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# Special Topics PHYS 8803

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This 0 page Class Notes was uploaded by Kylie Bartoletti DVM on Monday November 2, 2015. The Class Notes belongs to PHYS 8803 at Georgia Institute of Technology - Main Campus taught by Staff in Fall. Since its upload, it has received 29 views. For similar materials see /class/234286/phys-8803-georgia-institute-of-technology-main-campus in Physics 2 at Georgia Institute of Technology - Main Campus.

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Date Created: 11/02/15

Amplitude Equations for Superimposed Stripes To study periodic or quasiperiodic states given by a superposition of stripes that are oriented in different directions and modulations of these states we need to introduce more amplitudes one for each set of stripes For instance for the modulation of a lattice state based on stripes at wave vectors q1 and q2 both with magnitude go but with different directions we can write 5u A1 expl39q1 qrL ucz A2 expl39q2 XLucZ cc hot The amplitudes A1 and A2 corresponding to the two sets of stripes will again be slowly varying functions of space and time From the rotational symmetry of the physical system we know that the evolution will depend on the angle 6 between the wave vectors but not on the direction of q1 or q2 separately Also rotational symmetry tells us that the two sets of stripes have the same Zdependence ucZ and that only the nonlinear terms involving both A1 and A2 may lead to differences from the single amplitude case Then using the same type of symmetry arguments to restrict the possible terms as above we can argue phenomenologically that the amplitude equations should take the form 2 1061141 5141439 vozax1 ia A1 g0iA12 G6A22A1 f 2 70642 2 8A2 fozax2 ia A2 g0A22 G6A12A2 where derivatives are defined with respect to directions normal and parallel to a given set of Strlpesa e g a 1 q 3quot J qla 2 12 and 72 We have introduced a new parameter 66 that gives the coupling between stripes at relative orientation 6 The function 66 must satisfy 66 67r 6 since these two angles de ne the same relative orientation We have also assumed the most usual case of chiral symmetry ie the physical system is unchanged under reversing the sense of rotation 66 6 6 These amplitude equations can be generalized for any number of superimposed stripes with each amplitude A coupled to all the other amplitudes A with coupling coefficients 66lj depending on the relative orientation of the stripes A special case arises if three sets of stripes are mutually at 120 In this case a quadratic nonlinearity acting on the disturbance given by two sets of stripes can generate a disturbance along the third set of stripes This corresponds to additional quadratic terms in the amplitude equations for example 1061141 2 39 39 39 7142 143 where the represents the same terms as in and 7 is a real parameter with corresponding equations for A2 and A3 The form of the new term involving the compleX conjugate amplitudes and the fact that the coefficient 7 must be real follows from the translational invariance and parity symmetry Note that if the system has a u gt u symmetry so that the amplitude equations must be invariant under A gt Al the coefficient 7 must be zero Arguably the best known example showing the manifestation of the updown symmetry is provided by onset convection patterns For instance in the RayleighBe nard convection between two identical plates top and bottom the onset pattern is stripes 7 0 since uc z ucz For MarangoniBe39nard convection of a liquid layer whose top interface is open to the atmosphere asymmetrical top and bottom boundaries hence y at 0 the onset pattern is hexagons The coefficient G6 describes coupling between amplitudes of different sets of stripes thus determining the stability of patterns consisting of different combinations of stripes It can be shown quite generally that limG6 2 6a0 even though we might eXpect unity for this limit since that would reproduce the coefficient of the one amplitude nonlinearity This difference arises because of the interference effects that arise when 6 is identically zero These disappear for any nonzero 6 actually for any 6 2 814 The value of G6 for a given system can be deduced from a simplified calculation of the nonlinear saturation of the unmodulated state consisting of two sets of stripes at the critical wave number go and at a relative angle 6 In this method the solution to the amplitude equations A1 2A 1s g0 1G6 2 is compared with a Galerkin calculation of the nonlinear saturation of the superimposedstripe state Alternatively the values of G6 can be obtained