A geometric theory of selective decay in fluids with advected quantities

Transcription

1 A geometric theory of selective ecay in fluis with avecte quantities arxiv: v2 [physics.plasm-ph] 20 Oct 2018 François Gay-Balmaz 1 an arryl. Holm 2 PACS Numbers: Geometric mechanics Yy; Hamiltonian an Lagrangian mechanics Jj f; Fluis mathematical formulations A- Abstract Moifications of the equations of ieal flui ynamics with avecte quantities are introuce that allow selective ecay of either the energy h or the Casimir quantities C in the Lie-Poisson formulation. The issipate quantity (energy or Casimir respectively) is shown to ecrease in time until the moifie system reaches an equilibrium state consistent with ieal energy-casimir equilibria namely δ(h + C) = 0. The result hols for Lie-Poisson equations in general inepenently of the Lie algebra an the choice of Casimir. This selective ecay process is illustrate with a number of examples in 2 an 3 magnetohyroynamics (MH). Contents 1 Introuction Parameterizing subgriscale effects on macroscales Summary of key equations in Gay-Balmaz an Holm [2013] Comparison with previous approaches General energy issipative systems ouble-bracket formulations Comparison with ouble-bracket issipation Comparison with the triple bracket formalism Lagrange- Alembert variational principle Kelvin-Noether theorem Example: the rigi boy Laboratoire e Météorologie ynamique École Normale Supérieure/CNRS Paris France. gaybalma@lm.ens.fr 2 epartment of Mathematics Imperial College Lonon SW7 2AZ UK..holm@ic.ac.uk 1

2 Gay-Balmaz an Holm Casimir issipation in fluis with avecte quantities 2 2 Selective ecay on semiirect proucts Semiirect proucts Convergence to steay states of the unmoifie LP equations Examples Heavy top incompressible MH A general class Main Example: Selective ecay for MH Selective ecay for three-imensional MH homogeneous incompressible MH compressible isentropic MH Conclusions 32 References 33

3 Gay-Balmaz an Holm Casimir issipation in fluis with avecte quantities 3 1 Introuction Historically the hypothesis of selective ecay in MH turbulence assume that the total energy was to be minimize subject to the conservation of certain ieal invariants. This hypothesis was consistent with the observe long term evolution of freely ecaying MH turbulence at high magnetic Reynols number R m = UL/η where U is a typical velocity scale of the flow L is a typical length scale of the flow an η is the magnetic resistivity. This situation was calle an inverse cascae Frisch Pouquet Leorat an Mazure [1975] because the energy flux is preominantly towar small scales while the flux of the ieal invariant known as the magnetic helicity passes towar the larger scales an ostensibly creates the spiral structures observe in the ecay of MH turbulence. See Matthaeus an Montgomery [1980]; Montgomery an Bates [1999] for further historical iscussion of the selective ecay hypothesis for MH turbulence. The aim of this paper is to evelop a geometric theory of selective ecay of either the energy or the Casimirs (the invariants of the Lie-Poisson bracket of the ieal theory) an illustrate it for the MH Hamiltonian structure following the work of Gay-Balmaz an Holm [2013] for geophysical flui ynamics. We interpret the resulting moifications of the ieal MH equations as a means of ynamically an nonlinearly parameterizing the interactions between isparate scales by introucing new nonlinear pathways to issipation base on selective ecay. Remarkably the theory evelope here for selective ecay of either the energy h or the Casimir C contains the stanar energy-casimir equilibria of the ieal equations obtaine from a critical point of the sum δ(h + C) = 0 Holm et al. [1985]. Thus our selective ecay theory is always consistent with the energy-casimir equilibrium conitions in that the energy-casimir equilibria are also equilibria of the moifie equations. However the presence of the selective ecay terms allows a new balance that enlarges the class of asymptotic states beyon those that satisfy the energy- Casimir equilibrium conitions associate with δ(h + C) = 0. In particular the geometric selective ecay process introuce here may in some cases ten towars states that satisfy only a subset of the energy-casimir equilibrium conitions. This is explaine in the proof of the main result Theorem 2.3 an is illustrate for the cases of compressible an incompressible MH in Section 3. Casimirs A Poisson manifol is a manifol P with a Poisson bracket { } efine on the space of smooth functions on P ; see e.g. Marsen an Ratiu [1994]. A Poisson system with Hamiltonian h : P R yiels the time-evolution of any smooth function f : P R by computing the solution curves of the ynamical equation f/ = {f h}. Poisson systems often arise by Lie-group reuction of Hamiltonian systems with symmetry on Lie groups in which case the Poisson bracket is calle a Lie-Poisson bracket. Examples inclue the Euler equations of ieal incompressible flui ynamics for which the symmetry group is the particle-relabeling group Arnol an Khesin [1998] an the equations for a heavy top for which the symmetry group is the Eucliean group Holm [2011]. efinition 1.1 (Casimirs). Casimirs on a Poisson manifol (P { }) are functions C that satisfy {C h} = 0 for all h that is they are constant uner the flow generate by the Poisson bracket for any choice of the Hamiltonian. The existence of Casimirs is thus ue to the egeneracy of the Poisson bracket. In the case of reuction by symmetry on Lie groups this egeneracy arises

