Variational integrators for anelastic and pseudo-incompressible flows

3 MB·

by  Werner Bauer

Variational integrators for anelastic and
pseudo-incompressible flows
Werner Bauer1,2 and Franc¸ois Gay-Balmaz2
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1
0 Abstract
2
Theanelasticandpseudo-incompressibleequationsaretwowell-knownsoundproofap-
n
a proximations of compressible flows useful for both theoretical and numerical analysis in
J
meteorology, atmospheric science, and ocean studies. In this paper, we derive and test
0
structure-preserving numerical schemes for these two systems. The derivations are based
2
on a discrete version of the Euler-Poincar´e variational method. This approach relies on

] a finite dimensional approximation of the (Lie) group of diffeomorphisms that preserve
A
weighted-volume forms. These weights describe the background stratification of the fluid
N
and correspond to the weighed velocity fields for anelasic and pseudo-incompressible ap-
.
h proximations. In particular, we identify to these discrete Lie group configurations the
t
a associated Lie algebras such that elements of the latter correspond to weighted veloc-
m
ity fields that satisfy the divergence-free conditions for both systems. Defining discrete
[
Lagrangians in terms of these Lie algebras, the discrete equations follow by means of vari-

1 ational principles. We verify the structure-preserving nature of the resulting variational
v
integrators applying two test cases. The spectra of internal gravity waves, emitted by a
8
4 hydrostatic adjustment process, perfectly reflects the models’ dispersion relations. And,
4 shape and advection speed of a rising and a falling bubble match very well reference re-
6
sults in literature. Descending from variational principles, the schemes exhibit further
0
. a discrete version of Kelvin circulation theorem, are applicable to irregular meshes, and
1
show excellent long term energy behavior.
0
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1
: 1 Introduction
v
i
X
Numerical simulations of atmosphere and ocean on the global scale are of high importance in
r
a
the field of Geophysical Fluid Dynamics (GFD). The dynamics of these systems are frequently
modeled by the full Euler equations using explicit time integration schemes (see, e.g., [6]).
These simulations are however computationally very expensive. Besides highly resolved meshes
to capture important small scale features, the fast traveling sound waves have to be resolved
too, by very small time step sizes, in order to guarantee stable simulations [6]. As these sound
wave are assumed to be negligible in atmospheric flows, soundproof models, in which these fast
waves are filtered out, are a viable option that permits to increase the time step sizes and hence
to speed up calculations significantly.
1Imperial College London, Department of Mathematics, 180 Queen’s Gate, London SW7 2AZ, United King-
dom. [email protected]
2E´cole Normale Sup´erieure/CNRS, Laboratoire de M´et´eorologie Dynamique, Paris, France.
[email protected]
1
Frequently applied soundproof models are the Boussinesq, anelastic, pseudo-incompressible
approximations of the Euler equations [5, 11, 13]. There exist elaborated discretizations of
these equations in literature. However, these discretizations often do not take into account
the underlying geometrical structure of the equations. This may result in a lack of conserving
mass, momentum, energy, ortothefactthattheHelmholtzdecompositionofvectorfieldsorthe
Kelvin-Noether circulation theorem are not satisfied. Structure-preserving schemes descending
from Euler-Poincar´e variational methods [14], [7], [4] conserve these quantities, as they arise
from a Lagrangian formulation, in which these conserved quantities are given by invariants of
the Lagrangian under symmetries, [8].
With this paper, we contribute to develop variational integrators in the area of GFD by
including the anelastic and pseudo-incompressible schemes into the variational discretization
framework developed by [14]. To use this framework, we first have to describe the approxima-
tion of the Euler equations for a perfect fluid in terms of the Euler-Poincar´e variational method
[8]. The central idea is to use volume forms that are weighted by the corresponding back-
ground stratifications such that they match the divergence-free conditions of the correspond-
ingly weighed velocity fields associated to either the anelastic or the pseudo-incompressible
approximations. This will allow us to identify for these approximations the appropriate Lie
group configuration with corresponding Lie algebras. Using the latter to define appropriate
Lagrangian, the equations of motion follow by Hamilton’s variational principle of stationary
action.
