Lagrangian Averages, Averaged Lagrangians, and the Mean Effects of ﬂuctuations in ﬂuid Dynamics

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Lagrangian averages, averaged Lagrangians, and the mean effects of ﬂuctuations in ﬂuid dynamics

Article in Chaos (Woodbury, N.Y.) · July 2002 DOI: 10.1063/1.1460941 · Source: PubMed

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Citation: Chaos 12, 518 (2002); doi: 10.1063/1.1460941 View online: http://dx.doi.org/10.1063/1.1460941 View Table of Contents: http://chaos.aip.org/resource/1/CHAOEH/v12/i2 Published by the AIP Publishing LLC.

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Lagrangian averages, averaged Lagrangians, and the mean effects of fluctuations in fluid dynamics Darryl D. Holma) Theoretical Division and Center for Nonlinear Studies, Los Alamos National Laboratory, MS B284, Los Alamos, New Mexico 87545 ͑Received 29 October 2001; accepted 23 January 2002; published 20 May 2002͒ We begin by placing the generalized Lagrangian mean ͑GLM͒ equations for a compressible adiabatic fluid into the Euler–Poincare´ ͑EP͒ variational framework of fluid dynamics, for an averaged Lagrangian. This is the Lagrangian averaged Euler–Poincare´ ͑LAEP͒ theorem. Next, we derive a set of approximate small amplitude GLM equations ͑glm equations͒ at second order in the fluctuating displacement of a Lagrangian trajectory from its mean position. These equations express the linear and nonlinear back-reaction effects on the Eulerian mean fluid quantities by the fluctuating displacements of the Lagrangian trajectories in terms of their Eulerian second moments. The derivation of the glm equations uses the linearized relations between Eulerian and Lagrangian fluctuations, in the tradition of Lagrangian stability analysis for fluids. The glm derivation also uses the method of averaged Lagrangians, in the tradition of wave, mean flow interaction. Next, the new glm EP motion equations for incompressible ideal fluids are compared with the Euler-alpha turbulence closure equations. An alpha model is a GLM ͑or glm͒ fluid theory with a Taylor hypothesis closure. Such closures are based on the linearized fluctuation relations that determine the dynamics of the Lagrangian statistical quantities in the Euler-alpha equations. Thus, by using the LAEP theorem, we bridge between the GLM equations and the Euler-alpha closure equations, through the small-amplitude glm approximation in the EP variational framework. We conclude by highlighting a new application of the GLM, glm, and ␣-model results for Lagrangian averaged ideal magnetohydrodynamics. © 2002 American Institute of Physics. ͓DOI: 10.1063/1.1460941͔

This paper employs the Lagrangian-averaged Euler– in fluid mechanics that arise from Hamilton’s variational Poincare´ theorem to provide a bridge between the gener- principle, these conditions involve the relation between av- alized Lagrangian mean „GLM… equations and the Euler- eraged quantities in the Eulerian and Lagrangian descriptions alpha closure equations. The former „GLM… equations of fluid dynamics. An important advance was the develop- result from an exact theory of nonlinear waves on a La- ment of the generalized Lagrangian mean ͑GLM͒ equations grangian mean flow and are not closed, while the latter by Andrews and McIntyre,1 who introduced a slowϩfast de- „Euler-alpha… equations are closed via linearization that composition of the Lagrangian particle trajectory in general introduces a length scale ␣, which Taylor’s hypothesis form as renders constant if it is initially so. The Euler-alpha clo- ␰͑ ͒ϵ ͑ ͒ϩ␰͑ ͑ ͒ ͒ϭ⌶͑ ͑ ͒ ͒ ͑ ͒ sure equations with an additional Navier–Stokes viscous x x0 ,t x x0 ,t x x0 ,t ,t x x0 ,t ,t , 1.1 dissipation are of interest from the point of view of large where x is the Lagrangian mean trajectory of the particle eddy simulation and the object of the present work is to ␰ labeled by x0 , the fluctuation is and one may replace the show how they relate to a more general theory. To clarify summation in Eq. ͑1.1͒ by the more convenient composition this, we derive them from a small amplitude approxima- of functions ⌶ also introduced in Ref. 1. Andrews and McIn- tion to the GLM equations, the glm equations, by apply- tyre found that the Lagrangian mean of a fluid quantity ing Taylor’s hypothesis of frozen-in fluctuations. This evaluated at the mean particle position x is related to its derivation from first principles is general enough to in- Eulerian mean, evaluated at the displaced fluctuating posi- clude more physics, as we shall illustrate by deriving an tion x␰ by relation ͑2.7͒ below. This relation allows the GLM alpha model for Lagrangian-averaged ideal magnetohy- equations to be expressed directly in the Eulerian represen- drodynamics. tation. The Lagrangian-averaged Euler–Poincare´ ͑LAEP͒ theorem discussed here puts the averaged-Lagrangian approach I. INTRODUCTION and the method of GLM-averaged equations onto equal foot- The classical method of averages for dynamical systems ing. This is quite an extension for both approaches to mod- may be extended to the averaged Lagrangian method, but eling fluids. According to this extension, the averaged- only under certain conditions. For the nondissipative systems Lagrangian theory produces dynamics that can be verified directly by averaging the original equations. Likewise, the a͒Electronic mail: [email protected] GLM-averaged equations inherit the conservation laws and

Kelvin circulation transport properties that arise from the symmetries of the Lagrangian. Of course, this extension and these variational relationships are not possible with the Eu- lerian mean, because the Eulerian mean does not preserve the circulation transport properties of fluid mechanics. The resulting GLM equations are nonlinear and exact. However, they are not closed. The second part of the paper introduces the alpha model closure scheme in this variational framework for GLM. An alpha model is a closure of the GLM equations that is based on applying ͑1͒ Taylor’s fa- mous hypothesis of ‘‘frozen-in’’ turbulence to the Andrews FIG. 1. GLM theory factorizes the Lagrange to Euler map at a given time by and McIntyre slowϩfast decomposition and ͑2͒ Lagrangian first mapping the reference configuration to the mean position, then mapping the mean position to the current position. averaging in Hamilton’s principle. The alpha-model equations then result directly in the Eulerian representation as a 9 closure for GLM from the variational principle given by Chen et al. ͑Physically, alpha is the smallest active length Lagrangian-averaged Euler–Poincare´ theorem. scale participating in the nonlinear interactions—so scales smaller than alpha are swept along by the larger ones.͒ This A. A brief history of the Euler-alpha models is in contrast to the theory of second grade fluids, where The Euler-alpha equations for averaged incompressible alpha is a thermodynamic material parameter. See Foias ideal fluid motion were first derived in Holm, Marsden, and et al.,10,11 and references therein for additional discussions of Ratiu2,3 in the context of the Euler–Poincare´ theory for fluid the properties of alpha models. dynamics. ͑For an introduction to Euler–Poincare´ theory for classical mechanics on Lie groups, see Ref. 4.͒ That deriva- B. Outline of the paper tion proceeded essentially by choosing the kinetic energy to Section II lays out the notation and basic relations in the be the H1 norm of the Eulerian fluid velocity, rather than the GLM theory that we shall need to establish and apply the usual L2 norm. This choice generalized the unidirectional LAEP theorem in the sections that follow. Section III states 5 shallow water equation of Camassa and Holm from one and proves the Lagrangian averaged Euler–Poincare´ theorem dimension to three dimensions. The resulting n-dimensional ͑LAEP͒ theorem. Section IV explains how Lagrangian con- ⌬ϭ ϵ Ϫ␣2 ٌ ͑ Euler-alpha equation is with •u 0, v u u and con- siderations lead to geometric interpretations of the linearized ␣͒ stant length scale relations for spatial trajectory fluctuations in terms of the -displacement fluctuation, ␰. Section V discusses the linear ץ vϩu ٌvϩv ٌu jϩٌpϭ0. ͑1.2͒ ized glm approximation of the exact ͑but not closed͒ GLM t • jץ theory. Section VI then discusses modeling options for pro- This is the EP equation for the Lagrangian ducing a closed glm model, including modeling the pseudomomentum for the closed theory. Section VII concludes by 1 lϭ ͵ ͉u͉2ϩ␣2ٌ͉u͉2d3x, ͑1.3͒ highlighting the GLM, glm, and ␣-model sequence of results 2 in a new application—a Lagrangian averaged ␣—model clo- for a constant ␣ and divergenceless fluid velocity u. ͓For sure of incompressible ideal magnetohydrodynamics ϭ ␣ ͑MHD͒. Equation ͑7.15͒ provides an expression for the ٌ incompressible flow •u 0 and constant , one may re- pseudomomentum for this new closed theory of Lagrangian 2͉ ٌ͉ place u in this Lagrangian equivalently with 2 tr(e•e), .ϭ 1 ٌ ϩٌ T ͔ averaged ideal MHD and Eq. ͑7.18͒ provides its energetics where e 2( u u ) is the strain rate. Mathematically, this equation describes geodesic motion on the volume- 3 II. GENERALIZED LAGRANGIAN MEAN GLM preserving diffeomorphism group of R relative to the H1 „ … norm in a sense similar to the work of Arnold6 and Ebin and THEORY Marsden7 in which the Euler equations are shown to describe In ideal fluid dynamics, the Lagrange-to-Euler map gives geodesic motion on the same diffeomorphism group relative the current position of a fluid parcel that was initially at to the L2 kinetic energy norm. position x0 . This map is assumed to be a diffeomorphism Remarkably, the H1-geodesic Euler-alpha equation was g(t) ͑a smooth invertible map with a continuous inverse͒ later recognized as being identical to the inviscid limit of the parameterized by time t. Diffeomorphisms may be com- well-known second grade fluid equations introduced by Riv- posed, so the Lagrange-to-Euler map may be expressed, in lin and Erickson,8 although of course these equations were principle, as a composition of two other diffeomorphisms, derived from a completely different viewpoint. The differ- denoted as ences in their derivations imply corresponding differences in g͑t͒ϭ⌶͑t͒ ˜g͑t͒ ͑2.1͒ the interpretations of the solutions of these equations in each • of their contexts. In particular, the constant parameter alpha and subject to the chain rule under differentiation. See Fig. 1 ͑a length scale͒ is interpreted differently in the two theories. for a schematic representation of this composition of maps. In the Euler-alpha model, the parameter alpha is associated The GLM theory of Andrews and McIntyre1 applies an with the flow regime and, in numerical simulations, alpha averaging process to the Lagrange-to-Euler map that holds separates active and passive degrees of freedom, as shown in each fluid label x0 fixed. The averaging process (•) can be

