- Email: [email protected]
Phase dynamics with drift: boundary effects
Physica D 184 (2003) 259–265
Phase dynamics with drift: boundary effects Yves Pomeau a,b,∗ a
b
Laboratoire ASCI, Bat. 506, Orsay 91405 Cedex, France Department of Mathematics, University of Arizona, Tucson, AZ, USA
Abstract The description of patterns in nonequilibrium systems is a fascinating subject. It becomes somewhat untractable if one insists to keep the original equations in their complete form, as the Oberbeck–Boussinesq equations for instance in Rayleigh–Bénard convection. Various reduction scheme have been imagined, the ultimate one being the phase picture. I examine a simple version of the phase equation, relevant for systems of travelling rolls or for distributed self-oscillations. The boundary effects limit the ability of this phase to drift freely and yields a well-defined space structure, which can be analyzed in two different limits. © 2003 Elsevier B.V. All rights reserved. Keywords: Phase dynamics; Boundary; Oscillations
1. Introduction It is a great pleasure for me to contribute to the festschrift of Alan Newell, with whom I had years of fruitful exchanges and collaboration, both in France and at the University of Arizona in the US. One of the foremost contributions of Alan to nonlinear science has been the invention of the amplitude equations for patterns slightly above onset, like Rayleigh–Bénard thermal convection or in the buckling of long rectangular plates under in-plane stress. This concept of amplitude equation can be hardly overemphasized as a central idea in our science. A natural (although it took some time) extension of this idea is the one of phase motion: a wavelike pattern (or for that purpose any kind of pattern with some translational invariance or more generally any continuous symmetry) shows [1] a long wave-slow dynamical mode that degenerates in the infinite wavelength limit into immobile uniform translations. It is fair to say that an idea close to the one of phase dynamics was already present in the classical derivation of the equations of solid state elasticity by Cauchy in the first part of the 19th century: Cauchy derived the equations for a lattice (3D) of mass and springs in a way rather similar to the derivation of the slow phase dynamics of patterns. This phase equation can be written in a number of ways, including nonlinear effects, defects, etc. It is out of question to review here the rather large body of literature concerning this topic. The interested reader may find information on it in [3,5,6]. In this hommage to Alan Newell, I would like to present an extension of an application of this idea of phase dynamics that includes effects of boundary conditions (b.c.’s) and the possibility of nucleation of new “rolls” on one side of the “container” and their absorption on the other side. This shows that the phase ∗
Present address: Laboratoire ASCI, Bat. 506, Orsay 91405 Cedex, France. E-mail addresses: [email protected], [email protected] (Y. Pomeau). 0167-2789/$ – see front matter © 2003 Elsevier B.V. All rights reserved. doi:10.1016/S0167-2789(03)00224-0
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approach can be extended to situations where the total number of rolls is not conserved in the course of time. Some years ago, with Prost et al. [2], we looked at the following problem: let us suppose that a system with a phase, like a patterns of rolls, instead of being “naturally” in a state where this phase stays constant, shows a “trend” to constant phase slipping in space. If one thinks to a pattern of Rayleigh–Bénard rolls, this trend could be the result of a cross flow carrying the rolls in a certain direction. Other physical examples of this class of phenomena can be found in [2]. Another class of problems with the same mathematical structure is the one of extended time oscillations. There the phase variable is just the time phase of each local oscillator, that drifts “naturally” at constant rate. The interaction of neighboring oscillators can be represented by a regular diffusion term, if this interaction tends to synchronize neighbors. Indeed the phase picture is incomplete in general because it does not take into account variations in the amplitude itself, but this is at least a first approximation to space dependent phenomena. The constant phase drift, due to the conditions in the bulk, may go against the b.c.’s. In a system of rolls parallel to a boundary, the phase of the roll just touching this boundary is a relevant parameter in the b.c.’s. In a system with an optimization principle, the b.c. may tend to favor a specific value of the phase at the boundary, up to 2π, indeed. In a very simple mathematical formulation, I examine the effect of both such a pinning of the phase at boundary counteracting a bulk phase shift. The model under study does not claim to be realistic. In particular, it is not derived from any amplitude-like equation, although it could be. This paper intend to show that the balance between boundary pinning and bulk drift introduces two limits: the weak and strong limit, something that is perhaps of general interest. A simple form of phase equation with b.c.’s is introduced consistent with the requirement that phase is defined up to 2π. It is solved by asymptotic methods in two well-defined limits, showing how boundary effects change the simple picture of an uniformly drifting phase. The boundary effects ultimately defines the wavelength of the pattern. In the absence of any spatial structure, the equation for the phase ϕ with constant phase slip reads: dϕ = ω. dt
