Stretch fast dynamo mechanism via conformal mapping in Riemannian manifolds
aa r X i v : . [ m a t h - ph ] A ug Stretch fast dynamo mechanism viaconformal mapping in Riemannianmanifolds
L.C. Garcia de Andrade Abstract
Two new analytical solutions of self-induction equation, in Rie-mannian manifolds are presented. The first represents a twisted mag-netic flux tube or flux rope in plasma astrophysics, which shows thatthe depending on rotation of the flow the poloidal field is amplifiedfrom toroidal field which represents a dynamo. The value of the am-plification depends on the Frenet torsion of the magnetic axis of thetube. Actually this result illustrates the Zeldovich stretch, twist andfold (STF) method to generate dynamos from straight and untwistedropes. Motivated by the fact that this problem was treated usinga Riemannian geometry of twisted magnetic flux ropes recently de-veloped (Phys Plasmas (2006)), we investigated a second dynamosolution which is conformally related to the Arnold kinematic fastdynamo. In this solution it is shown that the conformal effect onthe fast dynamo metric only enhances the Zeldovich stretch, andtherefore a new dynamo solution is obtained. When a conformalmapping is performed in Arnold fast dynamo line element a uniformstretch is obtained in the original line element.
PACS numbers:02.40.Hw-Riemannian geometries Departamento de F´ısica Te´orica - Instituto de F´ısica - UERJRua S˜ao Fco. Xavier 524, Rio de Janeiro, RJMaracan˜a, CEP:20550-003 , Brasil.E-mail:[email protected].
Introduction
Geometrical tools have been used with success [1] in Einstein general rela-tivity (GR) have been also used in other important areas of physics, such asplasma structures in tokamaks as been clear in the Mikhailovskii [2] bookto investigate the tearing and other sort of instabilities in confined plasmas[2], where the Riemann metric tensor plays a dynamical role interacting withthe magnetic field through the magnetohydrodynamical equations (MHD).Recently Garcia de Andrade [3, 4] has also made use of Riemann metric toinvestigate magnetic flux tubes in superconducting plasmas. Thiffault andBoozer [5] following the same reasoning applied the methods of Riemann ge-ometry in the context of chaotic flows and fast dynamos. Yet more recentlyThiffeault [6] investigated the stretching and Riemannian curvature of mate-rial lines in chaotic flows as possible dynamos models. An interesting tutorialreview of chaotic flows and kinematical dynamos has been presented earlierby Ott [7]. Also Boozer [8] has obtained a geomagnetic dynamo from conser-vation of magnetic helicity. Actually as pointed out by Baily and Childress[9] have called the attention to the fact that the focus nowadays the focus ofkinematic dynamo theory is on the fast dynamos, specially in construction oncurved Riemannian manifolds to stretch, twist and fold magnetic flows fila-ments or tubes to generated dynamo solutions. This method was invented byZeldovich [10]. In this paper taking the advantage of the success of conformalRiemannian geometry techniques used to find out new solutions of Einsteingeneral relativistic field equations[1], we use this same conformal geometricaltechnique to find new solutions of the incompressible flows in Arnold metric[11, 12, 13]. To resume , finding and being able to recognize the existenceof dynamos or non-dynamos is not only important from physical and mathe-matical point of view, but also finding mathematical techniques which allowsus to obtain new dynamo solution from dynamos or either nondynamo met-rics such as the one we consider here as a flavour of the difficult to find adynamo solution in Riemannian manifolds. This also motivates us to showthat conformally Arnold fast dynamo metric, is also a dynamo solution. Re-cently Hanasz and Lesch [14] have used also a conformal Riemannian metricin E to investigate the galactic dynamo also using magnetic flux tubes. Alsorecently, Kambe [15] and Hattori and Zeitlin [16], have investigated the rateof stretching of Riemannian line elements of imcompressible fluids, in theframework of differential geometry of diffeomorphisms, in the spirit of Zel-2ovich stretching. Actually they considered the exponential stretching ofline