A large deviation approach to computing rare transitions in multi-stable stochastic turbulent flows Jason Laurie and Freddy Bouchet Laboratoire de Physique, ENS de Lyon, France ENS de Lyon, 13 June 2012
Bistability in Rotating Tank Experiment Transitions between blocked and zonal states Weeks, Tian, Urbach, Ide, Swinney, Ghil, Science, 1997 • Strong analogy to weather regimes in the Earth’s atmosphere
Bistability in the VKS Experiment Transitions in the polarization of the magnetic field Berhanu et al. EPL, 2007 • Transition trajectories may be concentrated around a single trajectory
Classical Bistability: Double-Well Potential x ( t ) = − dV � ˙ dx + k B T η ( t ) 1 1.5 1 < τ > = 36.2 1 0.8 0.5 0.1 PDF( τ / < τ >) 0.6 V(x) 0 x(t) 0.4 -0.5 0.01 0.2 -1 ∆ V 0 -1.5 0.001 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 98.6 98.8 99 99.2 99.4 99.6 99.8 100 0 2 4 6 8 10 x t/1000 τ / < τ > • Gradient system with a known energy landscape � � − ∆ V • Arrhenius law for transition rate: k = A exp Arrhenius 1889 k B T • Turbulent flows do not fall into this framework • Modern approaches include Freidlin–Wentzell theory (mathematics) and path integrals and instantons (physics)
Aim of this Talk • Large deviation of bistability in turbulent flows • We study the 2D stochastic Navier-Stokes equations (simplest turbulence model) • Computation of instantons with a minimum action method Differences to classical bistability phenomenon • Non-gradient dynamics, connected steady states, unknown steady states, complexity issues • Diffusion across steady states may prevent rare transitions, bistability and large deviation results
The 2D Stochastic Navier-Stokes Equations √ ∂ω ∂ t + v · ∇ ω = − αω + ν ∆ ω + 2 αη � �� � � �� � Dissipation Forcing ω = ( ∇ × v ) · e z , v = e z × ∇ ψ, ω = ∆ ψ • Stochastic white in time forcing: � η ( x , t ) η ( x ′ , t ′ ) � = C ( x − x ′ ) δ ( t − t ′ ) • Doubly periodic domain D • Consider the weak forcing and dissipation regime: ν ≪ α ≪ 1 • Timescale separation: τ energy = 1 ≪ 1 /α = τ dissipation
Leading Order Dynamics – The 2D Euler Equations ∂ω ∂ t + v · ∇ ω = 0 ω = ( ∇ × v ) · e z , = e z × ∇ ψ, ω = ∆ ψ v • The 2D Euler equations have an infinite number of steady states: v · ∇ ω = 0 ⇒ ω = f ( ψ ) • The flow self-organizes and converges toward steady states (attractors) • Robert–Miller–Sommeria equilibrium statistical mechanics predicts which states can be observed (what f ( · ) is selected) Bouchet and Venaille, Physics Reports, 2012
Bistability in the 2D Stochastic Navier-Stokes Equations Transitions between dipole and parallel flow states � • z 1 = D ω ( x , t ) exp ( iy ) d x Bouchet and Simonnet, PRL, 2009
The Onsager–Machlup Path Integral The transition probability Consider a transition from state ω 0 to state ω T in time T : � D [ ω ] e − 1 2 α A ( ω ) P ( ω 0 , 0; ω T , T ) = The action functional � T � 1 p ( x , t ) C − 1 ( x − x ′ ) p ( x ′ , t ) d x d x ′ d t A ( ω ) = 2 0 D = ω + v · ∇ ω + αω − ν ∆ ω ˙ p • Any deterministic trajectory ( p = 0) has zero action: A = 0
The Saddle-Point Approximation ( α ≪ 1) Which trajectory maximizes the transition probability P ? � D [ ω ] e − 1 2 α A ( ω ) P ( ω 0 , 0; ω T , T ) = • The most probable transition trajectory minimizes A ( ω ): ω ∗ = arg min A ( ω ) The Instanton Trajectory { ω | ω (0)= ω 0 , ω ( T )= ω T }
The Saddle-Point Approximation ( α ≪ 1) Which trajectory maximizes the transition probability P ? � D [ ω ] e − 1 2 α A ( ω ) P ( ω 0 , 0; ω T , T ) = • The most probable transition trajectory minimizes A ( ω ): ω ∗ = arg min A ( ω ) The Instanton Trajectory { ω | ω (0)= ω 0 , ω ( T )= ω T } Large Deviation Principle (same as Freidlin–Wentzell) α → 0 − α ln( P ) = A ( ω ∗ ) lim
Exact Results: Large Deviations for Rare States We can explicitly compute instantons for particular cases: • White in space forcing: C ( x − x ′ ) = δ ( x − x ′ ) • Parallel flows (flows with symmetry) • States that are eigenmodes of the Laplacian For the white noise case, we have the following large deviation result: D ω 2 d x Z→∞ e − 1 � P s ( ω ) ≃ 2 � D ω 2 d x is the enstrophy and P s = lim where Z = 1 T →∞ P 2