4 Gay-Balmaz an Holm Casimir issipation in fluis with avecte quantities 4 when passing from the canonical Hamiltonian formulation in terms of Lagrangian variables to the Lie Poisson formulation in terms of symmetry-reuce variables. An example is the reuction of the Hamiltonian escription of rigi boy ynamics from the siximensional phase space T SO(3) of the Euler angles for SO(3) rotations to the three-imensional space of angular momenta in so(3) R 3. For ieal fluis the reuction is from the Lagrangian variables to the Eulerian variables which are invariant uner relabeling of Lagrangian particles. Thus Casimir conservation is a property of the Lie Poisson bracket that results from the reuction by symmetry not the choice of Hamiltonian. Inee the Casimirs commute uner the Lie Poisson bracket with any Hamiltonian that is expresse in terms of the symmetry-reuce variables. This means the motion generate by the Lie Poisson bracket in the symmetry-reuce variables takes place along intersections of level sets of the Hamiltonian h an a Casimir C. Casimirs have been use in stability analyses of flui an plasma equilibria which exten traitional energy methos to the energy-casimir metho Arnol [1969]; Holm et al. [1985]. This metho applies in etermining the stability of a certain class of equilibrium solutions p e P uner the Poisson flow f/ = {f h} efine on the Poisson manifol P an generate by a Hamiltonian h. Namely the energy-casimir metho supposes that there is a function C the Casimir which is constant uner the flow generate by the Poisson bracket (since {C h} = 0 for all h) an that the equilibrium solution p e is a critical point of the sum h C := h + C so that C = h C (p e ) δp = 0 for a nonegenerate pairing. Linear Lyapunov stability follows if the critical point p e is a local extremum of h C that is if δ 2 h C (p e ) is positive efinite or negative efinite. This conition implies formal stability since δ 2 h C (p e ) is conserve by the linearize equations aroun the equilibrium solution p e an if it is either positive efinite or negative efinite then it efines a norm for Lyapunov stability see Holm et al. [1984]; Holm et al. [1985]. Nonlinear Lyapunov stability follows by a further argument which is available when the functional h C is convex in the neighborhoo of p e. One may use the energy-casimir metho to seek stable equilibrium states. Such equilibrium states are calle stable energy-casimir equilibria. Because of the exchange symmetry of h C := h + C uner C h these equilibrium states may be regare as either extrema of the energy h on a level set of a Casimir C or vice versa as extrema of a Casimir C on a level set of the energy h. Aim of the paper The aim of the present paper is to introuce a issipative moification of the Lie Poisson flow whose ynamics will ten towar conitions that inclue the energy-casimir equilibria of the ieal unmoifie equations starting from any initial state on P. In particular the present paper investigates the effects of imposing selective ecay of a certain Casimir while preserving the energy an vice versa imposing selective ecay of energy while preserving the chosen Casimir. This is accomplishe by using the Lie Poisson structure of the ieal theory an interpreting the resulting moifications of the equations as nonlinear pathways for selective issipation that parameterize the observe effects of the interactions among isparate scales of motion. This type of moification is computable at a single time scale so it may be useful in situations where it woul be computationally prohibitive to rely on the slower inirect effects of viscosity an other types of iffusivity which typically affect both the energy an the Casimir. In particular the present paper takes the Casimir issipation approach of Gay-Balmaz an Holm [2013] further by applying it to 2 an 3 compressible an incompressible magnetohyroynamics (MH).

5 Gay-Balmaz an Holm Casimir issipation in fluis with avecte quantities 5 The new feature of the present paper is its introuction of moifie Lie Poisson equations that escribe the selective ecay of ieal fluis with avecte quantities. Mathematically ieal flows that avect flui properties such as mass heat an magnetic fiel may be escribe by the combine actions of Lie groups on their ual Lie algebras an also on the vector spaces in which the avecte quantities are efine Holm Marsen an Ratiu [1998]. The Casimirs for such flows iffer from the Casimirs of simple ieal flui motion. These ifferences introuce by the avective flow of flui properties yiel new flui equilibria an new nonlinear mechanisms for selective ecay of either energy or Casimirs. In particular the proof of Theorem 2.3 of the present paper shows that uner the geometric selective nonlinear ecay process introuce here the flow tens towar conitions which inclue an exten the class of energy-casimir equilibria of the unmoifie flui equations satisfying δ(h + C) = 0 for extrema of the energy h on a level set of a Casimir C or vice versa for extrema of a Casimir C on a level set of the energy h. These extrema may be either maxima or minima epening on the choice of sign of a parameter (θ) an the sign of the Casimir appearing in the moifie equations. Various types of moifications of the Poisson bracket for Hamiltonian systems have been propose in the literature in orer to inclue issipation. This is usually accomplishe by aing a symmetric bilinear form to the Poisson bracket as initiate by Kaufman [1984]; Morrison [1984]; Grmela [1984]; Morrison [1986]. Specific classes of energy issipation were introuce later in Brockett [1991]; Kanrup [1991]; Bloch et al. [1996]; Holm Putkaraze an Tronci [2008]; Broy Ellis an Holm [2008] by application of a ouble bracket. See also Vallis Carnevale an Young [1989]; Shepher [1990] in which a moification of the transport velocity was use to impose energy issipation with fixe Casimirs in an incompressible flui. As the paper procees we will comment further on the relationships of the present results with those of previous theories. The theory we evelop here uses the Lie-Poisson Hamiltonian framework for ieal fluis to treat either Casimir ecay at fixe energy or energy ecay at fixe Casimirs. That is the same theory is use here to treat selective ecay of either quantity so that the choice of which mechanism to investigate can be mae motivate for example by the effects seen in large-scale numerical simulations of the fully issipative equations. The switch from ecay of the chosen Casimir C at fixe Hamiltonian h to ecay of the Hamiltonian h at fixe Casimir C is accomplishe by a simple exchange C h in one of the key formulas e.g. in equation (1.5). In fact either type of selective ecay leas to the same equilibrium conitions. Selective ecay of Casimirs. Selective ecay of Casimirs is an effect that was first observe in numerical simulations of 2 incompressible turbulence cascaes by Matthaeus an Montgomery [1980] as the rapi ecay of the enstrophy (turbulence intensity) while the energy staye essentially constant. This observe isparity in the time scales for ecay of the energy an the enstrophy in 2 turbulence has a profoun effect on its energy spectrum. In Gay-Balmaz an Holm [2013] selective ecay by Casimir issipation was introuce by moifying the vorticity equation base on the well-known Lie Poisson structure of the Hamiltonian formulation for vorticity ynamics in the case of 2 incompressible flows of ieal fluis Arnol [ ]; Holm Marsen an Ratiu [1998]. In this framework the earlier work of Vallis Carnevale an Young [1989]; Shepher [1990] on selective ecay of energy at fixe values of the Casimirs was recovere by the exchange of the Casimirs an the Hamiltonian in formula (1.5) for the moifie vorticity ynamics. This earlier work stuie selective ecay for the purpose of fining stable equilibrium states. As we show here imposing selective ecay in flui flows with avecte quantities may also enlarge the