reasonably arbitrary, except that it must be consistent with the diffeomorphism group, so that the Lagrange-to-Euler map for the average fluid trajectory is again expressible as a diffeomorphism, denoted ¯g(t). This is the key premise of ϭ⌶ GLM theory. Thus, in the decomposition g(t) (t)•˜g(t), the GLM averaging process may be expressed as ¯g(t) ϭ⌶ (t)•˜g(t) and consistency of GLM averaging with the diffeomorphisms allows one to choose ˜g(t)ϭ¯g(t). For this, GLM theory also requires the averaging process to satisfy ញ ϭ ϭ the projection property, so that g(t) gD (t) ˜g(t). Hence, a fluid parcel labeled by x0 has current position, ␰ ͒ϵ ͒ ϭ⌶ ͒ ͒ ͒ϭ⌶ ͒ ͒ x ͑x0 ,t g͑t x0 ͑t ͑˜g͑t x0 ͑x͑x0 ,t ,t , • • • ␰ FIG. 2. The GLM velocities u(x ,t) and ˜u(x,t)ϭ¯uL(x,t) are tangent to the ϭ ϭ ␰ ␰ and it has mean position x(x0 ,t) ˜g(t)•x0 x (x0 ,t). The current and mean trajectories, x and x, respectively. Lagrangian averaging process used in the WKB representation for fluid fluctuations in Gjaja and Holm12 satisfies both ␰ these key premises and may be represented this way. ϭ␹(⌶(x,t),t). Taking the LA material time derivative of ␹ L This section lays out the notation and basic relations in and using the definition of D /Dt in Eq. ͑2.5͒ yields the the GLM theory that we shall need to establish and apply the advective derivative relation, ␹ ␰ץ ␹ ␰ DLץ .LAEP theorem in the sections that follow ͩ ͪ ϩT␹ ⌶͑x,t͒ϭͩ ϩT␹ uͪ , ͑2.6͒ • tץ t • Dtץ A. GLM velocities and advective derivatives so DL␹␰/Dtϭ(D␹/Dt)␰.AsinEq.͑2.4͒ for the velocity, the ϭ⌶ ␹L ␹ The composition of maps g(t) (t)•˜g(t) yields via Lagrangian mean ¯ of a fluid quantity is defined as the chain rule the following velocity relation, ¯␹L͑x,t͒ϵ␹␰͑x,t͒ϭ␹͑x␰,t͒ϭ␹͑g͑t͒ x ,t͒. ͑2.7͒ ␰ • 0 x˙ ͑x ,t͒ϭg˙ ͑t͒ x ϭ⌶˙ ͑t͒ xϩT⌶ ͑g͑t͒ x ͒, ͑2.2͒ 0 • 0 • • 8 • 0 Taking the Lagrangian mean of Eq. ͑2.6͒ and again using its ⌶ ⌶ where T is the tangent of . The properties of the tangent projection property yields ␹ថ LϭDL¯␹L/Dtϭ(D␹/Dt)Lϭ¯␹˙ L. maps of diffeomorphisms that we shall need below are sum- Thus, as expected, the Lagrangian mean defined in ͑2.7͒ 13 marized, e.g., in the text by Abraham, Marsden, and Ratiu. commutes with the material derivative. ϭ Ϫ1 ␰ϭ Ϫ1 Invertibility implies the relation, x0 g (t)•x ˜g (t) •x, for the fluid parcel that initially occupies position x0 . B. Transformation factors of advected quantities Hence, one may define each fluid parcel’s velocity at its current position u(x␰,t) in terms of a vector field evaluated at its Advective transport laws for g(t) and ˜g(t) are defined mean position u␰(x,t)as by group actions from the right, ␰ Ϫ Ϫ ͑ ␰ ͒ϭ Ϫ1͑ ͒ ␰ϭ Ϫ1͑ ͒ ϵ ␰͑ ͒ a͑x ,t͒ϭa g 1͑t͒ and ã͑x,t͒ϭa ˜g 1͑t͒, u x ,t g˙ •g t •x g˙ •˜g t •x u x,t . 0• 0• ϭ ϭ ෈ The velocity relation ͑2.2͒ then implies where a0 a(x0,0) ã(x0,0), with a,ã V*. The factorization g(t)ϭ⌶(t) ˜g(t) implies ã(x,t)ϭa ⌶(x,t). Since a • • ⌶ץ ⌶ץ ␰͑ ͒ϭ ͑ ͒ϩ L͑ ͒ ͑ ͒ and ã refer to the same initial conditions, a0 , we have u x,t . 2.3¯• ץ x,t ץ u x,t t x Ϫ1͑ ͒ϭ ͑ ͒ϭ ⌶͑ ͒ϵF͑ ͒ ␰͑ ͒ ͑ ͒ a0•˜g t ã x,t a• x,t x,t •a x,t . 2.8 This is a standard velocity relation from GLM theory, in Note that the right-hand side of this equation is potentially which the Lagrangian mean velocity ¯uL is defined as rapidly varying, but the left-hand side is a mean advected L͑ ͒ϵ ␰͑ ͒ϭ Ϫ1͑ ͒ ϭ ͑ ͒ ͑ ͒Ϫ1 ͑ ͒ quantity. Here F(x,t) is the tensor transformation factor of ¯u x,t u x,t g˙g˜ t •x g8 t ˜g t •x. 2.4 the advected quantity a under the change of variables In the third equality one invokes the projection property of ␰ ⌶:x→x . For example, the density, D, transforms as the averaging process as ˜gϪ1(t)ϭ˜g(t)Ϫ1 and finds ¯g˙ ϭgថ ϭ ͑ ͒ L ϭ Ϫ1 ␰ ͑ ⌶͒͑ ͒ϭ ˜ ͑ ͒ F͑ ͒ϭ ͑ ⌶͒ ͑ ͒ g8 from Eq. 2.2 , so that ¯u (x,t) g8 (t)˜g(t) •x D det T x,t D x,t ; x,t det T , 2.9 ϵ˜u(x,t). Thus, the Lagrangian mean velocity ¯uL coincides ͒ ͑ ͒ ˜ ͑ ϭϪ ˜ ץ ˜ with ˜u, the vector field tangent to the mean motion associ- and D advects as tD div D˜u . 2.10 ated with ˜g(t). See Fig. 2 for a schematic representation of The transformation factors are 1, det(T⌶) and K this tangency property. ϵdet(T⌶)T⌶Ϫ1, for an advected scalar, density, and vector Hence, one may write the GLM velocity decomposition field, respectively. In each case, the corresponding transfor- L ͑2.3͒ in terms of the LA material time derivative D /Dt as mation factor F appears in a variational relation for an ad- ͑ ͒ DL vected quantity, expressed via Eq. 2.8 as ץ ͒ ͑ ͒ ␰͑ ͒ϭͩ ϩ L ٌͪ ⌶͑ ͒ϵ ⌶͑ ͒ ͑ u • x,t x,t . 2.5 ␦ ␰ϭ␦͑FϪ1 ͒ϭF Ϫ1 ␦ ϩ͑␦F Ϫ1͒¯ ץ u x,t t Dt a •ã • ã •ã. 2.11 For any other fluid quantity ␹ one may similarly define This formula is instrumental in establishing the LAEP theo- ␹␰ as the composition of functions ␹␰(x,t)ϭ␹(x␰,t) rem stated and proven next.

