complicated [6]. In the presence of an anisotropy this degeneracy is lifted and Eq.(1) is appropriate. Examples from hydrodynamics are convection inst

Transcription

1 Ordered defect chains in the 2D anisotropic complex Ginzburg-Landau equation Roland Faller and Lorenz Kramer Physikalisches Institut, Universitat Bayreuth, D Bayreuth, Germany November 21, 1997 Abstract. { In direct simulations of the anisotropic complex Ginzburg-Landau equation periodic defect chain structures are found. They develop spontaneously in a small parameter region, but once initialized their range of stable existence is rather large. We discuss dierent types found in dierent parameter regions. 1 Introduction The complex Ginzburg-Landau equation (CGLE) plays the role of a generalized normal form appropriate for a spatially extended medium in the vicinity of a forward bifurcation involving one oscillatory critical mode (Hopf bifurcation). Its applications extend from hydrodynamic instabilities [1, 2] and nonlinear optics [3] to chemical instabilities like the Belousov- Zhabotinsky reaction [4] or surface reactions [5]. For a general review see [6].The equation is investigated rather well in 1D [7, 8] and in 2D as far as the isotropic equation is concerned [9]. Also a number of results have been obtained in 3D [10]. Little is known about the anisotropic t A = [1 + (1 + ib 1 )@ 2 x + (1 + ib 2 )@ 2 y? (1 + ic)jaj 2 ]A (1) in 2D, which will be the subject of this paper. Here A is the complex amplitude modulating the critical mode in space and time. The usual dimensionless units are used. Actually the range of applicability of this equation is considerable. The isotropic case, i.e. Eq.(1) with b 1 = b 2, can essentially be applied only to isotropic systems undergoing a Hopf bifurcation with critical wavenumber q c = 0. When a nonzero wavenumber q c arises leading to traveling waves, as in many hydrodynamic instabilities, one has a continuous degeneracy of the critical mode (the orientation is arbitrary) which makes the situation more 1

2 complicated [6]. In the presence of an anisotropy this degeneracy is lifted and Eq.(1) is appropriate. Examples from hydrodynamics are convection instabilities in liquid crystals [12] and oscillatory boundary layers [11]. Also, of course, anisotropic systems with a q c = 0 bifurcation, as e.g. oscillatory surface reactions [5], require b 1 6= b 2. Taking into account linear transformations of x and y the term involving second derivatives is general. A linear group velocity term proportional to a rst space derivative has been removed by going into a moving frame. In case of degeneracy between left- or right-traveling waves, we assumed only one type to survive (which is often the case). Some aspects of Eq. (1) were considered in [13]. Let us list some simple properties. There are plane-wave solutions of the form A = F e i(qx+py?!t) ; F 2 = 1? p 2? q 2 ;! = c + (b 1? c)q 2 + (b 2? c)p 2 (2) which can be shown to be stable against (extended) long-wavelength modulations in a band of wavevectors q := (q; p) when c2 1 + b 1 c q c2 p 2 < 1 (3) 1 + b 1 c 1 + b 1 c > 0; 1 + b 2 c > 0 (4) holds. Therefore, existence of a stable band requires the Newell criteria (4) to hold. The boundaries determined by (3) are usually called the (generalized) Eckhaus instability. According to Eq.(3) the band-center solution with q = 0 is the last one to loose stability, i.e. the band shrinks to zero when a Newell criterion becomes violated. This is usually called the Benjamin-Feir instability. For q 6= 0 localized uctuations around the plane waves may be convected away, the group velocity being (@ p!) = 2((b 1? c)q; (b 2? c)p). Thus one may expect that plane waves with q 6= 0 can persist even beyond the "convective" Eckhaus stability limit, up to the boundary of "absolute stability" (see [14]). In particular for dimension D > 1 there exist other classes of long-time stable solutions, which were studied in the isotropic case. In 2D they often exhibit topological defects which here correspond to the simple zeros of A. It is easy to see that a simple zero of a complex eld in 2D in general survives a (small) variation of A (it only moves by a small amount). At the zeros the phase is no longer dened and the integral over the phase gradient around the defect varies by 2n with the topological charge n = 1. Defects can be created and annihilated inside the system only in pairs of opposite polarity, so the total topological charge of a system is a conserved quantity. Defects often emit spiral waves, which can extend over a large (ideally innitely large) region. The isolated spiral solution has a simple time dependence of the form exp(i!t) corresponding also to a rigid rotation of the object. Asymptotically the 2

