- Email: [email protected]
Magnetohydrodynamic convectons
4 October 1999
Physics Letters A 261 Ž1999. 74–81 www.elsevier.nlrlocaterphysleta
Magnetohydrodynamic convectons Sean 1 Blanchflower Department of Applied Mathematics and Theoretical Physics, UniÕersity of Cambridge, SilÕer Street, Cambridge CB3 9EW, UK Received 12 January 1999; received in revised form 5 August 1999; accepted 10 August 1999 Communicated by A.P. Fordy
Abstract In modelling two-dimensional nonlinear magnetoconvection a fundamental new spatially isolated convective pattern is found. The pattern consists of a single convective roll Ža ‘convecton’. that is able to expel practically all magnetic flux from its neighbourhood, whilst the rest of the computational domain contains an almost uniform magnetic field strong enough to suppress convection. q 1999 Published by Elsevier Science B.V. All rights reserved. PACS: 47.65; 47.54 Keywords: Localized patterns; Magnetoconvection
1. Introduction In some spatially extended systems the experimental region may be divided into two distinct domains; an isolated pattern occurs in a small region of the spatially homogeneous system, whilst the remainder usually contains an approximation to the trivial or basic state. In very wide boxes such localized solutions are generally associated with stable subcritical behaviour w18x. Stable examples of such localized states have been seen in a variety of physical and theoretical frameworks w13x. Particular study has been given to convection in binary-fluid mixtures in which isolated regions of travelling waves have been observed both experimentally w10x and theoretically w18x. More recently, pairs of isolated vortices have been seen in Taylor–Couette flow w6x, 1 Tel.: q44-1223-337900; fax: q44-1223-337918; e-mail: [email protected]
and in trays of small ball-bearings single standing waves Ž‘oscillons’. may be observed when vertical oscillation of the system induces the Faraday instability w19x. The phenomenon has also been investigated using low order amplitude equations, and isolated structures have been found in cases such as Swift–Hohenberg and reaction-diffusion systems w15,11x. In each of the above examples it is believed that a necessary though certainly not sufficient condition for localization is the bistability of the trivial and nonlinear states. There are examples of isolated waves in which the bifurcation to the extended waves is supercritical, for example in nematic electroconvection w4,14x, but they form a separate topic of study. There are also surprisingly few examples of such localized states being observed in two or three dimensions, as opposed to the usual single spatial dimension. This letter presents a novel example of a localized pattern in the case of magnetoconvection, shown in
0375-9601r99r$ - see front matter q 1999 Published by Elsevier Science B.V. All rights reserved. PII: S 0 3 7 5 - 9 6 0 1 Ž 9 9 . 0 0 5 7 3 - 3
S. Blanchflowerr physics Letters A 261(1999) 74–81
75
Fig. 1. An isolated convecton found in fully resolved Boussinesq magnetoconvection Ž R s 10 000, Q s 100 000..
Fig. 1. Flux expulsion w20x tends to create regions in which the magnetic field is separated from the fluid flow w17x, and under certain circumstances isolated convective states have been observed w1,5,21x. In particular, Fig.8d of w21x represents a pair of isolated rolls under the constraint of mirror symmetry. In retrospect, these results suggest that if the width of the box were increased and the imposed symmetry removed then an important development might occur, resulting in a single, fully isolated roll, unconstrained by the dimensions of the computational domain. Such a roll should be found in a box of arbitrary size, and several of them might coexist independently of one another. This is the first time such a state has been observed. The majority of the box remains in the basic state of strong, almost vertical magnetic field in which all convection is suppressed; however, a small region is formed from which the flow is able to expel virtually all the flux, creating an isolated island consisting of a single narrow convective roll Ža ‘convecton’.. Unlike the cases listed above in which isolated states were found either experimentally or in carefully chosen models, here such a state is located numerically for the full physical partial differential equations of magnetohydrodynamics. These studies are motivated by astrophysical considerations: in the dark umbrae at the centres of sunspots, where there are strong vertical magnetic fields, bright isolated features Ž‘umbral dots’. are observed, indicating localized areas of upflow w16x. The localization described above is more conveniently investigated through using a system which is simpler computationally, whilst still retaining the key features of the problem. In Section 2 we describe such a model in which the mhd equations are truncated vertically, and in Section 3 the results of the numerical experiments are presented. The experiments are then repeated in Section 4 for the case of
fully resolved Boussinesq magnetoconvection, and preliminary results are given.
2. A truncated model Consider magnetoconvection in a two-dimensional rectangular box of aspect ratio l. The equations of magnetoconvection under the standard Boussinesq approximation, with a uniform vertical background magnetic field, are w8x
Ev Et
q
E Ž c ,v .
Eu s s= 2v y s R
EŽ x , z .
y sz Q
Ex
E= 2A
ž
Ez
q
E Ž A,= 2A . EŽ x , z .
/
,
Ž 1. Eu Et
q
EA Et
q
E Ž c ,u . EŽ x , z .
Ec s = 2u q
E Ž c , A. EŽ x , z .
Ex
,
Ž 2.
