with angular brackets denoting averages primes the corresponding residuals, then eq. (2) can be separated into two coupled equations for the time evol

Transcription

1 This paper was published in Europhys. Lett. 27, 353{357, 1994 Current Helicity the Turbulent Electromotive Force N. Seehafer Max-Planck-Gruppe Nichtlineare Dynamik, Universitat Potsdam, PF , D Potsdam, Germany PACS { Magnetohydrodynamics electrohydrodynamics PACS { Magnetohydrodynamics; plasma ow Abstract In a magnetouid, magnetic velocity uctuations B 0 v 0 can generate a mean electromotive force along the mean magnetic eld, which has been termed the alpha eect. In the present paper the mathematical relationship between the alpha eect the mean current helicity of the uctuations, hb 0 curl B 0 i, hitherto proven for the case that (i) either the magnetic or the velocity uctuations are statistically homogeneous stationary, (ii) the rst-order smoothing approximation (FOSA) is valid, (iii) the mean ow, hvi, vanishes, is rederived, assuming the magnetic uctuations to be stationary the coupled magnetic-electric uctuations to be homogeneous, but without using FOSA allowing for a non-vanishing mean ow. In order to explain the origin of the cosmical magnetic elds, the theory of the turbulent dynamo has been developed [1, 2, 3]. The central mechanism in this theory is the alpha eect, namely the generation of a mean electromotive force (emf) along a mean, or large-scale, magnetic eld by turbulently uctuating, or small-scale, parts of velocity magnetic eld. The eect has also been invoked to underst the plasma behaviour in fusion experiments [4]. It has been found that kinetic magnetic helicities can enable a turbulent dynamo eect. With v, B A denoting uid velocity, magnetic eld a magnetic vector potential, the densities per unit volume of kinetic, magnetic current helicity are dened by H K = v curl v; H M = A B; H C = B curl B: (1) Let the evolution of the magnetic eld be described by the induction = curl(v B) + r2 B (2) with a constant magnetic diusivity = ( 0 ) 1, denoting the electrical conductivity. If, as usual in mean-eld electrodynamics, velocity magnetic elds are split up into mean uctuating parts according to v = hvi + v 0 ; B = hbi + B 0 ; (3) 1

2 with angular brackets denoting averages primes the corresponding residuals, then eq. (2) can be separated into two coupled equations for the time evolution of hbi B 0, namely = curl(hvi hbi) + r 2 hbi + curl E; 0 = curl(v 0 hbi) + curl(hvi B 0 ) + r 2 B 0 + curl G; (5) E = hv 0 B 0 i (6) G = v 0 B 0 hv 0 B 0 i: (7) E is the mean emf caused by the uctuations. Formally the averages can be understood as ensemble averages. They can however physically meaningful be dened as space, time, or space-time averages if the turbulence has the two-scale property, i.e. if the characteristic scales (space, time, or space-time) of the uctuations are much smaller than those of the mean elds, so that proper means can be obtained by averaging over intermediate scales. Usually E is evaluated by assuming the mean velocity to vanish, hvi = 0; (8) applying the rst-order smoothing approximation (FOSA), which consists in neglecting curl G in eq. (5). Then eq. (5) simplies 0 = curl(v 0 hbi) + r 2 B 0 ; (9) which with hbi taken as a constant (justied in the case of a space-time two-scale turbulence with h i dened as space-time average) can be easily solved by using Fourier transformation techniques, allowing for an expression of B 0 therefore also of E in terms of v 0. One arrives so at relations revealing a connection between the alpha eect the kinetic helicity HK 0 of v 0. As found by Keinigs [5], who used eq. (9) to express E in terms of B 0, the alpha eect is however more directly related to current helicity than to kinetic helicity, namely by the equation E hbi = hbi 2 hbi 2 hb0 curl B 0 i: (10) In deriving this Keinigs assumed the magnetic uctuations to be statistically stationary, homogeneous symmetric about the direction of hbi. Then E hbi are parallel. For the case that the velocity uctuations represent a turbulence that is stationary homogeneous, but not necessarily rotationally symmetric about any axis (or even fully isotropic), the scalar has consequently to be replaced by a tensorial quantity, Radler Seehafer [6] derived the relation a ij hb i ihb j i = hb 0 curl B 0 i; (11) where Cartesian coordinates x 1 ; x 2 ; x 3 have been used the tensor a ij is dened by E i = a ij hb j i: (12) 2

