Hopf bifurcations in Langmuir circulations

Physica D 59 (1992) 226-254 North-Holland

Hopf bifurcations in Langmuir circulations S t e p h e n M . C o x a ' , I S i d n e y Lelbovlch" a, I r e n e M . M o r o z b a n d A m i t T a n d o n a aSibley School of Mechanical and Aerospace Engineering, Upson Hall, Cornell University, Ithaca, N Y 14853-7501, USA bSchool of Mathematics, University of East Anglia, Norwich, Norfolk NR4 7TJ, UK •

Received 31 August 1991 Revised manuscript received 2 June 1992 Accepted 4 June 1992 Communicated by J. Guckenheimer

The equations governing two-dimensional Langmuir circulations in a continuously stratified layer of fluid possess 0(2) symmetry when laterally periodic boundary conditions are applied. In one limit, which is among those treated here, the mathematical problem is strictly analogous to a double-diffusive problem. Steady, oscillatory and multiple oscillatory states are all possible. The method of multiple scales is used to obtain evolution equations for the amplitudes of oscillatory convection when a single wavenumber is destabilized, and when two wavenumbers are simultaneously destabilized. In the process, this paper provides the first example of the derivation of amplitude equations for a double-Hopf bifurcation with symmetry in a fluid mechanical problem. Parallel numerical simulations of the full partial differential equations are carried out, and quantitative comparisons are made between the two methods, both when periodic boundary conditions are enforced, and when the more restrictive flux-free side-wall boundary conditions are enforced.

1. Introduction T h e objective o f this p a p e r is to m a k e the first quantitative c o m p a r i s o n b e t w e e n H o p f and m u l t i p l e - H o p f bifurcation analyses o f a physically significant fluid mechanical p r o b l e m in the p r e s e n c e o f s y m m e t r y , and predictions o b t a i n e d by direct numerical simulation o f the g o v e r n i n g fully n o n l i e n a r partial differential equations. T h e a s s u m e d s y m m e t r y with respect to spatial translations and reflections (or " O ( 2 ) " s y m m e t r y ) is associated with the imposition of periodic b o u n d a r y conditions in a h o m o g e n e o u s direction, and o u r p a p e r c o n c e r n s situations w h e r e such b o u n d a r y conditions are chosen, as well as w h e n a m o r e restrictive subset of these b o u n d a r y conditions is i m p o s e d which force the fluxes o f relevant quantities to vanish at multiples of the spatial period. T h e latter p r o b l e m does not have 0 ( 2 ) s y m m e t r y , but r a t h e r 7/2 s y m m e t r y (a fact that we do not explicitly exploit). This p a p e r also p r o v i d e s the first rigorous derivation of amplitude equations for a d o u b l e - H o p f bifurcation with s y m m e t r y in a p r o b l e m o f fluid mechanics. T h e physical p r o b l e m addressed is L a n g m u i r circulations ( L C ) in a layer o f density stratified fluid c o n t a i n e d b e t w e e n two horizontal planes. L a n g m u i r circulations are wind-induced c o u n t e r - r o t a t i n g vortices o b s e r v e d in the u p p e r layer of oceans and lakes for sufficiently strong winds, with their axes aligned r o u g h l y in the direction of the wind. Such m o t i o n s often are m a d e visible on the w a t e r surface by the lateral sweeping of flotsam, f o a m , or organic films into " w i n d r o w s " or streaks parallel to the 1Present address: Department of Applied Mathematics, The University of Adelaide, G.EO. Box 498, Adelaide 5001, South Australia. 0167-2789/92/$05.00 © 1992 - Elsevier Science Publishers B.V. All rights reserved

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surface wind direction. Langmuir circulations represent an important mechanism for the mixing of the surface layers of the oceans and of other natural bodies of water (see Leibovich [16] for a review), for the transfer of momentum and heat between the atmosphere and the ocean, and for marine biology. The first scientific observations of the phenomenon are due to Irving Langmuir [15]. According to the Craik-Leibovich theory [3,16] of Langmuir circulations, the mixing results from an instability of the water that is a consequence of the interaction between the surface water waves and the vorticity in the current generated by the applied wind stress. In the present problem, we adopt a model due to Leibovich [17], which isolates the so-called ocean mixed layer from the abyss below. With the supposition that a strong thermocline (density gradient induced by a temperature gradient) divides the two water regions, it is assumed that the stress applied at the lower boundary of the mixed layer is the same as that applied by the wind at the air-sea interface. Furthermore, the mixed layer itself is assumed to be stably density stratified by virtue of a temperature difference set by maintaining isothermal horizontal boundaries, with the temperature of the upper surface greater than that of the lower. Our interest here is in situations where the stratification is sufficiently stabilizing to lead to oscillatory (overstable) convection. We deal with this problem in two space dimensions, assuming that variations in the direction of the applied wind stress are ignorable. The latter assumption has strong justification from observations [16]. In one special limit of the problem outlined (which is included among those addressed here), the mathematical problem posed for LC is precisely analogous to double-diffusive thermohaline convection with constant values of the stabilizing diffusing quantity (say salt) on top and bottom and constant flux of the destabilizing diffusing quantity (say temperature) on these boundaries, when the Prandtl number is unity. This problem has been treated before [18,21], with no attention paid to the symmetry aspects. Some of the features imposed by the 0 ( 2 ) symmetry have been discussed in this problem by Moroz and Leibovich [22] (who treat the current problem, but adopt ad hoc choices for linear eigenvectors) and in [2], with focus on the double-zero (or Takens-Bogdanov) multiple bifurcation [5]. The features associated with 0 ( 2 ) symmetry have been extensively explored in the literature for other problems, notably "ideal double-diffusion" [11-13] (IDD, for example, thermohaline convection in which both temperature and solute are constant on the horizontal boundaries) and related problems of doublediffusive convection in which the destabilizing effects of temperature are balanced by the stabilizing effects of rotation or magnetic field rather than density [4,9,25], and the Taylor-Couette instability problem of fluid confined between differentially rotating co-axial circular cylinders [1,6,7]. In some of these studies, both direct numerical simulation based on the full partial differential equations and bifurcation analyses have been carried out. To the best of our knowledge, however, when comparisons have been made, they have been only qualitative. The sole exception is our companion paper [2], which gives a quantitative comparison such as done here, for the case of the Takens-Bogdanov bifurcation which also occurs in these problems. This lack of previous comparisons is due to the existence of a nonlinear degeneracy in the Hopf bifurcation analysis of "simple" problems such as IDD (where the linearized problem can be solved in closed form). This degeneracy occurs at the lowest nontrivial nonlinear order (cubic), and its consequence is that the dynamics are not fully specified until quintic terms are included. In the current case, this degeneracy does not arise. In all of these problems, 0 ( 2 ) symmetry arises when laterally periodic boundary conditions are imposed (in contrast to the more restrictive flux-free boundary conditions). When time-dependence sets in as overstable oscillations, the motions (restricted to two dimensions) may be in the form of waves which propagate horizontally in either direction (so-called right and left travelling waves) depending on

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initial conditions, or in the form of standing waves. It is not possible to have both stable travelling waves (TW) and stable standing waves (SW) coexisting: which one is observed is determined by the parameters of the problem. More specifically, 0 ( 2 ) symmetry expresses the invariance of the physical system to arbitrary translational shifts, y ~ y + 1, and reflections y ~ - y . In addition, the problem is invariant under a shift in the origin of time, t---~ t + ~b/to. The way that these invariances enter into a bifurcation problem may be traced as follows. If the partial differential equation Ut = F ( u , Uy, U y y L . . . ) ,

(1)

when linearized about an equilibrium at a point of bifurcation, admits solutions of the form u ( y , t) = UL(t) e i*y + uR(t ) e i~y + ( * ) ,

(2)

where UL(t ) and UR(t ) satisfy dUL/dt= itOUL, dUR/dt = --itouR, and (*) denotes the complex conjugate of all preceding terms, then u L and u R represent left and right travelling waves with frequencies Ito I; when lull = luRIa standing wave occurs. The invariances impose three mathematical constraints that (1), (2) and the nonlinear evolution equations for u L and u R must satisfy [11,23]: (i) invariance under translations: T(UL, dR) = eikt(UL, dR) ; (ii) invariance under reflections: R(UL, dR) ----(dR, UL) , (iii) invariance under phase shift: P(UL, dR) = (ei6uL, e-i~UR), so that d//L dt

=

du R dt

-

uLg(luLI2' ]UR]2)'

URg*(lu l=' lull=)'

(3)

where g is some complex function of lull and ]URIe, and the superscript asterisk denotes complex conjugation. Sets of equations such as (3) are derived here for LC in the vicinity of a Hopf bifurcation, and the generalization of (3) in the vicinity of a double-Hopf bifurcation, where the instabilities of two spatial modes coalesce. The dynamics of the LC problem in two dimensions are controlled by four parameters: a measure of the density stratification, S; an inverse Prandtl number, z; a parameter, A, that determines the depth of 2Az influence of the non-dimensional Stokes-drift gradient, which we take to be of the form h ( z ) = e (appropriate to monochromatic surface gravity waves, in which case, A is the wave number of the surface gravity waves made dimensionless with the layer depth), and a measure of the effect of wind-waves and current shear, R. Positive values of S correspond to a stabilizing density stratification, and positive values of R to a destabilizing effect of wind-wave forcing. We consider only the first (S, R) qudrant, and fix r = 1/6.7 ~ 0.15, which is the molecular value of this parameter for water. The linear stability characteristics of the rest state of no convection are identical for both flux-free and periodic lateral boundary conditions, but depend on the fundamental spatial period allowed (alternatively, the

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aspect ratio of the layer). In this paper, we consider (for reasons outlined in [2,18], and particularly to compare with and correct [18]) a quantization having dimensionless fundamental spatial period L = 4, corresponding to wavenumber k = ½"rr. A pair of curves (each nearly a straight line) is obtained in the (S, R)-plane for fixed ~" and h for the fundamental and each harmonic (integer multiple) of the fundamental wavenumber. One line gives the onset of steady convection while the other gives that of oscillatory convection: the stability boundary for oscillatory convection emanates from the steady convection boundary, which thereby loses its stability. The situation is sketched in fig. 1 for the fundamental wavenumber k = ½"rr, which is the case we consider. In this case the point of intersection, (S D, RD), for the ½"rrmode occurs at (395.2,823.5). The total stability boundary therefore comprises a set of approximately straight line segments, associated with the different wavenumbers, as illsutrated in the figure. Moreover it is possible to locate points on the boundary at which two oscillatory states become simultaneously excited, their wavenumbers being consecutive multiples of the fundamental. For A = 0 and motions with the fundamental wavenumber ½rr, the first possible intersection is with the mode with wavenumber k = "rr, and this occurs at (S, R) = (S HH, R HH) = (1078.878, 1613.307). The behaviour in the vicinity of such single and multiple Hopf bifurcations has been addressed previously by Lele [19], Leibovich et al. [18] and Moroz and Leibovich [22] (the latter addressing only the flux-free case). Those results in the first two of these references that stem from direct numerical simulation of the partial differential equations are incorrect due to an error in the numerical code. These errors are corrected in [2], and further elaboration will be provided here. In our numerical studies for moderate values of the stratification parameter, S, oscillatory convection is observed for small-to-moderate departures from the linear stability boundary; for larger supercriticalities, stability is rapidly lost to finite-amplitude steady convection. For larger values of S, however, steady convection is unstable, and oscillatory convection in the form of travelling waves is found regardless of the supercriticality [2]. In order to understand and model this behaviour, we have undertaken two studies. For both of these, parallel computations from bifurcation theory (with amplitude equations describing the system evolu-

1613.3--

823.5-741.6--

I

72.01

I

I

395.2

1078.9

S

)L

Fig. 1. Sketch of the linear stability boundary appropriate to the dimensionless wavenumber ½"rr. S1 and H1 mark the onset of steady and oscillatory convection, respectively, for the fundamental mode (that is, for the mode with dimensionless wavenumber k = ½ax). $2 and H2 mark the equivalent stability boundaries for the first harmonic (that is, for the mode with k = ~r).