Ec s z= 2A q
Ez
,
Ž 3.
where c is the stream function, v the vorticity, u the deviation from the uniformly stratified temperature, and A the perturbation to the magnetic flux function. A fixed temperature difference is maintained across the layer. The relevant dimensionless parameters are the Rayleigh number R, proportional to the superadiabatic temperature gradient, and the Chandrasekhar number Q, which measures the magnetic field strength. The ratio z of the magnetic and thermal diffusivities and the Prandtl number s are held constant at z s 0.1 and s s 1.0 throughout this letter. The surface and base of the box are impermeable but slippery, so that the normal component of the
76
S. Blanchflowerr Physics Letters A 261(1999) 74–81
velocity and the tangential component of the viscous stress both vanish. The magnetic field is held vertical at both horizontal boundaries. All quantities are constrained to be periodic in the x-direction, with period l. In order to allow a full sweep of the multi-dimensional parameter space, Eqs. Ž1. – Ž3. are truncated vertically. Previous studies show w1x that the vertical structure is comparatively simple, whereas full horizontal resolution is required if the solutions are to give some kind of isolated state. We expand
c s c 1 Ž x ,t . sinp z , u s u 1 Ž x ,t . sinp z q u 2 Ž x ,t . sin2p z ,
Ž 4.
A s A 0 Ž x ,t . q A1 Ž x ,t . cosp z , a generalisation of the fifth order system w8x in which truncation is also performed horizontally. Projecting onto the appropriate vertical modes gives five partial differential equations governing the evolution of each of the modes; these are solved using a pseudo-spectral method in x and a second-order Adams–Bashforth scheme for timestepping. The horizontal resolution is varied according to the aspect ratio, but 64 mesh intervals prove to be sufficient in the majority of cases. If we hold the Rayleigh number fixed at R s 20 000 and vary the field strength, as is more convenient in our system, linear theory predicts a supercritŽo . ical Hopf bifurcation Žfor l s 6. at Qmax s 24047 giving rise to stable oscillatory convection with m s 6, where m s lrL is the number of pairs of Žoscillatory. rolls, and L is the wavelength. The preferred number of pairs of rolls for steady convection is also m s 6 and the corresponding bifurcation occurs at Q6Že. s 1374, but these solutions are linearly unstable.
3. Numerical results Motivation for these studies came from work performed by Blanchflower et al. w1x who discovered a single large wavelength plume in wide two-dimensional boxes for the case of fully compressible magnetoconvection; flow is confined to an essentially field-free region whilst the remainder of the box contains strong, almost vertical magnetic field in which all convection is suppressed.
A key observation of w1x was the importance of using wide boxes; if the aspect ratio of the experimental domain is too narrow then the solution may not be representative of the choice of parameters. Only if the box is sufficiently wide is the solution effectively independent of its width. Accordingly, numerical experiments are first performed in a rectangular box of aspect ratio l s 6. The chosen value R s 20 000 is sufficiently large to allow fairly vigorous convection in the absence of a magnetic field. The field strength is then varied by increasing the Chandrasekhar number Q in small increments, and the computation run until a certain time has elapsed or a steady state reached. The computation for each value of Q is begun from the final state of the previous value. This approach of fixing R and increasing Q is essentially equivalent to the more usual method of holding Q fixed and decreasing R. As the field strength is increased the system bifurcates to states in which a permanent stagnant slab of flux is formed. These flux regions increase in size until they are strong enough to suppress one of the convective rolls. Increasing the field strength further causes weaker rolls to be overcome by the ever expanding flux regions, and some of the resulting sequence of states are shown in Fig. 2. In some cases this results in the suppression of all convection except for two adjoining rolls, analogous to the flux separated state of w1x. However further increases in Q result in the destruction of one of these leaving a new type of convection in the form of a single isolated roll or conÕecton. Ordinarily in thermal convection steady fluid flow occurs as a number of plumes which consist of fluid rising Žor falling. and moving horizontally outwards in both directions seen as two roughly circular rolls. They are characterised by horizontal mirror symmetry about their centre and examples may be seen in Fig. 2a,b. ConÕectons are a novel and important departure from this in that the system has undergone a bifurcation that removes the final reflection symmetry so that flow consists solely of a single roll, whilst retaining the usual Boussinesq point symmetry. Some results are developed in Fig. 3 which shows the sequence of bifurcations which may give rise to convectons. The stable branches were obtained by increasing and then decreasing Q in small steps until
S. Blanchflowerr physics Letters A 261(1999) 74–81
77
Fig. 2. The development of localized solutions as the magnetic field strength is increased, for the case of the truncated model with l s 6, R s 20 000. Ža. six steady rolls for Q s 3000, Žb. four rolls, Q s 8000, Žc. two isolated rolls, Q s 16 000, Žd. a single convecton, Q s 22 000. The shading shows deviations in temperature from the static state Ž u .. Also shown are velocity arrows, and magnetic field lines.
