) Considering (2.5) and (2.23), we can approximate this equation as follows: ρ0 Dc′ Dt = 0.(2.26) 2.4. General equation of heat transfer The general equation of heat transfer (see Landau & Lifshitz 1987, § 58) takes, when the conduction of heat is neglected, the following form: ρT Ds Dt +ρµDc Dt = 0.(2.27) To the first order of primed variables, this equation can be written as ρ0T0u· ∇s0+ρ0T0 Ds′ Dt +ρ0T′u· ∇s0+ρ0µ0 Dc′ Dt = 0,(2.28) where (2.23) has been used. Using (2.13), we can further rewrite s′in (2.28) as follows: s′= (∂s/∂φ)T,c|(φ0,T0,c0)φ′+ (∂s/∂T)φ,c|(φ0,T0,c0)T′+ (∂s/∂c)φ,T |(φ0,T0,c0)c′ =−β0φ′+ (cp0/T0−β0s0)T′+β0µ0−(∂µ/∂T )p,c|(φ0,T0,c0)c′.(2.29) Here the following notation has been introduced: cp0=cp(φ0, T0, c0).(2.30) 4 2.5. Alternative forms of the equation of motion We have thus fully formulated the isothermal anelastic approximation extended to a two-component fluid. For later reference, however, it is useful to rewrite the equation of motion (2.12) in somewhat different forms. We first note that, under the decomposition (2.5), the pressure pof the fluid can also be decomposed as follows: p=p0+p′,(2.31) where p0is defined by p0=p(φ0, T0, c0).(2.32) On the other hand, the following thermodynamic relations are obtained from (2.2): (∂p/∂φ)T,c =ρ, (∂p/∂T )φ,c =ρs, (∂p/∂c)φ,T =−ρµ. (2.33) Hence p′can be expressed, to the first order of φ′,T′, and c′, as follows: p′= (∂p/∂φ)T,c|(φ0,T0,c0)φ′+ (∂p/∂T)φ,c|(φ0,T0,c0)T′+ (∂p/∂c)φ,T |(φ0,T0,c0)c′ =ρ0φ′+ρ0s0T′−ρ0µ0c′.(2.34) This expression enables us to rewrite (2.12) in the following form: Du Dt =−∇(p′/ρ0) + T′∇s0−c′∇µ0.(2.35) The first term on the right-hand side, however, can further be rewritten as −∇(p′/ρ0) = −∇p′/ρ0+ (p′/ρ0)(∇ρ0/ρ0).(2.36) The substitution of (2.36) into (2.35) yields Du Dt =−∇p′/ρ0+ (p′/ρ0)(∇ρ0/ρ0) + T′∇s0−c′∇µ0.(2.37) Moreover, since ρ0is defined by (2.21), we can write ∇ρ0/ρ0= (∂ρ/∂φ)T,c|(φ0,T0,c0)∇φ0/ρ0=−(∂ρ/∂φ)T,c|(φ0,T0,c0)(g/ρ0)k.(2.38) Thus, in view of (2.34), the following expression for (p′/ρ0)(∇ρ0/ρ0) is obtained: (p′/ρ0)(∇ρ0/ρ0) = −(φ′+s0T′−µ0c′)(∂ρ/∂φ)T,c|(φ0,T0,c0)(g/ρ0)k =−(∂ρ/∂φ)T,c|(φ0,T0,c0)φ′ +s0(∂ρ/∂φ)T,c|(φ0,T0,c0)T′ −µ0(∂ρ/∂φ)T,c|(φ0,T0,c0)c′(g/ρ0)k.(2.39) The terms T′∇s0and −c′∇µ0in (2.37) can also be expressed as follows: T′∇s0=T′(∂s/∂φ)T,c|(φ0,T0,c0)∇φ0=−ρ0(∂s/∂φ)T,c|(φ0,T0,c0)T′(g/ρ0)k, −c′∇µ0=−c′(∂µ/∂φ)T,c|(φ0,T0,c0)∇φ0=ρ0(∂µ/∂φ)T,c|(φ0,T0,c0)c′(g/ρ0)k.(2.40) 5 Adding these expressions to (2.39), we get (p′/ρ0)(∇ρ0/ρ0) + T′∇s0−c′∇µ0 =−(∂ρ/∂φ)T,c|(φ0,T0,c0)φ′ +{∂(ρs)/∂φ}T,c|(φ0,T0,c0)T′ −{∂(ρµ)/∂φ}T,c|(φ0,T0,c0)c′(g/ρ0)k.(2.41) From (2.33), however, we observe that the following thermodynamic relations hold: {∂(ρs)/∂φ}T,c = (∂ρ/∂T )φ,c,− {∂(ρµ)/∂φ}T,c = (∂ρ/∂c)φ,T .(2.42) These relations enable us to rewrite (2.41) as follows: (p′/ρ0)(∇ρ0/ρ0) + T′∇s0−c′∇µ0 =−(∂ρ/∂φ)T,c|(φ0,T0,c0)φ′ + (∂ρ/∂T )φ,c|(φ0,T0,c0)T′ +(∂ρ/∂c)φ,T |(φ0,T0,c0)c′(g/ρ0)k.(2.43) On the other hand, ρ′in (2.20) is, to the first order of φ′,T′, and c′, given by ρ′= (∂ρ/∂φ)T,c|(φ0,T0,c0)φ′+ (∂ρ/∂T)φ,c|(φ0,T0,c0)T′+ (∂ρ/∂c)φ,T |(φ0,T0,c0)c′.