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tion derived by the method of multiple scales) and from numerical simulation of the partial differential equations are reported and compared. In the first study, we examine the evolution of an oscillatory state corresponding to a single spatial mode close to its stability threshold, both with periodic and flux-free lateral boundary conditions. The various coefficients that appear in the amplitude equations for this simple Hopf bifurcation have been evaluated using the computer algebra package Mathematica [26] to establish the preference for standing or travelling waves. Unlike the O(2)-symmetric Hopf bifurcation in IDD, no degeneracy in the TW solution arises at third order, so we are able to make direct quantitative comparisons between our bifurcation analyses and direct numerical integrations of the partial differential equations governing the Langmuir circulations. The flux-free boundary conditions do not admit travelling waves, so the invocation of these lateral boundary conditions amounts to an elimination of TWs and a stabilization of (supercritical) SWs where these may have been unstable in the 0 ( 2 ) problem. The second study proceeds along precisely the same lines, but addresses the situation near the double-Hopf point where the k = I rr spatial mode and its first harmonic k = ar are simultaneously destabilized and interact. In the range 0 <- A --<4 studied, a qualitative change is found by weakly nonlinear analysis to occur at A = Aa, where Aa is slightly greater than 2.525, so the specific values of A for which computed values of amplitude equation coefficients and system behaviour are reported are (0, 1, 2, 2.525, 3, 4). The l'rr TW, which is unstable for A < A a becomes stable for A > Aa. Other qualitative changes occur in the double-Hopf case as A varies, but these are not pursued here. Our numerical simulations are limited to the case A = 0. The numerical protocol holds S fixed at selected values, and varies R. If L -- 4, the ½or mode is destabilized first as R increases for S -< Sun, and the "rr mode for an undetermined interval of S (extending at least up to 40 000) beyond S = S HH. Nevertheless, we study not only the bifurcation of the l~r mode, but also the Hopf bifurcation of the ar mode, in order to correct the results of [18]. In [18], several apparently interesting forms of behaviour were noted, including (i) changes in the direction of bifurcation as S was increased along the "rr mode Hopf curve, and (ii) complex dynamics, including intermittently chaotic states. In the corrected study here, none of these features exist. For flux-free boundary conditions, we do find coexisting oscillatory stable solutions for S = 3000, as well mixed-mode (two-torus) solutions in a very narrow band in parameter space near (S, R) = (S HH, RHH). Nothing more exotic than these possibilities has been found with flux-free conditions. In the 0 ( 2 ) case, we find only TWs and steady states. Using both amplitude equations and simulation, we find a very rapid transition in the flux-free case, in which the convection cells split their structure. A single cell divides into two and then three vertically stacked cells, then back to a single cell with reversed flow directions: this sequence then repeats to complete the cycle. This process, which is accomplished in an extremely short fraction of the temporal period, has been seen before by Knobloch and Moore [14] and Sch6pf and Zimmermann [24] in linearized studies of binary convection. In the linearized case, the streamlines dividing stacked cells are horizontal; we show that nonlinear effects cause these bounding streamlines to slant, but that the splitting persists in the fully nonlinear calculations. As a general observation, we note that all qualitative behaviours found in the fully nonlinear simulations can be anticipated or inferred from the weakly nonlinear analysis, although similar events may occur at somewhat different values of the control parameters in the two methods. Some features, such as the cell splitting noted in the previous paragraph and the two-torus modulated wave solutions, occur in such small intervals of time or the control parameters that they are virtually certain not to have been found by simulation if their existence had not been revealed by the analytical treatment.

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231

Furthermore, the quantitative agreement between direct numerical simulation and weakly nonlinear theory is excellent provided we are sufficiently close to the stability margin. The dynamics of sets of partial differential equations have sometimes been studied by truncating a Galerkin representation at very low order. The resulting low-order dynamical system may, however, provide misleading results. For example, a low-order truncation introduced by Moroz and Leibovich [21] for the Langmuir circulation problem yielded degeneracies for isolated values of the wavenumber. These were artifacts of the drastic truncation. The multiple scales analysis described here is not subject to such qualitatively misleading results, provided one remains suitably close to the bifurcation points that are the subject of analysis. The paper is structured as follows. In section 2 we present the model and describe the relevant aspects of linear stability theory, noting how the underlying 0 ( 2 ) symmetry affects the form of the oscillations. Section 3 is devoted to the weakly nolinear analysis for a single oscillation and section 4 contains the bifurcation analysis of the evolution equations for the problem. The multiple Hopf bifurcation forms the basis of section 5. The bifurcations near the multiple Hopf point are discussed in section 6 when flux-free lateral boundary conditions are enforced, and in section 7 when periodic boundary conditions apply. In section 8 we compare the weakly nonlinear findings to results of the numerical integrations of the partial differential equations, and section 9 summarizes the paper and its physical implications.

2. The model and linear theory

Langmuir circulations are assumed to arise from an instability of a basic state which in dimensionless units corisists of a constant wind-driven shear, U ( z ) = Uo + z, and a linear temperature profile, T(z) = T O+ z. The fluid is assumed to be confined to a layer of finite depth, and density stratification is included through the Boussinesq approximation. When the velocity, temperature and pressure fields are independent of the x coordinate (in the direction of the wind), the non-dimensional equations describing the Langmuir circulations, which are perturbations to this basic state, take the form [16,19] ( 0 _V2) V2g, - R h ( z ) -~y 0u + S ~yy 00 = J ( , , V2~),

u - O--y= J(¢/' u),

~

Oy

where ~(y, z, t) is the streamfunction for the cross-wind velocity field, u(y, z, t) is the perturbation velocity in the wind direction, O(y, z, t) is the perturbation temperature field, h ( z ) = e 2a~ is the dimensionless Stokes-drift gradient, J ( f , g) is the Jacobian f y g z - f ~ g y and V2 is the Laplacian with respect to y and z (the cross-wind coordinates). Exact expressions for the parameters, ~-, R and S may be found in Moroz and Leibovich [21]. We suppose that a constant stress and constant temperature are applied on the upper and lower boundaries of the fluid layer, so that the velocity perturbations at the upper surface and the base of the fluid layer are stress-free and the temperature perturbation vanishes to give ~k=~z=O=uz=O

on z = 0 , - 1 .

(5)

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S.M. Cox et al. / H o p f bifurcations in Langmuir circulations

We explore the consequences of two choices for the lateral boundary conditions: the first where flow quantities are made periodic with period L in the horizontal, cross-wind direction (y), and the second where we impose flux-free lateral boundary conditions so that

~O=g,yy=Oy=uy=O o n y = 0 ,

L.

(6)

The set (4) may be written in compact notation as

(7)

[0,J + K ] ~ : N(qt, q¢), subject to the boundary condtions

sq,=o,

(8)

where = (~¢, U, 0 ) T

(9)

(where superscript T denotes the transpose) is the perturbation to the basic state, d and K are linear operators involving derivatives with respect to y and z but not t, N the vector of (nonlinear) Jacobians, and B the (linear) boundary condition operator. If we neglect the Jacobian terms in (4), then solutions satisfying periodic boundary conditions may be found in the form = I~(Z) e i(ky+'°t) ,

u = /~(z) e i(ky+~°t) ,

0 = / 9 ( z ) e i(ky+°~t) ,

(10)

where ~b(z), ti(z), O(z) satisfy (iw - I3) 13q~ - ikRh(z)fi + ikSO = 0,

(i~o - 13) t~ - ik6 = 0 ,

(i~o - r13) 0 - ik~b = 0

(11)

subject to ~b=D2~b=b=Da=0

onz=0,-1,

(12)

and we have introduced the operators D = d/dz and 13 = D 2 - k 2. Here, k is some integer multiple of the fundamental wavenumber, k b = 2"rr/L. The eigenvalue problem defined by (11), (12) is eighth order in ~b(z), ti(z) or t~(z) and requires a numerical approach for its complete determination. For the problem of ideal double diffusion ( I D D ) between flux-free horizontal boundaries, the form of the vertical eigenfunctions is known analytically. The analogous problem to (11), (12) then can readily be solved leading tc~ an algebraic characteristic equation for w [8,13]. Here we approximate each eigenfunction by a finite sum of Chebyshev polynomials, which transforms the ordinary differential eigenvalue problem (11), (12) into a generalized algebraic eigenvalue problem of the form [Ad(k) + K(k)] ~rt : 0.

(13)

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S.M. Cox et al. / Hopf bifurcations in Langmuir circulations

H e r e ~rt is a vector of Chebyshev coefficients for the eigenfunction (It(z) =- ((b(z), a(z), O(z)) x, and d(k) and K(k) are the matrices analogous to the operators d and K, with O/Oy replaced by ik. (For notational brevity we suppress the dependence of K(k; R, S, A, z) on R, S, A and ~'. Also, we note that information about the boundary conditions is incorporated in J(k) and K(k).) As we vary R (or fixed S, A and ~'), the onset of oscillatory convection occurs at R = R n (S, A, z) when all the eigenvalues of (13) have negative real part, except for one purely imaginary pair, A = - i t o , to real. In practice we use the computer algebra package Mathematica to compute the eigenvalues of (13), by solving the equivalent standard eigenvalue problem J-l(k)

K(k) ~rt = -A~rt.