III. LAGRANGIAN AVERAGED EULER–POINCARE´ THEOREM „LAEP… In this section, we state and prove the Lagrangian averaged Euler–Poincare´ theorem ͑LAEP͒ theorem given in Holm,14,15 following the framework of the Euler–Poincare´ ͑EP͒ theorem of Holm, Marsden, and Ratiu.2,3 We shall also illustrate the LAEP theorem by applying it to incompressible ideal fluids. This application will recover the familiar GLM motion equations in the geometrical framework of the LAEP theorem. Let the following list of assumptions hold:2,3 ͑a͒ There is a right representation of Lie group G on the vector space V and G acts in the natural way on the ϫ ϭ FIG. 3. The Lagrangian averaged Euler–Poincare´ ͑LAEP͒ theorem pro- right on TG V*:(vg ,a)h (vgh,ah). ͑ ͒ ϫ → duces a cube consisting of equivalence relations ͑i͒–͑iv͒ and ͑i͒–͑iv͒ on its b The real-valued function L:TG V* R is right left and right faces, respectively, and four commuting diagrams ͑one on each G-invariant. of its four remaining faces͒. The front face of the cube is in the Eulerian ͑ ͒ ෈ c In particular, if a0 V*, define the Lagrangian picture and the back face is in the Lagrangian picture. The top face relates L :TG→ by L (v )ϭL(v ,a ). Then L is right the Lagrangians and the bottom face relates their corresponding motion a0 R a0 g g 0 a0 equations. The left face is exact, the right face is Lagrangian-averaged. The invariant under the lift to TG of the right action of G bottom edge of the front face represents GLM theory. a0 on G, where G is the isotropy group of a . a0 0 ͑d͒ Right G-invariance of L permits one to define the La- t2 grangian on the Lie algebra g of the group G. Namely, ␦͵ l͑u͑t͒,a͑t͒͒dtϭ0 ͑3.3͒ ϫ → Ϫ1 Ϫ1 ϭ t l:g V* R by l(vgg ,a0g ) L(vg ,a0). Con- 1 versely, this relation defines for any l:gϫV*→R a holds, using the chain-rule induced variations, ϩ ͒␩͒⌶Ϫ1 ץ ϩ ͒␩ ϩ ⌶ ץ ϫ → ␦ ϭ right G-invariant function L:TG V* R. u ͑ t adu Ј ͑T ͑ t ad˜u ˜ , Ϫ • ͑e͒ For a curve g(t)෈G, let u(t)ϵg˙ (t)g(t) 1෈TG/G ␦ ϭϪ ␩ϭ␦͑F Ϫ1 ͒⌶Ϫ1 a a •ã Хg and define the curve a(t) as the unique solution of ϭϪ ␩ЈϪ͑FϪ1 ͑ ␩͒͒⌶Ϫ1 ͑ ͒ the linear differential equation with time dependent co- a • ã˜ , 3.4 efficients a˙ (t)ϭϪa(t)u(t), where the action of an el- where Lie derivatives of advected quantities by the ement of the Lie algebra u෈g on an advected quantity vector fields, Ϫ Ϫ a(t)෈V* is denoted by concatenation from the right. ␩Ј͑t͒ϵ␦⌶⌶ 1, ˜␩͑t͒ϵ␦˜g˜g 1, ϭ ෈ The solution with initial condition a(0) a0 V* can and be written as the advective transport relation, a(t) ␩͑t͒ϵ␦ggϪ1ϭ␩Јϩ͑T⌶ ˜␩͒⌶Ϫ1, ͑3.5͒ ϭ Ϫ1 • a0g(t) ; are indicated by concatenation on the right and these ͑ ͒ ϭ⌶ f The GLM factorization holds, g(t) (t)•˜g(t), in three vector fields all vanish at the end points; ϭ⌶ ϭ ͑ ͒ ϫ which the average defined as ¯g(t) (t)•˜g(t) ˜g(t) iv the Euler–Poincare´ (EP) equation holds on g V*, l ␦l␦ ץ .satisfies the projection property ˜¯g(t)ϭ˜g(t) as in Ref ͩ ϩad*ͪ ϭ छa, ͑3.6͒ t u ␦u ␦aץ .12 and the Lagrangian averaged Euler–Poincare´ ͑LAEP, After adding the last key assumption of factorization for ͒ ϫ˜ the GLM theory, we may state and prove the LAEP theorem. or EP equation holds on g˜ V*, l ␦l␦ ץ A. LAEP theorem Holm Refs. 14, 15 † „ …‡ ͩ ϩad*ͪͩ T⌶ͪ ϭͩ FϪ1 ͪ छã. ͑3.7͒ • t ˜u ␦u␰ • ␦a␰ץ The following are equivalent: ͑i͒ The averaged Hamilton’s principle holds Notation: In Eqs. ͑3.6͒ and ͑3.7͒, the operations ad* and छ ϭ͐ 3 t2 are defined by using the L2 pairing ͗ f ,g͘ fgd x as in ␦͵ L ͑g͑t͒,g˙͑t͒͒dtϭ0 ͑3.1͒ 2 a0 the EP theorem of Holm, Marsden, and Ratiu. The ad* op- t1 ͑ ͒ eration is defined as minus the L2 dual of the Lie algebra for variations ␦g(t) of g(t) vanishing at the end ␩ϭϪ ␩ operation, ad, or commutator, adu ͓u, ͔, for vector points; Ϫ͗ ␮ ␩͘ϵ͗␮ ␩͘ छ fields, adu* , ,adu . The diamond operation ͑ii͒ the averaged Euler–Lagrange (EL) equations for¯ L ͑ ͒ a0 is defined as minus the L2 dual of the Lie derivative, are satisfied on T*G˜ , namely, ͗bछa,␩͘ϵϪ͗b,L␩a͘ϭϪ͗b,a␩͘, where L␩a denotes the Lie derivative with respect to vector field ␩ of the ␦L ␦L a0 d a0 T⌶Ϫ T⌶ϭ0; ͑3.2͒ tensor a, and a and b are dual tensors under the L2 pairing. ␦g • dt ␦g˙ • See Fig. 3 for a pictorial interpretation of the LAEP ͑iii͒ the averaged constrained variational principle theorem.

Proof of the LAEP theorem: The equivalence of ͑i͒ and and ␩(t) is given by ␩(t)ϭ␦g(t)g(t)Ϫ1. The corresponding ͑ii͒ may be shown by a direct computation. To compute the statements also hold for the prime- and tilde-variables in the averaged Euler–Lagrange (EL) Eq. ͑3.2͒, we use the follow- variational relations ͑3.4͒ that are used in the calculation of ing variational relation obtained from the composition of the other equivalences. These observations show that ͑i͒ and ϭ⌶ ͑ ͒ ͑iii͒ are also equivalent, and this finishes the proof of the maps g(t) (t)•˜g(t), cf. the GLM velocity relation 2.2 : LAEP theorem. ␦g͑t͒ϭ␦⌶͑t͒ ˜g͑t͒ϩT⌶͑t͒ ␦˜g͑t͒. ͑3.8͒ • • The Lagrangian averaged Euler–Poincare´ ͑LAEP͒ theo- Hence, after integrating by parts and using the projection rem may be visualized as a cube of interlocking equivalence property we find that relations and commutative diagrams, as in Fig. 3. ␮ϭL ␮ Lie derivative vs ad*: The equality adu* u holds t2 0ϭ␦ ͵ L ͑g͑t͒,g˙ ͑t͒͒dt for any one-form density ␮ ͑such as ␮ϭ␦l/␦u, the varia- a0 t1 tional derivative͒. Thus, the EP motion Eq. ͑3.6͒ and the LAEP motion Eq. ͑3.7͒ may be written equivalently using t ␦L ␦L 2 a0 a0 Lie derivatives as, respectively, ϭ ͵ ͩ ␦gϩ ␦g˙ ͪ dt ␦g • ␦g˙ • l ␦l␦ ץ t1 ͩ ϩL ͪ ϭ छa, ͑3.12͒ t u ␦u ␦aץ ␦ ␦ t2 La d La ϭ ͵ ͩͩ 0Ϫ 0 ͪ ␦⌶͑ ͒ͪ ␦ ␦ • t •˜gdt l ␦l␦ ץ ˙t1 g dt g ͩ ϩL ͪͩ T⌶ͪ ϭͩ F Ϫ1 ͪ छã. ͑3.13͒ • t ˜u ␦u␰ • ␦a␰ץ t ␦L ␦L 2 a0 d a0 ϩ ͵ ͩ T⌶Ϫ T⌶ ͪ ␦˜gdt. ͑3.9͒ ␦g • dt ␦g˙ • • In this notation, the advection of mass by the LA motion ͒ ͑ ϩL ˜ ϭ ץ t1 takes the form ( t ˜u)D 0 and Eq. 3.13 immediately In the last equality, the first of the two integrals vanishes for implies the following corollary of the LAEP theorem. any ␦⌶, thereby ensuring that the Euler–Lagrange equations, ␦L ␦L B. LA Kelvin–Noether circulation theorem and a0 d a0 Ϫ ϭ0, pseudomomentum ␦g dt ␦g˙ The one form ˜vϵ((␦l/␦u␰) T⌶)/D˜ satisfies are satisfied before averaging is applied. Vanishing of the • second of these two integrals for arbitrary ␦˜g then yields the d 1 ␦l Ͷ ϭ Ͷ ͩ FϪ1 ͪ छ averaged Euler–Lagrange Eq. ͑3.2͒, in which the transforma- ˜v ␰ ã, dt c͑˜u͒ c͑˜u͒ ˜ ␦a • tion factor T⌶ is contracted with the Euler–Lagrange equa- D tions before averaging is applied. for any closed curve c(˜u) following the LA fluid motion. The equivalence of ͑iii͒ and ͑iv͒ in the LAEP theorem The quantity ˜v –˜u in the case when T⌶ϭIdϩٌ␰, for a now follows by substituting the variations ͑3.4͒ defined using vector field ␰ϭx␰Ϫx is the GLM pseudomomentum.1,16,17 the chain rule into ͑3.3͒ and integrating by parts to obtain This corollary follows from the LA motion equation written ˜ ϩL ץ ͒ ͑ as 3.13 and the LA mass conservation law, ( t ˜u)D t2 t2 ␦l ␦l 0ϭ␦ ͵ l͑u,a͒dtϭ ͵ ͳ ,␦uʹ ϩͳ ,␦aʹ dt ϭ0. Because of the equivalence relations in the LAEP theo- ␦u ␦a t1 t1 rem, the same result may be obtained by applying LA to Kelvin’s theorem for the exact EP motion Eq. ͑3.6͒, t2 ␦l ␦l ␦ ␦ ϩad*͒ Ϫ छa,␩Јʹ dt ͑3.10͒ ץ͑ ϭϪ͵ ͳ l u ␦u ␦a d 1 l 1 l t1 Ͷ ϭ Ͷ छa. dt c͑u͒D ␦u c͑u͒D ␦a l␦ ץ t2 Ϫ ͵ ͳ ͩ ϩad*ͪͩ T⌶ͪ This exact Kelvin’s theorem is easily derived from the form • t ˜u ␦u␰ץ t1 ͑3.12͒ of the exact motion equation, upon using exact mass ϩL )Dϭ0. We illustrate the LA ץ) conservation in the form ␦l t u Ϫͩ F Ϫ1 ͪ छã,˜␩ʹ dt. ͑3.11͒ Kelvin–Noether circulation theorem in Fig. 4. ␦a␰ • Thus, the independent variations ␩Ј in ͑3.10͒ and ˜␩ in ͑3.11͒ C. Applying the LAEP theorem to incompressible result in the EP motion Eq. ͑3.6͒ and the LAEP motion Eq. fluids ͑3.7͒, respectively. The Lagrangian averaged Euler ͑LAE͒ equations for an Finally we show that (i) and (iii) are equivalent. First incompressible fluid are derived from the LAEP theorem by note that the G-invariance of L:TGϫV*→R and the defini- using the reduced averaged Lagrangian in ͑iii͒ of the LAEP ϭ Ϫ1 ͑ ͒ tion of a(t) a0g(t) imply that the integrands in 3.1 and theorem, ͑3.3͒ are equal. In fact, this holds both before and after averaging. Moreover, all variations ␦g(t)෈TG of g(t) with ¯ϭ ͵ 3 ͓ 1 ˜ ͉ ␰͉2ϩ ␰͑ ⌶Ϫ ˜ ͔͒ ͑ ͒ l d x 2D u p det T D , 3.14 fixed end points induce and are induced by variations ϩ ␩ ␩ ץ ␩ץ෈ ␦ ϭ ␦ u(t) g of u(t) of the form u / t adu with (t) which is obtained as the LA of Hamilton’s principle for Eu- ෈g vanishing at the end points. The relation between ␦g(t) ler’s equations. The pressure constraint implies that the mean