(1)
In this equation ω represents the rate of change of the phase in the absence of boundary effects, wavelength changes, etc. This rate of change should be seen as proportional to the cross flow, for instance, if this is the source of phase drift. For a self-oscillating system, ω would be 2π multiplied by the natural frequency. In the amplitude equation approach, the rapid phase, spatial or temporal, is factored out. This would amount here to simply subtract from ϕ a constant ωt. However, this is not something satisfactory if one wants to look at physical b.c.’s: such b.c.’s are sensitive to the true physical phase, not to the one obtained after subtraction of a fast phase. This explains why I keep this fast phase. The next question is: what would happen to the phase if it is held in some sense by boundaries? This leads naturally to consider first the effect of possible space heterogeneities on the phase dynamics, something simple at least in principle, then to account for the finite length of the system. The corresponding equation is obtained by adding to (1) a space diffusion term. To make things simple, I shall consider the one-dimensional case only where the phase diffusion equation with drift reads: ∂ϕ ∂2 ϕ =ω+D 2. ∂t ∂z
(2)
In this equation, the coordinate z is taken perpendicular to the roll direction (if rolls are under consideration) and D (assumed positive) is the so-called parallel diffusion coefficient (if the drifting roll situation is considered, otherwise it would be the usual phase diffusion coefficient). The next step consists in the writing of the b.c. Indeed, this may be done quite freely in the present theoretical setting. However, there are some constraints: the b.c., imposed at z = 0 and L, cannot make any distinction between values of the phase differing by 2π, since they are physically equivalent. Let us consider a b.c. of the so-called mixed type, that mixes the function itself and its first space derivative. A
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simple form of this b.c. consistent with the constraint of periodicity in ϕ is a linear relation between (∂ϕ/∂z) and a circular function like sin (ϕ) α
∂ϕ + sin (ϕ) = 0. ∂z
(3)
This is to be imposed at the ends of the system. For one space dimension, this is to be imposed at z = 0 and L, with a change of sign of α from one end to the other. The b.c. (3) may look somewhat arbitrary. As said before, this arbitrariness is partly the consequence of the general character of the model under consideration. Being consistent with the elementary requirement of invariance under an addition of 2π to the phase, it shows an interesting balance between two effects that are to expected: first the system tends to have a phase as uniform in space as possible. This explains the contribution proportional to (∂ϕ/∂z) in Eq. (3), that becomes dominant at large α. On the other hand, one expects that, locally, it makes a difference for the phase to have one or another value at the boundary, within the constraint that phases differing by 2π are equivalent. Whence the circular function in Eq. (3). This point of view is perhaps made clearer when considering the variational formulation of the equation under consideration: the b.c. yields a boundary contribution to the energy. In Eq. (6) below this is the quantity Fboundary = −
D ( cos (ϕ)(z = 0) + cos (ϕ)(z = L)). α
This Euler–Lagrange energy is the lowest when the phase is equal to 0, mod 2π on the boundary if α is positive. Therefore, this model represents a balance between two conflicting tendencies at the boundary: to have there a well-defined value of the phase, mod 2π and to have a phase as uniform in space as possible. The conflict is there because, the phase tends to drift at a constant rate in the bulk, so that it cannot take a prescribed constant value, mod 2π, on the boundary, unless some nontrivial dynamics takes place there. This set of equation and b.c.’s define a gradient flow, although this is not a property I am going to use. Indeed, Eq. (2) can be written formally as: ∂ϕ ␦F =− . ∂t ␦ϕ
(4)
The “energy” functional F has a bulk contribution: L 2 D ∂ϕ Fbulk = dz − ωϕ 2 ∂z 0
(5)
plus another part that comes from the remainders of the partial integrations needed to get the functional derivative (␦F/␦ϕ) in the form of the original right-hand side of Eq. (2). This boundary contribution needed to cancel the leftover of the integration by part reads Fboundary = −
D ( cos (ϕ)(z = 0) + cos (ϕ)(z = L)). α
(6)
When writing this boundary term, one took care that α changes sign from one end to the other. Therefore, the full “energy” functional is F = Fbulk + Fboundary . Because it includes a term linear in ϕ, this energy has no minimum and the evolution is not, in general (at least if L is large enough), a relaxation to a steady rest state. The sequel is devoted to an analysis of solution of Eq. (2) with the b.c. (3).
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To make the analysis simpler, one gets rid of the parameters ω and D by rescaling. Take ω−1 as unit of time, √ √ D/ω as unit of length, and redefine a new α as the original α times ω/D. The equation under consideration becomes: ∂ϕ ∂2 ϕ =1+ 2 ∂t ∂z
(7)
with formally the same b.c. (3) as before. This rescaling has got rid of all parameters, except the total length L √ (measured in units D/ω) and the coefficient α in the b.c. Let us look first at time independent solutions ϕ|st = − 21 (z2 ) + az + b.
