MHD-Drift Equations: from Langmuir circulations to MHD-dynamo?
aa r X i v : . [ phy s i c s . f l u - dyn ] S e p MHD-Drift Equations: from Langmuircirculations to MHD-dynamo?
By V. A. V l a d i m i r o v
Dept of Mathematics, University of York, Heslington, York, YO10 5DD, UK(Received Sept 12th 2011)
We have derived the closed system of averaged MHD-equations for general oscillat-ing flows, which are purely oscillating in the main approximation. We have used themathematical approach which combines the two-timing method and the notion of thedistinguished limit. Properties of the commutators are used to simplify calculations. Thedirect connection with a vortex dynamo (or the Langmuir circulations) has been demon-strated and a conjecture on the MHD-dynamo has been formulated.
1. Introduction
This paper derives the averaged MHD-equations for oscillating flows. The resultingequations are similar to the original MHD-equations, but surprisingly (instead of com-monly expected Reynolds stresses) the drift velocity (or just the drift ) plays a part of anadditional advection velocity.It is known that the drift can appear from either Lagrangian or Eulerian consid-erations. The
Lagrangian drift appears as the average motion of Lagrangian particlesand its theory is often based on the averaging of ODEs, see Stokes (1847), Lamb (1932),Longuet-Higgins (1953), Batchelor (1967), Andrews & McIntyre (1978), Craik (1982), Yudovich (2006).In this paper we focus on the
Eulerian drift , which appears as the results of the Eulerianaveraging of related PDEs without addressing the motion of particles, see Craik & Leibovich(1976),Craik (1985), Riley (2001), Vladimirov (2010), Ilin & Morgulis (2011). The detailed ma-terials about the Eulerian drift can be found in Vladimirov (2010).To derive the averaged equations we employ the two-timing method, see e.g.
Nayfeh (1973),Kevorkian & Cole (1996). We expose it as an elementary, systematic, and justifiable pro-cedure that follows the form developed by Yudovich (2006), Vladimirov (2005), Vladimirov (2008),Vladimirov (2010). This mathematical procedure is complemented by a novel materialon the distinguished limit, which allows to find the proper slow time-scale.
2. Functions and operations
We introduce functions of variables x = ( x , x , x ), s , and τ , which in the text belowserve as dimensionless cartesian coordinates, slow time, and fast time. Definition 1.
The class H of hat-functions is defined as b f ∈ H : b f ( x , s, τ ) = b f ( x , s, τ + 2 π ) (2.1)where the τ -dependence is always 2 π -periodic; the dependencies on x and t are notspecified. V. A. VladimirovDefinition 2.
For an arbitrary b f ∈ H the averaging operation is h b f i ≡ π Z τ +2 πτ b f ( x , s, τ ) dτ, ∀ τ (2.2)where during the τ -integration s = const and h b f i does not depend on τ . Definition 3.
The class T of tilde-functions is such that e f ∈ T : e f ( x , s, τ ) = e f ( x , s, τ + 2 π ) , with h e f i = 0 , (2.3)The tilde-functions are also called purely oscillating functions ( T -function represents aspecial case of H -function with zero average). Definition 4.
The class B of bar-functions is defined as f ∈ B : f τ ≡ , f ( x , s ) = h f ( x , s ) i (2.4)(any H -function can be uniquely separated into its B - and T - parts with the use of (2.2)) Definition 5. T -integration (or tilde-integration) : for a given e f we introduce a newfunction e f τ called the T -integral of e f : e f τ ≡ Z τ e f ( x , s, σ ) dσ − π Z π (cid:16)Z µ e f ( x , s, σ ) dσ (cid:17) dµ (2.5)which represents the unique solution of a PDE ∂ e f τ /∂τ = e f (with an unknown function e f τ and a known function e f ) supplemented by the condition h e f i = h e f τ i = 0 (2.3).The τ -derivative of T -function always represents T -function. However the τ -integrationof T -function can produce an H -function. An example: e f = f sin τ where f is anarbitrary B : one can see that h e f i ≡
0, however h R τ e f ( x , s, σ ) dσ i = f = 0, unless f ≡
0. Formula (2.5) keeps the result of integration inside the T -class. T -integration isinverse to τ -differentiation ( e f τ ) τ = ( e f τ ) τ = e f ; the proof is omitted. Definition 6.