Proof: This LA momentum balance relation follows by applying Noether’s theorem to the reduced averaged La- grangian ͑3.14͒ for the LAE equations. See, Holm, Marsden, and Ratiu2,3 for discussions of Noether’s theorem in the EP context for continuum mechanics. Sections IV and V discuss the linearized glm approximation of the exact ͑but unclosed͒ GLM theory. Section VI discusses modeling options for producing a closed glm model.

IV. LINEARIZED FLUCTUATION FORMULAS A. Lagrangian and Eulerian pictures Lagrangian fluid trajectories are orbits under the action of the diffeomorphism group GϭDiff parameterized by time FIG. 4. The LA Kelvin–Noether circulation theorem describes the dynamics t, thus, ͛ Ϫ of the circulation integral ˜v•dx around the Lagrangian averaged fluid loop x͑t,x ͒ϭg͑t͒ x , x ϭg 1͑t͒ x͑t͒, g෈Diff. c(˜u) ͑dotted line͒. 0 • 0 0 • The corresponding velocity relations are ͒ϭ ͒ ϭ Ϫ1 ͒ ϭ ͒ x˙͑t,x0 g˙ ͑t x0 g˙g ͑t x u͑x,t . advected density is related to the mean fluid trajectory by • • This formula relates the Lagrangian and Eulerian definitions D˜ ϭdet T⌶, and, in general, D˜ Þconstant, so the LAE fluid of velocity. velocity has a nonzero divergence of order O(͉␰͉2).1 For variations, one introduces another parameter ⑀, and ˜ ͑ ͒ The GLM motion equation: For l in 3.14 the LAEP Eq. denotes ͑3.7͒ gives the LAE equation gץ gץ .͒⑀,g͑t,⑀͒: ϭg˙ ͑t,⑀͒, ϭgЈ͑t ץ ץ ץ ץ ⑀ץ tץ v ϩ˜u j ˜v ϩ˜v ˜u jϩ ˜␲ϭ0, ͑3.15͒˜ xiץ xiץx j i jץ t iץ Thus, the corresponding variational relations are ␲ with mean fluid quantities ˜vi and ˜ defined as Ј͑ ⑀ ͒ϭ Ј͑ ⑀͒ ϭ Ј Ϫ1͑ ⑀͒ ϭ␰͑ ⑀͒ x t, ,x0 g t, •x0 g g t, •x x,t, . ␦¯ 1 l ␰ ˜v ϭ ϭu ͑T⌶͒ j, This formula relates the Lagrangian and Eulerian definitions i i j i D˜ ␦˜u of spatial trajectory fluctuations. Thus, at spatial position x and time t, a given Lagrangian ␦¯l 1 trajectory has two tangent vectors: the Eulerian velocity, u ␰ 2 L ␲˜ ϭϪ ϭϪ ͉u ͉ ϩ¯p . ͑3.16͒ ϭg˙gϪ1, and the Eulerian fluctuation/variation, ␰(x,t) ␦ ˜ 2 D ϭgЈgϪ1. Here ¯pLϭp␰ is the Lagrangian mean pressure. When T⌶ In this section we show that these Lagrangian consider- -ϭIdϩٌ␰, for a vector field ␰ϭx␰Ϫx, one finds ations lead to geometric interpretations of the linearized re lations for spatial trajectory fluctuations in terms of the dis- L ␰ D placement fluctuation, ␰. vϭu ϩ ␰ ٌ␰ jϵ¯uLϪ¯p. ͑3.17͒˜ Dt j Hence, the LAEP Eq. ͑3.15͒ for the reduced averaged La- B. The uЈ-equation grangian ͑3.14͒ recovers the GLM motion equation in the Equality of cross derivatives in the difference, absence of rotation and buoyancy, with GLM pseudomomen- ␰ץ tum ¯pϭ(DL/Dt)␰ ٌ␰ j in this case.1,16,17 j uЈ͑x,t͒Ϫ ϭ͑g˙gϪ1͒ЈϪ͑gЈgϪ1͒ tץ When T⌶ϭId, so det T⌶ϭ1, we have ¯pϭ0, Dϭ1, v ϭu, ␲ϭϪ 1͉u͉2ϩp, and Eq. ͑3.15͒ reduces to the usual 2 gives the uЈ-equation, Euler equation for incompressible ideal fluid motion. ␰ץ ,Momentum balance: Following the EP theory of Holm ␰Ϫad␰u, ͑4.1͒ ץuЈ͑x,t͒ϭ ϩu ٌ␰Ϫ␰ ٌuϭ 2,3 t • • tץ -Marsden, and Ratiu leads to the momentum balance rela tion for the LAE Eqs. (3.15), which relates Eulerian velocity variations uЈ(x,t) and La- ␰ ץ ␰ 2 u ͉ grangian trajectory variation tendencies t (x,t). This is a͉ץ ˜D ץ ץ ͑D˜ ˜v ͒ϩ ͑D˜ ˜v ˜u jϩ¯pL␦ j͒ϭ ͯ , ͑3.18͒ xi key equation for making the glm approximations in the GLMץ x j i i 2ץ t iץ exp Lagrangian. The particular solutions of the uЈ-equation also where subscript exp refers to the explicit spatial dependence play a role as sources of inspiration for Taylor hypotheses in ␰ that yields a mean force arising from the ⌶-terms in ͉u ͉2 simplifying the glm equations to obtain the alpha models in ϭ͉DL⌶/Dt͉2 that appear in Eq. (2.3). Sec. VI.

