On geometry of the Lagrangian description of ideal fluids

0.16 MB·English

by  H. Gumral

On geometry of the Lagrangian description of ideal fluids
Hasan Gu¨mral
Department of Mathematics, Yeditepe University
Kayı¸sdag˘ı34750 I˙stanbul Turkey
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0 [email protected]
0 Nov 19, 2007
2

r
a Abstract The Euler equation for an inviscid, incompressible fluid in a three-
M
dimensional domain M in R3 implies that the vorticity is a frozen-in field. This can
beused to construct a symplectic structure on R×M. The normalized vorticity and
0
1 the suspended velocity fields are Hamiltonian with the function t and the Bernoulli
function,respectively. Thesymplecticstructureincorporatesthehelicityconservation

] law as an identity. The infinitesimal dilation for the symplectic two-form can be
h
p interpreted as a current vector for the helicity. The symplectic dilation implies the
- existence of contact hypersurfaces. In particular, these include contact structures on
h
thespaceofstreamlinesandtheBernoullisurfaces. ThesymplecticstructureonR×M
t
a can berealized as symplectisations of these through theEuler equation.
m
[

1
v
2
5 1 Introduction
4
1
The traditionaltheories for the problems offluid flow are statisticaland proba-
.
3
bilisticinnature[1],[2]. ThegeometricdescriptionsoftheEulerianequationsof
0
hydrodynamics in the frameworkof infinite dimensional Lie groups and Hamil-
8
0 tonian (Lie-Poisson) formulations [3]-[7] has been attracted the attention of
: scientists working on different aspects of fluid motions. On the other hand, the
v
need for some geometry, more specifically, the necessity of the symplectic ge-
i
X ometryfortacklingsomeopenproblemsofthe Eulerequationwasargued. The
r extensionofmethodsofsymplecticgeometrytoviscousflowswhichisdescribed,
a
inEuleriancoordinates,by the Navier-Stokesequation,wasalsoquestioned[2].
The bottomline ofthese argumentsmaybe the questionofincorporatingthose
geometric structures appropriate to and available from the Eulerian equations
into the traditional statistical descriptions of fluid flows [8]-[11],[1],[2].
Meanwhile, the relevance of geometric structures obtainable from the Eule-
rian equations to the statistical description of turbulence was discussed in Ref.
[1]. The Lagrangian description, that is, the description by the trajectories of
the velocity field of steady fluid motion in two and four dimensions was shown
to have the structure of a completely integrable Hamiltonian system [10],[11].
In [12], starting solely from the Euler equationof an ideal fluid we obtained
an infinitesimal symmetry, a symplectic structure on R×M for the suspended
1
velocity field and the conservation law for the helicity density. In [13] and [14],
we extended these intrinsic geometric structures to the study of kinematical
symmetries and generalized helicity conservation laws of three dimensional in-
compressible flows. In this work, we shall further exploit the Euler equation of
idealfluids to obtaingeometric structures relevant to a qualitative study of the
Lagrangiandescription of motion.
We shall introduce a current vector field governing the dynamics of the he-
licity density. This helicity current generates the dilation for the symplectic
structure. With reference to the Navier-Stokes equation we shall demonstrate
that the information content of the dynamical equations can be representedby
the helicity current. We shall also discuss some algebraic consequences of the
symplectic dilation. We shall then construct contact structures on the space of
streamlines and on the Bernoulli surfaces. We shall show that the symplectic
structure on the space of trajectoriesis symplectisations of these contact struc-
tures. For fluid dynamical content of this work we shall refer to Refs. [2],[15]
and the necessary mathematical background can be found in Refs. [16]-[23].
