Rational derivation of the Boussinesq approximation

Rational derivation of the Boussinesq approximation Kiyoshi Maruyama Department of Earth and Ocean Sciences, National Defense Academy, Yokosuka, Kanagawa 239-8686, Japan December 18, 2025 Abstract This note derives the Boussinesq approximation in a manner consistent with the conservation law of mass. It is shown that the governing equations of a fluid under the approximation can be obtained on the basis of the assumption that the fluid is incompressible, in the sense that the density of the fluid is constant. The equation of motion, in particular, can be formulated with the help of the conservation law of energy. The conditions for the approximation to be valid are also discussed. 1. Introduction The Boussinesq approximation is widely used to describe the motion of a fluid with a nonuniform temperature distribution. In deriving the approximation, it is traditionally assumed that the density of the fluid is a linear function only of the temperature of the fluid (see e.g. Landau & Lifshitz 1987, § 56; Mihaljan 1962). It can be shown, however, that this assumption violates the conservation law of mass (Maruyama 2014). This is a fatal defect of the traditional method for deriving the approximation. Thus, in view of the importance of the approximation, this note introduces a method to derive the approximation in a manner consistent with the conservation law of mass: the fluid is assumed to be incompressible in the sense that its density is constant. It is shown that the need for the buoyancy force in the approximation can be inferred on the basis of the conservation law of energy. The conditions necessary for the approximation to be valid (Spiegel & Veronis 1960) are also revised accordingly. 2. Governing equations under the Boussinesq approximation We consider the motion of a fluid in a uniform gravitational field; it is assumed that the fluid occupies a fixed domain Ω. We set up in the domain a system of rectangular coordinates (x1, x2, x3) with the x3-axis taken vertically upwards. The unit vectors in 1 the positive x1-, x2-, and x3-directions are respectively denoted by e1,e2, and e3. In the following, Latin indices are consistently used to represent the numbers 1, 2, and 3; the summation convention is also implied. 2.1. Basic assumptions First of all, we assume that the fluid is incompressible in the sense that the density ρ of the fluid is constant: ρ=ρ0.(2.1) The thermal expansion coefficient βof the fluid, however, is assumed not to vanish: β=v−1(∂v/∂T)p= 0,(2.2) where v=ρ−1is the specific volume of the fluid; Tand pare the temperature and the pressure of the fluid, respectively. We assume that Tcan be expressed in the form T=T0+T′,(2.3) where T0is a constant reference temperature, and T′the small deviation from T0. We also write p, denoting by p′the small perturbation pressure, as follows: p=p0+p′.(2.4) Here the hydrostatic pressure p0is defined by p0=−ρ0gx3+ constant,(2.5) in which gdenotes the acceleration due to gravity. 2.2. Equation of continuity Now, let us consider the equation of continuity ρ−1Dρ/Dt +∇ · u= 0,(2.6) in which D/Dt denotes the material derivative, and uis the velocity of the fluid. The substitution of (2.1) into (2.6) yields ∇ · u= 0.(2.7) This is the equation of continuity under the Boussinesq approximation. 2 2.3. Equation of motion Let us next consider the equation of motion: using (2.4), it can be written as ρDu Dt =−∇p′+∂τij ∂xj ei−(ρ−ρ0)ge3,(2.8) where τij are the components of the viscous stress tensor. Substituting (2.1) into (2.8), we obtain ρ0 Du Dt =−∇p′+∂τij ∂xj ei.