6.1. Geometric Description
We start with considering a quasi-neutral superfluid contained in a domain
and interacting with a “frozen” sourceless magnetic field
satisfying the superconductivity conditions
where
is the internal net superfluid electric field,
and
are the internal electric and magnetic fields, respectively, generated by the corresponding magnetic vector field potential
is the superfluid velocity and “×” denotes the usual vector product in the Euclidean space
The following natural boundary conditions
and
are imposed on the superfluid flow, where
is the vector normal to the boundary
which is considered to be almost everywhere smooth.
Then in adiabatic magnetohydrodynamics (MHD) quasi-neutral superconductive superfluid motion is described by the following system of evolution equations:
where, as before,
is the superfluid density,
is the “frozen” into the superfluid magnetic field,
is the internal liquid pressure and
is the specific superfluid entropy at time
The latter is related with the internal MHD superfluid specific energy function
owing to the first thermodynamic law:
satisfied for any admissible variations of the phase space parameters
where
is the internal absolute temperature in the superfluid for
The isentropic condition
where
for all
and the related to (
55) evolution diffeomorphism
entails the following expression for the specific internal energy
where
is the corresponding internal potential specific energy density and
is some still unknown function, depending in general on the imposed magnetic field
Let us now analyze, as before, the mathematical structure of quantities
as the
physical observables subject to their evolution (
55) with respect to the group diffeomorphisms
generated by the liquid motion vector field
where
denotes the corresponding Lie derivative with respect to the vector field
The relationships (
58) mean that the space of physical observables, being by definition, the adjoint space
to the extended configuration space is equal to
the tangent space at the identity
to the extended differential-functional group manifold
where we have naturally identified the abelian group product
with its direct tangent space sum
Consider now the constructed differential-functional group manifold
in Eulerian variables, on which one naturally acts the
-group
the standard way:
for
and any
Then, taking into account the suitably extended action (
59) on the differential-functional manifold
one can formulate the following easily checkable and crucial for what will follow further proposition.
Proposition 5. The differential-functional group manifold in Eulerian coordinates is a smooth symmetry Banach group equal to the semidirect product of the diffeomorphism group and the direct product of abelian functional density and one-form groups, endowed with the following group multiplication law in Eulerian variables:for arbitrary elements and Thus, one can proceed to studying the corresponding coadjoint action of the Lie algebra
on the adjoint space
As the Lagrangian configuration
and the entropy
are assumed to be invariant under the Banach diffeomorphism group action
the resulting group action can be reduced to the factor-group
action on the semidirect group product
Based on the multiplication law (
60) one easily calculates the following Lie algebra commutation relationships:
for any elements
and
The adjoint space to the semidirect product Lie algebra
can be, naturally, written symbolically as the space
where as before, the mapping
denotes the Hodge isomorphism. Then, taking into account the adjoint space
to the Lie algebra
is endowed with the following [
5,
6,
19,
33,
44,
45] canonical Lie-Poisson bracket
for any smooth functionals
on the adjoint space
where, as before, we denoted by
the specific momentum of the superfluid. The bracket (
62) naturally ensues from the canonical symplectic structure on the cotangent phase space
as it was before demonstrated in
Section 4.
Write down now the first two equations of the Euler MHD system (
55) as the local fluid mass and momentum conservation laws in the integral Ampere–Newton form
which is completely equivalent to the relationships (
58) and where
is the net internal superfluid pressure,
is the spatially distributed Lorentz force on the superfluid,
is the respectively oriented surface measure on the boundary
for the domain
and a smooth submanifold
D is chosen arbitrary. Taking into account that
for any
the second integral relationship (
63) becomes equivalent to the following:
where we have represented the internal superfluid pressure quantity as
for some mapping
strictly depending only on the internal liquid configuration
for all
Based on the Poisson bracket expression (
62), one can now easily determine the Hamiltonian function
corresponding to the Euler evolution Equation (
55) on the adjoint space
where the quantity
denotes the specific internal superfluid energy, modified by means of the “frozen” magnetic field
replacing the before defined in
Section 3 internal specified potential energy
by the shifted specified potential energy quantity
In particular, the Equation (
64) reduces to the equivalent Hamilton expression
for
and all
It is also seen that if
uniformly with respect to time
the internal energy expression (
67) brings about that (
40). Recall now that the following quasi-stationary second thermodynamic energy conservation law
holds for all admitted superfluid variations
and
As, by isentropic assumption,
for all
along fluid streamlines, for the internal pressure one easily obtains the expression
exactly coinciding with that of (
65).
The Hamiltonian function (
66) satisfies evidently the conservation condition
for all
To check this directly, it is enough to observe [
33] that the following differential relationship
holds for all
and whose integration over the domain
easily gives rise to the conservation of the Hamiltonian function (
66).
6.2. Magneto-Hydrodynamic Invariants and Their Geometry
The importance of spatial invariants describing the stability [
33] of MHD superfluid motion was previously stated long ago [
32,
33,
36,
46]. Based on the modern symplectic theory of differential–geometric structures on manifolds, we devise a unified approach to study MHD invariants of compressible superfluid flow, related with specially constructed symmetry structures and commuting to each other vector fields on the phase space.