(2.44) We see, therefore, that the equation of motion (2.12) can be expressed in the form Du Dt =−∇p′/ρ0−(ρ′g/ρ0)k.(2.45) However, it must be emphasized that, in spite of this result, the fluid density under the present approximation is given by ρ0, not by ρ0+ρ′. 2.6. Energetics of the extended approximation Let us next proceed to study, under the extended isothermal anelastic approximation, the energy balance of the fluid. We first consider the internal energy of the fluid. Let edenote the specific internal energy of the fluid. We then have the relation e=φ−p/ρ +Ts. (2.46) Accordingly, using (2.5), (2.8), (2.23), and (2.31), we obtain, to the first order of primed variables, the following expression for e: e= (φ0−p0/ρ0+T0s0)+(φ′−p′/ρ0+T0s′+s0T′).(2.47) Considering (2.34), however, we can further rewrite ein the following form: e= (φ0−p0/ρ0+T0s0) + T0s′+µ0c′.(2.48) 6 Taking the material derivative of (2.48), and multiplying the result by ρ0, we get ρ0 De Dt =ρ0u· ∇(φ0−p0/ρ0) + ρ0T0u· ∇s0+ρ0T0 Ds′ Dt +ρ0c′u· ∇µ0+ρ0µ0 Dc′ Dt .(2.49) This equation, by virtue of (2.24) and (2.28), reduces to ρ0 De Dt =∇·{ρ0(φ0−p0/ρ0)u} − ρ0T′u· ∇s0+ρ0c′u· ∇µ0.(2.50) However, since ρ0De/Dt =ρ0∂e/∂t +ρ0u· ∇e=∂(ρ0e)/∂t +∇ · (ρ0eu), we have ∂ ∂t(ρ0e) + ∇·{ρ0(e−φ0+p0/ρ0)u}=−ρ0T′u· ∇s0+ρ0c′u· ∇µ0.(2.51) Furthermore, in view of (2.48), this equation can be put into the following form: ∂ ∂t(ρ0e) + ∇ · ρ0{T0(s0+s′) + µ0c′}u=−ρ0T′u· ∇s0+ρ0c′u· ∇µ0.(2.52) Integrating (2.52) over the domain Ω containing the fluid, we obtain d dt ZΩ ρ0e dV =−ZΩρ0T′u· ∇s0−ρ0c′u· ∇µ0dV, (2.53) where it has been assumed that the normal component of uvanishes on the boundary of Ω. This is the equation for the rate of change of the internal energy of the fluid. The potential energy of the fluid, on the other hand, is invariable: d dt ZΩ ρ0gz dV = 0.(2.54) This is a logical consequence of the approximation (2.23). The equation representing the rate of change of the kinetic energy of the fluid can be derived from (2.35): taking the inner product of (2.35) with ρ0u, we get ∂ ∂t(1 2ρ0|u|2) + ∇ · ρ0(1 2|u|2+p′/ρ0)u=ρ0T′u· ∇s0−ρ0c′u· ∇µ0; (2.55) integrating (2.55) over the domain Ω, on the assumption that the normal component of uvanishes on the boundary of Ω, we obtain d dt ZΩ 1 2ρ0|u|2dV =ZΩρ0T′u· ∇s0−ρ0c′u· ∇µ0dV. (2.56) Maruyama (2021b) demonstrated that, under the isothermal anelastic approximation, the work done by the buoyancy force due to changes in temperature corresponds to the conversion between kinetic and internal energy. We see that this is also the case under the present extended approximation, comparing (2.56) with (2.53). Furthermore, it can be seen from the comparison of (2.56) with (2.53) that the work done by the buoyancy force due to changes in concentration also corresponds to the same energy conversion. Finally, adding (2.53), (2.54), and (2.56), we have d dt ZΩ ρ01 2|u|2+gz +edV = 0.(2.57) This equation shows that the total energy of the fluid is conserved. Hence the extended isothermal anelastic approximation is consistent with the conservation law of energy. 