Astronomical Notes. Astronomische Nachrichten Founded by H. C. Schumacher in 1821

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1 Astronomical Notes Astronomische Nachrichten Founded by H. C. Schumacher in 1821 Editors K. G. Strassmeier (Potsdam/Editor-in-Chief), G. Hasinger (Garching), R.-P. Kudritzki (Honolulu), T. Montmerle (Grenoble), H. W. Yorke (Pasadena) T N I R P RE

2 Astron. Nachr. / AN 331, No. 1, (2010) / DOI /asna Simple laminar dynamos D. Moss School of Mathematics, University of Manchester, Manchester M13 9PL, UK Received 2009 Sep 1, accepted 2009 Nov 19 Published online 2009 Dec 30 Key words magnetic fields magnetohydrodynamics (MHD) Although current interest in astrophysical dynamo theory is largely focussed on flows with both large- and small-scale motions, historically the study of dynamos driven by laminar flows has been important. Some classical laminar flow dynamos are reviewed. These results were obtained in an asymptotic regime corresponding to small values of system parameters. Numerical simulations have since been used to extend these results outside of these asymptotic regimes; the asymptotic results remain useful approximations well outside of their formal regions of validity. By changing slightly the system geometries some interesting new results have recently been obtained The latter include the very simple oneroll dynamo, with motions in a single meridional cell contained within a spherical volume of fluid, without differential rotation. 1 Introduction Much recent and current interest in dynamo theory, especially in astrophysics, has concentrated on the properties of complex flows with both large and small scale motions, for example as might occur in a rotating, stratified turbulent fluid. In this review I will briefly revisit some of the simple laminar flows that can also support dynamo action, discussing both the seminal early results from the 1960s and 1970s, and also reviewing selected recent work showing how numerical simulations of classical analytic results can reveal new modes of dynamo action. 2 Classical analytic results In much of the 1950s and even later the key question in dynamo theory was whether any motions in a multiply connected electrically conducting fluid were capable of maintaining a magnetic field (as opposed to the case of a technological dynamo, where the connectivity can be carefully constrained by metallic conductors). Already in the 1930s Cowling (1934) had established the first of a number of anti-dynamo theorems, showing that axisymmetric fluid motions could not sustain a steady axisymmetric magnetic field. Essentially, differential rotation can generate toroidal field from poloidal, but the difficulty arises in the converse step, the generation of poloidal field from toroidal. Both steps must necessarily occur, converting kinetic energy into magnetic, in order to compensate for the inevitable losses of both poloidal and toroidal magnetic energy from ohmic decay. Corresponding author: moss@ma.man.ac.uk The question was answered in the affirmative in two seminal papers. Herzenberg (1958) analysed a system consisting of two rotating conducting spheres embedded in a sphere of conducting fluid. By proceeding to the limit in which the radii of the spheres a d, their separation, he was able to show that steady dynamo action occurs for sufficiently rapid rotation if the angle φ between the rotation axes satisfies 90 <φ<270. The toroidal to poloidal difficulty mentioned above is overcome since some of the toroidal field generated by the rotational shear around one sphere appears as poloidal field when it has diffused to the location of the other sphere, and vice versa. This result was subsequently extended by several authors to more complex cases see Moffat (1978) for a review. Lowes & Wilkinson (1963) performed laboratory experiments which although perhaps rather ambiguous in how far they reproduced the details of the Herzenberg set up, did confirm dynamo action in a finite Herzenberg-like system. At about the same time Backus (1958) analysed a stasis dynamo, in which a short interval of strong differential rotation was followed by a period of stasis with zero velocities, then a short period of nonaxisymmetric poloidal motion, another period of stasis, and then a carefully chosen solid body rotation. The sequence can then be repeated. Backus showed that this procedure, although highly artificial, could lead to dynamo action. Again, Moffat (1978) gives an accessible discussion. The question of whether a fluid system could support dynamo action was now resolved. Gailitis (1970) considered a spherical volume of conducting fluid containing two toroidal vortices or rolls with motions in meridian planes, symmetrically positioned in northern and southern hemispheres. The radii a of the vortices are much smaller than both their separation 2d and of the major radii c of the tori,

