1Proceedings of 3ICAPM 3st International Conference on Applications of Porous Media May 29–June 3, 2006, Marrakech, MOROCCO Paper Number “put the paper number here”

CONVECTONS E. Knobloch*, knobloch@px1.berkeley.edu

O. Batiste**, oriol@fa.upc.edu A. Alonso***, arantxa@fa.upc.edu I. Mercader***, isabel@fa.upc.edu

* * *Professor, Department of Physics, University of California, Berkeley, CA 94720, USA ** Lecturer in Physics, Dep. Física Aplicada, Universitat Politècnica de Catalunya, Mòdul B4, C/ Jordi Girona 1-3, 08034 Barcelona,

Spain * * *Associate Professor, Dep. Física Aplicada, Universitat Politècnica de Catalunya, Mòdul B4, C/ Jordi Girona 1-3, 08034

Barcelona, Spain ABSTRACT

Multiple stable states of spatially localized steady convection are found in numerical simulations of water-ethanol and 3He-4He mixtures with negative separation ratio. These states are present in a regime where the conduction state is overstable. When the Rayleigh number is reduced these states are replaced by remarkable relaxation oscillations consisting of abrupt growth of large amplitude localized structures followed by their gradual erosion and collapse into a small amplitude time-dependent state. INTRODUCTION

Binary mixtures are characterised by cross-diffusion quantified by the separation ratio S. When S<0 the heavier

component (of concentration C) migrates up the temperature gradient. Thus in a layer heated from below the destabilizing temperature gradient competes with a stabilizing concentration gradient that develops in response to the heating. If this effect is strong enough convection sets in as growing oscillations once the Rayleigh number R exceeds a critical value Rc. In this paper we report on novel spatially localized steady states that are present in R>Rc. We call these states convectons [1-3].

The system is described by the dimensionless equations [4] ut + ( u ⋅∇)u = − ∇P + σR[(1+S)θ − S η] k + σ ∇2 u, (1) θt + (u ⋅∇)θ = w + ∇2 θ, (2)

ηt + (u ⋅∇)η = ∇2 θ + τ ∇2 η, (3) together with the incompressibility condition ∇⋅ u = 0 (4)

Here u ≡ (u,w) is the velocity field in (x,z) coordinates, P is the pressure, and θ denotes the departure of the temperature from its conduction profile, in units of the imposed temperature difference ∆T=T1−T0>0 across the layer. The variable η is defined such that its gradient represents the dimensionless convective mass flux. Thus η ≡ θ − Σ, where C = 1− z + Σ is the concentration of the heavier component in units of the concentration difference that develops across the layer as a result of cross-diffusion. The system is specified by four dimensionless parameters: the Rayleigh number R providing a dimensionless measure of the imposed temperature difference ∆T, the separation ratio S that measures the resulting concentration contribution to the buoyancy force due to cross-diffusion, and the Prandtl and Lewis numbers σ, τ, in addition to the aspect ratio Γ. For no-slip, fixed temperature, and no mass flux boundaries, u = θ = ηz = 0 on z=0,1;

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periodic boundary conditions are used in the horizontal, with period Γ. In the following we shall be interested in steady solutions of even and odd parity. satisfying, respectively {u(x,z), w(x,z), θ(x,z), η(x,z)}={-u(-x,z), w(-x,z), θ(-x,z), η(-x,z)}and {u(x,z), w(x,z), θ(x,z), η(x,z)}= -{u(-x,1−z), w(-x,1−z), θ(-x,1−z), η(-x,1−z)} relative to a suitable origin in x. These are the only steady solutions that can bifurcate from the conduction state

Odd convecton

(a)Θ

C

Even convecton

(b)Θ

C

Figure 1 shows a pair of spatially localized states obtained by solving the above equations in two dimensions for a water-ethanol mixture with S=-0.021, σ=6.22, τ =0.009, Γ =60 [5] when R=1775. Both states are numerically stable. This value of the Rayleigh number is above the onset of overstability which occurs at R= Rc ≈ 1760.8. In the following we seek to understand the origin of these states. For this purpose we have adapted a pseudo-spectral first-order time-stepping formulation to compute steady states via Newton's method [6], and implemented a continuation code to follow branches of stationary solutions [7] CONVECTONS Water-ethanol mixtures