(8)
In a finite system the constants a and b are determined by the two b.c.’s: sin (b) + αa = 0 and sin (− 21 (L2 ) + aL + b) = α(−L + a). This shows that there is no steady solution for L large. Because the sine function in the second b.c. is less than one in modulus it can balance the right-hand side only if a ≈ L. If this holds, the first b.c. cannot be satisfied. In order of magnitude estimates, no steady solution exist for L much bigger than 2π/α. Therefore, for L large enough, time dependent solutions exist only. It is likely that for intermediate values of L, that is of the order of magnitude of √ a few times the intrinsic length, D/ω, various dynamical behaviors are possible, that might have some interest. I shall consider the large L limit only, where the solution can be split into the solution near the ends and the one relevant for the bulk. As usual one needs next to match the “outer solution” (in the bulk) with the inner solution, valid near the ends. In next section, I deal with the inner solution, and match it with the outer one later on.
2. Boundary layer, inner and outer solution By definition the inner solution describes the behavior of the phase near the ends, where, presumably, the effect of the b.c.’s plays an important role. The classical amplitude equations of Newell and Whitehead [4], with the zero condition at the end has the familiar hyperbolic tangent behavior near the boundaries. To study the problem of a “long” geometry, one looks first at solutions of the phase equation in a semi-infinite layer, bounded at one end, say at z = 0 where the b.c. (3) applies. Far from the boundary, the solution is expected to tend to a simple solution of Eq. (7). This equation, being linear, can be solved in some form at least, given for instance, the boundary value of ϕ(z = 0, t). We shall limit ourselves to a few comments on the finite size solution. The outer/bulk solution is ϕ|as = t + qz + ϕ0 ,
(9)
where q and ϕ0 are two real numbers, arbitrary for the moment. Other solutions of the diffusion equation (9) decay far from the boundary and so cannot be taken as asymptotic condition. Wherever ϕ is the phase of a complex amplitude, the wavenumber is constrained to stay inside some finite band. Otherwise the amplitude itself cannot be computed. The theory presented below assumes that the wavenumber q selected by the b.c.’s is inside this band. The solution in Eq. (9) will be the asymptotic form imposed to the inner solution, itself being subject to the b.c. (3). Because there is an irreducible parameter, α, in the b.c. and because of the absence of any other parameter in
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this dimensionless form of the equation, it is worthwhile to look at the situation of small α and large α to get an idea of the general solution for an arbitrary α. As announced there is a regime of strong pinning, small α, and a weak pinning regime, large α, a noticeable feature of this model. 2.1. Large α limit Let us define ζ such that ϕ(z, t) = t + ζ(z, t). The equation for ζ is ∂ζ ∂2 ζ = 2 ∂t ∂z with the b.c. sin (t + ζ) + α
(10) ∂ζ . = 0 ∂z z=0
Notice that the algebra of the decomposition of the solution and of the b.c. does not depend on the value of α. The diffusion equation for ζ has the property of “mass” conservation. For a solution with well-defined time average properties, the time average of (∂ζ/∂z) is constant as a function of z. Therefore, the average “mass flux” of the outer solution, as given in Eq. (9), is the same as on the boundary: ∂ζ 1 q= = − sin (t + ζ) . (11) ∂z α z=0 z=0
In Eq. (11) the overbar is for the time average. To use this to compute q, one needs to get the value of ζ in the prescribed limit, α large. Let us assume that, in this limit, ζ is small, which is to be verified at the end to yield a consistent approximation. If this is true, the b.c. for ζ becomes a simple linear condition: ∂ζ sin (t) + α = 0 . (12) ∂z z=0 Let us write the solution of the inner equation as ζ(z, t) = A(z) eit + A∗ (z) e−it , where the complex valued function A∗ (z) is complex conjugate of A(z). From (12), the b.c. for A(z) is (∂A/∂z) = −(1/2iα)|z=0 , although A(z) is a solution of iA(z) =
d2 A . dz2
(13) √
Therefore, A(z) = A(0) e−(1+i)z/ 2 , which gives √ 2 ζ(z = 0) = ( cos t + sin t). 2α From (11), in the large α limit: √ √ 1 1 2 2 q = − sin (t + ζ) ≈ − cos (t) ( cos t + sin t) ≈ − 2 . α α 2α 4α z=0 This closes the case of large α.