automatically if the amplitude equations are formally derived using the method of multiple scales For instance applying the method of multiple scales to equations describing D Arcy porous medium convection yields 1 3 cos 61 cos 62 l 3 cos 61 cos 62 2 74cos6cos26 2 7 4cos6cos2 6 Notice that the general result has the correct 6 0 limit This G6 decreases monotonically from 6 0 to 6 7z2 where G7r2 l 37 143 This turns out to be important in deciding whether the crossroll instability or the Eckhaus instability provides the stability boundary for small 8 on the larger wave number side of the stability balloon G61 The reduction of amplitude equation for multiple stripes to a scaled form is not trivial This is because we cannot introduce the different 812 and 814 scalings of the physical space variables along a single pair of orthogonal directions to simultaneously eliminate the 8 dependence from the derivative terms appearing in all the equations Generally the correct procedure will be to introduce a single scaling of the space variable X s lZX 0 which eliminates all the derivative terms beyond second order The resulting scaled amplitude equations for superimposed stripes is then 2 2 2 1A 2 iA 6X1A W ZG6 AJ 4 jail with X the scaled coordinate along the stripe normal Note that the scaled equation no longer shows complete universality since the qualitative behavior depends on the details of the system through the interaction parameter 66 which is not universal As we have speculated during the discussion of nonlinear competition between the stripe and lattice patterns the eXistence of a potential may be extended to the case of superimposed stripes with or without the quadratic nonlinearity The potential does indeed eXist for the common case of a symmetric interaction parameter 66 G 6 at least for the case of periodic boundary conditions However now we see that the potential nature of the dynamics is a delicate feature of the physics of the system For example if a RayleighBenard system is rotated about a vertical aXis the amplitude equations continue to take the same form but with 36 i G 6 due to the breaking of chiral symmetry induced by the rotation This is sufficient to render the dynamics nonpotential and indeed chaos is observed in this system immediately at onset as we mentioned in lecture 3 Having constructed the amplitude equations for a superposition of stripes the following aspects of pattern formation can be treated The nonlinear saturation of and competition between the ideal structures consisting of stripes squares hexagons etc The stability of the ideal patterns to spatially dependent perturbations the stability balloon Some elementary examples of spatially dependent structures such as defects and the distortions induced by boundaries The development of patterns from an initial perturbation that is localized over some small part of the system so that the patterned state spreads over the system through the propagation of the boundary between the saturated state and the unstable uniform state known as a front Nonlinear Competition Between Stripes and Lattices A general and reliable way to investigate the nonlinear competition between different ideal solutions is stability analysis if out of the two states being compared one state is stable whereas the second is unstable we can focus our attention on the stable state and for most purposes ignore the unstable one If both states turn out to be stable the potential nature of amplitude equations can be used to determine the result of competition between the two patterns provided that these two states coeXist in the system Since the latter approach is based on amplitude equations it only works close to onset of the pattern At the level of the amplitude equation the competition between states of superimposed stripes at different orientations is captured through the pairwise interaction specified by the interaction parameter 66 Assuming that all wave numbers are on the critical circle 61 go so that we may drop the spatial derivative terms in the amplitude equations we obtain the following system 671 71I7LZZG67LIZZ77Lquot 71 ltgt j i Example Competition between stripes and lattices Consider the competition between the stripe state and a lattice states formed from superposition of two sets of stripes at an angle 6 The case 6 7r2 gives the square lattice otherwise the lattice is orthorhombic For simplicity we assume u gt u symmetry to exclude the quadratic nonlinearity The saturated solution of for a single set of stripes is XI 2 XS 2 1 AS oc J in the unscaled variables The linear stability to the growth of a second set of stripes of amplitude A2 at angle 6 is tested by linearizing