The definition of appropriate discrete diffeomorphism groups will be based on the idea to
use weighted meshes that provide discrete counterparts of the weighted volume forms. The
corresponding discrete Lie algebras will incorporate the required divergence-free conditions on
the weighed velocity fields. Defining appropriate weighted pairings required to derive the func-
tional derivatives of the discrete Lagrangians, the flat operator introduced in [14] is directly
applicable and we can thus avoid to discuss this otherwise delicate issue. Mimicking the con-
tinuous theory, the discretizations of anelastic and pseudo-incompressible equations follow by
variations of appropriate discrete Lagrangians.
Westructurethepaperasfollows. InSection2werecallthestandardformulationsofBoussi-
nesq, anelastic, and pseudo-incompressible approximations of the Euler equations for perfect
fluids. In Section 3 we show that these formulations follow for appropriate Langrangians from
the Euler-Poincar´e variational principle. In Section 4 we recall the variational discretization
framework introduced by [14] and extend it to suit anelastic and pseudo-incompressible equa-
tions. In Section 5 we derive corresponding discretizations on 2D simplicial meshes and provide
explicit formulations in terms of velocity and buoyancy or potential temperature fields. In
Section 6 we perform numerical tests to demonstrate the structure-preserving properties of the
variational integrators. In Section 7 we draw conclusions and provide an outlook.
2 Anelastic and pseudo-incompressible systems
In this section we review the three approximations of the Euler equations of a perfect gas
that will be the subject of this paper, namely, the Boussinesq, the anelastic, and the pseudo-
incompressible approximations (see, e.g., [6] for more details).
The Euler equations for the inviscid isentropic motion of a perfect gas can be expressed in
2
the form
1
∂ u+u·∇u+ ∇p = −gz, ∂ ρ+div(ρu) = 0, ∂ θ+u·∇θ = 0, (2.1)
t t t
ρ
where u is the three-dimensional velocity vector, ρ is the mass density, p is the pressure, g is
the gravitational acceleration, z is the unit vector directed opposite to the gravitational force.
The variable θ is the potential temperature, defined by θ = T/π, in which T is the temperature
and π is the Exner pressure
π = (p/p )R/cp,
0
with R is the gas constant for dry air, c is the specific heat at constant pressure, and p is a
p 0
constant reference pressure. Using the equation of state for a perfect gas, p = ρRT, we have
the relation
1
∇p = c θ∇π.
p
ρ
The equations (2.1) correspond to conservation of momentum, mass, and entropy, respectively.
Let us write
θ(x,y,z,t) = θ¯(z)+θ(cid:48)(x,y,z,t), π(x,y,z,t) = π¯(z)+π(cid:48)(x,y,z,t),
¯
in which θ(z) and π¯(z) characterize a vertically varying reference state in hydrostatic balance,
that is,
dπ¯
¯
c θ = −g. (2.2)
p
dz
In terms of the perturbations θ(cid:48) and π(cid:48), the equations (2.1) can be equivalently written as
θ(cid:48)
∂ u+u·∇u+c θ∇π(cid:48) = g z, ∂ ρ+div(ρu) = 0, ∂ θ+u·∇θ = 0. (2.3)
t p ¯ t t
θ
We introduce now three frequently applied approximations to these equations.
Boussinesq approximation. This approximation is obtained by assuming a nondivergent
flow and by neglecting the variations in potential temperature except in the leading-order
contribution to the buoyancy. We thus get, from (2.3), the system
θ(cid:48)
∂ u+u·∇u+c θ ∇π(cid:48) = g z, divu = 0, ∂ θ+u·∇θ = 0,
t p 0 t
θ
0
in which θ is a constant reference potential temperature. These equations can equivalently be
0
written as
∂ u+u·∇u+∇P(cid:48) = b(cid:48)z, divu = 0, ∂ b(cid:48) +u·∇b(cid:48) +N2w = 0,
t b t
where P(cid:48) = c θ π(cid:48), N2 = g ∂ θ¯ is the Brunt-Va¨is¨ala¨ frequency, and b(cid:48) = gθ(cid:48) is the buoyancy.
b p 0 θ0 z θ0
Making use of the full buoyancy b = g θ = gθ¯+θ(cid:48), we can write the system as
θ0 θ0
∂ u+u·∇u+∇P = bz, divu = 0, ∂ b+u·∇b = 0, (2.4)
t b t
where P := P(cid:48) + g (cid:82)zθ¯(z)dz.
b b θ0 0
3
The total energy is conserved since the energy density E = 1|u|2−bz = 1|u|2−g θ z verifies
2 2 θ0
the continuity equation
∂ E +div((E +P )u) = 0. (2.5)
t b
The requirement for nondivergent flow is easily justified only for liquids, and the errors incurred
approximating the true mass conservation relation by divu = 0 can be quite large in stratified
compressible flows. In this case, the anelastic and pseudo-incompressible models have to be
considered, which better approximate the true mass continuity equation.