3 waves approach locally a plane wave of the family (2). The selected frequency must be in the convectively, or possibly the only absolutely stable range!. Besides this "outer instability" the spiral can, for suciently large values of jbj, also undergo a core instability, which is connected with the fact that for jbj! 1 one has a Galilean invariance. This invariance is well known in the nonlinear Schrodinger limit of Eq.(1), which corresponds to jb 1 j?1 ; jb 2 j?1 ; jcj?1! 0. (Then one has in addition a scale invariance.) The Galilean invariance generates a family of moving spiral solutions. Breaking this invariance by a small jbj?1 leads to acceleration along the family [15]. In fact this core instability is on the linear level the analog of the meandering instability in excitable media [16]. An important aspect in discussing the defect solutions is their interaction. Interaction manifests itself in a (slow) drift of the spiral cores induced by the presence of other spirals (or other objects) as well as in a (small) shift of the frequency. This can be understood by noting that drift and frequency shift relate to the time change of the soft degrees of freedom of a spiral, namely position and phase. Qualitative aspects of the interaction properties can be understood from considering the spatial behavior of small (stationary) perturbations of the plane waves [17]. Of the four exponents characterizing the spatial behavior one is zero, corresponding to the translational mode. One is real and negative leading to decay of the mode away from the spiral, i.e. in the direction of the group velocity 2(b? c)q of the asymptotic plane wave. The remaining eigenvalues have positive real parts, i.e. the corresponding modes grow away from the spiral and are therefore relevant for matching to the external perturbation (like spiral waves emitted by an adjacent spiral). Depending on the parameters the eigenvalues are either real or complex conjugate. In the complex conjugate case the interaction of the defects is oscillatory, otherwise the interaction is (essentially) monotonic. The criterion for oscillatory interaction reads c? b 1 + bc > c crit 0:85 : (5) A quantitative analysis requires a proper asymptotic matching procedure [17]. In the oscillatory range one can have bound spiral pairs. However, it turns out that in that range there is also a spontaneous, initially very slow breaking of the symmetry between two spirals, which involves a dephasing (or desynchronisation), i.e. growth of a relative frequency shift. In this process the border where the waves emitted by the two spirals collide, which is characterized by a so-called shock structure, moves from its central position towards one spiral [18]. In ensembles of more spirals the eect manifests itself drastically by the fact that half of the spirals eventually become surrounded by shocks pushed out by the neighboring spirals. Then the trapped defects become passive objects sometimes called edge vortices. In the oscillatory and absolutely stable range one nds in suciently large systems so- 3

4 called vortex-glass [19, 20] or frozen [9] states. Here the positions of the spirals remains static but disordered in space. Ideally, the time dependence is of the form exp(i!t). This synchronization and spatial locking is apparently only possible in the range of oscillatory interaction of the spirals where one has edge vortices. The solution survives the introduction of anisotropy. Also states where the spirals are ordered in a simple lattice are possible [13, 18]. The lattices are quite dierent in the monotonic and in the oscillatory range. In the monotonic range one has a symmetric square lattice of spirals with alternating topological charge as in the simplest anti-ferromagnetic ordering. In the oscillatory range the symmetry between spirals of opposite topological charge is broken. In fact spirals of only one topological charge (either = +1 or?1) emit waves and organize the lattice. The other type becomes edge vortices trapped in the corners of the shocks. Weber et al. [13] presented simulations of a new class of solutions where the defects align spontaneously along chains. These chains, which bear some resemblance with chevron patterns observed in liquid crystal convection [21], will be discussed here in more detail. In addition some other types of solutions of Eq. (1) will be mentioned. 2 The ordered defect chain solutions (ODC) Direct simulations of the anisotropic CGLE using a pseudo-spectral method based on FFT were carried out on large vector computers. We concentrated on the plane b 2 =?c in parameter space. (Note that a change of the sign of all three parameters is a symmetry transformation of Eq. (1); we will always choose b 1 < 0; of course an exchange of b 1 and b 2 with simultaneous exchange of x and y is also allowed.) Ordered defect chains (ODC) do not develop easily out of disordered initial conditions. Very long runs with accurate time discretisation (4t 0:) are needed. We found parameters around b 1?3; c =?b 2 0:5 favorable, in agreement with [13]. However, once initialized properly, the solutions turn out to be stable in a rather large parameter range and insensitive to larger time steps. ODCs are in particular characterized by the fact that the distance between chains is signicantly larger than the spacing of defects within a chain. Within a chain the topological charges are the same, whereas they alternate from chain to chain. With the periodic boundary conditions used in the simulations the total topological charge in a simulation area is zero, so one always has an even number of chains (in our simulations two or four). We rst discuss stationary ODCs where the only time dependence is given by an overall phase factor exp(i!t). A typical picture of the modulus jaj of such an ODC is shown in Fig. 1a. The wave character of the ODC is brought out by the snapshot of Real(A) shown 4