For discussions of the linearized fluctuation relations ter taking variations͒ in the incompressible case, upon with applications to Lagrangian stability analysis with a invoking the preserved initial conditions D¯ ϭ1 and 18,19 similar geometric viewpoint, see Friedman and Schutz. div ␰ϭ0. ͑c͒ Substitute these relations into the second-variation La- C. Advected quantities grangian, to form ␰ ␰ϩ␰ ␰͔ 3 ץ␰ϩ ץ ␰ ץ͓͵For advected quantities the right action of the group, Љϭ l t •A• t t •B• •C• d x, denoted as where A, B, C are matrix operators involving the mean ͑ ⑀͒ϭ Ϫ1͑ ⑀͒ a t, a0g t, , fluid quantities and their gradients, i.e., the set " .͖ ¯implies the Eulerian advection ͑ at fixed x͒, ͕¯u,D¯ ,ٌ¯u,ٌD ͒ϭϪ Ϫ1 Ϫ1ϭϪ ϭϪL ͑d͒ Take the Eulerian mean to form the total mean La- a˙ ͑x a0g g˙g au ua, grangian and the Eulerian variation ͑Ј at fixed x͒, ¯lϭ¯l ϩ 1lЉ. ␦ ͒ϭ ͒ϭϪ Ϫ1 Ϫ1ϭϪ ␰ϭϪL 0 2 a͑x aЈ͑x a0g gЈg a ␰a. Remark: This is the general result. For an advected sca- Step 2: ␹ lar one finds ͑a͒ Derive the glm motion equation for barotropic com- ͒ ͑ ␹ЈϭϪL ␹ϭϪ␰ ٌ␹ ␰ • 4.2 pressible fluids by computing the EP equation, and for an advected density D¯ one finds d 1 ␦¯l 1 ␦¯l ␦¯l , ϩ ٌ¯u jϭٌ ␦ ␦ j DЈϭϪL␰D¯ ϭϪdiv D¯ ␰. ͑4.3͒ dt D¯ ¯u D¯ ¯u ␦D¯ We note that—from their definitions in terms of Taylor series for the total mean Lagrangian ¯l by taking its variations approximations—all of these linearized fluctuation relations ␦¯l ␦¯l introduce gradients of Eulerian mean fluid quantities. ␦¯lϭ͵ͫͩ ͪ ␦¯uϩͩ ͪ ␦D¯ ͬ d3x. • Thus, the smooth invertible ⑀-dependence representing ␦¯u ␦D¯ variations in the glm and GLM theories is generated by the These variational derivatives involve Eulerian means ␰ displacement vector field , just as the time dependence is of quadratic combinations of the Lagrangian fluctuat- ␰ ٌ␰ ץ generated by the fluid velocity. ␰ ing displacement , and its derivatives t and . For ␲ ٌ␰ j example, one combination that appears is j , ␰ϩ ␰ ץ ␰ ϭ ץ ␦ ␲ϭ 1␦ Љ ͦ␰ͦ2 where 2 l / ( t ) A. t B is the momen- V. DERIVING glm: THE ORDER O„ … GLM • EQUATIONS tum canonically conjugate to ␰. These Lagrangian quadratic statistical moments are unknown parameters in In this section, we shall obtain a set of Eulerian-mean the glm equations that must be independently specified, equations that approximate the GLM equations at second or modeled, in closing the equations. Thus, a number order in the displacement ␰. Following ideas familiar in La- of modeling decisions must be made in closing any glm grangian fluid stability analysis, we shall derive these approximate equations from a variational principle based on model. ͑ ͒ first taking the Eulerian mean of the second-variation of the b In Sec. VI, we shall discuss the various modeling pa- GLM Lagrangian and then using the EP formulation. Our rameters required to produce a closed glm model. This strategy for developing this order O(͉␰͉2) approximate Eu- will be done in the context of simplifying them and lerian mean counterpart for GLM involves the following constructing a more manageable class of closed two-step process: equations—the alpha models—obtained by using clo- Step 1: sures based on Taylor’s hypothesis of frozen-in turbulence. ͑a͒ Linearize the fluctuation relations to find Eqs. ͑4.1͒– ͑ ͒ ͑4.3͒, c The equations derived from this two-step procedure— being the small-amplitude approximation of the GLM ,DЈϭdiv D¯ ␰, ␹ЈϭϪ␰ ٌ¯␹ • equations—are called glm equations. They are derived ٌ ␰ϩ ٌ␰Ϫ␰ ץЈϭ u t ¯u• • ¯u. within the EP framework. These new equations de- ͑b͒ For the incompressible case, setting D¯ ϭ1 in the den- scribe the dynamics of Eulerian mean fluid quantities sity fluctuation equation gives influenced by small amplitude fluctuations. Being EP ͉ ϭϪ ␰ equations, they still retain the properties that result DЈ D¯ ϭ1 div . from particle relabeling symmetry. In particular, the Taking the divergence of the uЈ equation then yields glm equations retain the Kelvin–Noether circulation ␰͒ϩ ٌ͑ ␰͒ ͑ ץЈϭ ϭ div u 0 t div ¯u• div . theorem and its associated local conservation law for So div ␰ϭ0 is preserved, which means we may choose potential vorticity. initial conditions so that DЈϭ0. Thus, DЈ vanishes ͑af- Our next steps are:

͑1͒ Compute the mean momentum of the fluctuations, ␦ 1 mЉϭ ͩ lЉͪ . ␦¯u 2 ͑2͒ Write the EP glm equations for total momentum, ␦¯l m¯ ϭ , ¯l ϭ¯l ϩ͑ 1lЉ͒. ␦¯u 0 2 ͑3͒ Obtain a Kelvin circulation theorem for glm equations from their corresponding EP equations and the Kelvin– Noether theorem for these equations. ͑4͒ Derive the glm energy balance by Legendre transforming ¯l , the averaged Lagrangian. ͑ ͒ ¯ j 5 Derive the glm stress tensor Ti in the glm momentum FIG. 5. Illustrating the GLM/glm/Taylor hypothesis closure technique. i͉ ץ Lץjϭ ¯ ץϩ ץ balance law, tm¯ i jTi / x exp , including the ‘‘fluctuation stresses’’ by invoking Noether’s theorem again. from the Lagrangian statistics must be specified. Namely, the ͑ ͒ ͑6͒ Use the result in Sec. VI to interpret the Euler—␣ model quantity in the glm Kelvin circulation theorem 5.3 , ͒ ͑ ␰Ϫ ϫ␰͒͒ ץ stresses, circulation, and momentum in glm terms. ␰ϫ␻ ϭ␰ϫ ϭ␰ϫ Ј curl uЈ curl͑ t curl͑¯u , 5.4 -EP glm equations for incompressible mean flow: The must be specified in terms of ¯u, ٌ¯u, and ٌٌ¯u. This speci variational derivatives of the glm Lagrangian for incompress- fication is one of the objectives of our discussions in the next ible flow section. 1 ¯l͑¯u,D¯ ͒ϭ ͵ ͫ ¯D͉͑¯u͉2ϩ͉uЈ͉2͒ϩ¯p͑1ϪD¯ ͒ͬd3x, ͑5.1͒ VI. EULER-ALPHA MODELS 2 A. Opening remarks are obtained by using the linearized fluctuation relations We have seen that the use of Taylor expansions in the ͑4.1͒ and ͑4.3͒ as linearized fluctuation relations summons gradients of Eule- 1 rian mean fluid quantities into the mean second-variation La- ␦¯l͑¯u,D¯ ͒ϭ ͵ ͫ␦D¯ ͩ ͉͑¯u͉2ϩ͉uЈ͉2͒Ϫ¯pͪ 2 grangian. In turn, these gradients summon second-order spatial derivatives such as ٌٌ¯u into the glm motion equation ϩ␦ ͑ Ϫ ¯ ͒ϩ ¯ ␦ ͑ Ϫ␰ϫ Ј that results from the EP variational principle. ¯p 1 D D ¯u• ¯u curl u To achieve closure, even the incompressible glm case ϩٌ͑␰ uЈ͒͒ϩ␦¯u uЈ div ͑D¯ ␰͒ͬd3x. still requires an assumption to express the key element of the Lagrangian statistics ͑5.4͒ in terms of ¯u, ٌ¯u, and ٌٌ¯u.In • • this section we show that the linearized fluctuation equations We define the glm circulation velocity as themselves ͑relating the Eulerian and Lagrangian small dis- -ϵ Ϫ␰ϫ Јϩٌ͑␰ Ј͒ ͑ ͒ turbances͒ guide the formulation of the Euler-alpha approxi ¯v ¯u curl u •u . 5.2 mate closure assumptions. ͒ ϭ ٌ ͑ The corresponding EP motion equation with •¯u 0 is ex- Approach: The approach to the alpha-model equations is pressed as closely related to the glm approach, but with one important difference. Namely, the order is interchanged in the steps of ץ ϩ ٌ ϩ ٌ jϩٌ ϭ making approximations and varying the EP Lagrangian in¯ .v ¯u• ¯v ¯v j ¯u ¯p 0 ץ t Hamilton’s principle. This is the EP equation for the Lagrangian ͑5.1͒. It also has To obtain the glm equations: We expanded the Lagrang- the equivalent form, ian, took its Eulerian mean, then varied to obtain the equations of motion, and finally saw the need to approximate the closure. This could be done, in principle, by using the ץ .vϪ¯uϫcurl¯vϩٌ͑¯v ¯uϩ¯p͒ϭ0¯ t • uЈ-equation for the tendency of ␰ and the linearization of theץ GLM equations for the tendency of uЈ. We shall discuss a The Kelvin circulation theorem for the incompressible glm more direct approach to closure that yields the ␣-models. equations is simply To obtain the ␣-models: We shall expand the Lagrangian, d take its Eulerian mean, approximate the Lagrangian ͑by tak- Ͷ ͑¯uϪ␰ϫcurl uЈ͒ dxϭ0. ͑5.3͒ ing a particular solution of the uЈ-equation as a Taylor hy- dt ͑ ͒ • c u¨ pothesis͒, and then vary to find a closed set of EP motion ٌ ␰ϩ ٌ␰Ϫ␰ ץЈϭ Remark: We recall that u t ¯u• • ¯u. For the equations. This approach is illustrated in Fig. 5. ␰ ץϭ ٌ ␰ϭ Јϭ ٌ case that •¯u 0 and • 0, this becomes u t Remark: Because of the close relation between the ap- Ϫcurl(¯uϫ␰). Thus, to close the glm EP motion equation for proaches used in deriving these two sets of equations, one incompressible Eulerian mean flow, only one key element might hope for a bridge between them. For example, the glm