about this solution for the perturbation 71 A7 571 1622 to give the equations 615qu dt 2 Zl d5 dt 1 G95Z MIX31 quotXML 1H a mo 2 Rama L 23 CQogqugH gt QL A 21142HL0 s 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4pm 31ch 5cm 4414 Oujbtc Molina Wu Vow ea td Mwn agot aitnns quits HLJSLF TLLMT0A 0 quotEM 030mm 3 otwtnoilo EA WL m Jdu swam sew momk smug AME Jud W mikm Md will wnm breww aY in cumQ MM whim 2 601w SOHQD Q MLgfmes Koo ob kou m OJC 00 u LI 3 oh mmm quotV llt Tag I P Ch l C VV fc Let us next consider what kinds of derivatives of A can appear in the amplitude equation For instance a firstorder spatial derivative of the form 1A is not allowed by the above symmetries it is inconsistent with the parity symmetry AXL A XL Such a term would correspond to the motion of the pattern as a whole and thus represents a directed flux in the system Therefore such term cannot appear in the amplitude equation Which higherorder spatial derivatives can occur in the amplitude equation requires for the first time in our discussion acknowledgment of the facts that the system is near onset so that the parameter 8 is small and that the stripes have been arbitrarily aligned parallel to the y aXis which means that the x and y coordinates enter nonsymmetrically in the dynamics It then turns out that each x derivative applied to A acts like a small multiplicative factor of order 812 while each y derivative applied to A acts like a larger but still small multiplicative factor of order 814 A simple consequence is that the derivative axA has the same order of magnitude as aiA in the limit that 8 is sufficiently small This scaling can be obtained formally using the multiple scale analysis but is easier to determine by looking at the band of unstable modes in the Fourier space As the resulting pattern is composed of the unstable modes in the hatched region the amplitude will only contain wave numbers Q q qc inside the gray ellipsoid surrounding qc 6160 For small 8 the x and ysemiaxes of the ellipsoid scale as 812 and 814 and correspondingly the QC and Qy components scale likewise We can then argue as follows to deduce the form of the amplitude equation First we assume that it like most evolution equations that arise in a scientific context is smooth in that only a finite number of integer powers of the spatial partial derivatives appear Since we saw above that a first order derivative 3xA can be eliminated by a simple transformation the lowestorder x derivative that can appear is the secondorder term aiA which is also allowed by the various symmetries Once this term is included we ignore larger powers of xderivatives such as 6A as being higher order and so smaller in magnitude For the same reason we ignore nonlinear terms with spatial derivatives such as 6A 2 A since such a term is smaller than the eXisting cubic term IAIZA The strange way that the partial derivative ajA enters is a consequence of requ39ring that the 2 amplitude equation be invariant under a small rotation Indeed the eXpression 5x l39 2qc6 A is the generalization of aiA that includes yderivatives in such a way that the overall eXpression is invariant under the substitution A gt e39m 2q xA for an arbitrary small constant Finally there must be some kind of time derivative since this is an evolution equation and the simplest guess would be that a firstorder derivative 31A is sufficient This is allowed by the above symmetries but is also the simplest choice consistent with a symmetry not mentioned above but implicit in all drivendissipative pattemforming systems that the dynamics is not invariant under the timereversal symmetry t gt I so that there is a preferred direction of the dynamics in time The absence of timereversal symmetry rules out the possibility that the time derivative term is an even power such as 612A which is otherwise allowed by the spacerelated symmetries You should convince yourself that any other terms allowed by the above symmetries must be higher order than the terms already included Amplitude Equation Parameters Once we have accepted the form of the amplitude equation 2 roar1 5A ax 2Laj A go AFA go the unknown parameters 50 and To can be deduced from linear stability calculations Thus if we consider a small amplitude disturbance A 5A0 and linearize the amplitude equation about the zero solution A 0 we see that the time dependence is exponential with a growth rate 1618 02k2 But since u Axytuczeiq x cc this is the growth rate of a small physical perturbation 6u at wave vector q qc 16 and so must correspond to the growth rate 0q of the linear stability analysis for the uniform base state ub Thus we have 1 2 2 0qTo 8 0q qc for small 8 and small 61 go The