(14)

For the numerical integrations of the partial differential equations (4) 7 was chosen as (6.7) -1 ~ 0.15 (corresponding to the molecular Prandtl number for water), and A = 0. The cell aspect ratio is ½L = rr/kb, where k b is the smallest wavenumber which can be accommodated by the periodic domain. Disturbances with wavenumber k = nk b are possible for any integer n >- 1. For each n the linear stability boundary is a pair of nearly straight lines and the envelope of these lines for all n > 1 gives the stability boundary of the problem. Following Leibovich et al. [18] we choose k b --l~r. Fig. 1 depicts the typical marginal stability boundaries obtained for n = 1, 2 and for A < 5; $1 and H1 refer to the onset of steady and oscillatory convection for n = 1 and $2 and H2 the corresponding boundaries for n = 2. For example when h = 0, H1 and H2 intersect at (S Hri, RHH) ~ (1078.878, 1613.307) and both n = 1 (k = ½at) and n = 2 (k = "tr, for square cells) modes are therefore excited simultaneously. This is the multiple H o p f bifurcation we consider in this paper. We conclude this section with a remark related to the 0 ( 2 ) character of (4). These equations are invariant under the transformation (reflectional symmetry) (~, u, 0, y, z ) ~ ( - ~ , u, 0, - y , z), so that in particular if [~(z), fi(z), (9(z)]e i(ky+*t) is a solution of (11), (12), then so is [-t~(z), t~(z), 0(z)] e i(-ky+'°t). The solution to the linear stability problem is therefore [~b, u, 0] = a l i a ( z ) ,

t~(z), 0(z)] e i(ky+~°t) + a ~ [ - t~(z), a ( z ) ,

b(z)l e i(-ky+o't)

+ a~[~*(z), t~*(z), 0*(z)] e -i(ky+oJt) + a 2 [ - - ~ * ( z ) , /~*(z), 0*(z)] e -i(-ky+~°t) ,

(15)

which can be expressed as either [~b, u, 0] = a,[t~(z), t~(z),/)(z)] e i(kr+") + a ~ [ - ~(z), t~(z),/)(z)] e i(-ky+o't) + (,)

(16)

[~/,, u, 0] = a,[~(z), ~(z), 0(z)] e i(ky+'O + a 2 [ - t~*(z), ti*(z), 0*(z)] e i(ky-'') + (*).

(17)

or

Following Knobloch [11] we shall adopt the second representation (17) for our nonlinear analysis.

3. Hopf bifurcation with 0(2) symmetry Perturbing away from the line of oscillatory instability with S fixed, we have R = R H + E2~", and the linear growth rate for the eigenfunctions [~b,/~, t)] is ~7(~2). Accordingly we may introduce a slow time scale T = ~2t and expand ~b, u, 0 as power series in ~:

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234

[~t, U, 0] -~ E[I]t (1), U (1), 0 (I) ] -1- 2 [ ~ ( 2 )

U(2), 0(2)] _{_ E3[1~(3), U(3), 0(3)] - k ' ' ' ,

(18)

where [~(1), u(1), 0(1)] = [t~(z), tJ(z),/~(z)] a l ( T ) e i't'l + [ - t~*(z),/~*(z),/)*(z)] a2(T ) e i~2 + (*)

(19)

are the neutral modes. In (19) a l ( T ) is the amplitude of the left propagating wave with phase

qbI = ky + tot, a2(T ) that of the right propagating wave with phase ¢2 = ky - tot, and [t~(z), fi(z),/)(z)] satisfies (11), (12). In order to define the amplitudes a l ( T ) and az(T ) of the linear solution (19) unambiguously we must choose a normalization for the linear eigenfunction. This we do by analogy with IDD, where ~b = sin-trz, by setting f° 1 ]@(Z)I 2 d z = 1. With this normalization built in to our weakly nonlinear expansion, we have therefore been able to check our procedure directly against analytical results for IDD, by setting A = 0 and changing the boundary conditions at z = - 1 , 0. Upon substituting (18) into (4) and equating powers of e, we obtain a hierarchy of problems which must be solved iteratively for the [~O(j), u (j), 0 (j)] ( j = 1, 2 . . . . ). The ~?(e) problem reproduces the linear analysis of section 2 and yields (19) and (11), (12) for the eigenfunctions, with R = R", so that to is real. At 6 ( 2 ) we find that a particular solution exists in the form " + [-A~'(z), B~(z), D~'(z)] a 22 e 2'~2 " [qflz), u(2), 0(2)1 = [AI(Z) ' B,(z), DI(Z)] a 2I e 2'¢' + 1 {[A2(z), B2(z), D2(z)]]a,12 + [_A2(z), Be(z),

D~(z)lla212}

+ [A3(z), B3(z), D3(z)] a,a 2 e i(~'+4'2) + [0, B4(z), O4(z)] ala ~ e i(4''-~'2) + (*) (20) where Aj(z), Bj(z), Dj(z) satisfy the differential equations (52) listed in the appendix. Moving to t~(e 3) we find terms proportional to e i*~, e i4'2 in (4) which constitute forcing terms and must therefore be removed to maintain a uniformly valid expansion for q,, u and 0. If we write

[qJ(F3), U(F3), 0(F3)] = [~(Z, T), U(z, T), O(z, T)] e i4'1 + [-aP'*(z, T), U*(z, T), O*(z, T)] e i~2 + ( * ) (21) as the component of t~(e 3) which will eliminate these secular terms, then we obtain {ito - D} D q t - ikRHh(z) U + i k S " O = - O ~ a l r + ikr'h(z) fi - ik~'0 + alFl([al[ 2, la2] ~, z ) , (ito - I)) U - i k q t = - f l a i r + alF2([al[ 2, la2l 2, z ) , (ito - rI3) @ - i k q t = --(~a,T +

a,F3(la,] 2, la2l 2, z ) ,

(22)

and - ( - i t o - D) I ) ~ * - i k R n h ( z ) U* + ikS"®* = O(b*a2r + ikFh(z) •* - ik~'0* +

a2FT(la2] 2, lall 2, z ) ,

( - i t o - IS)) u * + ikT,* = -a*a2T + a2F~(la212,la,I 2, z ) , (-ito - r l ) ) ®* + ikqt* = -O*a2r + azF~(la2[ z, [a,[ 2, z ) ,

(23)

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235

where [~, tl, 01 are the linear eigenfunctions and the Fj are polynomial functions of la,I 2 and la212 whose coefficients depend upon q), a,/), A j ( z ) , Bj(z), Dr(z ) . Evolution equations for a l ( T ) and a2(T ) result when the inner product of both sides of (22) and (23) is taken with the eigenfunctions of the problem adjoint to (11), (12), Since our problem is reduced to solvability of an algebraic set, we utilize a more efficient method [20]. Using Mathematica (see the appendix for details) we obtain dq = a,(tx + iv + Pla~l 2 +

Qla212), G =

a2(/x - i u + Q*lal[ 2 +

e*[a212),

(24)

where the overdot represents the derivative with respect to T, p, and v are real linear functions of ~', and P and Q are complex constants. The form of (24) is the same as the O(2)-Hopf equations derived by Knobloch [11] for thermohaline convection. In that problem R e ( P ) = 0, indicating the presence of a degeneracy which is removed by computing higher order quintic terms. Such cubic-order degeneracies do not arise here and (24) provides a complete description of the O(2)-Hopf bifurcation for the Langmuir circulation problem.

4. Standing or travelling waves

If we write a j ( T ) = r j ( T ) e i~,(r) for j = 1, 2, then (24) decouples into modulus and phase equations: t:1 = rl( p, + prr21 + a r r ~ ) , ~1

=

1)

+ pir~ + Oir2 ,

t: = r2(/x + a~r~ + P'r]),

- ~ 2 = u + Qir~ + pir~ ,

(25) (26)

where the superscripts "r" and "i" refer to real and imaginary part, respectively. We may restrict our attention to (25), noting that nontrivial fixed points of (25) correspond to time-periodic solutions of the full system (24), by virtue of (26). The equilibrium solutions of (25) are (i) (rx, r2) = (0, 0); (ii) (rl, r 2 ) = (0, rT), or (rT, 0), where r T = [-/x[/a]l/2; (iii) (r 1, r2) = (rs, rs), where r s = [ - / z [ / ( a +/3)]1/2, where a = P~, /3 = Q~. Physically, (i) corresponds to the rest state, (ii) to travelling waves (TW) and (iii) to standing waves (SW). Of course (ii) and (iii) can only exist when the radicals are real. Following Knobloch [11] we can show that stable, nontrivial equilibrium states exist only when both T W and SW bifurcate supercritically. Travelling waves are realizable, that is, TW are stable and SW unstable, if a<0

and

/3-a<0

(27)

while SW are realizable if a+/3<0

and

/3-a>0.

(28)

We have used Mathematica to compute the coefficients in (25), (26) for k = ½~ and for k = "rr, SD(k) < S < 3000 and a = 0, 2 and 3. Here SD(k) is the smallest value of S for which the first bifurcation

S . M . Cox et al. / H o p f bifurcations in Langmuir circulations

236

of the basic state is to oscillatory rather than steady convection for disturbances of w a v e n u m b e r k ( S D ( k ) depends not only on k, but also on ~- and A). The results are displayed in tables 1 - 6 for A = 0, 2

and 3 respectively. Tables 1, 3, 5 give the data for w a v e n u m b e r k = 1"rr, and tables 2, 4, 6 for w a v e n u m b e r k = ~. Table 1 shows that for )t = 0 and k = x'rr criteria (27) and (28) are both violated, so that neither T W nor SW are stable for the range of S investigated. For k = rr, however, table 2 shows that (27) is n o w satisfied, and stable TWs exist for s D ( ' r r ) < S < 3 0 0 0 , where for A = 0 S°(w)~72.01. At S HH~- 1 0 7 8 . 8 7 8 - the value of S for which a multiple H o p f bifurcation occurs involving k = l~r and k = "rr- no stable oscillatory states are possible for k = ½rr but k = "rr is a stable TW. It is important to note that for S < S RH the k = ~r m o d e is not expected to be realizable because the basic state is linearly unstable to the k = ½"rrmode. For A --- 2 and 2, tables 3, 4 and 5, 6 show that the k = ½"rrm o d e can n o w bifurcate supercritically as a T W for S > So(A), where computations reveal that S¢(2)~-2425 > S HH and Sc(3 ) ~ 9 3 6 . 4 < S HH. The k = -rr T W bifurcates supercritically for all S. Therefore, when )t = 2 the only realizable state is expected Table 1 Coefficients of (25), and to for k = ½~r, ;t = 0, z ~ 0 . 1 5 , S

R 450

887.40

as functions of S and R, at the Hopf bifurcation point.