Figure 2 summarizes our results for water-ethanol mixtures in the form of a bifurcation diagram. The figure shows <N> -1 as a function of R where <N> is the (time-averaged) Nusselt number; this quantity provides a dimensionless measure of the strength of convection. The Hopf bifurcation at R=Rc leads to branches of both traveling waves (TW) and standing waves (SW), both spatially periodic and of uniform amplitude in 0 ≤ x <Γ. Since the TW branch is subcritical both branches are unstable near R= Rc [8]. The unstable TW branch can be followed numerically by seeking steady solutions to the equations in an appropriately moving frame; the speed of this frame is determined by fixing the phase of the solution [7]. The same numerical method also determines the branch of steady overturning convection (SOC). The results show that the SOC branch passes though a saddle-node bifurcation at R≈1743.3, but remains unstable until a parity-breaking bifurcation at R≈1747.0 where the TW branch terminates on the SOC branch. In the following it will be important that this point lies below Rc so that there is a large range of stable SOC states that coexist with a stable conduction state (R< Rc) and the small amplitude dispersive chaos state [9,5] that 'bifurcates' from Rc, apparently supercritically (Fig. 2).

As R increases beyond Rc the amplitude of the dispersive chaos gradually increases until it begins to be interrupted by

episodes in which an almost time-independent spatially localized state emerges. These states grow super-exponentially from the dispersive chaos state, but the saturated state is not stable and is eroded on a much longer time scale; once this

Fig. 1. (a) Odd and (b) even parity convectons at R=1775 in terms of contours of constant θ and C

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1740 1750 1760 1770 1780 17900

0.01

0.02

0.03

0.04

0.05

conv

ectiv

e in

stab

ility

abso

lute

inst

abili

ty

TW

SOC

Rayleigh number

Nus

selt

num

ber−

1

0 1000 2000 3000 4000 5000 6000 70000

0.01

0.02

time

Nus

selt

− 1

R=1774(a)

0 603050

3075

3100

3125

3150

3175

x

time

R=1774(c)

erosion is complete dispersive chaos is reestablished, but remains 'unstable' to the formation of new localized states, and the process therefore repeats in an irregular fashion (Fig. 3(a)). At larger values of R we find stable spatially localized states of both odd and even parity (Fig. 2), and refer to these states as convectons. Figure 4(a) shows the evolution of a small amplitude initial perturbation into an odd-parity

Fig 3. N(t) at R=1774 showing relaxation oscillations between dispersive chaos and localized steady convection. Space-time plots at two different time intervals in the time series (a), showing (b) the destruction of localized steady convection, and (c) the formation of localized steady convection from dispersive chaos. Time is in units of the thermal diffusion time in the vertical.

Fig. 2. Bifurcation diagram showing <N>-1 as a function of the Rayleigh number R when Γ=60. The conduction state loses instability at Rc=1760.8.Steady spatially periodic convection (SOC) acquires stability at a parity breaking bifurcation marking the destruction of a branch of spatially periodic traveling waves (TW). Above threshold small amplitude dispersive chaos is present (solid dots), which leads into the pinning region 1774<R<1781containing a multiplicity of stable localized states of both even and odd parity. This region lies below the onset of absolute instability

0 60100

500

900

x

time

R=1774(b)

4

0 601100

1200

1300

1400

1500

x

time

R=1775(a)

0 600

500

1000

1500

x

time

R=1779(b)

convecton when R=1775, Γ =60. In steady state the convecton consists of stationary rolls of uniform amplitude and wavelength, except for two 'front'-like structures at either end that separate it from the void.

Figure 5(a) shows that all odd parity convectons lie on a connected curve that snakes from small amplitude towards the amplitude of the coexisting spatially periodic roll state, while fig. 5(b) shows the corresponding results for even convectons. In each case stable convectons are found on portions of each branch between a left turning point and the next right turning point going up the snake.

