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

Ex ension o he iso he mal anelas ic
app oxima ion o a wo-componen luid
Kiyoshi Ma uyama
Depa men o Ea h and Ocean Sciences, Na ional De ense Academy,
Yokosuka, Kanagawa 239-8686, Japan
No embe 20, 2025
Abs ac
This pape ex ends a a ian o he anelas ic app oxima ion o a wo-componen
luid. The ene ge ics and he applicabili y o his ex ended app oxima ion a e also
discussed, oge he wi h he ela ion o he Boussinesq app oxima ion ex ended o
a wo-componen luid.
1. In oduc ion
Ogu a & Phillips (1962) de ised an app oxima ion called he anelas ic app oxima ion
in o de o s udy he mo ion unde g a i y o a deep laye o ideal gas wi h an isen opic
basic s a e. Ma uyama (2021a) econs uc ed he app oxima ion in such a manne ha
i can be applied o any kind o luid wi h an isen opic basic s a e.
Ma uyama (2021b) u he in oduced a a ian o he anelas ic app oxima ion which
deals wi h a luid ha ing an iso he mal basic s a e. As a limi ing case o his iso he mal
anelas ic app oxima ion, he well-known Boussinesq app oxima ion can be ob ained
The objec i e o his pape is o ex end he iso he mal anelas ic app oxima ion o a
wo-componen luid: he ene ge ics o he ex ended app oxima ion and he condi ions
o i s applicabili y a e also ully cla i ied; i s ela ion o he Boussinesq app oxima ion
ex ended o a wo-componen luid (Ma uyama 2019) is discussed as well.
2. Ex ended iso he mal anelas ic app oxima ion
We conside he mo ion o an in iscid luid consis ing o wo componen s, Aand B,
in a uni o m g a i a ional ield. The concen a ion o componen Ais deno ed by c; he
mass o Ain a uni olume o he luid is gi en by ρc, wi h ρbeing he densi y o he
luid. The luid is con ained in a ixed ini e domain Ω, and, in his domain, he z-axis
is aken e ically upwa ds. We deno e by k he uni ec o in he posi i e z-di ec ion.
1
2.1. Equa ion o mo ion
We i s w i e down he equa ion o mo ion o he luid as ollows:
Du
D =−∇p/ρ −gk,(2.1)
whe e D/D s ands o he ma e ial de i a i e, and udeno es he eloci y o he luid;
pis he p essu e o he luid, and g he accele a ion due o g a i y.
Le φdeno e he speci ic Gibbs ee ene gy o he luid: i sa is ies he ela ion
dφ =−sdT + dp +µdc, (2.2)
in which sand Ta e he speci ic en opy and he empe a u e o he luid, espec i ely;
= 1/ρ deno es he speci ic olume o he luid, and µis he chemical po en ial o he
luid (see Landau & Li shi z 1987,
§
58). In he ollowing, all he modynamic quan i ies
a e ega ded as known unc ions o φ,Tand c.
Using (2.2), we can ew i e ∇p/ρ as ollows:
∇p/ρ =∇φ+s∇T−µ∇c. (2.3)
The subs i u ion o (2.3) in o he equa ion o mo ion (2.1) yields
Du
D =−∇φ−s∇T+µ∇c−gk.(2.4)
Now, le us decompose φ,T, and cas ollows:
φ=φ0+φ′, T =T0+T′, c =c0+c′.(2.5)
He e φ0,T0, and c0a e de ined by
φ0=−gz +α1, T0=α2, c0=α3,(2.6)
wi h α1,α2, and α3being cons an s. Then (2.4) akes he ollowing o m:
Du
D =−∇φ′−s∇T′+µ∇c′.(2.7)
Unde he decomposi ion (2.5), sand µcan also be decomposed as ollows:
s=s0+s′, µ =µ0+µ′,(2.8)
whe e s0and µ0a e gi en by
s0=s(φ0, T0, c0), µ0=µ(φ0, T0, c0).(2.9)
We in oduce he e he ollowing assump ions:
|s′/s0| ≪ 1,|µ′/µ0| ≪ 1.(2.10)
2
Then he equa ion o mo ion (2.7) may be app oxima ed as
Du
D =−∇φ′−s0∇T′+µ0∇c′.(2.11)
Since s0∇T′=∇(s0T′)−T′∇s0and µ0∇c′=∇(µ0c′)−c′∇µ0, we ha e
Du
D =−∇(φ′+s0T′−µ0c′) + T′∇s0−c′∇µ0.(2.12)
This is he equa ion o mo ion unde he ex ended iso he mal anelas ic app oxima ion.