6 Gay-Balmaz an Holm Casimir issipation in fluis with avecte quantities 6 class of stable energy-casimir equilibrium states; see Theorem 2.3. In 3 incompressible flui turbulence the energy tens to ecay more rapily that the Casimirs o. This is contrary to selective ecay of turbulence in 2 so a ifferent moeling approach is require in 3. Thus a comprehensive theory must be capable of passing within the same framework from selective ecay of the Casimir in 2 to selective ecay of the energy in 3. This is accomplishe in the present theory by taking avantage of its exchange symmetry uner C h. 1.1 Parameterizing subgriscale effects on macroscales In a previous paper Gay-Balmaz an Holm [2013] the problem of parameterizing the interactions of isparate scales in flui flows was aresse by consiering a property of two-imensional incompressible turbulence. The property consiere was a type of selective ecay in which a Casimir of the ieal formulation (enstrophy in the case of 2 incompressible flows) was observe to ecay rapily in time compare to the much slower ecay of energy. That is the Casimir was observe to ecay while the energy staye essentially constant. The previous paper introuce a nonlinear flui mechanism that prouce the selective ecay by enforcing Casimir issipation at constant energy. This mechanism introuce an aitional geometric feature into the escription of the flow; namely it introuce a Riemannian inner prouct on the space of Eulerian flui variables. The resulting issipation mechanism base on ecay of enstrophy in 2 flows turne out to be relate to the numerical metho of anticipate vorticity iscusse in Saourny an Basevant [ ]. Several examples were given an a general theory of selective ecay was evelope that use the Lie Poisson structure of the ieal theory. A scale-selection operator allowe the resulting moifications of the flui motion equations to be interprete in these examples as parameterizing the nonlinear ynamical interactions between isparate scales. The type of moifie flui equations that was erive in the previous paper was also propose for turbulent geophysical flows where it is computationally prohibitive to rely on the slower inirect effects of a realistic viscosity such as in interactions between large-scale coherent oceanic flows an the much smaller eies. The selective ecay mechanism iscusse in the previous paper was base on Casimir issipation in the example of 2 incompressible flows treate as a ynamical parameterization of the interactions between isparate scales. Following that example the paper iscusse the general theory of selective ecay by Casimir issipation in the Lie algebraic context that unerlies the Lie Poisson Hamiltonian formulation of ieal flui ynamics as explaine in e.g. Holm Marsen an Ratiu [1998]. In particular it evelope the Kelvin circulation theorem an Lagrange- Alembert variational principle for Casimir issipation. In the Lagrange- Alembert formulation the moification of the motion equation to impose selective ecay was seen as an energy-conserving constraint force. Finally the previous paper extene the Casimir issipation theory to inclue fluis that possess avecte quantities such as heat mass buoyancy magnetic fiel etc. by using the stanar metho of Lie Poisson brackets for semiirect-prouct actions of Lie groups on vector spaces reviewe in Holm et al. [1985]. The main subsequent examples were the rotating shallow water equations an the 3 Boussinesq equations for rotating stratifie incompressible flui flows. Plan of the paper. The present paper pursues further the selective ecay approach base on Gay-Balmaz an Holm [2013] whose main results are reviewe in the remainer of this Introuction. The formulations of selective ecay of either Casimirs or energy on semiirect proucts is

7 Gay-Balmaz an Holm Casimir issipation in fluis with avecte quantities 7 summarize in Section 2. The applications to compressible an incompressible 2 an 3 magnetohyroynamics (MH) in Section 3 illustrate the use of the metho for flui ynamics. In particular we erive the moifie MH equations that enforce either selective ecay of Casimirs at fixe energy or vice versa. 1.2 Summary of key equations in Gay-Balmaz an Holm [2013] Let us recall that ieal incompressible 2 flui flows amit a Hamiltonian formulation in terms of a Lie Poisson bracket { } + given by Arnol [ ] [ f(ω) δf = {f h} + (ω) = ω δω ] { δf := ω δω δω } x y. (1.1) δω Here ω is the vorticity of the flow the bracket { } is the 2 Jacobian written as {f h} = J(f h) = f x h y h x f y an the angle bracket in (1.1) is the L 2 pairing in the omain of the (x y) plane. For convenience we shall take the omain to be perioic so we nee not worry about bounary terms arising from integrations by parts. Two types of conservation laws are associate with the Hamiltonian formulation. The first one is the conservation of energy i.e. the Hamiltonian h(ω). Conservation law of energy arises from the antisymmetry of the Lie Poisson bracket as h(ω) = {h h} + (ω) = 0 for any given choice of h. The secon type of conservation law arises because the Lie Poisson bracket has a kernel (i.e. is egenerate) which means there exist functions C(ω) for which C(ω) = {C h} + (ω) = 0 (1.2) for any Hamiltonian h(ω). Functions that satisfy this relation for any Hamiltonian are calle Casimir functions. (Lie calle them istinguishe functions accoring to Olver [2000].) For example the Casimirs for the Lie Poisson bracket (1.1) in the Hamiltonian formulation of 2 incompressible ieal flui motion are C Φ (ω) = Φ(ω) x y for any smooth function Φ Arnol [ ]. Ieal 3 fluis also amit this type of Lie Poisson bracket given by [ δf {f g} + (u) = u δu δg ] 3 x δu where u with iv u = 0 is the velocity an [ ] enotes the Lie bracket of vector fiels i.e. [u v] = v u u u. Lie Poisson brackets an Casimirs. In this paper we shall enote by g a Lie algebra with Lie brackets [ ] an by g a space in weak nonegenerate uality with g. That is there exists a bilinear map (calle a pairing) : g g R such that for any ξ g the conition