1775 1780 1785

0.005

0.01

0.015

0.02

Rayleigh number

Nus

selt

num

ber−

1

(a) odd branch

c

b a

1775 1780 1785

0.005

0.01

0.015

0.02

Rayleigh number

Nus

selt

num

ber−

1

(b) even branch

ab

c

Fig. 5. Branches of (a) odd and (b) even parity convectons in a Γ=60 domain as a function of the Rayleigh number R. Both branches exhibit snaking.

Fig 4. (a) The formation of an odd parity convecton when R=1775, Γ =60, starting from small amplitude initial conditions, in the form of a space-time plot of θ(x,z=1/2,t) with time increasing upward. (b) The destruction of this state when R is increased from R=1775 to R=1779. Both fronts advance at the same rate preserving the parity of the state.

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These turning points correspond to saddle-node bifurcations and hence to change in stability with respect to localized perturbations of the same parity as the convecton. Figure 6 shows sample convecton profiles at or near several turning points on each branch; these indicate that each convecton acquires a pair of rolls between corresponding turning points as one proceeds up each snake. As a result the convectons higher up each snake become broader and broader. On an infinite domain this process proceeds indefinitely as roll pairs are added and the convectons approach an infinitely extended SOC state. Note that the turning points at left and right line up exponential quickly, producing a well-defined pinning region in between.

The structure of the even convectons is greatly influenced by the rolls at either end. A roll rotating clockwise at the right end entrains higher concentration fluid into the convecton, while a counter-clockwise rotating roll entrains lower

concentration fluid. Since the addition of a roll pair changes the direction of the outermost rolls its effect manifests itself in the homogenized concentration within the convecton (Fig.6 (b)). In contrast, the rolls at the ends of an odd parity convecton always rotate in the same sense, resulting in the entrainment of higher concentration at one end and lower concentration at the other end. The net effect is that the homogenized concentration corresponds to the average concentration (Fig.6 (a)). It is clear, however, that the net effect of this process is to pump concentration through odd parity convectons, an effect that manifests itself in the inclination of the concentration contours in the 'conduction' part of odd parity states

(Fig. 6(a)). Thus all odd parity states pump flux horizontally.

Theoretical interpretation

In the following we identify the connection from the conduction state to the SOC state and back again with a pair of fronts separating the conduction state from the SOC state. The convectons are then to be thought of as bound states of these fronts. The origin of the multiplicity of stable convectons and of the associated snaking is easiest to understand in the context of variational systems like the Swift-Hohenberg equation [10-12,13]. Such systems have a conceptual advantage in having an 'energy' that is a nonincreasing function along temporal trajectories. In particular, one can locate parameter values at which pairs of states have identical energy. Such points are called Maxwell points. At a Maxwell point between two spatially homogeneous states fronts connecting such states will be stationary, but for any nearby parameter values such fronts will move, in such a way as to decrease energy. Thus the only stable state away from the Maxwell point is the lower energy state. If, however, one or both states are structured the fronts can be pinned to this structure, resulting in a broadening of the Maxwell point into a pinning region of finite width [14,15]. While the variational structure helps one to focus on the Maxwell point, there is nothing in the above description that requires variational structure per se, and indeed it is known that essentially identical behavior is found in nonvariational systems provided these are reversible in space, i.e., invariant under x → − x [10-12,13]. It is for this reason that we refer to the snaking region in fig. 2 as the pinning region. Theory [16] suggests that stable spatially localized states can exist only in the range of Rayleigh numbers below the onset of absolute instability, since otherwise the convectons inherit the instability of the conduction state. Detailed computations [3] confirm that this is indeed so. As a result when R is increased outside the pinning region the

c

b

a

(b) Even branch

Odd branch

c

b

a

(a)

Fig. 6. Convecton profiles on (a) the odd parity branch, and (b) the even parity branch at the turning points indicated in fig. 5(a,b) in terms of contours of constant C. Each wavelength contains a pair of rolls.