~o

~

~

/3 - a

u+/3

t~/7

0.677

5.420

-9.421

- 14.84

-4.001

0.0046

500

945.11

0.943

5.437

- 9.500

- 14.94

- 4.063

0.0046

1000

1522.27

2.291

5.618

-9.768

- 15.39

-4.150

0.0046

1078.878

1613.31

2.438

5.643

-9.759

- 15.40

-4.116

0.0046

1500

2099.28

3.115

5.808

- 9.633

- 15.49

- 3.825

0.0046

3000

3827.37

4.885

6.387

- 8.749

- 15.14

- 2.362

0.0045

~/7

Table 2 A s for table 1 but for k = ~r.

S

R

o~

a

/3

/3-a

a+/3

75

744.29

0.601

-0.831

-37.06

-36.23

-37.89

0.006

100

766.30

1.840

-0.834

-28.98

-28.14

-29.81

0.006

150

810.27

3.075

-0.830

-20.71

- 19.88

-21.55

0.006

350

985.42

5.826

-0.854

-10.81

-

-11.66

0.006

500

1115.98

7.249

-0.857

-

8.448

-

7.591

-

9.305

0.006

600

1202.64

8.067

-0.856

-

7.507

-

6.651

-

8.363

0.006

9.952

1000

1546.21

10.77

-0.840

-

5.548

-

4.708

-

6.388

0.006

1078.878

1613.31

11.24

-0.839

-

5.317

-

4.479

-

6.156

0.006

3000

3191.37

19.80

-0.871

-

3.179

-

2.308

-

4.050

0.005

Table 3 A s for table 1 but for )t = 2. S

R

to

a

/3

/3 - a

a +/3

/z/7

500

4753.81

0.559

3.661

- 17.70

-21.36

- 14.04

0.004

750 1000

6160.13 754'9.34

1.364 1.789

3.280 2.878

- 18.07 - 20.49

-21.35 - 23.37

- 14.79 - 17.62

0.004 0.004

2000

12950.4

2.580

0.994

- 34.47

- 35.64

- 33.47

0.004

2425

15176.0

2.715

0.100

-41.46

-41.56

-41.36

0.004

2500

15564.8

2.731

-0.175

-42.71

-42.54

-42.89

0.004

3000

18128.2

2.792

-0.145

-50.89

-50.74

-50.03

0.004

S.M. Cox et al. / H o o f bifurcations in Langmuir circulations

237

Table 4 As for table 2 but for A = 2. S

R

to

a

/3

/3-a

a+/3

Iz/F

500 1000 1500 3000

5609.49 7822.30 10049.1 16665.1

6.652 9.552 11.45 14.53

- 3.339 - 5.240 - 7.359 -11.73

-14.50 -13.76 -15.13 -25.65

-11.16 - 8.515 - 7.768 -13.92

-17.84 -19.00 -22.49 -37.37

0.006 0.005 0.005 0.005

Table 5 As for table 1 but for A = 3. S

R

~o

550 600 930 936.4 1000 1334.09 1500 3000

8900.74 9387.64 12544.9 12605.2 13202.6 16283.0 17785.5 30552.7

0.361 0.609 1.284 1.292 1.362 1.601 1.666 1.308

a

-

/3

/3-a

a+/3

t~/~

1.860 1.645 0.035 0.0001 0.355 2.587 3.667 15.99

-28.66 -27.39 -30.89 -31.02 -32.33 -40.14 -44.40 -90.75

-30.52 ~-29.04 -30.93 -31.02 -31.98 -37.56 -40.73 -74.76

-

0.004 0.004 0.004 0.004 0.004 0.004 0.004 0.003

/3

/3-a

a+/3

~/;

-20.75 -24.45 -29.02 - 31.55 -53.19

-15.42 -15.97 - 18.62 - 20.67 -48.41

-26.08 -32.93 - 39.42 - 42.43 -57.97

0.005 0.005 0.005 0.005 0.003

26.80 25.75 30.86 31.02 32.69 42.73 48.07 106.7

Table 6 As for table 2 but for A = 3. S

R

oJ

a

500 1000 1334.09 1500 3000

9903.06 13760.0 16283.0 17532.1 27822.4

6.181 8.649 9.684 10.09 12.24

-

t o b e t h e k = ar T W f o r S k = ' r r T W f o r S > S rill. The flux-free problem i. =

5.326 8.478 10.40 10.88 4.781

> S HH, b u t w h e n A = 3 t h e ~ r T W is r e a l i z a b l e f o r S c ( 3 ) < S < S Hn, a n d t h e

can be reconstructed

from the periodic problem

+

(29)

T h e s t a b i l i t y o f t h e o s c i l l a t o r y s t a t e s is t h e f f o r e d e t e r m i n e d bifurcation

b y s e t t i n g r~ = r 2 = r t o g i v e

by the sign of a +/3. When

is s u b c r i t i c a l ( a n d t h e S W is u n s t a b l e ) w h i l e f o r a + / 3 < 0 t h e b i f u r c a t i o n

( a n d t h e S W s t a b l e ) . T h e v a l u e o f or + / 3 is g i v e n i n t h e p e n u l t i m a t e

a +/3 > 0 the is s u p e r c r i t i c a l

columns of tables 1-6. We see that

b o t h k = ½~r a n d k

ar S W s a r e s t a b l e f o r all t h e e n t r i e s ( s e e a l s o [21] f o r a d i s c u s s i o n o f h = 0 ) . T h i s

conclusion, however,

depends on the enforcement

periodic boundary

of flux-free boundary

c o n d i t i o n s , a n d is n o t t r u e f o r

conditions.

5. Two interacting Hopf bifurcations with 0(2) symmetry In section simultaneously

2 we

saw that

when

the lines H1

and

H2

intersect,

two

destabilized. This constitutes a codimension-two-double-Hopf

oscillatory modes

become

bifurcation which we now

S.M. Cox et al. / Hopf bifurcations in Langmuir circulations

238

analyse. In order to unfold this bifurcation, we must now allow two parameters to vary independently, and so we set (S, R) = (S nil, R HH) + e2(~", F). Otherwise, the analysis proceeds as before, using multiple scales. This procedure automatically yields a reduced equation set in Birkhoff normal form, while a centre manifold approach requires subsequent coordinate transformations. Again, for our problem it is found that retention of cubic terms suffices to determine the dynamics; there is no need to extend the analysis to quintic terms. (Although, should this have been necessary, it appears to us that the method of multiple scales could again have been used, and that it again would recover the same reduction that a centre manifold-normal form analysis would produce.) The solution to the linear stability problem is now a combination of modes: [I//, /.~, 01 = [4/1' /'ll '

~}1]a1(T) ei'/" + [-"q*l,* ul,** t}~'] a2(T ) e i4"2 (30)

+[q'2, fie, 02)a3(T) e'~3 + [-4*5, ~ , 0~] a . ( r ) e i*" + ( * ) ,

where ~b1 = k l y + tolt, qb2 = k l y - tOlt, ~')3 = k2y + t°2t, q~4 = kzY - t°2t, and the frequencies to1, 602 have no low-order resonance. Here, k 1 = l"rr, k 2 = rr, and [~bj, fij, 0j] ( j = 1, 2) are the eigenfunctions of (11), (12). As in section 3 we seek a solution as a power series expansion in e (given by (18) where now [qj(1), uO), 0(1)] is given by (30)). Substituting (18) into (4), we obtain a system of inhomogeneous equations for the ~(E 2) and ff(e 3) problems. Their solution results in a coupled set of four nonlinear equations for the a t ( T ) which, because of the 0 ( 2 ) symmetry, take the form [10] a

2

a

2

,

dl = alg~(laal 2, ] 21, I 3l, 1a412),

G = a3gz(la, I2, la212, la

2

a2 = a2gl(la2l, la,l 2, laal 2, la312),

l z, 1a412), d4 a4g (I a 21,2 la,l 2, la412, la 12),

(31)

=

for some complex functions & ( j = 1, 2). The & can be found as in section 3 by numerically solving the If(e) and 6(e z) problems, in practice using computer algebra (see appendix), and applying the Fredholm alternative to the ~ ( 3 ) problem. Writing aj = Q e i~j as before, the modulus equations become, to cubic order, rl =

r 2

r 2

rl(J[/'l + Plrl + Qlr2 r 2

r 2

+

Ulr3

r

2

r 2 -t- V l r 4 )

r 2

r 2

,

?:3 = r 3 ( / - / ' 2 + Pzrl + Q2r2 + U2r3 + V2r4) ,

r2 =

r 2

r

2

r 2

r

2

r2(/~l -k- Q~r~ + P~r 2 + V l r 3 +

1;4 = r 4 ( / 2 2

r 2

Qzrl + Pzr2 + V2r3 +

+

r

2

r

2

U1F4) , U2r4)

-

(32)

r r r r Table 7 shows the coefficients P~, Q j, U~, Vj ( j = 1, 2). We define

Table

7

Coefficients

of the normal

form for the amplitudes

A

P~

Q~

U~

0

5.643

-

9.759

1

4.980

-12.84

2.525

0.009

-32.09

3

-2.587

-40.14

4

-7.842

-60.25

8.734

rl_ 4. V~

P2

Q~

-

11.96

2.581

-

10.12

-

16.60

1.810

32.35

-

55.78

0.938

-

80.95

51.33 131.1

- 171.3

1.103 -0.295

U~ 9.567

v~

-

0.839

-

5.317

-13.15

-

2.193

-

7.036

-30.19

-

8.786

-21.41

-32.56

-10.40

-29.02

-28.54

- 11.71

-44.86

S.M. Cox et al. / Hopf bifurcations in Langmuir circulations

239

Table 8 P a r a m e t e r values at the double-Hopf bifurcation point, (S an, R a n ) . A

R mt

Snn

P'I

0 1 2.525 3 4

1613.307 8998.48 12576.9 16283.0 25336.0

1078.878 1263.780 1313.59 1334.09 1324.97

0.0046~'0.0045~'0.0040~'0.0040~0.0035~'-

r

otj=Pj,

r

/3j=Qj,

r

3'j=Uj,

8j=Vj

r

/% 0.0053k" 0.0052~" 0.0047k" 0.0047~" 0.0043~"

0.0056~'0.0056~'0.0050~'0.0047~'0.0043~'-

0.0048g" 0.0049~" 0.0050~ 0.0050~ 0.01M7~"

(j=l,2),

and note that a 1,/31, 3'2, 82 are known from the bifurcation analysis for a single Hopf: a~ = a,/31 =/3 for k = ½,tr and 3'2 = a, 82 =/3 for k - "tr from (25). Table 8 shows the values of S mI, R Hn,/~1 and ~ at the double-Hopf bifurcation as a function of A. In the following two sections we discuss the dynamics of (32) for both the flux-free problem and the periodic problem.

