HAL Id: jpa-00247728
https://hal.archives-ouvertes.fr/jpa-00247728
Submitted on 1 Jan 1992
HAL is a multi-disciplinary open access archive for the deposit and dissemination of sci- entific research documents, whether they are pub- lished or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers.
L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.
Onset of zero Prandtl number convection
E. Knobloch
To cite this version:
E. Knobloch. Onset of zero Prandtl number convection. Journal de Physique II, EDP Sciences, 1992, 2 (5), pp.995-999. �10.1051/jp2:1992182�. �jpa-00247728�
Classification Physics Abstracts
47.20 47.25
Short Communication
Onset of zero Prandtl number convection
E. Knobloch(*)
Observatoire Midi-Pyren4es, 14 Avenue Edouard Belin, F31400 Toulouse, France
(Received 12 November 1991, accepted in final form 17 February 1992)
Abstract. The transition to convection in a zero Prandtl number fluid with stress-free and perfectly conducting boundaries differs significantly from finite Prandtl number convection, giving rise to a three-dimensional pattern. Two possible scenarios are described and compared
with recent numerical simulations by Thual Ii].
The Prandtl number of a fluid can approach zero in one of two ways either because the
viscosity is very small or because the thermal conduction is very efficient, and carries most of the heat. Both cases are of interest in astrophysics and geophysics. In the latter case, which is of interest here, the Oberbeck-Boussinesq equations reduce to [1, 2]
ut + u Vu = -Vp + V~u + R@e3, (la)
V u
= 0, (16)
0 = w + V~@, (lc)
where u e (u, v,w) is the velocity, and p are the temperature and pressure perturbations
and e3 is the unit vector in the vertical. The Rayleigh number is denoted by R. Following
Thual [I], we adopt stress-free fixed temperature boundary conditions:
uz=vz=w=@=0 on z=0,1. (Id)
The theoretical interest of these equations centers on the fact that the only nonlinear term vanishes for a pattern of parallel rolls. Thus the pattern expected for finite Prandtl number convection with the boundary conditions (Id), namely rolls, cannot saturate and so grows without bound. The question is: what happens? In recent simulations Thual [I] discovered finite amplitude periodic oscillations between orthogonal sets of rolls at a Rayleigh number
6.5 To above onset and proposed that such oscillations are the first state to appear. For larger
(* On leave from the Department of Physics, University of California, Berkeley, CA 94720, U-S-A-.
996 JOURNAL DE PHYSIQUE II N°5
ltayleigh numbers he found stable squares, a surprising result since such patterns are usually associated with poorly conducting boundary conditions [3]. In this letter we show that the first state may in fact be a time-independent mixed state consisting of perpendicular rolls in which
one set of rolls predominates, and discuss the mechanism by which squares become stable.
Some conjectures concerning the origin of Thual's oscillations are also presented.
By looking for square pattern solutions of the form [4]
w(z., y, z,t) = (ri(t) cos kz + r2(t) cos ky) sin ~z + (2)
with related expressions for the remaining variables, one can reduce equations (1) to the normal form
fi " Pri + brir( + cr)r( + drir( + (3a)
+2 " Pr2 + br)r2 + cr)r( + dr(r2 + (3b)
The form of these equations is dictated by symmetry considerations [5]. In particular transla- tion invariance in the horizontal plane is responsible for the decoupling of spatial phases. The equations describe the competition between rolls and squares near onset of convection which takes place at R
= Jlc e 27~~/4 with wavenumber k e ~/Vi. Note that for p
e R llc > 0
a pure roll solution (ri, r2) " (r,0) or (0,r) grows without saturation: r
= r(0)exp pt. Such
a solution is, however, unstable to perturbations in the form of orthogonal rolls. We set
(ri, r2) " (r,b), and linearize in b:
b = (p + br~ + dr~)b (4)
Thus even when b < 0 the rthogonal rolls grow apidly
whenever d >
original xponentially growing rolls break up efore they have time to
grow much. Note
that this observation requires the of the fifth order termsinuations (3). In the
ollowing we ssume that d is ufficiently large (or quivalently that b is sufficiently mall) that
the plitude of the rolls does not fall outside the alidity
of theuncation of equations (3).