(2.9) This equation, however, must contain an additional term in order to be consistent with the conservation law of energy; our aim in the following is to demonstrate this fact. To this end, we first need to consider the total energy of the fluid: ZΩ ρ01 2|u|2+gx3+edV. (2.10) Here edenotes the specific internal energy of the fluid. The conservation law of energy requires that this energy should satisfy the equation d dt ZΩ ρ01 2|u|2+gx3+edV =ZΣ uiτijnjdS −ZΣ q·ndS, (2.11) in which Σ denotes the boundary of Ω, and nthe unit outward normal on Σ; uiand nj are the components of uand n, respectively; qis the heat flux density due to thermal conduction. This equation states that the total energy of the fluid changes owing to the work done by the viscous force acting on Σ and owing to the heat transfer across Σ. On the other hand, it is obvious that d dt ZΩ ρ0gx3dV = 0.(2.12) Thus the potential energy of the fluid is invariable. We next proceed to derive, after Maruyama (2014), the equation that describes the rate of change of the internal energy of the fluid. This equation can be obtained from the general equation of heat transfer (see Landau & Lifshitz 1987, § 49): ρT Ds Dt =τij ∂ui ∂xj −∇·q,(2.13) where sdenotes the specific entropy of the fluid, and the first term on the right-hand side represents the heating due to viscous dissipation. Now, regarding sas a function of Tand p, we obtain Ds Dt =∂s ∂T p DT Dt +∂s ∂pT Dp Dt .(2.14) 3 Here, as is well known in thermodynamics, (∂s/∂T)pand (∂s/∂p)Tare given by ∂s ∂T p =cp T,∂s ∂pT =−∂v ∂T p =−vβ, (2.15) where cpis the specific heat at constant pressure. These coefficients may be evaluated at T=T0and p=p0, for Tand pdeviate only slightly from T0and p0. We then have, since v=v0=ρ−1 0, Ds Dt =cp0 T0 DT′ Dt −v0β0Dp0 Dt +Dp′ Dt ,(2.16) where (2.3) and (2.4) have been used; we have also introduced the notation cp0=cp(T0, p0), β0=β(T0, p0).(2.17) It should be noted that cp0and β0may depend on x3through p0. Similarly, when sis regarded as a function of eand v, we get Ds Dt =1 T0 De Dt +p0 T0 Dv Dt .(2.18) However, since Dv/Dt =Dρ−1/Dt = 0, it follows that Ds Dt =1 T0 De Dt .(2.19) As a consequence, using (2.1), (2.3), (2.16), and (2.19), we have ρT Ds Dt =ρ0T0 Ds Dt +ρ0T′Ds Dt =ρ0 De Dt +ρ0T′cp0 T0 DT′ Dt −v0β0Dp0 Dt +Dp′ Dt .(2.20) However, since Dp0/Dt =−ρ0gu3, (2.20) gives, to the first order of primed variables, ρT Ds Dt =ρ0 De Dt +ρ0β0T′gu3.(2.21) Substituting this into (2.13), we obtain ρ0 De Dt =τij ∂ui ∂xj −∇·q−ρ0β0T′gu3.(2.22) The integration of (2.22) over Ω yields d dt ZΩ ρ0edV =ZΩ τij ∂ui ∂xj dV −ZΣ q·ndS −ZΩ ρ0β0T′gu3dV. (2.23) This is the desired equation for the rate of change of the internal energy. 4 We can now derive the equation describing the rate of change of the kinetic energy of the fluid by subtracting (2.12) and (2.23) from (2.11): d dt ZΩ 1 2ρ0|u|2dV =ZΣ uiτijnjdS −ZΩ τij ∂ui ∂xj dV +ZΩ ρ0β0T′gu3dV. (2.24) We observe from this equation that (2.9) must contain the term ρ0β0T′ge3.(2.25) The force represented by this term is referred to as the buoyancy force. Thus we get ρ0 Du Dt =−∇p′+∂τij ∂xj ei+ρ0β0T′ge3.(2.26) This is the equation of motion under the Boussinesq approximation. In (2.26), τij are given, for example, by τij =η∂ui ∂xj +∂uj ∂xi,(2.27) in which ηis the dynamic viscosity. It should be emphasized that the work done by the buoyancy force corresponds to the conversion between kinetic and internal energy; this indicates that the origin of the buoyancy force is the pressure gradient force. Before closing this subsection, it is worth noting that, as shown in the appendix, the need for the buoyancy force can also be inferred from a simple thought experiment. 2.4. Temperature equation From (2.16) and (2.19), we have ρ0 De Dt =ρ0cp0 DT′ Dt −β0T0Dp0 Dt +Dp′ Dt .(2.28) The substitution of (2.28) into (2.22) yields the following temperature equation: ρ0cp0 DT′ Dt =τij ∂ui ∂xj −∇·q+β0T0Dp0 Dt +Dp′ Dt −ρ0β0T′gu3.