the solution in question became unstable. Very many more solution branches exist within the problem, but only a few key branches have been included. There are pitchfork bifurcations at Q1Že. s 190 and Q2Že. s 603 Žamong others. corresponding to steady convection with two Žwavenumber m s 1. and four Ž m s 2. rolls respectively. Due to the ability of a conducting fluid to expel magnetic flux, which leads to regions of reduced effective field strength, these branches show subcriticality w12x, and eventually gain stability at saddle-node bifurcations. The m s 1 branch results in two isolated convectons whose vorticities have opposite signs Žlabelled 2o., as in Fig. 2c. We also observe steady solutions with two isolated con-
vectons with vorticities of the same sign Ž2s. which result from a symmetry breaking of the m s 2 branch. The key bifurcation occurs as a symmetry breaking of the m s 1 branch, in which the Z2 reflection symmetry is lost, whilst retaining point symmetry. This branch gives rise to the single convecton solution and such a bifurcation has not been previously observed in studies of convection. Hitherto, all isolated states found in two-dimensional magnetoconvection have been part of a system of two or more dependent rolls. For example in w21x the imposed mirror Ž Z2 . symmetry of the problem means that Fig.8d of w21x shows a solution lying on the steady m s 1 branch, lacking the extra symmetry breaking
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S. Blanchflowerr Physics Letters A 261(1999) 74–81
Fig. 3. A schematic representation of part of the bifurcation sequence leading to the production of convectons for l s 6. Field strength is plotted horizontally and a measure of the amplitude of convection vertically. Solid lines represent observed stable solutions, and dashed lines are conjectural unstable branches. Squares, circles, and triangles show steady, Hopf, and symmetry-breaking bifurcations respectively. The number of convective rolls on each branch is shown.
required to produce a convecton. Branches arising via a primary pitchfork bifurcation from the trivial solution necessarily retain reflection symmetry w2x, so the branch which gives rise to the single convecton solution can only occur as a secondary bifurcation; Fig. 3 gives the simplest way in which this can occur. Finally, all three isolated steady branches Ž1, 2o, 2s. undergo Hopf bifurcations, as the reduction in effective field strength if Q is decreased allows weak oscillations to occur in the magnetic region. Examples of convectons may be seen in Fig. 2c,d which show the resulting patterns for two values of Q. In Fig. 2d Ž R s 20 000, Q s 22 000. the field strength is fairly close to the linear stability boundŽo . s 24047, yet flow is remarkably strong. ary at Q max The Nusselt number, a measure of convective efficiency defined so that N s 1 in the absence of all convection, is N s 1.32. The flow also expels flux from within the roll extremely efficiently; away from the edges the field strength is reduced to around a thousandth of its original value, whereas just outside the roll there is a sharp leap. In the magnetic region the field remains fairly uniformly vertical at a strength of 1.4 times its uniform value, and the temperature profile is also essentially uniform within the stagnant magnetic region. In addition there is a very narrow
current sheet near where the horizontal flow meets the magnetic region Žie near the surface on one side, and near the base on the other. where the flux is being swept by the convection, before diffusing outwards. The above solutions are associated with hysteresis and many types of solution coexist at identical parameter values. Examples of single convectons Žfor R s 20 000. were found for 13 700 F Q F 26 500, demonstrating that such flow is possible even when the purely-conducting state is stable to all small perturbations. The convecton in Fig. 2d has a width l f 1. However, this varies as the field strength is adjusted; with Q s 13 700 we obtain a width of l f 1.5, decreasing to l f 0.75 at Q s 26 500. Over the same range of Q the Nusselt number drops from 1.49 to 1.18. By way of comparison, if a solution is initiated from small perturbations to the trivial static solution then a different pattern is seen in all cases. For Q s 22 000 five pairs of oscillatory rolls are obtained that fill the box, as linear theory might suggest. The Žtime-averaged. Nusselt number is significantly lower Ž N s 1.08. signifying relatively feeble convection. Experiments were also carried out in boxes of different sizes. With smaller values of the aspect
S. Blanchflowerr physics Letters A 261(1999) 74–81
ratio the convecton solution was effectively observed down to l s 1; however it is only for l ) 2 that its nature is really clear, and a distinct region of high magnetic field can be seen. In very wide boxes, on the other hand, important and interesting behaviour is observed ŽFig. 4.. Fig. 4b shows two convectons with no reflection symmetry Žcorresponding to the 2s branch of Fig. 3., demonstrating that for l 4 1 such isolated rolls may coexist independently of one another, scattered across the box; the effects of flux expulsion allow a small convective roll to be formed having only a slight effect on the effective field strength of the magnetic region, with any viscous and magnetic stresses absorbed by the field. Fig. 4c shows a convectonranti-convecton pair retaining mirror symmetry about its centre, but having broken the point symmetry of a solution such as that shown in Fig. 2c. The mirror symmetry is then broken producing the single convecton of Fig. 4d. In boxes with fairly small aspect ratio single Žo . as in forcing all convectons are seen for Q - Qmax the magnetic field into one region the flow increases Žo . the effective field strength there so that Qeff ) Qmax . In arbitrarily large boxes convectons should still be Žo . possible, but will only be seen for Q ) Qmax , and Žo . indeed for l s 16 Ž Qmax s 24267. single convectons are only stable in the range 26 000 - Q - 30 000. This is because for l 4 1 a single convecton will only increase the effective field strength in the mag-
79
netic region by a very small amount, and so for Žo . this region will be unstable to perturbaQ - Qmax tions and oscillations will be seen. It is this bistability in very wide boxes, or coexistence of stable Žo . trivial and nonlinear solutions for Q ) Qmax , that is associated with isolated states. This argument carries to the earlier case of l s 6; if the effectiÕe field strength in the magnetic region is plotted on the horizontal axis instead of Q in Fig. 3 then the Hopf bifurcation at the lower end of the convecton branch Žo . occurs at Qmax , and convectons always ‘coexist’ Žin terms of effectiÕe Q . with a stable trivial solution.
