On convection in a horizontal magnetic field with periodic boundary conditions

Transcription

1 This article was downloaded by: [University of California, Berkeley] On: 02 December 2014, At: 16:08 Publisher: Taylor & Francis Informa Ltd Registered in England and Wales Registered Number: Registered office: Mortimer House, Mortimer Street, London W1T 3JH, UK Geophysical & Astrophysical Fluid Dynamics Publication details, including instructions for authors and subscription information: On convection in a horizontal magnetic field with periodic boundary conditions E. Knobloch a a Department of Physics, University of California, Berkeley, CA, 94720, U.S.A. Published online: 18 Aug To cite this article: E. Knobloch (1986) On convection in a horizontal magnetic field with periodic boundary conditions, Geophysical & Astrophysical Fluid Dynamics, 36:2, , DOI: / To link to this article: PLEASE SCROLL DOWN FOR ARTICLE Taylor & Francis makes every effort to ensure the accuracy of all the information (the Content ) contained in the publications on our platform. However, Taylor & Francis, our agents, and our licensors make no representations or warranties whatsoever as to the accuracy, completeness, or suitability for any purpose of the Content. Any opinions and views expressed in this publication are the opinions and views of the authors, and are not the views of or endorsed by Taylor & Francis. The accuracy of the Content should not be relied upon and should be independently verified with primary sources of information. Taylor and Francis shall not be liable for any losses, actions, claims, proceedings, demands, costs, expenses, damages, and other liabilities whatsoever or howsoever caused arising directly or indirectly in connection with, in relation to or arising out of the use of the Content.

2 This article may be used for research, teaching, and private study purposes. Any substantial or systematic reproduction, redistribution, reselling, loan, sub-licensing, systematic supply, or distribution in any form to anyone is expressly forbidden. Terms & Conditions of access and use can be found at

3 Ceophys. Asrrophys. Fluid Dynamics, 1986, Vol. 36, pp /86/ $18.50/ Gordon and Breach, Science Publishers, Inc. Printed in Great Britain On Convection in a Horizontal Magnetic Field with Periodic Boundary Conditions E. KNOBLOCH Department of Physics, University of California, Berkeley, CA 94720, U. S. A. (Recevied January 8, 1986; in final form February 20, 1986) The effect of an imposed horizontal magnetic field on two-dimensional convection in a plane layer heated from below is studied in the presence of periodic boundary conditions in the horizontal. An amplitude equation describing the transitions between travelling and standing waves and steady convection near a particular multiple bifurcation is derived. When the first instability is oscillatory, two solution branches bifurcate simultaneously from the conduction solution, corresponding to travelling and standing waves. The branch of travelling waves is stable throughout and terminates on the steady state branch in a steady state bifurcation as the Rayleigh number is increased. Beyond this point stable overturning convection is found. Standing waves are unstable. The results are indicative of the larger amplitude behavior of the system. I. DESCRIPTION OF THE PROBLEM Overstability (i.e., a Hopf bifurcation) in a spatially unbounded hydrodynamical system results in general in the appearance of propagating disturbances. In systems with periodic boundary conditions in the horizontal two types of patterns are found in the nonlinear regime near the bifurcation. We shall refer to these as travelling waves and standing waves. The travelling wave is a phase wave, and in systems with only one horizontal dimension and no distinction between left and right it can propagate in either direction. 161

4 162 E. KNOBLOCH In three-dimensional systems more complicated spatial patterns can also propagate (Swift, 1984). The standing wave is a particular superposition of travelling waves that produces a spatial pattern with nodes that remain fixed in space. In addition to these oscillatory states, many systems also admit nonlinear time-independent states, and one is naturally interested in studying the possible transitions in the nonlinear regime between oscillatory states and the time-independent states, that can occur as a parameter is varied. A problem of this type can be solved, often analytically, by studying the system in a neighborhood of a particular multiple bifurcation. In the present case the appropriate bifurcation occurs at critical parameter values for which an oscillatory and a steady-state bifurcation coincide. By studying the system for nearby parameter values the behavior of the system can be fully described. It is typically found that the behaviour identified near such a multiple bifurcation persists for parameter values substantially different from their critical values. In this way the study of multiple bifurcations provides a key to the understanding of the behaviour of a large variety of nonlinear systems (Knobloch and Proctor, 1981). In this paper we study two-dimensional nonlinear Boussinesq convection in an imposed horizontal magnetic field from this point of view. The system was first investigated by Arter (1983) in an attempt to elucidate some aspects of the dynamics associated with the solar magnetic field and is described by the nondimensionalized equations Vz$, + J($, V2$) = ORB, + ocqv2a, + gcqj(a, V A) + av4$, (1.1) where $(x, z, t) is the streamfunction, and O(x, z, t) and A(x, z, t) represent departures of the temperature magnetic vector potential from their conduction profiles. The dimensionless parameters appearing in these equations are the Rayleigh number R, the Chandrasekhar number Q, and the two Prandtl numbers CT= V/K, C=v]/ic, where v is the kinematic viscosity, v] the ohmic diffusivity,