6. Flux-free boundary conditions Moroz and Leibovich [22] computed the normal form equations for a double-Hopf bifurcation for an ad hoc approximation to the true eigenfunctions and demonstrated that stable periodic and quasiperiodic behaviour could be found close to the multiple bifurcation. This section extends this earlier study to the case of A ~ 0 and for the exact eigenfunctions, for the case of flux-free lateral boundary conditions. With r 1 = r 2 and r 3 ---- r 4 (32) becomes /:1 : rl[/d'l d- (orI q-/3,)r 2 + (3,1 + 81)r21,

1:3 = r3[/z2 + (a2 +/32) r2 + (3,2 + 82)r~1 •

(33)

Table 9 shows the values of the nonlinear coefficients appearing in (33) for various values of A. We note that all the coefficients are negative. Following Moroz and Leibovich [22] we rescale (34)

R3 = r31/-(3,2 + 82)

R , = r l ~ / - ( o q +/31),

Table 9 Coefficients of (33) as functions of A. A

al +/31

3'1 + 81

a2 +/32

"Y2+ 62

0 1 2.525 3 4

- 4.116 - 7.86 -32.08 -42.73 -68.09

- 3.226 - 6.48 -23.43 -29.62 -40.2

- 6.986 -11.34 -29.25 -31.46 -28.84

- 6.156 - 9.229 -30.20 -39.42 -56.57

240

s . M . Cox et al. / H o p f bifurcations in Langmuir circulations

to obtain /~1 =

Rx[/xl - R 2 + b R ~ ] ,

R~I,

(35)

c = -(o~ 2 +/32)/(~ a + / 3 , ) .

(36)

I~ 3 =

R3[/x2 + c R ~ -

where b = -(Yl + 61)/(Y2 + $2),

F r o m table 10 we see that b < - 0 . 5 and c < 0. There are four equilibrium states of (35): (a) Rest state (Rle , R3e ) --- (0, 0); (b) Pure SW,,/z (Rle, R3e ) = (x/--~, 0); (c) Pure SW,, (RI~, g3e )

~---

(0, V ~ ) ;

(d) (SW~/2,SW~)-mixed state (Rl~, R3~)=

---~c

/

'

-(--bc

]

"

H e r e the subscript 'e' denotes an equilibrium value. The rest state (0, 0) exists for all values of (/x 1, tz2), but is stable only for /~1, /xz < 0. SW~/2 exists for /x, > 0 with stability determined by the eigenvalues - 2 / % , t~2 + c ~ . Since -2/-h < 0 we r e q u i r e / z 2 + c/x~ < 0 for a stable SW,,/2. SW~ exists for /~z > 0 with stability given by eigenvalues - 2 / % , /z1 + b/x2; since - 2 / z 2 < 0 /x1 + bb~2 < 0 ensures the stability of SW,,. The stability of the mixed mode (R~e, R3e ) is determined by the eigenvalues A+_ = -(R2~ + R3Z~) -+ [(R~e + R~e) 2 - 4(1

-

(38)

b C ) R l 2~ R 3 2e ] 1 / 2 .

If 1 - b c > 0 then both eigenvalues have negative real parts and the mixed mode state is stable, existing for /x1 + b/x:,/x 2 + c/xI > 0. We observe stable quasi-periodic motion when the phase variation is incorporated. If 1 - b c < 0 then the mixed mode state is a saddle, existing f o r / x I + b/z 2,/x 2 + c/xI < 0. F r o m table 10 we see that both scenarios are possible, and with degeneracies appearing at cubic order for one value of A between 0 and 1, and for another value between 1 and 2.525. Figs. 2 and 3 depict the stability regions in unfolding space for both scenarios. When 1 - b c > 0 fig. 2 shows the existence of unique stability regions for each of the modes, whereas for 1 - b c < 0 fig. 3 shows that there is hysteresis. Either SW,~/2 or SW~ is possible when the mixed mode state is a saddle, depending on the initial state.

Table 10 Coefficients of (35) as functions of A. A

b

c

1 - bc

0 1 2.525 3 4

-0.524 -0.702 -0.776 -0.751 -0.711

-1.697 - 1.443 -0.912 -0.736 -0.424

0.111 -0.013 0.292 0.447 0.699

S.M. Cox et al. / H o p f bifurcations in Langmuir circulations

/ P'2

b IJ-2 + IJ. 1 = 0

. g 2 + el21 = 0

/

P-2 l.t2 + c p.

J

241

=0

b l.t2 + l.tI = 0

J ~tl

Fig. 2. Unfolding diagram for 1 - bc > 0, showing equilibria

)

Fig. 3. A s in fig. 2, but for 1 - bc < O.

and qualitative phase portraits. The arrow shows the sequence explored in section 8.

7. Periodic boundary conditions For the case of periodic boundary conditions the equilibrium states of a generalization of (32) have been classified by Chossat et al. [1] in terms of the isotropy subgroups, by identifying the maximal symmetry group of each subsystem of equations derived from (32), thereby determining the associated equilibrium solution(s) and so the most complicated temporal evolution which can be supported by that subsystem. This classification includes a degenerate quasi-periodic (rl, r2, 0, 0) state with r 1 # r2, as well as cases which cannot be distinguished by cubic-order normal forms, such as (rl, ---r~, r3, -----r3)states, or states which do not even exist, such as (r 1, r2, r3, r4) with all the rj distinct. The translation of this information for the Langmuir problem to cubic order is displayed in tables 11, 12, which we now Table 11 Primary equilibrium states and their eigenvalues. Case

Equilibrium state

Eigenvalues

Identification in

(rl e, r2e, r3 e, r4e)

A 1, A 2, A3, A 4

t h e full system

1

(0, 0, 0, 0)

/ ~ , #1,/~2, ~

rest state

2(a)

(~r-s-~l/al, 0, 0, 0)

- 2 / z l , / z ~ ( 1 - fl~/a~),

k = ½~r travelling wave

~[L2 -- Ot2~IA'1/O/1' /J"2 -- ~2~'~1/0/1

2(b)

(o, V-:--~,/~,, 0, o)

3

( V - m/(,~, + #,), V-~,/(,~, + ~), o, o)

- 2 / x 1, 2 / ~ 1 ( ~ 1

- ot,)/(~,

+ or1) ,

k = ½~r standing wave

/% - / x , ( a 2 + [32)/(a 1 + [31) (twice)

4(a)

(o, o, V:-~/~,2, o)

4(b)

(0, 0, 0, V - u~/v2)

5

(o, o, V- ~/(v2 + aO, V-~/(v2

- 2 / z 2 , / x 2 ( 1 - 82/,),2) ,

k = ~r travelling wave

~'1 -- '~1~1"~2/'Y2' ~L1 -- ~l ~'2/'Y2

+

a2))

-2~,2, 2re(a2 - v2)/(a2 + v2), It1 - t~(al + vl)/(a2 + 3'2) (twice)

k = ~r standing wave

S.M. Cox et al. / H o p f bifurcations in Langmuir circulations

242

Table 12 Secondary equilibrium states and their eigenvalues. Case

Equilibrium state (rle, rze, r3e, r4e )

Eigenvalues A~, A2, A3, A 4

Identification in the full system

6

\--~1~% -- °tzYl

( ~ p'2%-/x'T2-- ,0,

g , + / 3 t r , + 2 3,r3¢ ,2

two-torus formed from travelling waves for k = ½~r and k = ~r

]£20~1,

%

Ot2.yt [(a~r2~ + T2r23~)2 -- 4Armor,o] "~

0261 , 0, 0,

~1 + ~lrl2e "~- YlrZa,,

two-torus formed from travelling

Ix2 + azr~o + 6zrZ4e,

waves for k = ½Ir and k = ~r

(alr~e + y2r24e) ±

a~y2 B26~ /

[(alr~o + y2r24,)2 - 4Br~or~o]''2

(rl e, +rle, r3~, ±r3 e)

not computed,

two-torus formed from standing

where

A~A2A3A 4 < 0

waves for k = 21xr and k = rr

~1(~2 + ~2) --J~2(~1 + ~1) r.o = /~/(.,+~,)(~+~0_(.~+~0(% +~,)

summarize. This introduction of specific values corresponding to the physical system of interest greatly simplifies the Chossat et al. [1] zoology. The stability of each equilibrium state is obtained by considering exponentially growing or decaying solutions to ~# = # ~ r ,

(39)

where [.1,1 + 3ot, r~e +/3,r2~

2/3lraer2e

2Tlraer3e

261rler4e

tx 1 +/31r21~ + 3alr~e + 6,r23~ + ylr]¢ 2/32rzer3e

261rzer3e

2Ylrzer4e

2 + /32r~e P'2 + ctzrle

262r3~r4 ~

+ Tlr~ + ~lr2e 2/31rler2e

= 2Ot2rleg3e

+ 3"Y2r~e + 62rZe

2/32rl~r4e

2azrzcr4~

262r3er4~

2 /~2 + /32r]e + °tzrze

+ ~2r~o + 3r2r]~ _

(40)

and ~r = (~rl, ~r2, ~r3, ~ r 4 ) T represents perturbations to r = ( r l , r e, r3, r4) T from the equilibrium value r e. For ~rj oce At, (39) yields the quartic characteristic equation det[AI- #] =0, which may be solved for the eigenvalues Aj ( j = 1 , . . . , 4 ) . If R e ( A j ) < 0 for each j = 1 , . . . , 4 equilibrium state is stable, but if R e ( A , , ) > 0 for some m the solution is unstable.

(41) the

S.M. Cox et al. / Hopf bifurcations in Langmuir circulations

243

We now use (41) in conjunction with information on the coefficients of (32) to determine the stability of each equilibrium solution of (32). For all values of A investigated, ~1, f12, 3"2, ~1' (~2 < 0

However, for 0 < h < h

and

3"1> 0 .

(42)

a (where Aa~2.525 )

a l , a2 > 0 ,

(43)

for Aa < A < Ab (where Ab ~ 3.8) O~1 ( 0 ,

Of2 > 0 ,

while for A >

(44)

Ab

a l , a2 < 0 .

(45)

We therefore need to investigate three ranges of A. T h e asymptotic behaviour of solutions of (32) can be divided into a consideration of primary, secondary and tertiary branches of solutions. Chossat et al. [1] show that these branches are connected, and the primary branches (single-frequency motions comprising TWs or SWs) branch first from the rest state, secondary branches (with two-frequency mixed modes) branch from solutions on the primary branch, and tertiary branches (three independent frequencies) from secondary branches. For the sake of convenience, we group discussion of the stability of the rest state with the primary branches.