parameters 50 and To can now be simply read off from the growth rate 0q calculated from the linear instability analysis of the full evolution equations Finally the coefficient go which determines the saturation amplitude of the critical mode A 8g012 can be found from a Galerkin expansion calculation for the nonlinear saturation quotD ktc GamaHon I Amkm byWeep ofY o commo vz sawm cs ovaasan quotU1 spawn bdraan Uu 0 chl DoH OM ma a oY w convaon all cs wion OC H39 4 Focus Maui Srox aL cu Pea Ehka st byes 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mtrodmoc LL emew 9mm q IRMA wt welt umst a gb k dcgwm WR00JJV rs v VOL ochrL Jcbv x 4 comfonds 0 HUL MW i V quot r 549 MM39 H 39Bi O9 oD E 0x ka553 5th 37 oxfbuu 12 Cowfowr mdL ai QQW X COMeOMA t ah l Sw o cfmo n OCS BAL Q um omS wL JAIML miscaggwrz 2 Qavx 9on 339 mo How we MWLJdxm vorcion bai upqu Camgton ES kacS U4 3 MNme 05 m 041ch gunman amok ago3m mem frzsscbi cm OOAAH GW Bxun atdlbc a 9 M a u i g Mfg 01 ano g K O Oy 9amp9 Wag nitudw on is a me Mum bo GM is ho MM W94 Ci WLamp0 M Jco unmarwa Luzk Ajm hamchowjh bmwco b MWMM Lakxltr hm 7 639 Tilm m ak aka m biwge Sung Semen or CkL ag EWO Ljuou hiong mm Gofm ogctmgq ohm LBW6 6 mega eim we ammo M a bomolm valid Xmame OCBQ 61L9 o7 tag ti gt 91 R02 0 m sohhws Sam sfnwb W boa MU mums 03991 o 1 i1 mt 05020 WrorttsML 50 gm Rntwi wildx Um om 49c nk gods unsW id s E guBsHkuhpb 0 o i 3 C Lek 913cc 39 Aer m womng oJML Frauom was W gm R JVI39VQL 3 SefC ns ax A Mnm b CL39W WCX a Mo rle LIL awvw er 815 muesli dYL xql Ow wquok Lb kh M Ma s Wbcl j 0mm LL13 jigs 5 03 a WWW a lt11 mi q 39 3 4 Re 475 7 9 140 0amp0 3 1 2 5 6 39xz and so 1w cnshlozmb ca 15am 599 15 i wik nt w faded on MW 391 H unadrtan rfd kw m 117419 s z m W WC M Onsak kn Faw I ql in1 Duh m Wt Wug th 4 16W WfW RP W Mrim l mm a Luncth em L1 QowwA M u meandenssLbLanj can ammonj 01 T 1 JT Nonlinear saturation in the SwiftHohenberg equation As we have learned previously the SwiftHohenberg equation describes a typelS instability to a cellular structure in onedimension an 2 ru l62u u3 As r is increased from negative values the uniform state u 0 develops an instability at r 0 to a mode with critical wave number qc 1 For small positive r we can find a perturbation growing from very small amplitude ux t oc 6 cos x We wish to understand the nonlinear saturation of this state for small r A first guess might be to look for a solution which is just a saturation of the growing linear mode ux a1 cos x with al to be determined To see if this ansatz is a solution we substitute it into the equation for stationary solutions atu 0 3 l 0 ra1 a13 cos x a13 cos 3x 4 4 where we have rewritten cos3 x generated by the L13 term as a sum of cos x and cos 3x terms Now since the Fourier modes cos nx are linearly independent the coefficient on the right hand side of each mode must be zero For instance the coefficient of cos x determines a1 ral ia1320 gt a1Oiq43r12 However equating the coefficient of cos 3x to zero selects the wrong solution 1 Zal30 gt all 0 This inconsistency shows that the original guess was incomplete We rapidly realize however that the problem is removed by adding a small amplitude of cos 3x to the original ansatz so that now ux a1 cos x a3 cos 3x With this new ansatz we obtain the following equality 3 3 3 l 3 3 0 m1 Za13 Za12a3 Eala32cos x m3 64a3 Za13 Ea12a3 Za cos 3x 3 3 3 l a1a32 a12a3 cos 5x ala32 cos 7x a33 cos 9x 4 4 4 4 where again setting the coefficients of different harmonics cos nx to zero determines a1 and a3 As we are looking for a solution close to the bifurcation point where the effect of nonlinear terms is small a3 will be small compared to al Therefore setting the coefficient of cos x to zero gives 3 3 3 3 ra1 a13 a12a3 Eala32 N ml Za13 0 4 4 ie the same equation for all as before Similarly the coefficient of cos 3x gives 1 3 3 l l m 64a a3 a2a a3N 64a a3 0 gt a a3ocr32 3 3 4 l l 3 3 3 l 3 l 2 4 4 256 Now we can see that in both equations we kept the terms of order r 30 and neglected terms of order r 50 and higher which are indeed much smaller for r gt 0 Furthermore the coefficients of the higher harmonics cos 5x cos 7x and cos 9x are of order r 52 r 72 and r 92 respectively So at order r 30 they vanish and thus do not impose additional conditions on all and a3 To satisfy the equations obtained by setting the coefficients of higher harmonics to zero exactly at all orders in r one has to include all odd harmonics in the ansatz ux 2 an cos nx n odd with an oc rquot2as r approaches zero For small r the coefficients an can be calculated iteratively starting with al Close to onset of instability keeping only the first few terms usually gives a very accurate even though