A dimensionless function f = f ( x , s, τ ) belongs to the class O (1) f ∈ O (1) (2.6)if f = O (1) and all required partial x -, s -, and τ -derivatives of f are also O (1).Here we emphasize that through all the text below all large or small parameters arerepresented by various degrees of σ only; these parameters appear as explicit multipliersin all formulae containing tilde- and bar-functions; these functions always belong to O (1)-class.We also will use some properties of τ -derivatives such as b f τ = e f τ , h b f τ i = h e f τ i = 0 (2.7)The product of two T -functions e f and e g represents a H -function: e f e g ≡ b F , say. Separating T -part e F from b F we write e F = b F − h b F i = e f e g − h e f e g i = { e f e g } (2.8)where the notation {·} for the tilde-part is introduced to avoid two levels of tildes. Wewill use that the unique solution of a PDE inside the tilde-class is ∂ e f /∂τ = 0 ⇒ e f ≡ τ , then from the integration by parts we have h [ e a , e b τ ] i = −h [ e a τ , e b ] i = −h [ e a τ , b b ] i , h [ e a , e b τ ] i = −h [ e a τ , e b ] i = −h [ e a τ , b b ] i (2.10) HD-Drift Equations: from Langmuir circulations to MHD-dynamo? a , b ] stands for the commutator of two vector fields a and b which is antisymmetricand satisfies Jacobi’s identity for vector fields a , b , and c :[ a , b ] ≡ ( b · ∇ ) a − ( a · ∇ ) b , (2.11)[ a , b ] = − [ b , a ] , [ a , [ b , c ]] + [ c , [ a , b ]] + [ b , [ c , a ]] = 0 (2.12)A useful property of the commutator isdiv a = 0 , div b = 0 ⇒ div [ a , b ] = 0 (2.13)For any tilde-function e a and bar-function b (2.10),(2.12) give h [ e a , [ b , e a τ ]] i = [ b , V a ] where V a ≡ h [ e a , e a τ ] i (2.14)
3. Two-timing problem and distinguished limits
The governing equation for MHD-dynamics of a homogeneous inviscid incompressiblefluid with velocity field u ∗ , magnetic fields h ∗ , vorticity ω ∗ ≡ ∇ ∗ × u ∗ and current j ∗ ≡ ∇ ∗ × h ∗ is taken in the vorticity form ∂ ω ∗ ∂t ∗ + [ ω ∗ , u ∗ ] ∗ − [ j ∗ , h ∗ ] ∗ = 0 , in D ∗ (3.1) ∂ h ∗ ∂t ∗ + [ h ∗ , u ∗ ] ∗ = 0 , ∇ ∗ · u ∗ = 0 , ∇ ∗ · h ∗ = 0where asterisks mark dimensional variables, t ∗ -time, x ∗ = ( x ∗ , x ∗ , x ∗ )-cartesian coordi-nates, ∇ ∗ = ( ∂/∂x ∗ , ∂/∂x ∗ , ∂/∂x ∗ ), and [ · , · ] ∗ stands for the dimensional commutator(2.11). In this paper we deal with the transformations of equations, so the form of flowdomain D ∗ and particular boundary conditions can be specified at the later stages.We accept that the considered class of (unknown) oscillatory solutions u ∗ , h ∗ possessescharacteristic scales of velocity U , magnetic field H , length L , and high frequency σ ∗ U, H, L, σ ∗ ≫ /T ; T ≡ L/U (3.2)where T is a dependent time-scale. In the chosen system of units the dimensions of U and H coincide; we choose them being the same order U = H . The dimensionless variablesand frequency are x ≡ x ∗ /L, t ≡ t/T, b u ≡ b u ∗ /U, b h ≡ b h ∗ /U, σ ≡ σ ∗ T ≫ T slow (which can be differentfrom T ) and consider solutions of (3.1) in the form of hat-functions (2.1) u ∗ = U b u ( x , s, τ ) , h ∗ = U b h ( x , s, τ ); with τ ≡ σt, s ≡ Ω t, Ω ≡ T /T slow (3.4)Then the use of the chain rule and transformation to dimensionless variables give (cid:18) ∂∂τ + Ω σ ∂∂s (cid:19) b ω + 1 σ [ b ω , b u ] − σ [ b j , b h ] = 0 (3.5) (cid:18) ∂∂τ + Ω σ ∂∂s (cid:19) b h + 1 σ [ b h , b u ] = 0div b h = 0 , div b u = 0In order to keep variable s ‘slow’ in comparison with τ we have to accept that Ω /σ ≪ V. A. Vladimirov
Then eqn.(3.5) contains two independent small parameters: ε ≡ T σ ∗ = 1 σ , ε ≡ T slow σ ∗ ≡ Ω σ (3.6)Here we must make an auxiliary (technically essential) assumption: after the use of thechain rule (3.5) variables s and τ are (temporarily) considered to be mutually indepen-dent : τ, s − independent variables (3.7)From the mathematical viewpoint the increasing of the number of independent variablesin a PDE represents a very radical step, which leads to an entirely new PDE. This stepshould be justified a posteriori by the estimations of the error of the obtained solution(rewritten back to the original variable t ) substituted to the original equation (3.1).In a rigorous asymptotic procedure with σ → ∞ one has to consider asymptotic pathson the ( ε, ε )-plane such that ( ε, ε ) → (0 ,