(2.29) Here qis given, for example, by Fourier’s law: q=−k∇T′,(2.30) where kdenotes the thermal conductivity. This is the temperature equation under the Boussinesq approximation consistent with the conservation law of energy. 3. Summary and discussion The Boussinesq approximation has been reconstructed in a manner consistent with the conservation law of mass. It has been shown that the governing equations of a fluid under the approximation can be obtained on the basis of the following assumption: the fluid is incompressible in the sense that its density is constant. The equation of motion can in particular be formulated with the help of the conservation law of energy. 5 3.1. Applicability of the Boussinesq approximation We are now in a position to discuss what conditions are necessary for the Boussinesq approximation to be applicable to a specific motion of a compressible fluid. We assume that the motion possesses the following characteristic scales: a time scale τ, a velocity scale U, and a length scale L. Let also the scale of the temperature difference of the fluid be denoted by ∆T′. As in § 2, T0denotes a constant reference temperature; ρ0in the following should be interpreted as a constant reference density. We first recall (2.3). This expression states that the temperature of the fluid varies only slightly from T0. Hence ∆T′must be very small compared with T0: ∆T′/T0≪1.(3.1) In (2.4), on the other hand, the variation of the perturbation pressure p′is taken to be very small compared with that of the hydrostatic pressure p0. Thus we must have |∇p′|/|∇p0| ≪ 1.(3.2) Here, in view of (2.5), |∇p0|is given by |∇p0|=ρ0g. (3.3) As for |∇p′|, the following inequality can be obtained from (2.8): |∇p′| ≤ |ρ(u· ∇)u|+|ρ(∂u/∂t)|+|(∂τij/∂xj)ei|+|(ρ−ρ0)ge3|.(3.4) It therefore follows that, if each of the terms on the right-hand side of (3.4) is very small compared with ρ0g, (3.2) is satisfied. In the following, we examine each term in turn. We can estimate the first term on the right-hand side of (3.4) as follows: |ρ(u· ∇)u|=O(ρ0U2/L).(3.5) This term, therefore, is very small compared with ρ0gwhen U/(gL)1/2≪1.(3.6) The second term can also be estimated as |ρ(∂u/∂t)|=O(ρ0U/τ).(3.7) Hence this term is very small compared with ρ0gwhen, together with (3.6), (L/τ)/(gL)1/2≪1 (3.8) applies. Regarding the third term, we observe that, if (2.27) may be employed, |(∂τij/∂xj)ei|=O(ηU/L2).(3.9) 6 Then this term also is very small compared with ρ0gwhen (3.6) and ν/{(gL)1/2L} ≪ 1 (3.10) are satisfied. Here ν=η/ρ0is the kinematic viscosity. In order to estimate the last term of (3.4), we note here the thermodynamic relation dρ = (γ/a2)dp −ρβdT, (3.11) where γis the ratio of specific heats, and athe speed of sound. This allows us to write |ρ−ρ0|=O(γ∆p/a2) + O(ρ0β0∆T′).(3.12) Here ∆pdenotes the scale of the pressure variation. Now, let the vertical extent of the fluid be denoted by H. Then it is reasonable to put ∆p=ρ0gH. (3.13) In consequence, the last term of (3.4) can be estimated as follows: |(ρ−ρ0)ge3|=O(γρ0g2H/a2) + O(ρ0β0∆T′g).(3.14) Accordingly, since γ=O(1), this term is very small compared with ρ0gwhen (gH)1/2/a ≪1, β0∆T′≪1.(3.15) It is interesting to note that, for an ideal gas, the second condition in (3.15) is identical to (3.1). This is because β0=T−1 0for an ideal gas. The basic assumption (2.1) is also justifiable under (3.15); as is apparent from (3.12) and (3.13), |ρ−ρ0|/ρ0≪1 under (3.15). The remaining assumption (2.2) is commonly satisfied. Thus the conditions for the Boussinesq approximation to be applicable to the motion have all been formulated: (3.1), (3.6), (3.8), (3.10), and (3.15). 3.2. Further approximations to the temperature equation Although the temperature equation (2.29) is consistent with the conservation law of energy, it is somewhat hard to deal with. Thus, for practical purposes, it is desirable to introduce further approximations to the equation. Let us consider again the fluid motion in § 3.1. We can approximate the terms in the braces on the right-hand side of (2.29), ignoring terms containing primed variables, as β0T0Dp0 Dt +Dp′ Dt −ρ0β0T′gu3≈β0T0 Dp0 Dt =−ρ0β0T0gu3.