equations could potentially provide an Eulerian diagnostic A. Ideal GLM MHD for determining parameters in the alpha model from DNS of The Lagrangian averaged Euler ͑LAE͒ equations for in- the full Euler equations ͑or Navier–Stokes equations͒. The compressible ideal MHD are derived from the LAEP theo- glm equations form a systematic approximation for the origi- rem by subtracting the magnetic energy potential energy, nal GLM equations, within the EP framework. Thus, perhaps the GLM equations could be used to help answer questions 1 P ϭ ͵ ͉B␰͉2d3x, that may arise at the other levels of approximation in this mag 2 framework, particularly, in the alpha models. from the reduced averaged Lagrangian, ¯l (¯uL,D˜ )in͑3.14͒. Thus, the Lagrangian for GLM MHD in the Eulerian picture is B. Taylor hypothesis closure „THC… approach We shall use partial, or particular, solutions of the linear- ¯l ϭ ͵ ͓ 1D˜ ͉u␰͉2ϩp␰͑det T⌶ϪD˜ ͒Ϫ 1͉B␰͉2͔d3x. ͑7.1͒ ized velocity fluctuation Eq. ͑4.1͒, 2 2 ␰ϩ ٌ␰Ϫ␰ ٌ ͑ ͒ This Lagrangian is also obtained as the LA of Hamilton’s ץЈϭ u t ¯u• • ¯u, 6.1 principle for the exact incompressible ideal MHD equations. to guide certain choices of Taylor hypotheses. Two useful The variation of the GLM-averaged magnetic potential en- Taylor hypotheses for the uЈ-equation are: ergy gives THC #1: Neglect space and time derivatives of ␰ in the -␰ϩ¯u ٌ␰ϭ0. In the incom ץuЈ-equation, or, set d␰/dtϭ t • ␦ ϭ ͵ ␰ ␦ ␰ 3 ϭ ͵ ␰ Ϫ1 ␦˜ 3 ϩ ␦␰ pressible case, this yields the original Euler-alpha model of Pmag B • B d x B •K • Bd x terms in . Holm, Marsden, and Ratiu,2,3 in which one assumes uЈ ˜ ϭ ␰ ϭϪ␰ ٌ¯u. Here we recalled that B K B is a mean quantity and used • ␰ Ϫ • ␰ ϭ 1 ˜ THC #2: Assume that is frozen-in as a one-form. In the B K •B. Hence, the effect of GLM averaging in MHD is incompressible case, this yields the anisotropic alpha model to introduce a mean tensor permeability due to the fluctua- 20 ЈϭϪ ␰ tions as of Marsden and Shkoller, with u 2 •¯e, where ␰ ץ ϭ 1 ٌ ϩٌ T ¯e 2( ¯u ¯u ) is the mean strain rate tensor, and t ϩ ٌ␰ϭϪٌ T ␰ ␦P ␦¯l ¯u• ¯u • . ˜ ϵ mag ϭ ␰ Ϫ1ϭϪ H B •K . The Taylor hypotheses THC #1 and THC #2 are approxi- ␦B˜ ␦B¯ mate relations between Eulerian and Lagrangian statistics ͑namely, they are relations for uЈ as a function of ␰ and its Recall that GLM does not preserve incompressibility. derivatives͒ that yield closure when substituted into the av- The EP equation for ideal barotropic MHD is given in Holm, 2,3 eraged Lagrangian in Hamilton’s principle. We shall discuss Marsden, and Ratiu in the present notation as only the incompressible case, which is simpler than the com- l ␦¯l¯␦ ץ l¯␦ ץ .pressible case and has all the essential features ϩ ͩ ¯uLkͪ ϩ ٌ¯uLk L ␦ Lk ␦ ץ L ץ ץ Our discussion shall illustrate the Taylor hypothesis clo- t ¯u xk ¯u ¯u sure technique obtained by substituting THC #1 into the glm Lagrangian, before taking its variations. This approach re- ␦¯l ␦¯l . sults via the EP framework in closed equations based on the ϭD˜ ٌ ϩB¯ ϫcurl ␦ ˜ ␦¯ glm equations that retain their Kelvin circulation theorem D B and conservation properties. Among these closed equations Thus, the effect of the magnetic field on the GLM motion is for the incompressible case are variants of the Euler-alpha to add a ˜JϫB˜ force, with model ͑or, Lagrangian averaged Euler equations͒ that are also related to the theory of second grade fluids and have ˜ϭ ͑ ␰ Ϫ1͒ J curl B •K . been discussed as potential turbulence closure models when Navier–Stokes viscous dissipation is introduced, as in Chen The auxiliary GLM equations are the continuity equation for et al.9,21–23 We shall show how this approach also leads to a D˜ and the frozen-in flux rule for B˜ . That is, new generalization of the Euler-alpha model that includes .B˜ ϭcurl͑¯uLϫB˜ ͒ with div B¯ ϭ0 ץ ,D¯ ϩdiv D˜ ¯uLϭ0 ץ compressibility, elsewhere, in Holm.24 See Refs. 2, 3, 20, t t 25–27 for other developments of Taylor hypothesis closures Remarks: and further analysis of incompressible fluid dynamics in the ͑1͒ The GLM EP variational principle with Lagrangian Euler–Poincare´ framework. given by ͑7.1͒ yields the motion equation for the GLM description of ideal incompressible MHD in Cartesian coordinates as VII. GLM, glm, AND ␣-MODELS OF IDEAL MHD DL . ˜uLϪ¯p͒ϩ͑¯uLϪ¯p ٌ͒¯uLϩٌ␲˜ ϭcurl͑B␰ KϪ1͒ϫB¯͑ We conclude by highlighting the GLM, glm, and Dt k k k • ␣-model sequence of results for a new application—LA ap- ͑7.2͒ proximations of incompressible ideal MHD ͑magnetohydro- When T⌶ϭIdϩٌ␰, for a vector field ␰ϭx␰Ϫx, the ͒ ␲ dynamics . mean fluid quantities ˜vi and ˜ are defined as in Eq.

