Front propagation into unstable states

1. Front propagation intounstable states Wim van Saarloos Instituut-Lorentz Leiden University Physics Reports 386, 29-222 (2003)

2. Not today: turbulence without inertia “turbulence without inertia”Larson 2000 Weissenberg number 5 increasing extrusion speed Turbulence without inertia (Re<1) due to “viscoelastic effects” in polymers Our proposal: subcritical instability to weak turbulence in pipe

3. Fronts/interfaces between stable states The pattern dynamics is often governed by the motion of interfaces/fronts between to linearly stable states Chemical waves Turbulent flame front Turing patterns Mixed phase in type I superconductor 3He dendrite at 100mk (Rolley)

4. Front propagation into an unstable state Schumacher and Eckhardt Propagating Rayleigh-Taylor instability  time Propagating Rayleigh inst. Turbulent front (Couette) Limat et al. Powers et al. Taylor-Couette cell with throughflow Babcock et al. Discharge front Vitello et al.

5. Linearly unstable states or NB The instability need not be weak! No “near-threshold” assumption! Dissipative systems! 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

6. Linear spreading velocity v*  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* 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!

7. So only two classes of fronts! • 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) Only for “sufficiently localized” initial condition <exp(-*x) Pulled/pushed nomenclature: Stokes 1976; Goldenfeld et al. 1994 eithervas = v* “pulled” orvas = 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

8. Illustration of the scenario Uniformly translating Coherent Incoherent Fisher-KPP Swift-Hohenberg Kuramoto-Sivashinsky LinearPushed Pulled dynamics v* v>v*

9. The linear spreading speed v* t»1  (fixed) No growth/decay in co-moving frame =finite, t∞ gives Im[(k*)-v*k*]=0 at saddle point k*: Linearize about the unstable state =0. Eachfourier mode evolves as so  leading exponential x Gaussian correction

10. Exact results for pulled fronts and (for uniformly translating fronts) for the front profile: • 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. Ebert and vS, Physica D 146, 1 (2000)  Universal slow relaxation of front velocity and shape: x

11. Illustration of slow universal convergence Fisher-KPP: Asymptotic front solution with velocity v*=2 Actual time-dependent profile Main difference: relative shift

12. Illustration for pattern forming fronts Swift-Hohenberg eq. v(t)-v* (scaled) CGL Quintic CGL-eq.(chaotic) Swift-Hohenberg 1/time (scaled)

13. Origin of slow convergence 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! Remember asymptotic expression from linear analysis: Asymptotically: exponentialx Gaussian

14. Slow convergence: full matching analysis • In leading edge, writethen  obeys a diffusion equation • Write nonlinear front solution in terms of Xlarge X expansion is genericall as because of double root • Matchnonlinear front region and leading edge • Recognize that because of the logarithmic shift the variable should be used, with • Expand  in similarity solutions of the diffusion equation (Gaussian, etcetera)

15. 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 T. Powers et al.PRL 1997 E. Moses et al.PRL 1994

16. 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 fluctuations discharge front

17. Pushed fronts • Pushed if / “pulled unless”: • an exact uniformly translating or coherent front with v > v*anddecaying faster than exp(-*x) for x∞ “strongly heteroclinic orbit” - not along slowest eigendirection NB: No precise formulation known for incoherent fronts!

18. Postdocs Ute Ebert Judith Müller Deb Panja Andrea Rocco Goutam Tripathy Students Julien Kockelkoren Willem Spruijt Kees Storm Martin van Hecke Acknowledgement Thanks to many postdocs and students who were involved in this over a period of 13 years: ….and many senior colleagues!