elements in time, or dynamo action, in the context of negative curva-ture in turbulent flows. They also considered the concentration of vortexand magnetic flux tube. This provides a strong physical motivation to thepresent investigation. The paper is organized as follows: In section II thethe dynamo Riemann metric representing flux rope analytical solution of theself-induction equation in the case or zero resistivity is presented. In sectionIII the dynamo solution in Riemannian conformal geometry is given. In sec-tion IV Riemannian curvature of a particular conformal dynamo is computedand in section V conclusions are presented. II Thin flux rope dynamos in Riemannianmanifold
In this section we shall consider generalization of the Riemann metric ofa stationary twisted magnetic flux tube as considered by Ricca [17] in theRiemann manifold to address the nonstationary case where the toroidal andpoloidal magnetic fields, in principle, may depend on time. With this metricat hand , we are to solve analytical the self-induction magnetic flow equationto check for the dynamo existence. Let us now start by considering the MHDfield equations ∇ . ~B = 0 (1) ∂∂t ~B − ∇× [ ~u × ~B ] − η ∇ ~B = 0 (2)where ~u is a solenoidal field while η is the diffusion coefficient. Equation(2) represents the self-induction equation. The vectors ~t and ~n along withbinormal vector ~b together form the Frenet frame which obeys the Frenet-Serret equations ~t ′ = κ~n (3) ~n ′ = − κ~t + τ~b (4) ~b ′ = − τ~n (5)the dash represents the ordinary derivation with respect to coordinate s, and κ ( s, t ) is the curvature of the curve where κ = R − . Here τ represents the3renet torsion. The gradient operator becomes ∇ = ~t ∂∂s + ~e θ r ∂∂θ + ~e r ∂∂r (6)Now we shall consider the analytical solution of the self-induction magneticequation investigated which represents a non-dynamo thin magnetic fluxrope. Before the derivation of this result , we would like to point it outthat it is not trivial, since the Zeldovich antidynamo theorem states that thetwo dimensional magnetic fields do not support dynamo action, and here aswe shall see bellow, the flux tube axis possesses Frenet curvature as well astorsion and this last one cannot take place in planar curves. Let us nowconsider here the metric of magnetic flux tube ds = dr + r dθ R + K ( s ) ds (7)This is a Riemannian line element ds = g ij dx i dx j (8)if the tube coordinates are ( r, θ R , s ) [15] where θ ( s ) = θ R − R τ ds where τ isthe Frenet torsion of the tube axis and K ( s ) is given by K ( s ) = [1 − rκ ( s ) cosθ ( s )] (9)Since we are considered thin magnetic flux tubes, this expression is K ≈ ∇ in curvilinear coordinates [16] one obtains ∇ = 1 √ g ∂ i [ √ gg ij ∂ j ] (10)where ∂ j := ∂∂x j and g := detg ij where g ij is the covariant component of theRiemann metric of the flux rope. Here, to better compare the dynamo actiongeneration of toroidal field from poloidal fields we shall consider the that thetoroidal component of magnetic field B s ( s ) is given in the Frenet frame as ~B s = b ( s ) ~t (11)4ote also that we have considered that the flux rope magnetic field does notdepend on the r and θ R coordinates. While the poloidal magnetic field themagnetic field here can be expressed as ~B θ ( t, θ ) = e pt b ~e θ (12)Now let us substitute the definition of the poloidal plus toroidal magneticfields into the self-induction equation, which with the help of the expressions ~e θ = − ~nsinθ + ~bcosθ (13) ∂ θ ~e θ = − ~n [(1 + τ − κ ) sinθ + cosθ ] − ~b [ cosθ + sinθ ] (14)and ∂ t ~e θ = ω~e θ − ∂ t ~nsinθ + ∂ t ~bcosθ (15)Considering the equations for the time derivative of the Frenet frame givenby the hydrodynamical absolute derivative˙ ~X = ∂ t ~X + [ ~v. ∇ ] ~X (16)where ~X = ( ~t, ~n,~b ) represents the Frenet frame into the expressions for thetotal derivative of each Frenet frame vectors˙ ~t = ∂ t ~t + [ κ ′ ~b − κτ~n ] (17)˙ ~n = κτ~t (18)˙ ~b = − κ ′ ~t (19)one obtains the values of respective partial derivatives of the Frenet frame as ∂ t ~t = − τ κ [1 − κτ − v θ r ] ~n (20) ∂ t ~n = τ κ [1 − κ~τ − v θ r ] ~t + v θ r ~b (21) ∂~b = κτ − v θ r ~n (22)where we have used the hypothesis that ˙ ~b = 0 or κ ′ ( t, s ) = 0, which meansthat the curve