(3.16) The term representing the heating due to viscous dissipation, with τij given by (2.27), can also be ignored in comparison with the term ρ0β0T0gu3when the condition (1/β0T0)hν/{(gL)1/2L}i≪1 (3.17) 7 holds in addition to (3.6). In this case, (2.29) takes the form ρ0cp0 DT′ Dt =−∇ · q−ρ0β0T0gu3.(3.18) However, the last term of (3.18) can further be ignored when Γ0H/∆T′≪1,(3.19) where Γ0=β0T0g/cp0is the adiabatic lapse rate. We then obtain ρ0cp0 DT′ Dt =−∇ · q.(3.20) This is in fact the temperature equation in the original Boussinesq approximation. 3.3. Potential temperature under the Boussinesq approximation When the temperature equation is approximated by (3.18), we can employ, in place of T′, the potential temperature θdefined by θ=θ0+T′, θ0=T0+Zx3 ζa Γ0(x′ 3)dx′ 3.(3.21) Here the integral is taken from an arbitrary reference level x′ 3=ζa. In terms of θ, the temperature equation (3.18) is written as ρ0cp0 Dθ Dt =−∇ · q.(3.22) The equation of motion (2.26) can also be written as ρ0 Du Dt =−∇ˆp′+∂τij ∂xj ei+ρ0β0θge3,(3.23) where ˆp′is the modified perturbation pressure defied by ˆp′=p′+ρ0gZx3 ζa β0(x′ 3)θ0(x′ 3)dx′ 3.(3.24) Comparing (3.22) with (3.20), and (3.23) with (2.26), we observe that θplays formally the same role as T′in the original Boussinesq approximation. However, it is also important to pay attention to the difference between θand T′. Let us consider, as an example, Fourier’s law (2.30). It is expressed in terms of θas follows: q=−k∇θ+kΓ0e3.(3.25) Hence the heat flux density q, which is proportional to ∇T′, is not proportional to ∇θ. 8 This fact becomes significant, for example, when an adiabatic condition is imposed on a fixed boundary x3=ζb. Expressing the condition in terms of T′, we have (∂T′/∂x3)|x3=ζb= 0.(3.26) On the other hand, the same condition requires that (∂θ/∂x3)|x3=ζb= Γ0(ζb).(3.27) It therefore follows that θmust satisfy a condition formally different from that for T′. As is evident from the above discussion, θis not a mere substitute for T′. This is not surprising considering the following fact: potential temperature is a measure of specific entropy (see e.g. Gill 1982); it is, despite its name, a quantity essentially different from temperature. Appendix. Another derivation of the buoyancy force We consider the same physical situation as that in § 2, but the fluid is now assumed to be at rest. Let us first take a fluid element of unit volume in the fluid: it is referred to as element A, and its temperature is Ta. We next consider another fluid element of unit volume referred to as element B: its position relative to element A is eidxi, and its temperature is Tb. The fluid elements have the same mass ρ0, and Taand Tbare not so different. Our aim in this appendix is to analyze the quasistatic process of adiabatically interchanging the positions of the fluid elements in order to derive the buoyancy force in the Boussinesq approximation. Let us focus attention on element A. When its position is adiabatically interchanged with that of element B, the specific entropy sremains unchanged. Hence the change in temperature dTaof element A can be calculated from dTa=∂T ∂p s dpa.(A.1) Here dpadenotes the change in pressure experienced by element A when it is displaced vertically by dx3; since the pressure is hydrostatic, dpais given by dpa=−ρ0gdx3.(A.2) We can also write, using (2.15), the coefficient on the right-hand side of (A.1) as ∂T ∂p s =−∂s ∂pT∂s ∂T p =v0βTa cp ,(A.3) where βand cpare regarded as constant. Thus, from (A.1), (A.2), and (A.3), we have dTa=−βTag cp dx3.(A.4) 9