5 MAGNETOCONVECTION 163 and K the thermal diffusivity. With stress-free and fixed magnetic flux boundary conditions we require $=$ zz =8=A=O on z=o,l. ( 1.4) In addition, we shall impose periodic boundary conditions in the horizontal: where 21 = 2n/k is the spatial period. The linear stability of the pure conduction solution $ = 8 = A = 0 is described by a cubic eigenvalue equation. The oscillatory and steady state bifurcations coincide when there are simultaneously two zero eigenvalues (Knobloch and Proctor, 1981). This occurs when where p = k2 + n2. The system ( ) is equivariant under horizontal translations and under reflections in a vertical plane. As explained in the appendix the presence of these symmetries has two effects. On the one hand it doubles the number of zero eigenvalues to four thereby complicating the bifurcation structure. On the other it restricts the structure of the amplitude equation that describes the dynamics of the system ( ) in the vicinity of the multiple bifurcation to take the simple form (Dangelmayr and Knobloch, 1986): Y =~Y+Evu + A ~ Y + ~ ~C(u6 ~ u +6d) + EDIu~~Y + O(E ), (1.7) where EY is the (complex) amplitude of the streamfunction (E<< 1). The coefficients p, v are called unfolding parameters, and measure the departure of R, Q from their critical values (1.6). There are three nonlinear terms, whose coefficients must be computed before specific predictions for the system ( ) can be made. It is sufficient to compute these at the bifurcation (i.e., for p=v=o), and this calculation is carried out in the following section. It is helpful to point out that if the boundary conditions (1.5) are

6 164 E. KNOBLOCH replaced by Arter s boundary conditions $=$x,=t3,=a,=0 on x=0,21 (1.8) requiring no sideways fluxes at the boundaries, then no travelling waves are possible. The amplitude equation for this case is obtained from (1.7) by taking 0 to be real (see appendix): V = pu + FVV + Av3 + &MU2V + 0(&2), (1.9) where A4 = 2C + D. In contrast to (1.7) this amplitude equation contains only two nonlinear terms, and it describes the interaction between standing waves and steady states near the bifurcation (1.6) in systems with a Z(2) reflection symmetry (eg., Knobloch and Proctor 1981, Guckenheimer and Knobloch, 1983). Evidently, all solutions of (1.9) are also solutions of (1.7). However, in addition (1.7) enables us to establish the stability of these solutions with respect to disturbances in the form of travelling waves. This is of interest in horizontally infinite systems where the boundary conditions (1.8) are inappropriate, raising the possibility that the solutions computed by Arter are in fact unstable to such disturbances. In this paper we show that in the neighbourhood of the parameter values (1.6) the standing waves and some of the stationary solutions are indeed unstable. The remainder of this paper is organized as follows. In Section 2, we use the procedure first used by Knobloch and Proctor (1981) to derive the amplitude Eq. (1.7) from Eqs. ( ). A complete description of the dynamics described by (1.7) for small E has been given by Dangelmayr and Knobloch (1986), and we make use of this analysis to make specific predictions for the system ( ). Our results are summarized in the form of a bifurcation diagram. Brief conclusions follow in Section 4. For the interested reader the symmetry arguments that are responsible for the structure of the amplitude Eqs. (1.7) and (1.9) are summarized in the appendix. 2. DERIVATION OF THE AMPLITUDE EQUATION In this section we show how to derive the amplitude Eq. (1.7) from the basic equations. The method we use, due to Knobloch and