8 Gay-Balmaz an Holm Casimir issipation in fluis with avecte quantities 8 µ ξ = 0 for all µ g implies ξ = 0 an similarly for any µ g the conition µ ξ = 0 for all ξ g implies µ = 0. Recall that g carries a natural Poisson structure calle the Lie Poisson structure an given in terms of the pairing by [ δf {f h} + (µ) = µ ] (1.3) (see e.g. Marsen an Ratiu [1994]). Here f g F(g ) are real value functions efine on g an δf/ g enotes the functional erivative of f efine through the uality pairing by δf = ε f(µ + ε). ε=0 The Lie-Poisson bracket in (1.3) is obtaine by symmetry reuction of the canonical Poisson structure on the phase space T G of the Lie group G with Lie algebra g. The symmetry unerlying this reuction is given by right translation by G on T G. In the case of ieal flui motion this symmetry correspons to relabeling symmetry of the Lagrangian in Hamilton s principle. Lie Poisson (LP) equations. The Lie Poisson (LP) equations with Hamiltonian h : g R are by efinition the Hamilton equations associate to the Poisson structure (1.3) i.e. They are explicitly written as f = {f h} + for all f F(g ). (1.4) t µ + a µ = 0 where a ξ : g g is the coajoint operator efine by a ξ µ η = µ [ξ η]. One recalls that the coajoint operator is equivalent to the Lie erivative i.e. a ξ µ = ξ µ when µ g Ω 1 Vol is a 1-form ensity as occurs in the case when µ is the momentum ensity in ieal flui ynamics. Casimir functions. A function C : g R is calle a Casimir function for the Lie Poisson structure (1.3) if it verifies {C f} + = 0 for all functions f F(g ) or equivalently a δc µ = 0 for all µ g. A Casimir function C is therefore a conserve quantity for Lie Poisson equations associate to any choice of the Hamiltonian h. Symmetric bilinear form. Below we will enote by γ µ a (possibly µ-epenent µ g ) symmetric bilinear form γ µ : g g R. This form is sai to be positive if γ µ (ξ ξ) 0 for all ξ g. efinition 1.2 (Casimir-issipative LP equation). Given a Casimir function C(µ) for µ g a positive symmetric bilinear form γ µ an a real number θ > 0 we consier the following moification of the Lie Poisson (LP) ynamical equation (1.4) to prouce the Casimir issipative LP equation: f(µ) ([ δf = {f h} + θ γ µ ] [ δc ]) (1.5)

9 Gay-Balmaz an Holm Casimir issipation in fluis with avecte quantities 9 for arbitrary functions f h : g R. Equation (1.5) yiels the following equation for µ t µ + a µ = θ a [ δc ] (1.6) where : g g is the flat operator associate to γ µ that is for ξ g the linear form ξ g is efine by ξ η = γ µ (ξ η) for all η g. Note that the flat operator nee not be either injective or surjective. Note also that in equation (1.6) above the flat operator is evaluate at µ. It is important to observe that the moification term epens on both the given Hamiltonian function h an the chosen Casimir C. It is convenient to write (1.6) as [ t µ + a µ = 0 with moifie momentum µ := µ + θ δc ]. (1.7) The energy is preserve by the ynamics of equations (1.5) an (1.6) since we have ([ h(µ) = {h h} + θ γ µ ] [ δc ]) = 0. However when θ > 0 the Casimir function C is issipate since C(µ) ([ δc = {C h} + θ γ µ ] [ δc ]) [ = θ δc ] 2 (1.8) where ξ 2 γ := γ µ (ξ ξ) is the quaratic form (possibly egenerate) associate to the positive bilinear form γ µ. Remark 1.3 (Evolution of aitional Casimirs). If the Lie-Poisson bracket { } + in a given case amits an aitional Casimir C then C will evolve accoring to (1.5) as C(µ) ([ { } δ = C h θ γ C ] µ + [ δc ] ) ([ δ = θ γ C ] µ [ δc ] ) (1.9) { } where we have use C h + = 0 because C is a Casimir. Remark 1.4 (Left-invariant case). Recall that the Lie Poisson structure (1.3) is associate to right G-invariance on T G. We have mae this choice because ieal fluis are naturally rightinvariant systems in the Eulerian representation. Other systems such as rigi boies [ are ] left G-invariant. In this case one obtains the Lie Poisson brackets {f g} (µ) = µ an this leas to the following change of sign in the Casimir-issipative LP equation (1.6): t µ a µ = θ a γ δf [ δc ]. (1.10)

10 Gay-Balmaz an Holm Casimir issipation in fluis with avecte quantities 10 The moifie momentum is now an we have in comparison with (1.5) f(µ) [ µ := µ θ δc ] (1.11) ([ δf = {f h} θγ µ ] [ δc ]). Remark 1.5 (Relation with the metriplectic approach). By the following argument one may see that our selective ecay moel (1.5) fits into the framework of the metriplectic ynamics initiate in the work of Morrison [1984] Kaufman [1984] Grmela [1984] Morrison [1986]. Let (P { }) be a Poisson manifol let h C (P ) be the Hamiltonian of the system an C C (P ) a Casimir function. The metriplectic ynamics is formulate as follows (see e.g. Bloch et al. [2012]): f = {f h} + (f C) for all f C (P ) (1.12) where the bracket (f g) := f κ(g) is R-bilinear symmetric an positive (or negative) semiefinite with κ : T P T P a vector bunle map. Moreover it is assume that (f h) = 0 for all f C (P ). We shall now show that (1.5) fits into the context of the metriplectic ynamics. In our case the Poisson manifol is P = g enowe with the Lie Poisson bracket { } ±. Upon replacing C by an arbitrary function g C (g ) the issipative term in (1.5) reas ([ δf θγ µ ] [ δg ]) = θ δf a ( [ δg ] ) µ δf = κ µ where we have efine the vector bunle map κ µ : T µg T µ g by κ µ (ξ) := θ a ( ) δg = (f g) ( [ ξ ] ) µ (1.13) where µ : g g is the flat operator with respect to the pairing given by γ µ. For this vector bunle map the Lie-Poisson form (1.5) belongs to the class (1.12) of metriplectic systems. efinition 1.6 (Energy-issipative LP equation). As iscusse in Gay-Balmaz an Holm [2013] by simply exchanging h an C in the θ-term of equation (1.5) one obtains an energy-issipative LP equation that preserves the chosen Casimir C: f(µ) for arbitrary functions f h : g R. ([ δf = {f h} + θ γ µ δc ] [ δc ]) (1.14) Remark 1.7 (Energy-issipative formulation). In the energy-issipative formulation (1.14) we have C/ = 0 an energy ecay given by h(µ) ([ = {h h} + θ γ µ δc ] [ δc ]) = θ [ δc ] 2. (1.15) γ