approximate solution This procedure is called the Galerkin truncation The steady state solution we thus obtain consists of a dominant term ie the linear mode which can take either sign and grows in magnitude from threshold as r 12 consistent with a pitchfork bifurcation In addition there are spatial harmonic corrections with amplitudes that grow progressively more slowly with increasing order of the harmonic Example nonlinear saturation in d Arcy convection The linear instability analysis for d Arcy convection was conducted earlier and we found a simple sinusoidal dependence sinm7z l2 for the uid variables w E vz and 6 on the vertical coordinate The corresponding horizontal eigenfunctions are just cosmtx Note Close to threshold the instability boundaries can be more easily calculated using the amplitude equation formalism to be introduced later The derivation of the stability balloon suggests a quite limited applicability since it is constructed as the stability region of spatially periodic states in a laterally in nite system or for a system with periodic boundary conditions In practice it has been found to provide a useful guide to the behavior for a much Wider class of problems Often applying the stability criteria to a locally defined wave vector gives a good guide to the behavior even in small geometries and for disordered states The question of What stationary patterns might form can be crudely formulated in terms of What states may be generated from the range of wave vectors that fall Within the stability balloon The onset of time dependence can similarly often be associated with the local wave vector in some portion of the system passing outside of the stable band The local wave vector distribution in the pattern may sometimes be predicted from other considerations therefore providing a route to predicting the onset of time dependence in the system Thus the stability balloon is often a useful first guide to the range of behavior eXpected as the control parameter is varied Busse balloon for RayleighB nard convection The stability balloon for the ideal roll state in RayleighBe nard convection was worked out by Fritz Busse and coworkers in late 1960 s using the numerical Galerkin procedure described above They also performed extensive experiments to verify the predictions As we have seen the Bloch perturbation ansatz is characterized by a wave vector Q as well as by a function u1xi Z with the same periodicity as the base pattern Many different types of instabilities are encountered These are classified through properties of the Bloch wave vector Q whether the growth rate is real or complex and in addition through discrete parity and inversion properties of the function 111 These different types of instabilities will give different shapes for the growing perturbation and will appear experimentally as quite different geometrical structures The visual appearance of the perturbations has led to a rich nomenclature for the various instabilities found in the convection system Examples of the Busse balloon for RayleighBe nard convection for two different sets of uid parameters Prandtl numbers 071 and 70 are shown on the following pages There are a number of important features to learn from this type of diagram At each value of the control parameter the Rayleigh number in this case there is a range of wave numbers for stable stationary stripes This range is bounded by particular instability curves which tell us the nature of the instability including the shape of the perturbation that will grow and its dynamical properties It is convenient to use these eigenfunctions for the Galerkin expansion which can be written in the form 00 9 2 Hmquot I sinm7rz l 2cosn7rx mnl w 0 Wmquot I sinm7rz l2cosn x mnl u Z umquot I cosm7rz l2sinn x mnl where the horizontal component of the velocity u E vx can be found from the incompressibility condition The m n 1 components correspond to the linear mode obtained from linear stability analysis In principle an infinite number of modes will be generated by the nonlinear terms in the equations of motion and the sums should run over all modes In practice the sums are truncated at some level The above eXpansions are now substituted into the nonlinear equations 0 VH v 6 i 616 vV6 sz V26 V v 0 and the coefficients of each retained Fourier mode collected to give coupled dynamical equations for the mode amplitudes 6m WW1 and um The coefficients can be extracted using the orthogonality properties of the eXpansion functions We can illustrate this procedure truncating the sums at second order The nonlinear term V V6 acting on the linear modes generates the single term proportional to sin 2752 Thus the eXpansion to