5 in Fig. 1b. In the regions between the chains the phase velocity is pointing alternatively to the left and to the right, whereas near a defect it rotates around its center. One notes the chevron-like (or V-shaped) structure. The waves travel in the direction of convex curvature of the fronts. It is easy to understand from topological arguments that the wavelength in the plane-wave regions is twice the distance between defects along a chain: the two half waves on either side of a defect make up the 2 phase turn around the core. There exists a band of possible chain separations, but we have not traced it out. In general there is an asymmetry in the separation between neighboring chains, which may also vary (we remind of the periodic boundary conditions). It may be useful to interpret the ODCs in terms of waves emitted by the defects, which experience a nonlinear superposition. Shocks acting as strong sinks for the waves are situated inside the chains between the defects. Remarkably, in these shocks jaj becomes noticeably larger than one. Weak shocks are found between the chains where the wave crests are curved. Alternatively, the structure may be looked upon as an array of domain walls separating waves traveling in opposite directions. Note the shift in the defect position between neighboring chains. In the parameter range investigated the stationary ODCs were found mainly in two disconnected regions. Region I is located roughly within the rectangle 0:3 < c(=?b 2 ) < 0:8 and?5:8 < b 1 <?0:8, but actually it has a somewhat irregular shape (see Fig. 2). It is interesting to note that at about c = 0:7 Region I extends very near to the isotropic case. Region II is situated on the side of negative c and has a simpler shape. For b 1 <?3:5 it becomes a stripe extending between about c =?0:3 and c =?1:1. For larger b 1 the stripe tapers down to a tip at about b 1 =?1:7; c =?0:5. In the parameter range between Regions I and II, centered around the b 1 axis, we found another type of ODCs which drifts (very slowly) in the chain direction, see Figs. 3a,3b. Here A(x; y; t) is of the form exp(i!t)b(x? vt; y). The wave fronts between the chains are essentially plane and perpendicular to the chains, and no weak shocks are found here. Again, the waves travel alternatively in opposite direction. All drifting ODCs have an asymmetry in the chain separation. There exists a band of possible distances, but the equidistant ODCs do not drift at all. In traversing the b 1 axis, i.e. changing the signs of c? b 2, the direction of this drift is changed (actually the change of sign occurs slightly below the b 1 axis). One should note that hand in hand with the asymmetry in the chain spacings goes a (global) breaking of reection symmetry along x. In the drifting case this is more obvious, but it applies also to the stationary ODCs. There it can be seen in particular in the obliqueness of the ellipsoidal core regions. 5

6 Interestingly, the asymmetry vanishes spontaneously close to c = 0:3 where the stationary ODCs border on the drifting ones. The evolution from an asymmetric initial condition to the symmetric position is achieved by a small oblique drift of the defects. Drifting ODCs appear to exist essentially only in the region where perturbations of the plane waves between the defect chains behave monotonically in space [17]. The transition to stationary ODCs coincides rather accurately with the boundary to oscillatory behavior. This is quite analog to the transition from symmetric to symmetry-broken spiral lattices found in the isotropic case and presumably it is governed by the same mechanism [18]. At the border itself only the above-mentioned equidistant (symmetric) ODCs are found. We suspect that here a simple description in term of a superposition of left- and right-traveling waves is possible. In addition several solutions with smaller regions of stable existence were found. Close to the lower c-limit ODCs were found with an overall transverse motion of the whole structure. The distances between chains change in a somehow irregular fashion without distorting the chains themselves. Also, a rather irregular pattern of shocks (sinks) is found in the region between the chains. In another parameter range ODCs were found with small and repetitive, although probably not periodic, transverse excursions of the chains. Here not only the phase, but also the modulus jaj shows a chevron-type structure. Additionally in a very small range of parameter space another special solution was found, see Fig. 4a. Here the defect chains break up and rotate into an oblique position. These states decay into stationary or drifting ODCs in their respective range of stable existence. We cannot exclude that this solution is only a long-lived transient before ODCs break up into the band-center solution A = exp(?ict), which is here stable according to the Newell criterion (4). The exact type of solution presented in [13] could not be reproduced by us. There the direction of the drift of the ODCs was found to alter from chain to chain and thereby be related to the sign of the topological charge. Possibly these solutions were still in a transient state [22]. We found transients to last about ten times the maximum time simulated in that early work where large computers were not available. In the same parameter region investigated by Weber we nd the stationary ODCs. 3 Instabilities of the ODCs There are several possibilities for the ODCs to loose stability. One (rare) possibility was pointed out above. Another remarkable instability is found when the upper limit of b 1 is reached. Then the chains undergo long-wave undulations (Fig. 4b). Before the chains break 6