7 MAGNETOCONVECTION 165 Proctor (1981), is the simplest one, and the resulting amplitude equation finds a formal justification in center manifold theory (Guckenheimer and Knobloch, 1983) and in equivariant bifurcation theory (Dangelmayr and Knobloch, 1986). We begin by introducing the scaled parameters describing the departure of R, Q from their critical values (1.6), and a slow time We seek solutions of the form t = cpt. (2.2) II/ = Re {~(t, E)eikx sin xz + 0(c3)}, (2.3a) A=Re (czl(tf,c)eikxsinxz+c2zz(tf,e) sin2x2+o(e3)}. (2.3~) Substituting these expressions into ( ), and equating coefficients of like spatial eigenfunctions, we obtain the set of equations: ika ikal: EU = - ou - R,( 1 + E ~,)w, + ~ Q,( 1 + E ~Q)z~ P P EW;= ik ikx -w,+-u+&2-(vwz+uwz)+0(e4), P 2P ikx EW; = -WW,--(UW, -UW1) +0(&2), 4P (2.4a) (2.4b) (2.4~) ik ikx cz; = -l:z, + -u +E2--(UZ, +UZJ + O(E4), P 2P (2.4d)

8 - 166 E. KNOBLOCH ikrc Ed2 = -iwz2 - -(vz, -VZ1) + 0(&2), (2.4e) 4P where w = 4rc2/p (0 < w < 4), and the prime denotes differentiation with respect to the slow time t'. To solve these equations in powers of E, we note first that at leading order in E, Eqs. (2.4d, e) imply z2=--- ik z1 = - 2, + &g(&, t), ip 1 k2n 2 pp2w 1u12+Eh(E,t'), (2.5a) (2.5b) where g, h are both O( 1) at leading order. Substituting into (2.4d, e) we now obtain 1 lv12+&h+~h +0(c3), 1 ikn 1 k2n d h=---( vg-vg) IV 12 + O(E). 4 5PW 2 C3p2w2 dt' - +- ~ These equations can be solved iteratively in powers of E, and yield z2=--- 2 C2p2m k2rc [ lv12-& $~u~~--(u~t'+vv') & 2i 1 (2.6a) (2.6b) (2.7a) +O(c2), (2.7b) with similar expressions for wl, w2. Substituting these results into (2.4a), we find that all the terms of O(E') and O(E) vanish (this is because at the bifurcation the linear problem has two zero eigen-

9 MAGNETOCONVECTION values), leaving the amplitude Eq. (1.7) with 167 (2.8a) (1 +.)(w - A=-- 1-1) 8P k2 [ (1-C)A "'i (; :) c=--- [1+o+i+ai+i CPA (2.8b) (2.9a) (2.9~) and A=l+c+I;. In deriving these results we made use of the amplitude equation at leading order to eliminate u"'. The amplitude Eq. (1.7) admits travelling waves (TW), standing waves (SW), and steady states (SS), and describes all possible transitions among them near the multiple bifurcation. The steady states are given by p+ A I u12 =o. (2.10) For fixed Q, the bifurcation is supercritical if A<O. From (2.9a) we see that this occurs for 2 - c2 < w < 4. The value of w that minimizes R, is w=8/3. It can also be shown that the values of w that minimize the Rayleigh numbers for onset of both oscillatory and steady convection satisfy w > 8/3 (Q > 0). Thus A < 0 in the regime of interest. The travelling waves are given by v + D I u 1' = 0, (2.11) and for fixed Q are supercritical if D < 0. Again, this is satisfied in the regime of interest. Finally, the standing waves are supercritical if

10 168 E. KNOBLOCH M = 2C + D < 0, i.e., if +( 1 - i)a( 1 + (T + L ) (2.12) a condition that is again satisfied in the regime of interest. A complete description of the solutions of the amplitude Eq. (1.7) is given by Dangelmayr and Knobloch (1986). When A < 0 there are 18 regions in the (D,M) plane with different dynamics in the (p,v) plane (Figure 1). We do not repeat here this analysis, but note that FIGURE 1 The solutions of the amplitude Eq. (1.7) in the (D,M) plane for A<O. The plane splits into 18 regions. The present problem falls into region 11-.