4. Fully resolved Boussinesq case Having established the existence of isolated rolls in the truncated model described in Section 3, it is important to verify that they are not a result of the restricted number of modes. It is expected that the range of behaviour seen in Section 3 will be repeated, with only the bifurcation points shifted w1x. There are many examples of convective patterns or regimes in low order models of Rayleigh–Benard convection that do not translate to the fully resolved case w3x. Numerical experiments were thus repeated for the full nonlinear Boussinesq Eqs. Ž1. – Ž3. in an attempt to reproduce the above behaviour for the case l s 6. A pseudo-spectral code due to A.M.
Fig. 4. Some solution states of the truncated model for l s 16. Ža. Two isolated clusters of five and three rolls, for Q s 10 000, Žb. two independent isolated convectons, Q s 28 000, Žc. a convectonranti-convecton pair, also found at Q s 28 000, from different initial conditions Žd. a fully isolated convecton, Q s 29 000.
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S. Blanchflowerr Physics Letters A 261(1999) 74–81
Rucklidge was used with periodic horizontal boundary conditions, and the resolution was 256 = 33. Commencing with a relatively high field strength Q s 26 000, R s 20 000 the flow settled into a state with three convective rolls of differing sizes and an intermediate region of strong magnetic field, even when the earlier ‘convecton’ state was used as an initial condition. However, when this state was used as the starting point for a run with higher Rayleigh number and field strength, weaker rolls were subsumed into the magnetic region, and for R s 50 000,Q s 80 000 a single convecton was obtained. The convection is highly efficient at expelling flux; within the convecton the field strength is again reduced to around a thousandth of its original value. The Nusselt number in this case is 3.51 and the width l f 2.9, much larger than is possible in the truncated case. Increasing field strength further and reducing the Rayleigh number cause a reduction in the convecton size. The smallest roll obtained so far has width l f 0.75 for values of R s 10 000 and Q s 100 000, and is shown in Fig. 1. Further increases in field strength result in the flow tending rapidly to the trivial solution. Convectons in fully resolved Boussinesq magnetoconvection have also recently been found independently by P.C. Matthews Žprivate communication. for the case z s 1, but they are otherwise essentially identical to those found above. We may also be confident that such isolated rolls will be found in the case of two-dimensional fully compressible convection, especially for studies in a weakly stratified layer w7x, though it is expected that they will take the form of travelling waves w9x. The role of bistability in producing isolated states is still under study. Investigations of theoretical models of binary fluids w18x have found that bistability obtained via subcritical instabilities is necessary in order to produce localized solutions. In other such systems it is the interaction of the modes of the separate fields via gradient coupling that results in the observed localization. For magnetoconvection the physical mechanism that gives rise to such coupling is clear: the expulsion of magnetic flux via advection. In three dimensions it is easy to imagine an equivalent isolated axisymmetric cell in which fluid rises at the axis and moves radially outwards in all
directions, but other analogues may be sought in the form of single ‘rolls’ whose radius rises from and then decays back to zero in one horizontal direction. Bistability is also possible in other hydrodynamic systems; in thermosolutal convection, although the primary bifurcations are often subcritical it is unlikely that analogous isolated steady rolls should occur as localization of solutal gradients could prove impossible. In rotating convection on the other hand we would expect analogous cells to be found, with the role of the magnetic field being filled by angular momentum gradients. Certainly further investigation is warranted and work in this area is ongoing.
Acknowledgements I would like to thank Nigel Weiss, Michael Proctor and Alastair Rucklidge whose insight and erudition have advanced this letter greatly. Comments made by the referees have been particularly useful in the presentation of this work. The author is supported by a PPARC research studentship.
References w1x S.M. Blanchflower, A.M. Rucklidge, N.O. Weiss, MNRAS 301 Ž1998. 593. w2x J.D. Crawford, E. Knobloch, Ann. Rev. Fluid Mech. 23 Ž1991. 341. w3x J.H. Curry, J.R. Herring, J. Loncaric, S.A. Orszag, JFM 147 Ž1984. 1. w4x M. Dennin, D.S. Cannell, G. Ahlers, Phys. Rev. E 57 Ž1998. 638. w5x D.J. Galloway, N.O. Weiss, ApJ 243 Ž1981. 945. w6x A. Groisman, V. Steinberg, Phys. Rev. Lett. 78 Ž1997. 1460. w7x N.E. Hurlburt, M.R.E. Proctor, N.O. Weiss, D.P. Brownjohn, JFM 207 Ž1989. 587. w8x E. Knobloch, N.O. Weiss, L.N. da Costa, JFM 113 Ž1981. 153. w9x P.C. Matthews, M.R.E. Proctor, A.M. Rucklidge, N.O. Weiss, Phys. Lett. A 183 Ž1993. 69. w10x J.J. Niemela, G. Ahlers, D.S. Cannell, Phys. Rev. Lett. 64 Ž1991. 1365. w11x M. Or-Guil, M. Bode, C.P. Schenk, H.-G. Purwins, Phys. Rev. E 57 Ž1998. 6432. w12x M.R.E. Proctor, N.O. Weiss, Rep. Prog. Phys. 45 Ž1982. 1317. w13x H. Riecke, in: M. Golubitsky, D. Luss, S. Strogatz ŽEds..,
S. Blanchflowerr physics Letters A 261(1999) 74–81 Pattern Formation in Continuous and Coupled Systems, IMA Volume 1998. w14x H. Riecke, G.D. Granzow, Phys. Rev. Lett. 81 Ž1998. 333. w15x H. Sakaguchi, H.R. Brand, Europhys. Lett. 38 Ž1997. 341. w16x M. Sobotka, in: B. Schmieder, J.C. del Toro Iniesta, M. ŽEds.., Advances in the Physics of Sunspots, Vazquez ´ P.A.S.P. Conference Series, 1997, p. 155.