The second e m on he igh -hand side o (2.12) ep esen s he buoyancy o ce ha
a ises om changes in empe a u e. Rega ded as a unc ion o φ,T, and c,ssa is ies
(∂s/∂φ)T,c =−β,
(∂s/∂T )φ,c =T/cp−βs,
(∂s/∂c)φ,T =βµ −(∂µ/∂T)p,c,
(2.13)
whe e β= −1(∂ /∂T)p,c is he he mal expansion coe icien , and cp he speci ic hea
a cons an p essu e. Hence, in iew o (2.6) and (2.9), we can w i e
∇s0= (∂s/∂φ)T,c|(φ0,T0,c0)∇φ0=β0gk,(2.14)
in which he ollowing no a ion has been in oduced:
β0=β(φ0, T0, c0).(2.15)
The second e m on he igh -hand side o (2.12) can he e o e be ew i en as
T′∇s0=β0T′gk.(2.16)
The hi d e m on he igh -hand side o (2.12) ep esen s he buoyancy o ce due o
changes in concen a ion. The chemical po en ial µin his e m sa is ies he ela ions
(∂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)
whe e βc=ρ−1(∂ρ/∂c)T,p. Thus ∇µ0can be ew i en as
∇µ0= (∂µ/∂φ)T,c|(h0,s0,c0)∇φ0=βc0gk,(2.18)
whe e βc0=βc(φ0, T0, c0). As a esul , we ob ain he ollowing exp ession:
−c′∇µ0=−βc0c′gk.(2.19)
3
2.2. Equa ion o con inui y
Unde he decomposi ion (2.5), he densi y ρo he luid can be w i en as
ρ=ρ0+ρ′,(2.20)
whe e ρ0is de ined by
ρ0=ρ(φ0, T0, c0).(2.21)
We in oduce he e he ollowing assump ion:
|ρ′/ρ0| ≪ 1.(2.22)
On his assump ion, ρmay be app oxima ed as ollows:
ρ=ρ0.(2.23)
Subs i u ing (2.23) in o he equa ion o con inui y ∂ρ/∂ +∇ · (ρu) = 0, we ge
∇ · (ρ0u)=0.(2.24)
This is he equa ion o con inui y unde he ex ended app oxima ion.
2.3. Concen a ion equa ion
In he absence o di usion, he equa ion o he a e o change o he concen a ion c
o componen Ais gi en by (see Landau & Li shi z 1987,
§
58)
ρDc
D = 0.(2.25)
Conside ing (2.5) and (2.23), we can app oxima e his equa ion as ollows:
ρ0
Dc′
D = 0.(2.26)
2.4. Gene al equa ion o hea ans e
The gene al equa ion o hea ans e (see Landau & Li shi z 1987,
§
58) akes, when
he conduc ion o hea is neglec ed, he ollowing o m:
ρT Ds
D +ρµDc
D = 0.(2.27)
To he i s o de o p imed a iables, his equa ion can be w i en as
ρ0T0u· ∇s0+ρ0T0
Ds′
D +ρ0T′u· ∇s0+ρ0µ0
Dc′
D = 0,(2.28)
whe e (2.23) has been used. Using (2.13), we can u he ew i e s′in (2.28) as ollows:
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)
He e he ollowing no a ion has been in oduced:
cp0=cp(φ0, T0, c0).(2.30)
4
2.5. Al e na i e o ms o he equa ion o mo ion
We ha e hus ully o mula ed he iso he mal anelas ic app oxima ion ex ended o a
wo-componen luid. Fo la e e e ence, howe e , i is use ul o ew i e he equa ion o
mo ion (2.12) in somewha di e en o ms.