We start from a useful differential-geometric observation that the magneto-hydrodynamic Euler equations
action on the adjoint space to the Lie algebra
of the Banach group
generated by the following vector field differential relationship:
where
and
is an acceptable time-dependent vector field on the domain
describing the adiabatic superfluid and superconductive motion via the diffeomorphism subgroup mappings
Taking into account that the initial superfluid configuration
is fixed, one can define, following reasonings from [
47], a new differential relationship
on the domain
M with respect to the evolution variable
parameterized by the time parameter
where
is a
-independent vector field on
generating the diffeomorphism subgroup
commuting to that generated by the vector field (
71), i.e.,
for all
The action of the diffeomorphism subgroup
at any fixed time
can be naturally interpreted as rearranging the particle configurations in the superfluid not changing its other dynamic characteristics. If to denote the corresponding Lie derivatives with respect to the vector fields (
71) and (
72) by differential expressions
and
the commutation condition
for all
is equivalently rewritten as the operator commutator
Consider now an arbitrary integral invariant of the MHD superfluid, governed by the Euler system (
55):
generated by some specific density functional
and held over the domain
for any domain
corresponding to the diffeomorphism subgroup
generated by flow (
71). Taking into account that there holds the following density relationship
for any
one easily derives from (
74) and (
75) that also
for any
Thus, based on the commutation relationship (
73) one can formulate the following important lemma.
Lemma 1. Let vector fields (71) and (72) commute to each other and a density functional satisfies for all the conditionthen the following expressionsover the domain generated by the corresponding to the flow (71) diffeomorphism subgroup and arbitrary domain are for all integers the MHD invariants of the superfluid flow (55). Proof. A proof easily follows from the commutation condition (
73) and the superfluid density relationship (
75). □
As examples let us take, following [
33,
47], the vector field
commuting to the vector field
and
where the magnetic vector potential
satisfies the classical Maxwell relationships: the magnetic field
and the electric field
owing to the net electric field superconductivity (
54) condition
Really, the commutativity condition (
73) means that
which is satisfied, owing to the second and forth equations of the Euler MHD system (
55), as well as to the invariance
which holds owing to the algebraic relationship
commutativity of vector fields
and
and the integral relationship
equivalent to the condition
for all
The same statement we obtain from the slightly simpler reasoning:
following from the net electric field
superconductivity condition (
54) along the boundary
where
is the surface, generated by the diffeomorphism subgroup
and an arbitrarily chosen surface
The latter is, evidently, equivalent to the equality
modulo the gauge transformation
where
for some function
and all
Thus, one can formulate [
33,
47] the following proposition.
Proposition 6. The functionalsover the domain generated by the corresponding to the flow (71) diffeomorphism subgroup and arbitrary domain are for all integers the MHD invariants of the superfluid flow (55). In particular, the following relationships hold for all Remark 1. It is natural here to mention [33,35] that the specific entropy functional → satisfies the sufficient condition a priori generates for the superfluid flow (55) the infinite hierarchy of the MHD invariants over the domain generated by the corresponding to the flow (71) diffeomorphism subgroup and arbitrary domain To construct other MHD invariants, depending on the superfluid velocity
let us consider, following [
47], two differential one-forms
satisfying for all
the following identity:
for some function
where the vector field
is uniform with respect to the evolution parameter
and satisfies the following constraints:
and
at almost all points
for all evolution parameters
Then one can formulate the following general proposition.
Proposition 7. The following integral expressionsover the whole domain are for all integers the global MHD invariants. Proof. Consider, for example, a proof that
is an invariant: taking into account that
one obtains the expression:
for all
where we put, by definition,
denoted
the surface measure on the boundary
used the Cartan representation
and the natural boundary tangency condition
thus proving the proposition. Exactly similar calculations ensue for the next two invariant
on which we will not stop here. □
As a simple example, one can put
the vector field
and show by easy calculations, using the variational equality (
56) that
where, we have denoted the specific enthalpy [
41,
42,
43] function
As a consequence of equality (
91), under the spatial temperature constancy
condition for all
one obtains the following MHD superfluid invariant:
where
and
coinciding with the MHD invariant, presented before in [
33,
47]. If the above temperature condition does not hold, the equality (
91) reduces to the differential relationship
satisfied for all
and
Remark 2. It is worth to remark here that the following baroclinic relationshipholds for all and Similarly we also easily obtain the following invariant
coinciding exactly with the Hamiltonian function for the flow (
55) on the phase space
The third invariant is, eventually, closely related to the vorticity vector
and needs a more detail analysis.
It is instructive now to analyze the existence of integral invariants for the pure hydrodynamic case when the magnetic field
following the approach, devised before in [
47]. In particular, owing to the relationship (
94), there holds the following integral expression for the vorticity
and define the vector field
for some scalar smooth mapping
which we will choose from the assumed commutation condition
The latter gives rise to the equality
at any
or
where we took into account that
with respect the temporal parameter
From (
99) one obtains that the mapping
should satisfy the following constraints:
for some scalar smooth functions
and
It is easy to check that the system (
100) is compatible, i.e., the quasi-stationary thermodynamic relationship
jointly with Euler Equation (
10) make it possible to determine these unknown scalar smooth functions
and
for all
Consider now, following [
47], a slightly modified expression (
91) at the magnetic field
and calculate the related integral expression:
where we put, by definition, the function
If now to put that the mapping
satisfies for all
the constraint
the integral expression (
102) reduces to
where there is assumed the vorticity vector tangency
constraint. Thus, under conditions assumed above, the following vortex functional
persists to be conserved for all
If the function
being defined by relationships (
100), satisfies for all
the scalar constraint
one easily derives the following differential relationship:
or, equivalently, in the integral form
where we took into account that for the isentropic fluid flow under regard there holds the tangency
condition for all
If the right hand side of (
106) proves to be zero, i.e.,
this will mean that the constraint
for all
if
at
thus producing the vortex conservation functional (
104).