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w17x L. Tao, N.O. Weiss, D.P. Brownjohn, M.R.E. Proctor, ApJ 496 Ž1998. L39. w18x O. Thual, S. Fauve, J. Phys. France 49 Ž1988. 1829. w19x P.B. Umbanhowar, F. Melo, H.L. Swinney, Nature 382 Ž1996. 793. w20x N.O. Weiss, Proc. Roy. Soc. A 293 Ž1966. 310. w21x N.O. Weiss, JFM 108 Ž1981. 273.
Physics Letters A 261 Ž1999. 74–81 www.elsevier.nlrlocaterphysleta
Magnetohydrodynamic convectons Sean 1 Blanchflower Department of Applied Mathematics and Theoretical Physics, UniÕersity of Cambridge, SilÕer Street, Cambridge CB3 9EW, UK Received 12 January 1999; received in revised form 5 August 1999; accepted 10 August 1999 Communicated by A.P. Fordy
Abstract In modelling two-dimensional nonlinear magnetoconvection a fundamental new spatially isolated convective pattern is found. The pattern consists of a single convective roll Ža ‘convecton’. that is able to expel practically all magnetic flux from its neighbourhood, whilst the rest of the computational domain contains an almost uniform magnetic field strong enough to suppress convection. q 1999 Published by Elsevier Science B.V. All rights reserved. PACS: 47.65; 47.54 Keywords: Localized patterns; Magnetoconvection
1. Introduction In some spatially extended systems the experimental region may be divided into two distinct domains; an isolated pattern occurs in a small region of the spatially homogeneous system, whilst the remainder usually contains an approximation to the trivial or basic state. In very wide boxes such localized solutions are generally associated with stable subcritical behaviour w18x. Stable examples of such localized states have been seen in a variety of physical and theoretical frameworks w13x. Particular study has been given to convection in binary-fluid mixtures in which isolated regions of travelling waves have been observed both experimentally w10x and theoretically w18x. More recently, pairs of isolated vortices have been seen in Taylor–Couette flow w6x, 1 Tel.: q44-1223-337900; fax: q44-1223-337918; e-mail: [email protected]
and in trays of small ball-bearings single standing waves Ž‘oscillons’. may be observed when vertical oscillation of the system induces the Faraday instability w19x. The phenomenon has also been investigated using low order amplitude equations, and isolated structures have been found in cases such as Swift–Hohenberg and reaction-diffusion systems w15,11x. In each of the above examples it is believed that a necessary though certainly not sufficient condition for localization is the bistability of the trivial and nonlinear states. There are examples of isolated waves in which the bifurcation to the extended waves is supercritical, for example in nematic electroconvection w4,14x, but they form a separate topic of study. There are also surprisingly few examples of such localized states being observed in two or three dimensions, as opposed to the usual single spatial dimension. This letter presents a novel example of a localized pattern in the case of magnetoconvection, shown in
0375-9601r99r$ - see front matter q 1999 Published by Elsevier Science B.V. All rights reserved. PII: S 0 3 7 5 - 9 6 0 1 Ž 9 9 . 0 0 5 7 3 - 3
S. Blanchflowerr physics Letters A 261(1999) 74–81
75
Fig. 1. An isolated convecton found in fully resolved Boussinesq magnetoconvection Ž R s 10 000, Q s 100 000..
Fig. 1. Flux expulsion w20x tends to create regions in which the magnetic field is separated from the fluid flow w17x, and under certain circumstances isolated convective states have been observed w1,5,21x. In particular, Fig.8d of w21x represents a pair of isolated rolls under the constraint of mirror symmetry. In retrospect, these results suggest that if the width of the box were increased and the imposed symmetry removed then an important development might occur, resulting in a single, fully isolated roll, unconstrained by the dimensions of the computational domain. Such a roll should be found in a box of arbitrary size, and several of them might coexist independently of one another. This is the first time such a state has been observed. The majority of the box remains in the basic state of strong, almost vertical magnetic field in which all convection is suppressed; however, a small region is formed from which the flow is able to expel virtually all the flux, creating an isolated island consisting of a single narrow convective roll Ža ‘convecton’.. Unlike the cases listed above in which isolated states were found either experimentally or in carefully chosen models, here such a state is located numerically for the full physical partial differential equations of magnetohydrodynamics. These studies are motivated by astrophysical considerations: in the dark umbrae at the centres of sunspots, where there are strong vertical magnetic fields, bright isolated features Ž‘umbral dots’. are observed, indicating localized areas of upflow w16x. The localization described above is more conveniently investigated through using a system which is simpler computationally, whilst still retaining the key features of the problem. In Section 2 we describe such a model in which the mhd equations are truncated vertically, and in Section 3 the results of the numerical experiments are presented. The experiments are then repeated in Section 4 for the case of
fully resolved Boussinesq magnetoconvection, and preliminary results are given.