0) (3.8)Each such path can be prescribed by a particular function Ω( σ ). One may expectthat there are infinitely many different (although some of them can coincide) solutionsto (3.5) corresponding to different Ω( σ ). However for these equations (as well as formany others) a unique path can be found, which is called the distinguish limit (or the distinguished path ). The notion of the distinguish limit is practical and heuristic, seeNayfeh (1973), Kevorkian & Cole (1996); its definition can vary for different equationsand in different books and papers. For our problem we write that the distinguished limitis given by such a function Ω = Ω d ( σ ) that allows to build a self-consistent asymptoticsolution . Here the term self-consistent asymptotic solution means that the required suc-cessive approximations can be calculated. These calculations include the elimination ofthe reducible secular in s terms; the reducible secular terms are such terms which can beexcluded by increasing the slow time-scale. For instance, a non-secular term proportionalto sin s gives secular terms in its Taylor’s decomposition with respect to t = σs ). Belowwe show that the choiceΩ( σ ) = 1 /σ : τ = σt, s = t/σ, α = const (3.9)allows to build the distinguished limit solution. The uniqueness of such a path for theconsidered class of solutions can be proven but we avoid such details in this paper. Hencethe governing equations are b ω τ + ε [ b ω , b u ] − ε [ b j , b h ] + ε b ω s = 0 , ε ≡ /σ → b h τ + ε [ b h , b u ] + ε b h s = 0div b u = 0 , div b h = 0where the subscripts τ and s denote the related partial derivatives.
4. Derivation of the MHD-Drift averaged equation
Let us look for solutions of (3.10) in the form of regular series( b h , b u ) = ∞ X k =0 ε k ( b h k , b u k ); b h k , b u k ∈ H ∩ O (1) , k = 0 , , , . . . (4.1) HD-Drift Equations: from Langmuir circulations to MHD-dynamo? u ≡ , h ≡ b ω τ = e ω τ = 0; b h τ = e h τ = 0 (4.3)Their unique solution (2.9) is e ω ≡ e h ≡
0. Taking into account (4.2) we can write b ω ≡ , b h ≡ b ω τ = 0; b h τ = 0 (4.5)which have the unique solution e ω ≡ , e h ≡ , ω = ? , h = ? (4.6)where mean functions remain undetermined. The equations of second approximation thattake into account (4.2),(4.4),(4.6) are e ω τ + [ ω , e u ] = 0 , e ω τ + [ ω , e u ] = 0 (4.7)which after T -integration (2.5) yield e ω = [ e u τ , ω ] , e h = [ e u τ , h ] , ω = ? , h = ? (4.8)The equations of third approximation that take into account (4.2),(4.4),(4.6) are e ω τ + ω s + [ b ω , e u ] + [ ω , b u ] − [ j , h ] = 0 (4.9) e h τ + h s + [ b h , e u ] + [ h , b u ] = 0The bar-part (2.2) of this system is ω s + [ ω , u ] − [ j , h ] + h [ e ω , e u ] i = 0 (4.10) h s + [ h , u ] + h [ e h , e u ] i = 0which can be transformed with the use of (4.8) and (2.14) to the final form ω s + [ ω , u + V ] − [ j , h ] = 0 (4.11) h s + [ h , u + V ] = 0 V ≡ h [ e u , e u τ ] i (4.12)If one uses these equations as a closed mathematical model, then all the subscripts andbars can be deleted: ω s + [ ω , u + V ] − [ j , h ] = 0 (4.13) h s + [ h , u + V ] = 0 V ≡ h [ e u , e u τ ] i (4.14)One can see that: V. A. Vladimirov
1. The equation for the oscillating velocity e u in our consideration is absent, thereare only two restrictions: e u is incompressible and potential. Hence the drift velocity V (4.12) represents a function that is ‘external’ to the equations.2. The derived system of equations (4.11) looks similar to the original one (3.1). Onemay think that (4.11) describes ‘just’ an additional advection of vorticity and magneticfield by the drift. However, the fact that the averaged vorticity is transported with suchan additional velocity is highly non-trivial; in particular, it contains the possibility of the vortex dynamo or Langmuir circulations, which we consider below.