3 Astron. Nachr. / AN (2010) 89 Z Dobler, Shukurov & Brandenburg (2002) is a recent such study that provides a number of references. 3 Numerical simulations O 2a P During the 1970s and 1980s a number of simple flows in spheres, with meridional motions and differential rotation, were investigated numerically and shown to drive dynamos. A comprehensive review with some new results was presented by Dudley & James (1989), and Gubbins et al. (2000a, 2000b) also made an extensive survey. These results are not discussed here, rather I give some details of numerical simulations of dynamos inspired by the work of Herzenburg (1958) and Gailitis (1970). (a) (b) Fig. 1 (a) A cross-section in a meridian plane showing the configuration of the Gailitis model. In the text, a is the minor radius of the annuli centred on P and its reflection in the equator, c is the major radius, ie the distance of P from the axis OZ, and d is the distance of P from the equatorial plane. (b) Streamlines of a typical finite Gailitis flow. In these and other figures, the inner broken circle at fractional radius 0.2 indicates the inner limit of the computational domain, i.e. it the flow is confined to a deep spherical shell. i.e. a c, d see Fig. 1a. Gailitis showed that such a flow could support a steady nonaxisymmetric field, of odd (perpendicular dipole-like) symmetry with respect to the equatorial plane if the flow is in the sense shown in Fig. 1a, and of even symmetry (perpendicular quadrupole-like) if the sense is reversed. The last of the classical analytical results was provided by Ponomarenko (1971), who showed that a helical motion in an infinite medium could generate a (necessarily) nonaxisymmetric magnetic field. Ponomarenko-like flows appear particularly attractive for numerical simulation and laboratory experiment, especially in the form of Couette flows. 3.1 Herzenberg-like systems Brandenburg, Moss & Soward (1998) simulated a finite Herzenberg system, with the rotating spheres embedded in a cubical box of conducting fluid, the system being periodic in all three directions. Necessarily, for computational reasons the ratio of sphere radii to separation was not small, i.e. their system parameters were well outside the range of validity of the Herzenberg (1958) analysis. Herzenberg s analysis shows that the marginal magnetic Reynolds number satisfies Rmc 2 = 1 ( a ) 6 sin 2 φ cos φ (1) 4800 d (Moffat 1978). Brandenburg et al. s results, in their dependence of marginal dynamo number on a/d and φ, broadly resemble those of Herzenberg for a quite wide range of these parameters, in particular steady dynamo action was only found for the range of inclination angles 90 <φ< 270, with maximum growth rate at φ 125. A comparison between the result (1) and the numerical results is given in Fig. 2. For the first time a detailed visualization of the Herzenberg dynamo magnetic field was obtained; perhaps rather unsurprisingly with the advantage of hindsight it consisted predominantly of rolls surrounding each sphere, see Fig. 3. However a somewhat unexpected result was found when 0 < φ < 90.Nowunsteady dynamo action occurred, at rather larger marginal dynamo numbers than for the steady case. In fact, if unsteady dynamo action is explicitly sought, a modification of Herzenberg s analysis does yield a prediction for the marginal dynamo number and, again, the numerical results are broadly compatible with the asymptotic analysis (although far outside of its formal range of validity). 3.2 The Gailitis system Moss (2006) simulated an approximation to the Gailitis dynamo system, again necessarily for parameters outside of the range of validity of the asymptotic results of Gailitis