Anelastic approximation. The anelastic system approximates the continuity equation as
div(ρ¯u) = 0,
where ρ¯(z) is the vertically varying density of the reference state.
In the original anelastic system presented by [13], the reference state is isentropic so that
¯
θ(z) = θ is constant, which results in the approximation
0
θ(cid:48)
∂ u+u·∇u+c θ ∇π(cid:48) = g z, div(ρ¯u) = 0, ∂ θ+u·∇θ = 0. (2.6)
t p 0 t
θ
0
(cid:16) (cid:17)
(cid:0) (cid:1)
The energy density can be written as E = ρ¯ 1|u|2 −g θ z = ρ¯ 1|u|2 +c π¯θ , where π¯(z) =
2 θ0 2 p
− g z verifies the hydrostatic balance (2.2) for θ¯(z) = θ . The total energy is conserved since
cpθ0 0
E verifies the continuity equation
∂ E +div((E +P )u) = 0
t a0
with P := ρ¯(c θ π(cid:48) +gz)
a0 p 0
¯
In the subsequent work [15], the reference potential temperature θ was allowed to vary in
the vertical, leading to the momentum equation
θ(cid:48)
∂ u+u·∇u+c θ¯∇π(cid:48) = g z.
t p ¯
θ
The resulting system is however not energy conservative. In order to restore energy conserva-
tion, [11] considered the approximate momentum equation
θ(cid:48)
∂ u+u·∇u+∇(c θ¯π(cid:48)) = g z, div(ρ¯u) = 0, ∂ θ+u·∇θ = 0. (2.7)
t p ¯ t
θ
In this case, the energy density E = ρ¯(cid:0)1|u|2 +c π¯θ(cid:1), where π¯(z) is such that c ∂π¯ = −g,
2 p p∂z θ¯
satisfies the continuity equation
∂ E +div((E +P )u) = 0,
t a
with P := ρ¯(c θ¯π(cid:48) +gz).
a p
4
Pseudo-incompressible approximation. To obtain this approximation developed in [5],
one defines the pseudo-density ρ∗ = ρ¯θ¯/θ and enforces mass conservation with respect to ρ∗ as
∂ ρ∗ +div(ρ∗u) = 0. When combined with ∂ θ+u·∇θ = 0, it yields div(ρ¯θ¯u) = 0. These last
t t
twoequationscanbeusedwiththemomentumequationin(2.3)toyieldtheenergyconservative
system
θ(cid:48)
∂ u+u·∇u+c θ∇π(cid:48) = g z, div(ρ¯θ¯u) = 0, ∂ θ+u·∇θ = 0. (2.8)
t p ¯ t
θ
We note that the balance of momentum is equivalently written as ∂ u+u·∇u+c θ∇π = −gz,
t p
where π = π¯ + π(cid:48), with c ∂π¯ = −g. The energy density E = ρ∗(cid:0)1|u|2 +gz(cid:1) verifies the
p∂z θ¯ 2
continuity equation
∂ E +div((E +P )u) = 0
t pi
for P := c ρ∗θπ.
pi p
3 Variational formulation
We shall now formulate the anelastic and pseudo-incompressible equations in Euler-Poincar´e
variational form. Euler-Poincar´e variational principles are Eulerian versions of the classical
Hamilton principle of critical action. We refer to [8] for the general Euler-Poincar´e theory based
on Lagrangian reduction and for several applications in fluid dynamics. An Euler-Poincar´e
formulation for anelastic systems was given in [3]. We shall develop below a slightly different
Euler-Poincar´e approach, well-suited for the variational discretization, by putting the emphasis
on the underlying Lie group of diffeomorphisms associated to these systems.
As we have recalled above, the anelastic and pseudo-incompressible equations are based on
a constraint of the following type on the fluid velocity u(t,x):
div(σ¯u) = 0,
for a given strictly positive function σ¯(x) > 0 on the fluid domain D.
We assume that the fluid domain D is a compact, connected, orientable manifold with
smooth boundary ∂D. In our examples, D is a 2D domain in the vertical plane R2 (cid:51) x = (x,z)
or a 3D domain in R3 (cid:51) x = (x,y,z).