͑3.16͒ and ͑3.17͒, C. EP glm equations for incompressible ideal MHD 1 ␦¯l ␦¯l 1 The total mean Lagrangian for the incompressible glm ϭ ϭ ␰͑ ⌶͒j ␲ϭϪ ϭϪ ͉ ␰͉2ϩ L ˜vi uj T i, ˜ u ¯p . MHD flow is given by subtracting the average magnetic en- ˜ ␦˜ui ␦¯ 2 D D ergy from the glm Lagrangian ͑5.1͒ in the small amplitude Consequently, one again finds for the total momentum approximation, to find density, ␦¯ ¯ϭ ͵ ͓ 1 ¯ ͉͑ ͉2ϩ͉ Ј͉2͒ϩ ͑ Ϫ ¯ ͒Ϫ 1͉͑¯͉2ϩ͉ Ј͉2͔͒ 3 l l 2D ¯u u ¯p 1 D 2 B B d x. D˜˜vϭ ϭD˜ ͑u␰ϩul ٌ␰ j͒ϵD˜ ͑¯uLϪ¯p͒, ͑7.3͒ ␦˜u j ͑7.5͒ ϭϪ l ٌ␰ j where ¯p u j is the usual pseudomomentum for 1,16,17 The corresponding variational derivatives are given by, cf. GLM ideal fluids. Thus, in GLM ideal MHD the Eq. ͑5.2͒, magnetic field does not contribute in the definition of total momentum. 1 ␦¯l͑¯u,D¯ ,B¯ ͒ϭ ͵ ͫ␦D¯ ͩ ͉͑¯u͉2ϩ͉uЈ͉2͒Ϫ¯pͪ ͑2͒ The GLM averaging process preserves the transport 2 structure of the original ideal MHD equations. In par- ϩ␦ ͑ Ϫ ¯ ͒ϩ ¯ ␦ ͑ Ϫ␰ϫ Ј ticular, it also yields preserved linking numbers— ¯p 1 D D ¯u• ¯u curl u magnetic helicity and cross helicity—involving the ϩٌ͑␰ Ј͒͒ϩ␦ ͑ Ј ͑ ¯ ␰͒͒ frozen-in averaged magnetic field and the Lagrangian •u ¯u• u div D mean velocity. Thus, GLM averaging preserves not the Ϫ␦¯ ͑¯ Ϫ␰ϫ Ј͒ͬ 3 magnetic linking numbers themselves, but the property B• B curl B d x. that the GLM averaged dynamics has invariant linking numbers. On setting D¯ ϭ1 and div ␰ϭ0, we obtain the same glm cir- Specifically, the usual MHD calculation shows that the culation velocity as Eq. ͑5.2͒ in Sec. V A, namely, magnetic helicity ͐A˜ B˜ d3x, where curl A˜ ϭB˜ ,isanin- .vϵ¯uϪ␰ϫcurl uЈϩٌ͑␰ uЈ͒¯ • variant of GLM-MHD, by virtue of the frozen-in auxil- • ͒ ϭ ٌ ͑ iary relation for B˜ . The corresponding EP glm motion equation with •¯u 0 is ͐ LϪ ˜ 3 expressed as The cross helicity (¯u ¯p)•Bd x is also an invariant of ץ GLM-MHD. The latter arises from vϩ¯u ٌ¯vϩ¯v ٌ¯u jϩٌ␲¯ ϭ͑¯JϩJЉ͒ϫB¯ , ͑7.6͒¯ t • jץ ͔ ¯LϪ ͒ ˜ ͒ϭϪ ͓͑͑ LϪ ͒ ¯ ͒ ϩ⌸ ͑͑ ץ t ¯u ¯p •B div ¯u ¯p •B ˜u B . ϭϪ ␰ϫ ␲ϭ Ϫ 1 ͉ ͉2ϩ͉ ͉2 Thus, the topological invariants of knottedness for ideal with JЉ curl( curl BЈ) and ¯ ¯p 2( ¯u ¯uЈ ). MHD have corresponding invariants for GLM-MHD. This is the EP equation for the glm MHD Lagrangian ͑7.11͒. B. Ideal glm MHD This motion equation also has the equivalent form, ץ The linearized Eulerian/Lagrangian fluctuation relation ¯vϪ¯uϫcurl¯vϩٌ͑¯v ¯uϩ␲¯ ͒ϭ͑¯JϩJЉ͒ϫB¯ . ͑7.7͒ • tץ for a magnetic field is

ЈϭϪL ϭ ͑␰ϫ¯ ͒ϭϪ␰ ٌ¯ ϩ¯ ٌ␰Ϫ¯ ␰ B ␰B¯ curl B • B B• B div . D. Kelvin circulation theorem for the incompressible ͑7.4͒ glm MHD The Kelvin circulation theorem for the incompressible Hence, the glm ideal MHD energy variation may be com- glm MHD Eq. ͑7.7͒ is puted as d Ͷ ͑¯uϪ␰ϫcurl uЈ͒ dxϭ Ͷ ͑͑¯JϩJЉ͒ϫB¯ ͒ dx. 1 dt c͑¯u͒ • c͑¯u͒ • ␦ ͵ ͉BЈ͉2d3xϭϪ͵ ␦B¯ ͑␰ϫcurl͑␰ϫB¯ ͒͒d3x. 2 • ͑7.8͒ Thus, for incompressible glm MHD the additional statis- The motion equation for a glm theory of ideal MHD energy tical element required for closure is is obtained by including this variational derivative in the ¯J ϫB¯ force for the EP equation above, where ¯JЉϭϪcurl͑␰ϫBЈ͒ϭϪcurl͑␰ϫcurl͑␰ϫB¯ ͒͒ ϭϪ ͑ ¯ ͒ ͑ ͒ ¯Jϭcurl͑B¯ Ϫ͑␰ϫBЈ͒͒, ad␰ ad␰ B , 7.9 ϭ␰ ٌ¯ Ϫ¯ ٌ␰ ¯ where ad␰ B • B B• . with BЈϭcurl͑␰ϫB¯ ͒.

The latter contribution to the current density is also a famil- ´ iar expression from the theory of Lagrangian MHD stability E. Euler–Poincare equations for a Lagrangian averaged incompressible MHD model analysis, see, e.g., Bernstein et al.28–30 This was kindly pointed out to the author by Caramana ͑private communica- Dropping derivatives of ␰ in the linearized ﬂuctuation tion͒. relations gives, cf. Eqs. ͑4.1͒, ͑4.3͒, and ͑7.4͒,

ЈϭϪ␰ ٌ ЈϭϪ␰ ٌ ¯ ЈϭϪ␰ ٌ¯ These are, respectively, the continuity equation for D¯ , the u • ¯u, D • D, and B • B. ͑7.10͒ frozen-in flux rule for B¯ and passive advection of the La- grangian covariance statistics. The pressure constraint im- We substitute relations ͑7.10͒ into the averaged Lagrangian ¯ ϭ ϭ for incompressible MHD, poses incompressibility D 1, so that div ¯u 0. We impose initial condition div B¯ ϭ0 which is then preserved by conser- ¯ϭ ͵ L¯ 3 ϭ ͵ ͓ 1 ¯ ͉͑ ͉2ϩ͉ Ј͉2͒ϩ ͑ Ϫ ¯ ͒ vation of the flux of the average magnetic field B¯ . l d x 2D ¯u u ¯p 1 D

Ϫ 1͉͑B¯͉2ϩ͉BЈ͉2͔͒d3x, ͑7.11͒ F. Conservation laws for average energy and 2 momentum to find H¯ ϭ͐ 3 Ϫ¯ The Hamiltonian m¯ •¯ud x l in this case is the 1 Legendre transformation of the Lagrangian ¯l in ͑7.11͒ with ¯l ϭ ͵ ͓ D¯ ͉͑¯u͉2ϩ¯u ¯ui ␰k␰l͒ϩ¯p͑1ϪD¯ ͒ 2 i,k ,l momentum Ϫ 1͉͑B¯͉2ϩ¯B ¯Bi ␰k␰l͔͒d3x. ͑7.12͒ ␦¯l ͒ ͑ ͒ ץ␰k␰l ¯ ץi,k ,l ϭ ϭ͑ ¯ Ϫ 2 m¯ ␦ D lD k ¯u. 7.16 Here the Lagrange multiplier ¯p enforces incompressibility of ¯u the averaged flow. The Euler–Poincare´ equation describing Explicitly, this Hamiltonian is ideal barotropic MHD with an advected positive symmetric ͒ ¯ Ϫ1͑ ¯ Ϫ1 ͒Ϫ ͑ Ϫ͒ ץ␰k␰l ¯ ץarray ␰k␰l is given in Holm, Marsden, and Ratiu2 in the H¯ ϭ ͵ ͓ 1 ͑ Ϫ ¯ Ϫ1 2m¯ • 1 D lD k D m¯ ¯p 1 D present notation as ϩ 1͉B¯͉2ϩ 1B¯ B¯ ␰k␰l͔d3x. ͑7.17͒ l ␦¯l 2 2 ,k• ,l¯␦ ץ l¯␦ ץ ϩ ͩ ¯uk ͪ ϩ ٌ¯uk k When evaluated on the constraint surface D¯ ϭ1 this is the ␦ ␦ ץ ␦ ץ t ¯u xk ¯u ¯u conserved energy, namely, ␦¯l ␦¯l ␦¯l ϭ ¯ ٌ ϩ¯ ϫ Ϫ ٌ␰k␰l ͑ ͒ 1 D B curl . 7.13 ¯Eϭ ͵ ͓͉¯u͉2ϩ¯u ¯u ␰k␰lϩ͉B¯͉2ϩB¯ B¯ ␰k␰l͔d3x. ␦D¯ ␦B¯ ␦␰k␰l 2 ,k• ,l ,k• ,l ͑7.18͒ If ␰k␰lϭ␣2␦kl and ␣2ϭconstant ͑which is admitted by the ¯ ϭ͐ 3 Taylor hypothesis of passive advection for ␰͒ then the last Likewise, momentum M m¯ d x is conserved, since the term will vanish. Otherwise, as we shall see, it will create an Lagrangian ¯l in ͑7.11͒ is invariant under translations. A short additional stress due to gradients of the Lagrangian statistical calculation using the EP equation ͑7.13͒ yields the local con- covariance ␰k␰l. The effect of the magnetic field is to add a servation law for momentum, ¯ ϫ¯ ץ Jtot B force in the motion equation, in this case with m¯ ϭϪ T j , ͑7.19͒ ץ x j iץ t i ␦¯l ץ␰k␰l ץϭϪ ϭ ¯ Ϫ ⌬ˆ ¯ ⌬ˆ ϭ ¯ Jtot curl curl B curl B, and l k . with stress tensor ␦B¯ ¯Lץ L¯ ␦¯lץ ¯ ϭ␦¯ ␦¯ ϭ¯ Ϫ⌬ˆ ¯ jϭ jϪ k ϩ¯ j Ϫ ¯ k One may interpret H l / B B B as the average mag- Ti m¯ i¯u ¯u,i B B,i k ¯ץ uk ␦¯ i¯ץ netic induction field and the difference H¯ ϪB¯ as the magne- , j B B, j tization induced by Lagrangian averaging. For the con- ͑ ͒ ´ ͑ ͒ ␦¯l ␦¯l strained Lagrangian 7.11 , the Euler–Poincare Eq. 7.13 ϩ␦ jͩ L¯ ϪD¯ Ϫ¯Bk ͪ i with D¯ ϭ1 leads to the following motion equation: ␦D¯ ␦¯Bk ϩ ٌ ϩ ٌ jϩٌ␲ϭ͑¯ϩ Љ͒¯ ϩ¯⌫ ٌ␰k␰l ץ t¯v ¯u• ¯v ¯v j ¯u ¯ J J B kl , ϭm¯ ¯u jϪD¯¯u ␰ j␰l¯uk Ϫ¯B j͑¯B Ϫ⌬ˆ ¯B ͒ ͑7.14͒ i k,l ,i i i k j 1 1 i ,͒ ¯ϭ¯ϩ ϩ¯B ␰ j␰l¯B ϩ␦ ͑¯pϩ ͉B¯͉2Ϫ ¯B ¯B ␰k␰lϪB¯ ⌬ˆ B ¯ ץ␰k␰l ץϭ ˆ⌬ with definitions l k as above and Jtot J JЉ with k,l ,i i 2 2 i,k ,l • ͑ ͒ ϭ Ϫ⌬ˆ ␲ϭ Ϫ 1͉ ͉2Ϫ i ␰k␰l 7.20 ¯v ¯u ¯u, ¯ p 2 ¯u ¯ui,k¯u,l , ץ␰k␰l ץϭ ˆ⌬ where l k is the generalized Laplacian. ¯Jϭcurl B¯ , JЉϭϪcurl ⌬ˆ B¯ , G. Kelvin circulation theorem ¯⌫ ϭϪ 1¯u ¯ui ϩ 1¯B ¯Bi . kl 2 i,k ,l 2 i,k ,l The Kelvin circulation theorem for the incompressible The auxiliary equations are averaged MHD Eq. ͑7.14͒ is ϭ ͑ ϫ¯ ͒ d ¯ ץ ϩ ¯ ϭ ¯ ץ tD div D¯u 0, tB curl ¯u B , Ͷ ϭ Ͷ ¯ ϫ¯ ϩ Ͷ ¯⌫ ͑␰k␰l͒ ¯v dx Jtot B dx kld , ͑7.15͒ dt c͑¯u͒ • c͑u͒ • c͑¯u͒ ͒ ͑ ␰k␰lϩ ٌ␰k␰lϭ ץ and t ¯u• 0. 7.21