(14)
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2.2. Small α limit This limit is more conveniently studied by reducing the full problem to an integral equation for the boundary value of ζ(t, z = 0), which can be done in the following way. Let us begin with the formal solution ϕ = t + ζ(t, z), with the b.c. sin (t + ζ(t, z = 0)) + α(∂ζ/∂z)|z=0 = 0. This b.c. depends on two unknown functions of t, ζ(t, z = 0) and (∂ζ/∂z)|z=0 . These two functions are linked by the property that ζ(t, z) is a solution of the diffusion equation. Therefore, there is a linear relation between those two functions of t, which can be derived from the Green’s function solution of the heat equation t 1 −1/2 ∂ζ ζ(t, z = 0) = − √ dt (t − t ) (15) ∂z z=0 π −∞ or ∂ζ 1 (t, z = 0) = √ ∂z π
t
−∞
dt (t − t )−1/2
∂ζ (t , z = 0). ∂t
The b.c. becomes, as announced, an integral equation for the function ζ(t, z = 0) = ζ0 (t) t α dζ0 sin (t + ζ0 (t)) + √ dt (t − t )−1/2 = 0. dt π −∞
(16)
(17)
Formally, in the limit α tends to 0, the solution of (17) should become close to t + ζ0 = kπ, with k arbitrary integer. But this solution cannot be a valid starting point for an expansion near α = 0, since it has a constant time derivative that makes strongly divergent the integral term in the b.c. Therefore, the solution at the dominant order in α has to change from one value of k to another one to make converge the integral in (17). Therefore, the function ζ0 (t) must be almost constant and jumps in a short time by values of π at time intervals π too. The value of ζ0 inside the jump is given by the solution of a boundary layer problem, wherefrom the small parameter α can be scaled out. During this fast variation of ζ0 , one can neglect the time t in the argument of the sin function, so that the inner solution ζ0,i (t) valid during the jump is given by: t α dζ0,i sin (ζ0,i (t)) + √ dt (t − t )−1/2 = 0. (18) dt π −∞ Thanks to its simpler structure, the equation for the inner problem can be be made parameterless, as usual, by convenient rescaling (or by what is called sometimes a stretching transformation). In the present case, one gets rid of α by using ˜t = t/α2 as new variable (we shall drop the tilde below). This means in particular that the duration of the phase jump is of order α2 , a small quantity as expected. The b.c.’s for the inner solution are ζ0,i (t) tends to kπ as t tends to minus infinity and ζ0,i (t) tends to (k + 1)π as t tends to plus infinity. Beside a numerical study (in progress) nothing more can be said about this parameterless inner problem. There remains to find the value of q in this small α limit. One can still use the same general formula as in the opposite limit 1 q = − sin (t + ζ0 ). α
(19)
The function sin (t + ζ0 ) does vanish except during the short jumps from kπ to (k + 1)π, where it gets finite values. During those jumps, this function is of order 1 and is given by the solution of the inner Eq. (18). Since it has no reason to be symmetric around the middle of the jump, one expects that its contribution to the average in (19) is of order 1 times the duration of the jump, that is of order α2 . Therefore, in the small α limit, q behaves like α (this is the product of the duration of the jump, α2 , times the prefactor 1/α in the formula (19)). There is another contribution to the average sin (t + ζ0 ). It comes from the intervals where t + ζ0 is nearly kπ: there, from Eq. (17), t + ζ0 is not exactly kπ, because of the (small) integral term. A simple order of magnitude
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estimate is obtained by assuming that the time derivative of ζ0 is constant and equal to 1 in the integrand and that the integral extends approximately over one period π. The deviation of (t + ζ0 ) from kπ is of order α, so that the contribution to sin (t + ζ0 ) of one time period of ζ0 (t) (outside of the fast jumps) is of order α, as the one of the fast jumps. This shows that q ∼ α as α tends to 0. This closes our discussion of the half infinite solution of the equation for the drifting phase. In the case of the classical amplitude equation (without drift) it is not completely obvious to extrapolate the solution from the half infinite case to a long, but finite segment. In particular, it is necessary to match the phase from one side of the system to the other one to get a consistent solution. In the present case, this matching is made simpler because it involves a constant phase shift from one end to the other: the function ϕ(z, t) is a linear function of t and z in between the two ends, up to a constant ϕ0 . Because this constant is arbitrary (reflecting the arbitrariness in the choice of the origin of time in this autonomous problem), no constraint comes from the merging of the bulk solution with the two boundary layers at the ends, besides the one linked to the common value of q. This assumes however that α takes the same value (up to an obvious change of sign) at both ends of the system, so that the wavenumber selected by the boundary effect is the same at the two ends. It would be interesting to investigate what happens when this is not so. Moreover, finite size effects are to be expected when L is small enough to make it possible the existence of a stationary phase, consistent with the b.c.’s.
3. Summary and conclusion This essay intended to show how to analyze phase dynamics with drift in large systems. This belongs to the study of patterns in large systems, a field where Alan Newell and his co-workers made a decisive progress when introducing the idea of amplitude equation, that makes simple and amenable to a common rational inquiry many different physical problems, which is after all the fundamental purpose of Science: a scientific problem can often be seen as understood (or at least partially so) when put into a well-defined category. For instance, a PDE is either hyperbolic or elliptic, or solutions of an ODE have or not a simple attractor, etc. In this secular thrust to sort out various physical problems, the amplitude equation approach and its corollary, the phase equation, has been a major progress: its unique relevance is in its special location between useless generalities and narrowly specific models. References [1] [2] [3] [4] [5] [6]
Y. Pomeau, P. Manneville, J. Physique (Paris) L 40 (1979) L-609. J. Prost, Y. Pomeau, E. Guyon, J. Physique (Paris) II 1 (1991) 289–309. A.C. Newell, Y. Pomeau, Physica D 87 (1995) 216. A.C. Newell, J. Whitehead, J. Fluid Mech. 38 (1969) 203. M.C. Cross, P.C. Hohenberg, Rev. Mod. Phys. 65 (1993) 851. P. Manneville, Dissipative structures and weak turbulence, Academic Press, San Diego, 1990.