2 An infinitesimal symmetry
We shall begin with the Euler equation of ideal fluids
∂v
+v·∇v=−∇p (1)
∂t
for the divergence-free velocity field v tangent to the boundary of a connected
region M ⊂ R3 and the pressure function p. The identity v·∇v = 1∇|v|2−
2
v×(∇×v) can be used to bring the Euler equation (1) into the Bernoulli’s
form
∂v
−v×(∇×v)=∇α (2)
∂t
wherethefunction−α≡p+v2/2istheBernoullifunction[11],alsocalledtobe
the total (or stagnation) pressure [24]. In terms of the divergence-free vorticity
field w≡∇×v Eq. (2) gives
∂w
−∇×(v×w)=0. (3)
∂t
It follows from the identity
∇×(v×w)=(w·∇)v−(v·∇)w+(∇·w)v−(∇·v)w (4)
together with ∇·v=∇·w =0 that Eq. (3) is equivalent to
∂w
+[v,w]=0, [v,w]≡(v·∇)w−(w·∇)v (5)
∂t
whichmeansthatwisaninfinitesimaltime-dependentsymmetryofthevelocity
field v. That is, the time-dependent transformations generated by w on M
leaves the trajectories of the velocity field invariant.
2
3 A symplectic structure
In R3 a vector field w corresponds to a unique two-form
ω ≡w·(dx∧dx)=d (v·dx) (6)
M
where we let d ≡dx·∇ be the exteriorderivative onR3. If we replace d by
M M
the exteriorderivatived≡dt∂ +d onR×M anduse the Euler equation(2)
t M
to solve the time derivative v of the velocity field we obtain the two-form
,t
Ω =w·(dx∧dx)−(v×w+∇α)·dx∧dt, (7)
e
on R×M. By construction, Ω is closed on the space of solutions of the Euler
e
equation. In other words, the three-form
dΩ =(∇·w)·(dx·dx∧dx)+[w −∇×(v×w)]·dx∧dx∧dt (8)
e ,t
vanishes for divergence-free vector field w satisfying the Euler equation in the
vorticity form (3). Thus, we have the closed two-form Ω on R × M as an
e
extension
ω =d (v·dx)→Ω =d(v·dx)modEq.(2)
M e
ofω onM bytheEulerequationandthisisinducedbytheinvariantdifferential
operators
d →d≡dt∂ +d .
M t M
Under the condition that the Bernoulli function α is not the constant func-
tion the skew-symmetric matrix of the two-form Ω has a non-vanishing deter-
e
minant. Equivalently, this can be expressed as
1
Ω ∧Ω =−w·∇αdx·dx∧dx∧dt6=0 (9)
e e
2
whenever α 6= constant. It seems to be necessary to assume in addition that
w6=0. However,for a realisticfluid the validity ofthis conditionwas discussed
inRef. [25]. We thusobtainedfromthe Eulerequationanon-degenerateclosed
two-form Ω on R×M, that is, a symplectic structure [19]-[21],[23],[16]. The
e
non-zerofour-formin Eq. (9) is the symplectic or the Liouville volume element
on R×M. The symplectic two-form is exact
Ω =dθ modEq.(2), θ =v·dx+ψ(t)dt (10)
e e e
where ψ(t) is an arbitrary function.
4 Hamiltonian vector fields
The non-degeneracy of Ω means that given a one-form β ≡ β dxa on R×M
e a
with the local coordinates (xa)=(x0 =t,x) the equation
i(X)(Ω )=β , (Ω ) Xb =β (11)
e e ab a
3
has a unique solution for the vector field X = Xa∂ = X0∂ +X·∇ and vise
a t
verse. Here, i(X)(·) denotes the interior product or the contraction with the
vector field X [18], (Ω ) are the components of the skew-symmetric matrix
e ab
of the symplectic two-form Ω in the given coordinates and we employ the
e
summation over repeated indices.
For β being an exact one-form, namely, β = df for an arbitrary smooth
function f on R×M the solution
1 ∂ df
X = [w·∇f( +v·∇)− w·∇+(∇α×∇f)·∇] (12)
f w·∇α ∂t dt
ofEq. (11)issaidtobetheHamiltonianvectorfieldforΩ withtheHamiltonian
e
function f. From the skew-symmetry of (Ω ) we have the conservation law
e ab
0≡i(X )i(X )(Ω )=(Ω ) XaXb =Xaf =X0f +X·∇f =0 (13)
f f e e ab ,a ,t
fortheHamiltonianfunction. WeobservethatX reducesto−(w·∇α)−1w·∇
f
for the function f = t which obviously satisfies Eq. (13). In other words, the
normalized vorticity field is Hamiltonian
i(X )(Ω )=dt, X ≡−(w·α)−1w·∇ (14)
t e t
with the Hamiltonian function t.