curvature only depends on time. A simple example from solar5hysics, would be a flux tube curved and with torsion oscillating with fixedsun spots. Substitution of these vectorial expressions into expression (15)yields ∂ t ~e θ = [ ωcosθ − v θ r ] ~b − [ ωsinθ − τ − κ v θ r ] ~n + [ κτ (1 + τ − v θ r ) sinθ ] ~t (23)along with the equation ∂B θ ∂s = B θ rτ κ (24)and the fact that ∂B s ∂s = 0, together with the self-induction equation we obtainthe following system of equations, for a highly conductive fluid as our ownuniverse, with resistivity η = 0 ∂ t B θ + τ v θ sinθB θ = 0 (25) sinθB θ + B s sinθ [1 + τ − v θ r ] = 0 (26)To obtain these last two expressions we assume that v θ >>> v s and that ∂ s vs = 0 and that the continuity equation ∇ .~v = 0 (27)where we have considered that the flow is imcompressible which is a reason-able approximation in plasma physics. This expresion yields ∂v θ ∂s + v θ rτ κ = 0 (28)Equation (25) can be rewritten as[ p + τ v θ sinθ ] = 0 (29)which upon substitution on the equation (26) yields B θ B s sinθ [1 + τ − v θ r ] = 0 (30)which with the assumption that the flux tube has a small twist and v θ << B θ B s = τ ωrp (31)6ince B s ≈ constantb by hypothesis, we have that relation (31) tells usthat the relation τωrp > r > p ωτ . Thedivergence-free equations for the magnetic and flow fields, allows us to writedown the solutions for B θ B θ = B e pt − R rR cosθdθ (32)which if we recall the definition of the deviation of flat Riemann metric K ofthe tube above , we may express the integral in terms of K ( s ) as B θ = B e pt − R (1 − K ( s )) dθ (33)which shows that in the very thin flux rope dynamo in this solution the effectcurvature of the Riemannian tube is minor. Here we have used the fact thatthe external curvature of the rope is given by κ = κ . The solution fortorsion and velocity flow are essentially analogous. III Conformal dynamos on manifolds
Conformal mapping on a Riemannian line element can in general be repre-sented by ds = e λ ( ~x ) ( ds ) (34)Manifolds related in this manner, are said conformally related. Note alsothat this is intrinsically connected to the stretch part of STF mechanism togenerate rope dynamos. Actually in the previous section we considered aRiemannian metric for twisted flux tube which was stretched solely on theds element along the magnetic axis of the dynamo rope, through the factor K ( s ), of course when the flux rope is thin, the this stretch effect almost vanishthough twist and fold may still be kept. Therefore strictly speaking, this isnot a conformal mapping but only a stretch, therefore strictly speaking not allstretches are represented by conformal mapping but every conformal metricrepresents stretching in the Riemannian manifold. One of disadvantages ofconformal stretching is that the stretching is uniform as in the case of Arnold7ast dynamo metric. Conformal metric techniques have also been widely usedas a powerful tool obtain new solutions of the Einstein’s field equations of GRfrom known solutions. By analogy, here we are using this method to yield newsolutions of MHD dynamo from the well-known fast dynamo Arnold solution.We shall demonstrate that distinct physical features from the Arnold solutionmaybe obtained. Before that we just very briefly review the Arnold solution.The Arnold metric line element can be defined as [11] ds = e − λz dp + e λz dq + dz (35)which describes a dissipative dynamo model on a 3D Riemannian manifold.By dissipative here, we mean that contrary to the previous section, the re-sistivity η is small but finite. The flow build on a toric space in Cartesiancoordinates ( p, q, z ) given by T × [0 ,