7 2.7. Applicability of the extended approximation The present approximation was formulated on the assumptions (2.10) and (2.22). In the following, we examine under what conditions these assumptions are justifiable. We first focus attention on the assumption (2.22). When ρis regarded as a function of φ,T, and c, the following relations hold: (∂ρ/∂φ)T,c =ρ(γ/a2), (∂ρ/∂T )φ,c =ρs(γ/a2)−β, (∂ρ/∂c)φ,T =−ρρ(∂µ/∂p)T,c +µ(γ/a2), (2.58) where γdenotes the ratio of specific heats, and athe speed of sound. Substituting these relations into (2.44), we obtain ρ′=ρ0(γ0/a2 0)φ′−ρ0s0(γ0/a2 0)−β0T′ −ρ0ρ0(∂µ/∂p)T,c|(φ0,T0,c0)+µ0(γ0/a2 0)c′,(2.59) in which γ0and a0are defined by γ0=γ(φ0, T0, c0), a0=a(φ0, T0, c0).(2.60) Now, let the characteristic scales of φ′,T′, and c′be respectively denoted by ∆φ′, ∆T′, and ∆c′. Then, we can find the following estimate for |ρ′/ρ0|: ρ′/ρ0=Oγ0(gH/a2 0)(∆φ′/gH) +Oγ0(gH/a2 0)(s0∆T′/gH) +O(β0∆T′) +O(ρ0gH/µ0)(∂µ/∂p)T,c|(φ0,T0,c0)(µ0∆c′/gH) +Oγ0(gH/a2 0)(µ0∆c′/gH). (2.61) Here Hstands for the vertical extent of the domain Ω containing the fluid: we assume that Hsatisfies the conditions (gH)1/2/a0≤O(1),(ρ0gH/µ0)(∂µ/∂p)T,c|(φ0,T0,c0)≤O(1).(2.62) It then follows from (2.61) that, since γ0=O(1), (2.22) holds under the conditions ∆φ′/gH ≪1,(2.63) s0∆T′/gH ≪1,(2.64) β0∆T′≪1,(2.65) |µ0|∆c′/gH ≪1.(2.66) We can also find from (2.29) the following estimate for |s′/s0|: |s′/s0|=O{(cp0/s0)(Γ0H/∆T′)(∆T′/T0)(∆φ′/gH)} +O{(cp0/s0)(∆T′/T0)} +O(β0∆T′) +O{(cp0/s0)(Γ0H/∆T′)(∆T′/T0)(µ0∆c′/gH)} +O{(cp0/s0)(T0/µ0)(∂µ/∂T )p,c|(φ0,T0,c0)(µ0∆c′/cp0T0)}, (2.67) 8 where Γ0=β0T0g/cp0is the adiabatic lapse rate. As pointed out by Maruyama (2021b), it is reasonable to expect that the following conditions are met: cp0/s0≤O(1),Γ0H/∆T′≤O(1).(2.68) We also assume that (T0/µ0)(∂µ/∂T )p,c|(φ0,T0,c0)≤O(1).(2.69) Then the first assumption of (2.10) is justifiable when the conditions ∆T′/T0≪1,(2.70) |µ0|∆c′/cp0T0≪1 (2.71) hold together with (2.63), (2.65), and (2.66). On the other hand, using (2.17), we can express µ′as follows: µ′=ρ0(∂µ/∂p)T,c|(φ0,T0,c0)φ′ +{(∂µ/∂T )p,c|(φ0,T0,c0)+ρ0s0(∂µ/∂p)T,c|(φ0,T0,c0)}T′ +{(∂µ/∂c)T,p|(φ0,T0,c0)−ρ0µ0(∂µ/∂p)T,c|(φ0,T0,c0)}c′. (2.72) This expression yields the following estimate for |µ′/µ0|: |µ′/µ0|=O{(ρ0gH/µ0)(∂µ/∂p)T,c|(φ0,T0,c0)(∆φ′/gH)} +O{(T0/µ0)(∂µ/∂T )p,c|(φ0,T0,c0)(∆T′/T0)} +O{(ρ0gH/µ0)(∂µ/∂p)T,c|(φ0,T0,c0)(s0∆T′/gH)} +O{(gHµ−1 0/µ0)(∂µ/∂c)T,p|(φ0,T0,c0)(µ0∆c′/gH)} +O{(ρ0gH/µ0)(∂µ/∂p)T,c|(φ0,T0,c0)(µ0∆c′/gH)}. (2.73) Accordingly, the second assumption of (2.10) is justifiable when the condition (gHµ−1 0/µ0)(∂µ/∂c)T,p|(φ0,T0,c0)≤O(1) (2.74) is fulfilled in addition to the above conditions. It should be noted here, however, that the following inequality follows from (2.7): |∇φ′| ≤ |∂u/∂t|+|(u· ∇)u|+|s∇T′|+|µ∇c′|.(2.75) Let Ldenote the length scale characteristic of the motion of the fluid; we then have |∇φ′|=O(∆φ′/L),|∇T′|=O(∆T′/L),|∇c′|=O(∆c′/L).(2.76) Moreover, if the velocity scale characteristic of the motion is denoted by U, we get |∂u/∂t|=O(U/τ),|(u· ∇)u|=O(U2/L),(2.77) 9