A time-dependent conserved function for the velocity field can be found
again from the Euler equation. We recall that an energy consequence of the
Euler equation follows by taking dot product of its Bernoulli form with the
velocity field. The result is known as the Bernoulli equation [2],[24]
∂ 1
( v2)−v·∇α=0 (15)
∂t 2
which implies that if the pressure p does not depend explicitly on time, the
Bernoulli function α is a time-dependent conserved function
∂α 1
+v·∇α=0, −α(t,x)=p(x)+ v2(t,x) (16)
∂t 2
along the trajectories of the velocity field. Using this as the Hamiltonian func-
tionwecanwritethesuspendedvelocityfield∂ +v onR×M asaHamiltonian
t
vector field
i(∂ +v)(Ω )=dα. (17)
t e
The symplectic structure Ω onR×M induces a Lie algebraicstructure onthe
e
space of smooth functions on R×M with the Poisson bracket
∂f ∂g
{f,g} ≡ (Ω−1)ab = X (g) = Ω (X ,X ) (18)
e ∂xa e ∂xb f e f g
1 dg df
= [ w·∇f − w·∇g+(∇f ×∇g)·∇α] (19)
w·∇α dt dt
where(Ω−1)ab arethecomponentsoftheinverseofthematrixofthesymplectic
e
two-form and d/dt=∂/∂t+v·∇ is the convective derivative.
4
5 Conservation laws
We shall now discuss the differential invariants of the velocity field and obtain
theconservationlawforthehelicityasanidentityinvolvingtheinvariantforms.
Tothisend,werecallthatadifferentialp−formξissaidtobearelativeinvariant
for a vector field X if there exist a (p−1)−form ζ such that
L (ξ)≡i(X)dξ+di(X)ξ =dζ (20)
X
where L (·) is the Lie derivative. If ζ =0, ξ is said to be an absolute invariant
X
[21],[18]. Itimmediatelyfollowsfromthisdefinition, Eq. (17)anddΩ =0that
e
the symplectictwo-formisanabsoluteinvariantofthesuspendedvelocityfield.
In fact, this is true for any Hamiltonian vector field, so is for the normalized
vorticity field X . For the canonical one-form we compute the Lie derivative
t
1
L (θ )=dα+d(v2+ψ(t))=d(ψ(t)+ v2−p) (21)
∂t+v e 2
usingtheHamilton’sequation(17)forthefirstterminthedefinition(20). Thus,
θ isarelativeinvariant. Moreover,thederivationpropertyoftheLiederivative
e
implies the relative invariance
1
L (θ ∧Ω )=L (θ )∧Ω +θ ∧L (Ω )=d((ψ(t)+ v2−p)Ω ) (22)
∂t+v e e ∂t+v e e e ∂t+v e 2 e
of the three-form
θ ∧Ω = 2Hdx·dx∧dx+
e e
((v2+ψ)w−2Hv+v×∇α)·dx∧dx∧dt (23)
where the scalar component, namely, the coefficient of the term dx·dx∧dx, is
the helicity density
1 1
H≡ v·∇×v= v·w. (24)
2 2
Theabsoluteinvarianceofthefour-formΩ ∧Ω followsfromthesameargument.
e e
Moreover, from the right hand side of Eq. (9) we conclude that the absolute
invarianceofΩ ∧Ω isastatementforthe conservationofthe Liouvilledensity
e e
∂
(w·∇α)+v·∇(w·∇α)=0 (25)
∂t
along the trajectories of the velocity field.