11 so that MAGNETOCONVECTION 169 D 1 0 <- <-, M 3 m > 2. (2.13) Since D<O, the present system falls into region 11- in the (D,M) plane (see Figure 1). The regions in the (p, v) plane, in which different solutions are found, are indicated in Figure 2. The stability of the solutions is indicated by ( -, +) =(stable, unstable), for the relevant eigenvalues. Simple bifurcations occur along the lines that fan out from the origin. We briefly describe the nature of these bifurcations. FIGURE 2 The (p,v) plane in region 11- of Figure 1, showing TK SWand SS and their stability characteristics. Simple bifurcations occur along the lines separating the different regions (see text for description). The heavy line, traversed in the sense indicated, describes the sequence of transitions that occur when R is increased for fixed Q > Q,.

12 170 E. KNOBLOCH Along H, the original Hopf bifurcation from the conduction solution gives rise to TW and SW, while the steady states SS bifurcate along Lo. Along the line L, the TW terminate in a steady-state bifurcation on SS. The SW branch extends as far as the line of saddle-nodes SN, where the branch doubles back, producing an ambiclinic limit cycle at SL (cf., Knobloch and Proctor, 1981) and terminating on L, in a secondary Hopf bifurcation from the SS branch. Of particular interest is the succssion of transitions that occur as the Rayleigh number is increased for fixed Q > Q,. This corresponds to describing the line in the (p,v) plane in the direction shown in Figure 2. This information is summarized in the bifurcation diagram shown in Figure 3, in which the Nusselt number is plotted against the Rayleigh number. From these results we see that the Hopf bifurcation at gives rise to stable travelling waves, and unstable standing waves, both bifurcating supercritically, and that the TW branch terminates on the branch of steady overturning convection in a steady state N 1 FIGURE 3 The bifurcation diagram showing the Nusselt number against R for fixed Q>Qc along the section indicated in Figure 2. The TW branch is stable, and stability is transferred to the SS branch at RTW. The SWbranch is unstable.

13 MAGNETOCONVECTION 171 bifurcation. As this bifurcation is approached the drift speed of the TW decreases linearly to zero. At this point the travelling waves are superseded by stable steady overturning convection. This bifurcation is not hysteretic. The SW branch, on the other hand, is everywhere unstable, at least near this multiple bifurcation. However, in the absence of the translation symmetry this branch is stable (cf. Knobloch and Proctor, 1981), showing that in the presence of this symmetry the standing waves lose stability with respect to travelling wave disturbances. Note that the stable wave is the one that transports more heat across the layer (Knobloch, 1985). 3. DISCUSSION In this paper we have studied two-dimensional convection in an imposed horizontal magnetic field from the point of view of bifurcation theory. The theory emphasizes the study of multiple and degenerate bifurcations as the key to the understanding of complicated behaviour in dynamical systems. In the present problem we were able to establish the nature of the competition in the nonlinear regime between travelling and standing waves, and steady state convection by studying an appropriate multiple bifurcation. Such an analysis yields important clues as to the behaviour of the system at larger amplitudes, or equivalently for parameter values substantially far from their critical values. The basic reason for the success of this procedure is that near such multiple or degenerate bifurcations all the nonlinear behaviour of interest takes place at small amplitudes and hence becomes accessible to a perturbation analysis. Unless additional bifurcations occur this behaviour persists away from the multiple bifurcation. The amplitude equation we have derived owes its form to the symmetry properties of the system, and these in turn depend on the nature of the boundary conditions adopted. When periodic boundary conditions are used, the system becomes translationally invariant, and in consequence a Hopf bifurcation results not only in standing waves but also in travelling waves. In the present system there is no distinction between left and right, and hence for each left-travelling wave there is also a right-travelling wave; which wave occurs depends on the initial conditions. The translational symmetry has no