Phase dynamics with drift: boundary effects Yves Pomeau a,b,∗ a
b
Laboratoire ASCI, Bat. 506, Orsay 91405 Cedex, France Department of Mathematics, University of Arizona, Tucson, AZ, USA
Abstract The description of patterns in nonequilibrium systems is a fascinating subject. It becomes somewhat untractable if one insists to keep the original equations in their complete form, as the Oberbeck–Boussinesq equations for instance in Rayleigh–Bénard convection. Various reduction scheme have been imagined, the ultimate one being the phase picture. I examine a simple version of the phase equation, relevant for systems of travelling rolls or for distributed self-oscillations. The boundary effects limit the ability of this phase to drift freely and yields a well-defined space structure, which can be analyzed in two different limits. © 2003 Elsevier B.V. All rights reserved. Keywords: Phase dynamics; Boundary; Oscillations
1. Introduction It is a great pleasure for me to contribute to the festschrift of Alan Newell, with whom I had years of fruitful exchanges and collaboration, both in France and at the University of Arizona in the US. One of the foremost contributions of Alan to nonlinear science has been the invention of the amplitude equations for patterns slightly above onset, like Rayleigh–Bénard thermal convection or in the buckling of long rectangular plates under in-plane stress. This concept of amplitude equation can be hardly overemphasized as a central idea in our science. A natural (although it took some time) extension of this idea is the one of phase motion: a wavelike pattern (or for that purpose any kind of pattern with some translational invariance or more generally any continuous symmetry) shows [1] a long wave-slow dynamical mode that degenerates in the infinite wavelength limit into immobile uniform translations. It is fair to say that an idea close to the one of phase dynamics was already present in the classical derivation of the equations of solid state elasticity by Cauchy in the first part of the 19th century: Cauchy derived the equations for a lattice (3D) of mass and springs in a way rather similar to the derivation of the slow phase dynamics of patterns. This phase equation can be written in a number of ways, including nonlinear effects, defects, etc. It is out of question to review here the rather large body of literature concerning this topic. The interested reader may find information on it in [3,5,6]. In this hommage to Alan Newell, I would like to present an extension of an application of this idea of phase dynamics that includes effects of boundary conditions (b.c.’s) and the possibility of nucleation of new “rolls” on one side of the “container” and their absorption on the other side. This shows that the phase ∗
Present address: Laboratoire ASCI, Bat. 506, Orsay 91405 Cedex, France. E-mail addresses: [email protected], [email protected] (Y. Pomeau). 0167-2789/$ – see front matter © 2003 Elsevier B.V. All rights reserved. doi:10.1016/S0167-2789(03)00224-0
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approach can be extended to situations where the total number of rolls is not conserved in the course of time. Some years ago, with Prost et al. [2], we looked at the following problem: let us suppose that a system with a phase, like a patterns of rolls, instead of being “naturally” in a state where this phase stays constant, shows a “trend” to constant phase slipping in space. If one thinks to a pattern of Rayleigh–Bénard rolls, this trend could be the result of a cross flow carrying the rolls in a certain direction. Other physical examples of this class of phenomena can be found in [2]. Another class of problems with the same mathematical structure is the one of extended time oscillations. There the phase variable is just the time phase of each local oscillator, that drifts “naturally” at constant rate. The interaction of neighboring oscillators can be represented by a regular diffusion term, if this interaction tends to synchronize neighbors. Indeed the phase picture is incomplete in general because it does not take into account variations in the amplitude itself, but this is at least a first approximation to space dependent phenomena. The constant phase drift, due to the conditions in the bulk, may go against the b.c.’s. In a system of rolls parallel to a boundary, the phase of the roll just touching this boundary is a relevant parameter in the b.c.’s. In a system with an optimization principle, the b.c. may tend to favor a specific value of the phase at the boundary, up to 2π, indeed. In a very simple mathematical formulation, I examine the effect of both such a pinning of the phase at boundary counteracting a bulk phase shift. The model under study does not claim to be realistic. In particular, it is not derived from any amplitude-like equation, although it could be. This paper intend to show that the balance between boundary pinning and bulk drift introduces two limits: the weak and strong limit, something that is perhaps of general interest. A simple form of phase equation with b.c.’s is introduced consistent with the requirement that phase is defined up to 2π. It is solved by asymptotic methods in two well-defined limits, showing how boundary effects change the simple picture of an uniformly drifting phase. The boundary effects ultimately defines the wavelength of the pattern. In the absence of any spatial structure, the equation for the phase ϕ with constant phase slip reads: dϕ = ω. dt
(1)