Since Ω is closed and dθ = Ω these differential invariants of the velocity
e e e
field satisfy the relation
d(θ ∧Ω )−Ω ∧Ω ≡0 (26)
e e e e
identically. Using the expression (9) for the second term and computing the
derivative of the last term in the three-form (23) as
d[(v×∇α)·dx∧dx∧dt]=w·∇αdx·dx∧dx∧dt (27)
5
we obtain the divergence expression
∂H 1 1
+∇·(Hv− ( v2−p)w)=0 (28)
∂t 2 2
for the conservation law of the helicity density H. Note that the conserved
flux, which is the vector field under the spatial divergence, is independent of
the function ψ which we have introduced arbitrarily. We can take it to be
zero. Then, it immediately follows from Eqs. (21) and (22) that for v2/2−p=
constant,θ and θ ∧Ω become absolutelyinvariant. Moreover,fromEq. (28)
e e e
we see that on the level surfaces of the function v2/2−p the density H turns
into a conserved function of the velocity field.
We thus obtained the conservation law for the helicity density H≡v·w/2
associatedwiththevorticityfieldwwhichwasshowntobeaninfinitesimalkine-
maticalsymmetry. Thismeansthat,although,theconservationlawisgivenasa
divergenceexpressionthedensityisaconservedquantityoftheLagrangianmo-
tion. The symplectic structureΩ has aprominantroleinrelatingthe vorticity
e
field w to the helicity invariant.
We conclude that, apart from the assumption on the pressure function, the
Euler equation itself intrinsically contains a geometric structure in which we
can relate symmetries, invariant differential forms and conservation laws. In
the next section we shall examine the situation for the Navier-Stokes equation.
Weshallthencontinueourdiscussionofgeometricstructuresavailablefromthe
Euler equation.
6 Viscous flow
The Navier-Stokes equation for a viscous incompressible fluid in a bounded
domain M ⊂R3 is
∂v
+v·∇v=−∇p+ν∇2v (29)
∂t
where ν is the kinematic viscosity [2]. As in the case ofthe Euler equation, Eq.
(29) can be brought into the form
∂w
−∇×(v×w)=ν∇2w (30)
∂t
by which the two-form
Ω =w·(dx∧dx)−(v×w−ν∇×w)·dx∧dt (31)
ν
can be shown to be closed. For non-degeneracy we compute
1
Ω ∧Ω =2νH dx·dx∧dx∧dt6=0 (32)
ν ν w
2
where the density
1 1
H ≡ w·∇×w=− w·∇2v (33)
w
2 2
6
is known as the vortical helicity [2]. For viscous flows ∇2v 6= 0 and, for a
realistic fluid we have w 6= 0. Hence H 6= 0. This makes Ω non-degenerate.
w ν
Ω is also exact with the same canonical one-form θ
ν e
Ω =dθ modEq.(29) (34)
ν e
whichshowsthatthesymplecticstructureonthespaceoftrajectoriesmanifests
the dynamical properties of the fluid.
For an arbitrary smooth function f on R×M the Hamiltonian vector field
X defined by the symplectic two-form Ω is
f ν
1 ∂ df
X = [−w·∇f( +v)+ w+ν∇f ×∇×w)·∇] (35)
f
2νH ∂t dt
w
which gives (1/2νH )w·∇ for f =t. The suspended velocity field is, however,
w
not even locally Hamiltonian. These are vector fields obtained from Eq. (11)
for closed non-exact one-forms β. The symplectic two-form is invariant under
the flows of locally Hamiltonian vector fields because L (Ω ) = di(X)(Ω ) =
X ν ν
dβ ≡0whereweusedtheidentityL =i(X)◦d+d◦i(X)fortheLiederivative
X
and dΩ =0. We find, however,that the one-form
ν
i(∂ +v)(Ω )=ν∇2v·(dx−vdt) (36)
t ν
obtainedbycontractionwiththesuspendedvelocityfieldisnotevencloseddue
todiffusiveterm. Thatmeansthesymplectictwo-formisnotinvariantunderthe
Lagrangianmotion. Aconsequenceofthisnon-invarianceor,equivalently,ofthe
viscous diffusion is the non-conservation of the helicity density. Nevertheless,
using the symplectic form (31) and θ in the identity (26) we get
e
∂H 1
+∇·(Hv+νv×∇2v− v2w)=−2νH (37)
w
∂t 2
as the evolution equation for H.