11 Gay-Balmaz an Holm Casimir issipation in fluis with avecte quantities 11 By symmetry uner the exchange C h the two rates of ecay are the same in (1.8) an (1.15). An aitional Casimir C woul evolve accoring to equation (1.14) as C(µ) ([ δ = θ γ C ] µ δc [ δc ] ). (1.16) Equation (1.14) also leas to the following equation for µ t µ + a µ = θ a δc [ δc ] (1.17) where : g g is again the flat operator associate to γ µ. Of course these equations follow from (1.6) by exchanging C an h in the θ-term. Energy-Casimir equilibria an their linear stability analysis Theorem 1.8 (Energy-Casimir critical points). Energy-Casimir critical points µ e satisfying δ(h + C)(µ e ) = 0 are steay states of the moifie LP equations (1.6) an (1.17) for both the Casimir-issipative an the energy-issipative cases. Proof. The energy-issipative motion equation (1.17) is equivalent to [ δ(h + C) t µ + a δ(h+c) µ = θ a δc ] (1.18) an the Casimir-issipative motion equation (1.6) is equivalent to [ δ(h + C) t µ + a δ(h+c) µ = θ a ] (1.19) Hence the energy-casimir stationarity conition δ(h + C) = 0 prouces steay states t µ e = 0 of the moifie LP equations in both cases. Remark 1.9 (Linearize equations). The analysis of the linearize stability of energy-casimir equilibria in the energy-issipative case for example procees from the linearization of equation (1.18) aroun µ e for which δ(h + C)(µ e ) = 0. Setting =: µ an efining δ 2 (h + C) = µ 2 (h + C)(µ e ) µ =: µ µ allows the linearization of (1.18) to be written as t µ + a µ µ e = θ a δc e [ µ ]. (1.20) e The linearize equation in the neighborhoo of µ e for the corresponing Casimir-issipative case is obtaine from exchanging h C. The presence of the θ-term on the right han sie of (1.20) alters the linearise spectrum of the energy-casimir equilibria of the unmoifie LP equations. An investigation of the effects of selective ecay on the linear stability properties of the energy-casimir equilibria is quite likely to be interesting. However we shall efer this investigation to another work.

12 Gay-Balmaz an Holm Casimir issipation in fluis with avecte quantities Comparison with previous approaches General energy issipative systems Note that given a Poisson manifol (P { }) we can formulate the following energy issipative system f = {f h} ((f h)) for all f C (P ) (1.21) where the bracket ((f g)) = f Σ(g) is R-bilinear symmetric an positive semiefinite with Σ : T P T P a vector bunle map. We can impose that a given function C C (P ) is preserve by the system by assuming ((f C)) = 0 for all f C (P ). Our issipative system (1.14) fits into this general picture by choosing ([ δf ((f g)) = θ γ µ δc ] [ δg δc ]) ( [δc ] ) µ i.e. Σ µ (ξ) = θ a δc ξ corresponing to the exchange h C in equation (1.13) ouble-bracket formulations There are apparent similarities in the Lie algebraic formulations of the present energy issipation an the ouble-bracket formulations mentione in the Introuction an reviewe for example in Bloch et al. [2012]. Inee for general Lie algebras the ouble-bracket issipation equations can be written as f(µ) ) = {f h} + (µ) θγ (a δk µ a δf µ (1.22) (compare with equation (1.14) to see the ifferences) where γ is a inner prouct on g γ is the inner prouct inuce on g an k : g R is a given function. One reaily checks that Casimirs are preserve while in the special case k = h the energy issipates. In that case the equation of motion arising from (1.22) is given by t µ + a µ = θ a ( ) µ (1.23) a µ where : g g is the sharp operator associate to γ. See Bloch et al. [1996]; Holm Putkaraze an Tronci [2008] for iscussions of ouble-bracket issipation of energy. Note that (1.22) with h = k fits into the framework of (1.21) Comparison with ouble-bracket issipation Formula (1.23) for ouble-bracket issipation coincies in some particular cases with equations (1.17) obtaine from our approach after exchanging the functions C an h in (1.5). More precisely this coincience occurs in the special case of quaratic Lie algebras i.e. Lie algebras that amit an a-invariant inner prouct γ for example semisimple Lie algebras. In this special case taking the quaratic Casimir C(µ) = 1 γ(µ µ) our equation (1.14) an the ouble bracket equation (1.23) 2

13 Gay-Balmaz an Holm Casimir issipation in fluis with avecte quantities 13 with k = h coincie. In that case (a ξ µ) = a ξ µ an equation (1.23) takes the ouble-bracket form [[ ] ] t µ a µ = θ a ( ) µ = θ a µ µ µ. (1.24) This iscussion generalizes to µ-epenent a-invariant inner proucts γ µ. Examples. Examples of ouble-bracket formulations in flui ynamics inclue Vallis Carnevale an Young [1989]; Shepher [1990] in which a moification of the transport velocity was use to impose energy issipation with fixe Casimirs. The Lie algebraic nature of this moification of the transport velocity becomes clear by rewriting the special case (1.23) of the ouble bracket motion equation (1.22) as t µ + a v µ = 0 with transport velocity v = θ( a µ ). (1.25) Comparison with the triple bracket formalism Let us consier the following general form of a triple bracket on a manifol P {f g h} = C(f g h) f g h C (P ) where C is an antisymmetric 3-contravariant tensor fiel on P. Such a triple bracket can be use to formulate the following energy-issipative system on the Poisson manifol (P { }) f = {f h} ((f h)) for all f C (P ) with ((f g)) := γ (C( C f) C( C g)) (1.26) Here the expression C( C f) enotes the vector (i.e. linear form on T P ) efine by α T P C(α C f) R For example in local coorinates C( C f) is the vector C ijk j C k f an the energy-issipative bracket becomes ((f g)) := γ (C( C f) C( C g)) = γ il C ijk j C k fc lmn m C n g where γ is a (possibly egenerate) metric. In the particular case when P is a quaratic Lie algebra g with a-invariant inner prouct γ then C(ξ η ζ) := γ(ξ [η ζ]) is antisymmetric. Ientifying g with g with the help of γ the triple bracket reas {f g h} = γ( f [ g h]) where f = (f) is the graient of f relative to γ. This is the Lie algebraic generalization of the Nambu bracket (Nambu [1973]) given in Bialynicki-Birula an Morrison [1991]. In this case we have ((f h)) = γ([ C f] [ C h]) which coincies with the issipation term in our system (1.14) in the particular case of quaratic Lie algebras. This remark generalizes easily to a µ-epenent a-invariant inner prouct γ µ. In general however our system (1.14) is not a special case of a triple bracket construction (1.26).