2. A truncated model Consider magnetoconvection in a two-dimensional rectangular box of aspect ratio l. The equations of magnetoconvection under the standard Boussinesq approximation, with a uniform vertical background magnetic field, are w8x
Ev Et
q
E Ž c ,v .
Eu s s= 2v y s R
EŽ x , z .
y sz Q
Ex
E= 2A
ž
Ez
q
E Ž A,= 2A . EŽ x , z .
/
,
Ž 1. Eu Et
q
EA Et
q
E Ž c ,u . EŽ x , z .
Ec s = 2u q
E Ž c , A. EŽ x , z .
Ex
,
Ž 2.
Ec s z= 2A q
Ez
,
Ž 3.
where c is the stream function, v the vorticity, u the deviation from the uniformly stratified temperature, and A the perturbation to the magnetic flux function. A fixed temperature difference is maintained across the layer. The relevant dimensionless parameters are the Rayleigh number R, proportional to the superadiabatic temperature gradient, and the Chandrasekhar number Q, which measures the magnetic field strength. The ratio z of the magnetic and thermal diffusivities and the Prandtl number s are held constant at z s 0.1 and s s 1.0 throughout this letter. The surface and base of the box are impermeable but slippery, so that the normal component of the
76
S. Blanchflowerr Physics Letters A 261(1999) 74–81
velocity and the tangential component of the viscous stress both vanish. The magnetic field is held vertical at both horizontal boundaries. All quantities are constrained to be periodic in the x-direction, with period l. In order to allow a full sweep of the multi-dimensional parameter space, Eqs. Ž1. – Ž3. are truncated vertically. Previous studies show w1x that the vertical structure is comparatively simple, whereas full horizontal resolution is required if the solutions are to give some kind of isolated state. We expand
c s c 1 Ž x ,t . sinp z , u s u 1 Ž x ,t . sinp z q u 2 Ž x ,t . sin2p z ,
Ž 4.
A s A 0 Ž x ,t . q A1 Ž x ,t . cosp z , a generalisation of the fifth order system w8x in which truncation is also performed horizontally. Projecting onto the appropriate vertical modes gives five partial differential equations governing the evolution of each of the modes; these are solved using a pseudo-spectral method in x and a second-order Adams–Bashforth scheme for timestepping. The horizontal resolution is varied according to the aspect ratio, but 64 mesh intervals prove to be sufficient in the majority of cases. If we hold the Rayleigh number fixed at R s 20 000 and vary the field strength, as is more convenient in our system, linear theory predicts a supercritŽo . ical Hopf bifurcation Žfor l s 6. at Qmax s 24047 giving rise to stable oscillatory convection with m s 6, where m s lrL is the number of pairs of Žoscillatory. rolls, and L is the wavelength. The preferred number of pairs of rolls for steady convection is also m s 6 and the corresponding bifurcation occurs at Q6Že. s 1374, but these solutions are linearly unstable.
3. Numerical results Motivation for these studies came from work performed by Blanchflower et al. w1x who discovered a single large wavelength plume in wide two-dimensional boxes for the case of fully compressible magnetoconvection; flow is confined to an essentially field-free region whilst the remainder of the box contains strong, almost vertical magnetic field in which all convection is suppressed.
A key observation of w1x was the importance of using wide boxes; if the aspect ratio of the experimental domain is too narrow then the solution may not be representative of the choice of parameters. Only if the box is sufficiently wide is the solution effectively independent of its width. Accordingly, numerical experiments are first performed in a rectangular box of aspect ratio l s 6. The chosen value R s 20 000 is sufficiently large to allow fairly vigorous convection in the absence of a magnetic field. The field strength is then varied by increasing the Chandrasekhar number Q in small increments, and the computation run until a certain time has elapsed or a steady state reached. The computation for each value of Q is begun from the final state of the previous value. This approach of fixing R and increasing Q is essentially equivalent to the more usual method of holding Q fixed and decreasing R. As the field strength is increased the system bifurcates to states in which a permanent stagnant slab of flux is formed. These flux regions increase in size until they are strong enough to suppress one of the convective rolls. Increasing the field strength further causes weaker rolls to be overcome by the ever expanding flux regions, and some of the resulting sequence of states are shown in Fig. 2. In some cases this results in the suppression of all convection except for two adjoining rolls, analogous to the flux separated state of w1x. However further increases in Q result in the destruction of one of these leaving a new type of convection in the form of a single isolated roll or conÕecton. Ordinarily in thermal convection steady fluid flow occurs as a number of plumes which consist of fluid rising Žor falling. and moving horizontally outwards in both directions seen as two roughly circular rolls. They are characterised by horizontal mirror symmetry about their centre and examples may be seen in Fig. 2a,b. ConÕectons are a novel and important departure from this in that the system has undergone a bifurcation that removes the final reflection symmetry so that flow consists solely of a single roll, whilst retaining the usual Boussinesq point symmetry. Some results are developed in Fig. 3 which shows the sequence of bifurcations which may give rise to convectons. The stable branches were obtained by increasing and then decreasing Q in small steps until
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Fig. 2. The development of localized solutions as the magnetic field strength is increased, for the case of the truncated model with l s 6, R s 20 000. Ža. six steady rolls for Q s 3000, Žb. four rolls, Q s 8000, Žc. two isolated rolls, Q s 16 000, Žd. a single convecton, Q s 22 000. The shading shows deviations in temperature from the static state Ž u .. Also shown are velocity arrows, and magnetic field lines.