These are properties of the Lagrangian motion reflected by the dynamical
(or Eulerian) equations. We shall continue our discussion of the geometry of
the Lagrangianmotion in the framework of the Euler equation. We shall occa-
sionally present the results for the Navier-Stokes equation in order to see how
the dynamics is encoded into geometric objects.
7 A symplectic dilation
For our discussion of the geometric structure of the Lagrangian motion of the
Euler flow a particularly interesting solution of Eq. (11) is obtained when we
let the one-form β be the canonical one-form θ . The vector field J satisfying
e
the equation
i(J)(Ω )=θ (38)
e e
7
can be uniquely determined to be
1
J = [2H(∂ +v·∇)+(ψ(t)−v2)w·∇+v×∇α·∇] (39)
w·∇α t
for the Euler equation and, for the Navier-Stokes equation we find
−1 1
J = [2H(∂ +v)+(p− v2)w−νv×∇×w·∇]. (40)
ν t
2νH 2
w
It follows from Eq. (38) and dΩ =0 that J fulfills the condition
e
L (Ω )=di(J)(Ω )=dθ =Ω (41)
J e e e e
of being an infinitesimal symplectic dilation for Ω [26]. As a consequence of
e
Eq. (41)andthederivationpropertyoftheLiederivativeweseethatJ expands
the Liouville volume in Eq. (9). J is also saidto be the Liouville vector field of
Ω [21].
e
WeshallnowgiveaninterpretationofthesymplecticdilationJ inconnection
with the helicity conservation. We first compute its divergence with respect to
the symplectic volume [18],[19],[21],[23]
1 1
(div J)( Ω ∧Ω )=L ( Ω ∧Ω )=Ω ∧Ω (42)
Ωe 2 e e J 2 e e e e
where we used Eq. (41). By the same equation we see that the second equality
is the identity in Eq. (26) resulting in the helicity conservation. Thus, the
conservation law (28) can be written as the equation
div J −2≡0 (43)
Ωe
involving the symplectic-divergence of the symplectic dilation. With this inter-
pretationweintendtocallJ tobe the currentassociatedwiththe helicity. The
dynamical content of the helicity current can be revealedfrom a comparisonof
the symplectic structures obtained from the Navier-Stokes and the Euler equa-
tions. The canonical one-forms are the same. So, the dynamics is encoded into
the symplectic two-forms. They define the current vectors by Eq. (38) for the
same canonical one-form. With this definition, the dynamical properties of the
fluid, such as viscosity, become implicit in the helicity current. Thus, we can
think of the pair (θ ,J) as a geometric representative of the dynamics of fluid
e
motion on the space of trajectories.
8 Algebraic consequences
The helicity current is not a Hamiltonian vector field. However, it takes a
Hamiltonian vector field into a Hamiltonian vector field by its action via Lie
derivative. To see this, we compute
i([J,X ])(Ω ) ≡ L (i(X )(Ω ))−i(X )(L (Ω )) (44)
f e J f e f J e
= −d(J(f)−f) (45)
8
where Eq. (44) is an identity [23],[18] and we used Eq. (41). Thus, [J,X ] is
f
Hamiltonianwiththe functionJ(f)−f. ReplacingX with [J,X ] inEq. (44)
f f
and using Eq. (45) we get
i([J,[J,X ]])(Ω ) ≡ L (i([J,X ])(Ω ))−i([J,X ])(L (Ω )) (46)
f e J f e f J e
= −d(J(J(f))−2J(f)+f) (47)
which is also Hamiltonian. Thus, by repeated applications of the Lie derivative
with respect to the symplectic dilation J one cangenerate an infinite hierarchy
of Hamiltonian vector fields (L )k(X ) anchored to X .
J f f
In particular,we let f =t so that X ≡−(w·∇α)−1w·∇ and consider the
t
Hamiltonian vector fields (L )k(X ) for k ≥0. The identity [23],[18]
J t
L ≡L ◦L −L ◦L (48)
[J,Xt] J Xt Xt J
evaluated on the suspended velocity field ∂ +v gives
t
L (∂ +v)=−L ([J,∂ +v]) (49)
[J,Xt] t Xt t
where we used [∂ +v,X ]=0. The vector field [J,∂ +v] is Hamiltonian with
t t t
the function J(α)−α= ψ−v2. So, we can express the right hand side of Eq.