3 Eq. (10) has been written in such a way that also this slightly more general case is covered. In the following the relation (10) is rederived without applying FOSA without using eq. (8): Let the magnetic uctuations be statistically stationary the coupled magnetic-electric uctuations be statistically homogeneous. To formulate the actually needed assumption on the spatial symmetry more precisely: The mixed magnetic-electric two-point correlation tensor Q ij (x; r) = hbi(x)e 0 j(x+r)i, 0 E 0 denoting the uctuating part of the electric eld E, is supposed to be independent of x. Our rederivation of eq. (10) will be essentially based on a consideration of the mean value of the uctuating part of the magnetic helicity, ha 0 B 0 i. Now a statistically homogeneous eld is by denition of innite spatial extent, so that it is dicult to dene a proper vector potential for B 0. This diculty can be overcome by working with relative magnetic helicity, H R, as dened by Berger Field [7]. H R is not a helicity density, as e.g. H M, but measures the total, i.e. volume-integrated helicity of a magnetic eld in a (in general nite) volume V, namely by comparing it to the current-free eld with the same normal component on. More explicitly: The eld B given in V is extended into the exterior ^V of V by the current-free eld in ^V with the same normal component on as the given eld. From the total magnetic helicity of the entire eld dened in all space one then subtracts that of the eld B p which is current-free on both sides of whose normal component on is again that of the given eld (B p is thus generated by a current-sheet on ). H R depends only on the eld B in V. Let V be a sphere cut out of our innitely extended eld. The rate of change of H R in V is given by [7] dh R (V ) = 2 E B dv + 2 (A p E) n ds: (13) dt V Here n is the exterior unit normal on A p the Coulomb vector potential (satisfying div A p = 0) of B p. The volume integral on the right of eq. (13) is due to the internal dissipation of magnetic helicity, the surface integral describes the ow of magnetic helicity through. For the uctuating part HR 0 of H R one nds a relation fully analogous to eq. (13), which after averaging reads dhhr 0 (V )i = 2 he 0 B 0 i dv + 2 ha 0 p E 0 i n ds: (14) dt V A coupling between uctuating mean elds does not occur here, since only linear equations are used to dene H R to calculate its rate of change. So eq. (14) is derived from Maxwell's equations div B 0 = 0 = = curl E 0 B 0 = curl A 0. Because of the assumed statistical stationarity of the magnetic uctuations, the left-h side of eq. (14) must vanish. The surface integral on the right-h side vanishes as a consequence of the spatial homogeneity of the correlation between magnetic electric uctuations: A 0 p can be represented in the form A 0 p(x) = 0 4 J 0 p(x a ) jx x a j ds a; (15) where J 0 p is the density of the sheet current on that generates B 0 p. It is given by the jump 4B 0 p of the tangential component of B 0 p across, 0 J 0 p(x a ) = n a 4B 0 p(x a ): (16) 3