¯ ϭ ¯ Ϫ ⌬ˆ ¯ ¯ ϫ¯ k l 2 kl 2 where Jtot curl B curl B. Thus, the Jtot B force and the ␰ ␰ ϭ␣ ␦ and ␣ ϭconstant, the Euler–Poincare´ Eq. ␰k␰l gradients of the Lagrangian covariance can both gener- ͑7.13͒ with D¯ ϭ1 leads to the following motion equation: ate circulation in Lagrangian averaged MHD at this level of ͒ ͑ ¯ϩ ٌ ϩ ٌ jϩٌ␲ϭ͑¯ϩ Љ͒ϫ ץ .closure t¯v ¯u• ¯v ¯v j ¯u ¯ J J B, 7.24 with ¯J ϭ¯JϩJЉ and H. Topological conservation laws for incompressible tot averaged MHD 1 ␣2 ,vϭ¯uϪ␣2⌬¯u, ␲¯ ϭpϪ ͉¯u͉2Ϫ ٌ͉¯u͉2¯ These equations in Lie-derivative form are 2 2 ץ ¯ϭ ¯ ЉϭϪ ␣2⌬¯ ͩ ϩL ͪ ͑¯v dx͒ϭϪd␲ϩ͑¯J ϫB¯ ͒ xϩ¯⌫ d͑␰k␰l͒, J curl B, J curl B. t u¨ • tot • klץ The auxiliary equations are ץ ͩ ϩL ͪ ͑¯ ͒ϭ ϭ ¯ ϭ ͒ ͑ ͒ ¯ϭ ͑ ϫ ¯ ץ ϩ ¯ ϭ ¯ ץ ,B dS 0, div ¯u 0, div B 0 ¯ t u • tD div D¯u 0 and ␫B curl ¯u B . 7.25ץ L ¯ where ¯u is the Lie derivative with respect to the Lagrangian These are the continuity equation for D and the frozen-in mean velocity, ¯u. Consequently, one finds the expected con- flux rule for B¯ . The pressure constraint imposes incompress- ͐ ¯ ¯ 3 ¯ ¯ ϭ ϭ servation of the magnetic helicity A•Bd x, where curl A ibility D 1, so that div ¯u 0. We impose initial condition ϭ¯ ¯ ϭ¯ ¯ ϭ B, or equivalently, d(A•dx) B•dS. This is a conse- div B 0 which is then preserved by conservation of the flux quence of the frozen-in motion for the flux of averaged mag- of the Lagrangian averaged magnetic field B¯ . netic field. Remarks: In contrast, one finds that gradients in the Lagrangian ͑ ͒ covariance ␰k␰l can prevent conservation of the cross helicity 1 In this subcase, constancy of the Lagrangian covariance ␰k␰lϭ␣2␦kl ͐¯v B¯ d3x, since we have allows conservation of both the helicity of • the average magnetic field and also the cross helicity of ͒ ͑ ϭϪ ͑͑ ¯ ͒ ϩ␲¯ ͒ϩ¯⌫ ¯ ٌ͑␰k␰l͒͒ ¯ ͑ ץ t ¯v•B div ¯v•B ¯u B klB• . 7.22 the average magnetic field and average velocity. By Eq. ͑7.22͒, for this case we have, Thus, the topological invariants of knottedness for ideal ͒ ͑ ͒ ¯ϭϪ ͑͑ ¯ ͒ ϩ␲͒ ¯ ͑ ץ MHD have corresponding invariants for the Lagrangian av- t ¯v•B div ¯v•B ¯u B . 7.26 eraged incompressible MHD model, only provided the last ͑2͒ The energy conservation law ͑7.23͒ sets the stage for term in Eq. ͑7.22͒ vanishes. proving global existence, uniqueness and regularity results for the incompressible MHD-␣ model when dissi- I. Incompressible MHD-␣ model pation is added. ͑3͒ Other incompressible averaged MHD closure options are ͑ ͒ The conserved energy 7.18 becomes the H1 norm in available, corresponding to other Taylor hypotheses than ϩ ٌ␰ϭ ␰ ץ ␰ץ both ¯u and B¯ , when one chooses the subcase ␰k␰lϭ␣2␦kl passive advection / t ¯u• 0 for and other ap- and ␣2ϭconstant, which is admitted by Taylor hypothesis proximations for uЈ and BЈ. These other closure options closure THC #1 that the Lagrangian fluctuation ␰ undergoes all allow straightforward generalizations of the case con- ϩ ٌ␰ϭ ץ ␰ץ passive advection / t ¯u• 0. Thus, in this subcase sidered here. when the Lagrangian covariance is isotropic and homoge- ͑4͒ The incompressible averaged MHD case may be modi- neous, the conserved energy is fied to include compressibility by following the steps for 1 the compressible case without magnetic fields discussed Eϭ ͵ ͓͉¯u͉2ϩ␣2ٌ͉¯u͉2ϩ͉B¯͉2ϩ␣2ٌ͉B¯͉2͔d3x, ͑7.23͒ in Ref. 24. Modifications to include Lagrangian averaged¯ 2 descriptions of Hall MHD and reduced MHD are also and the presence of ␣2Þ0 regularizes both the mean veloc- straightforward. ity and the mean magnetic field. Thus, for a given total en- ͑5͒ The Hamiltonian H¯ in Eq. ͑7.17͒ generates the equation ergy, the averaged solutions are more regular than the corre- of motion ͑7.14͒ and auxiliary Eq. ͑7.25͒ via the Lie– 2 sponding exact MHD solutions for which the terms in ␣ in Poisson bracket for ideal MHD discussed in Ref. 31. The 2 the energy are absent. The additional terms in ␣ can be methods of Holm, Marsden, Ratiu, and Weinstein31 can interpreted as proportional to the kinetic and magnetic en- also be used to classify the equilibrium solutions for this ergy dissipated by the exact MHD solutions in the time pe- new averaged MHD theory and determine their linear 2 2 riods, ␣ /␯ for viscosity and ␣ /␩ for resistivity. These and nonlinear stability conditions. terms represent the energy associated with turbulence and high field curvature at length scales less than ␣ that become Thus, the glm theory provides a bridge that spans from unresolvable as a result of the Lagrangian averaging process. the exact nonlinear ͑but unclosed͒ GLM theory to the alpha- In principle, one could choose different length scales for the model closures in the framework of Lagrangian averaging kinetic and magnetic energy modifications due to the aver- and averaged Lagrangians, as illustrated here in detail for aging, However, we shall introduce only ␣. ideal MHD. We hope this bridge will be useful in answering For the constrained averaged Lagrangian ͑7.11͒ with ho- questions that arise in the context of the alpha models and mogeneous isotropic Lagrangian fluctuation statistics with other turbulence closure models.

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