(49) as a Hamiltonian vector field. To find the Hamiltonian function we shall
use the Lie algebra isomorphism
[X ,X ]=−X (50)
f g {f,g}
defined by the symplectic structure Ω [23],[21]. Taking f = t and g = ψ−v2
e
we compute the Poissonbracket
w·∇v2
{t,ψ−v2}=X (ψ−v2)= (51)
t w·∇α
whichgivestheHamiltonianfunctionuptoaconstant. ItfollowsfromEq. (49)
that on the level surfaces defined by the constant values of the Hamiltonian
function (51) of L ([J,∂ +v]) we have [[J,X ],∂ +v]=0. Since w 6=0, this
Xt t t t
meansthat w lies onthe two-parameterfamily p+av2/2=b,a,b=constants,
of surfaces. In this case, the hierarchy of Hamiltonian vector fields anchored
to X becomes infinitesimal symmetries of the velocity field. In particular, this
t
family of surfaces includes the Bernoulli surfaces for a= 1. In other words, on
the level sets −α = b of the Hamiltonian function, ∂ +v admits the infinite
t
hierarchy (L )k(X ) of time-dependent infinitesimal symmetries.
J t
9 Contact structures
Acontactstructureonathreedimensionalmanifoldisafieldofnon-integrable,
two-dimensional hyperplanes in its tangent spaces. Locally, this may be de-
scribed as the kernel of a one-form ω satisfying ω ∧dω 6= 0 everywhere. The
contact form ω determines a unique vector field E by the conditions
i(E)(ω)=1, i(E)(dω)=0 (52)
9
which is called the Reeb vector field. [20]-[22].
A geometric consequence of the existence of a symplectic dilation is that
the canonical one-form θ can be related to a contact structure on some three
e
dimensional domain in R×M because from Eq. (41) we have
1
θ ∧dθ =θ ∧Ω =i(J)(Ω )∧Ω =i(J)( Ω ∧Ω )6=0. (53)
e e e e e e e e
2
According to the definition of Ref. [28] a hypersurface in a symplectic man-
ifold admits a contact one-form if and only if there exists a symplectic dilation
which is defined on its neighborhood and is transversal to the hypersurface.
It is also remarked that such hypersurfaces arise as level sets of Hamiltonian
functions of Hamiltonian vector fields [26]-[28],[22]. In this section we shall de-
scribe two examples of contact hypersurfaces in R×M. In the next section we
shall consider their relation with the symplectic structure Ω and discuss some
e
physical significance.
9.1 Spatial hypersurfaces
First, we consider spatial hypersurfaces M = {(t,x) ∈ R × M | t = c =
c
constant}aslevelsetsofthefunctiont. RecallthattistheHamiltonianfunction
for the normalized vorticity field X . Then, the transversality condition
t
2H
i(J)(i(X )(Ω ))=i(J)(dt)= 6=0 (54)
t e w·∇α
for M holds for non-vanishing values of the helicity density. In other words,
c
for H=6 0 the helicity current is not contained in the tangent spaces to M . In
c
this case, the contact form on M is obtained as follows. Let
c
i:M →R×M :(t=c,x)7→(t,x) (55)
c
be the inclusion of time slices M into space-time. A function on R×M gives
c
a functiononM when composedby i. This operationcanthen be extendedto
c
differential forms. If σ =f (x)dxa is a one-form on R×M its pull-back i∗σ to
a
M by the inclusion map i is defined to be
c
i∗σ =i∗(f (x)dxa)=i∗(f (x))i∗(dxa)=(f ◦i)(x)di(xa) (56)
a a a
where we used the commutativity of the operators d and i∗ [18]. In particular,
we compute
ω ≡ i∗θ = v(t=c,x)·dx (57)
e
dω = di∗θ = i∗dθ =i∗Ω =w(t=c,x)·dx∧dx (58)
e e e
and it follows that ω is a contact form on M
c
ω∧d ω =i∗θ ∧i∗Ω =i∗(θ ∧Ω )=2Hdx·dx∧dx6=0 (59)
M e e e e
10