7.1. Primary branches Case 1. (r~, r 2, r3, r4) = (0, O, O, 0). From table 11 the rest state exists for all choices of ~1 and/z2, but is stable only when both/~1,/~2 < 0. Case 2. ( r l , O, O, O) or (0, r2, O, 0). The T W associated with k = ½ax exists provided i~i/a 1 < 0. Because of (43), (44), (45) we need to consider separately the cases a 1 > 0 (so that/~1 < 0) and a 1 < 0 (so that jlJ.1 ~> 0). In the first case the first eigenvalue A 1 > 0 and TWo/2 bifurcates subcritically, and is unstable. In the second case TW,/2 bifurcates supercritically and is stable provided A3, A 4 < 0 . Secondary bifurcations occur when either A 3 o r A 4 vanishes, and are dealt with in the next subsection. Case 3. (rl, r l , O, 0). When r 1 and r 2 are both non-zero, but r 3 = r 4 = 0, we must take r~ = r2, and we obtain a SW associated with k = ½~r. This exists provided/X~/(al +/31) < 0. Since al +/31,/31 - oq < 0 by table 1, we must take/~1 > 0, giving A 2 > 0. SW~/2 therefore bifurcates supercritically but is unstable. T h e vanishing of A 3 = A 4 yields secondary branches. Case 4. (0, O, r3, O) or (0, O, O, r 4 ). When the only nonzero mode is r 3 or r 4 we obtain a T W associated with k = ~r, which exists when I-~2/3"2< 0. Since 3'2 < 0 for all A, we r e q u i r e / z 2 > 0. The T W . bifurcates supercritically and is stable provided A3, A 4 < 0. Secondary mixed travelling mode solutions appear when either A 3 o r A 4 vanishes.

S.M. Cox et al. / Hopf bifurcations in Langmuir circulations

244

Case 5. (0, O, r3, r3). The SW associated with k = ~ r exists provided /22/(~2 + 6 2 ) < 0 , which means, from (42), that ~2 > 0. S W therefore bifurcates supercritically, but since A 2 > 0, it is unstable. Mixed SW solutions arise when A 3 = A 4 = 0. 7.2. Secondary branches Table 12 summarizes the secondary branches. Case 6. (r~, O, r 3, O) or (0, re, O, r4). This mixed mode T W state exists provided A ~- OtlT2 -

a2"Yl > 0 ,

]J~2"Y1 -

~/q ")/2 > 0 ,

]./q 0/2 -

jLb2a l > 0

(46)

holds, or (46) with all the inequalities reversed. For 0 < A < Ac ~ 4, A < 0, while for A > Ac, A > 0 so both cases require investigation. The analysis of the primary modes shows that (rj, 0, r3, 0) bifurcates from the TW~ mode on the line/z2y ~ - / x ~ y 2 = 0 and from the TWo/2 mode on the line tz~a 2 - tz2a ~ = O. When A < 0, A~ > 0 and the mixed mode is unstable, but when A > 0 we have stability provided A3, A 4 < 0. For A < 0 there is a considerable region in (/~,/x2)-space for which no stable states exist, while this region is replaced by one where a mixed mode state is stable for A > 0. Case 7. ( r l , O, O, r 4) or (0, r2, r 3 , 0 ) . Case 7 also represents a mixed T W state, but leads to different dynamics. For all values of A investigated case 7 exists provided B~a,

y z - /326~ < O ,

tzz6~- th6z
/.q/32-/22/3,<0.

(47)

The first inequality holds for all values of h we have investigated. However, the eigenvalue A~ > 0 so that this mixed-mode T W state is always unstable. Moreover, hysteresis is now possible for ot~1 <~ 0 (i.e., for h > A~ ~ 2.525). Case 8. ( r , , +--r1, r3, +--r3). When r~ = r~ and r~ = r 2 the fixed points of (32) admit three solutions: (r 1, r x, r3, r3), (rl, - r t , r3, r3), (r l, rt, r3, - r 3 ) . To cubic order in the normal form expansion, the three solutions have identical characteristics, and so are investigated together. Bifurcations occur on /x2(y~ + 61) =/.h(~/2 + 62) to a SW,, mode and o n / z ~ ( a 2 + 132) =/a~(a~ +/31) to a SW~/2 mode; both of these primary branches are unstable. An explicit evaluation of the four eigenvalues determining the stability of the mixed SW state is a formidable undertaking and has not been done. Instead we have computed the product of the eigenvalues for 0 < h ~ 4. We find that A~A2A3A 4 < 0 so that either one or three of the eigenvalues are positive, yielding an unstable mixed SW state. The only stable two-frequency mixed-mode state is therefore case 6 for h-> 4, since mixed modes correspond to quasi-periodic flows when the phase information is restored. 7.3. Tertiary branches We conclude our bifurcation analysis with a discussion of the tertiary branches states of (32). Case 9. ( r l , r2, O, r4), plus three others. As a typical three-mode state we shall consider the (r l, r z, O, r4)-mode; the other three follow in an analogous manner.

S.M. Cox et al. / H o p f bifurcations in Langmuir circulations

245

Because of the non-degeneracy in the ~(E 3) normal form equations, the only permissible fixed points of the three-mode system that result from setting r 3 t o zero in (32) are (compare with [1]): (0, 0, 0, 0), (r 1, 0, 0, 0), (0, r2, 0, 0), (0, 0, 0, r4), (r 1, rl, 0, 0), (rl, 0, 0, r4), (rl, r E, 0, r4). The fixed point ( r l , rE, 0 , 0) can exist only in degenerate cubic systems [11] and (0, rE, 0 , r 4 ) is precluded. Since we do not require explicit expressions for rx, r 2 and r4, we omit them here. Stability of the tertiary mode is determined from the solutions to the cubic characteristic equation (48)

A 3 + c2 A 2 + C l A + c o = O ,

where c2 =

-2[~1(r] + n 2 2 2r

r 2) + ~/2r24], c~ = -4[(/321 z

-- O t2, l ) r l2r 2 2 -- B r l2r 42 - A r 22r 42] ,

2

Co = --srlrEr4lTEtOtl -- fl~) + Yl(fllfl2 -- °tla2) + ~l(fll °t2

°tlfl2)] •

(49)

As in case 8, we do not explicitly determine the eigenvalues but instead examine the sign of c o to test for stability. Computations show that c o < 0 for all values of A in table 7, so that A1AEA 3 > 0, and at least one eigenvalue is positive. The tertiary mode (r 1, r2, 0,r4) is therefore unstable. Similar results obtain for the other three-mode systems. Consequently, no stable three-frequency flow is possible. Case 10. Full system. Finally we consider the full system (r~, r2, r3, r4), which, according to Chossat et al. [1], corresponds to a four-frequency flow. A simple calculation shows that such a mode can exist only provided (Or2 -- J~2)('Yl -- ~1) = (0~1 -- J~l)(~/2 -- ~ 2 ) '

(50)

which is not satisfied for any of the values of A investigated. Case 10 therefore reduces to case 8 for eq. (32), namely an unstable mixed-mode SW state.

8. Comparison of partial differential equations with weakly nonlinear theory In this section we compare the predictions of the weakly nonlinear theory described above with numerical integrations of the full partial differential equations for A = O; we also re-interpret the unfolding parameters in terms of physical parameters. 8.1. Flux-free problem. Single H o p f bifurcation

For S = 350 and k = ar (L = 2), table 13 shows the comparison of Nusselt numbers and frequencies between the integrations of the partial differential equations and the weakly nonlinear theory. The Nusselt number is given by

(0 ,y 0 t,) where the angle brackets indicate a horizontal (y) average, and represents the dimensionless heat flux through the upper surface, z = 0. The agreement is seen to be remarkably good, but with the weakly nonlinear theory underpredicting the Nusselt number.

S.M. Cox et al. / Hopf bifurcations in Langmuir circulations

246

Table 13 Results of numerical integrations of the partial differential equations with flux-free boundary conditions, and corresponding weakly nonlinear (w.n.) predictions. Here the solution is the SW for k = "rr at S = 350.0, for which the H o p f bifurcation occurs at R H = 985.42. The frequency of the motions is to, and the Nusselt number is Nu. The maximum and minimum values of Nu over the SW cycle are denoted by NUmax and NUm~,, respectively; Nu - 1 denotes the time average of Nu - 1 over the SW cycle. R

987 990 995

Nu - 1

NUma x -

Nu,.~.

w

numerical

w.n.

numerical

w.n.

numerical

w.n.

0.0076 O.0249 0.0488

0.0074 O.0216 0.0452

0.0155 O.0499 0.0976

0.0153 O.0444 0.0928

5.818 5.764 5.712

5.814 5.791 5.753

The structure of the SW is of some interest. Knobloch and Moore [14] have shown that there are linear SW for binary fluid convection problems, under appropriate boundary conditions, for which there is no instant in time at which the fluid is completely at rest. Further, they showed in a "snapshot" of the cell at a particular time near the changeover between clockwise and counter-clockwise rotation that the cell may split into two or three vertically stacked cells. Sch6pf and Z i m m e r m a n n [24] analysed the linear SW in more detail, and illustrated the passage of each convection cell through one, three and two vertically stacked cells, as the direction of convection in the cell reversed. Each reversal took approximately five percent of the total period of the SW. The linear theory described by Knobloch and Moore, and by Sch6pf and Z i m m e r m a n n predicts that the boundaries between the stacked cells are exactly horizontal. Our weakly nonlinear expansion shows how these horizontal boundaries become tilted, and we have compared our weakly nonlinear predictions with direct numerical integrations of the partial differential equations describing Langmuir circulations. As noted by Knobloch and Moore [14], the cell-splitting occurs near the minimum of the kinetic energy, but there is also a re-adjustment of the thermal structure, which occurs at a different time, near the minimum of the Nusselt number. Here, the thermal plumes reverse vertical direction. This reversal takes place during a small interval of time when the convective motions are least efficient at heat transport. Indeed, the convection is briefly less efficient at heat transport than the pure conduction solution, so that N u - 1 becomes negative. For example, both weakly nonlinear theory and direct numerical simulation predict that, close to the H o p f bifurcation point, Nu - 1 becomes negative for a small part of the cycle at S = 350, with weakly nonlinear theory giving Nu - 1 = 0.0047(1 + 1.027 sin 2oJt)(R - R H) + 6((R - RH)2). Further from the bifurcation point, numerical integrations show that N u - 1 remains positive throughout the SW cycle. For R = 1160, Fig. 4 shows the sequence of streamline contours at various times in the SW cycle, for the weakly nonlinear predictions. Also shown is the variation of the kinetic energy and the Nusselt number with time over one period of the SW cycle. Fig. 5 shows the similar sequence for the numerical integrations. Note that in this comparison, R - R H ~ 175, so good quantitative agreement is not to be expected. For S D (½-rr)< S < S m~ and with L = 4 we expect and we observe SW~/:.