14 Gay-Balmaz an Holm Casimir issipation in fluis with avecte quantities 14 Note that the triple bracket formulation can also be use to formulate a metriplectic ynamics f = {f h} + (f C) for all f C (P ) with (f g) := γ (C( h f) C( h g)). (1.27) This construction was mae in [Bloch et al ] for the special case when P is a quaratic Lie algebra g with a-invariant inner prouct γ an with C(ξ η ζ) := γ(ξ [η ζ]). In this particular case we have (f g) = γ([ h f] [ h g]) an (1.27) coincies with our equation (1.5). However this coincience oes not hol in general. Outlook. In the remainer of this paper we will first concentrate on Casimir issipation at constant energy an then treat the opposite case of energy issipation at a constant Casimir by simply switching h an C as in passing from equation (1.5) to equation (1.14). After this switch we can reuce further to the ouble bracket form seen previously in the literature for the case of a quaratic Casimir an an A-invariant inner prouct on the Lie algebra. To illustrate the metho the explicit formulas for selective energy ecay at fixe values of the Casimir will be iscusse in etail for MH an compare with historical treatments of the selective ecay hypothesis for MH such as Brown Canfiel an Pertsoy [1999] Lagrange- Alembert variational principle Equations (1.6) an (1.17) provie the constraint forces that will guie the ieal MH system into a particular class of equilibria by ecreasing respectively either a particular choice of Casimir at constant energy or vice versa. The balance between the Casimir an energy that occurs at a critical point of their sum etermines the class of equilibria that is achievable by a given choice of constraint force. The existence of a constraint force that will ynamically guie an MH system into a certain class of equilibria (or preserve it once it has been obtaine) may be useful in the esign an control of magnetic confinement evices. The Lagrange- Alembert variational principle extens Hamilton s principle to the case of force systems incluing nonholonomically constraine systems (Bloch [2004]). We now explain following Gay-Balmaz an Holm [2013] how the Casimir-issipative LP equations (1.6) an energyissipative LP equations (1.17) can be obtaine from the Lagrange- Alembert principle. Consier the Lagrangian l : g R relate to h via the Legenre transform that is we have h(µ) = µ ξ l(ξ) µ := δl δξ where we have assume that the secon relation yiels a bijective corresponence between ξ an µ. In terms of l equation (1.6) for Casimir issipation reas 0 t µ + a ξ µ = θ a ξ [ ] δc ξ µ := δl δξ. (1.28) These equations can be obtaine by applying the Lagrange- Alembert variational principle [ T ] T ([ ] ) δc δ l(ξ) + θ γ 0 ξ [ξ ζ] = 0 for variations δξ = t ζ [ξ ζ]

15 Gay-Balmaz an Holm Casimir issipation in fluis with avecte quantities 15 where ζ g is an arbitrary curve vanishing at t = 0 T. Thus in the Lagrange- Alembert formulation the moification of the motion equation to impose selective ecay of the Casimir is seen as an energy-conserving constraint force. Remark Similarly the energy-issipative LP equation (1.17) amits the variational formulation [ T ] T ([ δ l(ξ) + θ γ ξ δc ] [ ]) δc 0 0 ζ = 0 for variations δξ = t ζ [ξ ζ] where ζ g is an arbitrary curve vanishing at t = 0 T Kelvin-Noether theorem The well-known Kelvin circulation theorems for fluis can be seen as reformulations of Noether s theorem an therefore they have an abstract Lie algebraic formulation (the Kelvin-Noether theorems) see Holm Marsen an Ratiu [1998]. We now iscuss the abstract Kelvin circulation theorem for Casimir-issipative LP equation (1.6). In orer to formulate the Kelvin-Noether theorem one has to choose a manifol C on which the group G acts on the left an consier a G-equivariant map K : C g i.e. K(gc) A g ν = 1 K(c) ν g G. Here gc enotes the action of g G on c C an A g enotes the coajoint action efine by A g µ ξ = µ A g ξ where µ g ξ g an A g is the ajoint action of G on g. Given c C an µ g we will refer to K(c) µ as the Kelvin-Noether quantity (Holm Marsen an Ratiu [1998]). In application to fluis C is the space of loops in the flui omain an K is the circulation aroun this loop namely K(c) u x := u x. The Kelvin-Noether theorem for Casimir-issipative LP equations is formulate as follows. Proposition Fix c 0 C an consier a solution µ(t) of the Casimir-issipative LP equation (1.6). Let g(t) G be the curve etermine by the equation = ġg 1 g(0) = e. Then the time erivative of the Kelvin-Noether quantity K(g(t)c 0 ) µ(t) associate to this solution is [ δc K(g(t)c 0) µ(t) = θ K(g(t)c 0 ) a ]. Note that g(t) G is the motion in Lagrangian coorinates associate to the evolution of the momentum µ(t) g in Eulerian coorinates. The θ term is an extra source of circulation with a ouble commutator. This term is absent in the orinary Lie Poisson case (i.e. for θ = 0) an therefore in this case the Kelvin-Noether quantity K(g(t)c 0 ) µ(t) is conserve along solutions. Corollary In the case of the energy-issipative LP equation (1.17) the Kelvin-Noether theorem is foun from the exchange C h to be [ K(g(t)c 0) µ(t) = θ K(g(t)c 0 ) a δc δc ]. c

