Front propagation intounstable states

Wim van SaarloosInstituut-LorentzLeiden University

Physics Reports 386, 29-222 (2003)

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Not today: turbulence without inertia

increasing extrusion speed Weissenberg number

“turbulence without inertia” Larson 2000

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Turbulence without inertia (Re<1) due to “viscoelastic effects” in polymers

Our proposal: subcritical instability to weak turbulence in pipe

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Fronts/interfaces between stable states

3He dendrite at 100mk (Rolley)

Turing patterns Chemical waves

The pattern dynamics is often governed by the motion of interfaces/fronts between to linearly

stable states

Mixed phase in type I superconductor

Turbulent flame front

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Front propagation into an unstable state

Propagating Rayleigh inst.

Propagating Rayleigh-Taylor instability

Discharge front

Turbulent front (Couette)

Taylor-Couette cell with throughflow

Powers et al.

Limat et al.

Schumacher and Eckhardt

Vitello et al.

Babcock et al.

t

ime

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Linearly unstable states

We consider situations in which we have a large domain in which the dynamical state is linearly unstable so that for perturbations about the unstable state

or

NB The instability need not be weak!

No “near-threshold” assumption! Dissipative systems!

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Linear spreading velocity v*

Asymptotic velocity: v* (“linear spreading velocity”)plasma physics, about 1960; “absolute versus convective

instabilities”Lifshitz and Pitaevskii, “Physical Kinetics” (Landau-Lifshitz vol.

10)

According to the linear dynamics already a small perturbation grows out and spreads!

V*t

x

Implication: front solutions with asymptotic speed vas<v* are unstable and hence dynamically irrelevant!

Dynamically relevant (“selected”) front solutions must have vas≥v*

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So only two classes of fronts!

either vas = v* “pulled”

or vas = v > v* “pushed” • The typical case if the nonlinearities enhance growth

• Determined by all nonlinearities; “nonlinear eigenvalue problem” associated with strongly heteroclinic orbit

• Each case is different

• Responds like interfaces between (meta)stable states

• The typical case if the nonlinearities don’t enhance growth

• All relevant quanties can be calculated!

• Universal exact results for slow convergence

• “selection”; no nonlinear eigenvalue problem

Ebert+vS (2000)

Pulled/pushed nomenclature: Stokes 1976; Goldenfeld et al. 1994

Only for “sufficiently localized” initial condition <exp(-*x)

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Illustration of the scenario

Lin

ear

Pu

shed

Pu

lled

d

yn

am

ics

Uniformly translating Coherent Incoherent Fisher-KPP Swift-Hohenberg Kuramoto-Sivashinsky

v*

v>

v*

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The linear spreading speed v*

Linearize about the unstable state =0. Eachfourier mode evolves as so

No growth/decay in co-moving frame =finite, t∞ gives

Im[(k*)-v*k*]=0 at saddle point k*:

t»1 (fixed)

leading exponential x Gaussian correction

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Exact results for pulled fronts

Universal slow relaxation of front velocity and shape:

x

and (for uniformly translating fronts) for the front profile:

Ebert and vS, Physica D 146, 1 (2000)

• Independent of initial conditions provided <exp(-*x)• Independent of at what level one tracks front position• Independent of form of the nonlinearities and form of

the equation (as long as front is pulled)• v(t) approaches v* always slowly from below• Also holds for pattern forming fronts and difference

eqs.

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Illustration of slow universal convergence

Fisher-KPP:

Asymptotic front solution with velocity v*=2

Actual time-dependent profile

Main difference: relative shift

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Illustration for pattern forming fronts

v(t)-v* (scaled)

Swift-Hohenberg eq.

Quintic CGL-eq.(chaotic)

1/time (scaled)

Swift-Hohenberg

CGL

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Origin of slow convergence

Remember asymptotic expression from linear analysis:Asymptotically: exponential x Gaussian

So to leading order for a position where =const.: •Logarithmic shift from diffusive

dynamics, but prefactor wrong! (1/2 instead of 3/2)

•Logarithm originates ahead of front, but is dominant term througout!

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Slow convergence: full matching analysis

• Recognize that because of the logarithmic shift the variable should be used, with

• Expand in similarity solutions of the diffusion equation (Gaussian, etcetera)

• In leading edge, writethen obeys a diffusion equation

• Write nonlinear front solution in terms of X large X expansion is genericall as because of double root

• Match nonlinear front region and leading edge

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Summary “pulled” fronts

• Ubiquitous: “the default case” if nonlinearities are saturating

• Assuming the front is pulled, we can calculate all relevant quantities (including pattern wavelength), even if the fullnonlinear dynamics is very complicated

• Matching analysis underlyingthe exact results for the slow convergence gives strong support for the general scenario

E. Moses et al.PRL 1994

T. Powers et al.PRL 1997

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Implications of slow 1/t convergence• Exact results also apply to difference equations, or equations with a kernel

• Numerically, one finds too low velocity unless one extrapolates with 1/t term

• No moving boundary formulation forpattern-forming pulled fronts!Ebert and vS (2000)

• Slow convergence can plague experiments!Taylor-Couette fronts: Ahlers and Cannell, PRL (1983)Rayleigh-Bénard fronts: Fineberg and Steinberg, PRL (1987)

• Pulled fronts very sensitive to noise and fluctuationsdischarge front

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Pushed fronts

Pushed if / “pulled unless”:

an exact uniformly translating or coherent front with

v > v* and decaying faster than exp(-*x) for x∞

“strongly heteroclinic orbit” - not along slowest eigendirection

NB: No precise formulation known for incoherent fronts!

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Acknowledgement

Postdocs• Ute Ebert• Judith Müller • Deb Panja• Andrea Rocco• Goutam Tripathy

Students• Julien Kockelkoren• Willem Spruijt• Kees Storm• Martin van Hecke

Thanks to many postdocs and students who were involved in this over a period of 13 years:

….and many senior colleagues!