8.2. Flux-free problem. Double-Hopf bifurcation We now translate the unfolding pictures in figs. 2 and 3 for the double-Hopf bifurcation involving k = ½w and k = w into physical parameter space.

247

S.M. Cox et al. / H o p f bifurcations in Langmuir circulations

I@@1

HH

@@ CZZ (a)

(a)

0.3-

2-

E°Z°

0.2-

0.1-

0

t

0.5

1.0

(b)

C

! 0.5

1.0

(b)

Fig. 4. (a) Streamline sequence from weakly nonlinear theory over approximately half of a cycle for SW at S = 350, R---1160. The figures shown are not at equal intervals of time. Setting the origin of time to be at a minimum in the (cross-stream) kinetic energy, the figures in the panel occur at t / ~ = (-0.1, -0.005, -0.003, -0.002, -0.001, 0, 0.001, 0.004, 0.1, 0.5), where ff is the period. (b) Nusselt number (-1) and (cross-stream) kinetic energy as functions of time.

Fig. 5. As in fig. 4, but computed directly from the partial differential equations. (a) The times for each figure in the panel differ from those of fig. 4a, and are given by t / ~ = (-0.004, -0.003, -0.0009, -0.0006, -0.0003, 0, 0.0014, 0.0035, 0.0058, 0.5), where ~- is the period. (b) As in fig. 4b.

W h e n h = 0 the r e l a t i o n b e t w e e n the u n f o l d i n g a n d the physical p a r a m e t e r s is /" = - 6 0 8 / . q + 677p~2 ,

~'= - 7 1 5 / ~ + 586/%.

(51)

T h e l i n e a r stability b o u n d a r i e s P-1 = 0 a n d p~ = 0 b e c o m e ? = 1.154~" a n d ? = 0.8505~', respectively. T h e line bp,2 +/xl. --- 0 c o r r e s p o n d s to ~'= 1.69~', a n d the l i n e / z : + c/x1 = 0 c o r r e s p o n d s to ? = 1.93~', so t h a t

248

S.M. Cox et al. / Hopf bifurcations in Langmuir circulations

the mixed mode (SW~/2, SW,~) occurs in the wedge 1.69J'< F < 1.93£'. Fig. 6 gives the linear and nonlinear stability boundaries in the (J', F) plane for the various nonlinear oscillatory modes. We have chosen to explore the outcomes of numerical experiments at a sequence of F points along the path shown in fig. 6 (which corresponds to that in fig. 2) for ~'= 10, which is slightly above the value of S = S nn. This gives some prospect for comparison with weakly nonlinear theory. In all the numerical simulations close to the double-Hopf point, appropriate initial conditions were provided by the tT(e) and tT(e2) components of the velocity and temperature fields, derived by our weakly nonlinear analysis. At F= 11.7, weakly nonlinear theory predicts a stable SW~ mode with frequency to = 11.28, and the numerical experiments find the same behaviour with to = 11.27. Fig. 7 shows the time trace of the Nusselt number for this SW. Between F= 16.9 and 19.3, weakly nonlinear theory predicts a stable mixed mode. Note that the range in R for which this is predicted is very small, from R = 1630.2 to 1632.6, forming a wedge of about 3° centred on (S un, RHH). At F= 17.7, in the middle of this wedge, numerical ~imulations show what first appears to be a mixed mode, but when integrated for a sufficiently long time (250 000 time steps, amounting to 625 units of time or about 240 cycles of the slowest mode), the ½"rr contribution slowly decays, leaving a SW,, mode. This led us to search for a mixed mode at larger F, recognizing that the domain for mixed modes for the partial differential equation, if it is not vanishingly small, may be a distorted version of the weakly nonlinear wedge. At F= 19.3, which is at the top border of the weakly nonlinear wedge, the numerical simulations show two-frequency flow, with a slow decrease of the SW,,/2 contribution for first 400 time units and then an increase in the subsequent 1000 time units. The time trace for N u - 1 in fig. 8 clearly shows a mixed SW state, with frequencies to~/2 ~ 2 . 5 and to=-~ 11.3. This flow is slowly continuing to evolve, however, and after the solution has been tracked for 560 000 time steps, the trend leads us to believe that eventually it will end up as a single-frequency SW,,/2 mode.

la.2+cl.t~-,,,x~ I --0 b~t2+l.t I =0 ~"-,~:,...,,

., ~t ~

=

o

.::.~?.:.:...~-~.,.'..,;.::

:;'~,;~"

'g

2= 0

~..'..',":~_';i

~,~,.,:.,,.,.,

sw,, ~i';:i[ mixedmode

sw~2 )_

V Fig. 6. Unfoldingdiagram translated to (~',F) parameter space for the case of the flux-freeproblem.

S.M. Cox et al. / Hopf bifurcations in Langmuir circulations

249

NU-I

NU-I

0.080.060.040.02Q

•

•

•

0

Jt

IV 1

Fig. 7. Time trace of Nu - 1 for the SW~ mode at ~'= 11.7. Note that the Nusselt number is less than 1 for part of the SW cycle.

0 882.5

885.0

887.5

890.0

Fig. 8. Time trace of N u - 1 for the quasi-stable (SW~, SW~2) mode at 7 = 19.3. Note that the Nusselt number is less than 1 for part of the SW cycle.

Fig. 9 shows a phase portrait in the (w, 0) plane measured at point ( y , z ) = ( 0 , - ½ ) , and the corresponding Poincar6 section, defined for crossings of N u - 1 = 0.04. (Here, w =-O~b/Oy is the vertical velocity.) There may be stable mixed-mode (quasiperiodic) states somewhere in the range ~'= 17.7 to 19.3 for ~'= 10 as predicted by weakly nonlinear theory, but we have not explored this further. We think it fair to say, however, that the simulations done, while not definitively confirming the weakly nonlinear predictions of stable mixed modes, certainly indicate the presence of mixed modes that are at least quasi-stable. At ~'= 21.7, where the weakly nonlinear theory predicts a stable SW~/2 mode with to = 2.23, the numerical simulations agree with to = 2.18. Fig. 10 shows the time trace of the Nusselt number for this SW. Finally, at 7 = 36.7, where weakly nonlinear theory continues to predict a stable SW~/2 mode. we are able to find only a ½xr steady state, with Nu - 1 = 2.73. Our results are in marked contrast to [18]. For example, they report calculations at S = 1200, in an investigation of the behaviour near the double-Hopf point. In a traverse with R increasing, they first find stable SW~ modes, then a frequency-locked periodic solution, then a quasi-periodic state analogous to our mixed modes, then a SW~/2 mode, and then finally a k = ½~r steady state. Although this sounds rather like the possibilities we would have expected (except for the frequency-locked state), in fact all of (a)

(b)

W

1~, \ . . . . ~ . ~..... :::-

W

0.5-

-ol.1 -0!05 -1

0.105

0!1

theta

-0.5-

I -0.2

I -O.1

I 0

I 0.1

I 0.2

-1-

.:::-~;~ ........

thetl:a Fig. 9. (a) Phase portrait in the (w, 0) plane, where w and 0 are the vertical velocity and the perturbation temperature, measured at the point (y, z) = (0, - ½). Here, ~'= 19.3. (b) Poincar6 section of the torus in (a), with points plotted whenever Nu - 1 takes the value 0.04.

250

S.M. Cox et al. / Hopf bifurcations in Langmuir circulations Nu-I 0.2-

0.i-

I

0

o

V

V

V

: r

V

V

VA

Fig. 10. T i m e trace of Nu - 1 for the SW~/2 m o d e at F = 21.7. Note that the Nusselt n u m b e r is less than 1 for part of the SW cycle.

the time-dependent solutions that they report were found below the wedge in which weakly nonlinear theory predicts mixed modes, in fact, below the line /x~ = 0 in fig. 6. Numerical simulation results for flux-free conditions at S = 3000 are given in [2], and we refer to that paper for details. We note here simply that the complex sequence of bifurcations on two coexisting stable branches previously reported in corresponding calculations by Leibovich et al. [18] is not found. Instead, the possibilities found consist of stable SW=/2 and SW~ modes, including a band of R for which both are stable.

8.3. Periodic problem The picture with periodic boundary conditions is somewhat simpler. For a single Hopf bifurcation of the k = 7r mode at S = 350, the weakly nonlinear analysis predicts at R = 987 that Nu - 1 = 0.0509, while the partial differential equations give Nu - 1 = 0.0510. For R = 990 the corresponding values are respectively 0.1475 and 0.1242. Thereafter the values diverge, with the weakly nonlinear theory overpedicting Nu - 1. For example, at R = 1000 we obtain 0.4694 from weakly nonlinear theory, and 0.3419 from numerical simulations. Close to the double-Hopf bifurcation the conditions for the existence and stability of the T W mode become 0.8505~'< F<0.873~', which is a painfully small region in parameter space. Outside of this region there are no stable oscillatory modes predicted by the weakly nonlinear analysis. Indeed, in numerical simulations TW~ is never observed; instead we find the k = ½~r steady state. We presume, while aware that the presumption may prove false, that given sufficient numerical precision and patience, the TW~ could be found.

9. Summary This paper has accomplished the following objective: (1) we have derived the first example of a double-Hopf bifurcation with 0(2) symmetry in a problem in fluid mechanics; (2) we have given the first quantitative comparison of the performance of weakly nonliner theory and numerical solutions of the full partial differential equations in problems having 0(2) symmetry; (3) similar comparisons are given for flux-free boundary conditions; and (4) we have corrected earlier [18] numerical results for Langmuir circulations and its double-diffusion analogue.

S.M. Cox et al. / Hopf bifurcations in Langmuir circulations

251

We find that the weakly nonlinear theory is consistently a good guide, and is quantitatively accurate at points not too distant from the stability margin. Using symbolic computation, weakly nonliner analyses can be carried out economically to give a good qualitative picture of the dynamics of this kind of system over substantial regions of parameter space. The resulting picture will be quantitatively accurate sufficiently close to the stability boundary. More importantly, the weakly nonlinear analyses predict behaviour that may be found by direct simulation only with great difficulty. To the extent that the horizontal boundary conditions used here are capable of modelling Langmuir circulations, the results of this paper suggest that the only robustly observable motions that should be anticipated are travelling waves and steady states. Both of these possibilities in fact have been observed, if one attributes the transverse motion of surface windrows to the existence of travelling waves.