7 up the defects become surrounded by shocks. After leaving the chains, defects either collide and annihilate or persist in irregular motion, which probably forms a type of vortex uid [23]. Another instability occurs at the upper limit of c. Then the system either evolves into a lattice with edge vortices or into a vortex glass. Also a direct transition into defect chaos can be observed in a range where the upper c limit is close to the boundary of absolute stability of plane waves traveling along the chains with the observed wavelength c 0:8). The two latter instabilities were already reported by Weber et al. [13]. 4 Conclusions One can say conclusively that the 2D anisotropic CGLE shows new classes of solutions, which are unknown in the isotropic case. When simulated in quadratic systems with periodic boundary conditions the ordered defect chains have a rather large range of stable existence, although they do not develop very easily spontaneously. There are two main types of ODCs: stationary and drifting ones. At present the analytic understanding of these solutions is limited to qualitative arguments. More work, both in simulation and analysis remains to be done. In particular, we believe that an understanding of the transition between the two types of ODCs should be possible. Let us mention three more interesting solution classes which we found in the anisotropic CGLE. In some parameter ranges there exist slowly drifting versions of the previously mentioned vortex-glass (or frozen) states. The drift can be in any direction, but its magnitudes, for given parameters, are similar for dierent realisations of such states. Moreover, when the parameters are chosen such that one has "strong" Benjamin-Feir stability in one direction (e.g. b 2 = c) and weak instability in the other direction (b 1 c slightly below?1) one nds two kinds of phase chaos, i.e. chaos without defects. One type ("PCII") is the analog of the phase chaos found in 2D [24]. However, it exists in a larger parameter range and represents there the global attractor of the system (i.e. with stochastic initial conditions one ends in this solution). Finally, in a considerably smaller parameter range we also found a type of phase chaos with spatial variations only in the unstable direction ("PCI"), i.e. a quasi-1d phase chaos. This demonstrates that Eq.(1) contains a multitude of unexpected solution classes. More details will be published elsewhere 7

8 5 Acknowledgments We have beneted from discussions with W. Pesch, J. Neubauer and A. Roberg. Extensive use of high-performance computer facilities at the LRZ Munchen (Cray T90) and the HLRS Stuttgart (NEC SX4), as well as nancial support by DFG (Kr690/4) are gratefully acknowledged. References [1] W. Schopf, W. Zimmermann. Convection in binary uids: codimension-2-bifurcation and thermal uctuations, Phys. Rev. E47, (1993) [2] M. Treiber, N. Eber, A. Buka, L. Kramer. Travelling Waves in Electroconvection of the Nematic Phase 5: A Test of the Weak Electrolyte Model, J. de Physique II 7, (1997) [3] P. Coullet, L. Gil, P. Rocca. Optical vortices, Optics Comm. 73, (1989) [4] F. Hynne, P. Graae Srenson, T. Mller. Complete optimization of models of the Belousov- Zhabotinsky reaction at a Hopf bifurcation, J. Chem. Phys. 98, (1993) [5] M. Baer, M. Hildebrand, M. Eiswirth, M. Falcke, H. Engel, M. Neufeld. Chemical turbulence and standing waves in a surface reaction model: The inuence of global coupling and wave instabilities, Chaos 4, (1994) [6] M.C. Cross, P.C. Hohenberg. Pattern formation outside of equilibrium, Rev. Mod. Phys. 65, (1993) [7] W.v. Saarlos, P.C. Hohenberg. Fronts, pulses, sources and sinks in generalized complex Ginzburg-Landau equations, Physica D56, (1992) [8] S.Popp, O.Stiller, I.S.Aranson und L.Kramer. Hole solutions in the 1d complex Ginzburg- Landau Equations, Physica D87, (1995) [9] H. Chate, P. Manneville. Phase Diagram of the Two-Dimensional complex Ginzburg- Landau equation, Physica A224, (1996) [10] I. Aranson, A. Bishop. Stretching of vortex lines and generation of vorticity in the three-dimensional complex Ginzburg-Landau equation, Phys. Rev. Letters, in press. [11] T. M. Haeusser, S. Leibovich. Amplitude and mean drift equations for the oceanic Ekman layer, Phys. Rev. Letters 79, (1997) 8