We fix a volume form µ on D, i.e., an n-form, n = dimD, with µ(x) (cid:54)= 0, for all x ∈ D . If
D is a domain in R3 (cid:51) (x,y,z), one can take µ = dx∧dy ∧dz to be the standard volume of
R3 restricted to D. We shall denote by div (u) the divergence of u with respect to the volume
µ
form µ. Recall that the divergence is the function div (u) defined by the equality
µ
£ µ = div (u)µ,
u µ
in which £ denotes the Lie derivative with respect to the vector field u, see, e.g., [1]. When
u
µ is the standard volume, one evidently recovers the usual divergence operator div on vector
fields.
Diffeomorphism groups. Let us denote by Diff (D) the group of all smooth diffeomor-
µ
phisms ϕ : D → D that preserve the volume form µ, i.e., ϕ∗µ = µ. The group structure on
Diff (D) is given by the composition of diffeomorphisms. The group Diff (D) can be endowed
µ µ
with the structure of a Fr´echet Lie group, although in this paper we shall only use the Lie group
5
structure at a formal level. The Lie algebra of the group Diff (D) is given by the space X (D)
µ µ
of all divergence free (relative to µ) vector fields on D, parallel to the boundary ∂D:
X (D) = {u ∈ X(D) | div (u) = 0, u(cid:107)∂D}.
µ µ
Given the strictly positive function σ¯ > 0 on D, we consider the new volume form σ¯µ with
associated diffeomorphism group and Lie algebra denoted Diff (D) and X (D) = {u ∈ X(D) |
σ¯µ σ¯µ
div (u) = 0, u(cid:107)∂D}, respectively. InthenextLemma, werewritetheconditiondiv (u) = 0
σ¯µ σ¯µ
by using exclusively the divergence operator div associated to the initial volume form µ.
µ
Lemma 3.1 Let D be a manifold endowed with a volume form µ and let σ¯ > 0 be a strictly
positive smooth function on D. Then we have
div (σ¯u) = σ¯div (u).
µ σ¯µ
Proof. We will use the following properties of the Lie derivative £ , the exterior differential
u
d, and the inner product i on differential forms (see, e.g., [1]): for a k-form α, an n-form β,
u
and a vector field u, we have
£ α = d(i α)+i dα, d(α∧β) = dα∧β +(−1)kα∧dβ,
u u u
i (α∧β) = i α∧β +(−1)kα∧i β.
u u u
On the one hand, we have
div (σ¯u)µ = £ µ = d(i µ) = d(σ¯i µ) = dσ¯ ∧i µ+σ¯d(i µ)
µ σ¯u σ¯u u u u
= (i dσ¯)µ−i (dσ¯ ∧µ)+σ¯div u = (dσ¯ ·u)µ+σ¯div u.
u u µ µ
On the other hand, we have
σ¯div (u)µ = £ (σ¯µ) = (dσ¯ ·u)µ+σ¯£ µ = (dσ¯ ·u)µ+σ¯div (u).
σ¯µ u u µ
This proves the result. (cid:4)
From this Lemma, we deduce that the appropriate Lie groups associated to the anelastic
and pseudo-incompressible systems are given by
G = Diff (D) and G = Diff (D),
ρ¯µ ρ¯θ¯µ
respectively. Indeed, from the preceding Lemma, it follows that the Lie algebras of these groups
can be written as
X (D) = {u ∈ X(D) | div (ρ¯u) = 0, u(cid:107)∂D} and
ρ¯µ µ
¯
X (D) = {u ∈ X(D) | div (ρ¯θu) = 0, u(cid:107)∂D},
ρ¯θ¯µ µ
respectively. They correspond to the anelastic and pseudo-incompressible constraints on the
fluid velocity. We will continue to uses the subscript σ¯µ when referring to both Lie groups and
both Lie algebras.
6
Euler-Poincar´e variational principles. The diffeomorphism group Diff (D) plays the
σ¯µ
role of the configuration manifold for these fluid models. The motion of the fluid is completely
characterized by a time dependent curve ϕ(t, ) ∈ Diff (D): a particle located at a point
σ¯µ
X ∈ D at time t = 0 travels to x = ϕ(t,X) ∈ D at time t. Exactly as in classical mechanics,
the Lagrangian of the system is defined on the tangent bundle T Diff (D) of the configuration
σ¯µ
manifold. We shall denote it by L : T Diff (D) → R. The index Θ indicates that this
Θ0 σ¯µ 0
Lagrangian parametrically depends on the potential temperature Θ (X) that is expressed here
0
in the Lagrangian description.