In this equation ω represents the rate of change of the phase in the absence of boundary effects, wavelength changes, etc. This rate of change should be seen as proportional to the cross flow, for instance, if this is the source of phase drift. For a self-oscillating system, ω would be 2π multiplied by the natural frequency. In the amplitude equation approach, the rapid phase, spatial or temporal, is factored out. This would amount here to simply subtract from ϕ a constant ωt. However, this is not something satisfactory if one wants to look at physical b.c.’s: such b.c.’s are sensitive to the true physical phase, not to the one obtained after subtraction of a fast phase. This explains why I keep this fast phase. The next question is: what would happen to the phase if it is held in some sense by boundaries? This leads naturally to consider first the effect of possible space heterogeneities on the phase dynamics, something simple at least in principle, then to account for the finite length of the system. The corresponding equation is obtained by adding to (1) a space diffusion term. To make things simple, I shall consider the one-dimensional case only where the phase diffusion equation with drift reads: ∂ϕ ∂2 ϕ =ω+D 2. ∂t ∂z
(2)
In this equation, the coordinate z is taken perpendicular to the roll direction (if rolls are under consideration) and D (assumed positive) is the so-called parallel diffusion coefficient (if the drifting roll situation is considered, otherwise it would be the usual phase diffusion coefficient). The next step consists in the writing of the b.c. Indeed, this may be done quite freely in the present theoretical setting. However, there are some constraints: the b.c., imposed at z = 0 and L, cannot make any distinction between values of the phase differing by 2π, since they are physically equivalent. Let us consider a b.c. of the so-called mixed type, that mixes the function itself and its first space derivative. A
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simple form of this b.c. consistent with the constraint of periodicity in ϕ is a linear relation between (∂ϕ/∂z) and a circular function like sin (ϕ) α
∂ϕ + sin (ϕ) = 0. ∂z
(3)
This is to be imposed at the ends of the system. For one space dimension, this is to be imposed at z = 0 and L, with a change of sign of α from one end to the other. The b.c. (3) may look somewhat arbitrary. As said before, this arbitrariness is partly the consequence of the general character of the model under consideration. Being consistent with the elementary requirement of invariance under an addition of 2π to the phase, it shows an interesting balance between two effects that are to expected: first the system tends to have a phase as uniform in space as possible. This explains the contribution proportional to (∂ϕ/∂z) in Eq. (3), that becomes dominant at large α. On the other hand, one expects that, locally, it makes a difference for the phase to have one or another value at the boundary, within the constraint that phases differing by 2π are equivalent. Whence the circular function in Eq. (3). This point of view is perhaps made clearer when considering the variational formulation of the equation under consideration: the b.c. yields a boundary contribution to the energy. In Eq. (6) below this is the quantity Fboundary = −
D ( cos (ϕ)(z = 0) + cos (ϕ)(z = L)). α
This Euler–Lagrange energy is the lowest when the phase is equal to 0, mod 2π on the boundary if α is positive. Therefore, this model represents a balance between two conflicting tendencies at the boundary: to have there a well-defined value of the phase, mod 2π and to have a phase as uniform in space as possible. The conflict is there because, the phase tends to drift at a constant rate in the bulk, so that it cannot take a prescribed constant value, mod 2π, on the boundary, unless some nontrivial dynamics takes place there. This set of equation and b.c.’s define a gradient flow, although this is not a property I am going to use. Indeed, Eq. (2) can be written formally as: ∂ϕ ␦F =− . ∂t ␦ϕ
(4)
The “energy” functional F has a bulk contribution: L 2 D ∂ϕ Fbulk = dz − ωϕ 2 ∂z 0
(5)
plus another part that comes from the remainders of the partial integrations needed to get the functional derivative (␦F/␦ϕ) in the form of the original right-hand side of Eq. (2). This boundary contribution needed to cancel the leftover of the integration by part reads Fboundary = −
D ( cos (ϕ)(z = 0) + cos (ϕ)(z = L)). α
(6)
When writing this boundary term, one took care that α changes sign from one end to the other. Therefore, the full “energy” functional is F = Fbulk + Fboundary . Because it includes a term linear in ϕ, this energy has no minimum and the evolution is not, in general (at least if L is large enough), a relaxation to a steady rest state. The sequel is devoted to an analysis of solution of Eq. (2) with the b.c. (3).
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To make the analysis simpler, one gets rid of the parameters ω and D by rescaling. Take ω−1 as unit of time, √ √ D/ω as unit of length, and redefine a new α as the original α times ω/D. The equation under consideration becomes: ∂ϕ ∂2 ϕ =1+ 2 ∂t ∂z
(7)
with formally the same b.c. (3) as before. This rescaling has got rid of all parameters, except the total length L √ (measured in units D/ω) and the coefficient α in the b.c. Let us look first at time independent solutions ϕ|st = − 21 (z2 ) + az + b.
(8)
In a finite system the constants a and b are determined by the two b.c.’s: sin (b) + αa = 0 and sin (− 21 (L2 ) + aL + b) = α(−L + a). This shows that there is no steady solution for L large. Because the sine function in the second b.c. is less than one in modulus it can balance the right-hand side only if a ≈ L. If this holds, the first b.c. cannot be satisfied. In order of magnitude estimates, no steady solution exist for L much bigger than 2π/α. Therefore, for L large enough, time dependent solutions exist only. It is likely that for intermediate values of L, that is of the order of magnitude of √ a few times the intrinsic length, D/ω, various dynamical behaviors are possible, that might have some interest. I shall consider the large L limit only, where the solution can be split into the solution near the ends and the one relevant for the bulk. As usual one needs next to match the “outer solution” (in the bulk) with the inner solution, valid near the ends. In next section, I deal with the inner solution, and match it with the outer one later on.