4 90 D. Moss: Simple laminar dynamos Table 1 The Gailitis system: estimates of marginal magnetic Reynolds number, R mcn, from the simulations of Moss (2008) compared to those, R mc, from Eq. (2). a eff is an estimate of the half-width of the meridional cells in Moss (2008) and c and d are defined in the caption to Fig. 1. a eff c d R mcn R mc a, and steady fields of even symmetry were found when the flows were reversed. In detail, Gailitis s analysis gives for the marginal magnetic Reynolds number ( c ) ( ) 2 d R mc = T m, (2) a c Fig. 2 Marginal values of R m (i.e. R mc, Eq. 1) versus a/d for various values of a and d, ϕ = 125. The solid line represents the asymptotic solution, with slope 3. From Brandenburg et al. (1998). where T m = O(1) and is given in Moffat (1978). In Table 1 estimates of marginal Reynolds number for the numerical results, R mcn, are compared with those from (2). Only in the third entry, where a/d 0.5 (i.e. the rolls are close to the equatorial plane), is the analytical estimate bad. Note that Moss (1990) also investigated a flow quite closely related to that of the Gailitis system, in the context of rotationally driven meridional circulation in rapidly rotating Ap/Bp stars, finding results that were consistent with those of Moss (2008). Moss (2006) asked a further question, what happens when the flow in one of the hemispheres of Figs. 1a or 1b is reversed? see Fig. 4a. Then an unsteady dynamo was found to be excited, at significantly larger marginal dynamo numbers than in the previous case. This was a quite new and unexpected result. A cursory inspection of the dynamo equations shows that such a flow must mix symmetries, and indeed the eigenfunctions have oscillating parities. Fig. 3 (online colour at: ) A threedimensional view of the magnetic field vectors in a finite Herzenberg system. Vectors are plotted only where the field strength exceeds 80% of the maximum. Each sphere is encircled by two flux rings, in opposite senses. The long arrows indicate the spin axes of the spheres. a =0.25, d =0.40, R m = 633, ϕ = 125.(From Brandenburg et al. 1998). (1970). Now for computational reasons the fluid flow extends throughout each hemisphere (the vortices are softened ), but is concentrated within a half-radius a eff, in contrast to the spatially localized vortices of Gailitis. The Gailitis system is sketched in Fig. 1a. Moss (2006) showed that the numerical results were (again...) loosely consistent with the small-parameter analysis. (Some of these results had previously appeared in Gailitis (1993).) In particular, steady fields of odd symmetry with respect to the equatorial plane were excited when the flow was in the sense shown in Fig. 3.3 The one-roll dynamo Moss (2008) observed that as the neutral lines of the antisymmetric flow of of Fig. 4a move closer to the equator, the flows adjacent to the plane being in opposite senses will in some sense cancel out. A limit of this process could be taken to be a single axisymmetric circulation in meridian planes, centred on the equatorial plane and linking the hemispheres see, e.g., the streamlines in Fig. 4b. Can such a flow as shown in Fig. 4b (note, without any differential rotation) excite a dynamo? Moss s simulations showed that a dynamo could indeed be excited; at marginal excitation it is steady and of intermediate parity (i.e. of mixed symmetry). Note that there is no differential rotation present. For large enough velocities/dynamo number there is a bifurcation to oscillating eigenfunctions. Moss termed this a one roll dynamo. An interesting point is that Dudley & James (1989) investigated a superficially similar meridional flow, but in the presence of a differential rotation. Their system was shown

5 Astron. Nachr. / AN (2010) 91 Moss (2008) also showed that if the centre of the single meridional cell of Fig. 4b were displaced from the equatorial plane, then a steady dynamo continues to be excited. 3.4 The Möbius dynamo Shukurov, Stepanov & Sokoloff (2008) investigated numerically flows of conducting fluid parallel to a Möbius surface and related surfaces, finding that such flows could produce dynamo action, at relatively low marginal magnetic Reynolds numbers. The flow has analogies with the Ponomarenko flow, and oscillatory fields are again produced, localized near the Möbius surface. 