the solution in question became unstable. Very many more solution branches exist within the problem, but only a few key branches have been included. There are pitchfork bifurcations at Q1Že. s 190 and Q2Že. s 603 Žamong others. corresponding to steady convection with two Žwavenumber m s 1. and four Ž m s 2. rolls respectively. Due to the ability of a conducting fluid to expel magnetic flux, which leads to regions of reduced effective field strength, these branches show subcriticality w12x, and eventually gain stability at saddle-node bifurcations. The m s 1 branch results in two isolated convectons whose vorticities have opposite signs Žlabelled 2o., as in Fig. 2c. We also observe steady solutions with two isolated con-
vectons with vorticities of the same sign Ž2s. which result from a symmetry breaking of the m s 2 branch. The key bifurcation occurs as a symmetry breaking of the m s 1 branch, in which the Z2 reflection symmetry is lost, whilst retaining point symmetry. This branch gives rise to the single convecton solution and such a bifurcation has not been previously observed in studies of convection. Hitherto, all isolated states found in two-dimensional magnetoconvection have been part of a system of two or more dependent rolls. For example in w21x the imposed mirror Ž Z2 . symmetry of the problem means that Fig.8d of w21x shows a solution lying on the steady m s 1 branch, lacking the extra symmetry breaking
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Fig. 3. A schematic representation of part of the bifurcation sequence leading to the production of convectons for l s 6. Field strength is plotted horizontally and a measure of the amplitude of convection vertically. Solid lines represent observed stable solutions, and dashed lines are conjectural unstable branches. Squares, circles, and triangles show steady, Hopf, and symmetry-breaking bifurcations respectively. The number of convective rolls on each branch is shown.
required to produce a convecton. Branches arising via a primary pitchfork bifurcation from the trivial solution necessarily retain reflection symmetry w2x, so the branch which gives rise to the single convecton solution can only occur as a secondary bifurcation; Fig. 3 gives the simplest way in which this can occur. Finally, all three isolated steady branches Ž1, 2o, 2s. undergo Hopf bifurcations, as the reduction in effective field strength if Q is decreased allows weak oscillations to occur in the magnetic region. Examples of convectons may be seen in Fig. 2c,d which show the resulting patterns for two values of Q. In Fig. 2d Ž R s 20 000, Q s 22 000. the field strength is fairly close to the linear stability boundŽo . s 24047, yet flow is remarkably strong. ary at Q max The Nusselt number, a measure of convective efficiency defined so that N s 1 in the absence of all convection, is N s 1.32. The flow also expels flux from within the roll extremely efficiently; away from the edges the field strength is reduced to around a thousandth of its original value, whereas just outside the roll there is a sharp leap. In the magnetic region the field remains fairly uniformly vertical at a strength of 1.4 times its uniform value, and the temperature profile is also essentially uniform within the stagnant magnetic region. In addition there is a very narrow
current sheet near where the horizontal flow meets the magnetic region Žie near the surface on one side, and near the base on the other. where the flux is being swept by the convection, before diffusing outwards. The above solutions are associated with hysteresis and many types of solution coexist at identical parameter values. Examples of single convectons Žfor R s 20 000. were found for 13 700 F Q F 26 500, demonstrating that such flow is possible even when the purely-conducting state is stable to all small perturbations. The convecton in Fig. 2d has a width l f 1. However, this varies as the field strength is adjusted; with Q s 13 700 we obtain a width of l f 1.5, decreasing to l f 0.75 at Q s 26 500. Over the same range of Q the Nusselt number drops from 1.49 to 1.18. By way of comparison, if a solution is initiated from small perturbations to the trivial static solution then a different pattern is seen in all cases. For Q s 22 000 five pairs of oscillatory rolls are obtained that fill the box, as linear theory might suggest. The Žtime-averaged. Nusselt number is significantly lower Ž N s 1.08. signifying relatively feeble convection. Experiments were also carried out in boxes of different sizes. With smaller values of the aspect
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ratio the convecton solution was effectively observed down to l s 1; however it is only for l ) 2 that its nature is really clear, and a distinct region of high magnetic field can be seen. In very wide boxes, on the other hand, important and interesting behaviour is observed ŽFig. 4.. Fig. 4b shows two convectons with no reflection symmetry Žcorresponding to the 2s branch of Fig. 3., demonstrating that for l 4 1 such isolated rolls may coexist independently of one another, scattered across the box; the effects of flux expulsion allow a small convective roll to be formed having only a slight effect on the effective field strength of the magnetic region, with any viscous and magnetic stresses absorbed by the field. Fig. 4c shows a convectonranti-convecton pair retaining mirror symmetry about its centre, but having broken the point symmetry of a solution such as that shown in Fig. 2c. The mirror symmetry is then broken producing the single convecton of Fig. 4d. In boxes with fairly small aspect ratio single Žo . as in forcing all convectons are seen for Q - Qmax the magnetic field into one region the flow increases Žo . the effective field strength there so that Qeff ) Qmax . In arbitrarily large boxes convectons should still be Žo . possible, but will only be seen for Q ) Qmax , and Žo . indeed for l s 16 Ž Qmax s 24267. single convectons are only stable in the range 26 000 - Q - 30 000. This is because for l 4 1 a single convecton will only increase the effective field strength in the mag-