14 172 E. KNOBLOCH effect on the steady state bifurcation. In contrast, with boundary conditions that break the translational symmetry, such as those requiring horizontal fluxes to vanish on fixed vertical boundaries (Arter, 1983), travelling waves are no longer possible, and only standing waves are found. Since in the present problem the standing waves are unstable, we see that the choice of boundary conditions can be crucial in determining the stability of a given solution. In particular, in modelling spatially infinite systems undergoing a Hopf bifurcation it is necessary to use periodic boundary conditions, and not to artificially constrain the system by no horizontal flux boundary conditions. A detailed discussion of the symmetries associated with periodic boundary conditions, and their influence on the dynamics, is given by Knobloch (1986). Travelling waves are readily found in numerical calculations modelling thermosolutal convection (Knobloch et al., 1986), and in certain circumstances can exhibit secondary bifurcations to modulated waves. While this occurs in other systems (eg., Knobloch, 1986), it does not occur near the particular multiple bifurcation we have studied here, at least for the physically interesting case za > 2 (horizontally elongated cells). Much of the possible dynamical behaviour can be established on the basis of group-theoretic arguments (cf. Dangelmayr and Knobloch, 1986; Knobloch et al., 1986), and analyzed within this framework. To apply the results to specific problems typically necessitates the calculation of a few coefficients from the basic equations describing the system, in order to determine which of the possible cases is actually realized. The method we have used in this paper is the most direct one, and turns the application of the general theory into a simple exercise. References Arter, W., Nonlinear convection in an imposed horizontal magnetic field, Geophys. Asirophys. Fluid Dyn. 28, (1983). Dangelmayr, G. and Knobloch, E., The Takens-Bogdanov bifurcation with O(2) symmetry, Phil. Trans. Roy, Soc., in press (1986). Guckenheimer, J. and Knobloch, E., Nonlinear convection in a rotating layer: Amplitude expansions and normal forms, Geophys. Asirophys. Fluid Dyn. 23, (1983). Knobloch, E., Doubly diffusive waves, Proc. Joint ASCE-ASME mech. conf. (N. E. Bixler and E. A. Spiegel, eds.), FED vol. 24, (1985).

15 MAGNETOCONVECTION 173 Knobloch, E., Oscillatory convection in binary mixtures, Phys. Reo. A, in press. Knobloch, E., Deane, A., Toomre, J. and Moore, D. R. Doubly diffusive waves. In: Multiparameter Bifurcation Theory. Contemporary Mathematics 56 (M. Golubitsky and J. Guckenheimer, eds.) American Mathematical Society, Providence, in press (1986). Knobloch, E. and Proctor, M. R. E., Nonlinear periodic convection in doublediffusive systems, J. Fluid Mech. 108, (1981). Swift, J. W., Bifurcation and Symmetry in Convection. Ph.D. thesis, University of California, Berkeley, 283 pp (1984). Appendix: Symmetries and the Amplitude Equation (1.7) In this appendix we explain how symmetry arguments can be used to derive the structure of an amplitude equation describing the dynamics of a system near a bifurcation. Throughout the discussion we shall emphasize the application of such arguments to systems of partial differential equations. We begin by observing that the basic Eqs. ( ) with the boundary conditions (1.4, 1.5) are equivariant under translations x+x+ 1, and reflections x+ --x. We denote these operations by T and R. Thus T(x, Z) =(x + 1, z), R(x, Z) =( -x,z). (A.1) If we write Eqs. ( ) symbolically in the form LY(x, z, t) = 0, Y = ($, 8, A), (A4 equivariance with respect to T and R implies that the operator L together with the boundary conditions (1.4, 1.5) commutes with these operations: TL = LT, RL = LR. (A.3) These equations state that translating and reflecting the system is the same as translating and reflecting the fields Y. Similarly, the fields Y are equivariant under T and R if TY=YT, RY=YR, (A.4) i.e., translating and reflecting the fields is the same as translating and reflecting the observer. As an example, note that since the horizontal

16 174 E. KNOBLOCH velocity component u = $, changes sign under reflection Hence II/ is equivariant under reflections if For the present system it is easy to show that (A.3) holds. This does not imply that all the solutions of (A.2) are equivariant, in the sense (A.4). However, there is one solution of (A.2), the conduction solution Y =0, which is. Other solutions of (A.2) break one or other of the symmetries. In particular this is so for the solutions that bifurcate from the conduction solution. If we look for solutions that are spatially periodic, with period 271/k, we can identify x with x + 271/k. This turns x into an angle-like coordinate, and the translations through 1 can be identified with rotations through kl (mod 271). The group of rotations and reflections of a circle is called O(2). It is the symmetry of Eqs. ( ) in the sense that (A.3) holds for all elements in this group. The group has two subgroups that are of interest to us: the group of reflections 272) consisting of two elements, and the group of rotations SO(2) which is a continuous group. We now turn to the bifurcation problem and linearize (A.2) about the conduction solution Y = 0: From (A.3) it follows that DL(0, R, Q) also commutes with the operations T and R: Suppose now that at R=R, and Q=Q,, DL(O,R,,Q,) has two zero eigenvalues with corresponding real eigenfunctions Y and Y2. Then from (A.8) it follows that TYl, TY, are also eigenfunctions of DL(0, R,, Q,) with zero eigenvalues. The original and the "rotated" eigenfunctions are linearly independent of one another, since sin kx and sink(x+z,) are linearly independent for any 1, such that