second order is given by taking 6 87rYt cos 7rz cos 7m 47rZt sin 27m w 47rXt cos 7rz cos 7m u 47rXt sin 7rz sin 7m where we have written w11 47X 611 8751 and 620 47Z with the numerical prefactors chosen for later convenience The horizontal velocity u is calculated from the incompressibility condition We now substitute these eXpansions into the evolution equations and collect terms of the form sinm7tz l2 cosmtx For example in the heat equation that we write in the form at6vva sz V26 2 0 we get SnatY l67r3XZ 47rRX l67r3Y cos 7rz cos 7m 476 167z3z 167r3XYsin 272 0 In the above equation we have only kept the terms of the same order as those in the eXpansion of the fields at the top of the page The higher order terms indicated by are ignored in the truncation they should be balanced by higher order terms in the mode eXpansion We have also kept the time derivatives as we are interested in the dynamics of the leading order terms in the solution not just in their asymptotic magnitudes Since the Fourier modes are orthogonal we should set the coefficient of each mode to zero Thus we get QzQJQanY Y XZ az54aZmXY ZL where r RRc R47t2 The d Arcy equation is simple balancing terms involving cos th cos TEX gives 0 X Y Rescaling the time variable by 272 we obtain three coupled nonlinear ODEs determining the dynamics of amplitudesX Y and Z OzX Y anY Y XZ D Zz XY ZL For small r l the stationary solutions of this system are erzia w Z r 1 again showing the characteristic behavior of a supercritical pitchfork bifurcation inX and Y 1 The neglected terms in the heat equation are of order r l 2 with n 3 4 5 so that the strength of the higher order Fourier modes decreases with the order for small r l and the mode truncation scheme gives a systematic eXpansion in integral and halfintegral powers of the distance to the bifurcation r l It should also be noted that the system of equations 2 is analogous to the famous Lorenz equations In early 1960s Edward Lorenz followed a similar analysis for RayleighBe39nard convection with no porous medium but with the same physically artificial boundary condition on the horizontal velocity freeslip rather than the physical noslip used by Rayleigh This leads to an analysis similar to the one that we have done except that in the Lorenz case the first equation is also dynamic involving X and various coefficients and the time scaling in the equations are different The Lorenz equations are X oX Y Y rX Y XZ z39 bXY Z In the Lorenz equations 0 corresponds to the uid Prandtl number that Lorenz took to be 10 in his original simulations and b 83 Lorenz investigated these three coupled nonlinear ordinary differential equations and uncovered the phenomenon of chaos that is complicated aperiodic motion highly sensitive to the choice of initial conditions On the other hand Eqs 2 can be reduced to just two dynamical equations in Y and Z which eliminates the possibility of chaos by the PoincareBendixson theorem 1 For large r l the truncation of the field expansions at low order does not give a good approximation to the true dynamics of the equations for example the Lorenz model at r 28 shows chaos whereas if enough modes are retained to give a good numerical approximation scheme while keeping the same periodicity no chaos is found Universality and Scales In our former discussion of the amplitude equation we found that we could eliminate the scale factors To 50 and go from the amplitude equation by transforming time space and magnitude variables We can modify this transformation of variables as follows 12 l2 l2 amp A X 8 x Y 814amp y Tz t 60 60 TO 8 to obtain the scaled amplitude equation from which all the parameters have been removed 2 2 62 iZ 6X 6 Z Z2A7 The positive sign for the first term on the right hand side corresponds to 8 gt 0 and the negative sign to 8 lt 0 In this equation there are no parameters that depend on the physical nature of the system This dramatically demonstrates the universality of pattern forming phenomena near onset when the amplitude equation is a good description since we can analyze the behavior of amplitude equation without referring back to the physical nature of the system This shows us that all stripe states near threshold in a rotationally invariant system will have the same properties The absence of any explicit dependence on the small parameter 8 in the scaled amplitude equations immediately tells us the scaling behavior with small 8 of the physical length x time I and pattern intensity A For example we expect the time dependence of the solution to the scaled amplitude equation to occur on an 01 time scale with respect to the variable T The relationship to the physical time scale then shows us