Chemical front propagation in cellular flows: The role of large - PowerPoint PPT Presentation

Fronts in Flows The large- t limit Regimes I & II Regime III Summary FKPP vs G Conclusions Chemical front propagation in cellular flows: The role of large deviations Alexandra Tzella University of Birmingham Irregular transport: analysis and applications, Basel, Switzerland

Fronts in Flows The large- t limit Regimes I & II Regime III Summary FKPP vs G Conclusions Reactive fronts in environmental flows Phytoplankton bloom off the coast of Alaska (NASA’s Goddard Space, Sept. 22, 2014).

Fronts in Flows The large- t limit Regimes I & II Regime III Summary FKPP vs G Conclusions Reactive fronts in experimental flows Cellular vortex flow Random flow Pocheau & Harambat (2008) Haslam & Ronney (1995)

Fronts in Flows The large- t limit Regimes I & II Regime III Summary FKPP vs G Conclusions Reactive fronts in experimental flows Cellular vortex flow Random flow Pocheau & Harambat (2008) Haslam & Ronney (1995) How do heterogeneities influence the front? e.g. speed, shape etc Xin (2000,2009), Berestycki (2003)

Fronts in Flows The large- t limit Regimes I & II Regime III Summary FKPP vs G Conclusions FKPP fronts in the absence of a flow Reaction-diffusion with FKPP nonlinearity � 1 if x ≥ 0 ∂ t θ ( x , t ) = κ ∆ θ ( x , t ) + 1 τ θ (1 − θ ) , θ ( x , 0) = 0 if x < 0 . and front-like conditions in x and no-flux boundary conditions in y . At large times, a front is established: θ ( x , t ) → Θ( x − ct ) , when t ≫ 1 .

Fronts in Flows The large- t limit Regimes I & II Regime III Summary FKPP vs G Conclusions FKPP fronts in the absence of a flow Reaction-diffusion with FKPP nonlinearity � 1 if x ≥ 0 ∂ t θ ( x , t ) = κ ∆ θ ( x , t ) + 1 τ θ (1 − θ ) , θ ( x , 0) = 0 if x < 0 . and front-like conditions in x and no-flux boundary conditions in y . At large times, a front is established: θ ( x , t ) → Θ( x − ct ) , when t ≫ 1 . 10 κ =1 1 κ 1 τ 0 . 1 θ ( x , t ) = 1 θ ( x , t ) = 0

Fronts in Flows The large- t limit Regimes I & II Regime III Summary FKPP vs G Conclusions FKPP fronts in cellular flows Reaction-diffusion-advection with FKPP nonlinearity ∂ t θ ( x , t ) + u ( x ) · ∇ θ ( x , t ) = Pe − 1 ∆ θ ( x , t ) + Da θ (1 − θ ) , where Pe = U ℓ/κ and Da = ℓ/ U τ with streamfunction u = ∇ ⊥ ψ with ψ = − sin( x ) sin( y ) .

Fronts in Flows The large- t limit Regimes I & II Regime III Summary FKPP vs G Conclusions FKPP fronts in cellular flows Reaction-diffusion-advection with FKPP nonlinearity ∂ t θ ( x , t ) + u ( x ) · ∇ θ ( x , t ) = Pe − 1 ∆ θ ( x , t ) + Da θ (1 − θ ) , where Pe = U ℓ/κ and Da = ℓ/ U τ with streamfunction u = ∇ ⊥ ψ with ψ = − sin( x ) sin( y ) . − + − + − +

Fronts in Flows The large- t limit Regimes I & II Regime III Summary FKPP vs G Conclusions FKPP fronts in cellular flows At large times, a pulsating front is established: θ ( x , y , t ) → Θ( x − ct , x , y ) , when t ≫ 1 . where Θ is 2 π -periodic in the second variable. Berestycki & Hamel (2002)

Fronts in Flows The large- t limit Regimes I & II Regime III Summary FKPP vs G Conclusions FKPP fronts in cellular flows At large times, a pulsating front is established: θ ( x , y , t ) → Θ( x − ct , x , y ) , when t ≫ 1 . where Θ is 2 π -periodic in the second variable. Berestycki & Hamel (2002) Examples obtained for varying Da and Pe = 250

Fronts in Flows The large- t limit Regimes I & II Regime III Summary FKPP vs G Conclusions FKPP fronts in cellular flows At large times, a pulsating front is established: θ ( x , y , t ) → Θ( x − ct , x , y ) , when t ≫ 1 . where Θ is 2 π -periodic in the second variable. Berestycki & Hamel (2002) Examples obtained for varying Da and Pe = 250

Fronts in Flows The large- t limit Regimes I & II Regime III Summary FKPP vs G Conclusions FKPP fronts in cellular flows At large times, a pulsating front is established: θ ( x , y , t ) → Θ( x − ct , x , y ) , when t ≫ 1 . where Θ is 2 π -periodic in the second variable. Berestycki & Hamel (2002) Examples obtained for varying Da and Pe = 250 Da = 4 × 10 − 2 Da = 4 × 10 − 1 Da = 4