16 Gay-Balmaz an Holm Casimir issipation in fluis with avecte quantities Example: the rigi boy The Lie Poisson bracket on the ual Lie algebra of so(3) may be written on R 3 as {F h} (Π) = Π δf δπ δπ (1.29) with Π R 3. The corresponing Lie Poisson motion equation is for the left-invariant case Π Π δπ = 0. This equation escribes the motion of a rigi boy with Hamiltonian h(π) = 1Π 2 I 1 Π an symmetric positive-efinite moment of inertia tensor I whose principle moments are assume to be orere as I 1 > I 2 > I 3. The Casimir for this Lie Poisson bracket is C(Π) = 1 2 Π 2 with δc = Π δπ an one may check that the Lie-Poisson bracket (1.29) yiels {C h} = 0 for any Hamiltonian h. Selective Casimir ecay for the rigi boy. The moifie momentum inuce by the principle of selective ecay of Casimirs is foun from equation (1.11) in this case to be ( δc Π = Π + θ δπ ) ( = Π + θ Π ). δπ δπ The angular velocity of the rigi boy is given by = Ω = δπ I 1 Π. Choosing the usual inner prouct on R 3 for the bilinear form γ Π yiels = I which implies from equation (1.10) that Π Π Ω = θ(π Ω) Ω so that 1 2 Π2 = θ Ω Π 2 0. (1.30) One might also have chosen the inner prouct γ I associate with the inertia tensor I in which case 1 Π Π Ω = θi(π Ω) Ω 2 Π2 = θi(ω Π) (Ω Π) 0. Selective energy ecay for the rigi boy. Upon choosing instea to issipate the energy at a fixe value of the Casimir an taking the usual inner prouct on R 3 for the bilinear form γ Π so that = I the moifie Lie Poisson motion equation for selective ecay of energy is foun from equation (1.17) written in the left-invariant case. Thus for the rigi boy Hamiltonian h(π) = 1Π 2 I 1 Π an Casimir C(Π) = 1 2 Π 2 this becomes Π + Ω Π = θ Π (Π Ω) (1.31) which is the Lanau-Lifshitz equation for spatially homogeneous ynamics of magnetization (Π) at the microscopic scale Lanau an Lifshitz [1935]. Consequently for this choice of the inner prouct given by the bilinear form γ Π the rigi boy energy ecays as ( ) 1 2 Ω Π = Ω Π = θ Ω Π 2 0. Remark By the exchange symmetry of the ynamics of (1.5) an (1.14) uner h C the energy of the rigi boy ecays at constant Casimir at the same rate as its Casimir ecays at constant energy. The ecay of either the energy or the Casimir ens at an equilibrium of the rigi boy flow at which the angular frequency Ω an the Π angular momentum are aligne so that Ω Π = 0 as expecte from applying the energy-casimir stability metho in the example of the rigi boy flow Holm et al. [1984].

17 Gay-Balmaz an Holm Casimir issipation in fluis with avecte quantities 17 Figures for selective ecay of rigi-boy energy at constant Casimir. Equation (1.31) governs energy ecay of the rigi boy flow while preserving the Casimir whose level set efines angular momentum spheres in R 3. The basins of attractions for the two North (green) an South (blue) least energy states are shown in Figures 1.1 for two ifferent values of θ. Along the basin bounaries in these figures a slight change in the initial conitions may result in approaches to iametrically opposite equilibrium states asymptotically in time Π3 0.0 Π Π Π Π Π 1 Figure 1.1: Left: For the solution curves of (1.31) with θ = 0.1 this Figure shows the basins of attraction of the North (green) an South (blue) least energy states (of longest principle axis) lying at opposite points on the angular momentum sphere. Initial conitions starting in the blue (resp. re) region stay in the blue (resp. green) region. Along the basin bounaries a slight change in the initial conitions may result in asymptotic approaches to iametrically opposite equilibrium states. Right: For the solution curves of (1.31) with θ = 0.3 this Figure shows the basins of attraction of the North (green) an South (blue) least energy states (of longest principle axis) lying at opposite points on the angular momentum sphere. Initial conitions starting in the blue (resp. green) region stay in the blue (resp. re) region. Along the basin bounaries a slight change in the initial conitions may result in asymptotic approaches to iametrically opposite equilibrium states. 2 Selective ecay on semiirect proucts 2.1 Semiirect proucts The Hamiltonian structure of fluis that possess avecte quantities such as heat mass buoyancy magnetic fiel etc. can be unerstoo by using Lie Poisson brackets for semiirect-prouct Lie algebras Marsen Ratiu an Weinstein [1984]. In this setting besies the Lie group configuration space G one nees to inclue a vector space V on which G acts linearly. Its ual vector space V contains the avecte quantities. One then consiers the semiirect prouct G S V with Lie algebra g S V an the Hamiltonian structure is given by the Lie Poisson bracket (1.3) written on (g S V ) instea of g. We refer to Marsen Ratiu an Weinstein [1984] Holm Marsen an Ratiu [1998] for a etaile treatment. Given

18 Gay-Balmaz an Holm Casimir issipation in fluis with avecte quantities 18 a Hamiltonian function h = h(µ a) with h : (g S V ) R one thus obtains the Lie Poisson equations t (µ a) + a ( (µ a) = 0 (2.1) δa) for µ(t) g an a(t) V. More explicitly making use of the expression for the a -operator in the semiirect prouct case these equations rea t µ + a µ + δa a = 0 where the operator : V V g is efine by ta + a = 0 (2.2) v a ξ := aξ v for all v V a V an ξ g (2.3) an aξ V enotes the (right) Lie algebra action of ξ g on a V. Casimir issipation for semiirect proucts. From the Lie algebraic point of view the irect generalization of (1.6) to semiirect prouct Lie groups woul be [( δc t (µ a) + a ( δa) (µ a) = θ a ( δa) δc ) ( δa )] (2.4) δa where the flat operator : g V g V is associate to a positive symmetric bilinear map γ (µa) : (g V ) (g V ) R. Using the expression [(ξ v) (η w)] = ([ξ η] vη wξ) for the Lie bracket on g S V we can write (2.4) as in which both µ an a are moifie as ( µ ã) = (µ a) + θ t (µ a) + a ( ( µ ã) = 0 (2.5) δa) ([ δc ] δc δa δc δa ). (2.6) By using the formula a (ξv)(µ a) = (a ξ µ + v a aξ) in equation (2.5) one fins the explicit Casimir-issipative system t µ + a µ + δa ã = 0 ta + ã = 0. (2.7) When γ is iagonal on the Cartesian prouct g V we can write (ξ v) = (ξ v ) an (2.7) can be written explicitly as t µ + a µ + [ δa a + θ a δc ] + θ ( δa δc δa δc ) = 0 δa t a + a ( + θ δc δa δc ) (2.8) δa = 0. One may verify that the moifie semiirect-prouct Lie Poisson system (2.4) issipates the Casimir C while keeping energy conserve uner the moification of both µ an a. Namely one computes the Lie Poisson form ([( f(µ a) δf = {f h} + (µ a) θγ δf ) ( δa )] [( δc δa δc ) ( δa )]) (2.9) δa