that the physical time scale for example for the growth of a small initial perturbation from the spatially uniform state will be 13908 1 which diverges toward threshold as 8 1 with 8 the small parameter that measures the distance of the control parameter from threshold and goes to zero at threshold ie the reduced bifurcation parameter Similarly the length scale over which the intensity of the pattern grows from a suppressed value near a boundary or the core of a defect will be 508 12 for the direction perpendicular to the stripes and 50 qc128 4 along the stripes again with both diverging toward threshold It is in fact variations on these scales that are slow enough to be captured by the amplitude equation Note that these length scales for the spatial variation correspond in Fourier space to a wave number deviation 6 qc 081250 1 This is the order of the width Ag of the unstable band near threshold so that all of these states are accessible to the amplitude equation treatment Finally the amplitude of the pattern will have the characteristic square root dependence proportional to 812 or more precisely to slZgo 12 that we have predicted for a general supercritical pitchfork bifurcation These power law dependencies are quite analogous to the rather common divergences such as critical slowing down that occur at a second order phase transition and indeed have the same parameter dependence as in the approximate mean field description of this phenomenon Similar universal amplitude equations and conclusions hold for oscillatory instabilities and typell and typeIII instabilities a 3 b G 5 47 1 v 5 1 E24 quotf K r 3 quot norm a Ian I 139 H 39 39 A gaa 2 3h n3939 l 1 39 39I 24 i 39 s047 7 O t o 4 a 12 is x mm Liquid crystal convection a Velocity of uid measured with tracer particles b Variation of maximum of velocity with time for two different values of 8 showing the longer time scale at smaller 8 with 1 0c 5 1 scaling c Spatial variation of velocity field and d of the envelope A for two different values of 8 with x 0c 5 scaling Low Prandtl numbers most gases For lower Prandtl number uids Eckhaus instability becomes the dominant low wavenumber instability it is nevertheless preempted by the zigzag instability just above onset On the other hand an oscillatory instability the line denoted by the symbol 0 in the gure below becomes important for larger values of R replacing the cross roll instability As the name implies the oscillatory instability is of oscillatory type Ima 7 O The stability balloon gives us a quite complete understanding of the ideal spatially periodic states including both the existence of saturated steady solutions and the stability of these states to further perturbations Few experimental systems show the ideal straight parallel roll state because of the effects of boundaries or the way the pattern forms from initial conditions However many experiments in convection have shown that the stability balloon gives a useful phenomenological understanding of approximately straight roll regions in the disordered states encountered in most experimental geometries Furthermore the types of dynamics found when the pattern is stressed beyond stability boundaries are predicted quite well by the linear stability analysis 105 104 Rayleigh Number R 10 20 30 40 Wavevector q Stability diagram for the ideal roll states in RayleighBe39nard convection for P 071 Theory of Amplitude Equations Previously we have discussed the evolution of infinitesimal perturbations of a uniform state into saturated stationary spatially periodic solutions By restricting attention to these simple solutions it was straightforward to formulate the effects of the nonlinearities using analytical methods near threshold and fairly simple numerical methods further from threshold However most realistic geometries do not permit spatially periodic solutions since these are usually not compatible with the boundary conditions at the lateral walls Even if periodic solutions are consistent with some finite domain they do not exhaust all the possibilities More typically patterns have an ideal form stripes hexagons etc over small regions and these ideal forms are distorted over long length scales disrupted in localized regions by defects and these distortions and defects are timedependent The amplitude equation formalism provides a method to study such effects Amplitude equations capture three basic ingredients of pattern formation the growth of the perturbation about the spatially uniform state the saturation of the growth by nonlinearity and what we will loosely call dispersion namely the effect of spatial distortions In a Fourier decomposition dispersion corresponds to a