The equations of motion in the Lagrangian description follow from the Hamilton principle
(cid:90) T
δ L (ϕ,ϕ˙)dt = 0, (3.1)
Θ0
0
over a time interval [0,T], for variations δϕ with δϕ(0) = δϕ(T) = 0.
In the Eulerian description, the variables are the Eulerian velocity u(t,x) and the potential
temperature θ(t,x). They are related to ϕ(t,X) and Θ (X) as
0
u(t,ϕ(t,X)) = ϕ˙(t,X) and θ(t,ϕ(t,X)) = Θ (X). (3.2)
0
We assume that the Lagrangian L can be rewritten exclusively in terms of these two Eulerian
Θ0
variables, and we denote it by (cid:96)(u,θ). This assumption means that L is right-invariant with
Θ0
respect to the action of the subgroup
Diff (D) = {ϕ ∈ Diff (D) | Θ (ϕ(X)) = Θ (X), ∀X ∈ D}
σ¯µ Θ0 σ¯µ 0 0
of all diffeomorphisms that keep Θ invariant.
0
By rewriting the Hamilton principle (3.1) in terms of the Eulerian variables u and θ, we get
the Euler-Poincar´e variational principle
(cid:90) T
δ (cid:96)(u,θ)dt = 0, for variations δu = ∂ v+[u,v], δθ = −dθ·v, (3.3)
t
0
where v(t,x) is an arbitrary vector field on D parallel to the boundary and with div(σ¯v) = 0,
(i.e., v ∈ X (D) by Lemma 3.1), and with v(0,x) = v(T,x) = 0. The bracket [u,v], locally
σ¯µ
given by [u,v]i := uj∂ vi −vj∂ ui, is the Lie bracket of vector fields.
j j
The expressions for δu and δθ in (3.3) follow by taking the variation of the first and second
equalities in (3.2) and defining v(t,x) as v(t,ϕ(t,X)) = δϕ(t,X). A direct and efficient way
to obtain these expressions, or the variational principle (3.3), is to apply the general theory of
Euler-Poincar´e reduction on Lie groups, see [8].
In order to compute the associated equations, one needs to fix an appropriate space in
nondegenerate duality with X (D). This is recalled in the next Lemma, which follows from
σ¯µ
the Hodge decomposition and shall play a crucial role in the discrete setting later. Recall that
if V is a vector space, a space in nondegenerate duality with V is a vector space V(cid:48) to together
with a bilinear form (cid:104) , (cid:105) : V(cid:48) ×V → R such that (cid:104)α,v(cid:105) = 0, for all v ∈ V, implies α = 0 and
(cid:104)α,v(cid:105) = 0, for all α ∈ V(cid:48), implies v = 0.
Lemma 3.2 The space Ω1(D)/dΩ0(D) of one-forms modulo exact forms is in nondegenerate
duality with the space X (D), the Lie algebra of Diff (D). The nondegenerate duality pairing
σ¯µ σ¯µ
is given by
(cid:90)
(cid:104) , (cid:105) : Ω1(D)/dΩ0(D)×X (D) → R, (cid:104)[α],u(cid:105) := (α·u)σ¯µ, (3.4)
σ¯ σ¯µ σ¯
D
7
where [α] denotes the equivalence class of α modulo exact forms.
Proof. It is well-known that if g is a Riemannian metric, with µ the associated volume form
g
on D, then
(cid:90)
(cid:104) , (cid:105) : Ω1(D)/dΩ0(D)×X (D) → R, (cid:104)[α],v(cid:105) = (α·v)µ ,
µg g
D
is a nondegenerate duality pairing, see e.g., [12, §14.1]. This result follows from the Hodge
decomposition of 1-forms, which needs the introduction of a Riemannian metric g.