Equations (3) also
iven by
Their stability with respect to amplitude perturbations is given by the growth rate si "
-4p 2bp~; that with respect to rolls is given by s2 " 2p + 2(c d)p~. Thus the small
amplitude squares
,
p~ +~ -p/b, are supercritical (p > 0) whenever < 0, and are then stable with respect to amplitude perturbations but unstable with respect to perturbations in the form of rolls. When c- d < 0 the opposite is the case for the large amplitude solution p~
+~ -b /(c+d),
present whenever c+ d > 0. It must be emphasized, however, that unless d is large (orb small)
this fixed point is an artifact of the truncation of equations (3). This is also true of the two mixed mode states (ri, r2), (r2, ri) with ri # r2, hereafter referred to as cross-rolls, satisfying
b + d(r) + r()
= 0, (7a)
ri = Pd/b(C d) + O(P~)> (7b)
and present for b < 0 provided d > 0, c d < 0. When p is small these fixed points describe
nearly pure parallel roll patterns, with the contribution of the competing rolls increasing with p. Note that the two fixed points correspond to identical cross-roll patterns related by rotation
through 90°. When c < 0, both are stable foci, with O(p) decay rate and O(p~/~) frequency;
when c > 0 both are unstable. When is small these stability assignments are completely determined by the truncation of equations (3) at fifth order [6]. With increasing Rayleigh
number r] reaches -b/2d and the mixed modes become squares. At this point the eigenvalue
s2 of the squares passes through zero, and for (d= c)b~/4d~ < p < b~/4(c + d) the squares are
stable, with the mixed modes absent. Observe finally that the line ri
" r2 containing the two
square patterns and bisecting the (ri > 0,r2 > 0) quadrant is invariant under the dynamics
of equations (3). Hence no trajectory can cross from one side to the other. This is so even when the equations are not truncated, and is a consequence of the fact that near enough to the threshold only two modes partake in the dynamics.
Figure la shows a sketch of the resulting (ri, r2) Plane for the truncated equations for small p and c < 0. The cross~rolls are stable and nearly all initial conditions are attracted to
them. In contrast, in figure 16, drawn for small p and c > 0, the cross-rolls are unstable,
and nearly all initial conditions are attracted to one of the two stable finite amplitude limit cycles surrounding them. Since each limit cycle must pass through the gap (when p is small)
between the unstable focus and the nearby axis the resulting solution looks like a pure roll for part of its period. The rest of the time it spends near the two fixed points corresponding to squares and following the line ri " r2 between them. During this phase it therefore looks like
a square. Provided d is large enough (or b small enough) these oscillations are fully described
by the truncated equations. They are not, however, of the type found by Thual.
rz j
Mj
, , '
,
' ' '
~l',
', , ~f~
, fit '
,
~ '
' q q
a) b)
Fig. i. (a) The (ri, r2) Plane for the truncated equations (3) with b < 0, c < o, d » i. Fixed points corresponding to squares (Si,52) and to the mixed modes (Mi,M2) are indicated. (b) The same but for c > 0 showing the two limit cycles. The dashed line indicates the boundary of the region in which the analysis holds when d = O(i), d c > 0.
998 JOURNAL DE PHYSIQUE II N°5
The calculation of the coefficients is routine (cf. [4]). The one distinction is that vertical
vorticity modes contribute to the coefficients c and d, and these must be properly taken into
account. This is not necessary for the third order terms (see, e.g., [7]). One obtains
b = -1.408453, c = -0.000883, d
= 0.012395. (8)
Consequently, the scenario depicted in figure la applies, and the first convective state is a finite amplitude cross-roll pattern. With increasing Rayleigh number the cross-rolls transfer their
stability to the squares which remain stable until the saddle-node bifurcation at p
= b~/4(c+d).