range of component modes with wave vectors centered about the critical wave vector qc The interplay of these three effects lies at the heart of pattern formation and amplitude equations have yielded many useful quantitative insights In addition amplitude equations provide a natural extension of the classification based on the linear instability into the weakly nonlinear regime We will see that the form of an amplitude equation is dictated by the linear classification typeIS typeIlla etc together with some simple assumptions about the effect of nonlinearity Amplitude Eguation for Stripe States Origin and Meaning of the Amplitude In analyzing the linear instability of a spatially uniform state we looked at the dynamics of monochromatic disturbances described by a single Fourier mode du uqx eiq39xieaqt co This ansatz may be appropriate for laterally finite systems with periodic boundary conditions close to onset of instability but in the more general case eg for infinite periodic boundary conditions we have to consider a superposition of all unstable modes i39xl th ic39xr o39uIqugtanuqxHeq e dqCCAXLtllcXH q cchot The complexvalued amplitude Aopoj represents slow modulation of the pattern corresponding to the critical onset mode ucXHei 39Xi Indeed the integral is composed of spatial Fourier modes which vary slowly in space just above onset since lq qc OC 8 where 8 p pcpc is the reduced bifurcation parameter Similarly the amplitude varies very slowly in time as the growth rates for the unstable modes are small Re oquotq 0C 8 just above onset aqeimiq Xi eaq dq Reo q gt0 There are three sources generating the higher order terms hot in the expansion of 6u First of all they will come from spatial harmonics generated by the nonlinearities Corrections will also arise at the linear level since for a spatially varying amplitude the structure ucXH in the confined direction will not give the precise solution to the evolution equations in other words uq XH CXH Finally the mode structure is perturbed if the control parameter is not exactly equal to its threshold value Using the rotational isotropy of the system we can align qc arbitrarily but conveniently in the x direction in the following we assume X L x y XH Z so that the fast oscillation is described by an eXponential function of a single coordinate eXpz39qcx to simplify further calculations The eXpression 6u Axytuczeiq x cc then represents a distorted set of stripes that are approximately parallel to the y aXis The function ucZ describes the structure of the critical unstable mode in the confined direction and is known to us by some previous linear stability calculation zqcx cc is A simple illustration of a onedimensional modulated periodic state ux Axe shown in the gure below Take for instance a modulation function Ax 051 02 cos01xe slt 3 An obvious feature of the resulting pattern is the modulation of the magnitude of the sinusoid but if you look carefully you will also see that the local periodicity of ux ie the distance between two adjacent zero crossings is no longer constant so that the local wave number defined as e g l lA dAdx l also has a slow spatial modulation It is useful to express the amplitude Axyt in polar form A Lie where axyt is its realvalued magnitude and x y t is its realvalued phase The magnitude and phase then play different roles in the dynamics of A The magnitude gives the size of the perturbation 5n near onset and typically evolves quickly eg exponentially rapidly in the linear approximation On the other hand if we write the expansion in the form 5n 2auc z cosqcx hot assuming ucz to be real for simplicity we see that the phase sets the position of the growing stripes e g a constant phase 1 translates the field 5n by a distance 7 qc 1 0 in the x direction Because of its link to translational and rotational symmetries of the system and to Goldstone modes the phase generally evolves more slowly than the magnitude and its dynamics can often be isolated and studied separately Agucox ons 0 Lirumm SWCQA IB Ma qscs W WWW E W ag i In W yamous Laban s w WL awzbofzd 0L ormak tackqu 60w swomzks M413ch Unwrg onL a datsz Lam m emi lhss shim a ManmQobrum 535m pzoome maskML to McFoNniccal abroadk 09 JltYWL mg WnaluL alga adj003w us 0 09d an M sfmbd and Moral Scams 0 L5 twang shutWe ERQow ma m va co and elaswussm Mnumr LVoMtan chLocHons and 39Uw lr LEW sinch Malgstg gar WW5 W mot Cvnd mm in sow Ms Hcslro wUZMSuck a LOw gmwag Was was Q00er cacmd cu b3 Atom Thing in QC39L RLSu chshA wwwkiml MA stimulq g we nd Lad con amok Waco k9 WMcMgt maid ewcn WPazmg55 vow btoMStmL sea Kwms M31 and AWDLOP Auxcr amouer 0amp om OCQGAMSVH Ama 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