2. Boundary layer, inner and outer solution By definition the inner solution describes the behavior of the phase near the ends, where, presumably, the effect of the b.c.’s plays an important role. The classical amplitude equations of Newell and Whitehead [4], with the zero condition at the end has the familiar hyperbolic tangent behavior near the boundaries. To study the problem of a “long” geometry, one looks first at solutions of the phase equation in a semi-infinite layer, bounded at one end, say at z = 0 where the b.c. (3) applies. Far from the boundary, the solution is expected to tend to a simple solution of Eq. (7). This equation, being linear, can be solved in some form at least, given for instance, the boundary value of ϕ(z = 0, t). We shall limit ourselves to a few comments on the finite size solution. The outer/bulk solution is ϕ|as = t + qz + ϕ0 ,
(9)
where q and ϕ0 are two real numbers, arbitrary for the moment. Other solutions of the diffusion equation (9) decay far from the boundary and so cannot be taken as asymptotic condition. Wherever ϕ is the phase of a complex amplitude, the wavenumber is constrained to stay inside some finite band. Otherwise the amplitude itself cannot be computed. The theory presented below assumes that the wavenumber q selected by the b.c.’s is inside this band. The solution in Eq. (9) will be the asymptotic form imposed to the inner solution, itself being subject to the b.c. (3). Because there is an irreducible parameter, α, in the b.c. and because of the absence of any other parameter in
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this dimensionless form of the equation, it is worthwhile to look at the situation of small α and large α to get an idea of the general solution for an arbitrary α. As announced there is a regime of strong pinning, small α, and a weak pinning regime, large α, a noticeable feature of this model. 2.1. Large α limit Let us define ζ such that ϕ(z, t) = t + ζ(z, t). The equation for ζ is ∂ζ ∂2 ζ = 2 ∂t ∂z with the b.c. sin (t + ζ) + α
(10) ∂ζ . = 0 ∂z z=0
Notice that the algebra of the decomposition of the solution and of the b.c. does not depend on the value of α. The diffusion equation for ζ has the property of “mass” conservation. For a solution with well-defined time average properties, the time average of (∂ζ/∂z) is constant as a function of z. Therefore, the average “mass flux” of the outer solution, as given in Eq. (9), is the same as on the boundary: ∂ζ 1 q= = − sin (t + ζ) . (11) ∂z α z=0 z=0
In Eq. (11) the overbar is for the time average. To use this to compute q, one needs to get the value of ζ in the prescribed limit, α large. Let us assume that, in this limit, ζ is small, which is to be verified at the end to yield a consistent approximation. If this is true, the b.c. for ζ becomes a simple linear condition: ∂ζ sin (t) + α = 0 . (12) ∂z z=0 Let us write the solution of the inner equation as ζ(z, t) = A(z) eit + A∗ (z) e−it , where the complex valued function A∗ (z) is complex conjugate of A(z). From (12), the b.c. for A(z) is (∂A/∂z) = −(1/2iα)|z=0 , although A(z) is a solution of iA(z) =
d2 A . dz2
(13) √
Therefore, A(z) = A(0) e−(1+i)z/ 2 , which gives √ 2 ζ(z = 0) = ( cos t + sin t). 2α From (11), in the large α limit: √ √ 1 1 2 2 q = − sin (t + ζ) ≈ − cos (t) ( cos t + sin t) ≈ − 2 . α α 2α 4α z=0 This closes the case of large α.