19 Gay-Balmaz an Holm Casimir issipation in fluis with avecte quantities 19 which for f = h shows that the energy is conserve while for f = C shows that the Casimir issipates. Remark 2.1 (A simplification for δc = 0). Note that the moification of µ in the system (2.7) implies a moification of the µ-equation only. However a moification of a alone will yiel a moification of both the µ- an a-equations. For example if δc = 0 then equation (2.6) reuces to ( ) δc µ = µ ã = a θ (2.10) δa an equation (2.8) simplifies to t µ + a µ + δa a = θ ( δc δa δa ) t a + a ( δc = θ δa ). (2.11) Energy issipation for semiirect proucts. Exchanging the role of h an C in the θ-term of (2.4) we get the energy-issipative LP equation which preserves the Casimir C for semiirect prouct Lie groups t (µ a) + a ( δa) (µ a) = θ a ( δc δc δa ) In Lie Poisson form this becomes f(µ a) = {f h} + (µ a) θ γ ([ δc ] δc δa δc δa ) (2.12) ([( δf δf ) ( δc δa δc )] [( δa ) ( δc δa δc )]) (2.13) δa which for f = h shows that the energy issipates as h(µ a) [ = θ δc ] 2 θ δc δa δc δa γ 2 γ (2.14) while for f = C equation (2.13) shows that the Casimir is conserve uner the ynamics of (2.13). After using the formula a (ξv)(µ a) = (a ξ µ + v a aξ) for the coajoint operator of the semiirect prouct g S V an assuming that γ is iagonal on the Cartesian prouct g V the system (2.12) is explicitly given by t µ + a µ + t a + a + θ δa a + θ a δc ( δc δa δc δa Remark 2.2 (Simplifications for δc simplifies to [ δc ) δc = 0. = 0). When δc t µ + a µ + δa a = θ δc ( δc δa δa ] + θ δc δa ( δc δa δc δa ) = 0 (2.15) = 0 the energy-issipative system (2.15) ) t a + a = 0 (2.16)

20 Gay-Balmaz an Holm Casimir issipation in fluis with avecte quantities 20 cf. equation (2.11) for the corresponing simplification in the Casimir-issipative case. Note that contrary to the Casimir issipative case (Remark 2.1) the avection equation is left unchange. The LP form of (2.16) may be obtaine by substitution to fin f = δf tµ + δf δa ta = {f h} + (µ a) θ γ ( δc δa δc δa ) δf. (2.17) Setting f = h in the final equation of (2.17) gives the energy issipation equation h(µ a) ( δc = θ γ δa δc ) = θ δa δc δa 2 γ (2.18) which may also be obtaine by setting f = h an δc equation (2.13) to fin h(µ a) ( = θ 0 δc δa = 0 in the moifie energy-issipative LP ) 2. (2.19) γ Finally setting f = C an using δc = 0 in the final equation of (2.17) shows that the energyissipative system (2.16) preserves the Casimir C. 2.2 Convergence to steay states of the unmoifie LP equations In the iscussions below we shall assume that the solutions of the moifie (issipative) equations possess long-time existence. That is we shall work formally from the viewpoint of mathematical analysis an ignore the possibility of blow up in finite time. Theorem 2.3 (Steay states). For either Casimir-issipative or energy-issipative LP equations for semiirect prouct Lie groups uner the moifie ynamics (2.4) or (2.12) the issipate quantity (Casimir or energy respectively) assume to be positive 1 ecreases in time until the moifie system reaches a set of states that inclue the energy-casimir equilibria associate to the Casimir C namely δ(h + C) = 0 inepenently of the Lie algebra an the choice of Casimir. Proof. Although a shorter proof of this theorem can be given we choose to present it in three ifferent cases epening on how the avecte variables are treate. This allows us to make several relevant comments in the proof. These cases are the following: (I) the avecte variables a are absent; (II) all of the variables µ a are moifie; an (III) the avecte variables a are present but are left unmoifie. (I) The first class is the case in which the avecte variables a are absent so that h = h(µ). In this case for (1.6) resp. (1.17) we have C(µ) = θ [ δc ] 2 γ resp. h(µ) = θ [ δc 1 As iscusse in Gay-Balmaz an Holm [2013] one may assume C 0 knowing that if C 0 is inefinite one may replace it in these formulas by its square C C 2 since the squares of Casimirs are still Casimirs. ] 2 γ.

21 Gay-Balmaz an Holm Casimir issipation in fluis with avecte quantities 21 Thus if h C 0 an γ is nonegenerate both solutions converge to an asymptotic state with [ δc ] = 0. (2.20) This conition hols for steay states µ e that satisfy the energy-casimir equilibrium conition δ(h + C)/ = 0 at µ = µ e inepenently of the Lie algebra an the choice of Casimir. Note also that (2.20) means that the θ-term in the moifie equation (2.17) tens to zero. (A) In the special case when an a-invariant pairing κ exists (e.g. if g is semisimple) then a ξ µ = [µ ξ] an if we choose the Casimir C(µ) = 1 κ(µ µ) then 2 [ δc ] = a µ an in this case the solutions of both the Casimir issipative an energy issipative LP equations converge to a steay state for any choice of the Hamiltonian h an for all equilibria not just for energy-casimir equilibria. This is the case for the rigi boy an for the 2 ieal flui. (B) The above setting is not the only one in which this occurs. For example for ieal incompressible 3 fluis with the helicity Casimir C = u curl u 3 x the conition (2.20) becomes [u curl u] = 0 i.e. curl(u curl u) = 0 i.e. u ω = p. These equilibria are the steay Lamb flows in which the level sets of pressure p form symplectic manifols Arnol an Khesin [1998]. In this situation both the energy-issipative an Casimirissipative LP equations converge to a steay state of the unmoifie equations. The latter hols for the case that the Casimir is taken to be helicity-square cf. the footnote above. (II) For a semiirect prouct LP system in which all variables are moifie as in (2.4) an (2.12) an if γ is nonegenerate on g V the equations converge as in (2.14) to a state with both [ δc ] = 0 an δc δa δc δa = 0. (2.21) These conitions mean that the θ-term in the moifie equation (2.17) tens to zero. Again this pair of conitions is satisfie for steay states (µ e a e ) that satisfy the energy-casimir equilibrium conition δ(h+c)/δ(µ a) = 0 at (µ a) = (µ e a e ) inepenently of the Lie algebra an the choice of Casimir provie δc 0. When δc = 0 conition (2.21) reuces to 0 = δc δa (2.22) which is an equilibrium state of the selective ecay equation of energy (resp. Casimir) for δc = 0 as given in (2.18). The requirement δc = 0 restricts the choice of Casimirs for either the moifie LP equations or the energy-casimir equilibrium conitions of the unmoifie equations. However this case still