Non-Isolated Steady States Lead to Non-Standard Large Deviations Attractors of the 2D Euler equations (equilibrium) • The 2D Euler equations contain non-isolated attractors • Any steady state ω is connected to zero through a continuous path of steady states: s ω ( st ) , 0 ≤ s ≤ 1 • Therefore, any two steady states, ω 1 and ω 2 can be connected through a continuous path of steady states (attractors are non-isolated)
Non-Isolated Steady States Lead to Non-Standard Large Deviations Attractors of the 2D Euler equations (equilibrium) • The 2D Euler equations contain non-isolated attractors • Any steady state ω is connected to zero through a continuous path of steady states: s ω ( st ) , 0 ≤ s ≤ 1 • Therefore, any two steady states, ω 1 and ω 2 can be connected through a continuous path of steady states (attractors are non-isolated) 2D Navier-Stokes equations (non-equilibrium) • Dynamics can slowly diffuse across steady states: τ ∼ 1 /α • For transitions between steady states: A ( ω ∗ ) → 0 as α → 0 Transition is not rare! No large deviation and no bistability
The Importance of Degenerate Forcing Strategy: If we can prevent diffusion across steady states, then transitions between two steady states will become a rare event � � η ( x , t ) η ( x ′ , t ′ ) = C ( x − x ′ ) δ ( t − t ′ ) Force Correlation: � • Definition: C k = D C ( x ) exp( i k · x ) d x , if C k = 0 for some k , the force is called degenerate, otherwise non-degenerate • If the forcing is non-degenerate, the dynamics can diffuse across continuous sets of steady states ( A → 0) Then there is no large deviation and no bistability • What about if we set C k = 0 at the largest scales (the scale of the attractors)?
The Importance of Degenerate Forcing Strategy: If we can prevent diffusion across steady states, then transitions between two steady states will become a rare event � � η ( x , t ) η ( x ′ , t ′ ) = C ( x − x ′ ) δ ( t − t ′ ) Force Correlation: � • Definition: C k = D C ( x ) exp( i k · x ) d x , if C k = 0 for some k , the force is called degenerate, otherwise non-degenerate • If the forcing is non-degenerate, the dynamics can diffuse across continuous sets of steady states ( A → 0) Then there is no large deviation and no bistability • What about if we set C k = 0 at the largest scales (the scale of the attractors)? The transition at the largest scale will have to be excited via nonlinear interactions
Bistability with Degenerate Forcing 0.7 2 ≤ | k | ≤ 7 0.6 0.5 2−7 |z 1 | 0.4 0.3 0.2 0.1 Increasing Degeneracy 50 100 150 200 250 300 time* ν 0.6 3 ≤ | k | ≤ 7 − 0.5 0.4 3−7 ← |z 1 | 0.3 Bouchet and Simonnet, PRL, 2009 0.2 � 0.1 • z 1 = D ω ( x , t ) exp ( iy ) d x 0 50 100 150 200 250 300 time* ν • Bistability becomes more 0.6 4 ≤ | k | ≤ 7 apparent as forcing becomes 0.5 0.4 more degenerate 4−7 |z 1 | 0.3 0.2 0.1 0 50 100 150 200 250 300 time* ν
Numerical Computation of Instantons • We implement a variational approach to determine the instanton trajectory by minimizing A ( ω ) (minimum action method) E, Ren, Vanden-Eijnden, 2004 • The initial and final states are fixed throughout the minimization • We iteratively minimize an initial guess, simultaneously over space and time, in a descent direction d n : ω n +1 = ω n + l n d n • Newton or quasi-Newton methods (BFGS) are too expensive to implement • We utilize a nonlinear conjugate gradient method with central differencing scheme in time and pseudo-spectral in space
Numerical Instantons: Non-Degenerate vs. Degenerate Transition between a parallel flow and dipole
Conclusions • The 2D stochastic Navier-Stokes equations are a non-gradient system with non-isolated steady states • Because the set of attractors are connected, the classical phenomenology may not hold • Feasible to numerically compute instantons using a minimum action method • No bistability for non-degenerate forcing • We have explicit large deviation predictions for rare stationary probabilities
Recommend
More recommend