5. Stokes drift
Our description of the drift differs from a classical one, therefore we first demonstratethe match of (4.12) with the classical Stokes drift. Let velocity field e u and e ξ ≡ e u τ be e u ( x , s, τ ) = p ( x , t ) sin τ + q ( x , t ) cos τ (5.1) e ξ ( x , t, τ ) = − p ( x , t ) cos τ + q ( x , t ) sin τ with arbitrary B -functions p and q . Straightforward calculations yield[ e u , e ξ ] = [ p , q ] (5.2)hence the commutator is surprisingly not oscillating. The drift velocity (4.12) is V = 12 h [ e u , e ξ ] i = 12 [ p , q ] (5.3)The dimensional solution for a plane potential harmonic travelling wave is b u ∗ = U e u , , e u = exp( k ∗ z ∗ ) (cid:18) cos( k ∗ x ∗ − τ )sin( k ∗ x ∗ − τ ) (cid:19) (5.4)where ( x ∗ , z ∗ ) are cartesian coordinates and k ∗ = 1 /L is a wavenumber. In Stokes (1847),Lamb (1932), Debnath (1994) one can see that U = k ∗ g ∗ a ∗ /σ ∗ where a ∗ and g ∗ aredimensional spatial wave amplitude and gravity; however these physical details are ex-cessive for our analysis. The dimensionless velocity field (5.4) and e ξ are e u = e z (cid:18) cos( x − τ )sin( x − τ ) (cid:19) , e ξ = e z (cid:18) − sin( x − τ )cos( x − τ ) (cid:19) (5.5)where both fields (5.5) are unbounded as z → ∞ , but it is not essential for our purposes.Fields p ( x, z ), q ( x, z ) (5.1) are p = Ae z (cid:18) sin x − cos x (cid:19) , q = Ae z (cid:18) cos x sin x (cid:19) (5.6)The calculations (with the use of (5.2)) yield V = e z (cid:18) (cid:19) (5.7)The dimensional version of (5.7) is V ∗ = U k ∗ σ ∗ e k ∗ z ∗ (cid:18) (cid:19) (5.8)which coincides with the classical expression for the drift velocity given by Stokes (1847),Lamb (1932), Debnath (1994). To obtain (5.8) one should take into account the differencebetween the time t and s = t/σ (3.9). HD-Drift Equations: from Langmuir circulations to MHD-dynamo?
6. The averaged Euler’s equations and vortex dynamo
A special case of (4.11),(4.13) without a magnetic field is ω s + [ ω , u + V ] = 0 . div u = 0 (6.1)Different s -independent versions of eqn.(4.11) were derived in the studies of Langmuir cir-culation by Craik & Leibovich(1976) and for the steady streaming problems by Riley (2001),Ilin & Morgulis (2011); the methods employed by these authors are different and morecumbersome than our method. In order to demonstrate the possibility of vortex dynamowe first notice that eqn.(6.1) can be integrated (in space) as u s + ( u · ∇ ) u + ω × V = −∇ p, ∇ · u = 0 (6.2)where p is a function of integration and the second equation follows from the continuityequation in (3.1). Let the zero approximation (4.4) represent the plane potential travel-ling gravity wave (5.5) with the drift velocity (5.7). Let cartesian coordinates ( x, y, z ) besuch that V = ( U, , U = e z , u = ( u, v, w ) where all components are x -independent(translationally-invariant), and x, z -variables coincide with ones in (5.5). Then the com-ponent form of (6.2) is u s + vu y + wu z = 0 v s + uv y + wv z − U u y = − p y w s + vw y + ww z − U u z = − p z v y + w z = 0which can be rewritten as (see Vladimirov (1985), Vladimirov (1985a)) v s + vv y + wv z = − P y − ρ Φ y (6.3) w s + vw y + ww z = − P z − ρ Φ z v y + w z = 0 ρ s + uρ x + vρ y = 0where ρ ≡ u , Φ ≡ U = e z , and P is a modified pressure. One can see that (6.3) ismathematically equivalent to the system of equations for an incompressible stratifiedfluid, written in Boussinesq’s approximation. The effective ‘gravity field’ g = −∇ Φ =(0 , , − e z ) is non-homogeneous that makes the analogy with a ‘standard’ stratified fluidnon-complete. Nevertheless one can see that any increasing function u ( z ) ≡ ρ ( z ) (takenfrom the shear flow ( u, v, w ) = ( u ( z ) , ,
7. Discussion