1] of the two dimensional torus. The co-ordinates p and q are build as the eigenvector directions of the toric cat mapin R which possesses eigenvalues as χ = (3+ √ > χ = (3 −√ < ds = e λz + λ ) dp + e − λz − λ ) dq + e λ dz (36)which represents a simple global translation and is not changed at everypoint in the manifold. Let us now recall the Arnold et al [13] definition of aorthogonal basis in the Riemannian manifold M ~e p = e λz ∂∂p (37) ~e q = e − λz ∂∂q (38) ~e z = ∂∂q (39)Assume a magnetic vector field ~B on M ~B = B p ~e p + B q ~e q + B z ~e z (40)The vector analysis formulas in this frame are ∇ f = [ e λz ∂ p f, e − λz ∂ q f, ∂ z f ] (41)8here f is the map function f : R → R . The Laplacian is given by∆ f = ∇ f = [ e λz ∂ p f + e − λz ∂ q f + ∂ z f ] (42)while the divergence is given by ∇ . ~B = div ~B = div [ B p ~e p + B q ~e q + B z ~e z ] = [ e λz ∂ p B p + e − λz ∂ q B q + ∂ z B z ] (43)In particular one may write div~e p = div~e q = div~e z = 0 (44)in turn the curl is written as curl ~B = curl [ B p ~e p + B q ~e q + B z ~e z ] (45)where curl p ~B = e − λz ( ∂ q B z − ∂ z ( e λz B q )) (46) curl q ~B = − e λz ( ∂ p B z − ∂ z ( e − λz B p )) (47) curl z ~B = e λz ∂ p B q − e − λz ∂ q B p (48)and curl~e p = − λ~e q (49) curl~e q = − λ~e p (50) curl~e z = 0 (51)The Laplacian operators of the frame basis are∆ ~e p = − curlcurl~e p = − λ ~e p (52)∆ ~e q = − curlcurl~e q = − λ ~e q (53)∆ ~e z = 0 (54)from these expressions Arnold et al [13] were able to build the self-inducedequation in this Riemannian manifold as ∂ t B p + v∂ z B p = − λvB p + η [∆ − λ ] B p − λe λz ∂ p B z (55) ∂ t B q + v∂ z B q = + λvB q + η [∆ − λ ] B p − λe − λz ∂ q B z (56)9 t B z + v∂ z B z = η [∆ − λ∂ z ] B z (57)Decomposing the magnetic field on a Fourier series, Arnold et al were ableto yield the following solution b ( p, q, z.t ) = e λvt b ( p, q, z − vt,
0) (58)where B ( x, y, z, t ) = b ( p, q, z, t ) and the fast dynamo limit η = 0 was used.Now with these formulas , we are able to compute the solution of the self-induced magnetic equation in the background of conformal Riemannian lineelement ds = Ω( z )[ e − λz dp + e λz dq + dz ] (59)The reason for using the general conformal stretching factor Ω( z ) instead ofthe previous exponential stretching , is to show that the dynamo obtained isnot only due to the exponential stretching but conformal dynamos, allow forthe existence of more general conformal stretching. A far obvious, thoughimportant observation here is the fact that from the equations bellow, werecover the Arnold et al [13] if we simply make the conformal factor Ω := 1.Denoting the dual one form for Arnold basis as φ p = e − λz dp (60) φ q = e λz dq (61) φ z = dz (62)one obtains the Arnold fast dynamo metric as ds = φ p + φ q + φ z (63)Thus the conformal one form dual basis can be expressed as φ pC = Ω φ p (64) φ qC = Ω φ q (65) φ zC = Ω φ z (66)On the other hand the vector field basis in conformal metric becomes ~e p = Ω − e λz ∂∂p (67)10 e q = Ω − e − λz ∂∂q (68) ~e z = Ω − ∂∂z (69)Let us now repeat some of the fundamental vector analysis relations abovein the conformal geometry. The first is the Laplacian of ~B ∆ C ~B = Ω − ∆ ~B −
12 Ω − [ ∂ z Ω] ∂ z ~B (70)A fundamental change here in the conformal stretching in Riemannian ge-ometrical dynamos, is that of the velocity flow. In Arnold fast dynamoexample, the flow is a very simple one which is given by (0 , , v ) where v isconstant. Here the dynamo flow is effectively in the dynamo equation by aterm of the form ( ~v. ∇ ) B p = Ω − v∂ z B p (71)which shows that a nonconstant effective velocity flow such as v eff = Ω − v would act in conformal dynamos with respect to the previous Arnold exam-ple. The other fundamental component of the curl [ ~v × ~B ] given by( ~B. ∇ ) ~v = Ω − v [ ∂ z Ω] ~e z (72)With these expressions from conformal geometry in hand, we are now ableto express the Arnold et al dynamo equations are ∂ t ~B + ( ~v. ∇ C ) ~B = ( ~B. ∇ C ) ~v + η ∆ C ~B (73)In terms of components this conformal self-induced equation in the Rieman-nian manifold can be expressed as ∂ t B p +Ω − v∂ z B p = − λ Ω − vB p + η [(∆ − λ ) B p − λe λz ∂ p B z ] − η − ( ∂ z Ω)( ∂ z B p + λB p )(74)The equation for q component can be obtained from p one by simply per-forming the substitution λ → − λ . The expression for component-z is ∂ t B z + Ω − v∂ z B z = η [∆ − λ∂ z −
12 Ω − ∂ z Ω ∂ z ] B z (75)11ecomposing again the magnetic field on a Fourier series now in conformalgeometry, yields the following solution b ( p, q, z.t ) = e λ Ω − vt b ( p, q, z − vt,
0) = e λv eff t b ( p, q, z − vt,
0) (76)where is the conformal Riemannian fast dynamo solution. As given explicitlyin this solution the basic effect of the conformal geometry in fast dynamos ison the speed of dynamo which is an important physical effect.