In our case, the volume forms µ and σ¯µ are not necessarily associated to a Riemannian
metric. We shall thus introduce a Riemannian metric g uniquely for the purpose of this proof,
with associated Riemannian volume form µ . Let f be the function defined by σ¯µ = fµ . Since
g g
D is orientable and connected, we have either f > 0 or f < 0 on D. We can rewrite the duality
pairing (3.4) as
(cid:90) (cid:90)
(α·u)σ¯µ = (α·(fu))µ . (3.5)
g
D D
By successive applications of Lemma 3.1, we have
f
div (fu) = div (fu) = f div (u) = div (σ¯u) = 0,
µg σf¯µ σ¯µ σ¯ µ
where the last equality follows since u ∈ X (D). This proves that v = fu ∈ X (D). We
σ¯µ µg
can thus write the duality pairing (cid:104) , (cid:105) in terms of the nondegenerate duality pairing (3.5) as
σ¯
(cid:104)[α],u(cid:105) = (cid:104)[α],fu(cid:105), which proves that it is nondegenerate. (cid:4)
σ¯
In a similar way to (3.4), we shall identify the dual to the space of functions F(D) with
itself by using the nondegenerate duality pairing
(cid:90)
F(D)×F(D) → R, (cid:104)h,θ(cid:105) = (hθ)σ¯µ. (3.6)
σ¯
D
Given a Lagrangian (cid:96) : X (D)×F(D) → R, the functional derivatives of (cid:96) are defined with
σ¯µ
respect to the parings (3.4) and (3.6) and denoted
(cid:20) (cid:21)
δ(cid:96) δ(cid:96) δ(cid:96)
∈ Ω1(D)/dΩ0(D), for ∈ Ω1(D), and ∈ F(D).
δu δu δθ
Proposition 3.3 The variational principle (3.3) yields the partial differential equation
δ(cid:96) δ(cid:96) δ(cid:96)
∂ +£ + dθ = −dp, with div (σ¯u) = 0, u(cid:107)∂D, (3.7)
t u µ
δu δu δθ
where £ denotes the Lie derivative acting on one-forms, given by £ α = d(i α)+i dα. This
u u u u
equation is supplemented with the advection equation
∂ θ+dθ·u = 0,
t
which follows from the definition of θ in (3.2).
8
Proof. By definition of the functional derivatives, we have
(cid:90) T (cid:90) T (cid:90) δ(cid:96) (cid:90) T (cid:90) δ(cid:96)
δ (cid:96)(u,θ)dt = ·δuσ¯µdt+ ·δθσ¯µdt.
δu δθ
0 0 D 0 D
Using the expression for δu in (3.3), integrating by parts, and using the equalities £ v = [u,v]
u
and d(α·v)·u = (£ α)·v+α·(£ v), the first term reads
u u
(cid:90) T (cid:90) (cid:18) δ(cid:96) δ(cid:96)(cid:19) (cid:90) T (cid:90) (cid:18)δ(cid:96) (cid:19)
− ∂ +£ ·vσ¯µdt+ d ·v ·uσ¯µdt.
t u
δu δu δu
0 D 0 D
(cid:0) (cid:1) (cid:0) (cid:1) (cid:0) (cid:1)
Wecanwrited δ(cid:96) ·v ·u = div δ(cid:96) ·vu −δ(cid:96)·vdiv (u) = div δ(cid:96) ·vu , sincediv (u) =
δu σ¯µ δu δu σ¯µ σ¯µ δu σ¯µ
0. Then, by the Gauss Theorem,
(cid:90) (cid:18) (cid:19) (cid:90)
δ(cid:96) δ(cid:96)
div ·vu σ¯µ = ·vi µσ¯ = 0,
σ¯µ u
δu δu
D ∂D
since u(cid:107)∂D. Combining these results, we thus get
(cid:90) T (cid:90) (cid:18) δ(cid:96) δ(cid:96) δ(cid:96) (cid:19)
∂ +£ + dθ ·vσ¯µdt = 0,
t u
δu δu δθ
0 D
for all v ∈ X (D). By Lemma 3.2, it follows that the one-form ∂ δ(cid:96) +£ δ(cid:96) + δ(cid:96)dθ is exact,
σ¯µ tδu uδu δθ
i.e., there exists a function p such that this expression equals dp. (cid:4)
Remark 3.4 We note that the statement of Proposition 3.3 does not need the introduction of
a Riemannian metric g on D. Only a volume form µ is fixed, together with a strictly positive
function σ¯. It can be however advantageous to formulate the equations (3.7) in terms of a
Riemannian metric g (note that we do not suppose that µ or σ¯µ equals µ ). In this case,
g
identifying one-forms and vector fields via the flat operator u ∈ X(D) → u(cid:91) = g(u, ) ∈ Ω1(D),
the space Ω1(D)/dΩ0(D) can be identified with the space of vector fields X(D) modulo gradient
(with respect to g) of functions. The nondegenerate duality pairing (3.4) thus reads
(cid:90)
(cid:104)[v],u(cid:105) = g(v,u)σ¯µ. (3.8)
σ¯
D
In terms of this duality pairing, the equations (3.7) are equivalently written as
δ(cid:96) δ(cid:96) δ(cid:96) δ(cid:96)
∂ +u·∇ +∇uT · + ∇θ = −∇p, (3.9)
t
δu δu δu δθ
where ∇ acting on a vector field is the covariant derivative associated to the Riemannian metric
g, ∇ acting on a function is the gradient relative to g, and ∇uT denotes the transpose with
respect to g.