We i s no e ha , unde he decomposi ion (2.5), he p essu e po he luid can also
be decomposed as ollows:
p=p0+p′,(2.31)
whe e p0is de ined by
p0=p(φ0, T0, c0).(2.32)
On he o he hand, he ollowing he modynamic ela ions a e ob ained om (2.2):
(∂p/∂φ)T,c =ρ, (∂p/∂T )φ,c =ρs, (∂p/∂c)φ,T =−ρµ. (2.33)
Hence p′can be exp essed, o he i s o de o φ′,T′, and c′, as ollows:
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 exp ession enables us o ew i e (2.12) in he ollowing o m:
Du
D =−∇(p′/ρ0) + T′∇s0−c′∇µ0.(2.35)
The i s e m on he igh -hand side, howe e , can u he be ew i en as
−∇(p′/ρ0) = −∇p′/ρ0+ (p′/ρ0)(∇ρ0/ρ0).(2.36)
The subs i u ion o (2.36) in o (2.35) yields
Du
D =−∇p′/ρ0+ (p′/ρ0)(∇ρ0/ρ0) + T′∇s0−c′∇µ0.(2.37)
Mo eo e , since ρ0is de ined by (2.21), we can w i e
∇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 iew o (2.34), he ollowing exp ession o (p′/ρ0)(∇ρ0/ρ0) is ob ained:
(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 e ms T′∇s0and −c′∇µ0in (2.37) can also be exp essed as ollows:
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 hese exp essions o (2.39), we ge
(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)
F om (2.33), howe e , we obse e ha he ollowing he modynamic ela ions hold:
{∂(ρs)/∂φ}T,c = (∂ρ/∂T )φ,c,− {∂(ρµ)/∂φ}T,c = (∂ρ/∂c)φ,T .(2.42)
These ela ions enable us o ew i e (2.41) as ollows:
(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 he o he hand, ρ′in (2.20) is, o he i s o de o φ′,T′, and c′, gi en 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, he e o e, ha he equa ion o mo ion (2.12) can be exp essed in he o m
Du
D =−∇p′/ρ0−(ρ′g/ρ0)k.(2.45)
Howe e , i mus be emphasized ha , in spi e o his esul , he luid densi y unde he
p esen app oxima ion is gi en by ρ0, no by ρ0+ρ′.
2.6. Ene ge ics o he ex ended app oxima ion
Le us nex p oceed o s udy, unde he ex ended iso he mal anelas ic app oxima ion,
he ene gy balance o he luid. We i s conside he in e nal ene gy o he luid.
Le edeno e he speci ic in e nal ene gy o he luid. We hen ha e he ela ion
e=φ−p/ρ +Ts. (2.46)
Acco dingly, using (2.5), (2.8), (2.23), and (2.31), we ob ain, o he i s o de o p imed
a iables, he ollowing exp ession o e:
e= (φ0−p0/ρ0+T0s0)+(φ′−p′/ρ0+T0s′+s0T′).(2.47)
Conside ing (2.34), howe e , we can u he ew i e ein he ollowing o m:
e= (φ0−p0/ρ0+T0s0) + T0s′+µ0c′.(2.48)
6
Taking he ma e ial de i a i e o (2.48), and mul iplying he esul by ρ0, we ge
ρ0
De
D =ρ0u· ∇(φ0−p0/ρ0) + ρ0T0u· ∇s0+ρ0T0
Ds′
D +ρ0c′u· ∇µ0+ρ0µ0
Dc′
D .(2.49)
This equa ion, by i ue o (2.24) and (2.28), educes o
ρ0
De
D =∇·{ρ0(φ0−p0/ρ0)u} − ρ0T′u· ∇s0+ρ0c′u· ∇µ0.(2.50)
Howe e , since ρ0De/D =ρ0∂e/∂ +ρ0u· ∇e=∂(ρ0e)/∂ +∇ · (ρ0eu), we ha e
∂
∂ (ρ0e) + ∇·{ρ0(e−φ0+p0/ρ0)u}=−ρ0T′u· ∇s0+ρ0c′u· ∇µ0.(2.51)
Fu he mo e, in iew o (2.48), his equa ion can be pu in o he ollowing o m:
∂
∂ (ρ0e) + ∇ · ρ0{T0(s0+s′) + µ0c′}u=−ρ0T′u· ∇s0+ρ0c′u· ∇µ0.(2.52)
In eg a ing (2.52) o e he domain Ω con aining he luid, we ob ain
d
d ZΩ
ρ0e dV =−ZΩρ0T′u· ∇s0−ρ0c′u· ∇µ0dV, (2.53)
whe e i has been assumed ha he no mal componen o u anishes on he bounda y
o Ω. This is he equa ion o he a e o change o he in e nal ene gy o he luid.
The po en ial ene gy o he luid, on he o he hand, is in a iable:
d
d ZΩ
ρ0gz dV = 0.(2.54)
This is a logical consequence o he app oxima ion (2.23).