The above conclusions apply, however, only if d is large enough (or b small enough) that the truncated equations correctly capture all the dynamics. When and d are both O(I) a
description of zero Prandtl number convection in terms of equations (3) is restricted to the small amplitude region of the phase plane, as indicated in figure I. The cross-roll fixed points
are now affected by higher order terms and may become unstable. If this were the case then
a modified version of the scenario depicted in figure 16 would apply. The large amplitude
squares, if present at all, would acquire a higher-dimensional stable manifold. In such a case
a limit cycle alternating between the two types of rolls (r,0), (0,r) could be present but would require a higher-dimensional system for its description. Such a description would in any
case be inevitable since the higher harmonics of the pattern (2) are now also excited to O(I) amplitudes. An identical problem arises in attempts to explain the nature of the oscillations in binary fluid convection with positive separation ratio that are found between the Soret regime (squares) and the Rayleigh regime (rolls) [8].
Thual's simulations, and the small value of the coefficient d, both lend support to the latter
scenario, although the scenario in figure la cannot be excluded: owing to their degeneracy the normal form equations (3) apply in a smaller neighborhood of p
= 0 than the usual Landau
equation (roughly defined by the condition p~/~ < l instead of p~/~ « l). Thual's simulations do not fall into this regime.
A similar analysis is readily carried out for the competition between rolls and patterns with
hexagonal symmetry. The amplitude equations now take the form
ii = pri + b(r( + r()ri + c(r( + r()r) + d(r( + r()ri + er(r(ri, (9a)
+2 = pr~ + b(r( + r))r2 + c(r( + r))r( + d(r( + r()r2 + er(r)r2, (9b)
~3 " @~3 + b(~~ ~ ~()~3 + C(~~ ~ ~()~( ~ ~(~~ ~ ~~)~3 + er~~(~3> (~C)
with different values of the coefficient e for hexagons and regular triangles [9]. Although the coefficients b,c, d, e have not been calculated, it is easy to show that (a) a growing roll pattern
(r, 0,0) is strongly unstable to rolls at 120° (for example, (0,b,0)) provided d > 0 even when b < 0, and (b) that fixed points corresponding to hexagons, regular triangles and the patchwork quilt [9] are all unstable for small p > 0. In Thual's simulations competition between rolls and the above patterns could not be directly studied because of his use of a square periodic box.
Acknowledgements.
The author wishes to thank the referee for sending him a related work [10], in which equations (3) are also obtained but the coefficients of the quintic terms are not calculated. In addition he is grateful to T. Clune for checking the computation of the normal form coefficients, and to O. Thual and J.-P. Zahn for discussions. This work was supported by a CNRS visiting professorship at the Observatoire Midi-Pyrendes.
References
[1]Thual O., (1991) Preprint.
[2]Spiegel E-A-, J. Gecphys. Res. 67 (1962) 3063.
[3] Busse F-H- and Riahi N. J. Fluid Mech. 96 (1980) 243;
Proctor M.R.E., J. Fluid Mech. 113 (1981) 469.
[4] Knobloch E., Physica D 41 (1990) 450.
[5] Crawford J.D. and Knobloch E., Ann. Rev. Fluid Mech. 23 (1991) 341.
[6] Knobloch E., Contemp. Math. 56 (1986) 193.
[7] Clune T. and Knobloch E., Phys. Rev. A 44 (1991) 8084.
[8] Le Gal P., Pocheau A. and Croquette V., Phys. Rev. Lent. 54 (1985) 2501;
Moses E. and Steinberg V., Phys. Rev. Lett. 57 (1985) 2018;
Bigazzi P., Ciliberto S. and Croquette V., J. Phys. France 51 (1990) 611;
Moses E. and Steinberg V., Phys. Rev. A 43 (1991) 707.
[9] Golubitsky M., Swift J-W- and Knobloch E., Physica D lo (1984) 249.
[10] Kumar K., Convective patterns at zero Prandtl number, in 1991 Summer Study Program in
Geophysical Fluid Dynamics, (Woods Hole Oceanog. lust. Tech. Rept., WHOI-92, 1991), to appear.