4 By using appropriate Green's functions for the potential-eld problems in V its complement ^V, 4B 0 p in turn admits of a representation 4B 0 p(x a ) = (B 0 (x b ) n b ) G(x a ; x b ) ds b : (17) The detailed form of the vector-valued function G(x a ; x b ) is not needed here. We shall only use the fact that A 0 p at a given position on is obtained as the superposition of the inuences of the values of B 0 n at all other points. Let r; ; be spherical polar coordinates consider that part of the helicity ux at the north pole, = 0, of our sphere which is determined by the value of B 0 n at the point ( 0 ; 0 ). Compare this with the helicity ux at ( 0 + ; 0 ) due to the inuence of the south pole, =. The vector r from ( 0 ; 0 ) to the north pole is equal to that from the south pole to ( 0 + ; 0 ). So the two-point correlation between uctuating magnetic electric elds is the same for both point pairs the considered parts of the helicity uxes at the north pole at the point ( 0 + ; 0 ) cancel one another. The latter can perhaps be seen more clearly from the fact that the relevant quantities for the second point pair are obtained from those of the rst one by a mirror reexion in the plane through the center of the sphere perpendicular to the radius vector of the point ( 0 =2; 0 ). The helicity ow as a pseudo-scalar quantity must change sign under such a reexion. Since in this way for any contribution to the helicity ow another one just cancelling it can be found, the total ow of relative magnetic helicity through must vanish. Letting now V! 0, we arrive at he 0 B 0 i = 0: (18) Next we derive an expression for he 0 B 0 i following from Ohm's law. Using 0 j = curl B eq. (6), the unaveraged averaged forms of Ohm's law can be written as hji + j0 E = (hvi + v 0 ) (hbi + B 0 ) = curlhbi + curl B 0 hvi hbi hvi B 0 v 0 hbi v 0 B 0 (19) For their dierence we then nd E 0 = E hei hei = curlhbi hvi hbi E: (20) = curl B 0 hvi B 0 v 0 hbi v 0 B 0 + E; (21) which by scalar multiplication with B 0 subsequent averaging gives E 0 B 0 = curl B 0 B 0 + (v 0 B 0 )hbi + E B 0 (22) he 0 B 0 i = hcurl B 0 B 0 i + E hbi: (23) Note that eq. (23) has been derived without any assumption except for Ohm's law in the form j = (E + v B). It shows that, as already noticed by Keinigs Gerwin [8] for the case of hvi = 0, eq. (10) is equivalent to eq. (18). We have, however, derived eq. (18) independently therefore achieved an independent derivation of eq. (10). 4

5 Most remarkable in our rederivation seems to be the exemption from FOSA. Our presuppositions concerning the temporal spatial symmetry of the turbulence dier slightly from those of Keinigs [5], who assumed the magnetic-magnetic (instead of the magneticelectric) correlations to be independent of position. Nothing has been assumed about the mean velocity. So we have in particular dispensed with the usual assumption hvi = 0. If hvi is uniform, then one can change to a coordinate system moving with the velocity hvi with respect to the original one. A non-uniform mean velocity, on the other h, has represented a considerable diculty in traditional dynamo theory has, for the calculation of the turbulent emf, been taken into account at most approximately (see ref. [1], Chapter 8). The component of the turbulent emf in the direction of hbi, hbi, can lead to an increase as well as to a decrease of the energy contained in the mean magnetic eld. In the rst case it acts as a dynamo. Eq. (10) can be used to distinguish between both possibilities [8, 6, 9]: To be able to pick up energy from the uctuations, the mean magnetic eld must possess a current helicity, hbi curl hbi, whose sign is opposite to that of the mean current helicity of the uctuations. * * * It has been pointed out by an anonymous referee that, instead of working with relative magnetic helicity, one can alternatively dene A 0 via the relation between the Fourier transforms (considered as generalised functions) of A 0 B 0. Then eq. (18) is readily obtained using normal Coulomb gauge for A 0. References [1] Krause F. Radler K.-H., Mean-Field Magnetohydrodynamics Dynamo Theory (Akademie-Verlag, Berlin) [2] Moffatt H. K., Magnetic Field Generation in Electrically Conducting Fluids (Cambridge University Press, Cambridge) [3] Parker E. N., Cosmical Magnetic Fields (Clarendon Press, Oxford) [4] Strauss H. R., Phys. Fluids, 28 (1985) [5] Keinigs R. K., Phys. Fluids, 26 (1983) [6] Radler K.-H. Seehafer N., in Topological Fluid Mechanics, edited by H. K. Moatt A. Tsinober (Cambridge University Press, Cambridge) 1990, p [7] Berger M. A. Field G. B., J. Fluid Mech., 147 (1984) 133. [8] Keinigs R. Gerwin R. A., IEEE Transact. Plasma Sci., PS-14 (1986) 858. [9] Seehafer N., Astron. Astrophys., 284 (1994)