Acknowledgements This work was supported by NSF OCE-9017882, NSF AM-88-14553, and AFOSR-89-0226. This research was conducted using the computational resources of the Cornell Theory Center, which receives major funding from the National Science Foundation and IBM Corporation, with additional support from New York State Science and Technology Foundation and members of the Corporate Research Institute. SL and IMM also benefitted from a NATO Collaborative Research Grant.

Appendix The functions A j, Bj, Dj of (20) satisfy (2ito - I7)2) I52A ~ - 2 i k R " h ( z ) B~ + 2ikSHO~ = - i k W ( ~ , I)~) , (2ito - I)2) B~ - 2ikA~ = -ikW((b, dr), D4A2 = ik[O(6* I36) - D(4, I36")], 7D232 =

(2ito - ~-I)2)O ~ - 2ikA~ = -ikW(~b, 0) ,

3232 = ik[D(6*a) - D(6a*)],

ik[D(6*O) - D(60*)],

(I)2)2A 3 + 2ikRHh(z) B 3 - 2ikSHO3 =

(52)

-ik[W(&*, f~&) + W(&, t3&*)],

I)2B 3 + 2ikA 3 = ik[W(&*, C,) - W(&, a*)], (2ito - D2)B4 = 2ik D(~b~),

• 152B3 + 2ikA3 = ikIW(&*, ~) - W(&, ~*)],

(2ito - T D2)D4 = 2ik O(~b/)),

where D = d / d z , 15 = D 2 - k 2, I) 2 = D 2 - 4k 2, W ( f , g) = f z g - f g z . It is straightforward to show that A2, B2, D2, B 3 and D 3 are real, while A 3 is pure imaginary. To solve these differential equations in practice, we approximate the functions of z by finite sums of Chebyshev polynomials, and solve algebraic problems for the coefficients in these sums. Each of the four sets of ordinary differential equations in (52) is equivalent to a vector equation of the form [imtod(nk) + Ko(nk)] A = N ,

(53)

S.M. Cox et al. / Hopf bifurcations in Langmuir circulations

252

where m and n are integers (in this case - 2 , 0 or 2); ordinary differential operators D

J=

K0 =

-

d(nk)

and K0(nk ) are matrices that represent the

n2k 2 0 ! ) 0 1 , 0 0

( - ( D 2 - n2k2)2 -inkRHh(z) -ink - ( D e - n2k2) -ink

0

inkS n

] (54)

- r ( D 2 - n2k 2) ]

together with appropriate boundary conditions; A is a vector of Chebyshev coefficients that approximates, for example, the triple [Al(Z), B~(z), D 1(z)], and N is a vector of coefficients that represents the nonlinear terms. Equations such as (53) are readily solved by inverting the matrix mitoJ(nk) + K0(nk ) to give

A = [imtod(nk) + Ko(nk)]

1 N.

(55)

This inversion is possible in general provided there are no low-order resonances, so that - i m t o is not an eigenvalue of d-l(nk) Ko(nk ). Thus all the terms of ¢~(e) in the expansion of qt can be computed. The amplitude equations for a s and a 2 that arise from the equations governing the terms proportional to 63 e i'~1 and 63 e i62 are found as follows. Consider first the vector equation that specifies the terms proportional to 6 3 e idol, which is of the form [itod(k) + Ko(k)] ~(3) = R(3),

(56)

where R (3) contains terms proportional to & l , s'al, a~la~l2, aala2l 2 and da~/dT, and where q,(3) = [0(3), u <3), 0<3)]. The matrix i~od(k)+ Ko(k ) is singular (or at least approximately so, to within a small numerical tolerance) and so we must ensure that R <3) lies in its range for a solution ~(3) to (56) to exist. Following Mahalov and Leibovich [20], this condition is achieved numerically using a by-product of the linear eigenvalue problem for oJ, which is the decomposition J -=(k) Ko(k) = VTA(V T)-' ,

(57)

where A is a diagonal matrix of eigenvalues of d-~(k) K0(k ), and V is a matrix whose rows are the corresponding eigenvectors. Using this decomposition, we rewrite (56) as (i~ol + A ) ( v T ) -= t/p(3) = (V T) zJ-~(k) R (3) ,

(58)

where I is the identity matrix. The solvability condition is obtained by noting that ito is an eigenvalue of (11) and so - i t o is an eigenvalue of J-~(k) K0(k ). Therefore -ito is a diagonal element, the pth, say of A, and the p t h row of the left-hand side of (58) is zero. Setting the pth row of the right-hand side of (58) to zero gives the amplitude equation for a I . The amplitude equation for a 2 follows from a similar consideration of the terms proportional t o 6 3 e i~'2. At the double-Hopf bifurcation point, J - l ( k ) K0(k ) has two pairs of purely imaginary eigenvalues:

S.M. Cox et al. / Hopf bifurcations in Langmuir circulations

253

A = +-iron, corresponding to k 1 = ½~r, and A = -+---ito2 for k 2 = "rr. We do not list here the equations to be solved at ~7(e2) f o r the weakly nonlinear analysis of the d o u b l e - H o p f bifurcation, but in our c o m p u t e r algebra they take the form [(irn~to 1 + im2to2)

d(nlk ~ + n2k2) + Ko(n~k 1 + n2k2) ] A = N,

(59)

where Imp.21, [n~,21 --- 2; m 1 + m 2 and n 1 + n 2 are each - 2 , 0 or 2. We assume no low-order resonances, so that the matrix in (59) is invertible, and we can compute each c o m p o n e n t of ~(2). N o w to find the amplitude equations for at(T ) we consider terms proportional to e 3 e i~'j. As for the single H o p f we arrive at a matrix equation of the form

(At| +

A ) ( v T ) -1 1~ (3) _~ (VT)-'d-X(k)

R(3) '

(60)

where Aj = itol, - i t o l , ito2, --ito 2 and k = k 1, k 1, k2, k 2, as j = 1, 2, 3, 4. Since each At is an eigenvalue of the linear problem, then - A t is an element of the diagonal matrix A, and the appropriate row of (60) must vanish. T h e four solvability conditions follow, and give the evolution equations for the four amplitudes al_ 4.

References [1] P. Chossat, M. Golubitsky and B.L. Keyfltz, Hopf-Hopf mode interactions with 0(2) symmetry, Dynam. Stab. Syst. 1 (1986) 255-292. [2] S.M. Cox, S. Leibovich, I.M. Moroz and A. Tandon, Nonlinear dynamics in Langmuir circulations with 0(2) symmetry, J. Fluid Mech. 241 (1992) 669-704. [3] A.D.D. Craik and S. Leibovich, A rational model for Langmuir circulations, J. Fluid Mech. 73 (1976) 401-426. [4] G. Dangelmayr and E. Knobioch, Interaction between standing and travelling waves and steady states in magnetoconvection, Phys. Lett. A 117 (1986) 394-398. [5] G. Dangelmayr and E. Knobloch, The Takens-Bogdanov bifurcation with O(2)-symmetry, Phil. Trans. R. Soc. London A 322 (1987) 243-279. [6] M. Golubitsky and I. Stewart, Symmetry and stability in Taylor-Couette flow, SIAM J. Math. Anal. 17 (1986) 249-288. [7] M. Golubitsky, I. Stewart and D.G. Schaeffer, Singularities and groups in bifurcation theory, Vol. II (Springer, Berlin, 1988). [8] H.E. Huppert and D.R. Moore, Nonlinear double-diffusive convection, J. Fluid Mech. 78 (1976) 821-852. [9] N.E. Hurlburt, M.R.E. Proctor, N.O. Weiss and D.P. Brownjohn, Nonlinear compressible magnetoconvection Part 1. Travelling waves and oscillations, J. Fluid Mech. 207 (1989) 587-628. [10] G. Iooss, Recent progress in the Couette-Taylor problem, preprint (1985). [11] E. Knobloch, Doubly diffusive waves, in: Proc. Joint ASCE-ASME Mech. Conf., eds. N.E. Bixler and E.A. Spiegel, Vol. 24 (1985) pp. 17-22. [12] E. Knobloch, On the degenerate Hopf bifurcation with 0(2) symmetry, in: Multiparameter Bifurcation Theory, eds. M. Golubitsky and J. Guckenheimer (Am. Math. Soc. Providence, RI, 1986). [13] E. Knobloch, A. Deane, J. Toomre and D.R. Moore, Doubly diffusive waves, in; Multiparameter Bifurcation Theory, eds. M. Golubitsky and J. Guckenheimer (Am. Math. Soc. Providence, RI, 1986). [14] E. Knobloch and D.R. Moore, Linear stability of experimental Soret convection, Phys. Rev. A 37 (1988) 860-870. [15] I. Langmuir, Surface motion of water induced by wind, Science 87 (1938) 119-123. [16] S. Leibovich, The form and dynamics of Langmuir circulations, Ann. Rev. Fluid Mech. 15 (1983) 391-427. [17] S. Leibovich, Dynamics of Langmuir circulations in a stratified ocean in: The Ocean Surface, Proc. Symp. of Wave Breaking, Turbulent Mixing and Radio Probing of the Ocean Surface (Tohoku University, Sendal, Japan, July 1984), eds. Y. Toba and H. Mitsuyasu (Reidel, 1985) pp. 457-464. [18] S. Leibovich, S.K. Lele and I.M. Moroz, Nonlinear dynamics in Langrnuir circulations and in thermosolutal convection, J. Fluid Mech. 198 (1989) 471-511.

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S.M. Cox et al. / H o p f bifurcations in Langmuir circulations

[19] S.K. Lele, Some problems in hydrodynamic stability arising in geophysical fluid dynamics (Ph.D. thesis, Cornell University, 1985). [20] A. Mahalov and S. Leibovich, On the calculation of coupling coefficients in amplitude equations, J. Comput. Phys. 101 (1992) 441-444. [21] I.M. Moroz and S. Leibovich, Competing instabilities in a nonliner model for Langmuir circulations, Phys. Fluids 28 (1985) 2050-2061. [22] I.M. Moroz and S. Leibovich, Codimension-two Hopf bifurcations in a model for Langmuir circulations, in: Proc. Intern. Conf. on Fluid Mechanics (Peking U. P., Peking, 1987). [23] D.H. Sattinger, Branching in the presence of symmetry, CBMS-NSF Conf. Series in Appl. Math. Vol. 40 (SIAM, Philadelphia, 1983). [24] W. Sch6pf and W. Zimmermann, Results on wave patterns in binary fluid convection, Phys. Rev. A 41 (1990) 1145-1148. [25] M. Silber and E. Knobloch, Travelling wave convection in a rotating layer, Geophys. Astrophys. Fluid Dyn. 51 (1990) 195-209. [26] S. Wolfram, Mathematica. A system for doing mathematics by computer (Addison-Wesley, Reading, MA, 1988).