The equa ion ep esen ing he a e o change o he kine ic ene gy o he luid can be
de i ed om (2.35): aking he inne p oduc o (2.35) wi h ρ0u, we ge
∂
∂ (1
2ρ0|u|2) + ∇ · ρ0(1
2|u|2+p′/ρ0)u=ρ0T′u· ∇s0−ρ0c′u· ∇µ0; (2.55)
in eg a ing (2.55) o e he domain Ω, on he assump ion ha he no mal componen o
u anishes on he bounda y o Ω, we ob ain
d
d ZΩ
1
2ρ0|u|2dV =ZΩρ0T′u· ∇s0−ρ0c′u· ∇µ0dV. (2.56)
Ma uyama (2021b) demons a ed ha , unde he iso he mal anelas ic app oxima ion,
he wo k done by he buoyancy o ce due o changes in empe a u e co esponds o he
con e sion be ween kine ic and in e nal ene gy. We see ha his is also he case unde
he p esen ex ended app oxima ion, compa ing (2.56) wi h (2.53). Fu he mo e, i can
be seen om he compa ison o (2.56) wi h (2.53) ha he wo k done by he buoyancy
o ce due o changes in concen a ion also co esponds o he same ene gy con e sion.
Finally, adding (2.53), (2.54), and (2.56), we ha e
d
d ZΩ
ρ01
2|u|2+gz +edV = 0.(2.57)
This equa ion shows ha he o al ene gy o he luid is conse ed. Hence he ex ended
iso he mal anelas ic app oxima ion is consis en wi h he conse a ion law o ene gy.
7
2.7. Applicabili y o he ex ended app oxima ion
The p esen app oxima ion was o mula ed on he assump ions (2.10) and (2.22). In
he ollowing, we examine unde wha condi ions hese assump ions a e jus i iable. We
i s ocus a en ion on he assump ion (2.22).
When ρis ega ded as a unc ion o φ,T, and c, he ollowing ela ions 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)
whe e γdeno es he a io o speci ic hea s, and a he speed o sound. Subs i u ing hese
ela ions in o (2.44), we ob ain
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 a0a e de ined by
γ0=γ(φ0, T0, c0), a0=a(φ0, T0, c0).(2.60)
Now, le he cha ac e is ic scales o φ′,T′, and c′be espec i ely deno ed by ∆φ′, ∆T′,
and ∆c′. Then, we can ind he ollowing es ima e o |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)
He e Hs ands o he e ical ex en o he domain Ω con aining he luid: we assume
ha Hsa is ies he condi ions
(gH)1/2/a0≤O(1),(ρ0gH/µ0)(∂µ/∂p)T,c|(φ0,T0,c0)≤O(1).(2.62)
I hen ollows om (2.61) ha , since γ0=O(1), (2.22) holds unde he condi ions
∆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 ind om (2.29) he ollowing es ima e o |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
whe e Γ0=β0T0g/cp0is he adiaba ic lapse a e. As poin ed ou by Ma uyama (2021b),
i is easonable o expec ha he ollowing condi ions a e me :
cp0/s0≤O(1),Γ0H/∆T′≤O(1).(2.68)
We also assume ha
(T0/µ0)(∂µ/∂T )p,c|(φ0,T0,c0)≤O(1).(2.69)
Then he i s assump ion o (2.10) is jus i iable when he condi ions
∆T′/T0≪1,(2.70)
|µ0|∆c′/cp0T0≪1 (2.71)
hold oge he wi h (2.63), (2.65), and (2.66).
On he o he hand, using (2.17), we can exp ess µ′as ollows:
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 exp ession yields he ollowing es ima e o |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)
Acco dingly, he second assump ion o (2.10) is jus i iable when he condi ion
(gHµ−1
0/µ0)(∂µ/∂c)T,p|(φ0,T0,c0)≤O(1) (2.74)
is ul illed in addi ion o he abo e condi ions.
I should be no ed he e, howe e , ha he ollowing inequali y ollows om (2.7):
|∇φ′| ≤ |∂u/∂ |+|(u· ∇)u|+|s∇T′|+|µ∇c′|.(2.75)
Le Ldeno e he leng h scale cha ac e is ic o he mo ion o he luid; we hen ha e
|∇φ′|=O(∆φ′/L),|∇T′|=O(∆T′/L),|∇c′|=O(∆c′/L).(2.76)
Mo eo e , i he eloci y scale cha ac e is ic o he mo ion is deno ed by U, we ge
|∂u/∂ |=O(U/τ),|(u· ∇)u|=O(U2/L),(2.77)
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