(14)
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2.2. Small α limit This limit is more conveniently studied by reducing the full problem to an integral equation for the boundary value of ζ(t, z = 0), which can be done in the following way. Let us begin with the formal solution ϕ = t + ζ(t, z), with the b.c. sin (t + ζ(t, z = 0)) + α(∂ζ/∂z)|z=0 = 0. This b.c. depends on two unknown functions of t, ζ(t, z = 0) and (∂ζ/∂z)|z=0 . These two functions are linked by the property that ζ(t, z) is a solution of the diffusion equation. Therefore, there is a linear relation between those two functions of t, which can be derived from the Green’s function solution of the heat equation t 1 −1/2 ∂ζ ζ(t, z = 0) = − √ dt (t − t ) (15) ∂z z=0 π −∞ or ∂ζ 1 (t, z = 0) = √ ∂z π
t
−∞
dt (t − t )−1/2
∂ζ (t , z = 0). ∂t
The b.c. becomes, as announced, an integral equation for the function ζ(t, z = 0) = ζ0 (t) t α dζ0 sin (t + ζ0 (t)) + √ dt (t − t )−1/2 = 0. dt π −∞
(16)
(17)
Formally, in the limit α tends to 0, the solution of (17) should become close to t + ζ0 = kπ, with k arbitrary integer. But this solution cannot be a valid starting point for an expansion near α = 0, since it has a constant time derivative that makes strongly divergent the integral term in the b.c. Therefore, the solution at the dominant order in α has to change from one value of k to another one to make converge the integral in (17). Therefore, the function ζ0 (t) must be almost constant and jumps in a short time by values of π at time intervals π too. The value of ζ0 inside the jump is given by the solution of a boundary layer problem, wherefrom the small parameter α can be scaled out. During this fast variation of ζ0 , one can neglect the time t in the argument of the sin function, so that the inner solution ζ0,i (t) valid during the jump is given by: t α dζ0,i sin (ζ0,i (t)) + √ dt (t − t )−1/2 = 0. (18) dt π −∞ Thanks to its simpler structure, the equation for the inner problem can be be made parameterless, as usual, by convenient rescaling (or by what is called sometimes a stretching transformation). In the present case, one gets rid of α by using ˜t = t/α2 as new variable (we shall drop the tilde below). This means in particular that the duration of the phase jump is of order α2 , a small quantity as expected. The b.c.’s for the inner solution are ζ0,i (t) tends to kπ as t tends to minus infinity and ζ0,i (t) tends to (k + 1)π as t tends to plus infinity. Beside a numerical study (in progress) nothing more can be said about this parameterless inner problem. There remains to find the value of q in this small α limit. One can still use the same general formula as in the opposite limit 1 q = − sin (t + ζ0 ). α
(19)
The function sin (t + ζ0 ) does vanish except during the short jumps from kπ to (k + 1)π, where it gets finite values. During those jumps, this function is of order 1 and is given by the solution of the inner Eq. (18). Since it has no reason to be symmetric around the middle of the jump, one expects that its contribution to the average in (19) is of order 1 times the duration of the jump, that is of order α2 . Therefore, in the small α limit, q behaves like α (this is the product of the duration of the jump, α2 , times the prefactor 1/α in the formula (19)). There is another contribution to the average sin (t + ζ0 ). It comes from the intervals where t + ζ0 is nearly kπ: there, from Eq. (17), t + ζ0 is not exactly kπ, because of the (small) integral term. A simple order of magnitude
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estimate is obtained by assuming that the time derivative of ζ0 is constant and equal to 1 in the integrand and that the integral extends approximately over one period π. The deviation of (t + ζ0 ) from kπ is of order α, so that the contribution to sin (t + ζ0 ) of one time period of ζ0 (t) (outside of the fast jumps) is of order α, as the one of the fast jumps. This shows that q ∼ α as α tends to 0. This closes our discussion of the half infinite solution of the equation for the drifting phase. In the case of the classical amplitude equation (without drift) it is not completely obvious to extrapolate the solution from the half infinite case to a long, but finite segment. In particular, it is necessary to match the phase from one side of the system to the other one to get a consistent solution. In the present case, this matching is made simpler because it involves a constant phase shift from one end to the other: the function ϕ(z, t) is a linear function of t and z in between the two ends, up to a constant ϕ0 . Because this constant is arbitrary (reflecting the arbitrariness in the choice of the origin of time in this autonomous problem), no constraint comes from the merging of the bulk solution with the two boundary layers at the ends, besides the one linked to the common value of q. This assumes however that α takes the same value (up to an obvious change of sign) at both ends of the system, so that the wavenumber selected by the boundary effect is the same at the two ends. It would be interesting to investigate what happens when this is not so. Moreover, finite size effects are to be expected when L is small enough to make it possible the existence of a stationary phase, consistent with the b.c.’s.
3. Summary and conclusion This essay intended to show how to analyze phase dynamics with drift in large systems. This belongs to the study of patterns in large systems, a field where Alan Newell and his co-workers made a decisive progress when introducing the idea of amplitude equation, that makes simple and amenable to a common rational inquiry many different physical problems, which is after all the fundamental purpose of Science: a scientific problem can often be seen as understood (or at least partially so) when put into a well-defined category. For instance, a PDE is either hyperbolic or elliptic, or solutions of an ODE have or not a simple attractor, etc. In this secular thrust to sort out various physical problems, the amplitude equation approach and its corollary, the phase equation, has been a major progress: its unique relevance is in its special location between useless generalities and narrowly specific models. References [1] [2] [3] [4] [5] [6]
Y. Pomeau, P. Manneville, J. Physique (Paris) L 40 (1979) L-609. J. Prost, Y. Pomeau, E. Guyon, J. Physique (Paris) II 1 (1991) 289–309. A.C. Newell, Y. Pomeau, Physica D 87 (1995) 216. A.C. Newell, J. Whitehead, J. Fluid Mech. 38 (1969) 203. M.C. Cross, P.C. Hohenberg, Rev. Mod. Phys. 65 (1993) 851. P. Manneville, Dissipative structures and weak turbulence, Academic Press, San Diego, 1990.