9 [12] L. Kramer and W. Pesch, in Pattern formation in liquid crystals, A. Buka and L. Kramer, eds. (Springer, New York, 1996) p. 221 [13] A. Weber, E.Bodenschatz, L. Kramer. Defects in continuous Media, Advanced Materials 3, (1991) [14] I. Aranson, L. Aranson, L. Kramer, A. Weber. Stability of spirals and travelling waves in nonequilibrium media, Phys. Rev. A46, 2992(1992); A. Weber, L. Kramer, I. S. Aranson und L. B. Aranson. Stability Limits of Traveling Waves and the Transition to Spatiotemporal Chaos in the Complex Ginzburg-Landau Equation, Physica D61, (1992) [15] I. Aranson, L. Kramer, A. Weber. The core instability and spatio-temporal intermittency of spiral waves in oscillatory media, Phys. Rev. Letters 72, (1994) [16] D. Barkley. Euclidean symmetry and the dynamics of rotating spiral waves, Phys. Rev. Letters 72, (1994). [17] I. S. Aranson, L. Kramer, A. Weber. The theory of Interaction and Bound States of Spiral Waves in Oscillatory Media, Phys. Rev. E47, 3231(1993); [18] I. Aranson, L. Kramer, A. Weber. The Formation of Asymmetric States of Spiral Waves in Oscillatory Media, Phys. Rev. E48, R9-R12(1993) [19] G. Huber, P. Alstrm, T. Bohr. Nucleation and Transients at the Onset of Vortex Turbulence, Phys. Rev. Letters 69, (1992) [20] T. Bohr, G. Huber, E.Ott. The structure of spiral-domain patterns and shocks in the 2D complex Ginzburg-Landau equation, Physica D106, (1997) [21] A. G. Roberg, L. Kramer. Pattern formation from Defect Chaos - A Theory of Chevrons, Physica D, in press [22] A. Weber. private communication [23] G. Huber. Vortex Solids and Vortex Liquids in a Complex Ginzburg-Landau System in Spatio-Temporal Patterns in Nonequilibrium Complex Systems, P. E. Cladis and P. Paly- Muhoray, eds., SFI Studies in the Sciences of Complexity, Addison-Wesley, Reading MA 1995 [24] P. Manneville and H. Chate Physica D96, (1996) 9

10 Figure Captions Fig. 1: Stationary ODC at b 1 =?3:3 ; c =?b 2 = 0:5. (a): modulus of A. (b): real part of A. Here, as in all the gures of jaj, black means zero, the colors red, yellow and green express intermediate values (in increasing order), light blue and medium blue is about one and dark blue denotes values larger than one. The simulations were performed on quadratic domains of length 100 with periodic boundary conditions and we used Fourier modes. After initialisation with an ODC the runs lasted iterations with a time-step of t = 0:1. 10

11 11 Fig. 2: Regions of stable existence of dierent types of solutions in the b1; b2 plane (c =?b2) shock DC vortex glass plain wave oscillatory shock with wave par. shocks -2.0 undulational instability. oblique shock DC edge lattice vortex fluid c=-b constant solution stationary chevrons chevrons drifting along the chain equidistant chains chevrons drifting perpemdicular to the chain no chevrons (see text) DC defect chaos convective instability for waves along the chains absolute instabilty for waves along the chains BF-line isotropic casen

12 Fig. 3: Drifting ODC at b 1 =?2:0 ; c =?b 2 = 0. (a): modulus of A. (b): real of part of A. Fig. 4: (a): Oblique ODC at b 1 =?1:8; c =?b 2 =?0:5. (b):the undulational instability at b 1 =?1:5; c =?b 2 = 0:3. 12