17 MAGNETOCONVECTION 175 O<E,<2n. Thus the presence of the 0(2) symmetry doubles the number of eigenfunctions with zero eigenvalue. This is a common feature of bifurcations in the presence of symmetry. We can choose any two such sets of independent eigenfunctions as the basis for the null eigenspace and write the streamfunction at the bifurcation in the form $(x, z, t) = Re { ZieikX + weik(x+lo) ] sin zz, (A.9a) where (u, w) are complex amplitudes. The phase kl, can be absorbed in the definition of the amplitude w, yielding $(x, z, t) = Re { ueikx + weikx} sin TCZ. (A.9b) We can see that the translations and reflections of x can be thought of as transformations on the amplitudes (0, w): T(u, w) = eikr(u, w), R(u, w) = -(6, W). (A.lOa) (A.lOb) The transformations (A.10) represent the action of the group 0(2) on the original system. Since this system is equivariant with respect to the group O(2) (cf. A.3), equations for the amplitudes (u, w) must be equivariant with respect to the representation (A.lO) of O(2). The amplitude equations will take the form of two complex first order ordinary differential equations for the amplitudes u, w whose linearization about (u, w) = (0,O) will have two zero eigenvalues, corresponding to four zero eigenvalues in real variables. To construct these equations we first write down the invariants of (A.lO): Next, we write down the most general equivariant vector field. This assures that both sides of the amplitude equations transform the same way under (A.lO). We obtain (A.12)

18 176 E. KNOBLOCH where the functions gj(ol, oz, 03), j= 1,2,3,4, are real functions of the invariants (A.ll) and are assumed to be infinitely differentiable at the origin. At this point it is convenient to choose the coordinates (u, w) to diagonalize the linearization of these equations: (i) = (: A)(i) + nonlinear terms. (A. 13) The nonlinear terms can be simplified by an equivariant nearidentity nonlinear coordinate change of the form (cf. A.12) to yield the simpler system (A.14) W=[AIU~~+BIW~~+C(UW+VW)]D+D~U~~W. (A.15) Here the coefficients A, B, C, D are real and only the leading order nonlinear terms have been retained. We can now include small linear terms that are present when (R - RJR, and (Q - Q,)/Q, are small but nonzero and obtain equations that by an additional small rotation of (u, w) can be put into the convenient form d=w, W=~U+VW + [Al~l +BIwI2+ C(uG+fiw)]u+ DIuI2w, (A.16) where p,v are linearly related to (R-RJR, and (Q-Q,)/Q,. The introduction of such terms changes the coeficients of the nonlinear terms by a small amount, but has no effect provided certain nondegeneracy conditions are satisfied. In particular it is necessary that A#O and that the coefficients M=2C+D and D do not lie on any of the straight lines separating the different regions in Figure 1. In this case it is sufficient to calculate the coefficients at the multiple bifurcation. Finally, we blow up the neighbourhood of the multiple bifurcation by introducing a small parameter E (O<E<<~) and re-

19 MAGNETOCONVECTION 177 scaling the variables and unfolding parameters as follows: The resulting Eq. (1.7) is the desired normal form of the amplitude equation describing the dynamics in the neighbourhood of the multiple bifurcation. From the above discussion it is clear that if the translation symmetry is absent only two zero eigenvalues will be present. Because of the way u,w were chosen in (A.13) we must now take both u and w to be real. It follows that this problem is described by the normal form (1.9). In such a system travelling waves are absent. The calculation described in Section 2 is remarkable because none of the successive coordinate changes mentioned above are required in order to cast the amplitude equations into normal form.