4 Discussion and conclusions (a) (b) Fig. 4 (a) An example of a finite reversed-gailitis flow. (b) As the flow of panel (a) approaches the equator and the cells merge, the limit is a one-roll flow. to excite a unsteady dynamo at marginal excitation. Moss found that the meridional part of the Dudley & James flow could not be shown to excite a dynamo, i.e. that a toroidal flow was essential. Possibly very much larger velocities than accessible numerically are needed, but it appears superficially that near-surface velocity shear was choking the system (Moss s meridional velocities were concentrated well away from the surface of the fluid sphere, those of Dudley & James were not.) Alternatively, it could be that the form of the Dudley & James meridional flow is itself sufficiently different to that of Moss (2008) so as not to produce dynamo action. Gubbins et al. (2000a, 2000b) reported on similar instances. In this short review I have attempted to demonstrate that numerical simulation of classical simple dynamo systems can produce perhaps surprising new results. Also, the asymptotic results of Herzenberg (1958) and Gailitis (1970) remain useful estimates for marginal dynamo numbers that are far outside of the nominal range of validity of their respective analyses. Further, numerical simulations have made it possible to produce visualizations of the magnetic fields of these dynamos. Changes in the geometries of the Herzenberg and Gailitis systems (different range of inclinations of rotation axes and reversal of one meridional roll, respectively) produce novel and unanticipated results. The antisymmetric Gailitis flow (Fig. 4a) leads quite naturally to the one roll dynamo of Moss (2008), which is a quite new result and has a claim to be the simplest possible dynamo flow in a spherical volume. An obvious question is, do these results have any physical applications? Dolginov & Urpin (1979) attempted to apply the Herzenberg concept to the generation of magnetic fields in binary stars systems, concluding that in some nova and symbiotic star systems, given favourable assumptions growth times for magnetic fields could be as short as O(10 3 ) yr. Although there are some unresolved questions about this work, it might be fruitful to pursue these ideas via numerical simulations similar to those of Brandenburg et al. (1998) with parameters chosen to be astrophysically relevant. Bigazzi, Brandenburg & Moss (1999) tried to use the Herzenburg dynamo concept to model magnetic fields localized near pairs of rotating vortex tubes found in some simulations of compressible magnetoconvection. Moss (1990) did show that a sufficiently rapidly rotating A star might just excite a Gailitis-like dynamo with a field of perpendicular dipole-like symmetry, but the rotational periods required are so short that the result is unlikely to be of general significance to the magnetic A star problem. It has been suggested that the one-roll dynamo of Sect. 3.3 might have application to dynamo experiments where the flow is driven by a propeller at one end of a cavity.

6 92 D. Moss: Simple laminar dynamos Finally it is worth commenting that aspects of the physics of these laminar dynamos can still pose tricky problems. For example, whereas asymptotic analysis can throw light on finite Ponomarenko-type dynamos, it has so far proved impossible to understand significant aspects of the Möbius dynamo (D. Sokoloff, private communication). In their simplicity, the systems described in this review touch on some fundamental areas of modern dynamo theory, but there are also areas of both astrophysics and laboratory experiments where they may find some application. Certainly it is clear that study of simple dynamo systems had not been exhausted as early as some might have believed! Acknowledgements. The author thanks the organizers of the meeting Astrophysical Magnetohydrodynamics (Kiljavanranta, Finland, April 2009) for supporting his attendance at the meeting, and thus encouraging him to assemble this review. He is grateful to D. Sokoloff and to the anonymous referee for helpful comments which improved the paper. References Backus, G.E.: 1958, AnP 4, 372 Bigazzi, A., Brandenburg, A., Moss, D.: 1999, PhPl 6, 72 Brandenburg, A., Moss, D., Soward, A.: 1998, Proc. R. Soc. Lond. A 454, 1283 Cowling, T.G.: 1934, MNRAS 94,39 Dobler, W., Shukurov, A., Brandenburg, A.: 2002, Phys Rev E 65, Dolginov, A.Z., Urpin, V.A.: 1979, A&A 79, 60 Dudley, Dudley, M.L., James, R.W.: 1989, Proc. R. Soc. Lond. A. 425,407 Gailitis, A.: 1970, MagGi 6, 19 Gailitis, A.: 1993, MagGi 29, 3 Gubbins, D., Barber, C.N., Gibbons, S., Love, J.J.: 2000a, Proc. R. Soc. Lond. A 456, 1337 Gubbins, D., Barber, C.N., Gibbons, S., Love, J.J.: 2000b, Proc. R. Soc. Lond. A 456, 1669 Herzenberg, A.: 1958, Phil. Trans. R. Soc. Lond. A 250, 543 Lowes, F.J., Wilkinson, I., 1963, Nature 198, 1158 Moffatt, H.K.: 1978, Magnetic Field Generation in Electrically Conducting Fluids, CUP, Cambridge Moss, D.: 1990, MNRAS 243, 537 Moss, D.: 2006, GApFD 100, 49 Ponomarenko, Y.B.: 1973, JAMTP 14, 775 Shukurov, A., Stepanov, R., Sokoloff, D.: 2008, Phys Rev E 78,