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behavior remains simple: the bounding fronts depin, and new rolls nucleate as the fronts move apart, ultimately filling the domain with a periodic SOC state (Fig. 4(b)).

3He - 4He mixtures

The situation for 3He-4He mixtures (S=-0.5, σ=0.6, τ=0.03) differs in an important respect. Here the pinning region containing stable convectons is centered around R=2800 but extends into the region of absolute instability. As a result when R increases the convectons lose stability before the bounding fronts depin. When this happens the 'void' between adjacent convectons fills with waves (Fig. 7) but the localized structure remains [1,2]. These waves produce irregular oscillations in the front location, and these are in turn responsible for the fluctuations in the number of rolls in the convecton, as revealed by N(t) (Fig. 7(c)).

When R is decreased below the pinning region the convecton is gradually eroded as rolls are destroyed pairwise at either end, and the convecton decays back to a small amplitude state, before a new convecton regrows (Fig.8). As before the convectons created in this nucleation process may have different numbers of rolls (Fig.8(a)). In contrast, if R is decreased by an

insufficient amount to unpin the fronts the convecton stabilizes, perhaps with a smaller number of rolls (Fig. 8(b)).

−0.2

0

0.2

θ

σ= 0.6 τ= 0.03 S=−0.5 R=2750 Γ=30

−0.4−0.200.20.4

C

0 20 40 60 80 1000.33

0.34

0.35

Nu−

1

time

Fig 8. The evolution of an odd parity convecton when R is decreased from R=2800 to (a) R=2700, (b) R=2750, for Γ=30. The top panels show the state at the final instant; the lower panels show the evolution of the Nusselt number N.

−0.2

0

0.2

θ

σ= 0.6 τ= 0.03 S=−0.5 R=2900 Γ=30

−0.4−0.200.20.4

C

0 100 200 300 400 5000.2

0.3

0.4

Nu−

1

time

Fig 7. Confined convecton-like state maintained by traveling waves in the void region when R=2900, Γ=30 and t=500. The third panel shows the fluctuations in the Nusselt number arising from fluctuations in the number of rolls in the structure.

−0.02

0

0.02

θ

σ= 0.6 τ= 0.03 S=−0.5 R=2700 Γ=30

−0.5

0

0.5

C

0 200 400 600 800 1000 1200 14000

0.5

Nu−

1

time

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CONCLUSIONS

In this paper we have examined the origin and structure of spatially localized steady convection in binary fluid mixtures heated from below. We have seen that past the primary Hopf bifurcation the system settles into a state of dispersive chaos, much as in the experiments. This state is characterized by repeated growth and collapse of spatially localized structures [17]. We have seen that with increasing Rayleigh number the character of this state changes with the occasional formation of localized but almost stationary structures, much as observed in water-ethanol mixtures for slightly different parameter values [18]. Our computations indicate that the localized steady states we have called convectons emerge from this state. Although branch following techniques indicate that the convectons appear via saddle-node bifurcations little or no hysteresis has been observed in this transition.

We have seen that the convectons are organized into a pair of branches that snake towards the spatially periodic SOC

state. As this occurs the convectons add rolls in a pairwise fashion, thereby becoming broader and broader. We have attributed this behavior to the presence of a pinning interval in the Rayleigh number characterized by fronts separating the conduction and convection states, that are locked to the latter. Although we have not studied the stability of these states in detail we have confirmed via time-dependent integration the existence of multiple coexisting stable convectons within the pinning region. This multiplicity of states has thus far not been seen in experiments possibly because self-trapping [19] precludes a change in the convecton size in any given experiment. The experiments do reveal, however, convectons of increasing width with increasing Rayleigh number [18,20], presumably reflecting changes in the basin of attraction of the different convecton states. This in turn may be due to the ability of more vigorous convection to excavate a broader trough in the concentration distribution.

ACKNOWLEDGMENTS

This work was supported in part by DGICYT under grant BMF2003-00657, AGAUR grant 2005SGR-00024 and by the National Science Foundation under grant DMS-0305968. We are grateful to J. Burke and A. Champneys for helpful discussions.

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