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netic region by a very small amount, and so for Žo . this region will be unstable to perturbaQ - Qmax tions and oscillations will be seen. It is this bistability in very wide boxes, or coexistence of stable Žo . trivial and nonlinear solutions for Q ) Qmax , that is associated with isolated states. This argument carries to the earlier case of l s 6; if the effectiÕe field strength in the magnetic region is plotted on the horizontal axis instead of Q in Fig. 3 then the Hopf bifurcation at the lower end of the convecton branch Žo . occurs at Qmax , and convectons always ‘coexist’ Žin terms of effectiÕe Q . with a stable trivial solution.
4. Fully resolved Boussinesq case Having established the existence of isolated rolls in the truncated model described in Section 3, it is important to verify that they are not a result of the restricted number of modes. It is expected that the range of behaviour seen in Section 3 will be repeated, with only the bifurcation points shifted w1x. There are many examples of convective patterns or regimes in low order models of Rayleigh–Benard convection that do not translate to the fully resolved case w3x. Numerical experiments were thus repeated for the full nonlinear Boussinesq Eqs. Ž1. – Ž3. in an attempt to reproduce the above behaviour for the case l s 6. A pseudo-spectral code due to A.M.
Fig. 4. Some solution states of the truncated model for l s 16. Ža. Two isolated clusters of five and three rolls, for Q s 10 000, Žb. two independent isolated convectons, Q s 28 000, Žc. a convectonranti-convecton pair, also found at Q s 28 000, from different initial conditions Žd. a fully isolated convecton, Q s 29 000.
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Rucklidge was used with periodic horizontal boundary conditions, and the resolution was 256 = 33. Commencing with a relatively high field strength Q s 26 000, R s 20 000 the flow settled into a state with three convective rolls of differing sizes and an intermediate region of strong magnetic field, even when the earlier ‘convecton’ state was used as an initial condition. However, when this state was used as the starting point for a run with higher Rayleigh number and field strength, weaker rolls were subsumed into the magnetic region, and for R s 50 000,Q s 80 000 a single convecton was obtained. The convection is highly efficient at expelling flux; within the convecton the field strength is again reduced to around a thousandth of its original value. The Nusselt number in this case is 3.51 and the width l f 2.9, much larger than is possible in the truncated case. Increasing field strength further and reducing the Rayleigh number cause a reduction in the convecton size. The smallest roll obtained so far has width l f 0.75 for values of R s 10 000 and Q s 100 000, and is shown in Fig. 1. Further increases in field strength result in the flow tending rapidly to the trivial solution. Convectons in fully resolved Boussinesq magnetoconvection have also recently been found independently by P.C. Matthews Žprivate communication. for the case z s 1, but they are otherwise essentially identical to those found above. We may also be confident that such isolated rolls will be found in the case of two-dimensional fully compressible convection, especially for studies in a weakly stratified layer w7x, though it is expected that they will take the form of travelling waves w9x. The role of bistability in producing isolated states is still under study. Investigations of theoretical models of binary fluids w18x have found that bistability obtained via subcritical instabilities is necessary in order to produce localized solutions. In other such systems it is the interaction of the modes of the separate fields via gradient coupling that results in the observed localization. For magnetoconvection the physical mechanism that gives rise to such coupling is clear: the expulsion of magnetic flux via advection. In three dimensions it is easy to imagine an equivalent isolated axisymmetric cell in which fluid rises at the axis and moves radially outwards in all
directions, but other analogues may be sought in the form of single ‘rolls’ whose radius rises from and then decays back to zero in one horizontal direction. Bistability is also possible in other hydrodynamic systems; in thermosolutal convection, although the primary bifurcations are often subcritical it is unlikely that analogous isolated steady rolls should occur as localization of solutal gradients could prove impossible. In rotating convection on the other hand we would expect analogous cells to be found, with the role of the magnetic field being filled by angular momentum gradients. Certainly further investigation is warranted and work in this area is ongoing.
Acknowledgements I would like to thank Nigel Weiss, Michael Proctor and Alastair Rucklidge whose insight and erudition have advanced this letter greatly. Comments made by the referees have been particularly useful in the presentation of this work. The author is supported by a PPARC research studentship.
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