IV Riemann curvature of conformal dynamos
The important role of negative curvature of geodesic flows in dynamos havebeen investigated by Anosov [19]. These Anosov flows, though somewhat ar-tificial, provide excelent examples for numerical computation experiments infast dynamos [20]. Within this motivation we include here a simple exampleof conformal spatially stretching, where Ω := e λz . This conformal stretchingapplied to Arnold metric yields the conformal metric as ds = dp + e λz dq + e λz dz (77)or in terms of the frame basis form ω i ( i = 1 , ,
3) is ds = ( ω p ) + ( ω q ) + ( ω z ) (78)The basis form are write as ω p = dp (79) ω q = e λz dq (80)and ω z = e λ z dq (81)By applying the exterior differentiation in this basis form one obtains dω p = 0 (82) dω z = 0 (83)and dω q = λe − λ z ω z ∧ ω q (84)12ubstitution of these expressions into the first Cartan structure equationsone obtains T p = 0 = ω pq ∧ ω q + ω pz ∧ ω z (85) T q = 0 = λe − λ z ω z ∧ ω q + ω qp ∧ ω p + ω qz ∧ ω z (86)and T z = 0 = ω zp ∧ ω p + ω zq ∧ ω q (87)where T i are the Cartan torsion 2-form which vanishes identically on a Rie-mannian manifold. From these expressions one is able to compute the con-nection forms which yields ω pq = − αω p (88) ω qz = λe − λ z ω q (89)and ω zp = βω p (90)where α and β are constants. Substitution of these connection form into thesecond Cartan equation R ij = R ijkl ω k ∧ ω l = dω ij + ω il ∧ ω lj (91)where R ij is the Riemann curvature 2-form. After some algebra we obtain thefollowing components of Riemann curvature for the conformal antidynamo R pqpq = λe − λ z (92) R qzqz = 12 λ e − λz (93)and finally R pzpq = − αλe − λ z (94)We note that only component to which we can say is positive is R pzqz whichturns the flow stable in this q-z surface. This component also dissipates awaywhen z increases without bounds, the same happens with the other curvaturecomponents [21]. 13 Conclusions
In conclusion, we have used a well-known technique to find solutions of Ein-stein’s field equations of gravity namely the conformal related spacetimemetrics to find a new anti-dynamo solution in MHD three-dimensional Rie-mannian nonplanar flows. Examination of the Riemann curvature [21] com-ponents enable one to analyse the stretch and compression of the dynamoflow. New conformal fast dynamo metric are obtained from the conformallyself-induced equation.It is shown that in the effect of conformal mappingin Riemannian dynamo flow is to change the fast dynamo speed. Futureperspectives includes the investigation of homological obstructions in theconformal geodesic flows from Anosov flows generalizing the investigation ofVishik [22] and Friedlander and Vishik [23].
Acknowledgements
I would like to dedicate this paper to Professor Vladimir I. Arnold on theocasion of his senventh birthday. I would like also to thank CNPq (Brazil)and Universidade do Estado do Rio de Janeiro for financial supports.14 eferenceseferences