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SN~ (~6) lff 2It 3k_~ , 5 ",,x_J {b)

SN~ (~6) lff 2It 3k_~ , 5 ",,x_J {b)

74 Hence odd modes in y

74 Hence odd modes in y bifurcate in a pitchfork because they break ~, while the even modes do so because the translation analogous to (1.7), y ~ y+2rc/2n, acts by -I. If the boundary conditions in y do not extend to PBC we expect only the odd modes to undergo a pitchfork bifurcation. These results explain why Hall and Walton [1977] find a pitchfork in the Rayleigh-Bfnard problem with the boundary conditions (4.1b) and (4.6) for both odd and even eigenfunctions in the horizontal direction. 5 The Faraday Experiment In the Faraday experiment a fluid layer is subjected to a vertical oscillation at frequency co. When the forcing amplitude A is small, the fluid surface remains essentially fiat, but waves are parametrically excited when the amplitude is increased. Indeed, in careful experiments, for most frequencies of vibration the initial transition from the flat surface is to a standing wave at 0)/2, half the driving frequency. In this section we discuss results of Gollub and coworkers using containers with differing geometry, and hence different symmetries; and we explain how these symmetries, when coupled with boundary conditions, affect the analysis of these parametric instabilities. We focus on the experiments of Ciliberto and Gollub [1985a] and Gollub and Simonelli [1989] which employ containers of circular and square cross-section, respectively. The experiments were performed by fixing the forcing frequency 0), slowly varying the amplitude A, and observing the asymptotic behaviour of the surface. For a circular vessel, Ciliberto and Gollub [1985a] find that for most frequencies the initial transition is to a standing wave with azimuthal mode number m; that is, the spatial pattern is invariant under rotations by 2~x/m. There is also a radial index for the number of radial modes, but the radial structure does not play a significant role. The existence of well-defined modes is not surprising, given the 0(2) symmetry of the apparatus. Further, the experiments show that different choices of 0) lead to standing waves with different azimuthal mode numbers, and hence that there exist isolated values of co at which the primary transition from a flat surface occurs by the simultaneous instability of two modes with unequal azimuthal mode numbers m and n. Ciliberto and Goltub [1985a] studied such a codimension two instability for modes with m = 4 and n = 7. Near the point of multiple instability they observed complicated dynamics, including quasiperiodic and chaotic motion. The multiple instability has been analysed by numerous authors using a variety of approximation techniques; see Ciliberto and Gollub [1985b], Meron and Procaccia [1986], and Umeki and Kambe [1989]. We focus here on the approach of Crawford, Knobloch, and Riecke [1989]. Since the forcing is periodic, it is natural to consider the stroboscopic map $ which takes the fluid state at time t to its state one period later, at time t+2~/0). Indeed the experimental measurements provide essentially a reconstruction of the dynamics of 5 2, the twice iterated map. The construction of 5 from 'first principles' would require integrating the Navier-Stokes equations. The idea of Crawford et al. [1989] is to develop a description of 5 by appealing to symmetry and genericity. First, note that the flat surface F is a fixed point of 5 and that the 0)/2 standing wave is a 2-cycle; hence the parametric instability can be identified as a period- doubling bifurcation for S. At the period-doubling bifurcation point the linearization

75 (dS) F of S at F has an eigenvalue -1. This conclusion is supported by the linear analysis of Benjamin and Ursell [ 1954]. Any model of this experiment will be O(2)-symmetric; hence the generalized eigenspace V of (dS) F for eigenvalue -1 is O(2)-invariant, Now we impose genericity. For a nonzero azimuthal mode number we expect V to be 2-dimensional. The reason is that generically the action of 0(2) is irreducible, hence of dimension < 2; but if the dimension is 1 then the period-doubled state has rotational symmetry, and hence has azimuthal mode number zero. So we may identify V with ¢ and write the action of 0 e S0(2) on V as z ~ emi0z. (5.1) Crawford et al. [1989] analyse the interactions of modes with mode numbers n > m >_ 1 as follows. They assume that a centre manifold reduction has been performed to yield an O(2)-equivariant mapping S: 2~C 2, S(0)=0, (dS) 0=-I (5.2) whose asymptotic dynamics is equivalent to that of S. The action of 0(2) on ~2 is determined by the mode numbers and is 0"(Zl'Z2) --- (emi0zl' eni0z2) (5.3) K(Zl,Z2) = (21,~2). Since m 4 n the representations of 0(2) on the two coordinates zj are distinct. Thus (dS) 0 cannot be nilpotent. Crawford et al. show that there exist appropriate choices for the low order terms of S for which the dynamics of S corresponds approximately to that of S observed in experiments. Some questions remain open, but their resolution requires further experiments and will not be discussed here. In contrast, the experiments of Simonelli and Gollub [1989] are performed in a square container. Here mode numbers m and n corresponding to the two horizontal directions x and y are also observed. Not surprisingly, whenever mode (m,n) is observed the surface can be perturbed to a new surface with mode numbers (n,m), corresponding to the reflectional symmetry of the square about a diagonal. When m 4 n several authors have analysed the initial transition. In particular Silber and Knobloch [1989] describe the stroboscopic map, but with the symmetry modified from 0(2) to D 4 to correspond to the symmetry of the container. Feng and Sethna [1989] perform an asymptotic analysis of the Navier-Stokes equations to study the nonlinear behaviour of the standing waves in a nearly square cross-section. Boundary conditions play a more subtle role in the square case. Let the fluid state be specified by the surface deformation ~(x,y) and the fluid velocity field u(x,y,z), where 0 < x,y < n are the horizontal coordinates and z is the vertical coordinate. The realistic no-slip boundary conditions at the sidewalls require u(0,y,z) = u(n,y,z) = 0, u(x,0,z) = u(x, n, z) = 0; while the ~ field may satisfy either NBC or DBC depending on the experimental arrangement: a; a; a; a'-~- (O,y)= ~-x (n,y)= ~ (x,O)= ~-y (n,O)= O, or ~(O,y) = ~(~,y) = ~(x,O) = ~(y,O) = O.

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