We shall now apply this setting to the anelastic and the pseudo-incompressible equations.
The fluid domain D is a subset of the vertical plane R2 (cid:51) (x,z) or of the space R3 (cid:51) (x,y,z),
and has a smooth boundary ∂D. We fix a volume form µ on D.
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1) Anelastic equations. For the anelastic equation, we take σ¯(z) = ρ¯(z), the reference mass
density. The Lagrangian is given by
(cid:90) (cid:18) (cid:19)
1
(cid:96)(u,θ) = |u|2 −c π¯θ ρ¯µ, u ∈ X (D), (3.10)
p ρ¯µ
2
D
where π¯(z) is such that c ∂π¯ = −g and the norm is computed relative to the standard inner
p∂z θ¯
product on R2 or R3.
Relative to the pairings (3.8) and (3.6) we get
δ(cid:96) δ(cid:96)
= u and = −c π¯, (3.11)
p
δu δθ
so that the Euler-Poincar´e equations (3.9) read ∂ u + u · ∇u + ∇uT · u − c π¯∇θ = −∇p, in
t p
terms of the pressure p. To permit a comparison of these anelastic equations with those given
in the standard form of (2.7) in terms of Exner pressure π(cid:48), we note that ∇uT·u = 1∇|u|2 and
2
that −c π¯∇θ differs from −gθ(cid:48)z by a gradient term, indeed:
p θ¯
θ θ(cid:48)
−c π¯∇θ = −c ∇(π¯θ)+c (∇π¯)θ = −c ∇(π¯θ)−g z = −∇(c π¯θ+gz)−g z.
p p p p ¯ p ¯
θ θ
Therefore, with π(cid:48) defined in terms of p by the equality c θ¯π(cid:48) = p + 1|u|2 − gz − c π¯θ, the
p 2 p
Euler-Poincar´e equations yield the anelastic equations (2.7).
¯
2) Pseudo-incompressible equations. In this case we take σ¯(z) := ρ¯(z)θ(z) and the La-
grangian is given by
(cid:90) (cid:18) (cid:19)
1 1
(cid:96)(u,θ) = |u|2 −gz ρ¯θ¯µ, u ∈ X (D). (3.12)
θ 2 ρ¯θ¯µ
D
As before, the kinetic energy is computed relative to the standard inner product on R2 or R3.
Relative to the pairings (3.8) and (3.6) we get
(cid:18) (cid:19)
δ(cid:96) 1 δ(cid:96) 1 1
= u and = − |u|2 −gz , (3.13)
δu θ δθ θ2 2
so that the Euler-Poincar´e equations (3.9) read
(cid:18) (cid:19) (cid:18) (cid:19) (cid:18) (cid:19)
1 1 1 1 1
∂ u +u·∇ u +∇uT · u− |u|2 −gz ∇θ = −∇p.
t θ θ θ θ2 2
After some computations, using the relation −g = c ∂ π¯, these equations recover the pseudo-
θ¯ p z
incompressible system (2.8) with c (π¯ +π(cid:48)) = p+ 1(1|u|2 −gz).
p θ 2
Based on these results, we can formulate the following statement that will allow us to
derive the variational discretization of these two models by the discrete diffeomorphism group
approach.
Theorem 3.5 Consider a domain D with smooth boundary ∂D and volume form µ. The
anelastic system with reference density ρ¯, resp., the pseudo-incompressible system with refer-
¯
ence density ρ¯ and reference potential temperature θ can be derived from an Euler-Poincar´e
variational principle for the Lie group
G = Diff (D) resp. G = Diff (D), (3.14)
ρ¯µ ρ¯θ¯µ
with Lagrangian (3.10), resp., (3.12).
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