1. The consideration of this paper is based on the assumption that the enforced fre-quency σ ∗ (3.3) of oscillations is higher than all intrinsic frequencies. This frequencyappears in our theory via the prescribed potential velocity e u (4.3).2. The prescribed oscillatory velocity e u can be caused by different factors. For exam-ple, it can be produced by oscillations of boundaries or appear in full viscous theory afterthe matching of external flow with boundary-layer solution. The latter option is oftenconsidered, see Riley (2001), Vladimirov (2008), Ilin & Morgulis (2011).3. To justify the distinguished limit (3.9) mathematically, one should prove that any V. A. Vladimirov different path Ω( σ ) produces an asymptotic solution which contains terms secular in s ordoes not produce any asymptotic solution at all. The following statement can be proven:for V = 0 (4.12) and the function Ω( σ ) = 1 /ω α (with a constant α > −
1) all solutionswith α < α > V ≡ T slow = σ ∗ T in the system.5. The consideration of translationally-invariant MHD-motion in (4.11) is possible inthe spirit of the analogy between MHD flows and stratified flows, see Vladimirov, Moffatt and Ilin (1996).6. The mathematical justification of the equation (4.11) by the estimation of the errorin the original equation (3.1) is easily achievable.7. The higher approximations of the averaged equation (4.11) can be derived. Theyare especially useful for the study of motions with V ≡
0. In particular, one can showthat in this case Langmuir circulations cab be still generated by a similar mechanism.8. The viscosity and diffusivity can be routinely incorporated in (4.11) as the RHS-terms ν ∇ ω and κ ∇ h . Accordingly, viscous and diffusion terms will appear in theequations (6.2) and (6.3). At the same time after the incorporation of viscosity onemore small parameter appears in the list (3.6), and the distinguished limit should bereconsidered.9. The incorporation of the density stratification and gravity field into presented theory(or as a separate theory) is straightforward.10. The abolishing of the requirement of a vanishing mean flow in zeroth approximation(4.2) is also straightforward. However in this case the distinguished limit (3.9) is differentand the resulted averaged equations are more complicated than (4.11).11. For the finite and time-dependent flow domain D ( t ) the definition of average (2.2)directly works only if x ∈ D at any instant. If at some instant x / ∈ D then the theoryshould include a ‘projection’ of the boundary condition on the ‘undisturbed’ boundary.Such a consideration requires the smallness of the amplitude a ∗ /L of spatial oscillationsof fluid particles. However one can see that a ∗ ∼ u ∗ /σ ∗ and hence a ∗ /L ∼ /σ ≡ ε (3.6).Therefore the considering of a time-dependent domain does not introduce any new smallparameter, and the distinguished limit (3.9) will stay the same. In particular, it meansthat our small parameter ε is the same as the dimensional slope of free surface in thetheory of Langmuir circulations by Craik & Leibovich(1976).12. The determining of the function u ( z ) ≡ ρ ( z ) in (6.3) (for real Langmuir circulations)requires an additional theory which includes viscosity or turbulent tangential stresses atthe free surface. It is interesting, that from this viewpoint an ‘unstable stratification’ canbe continuously generated and amplified by tangential stresses applied at free surface.We do not compete here with the theories by Craik & Leibovich(1976), Leibovich (1983),Thorpe (2004), which describe this phenomenon well.13. One can suggest that since the equations (4.11) for h ≡ h = 0 is similar, then these full equations could also describe a possiblemechanism of MHD-dynamo, such as the generation of the magnetic field of the Earth.The author thanks the Department of Mathematics of the University of York for theresearch-stimulating environment. The author is grateful to Profs. A.D.D.Craik, K.I.Ilin,S.Leibovich, H.K.Moffatt, A.B.Morgulis, N.Riley, and V.A.Zheligovsky for helpful dis-cussions. HD-Drift Equations: from Langmuir circulations to MHD-dynamo? REFERENCESAndrews, D. and McIntyre, M.E.
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