Fronts in Flows The large- t limit Regimes I & II Regime III Summary FKPP vs G Conclusions FKPP fronts in cellular flows At large times, a pulsating front is established: θ ( x , y , t ) → Θ( x − ct , x , y ) , when t ≫ 1 . where Θ is 2 π -periodic in the second variable. Berestycki & Hamel (2002) Examples obtained for varying Da and Pe = 250 Da = 4 × 10 − 2 Da = 4 × 10 − 1 Da = 4 What is the front speed c as a function of Pe and Da? (when Pe ≫ 1)

Fronts in Flows The large- t limit Regimes I & II Regime III Summary FKPP vs G Conclusions Outline of asymptotic regimes for Pe ≫ 1 Regime Da front speed approach Pe − 3 O (Pe − 1 ) 4 C 1 (PeDa) I boundary-layer analysis O ((log Pe) − 1 ) (log Pe) − 1 C 2 (Da log Pe) II boundary-layer analysis III O (Pe) C 3 (Pe / Da) WKB analysis

Fronts in Flows The large- t limit Regimes I & II Regime III Summary FKPP vs G Conclusions Outline of asymptotic regimes for Pe ≫ 1 Regime Da front speed approach Pe − 3 O (Pe − 1 ) 4 C 1 (PeDa) ⋆ I boundary-layer analysis O ((log Pe) − 1 ) (log Pe) − 1 C 2 (Da log Pe) II boundary-layer analysis III O (Pe) C 3 (Pe / Da) WKB analysis ⋆ for fixed values of Pe Da, it recovers the scaling prediction by Audoly et al. (2000), Novikov & Ryzhik (2007) . ∗ based on work by Haynes & Vanneste (2014) on the dispersion of non-reacting tracers in 2D cellular flows.

Fronts in Flows The large- t limit Regimes I & II Regime III Summary FKPP vs G Conclusions Outline of asymptotic regimes for Pe ≫ 1 Regime Da front speed approach Pe − 3 O (Pe − 1 ) 4 C 1 (PeDa) ⋆ boundary-layer analysis ∗ I O ((log Pe) − 1 ) (log Pe) − 1 C 2 (Da log Pe) boundary-layer analysis ∗ II III O (Pe) C 3 (Pe / Da) WKB analysis ⋆ for fixed values of Pe Da, it recovers the scaling prediction by Audoly et al. (2000), Novikov & Ryzhik (2007) . ∗ based on work by Haynes & Vanneste (2014) on the dispersion of non-reacting tracers in 2D cellular flows.

Fronts in Flows The large- t limit Regimes I & II Regime III Summary FKPP vs G Conclusions Outline of asymptotic regimes for Pe ≫ 1 Regime Da front speed approach Pe − 3 O (Pe − 1 ) 4 C 1 (PeDa) ⋆ boundary-layer analysis ∗ I O ((log Pe) − 1 ) (log Pe) − 1 C 2 (Da log Pe) boundary-layer analysis ∗ II O (Pe) † III C 3 (Pe / Da) WKB analysis ⋆ for fixed values of Pe Da, it recovers the scaling prediction by Audoly et al. (2000), Novikov & Ryzhik (2007) . ∗ based on work by Haynes & Vanneste (2014) on the dispersion of non-reacting tracers in 2D cellular flows.

Fronts in Flows The large- t limit Regimes I & II Regime III Summary FKPP vs G Conclusions The eigenvalue problem for the speed Linearising around the tip of the front, θ ≈ 0 ∂ t θ ( x , t ) + u ( x ) · ∇ θ ( x , t ) = Pe − 1 ∆ θ ( x , t ) + Da θ ////// (1 − θ ) . For t ≫ 1 we employ the long-time large-deviation form which at leading order is where c = x θ ( x , t ) ≈ e − t ( g ( c ) − Da ) t = O(1) � for c < g − 1 (Da) ∞ , = for c > g − 1 (Da) 0 , The front speed is given by c = g − 1 (Da). G¨ artner & Freidlin (1979)

Fronts in Flows The large- t limit Regimes I & II Regime III Summary FKPP vs G Conclusions The eigenvalue problem for the speed Linearising around the tip of the front, θ ≈ 0 ∂ t θ ( x , t ) + u ( x ) · ∇ θ ( x , t ) = Pe − 1 ∆ θ ( x , t ) + Da θ ////// (1 − θ ) . For t ≫ 1 we employ the long-time large-deviation form which at leading order is where c = x θ ( x , t ) ≈ e − t ( g ( c ) − Da ) t = O(1) � for c < g − 1 (Da) ∞ , = for c > g − 1 (Da) 0 , The front speed is given by c = g − 1 (Da). G¨ artner & Freidlin (1979)

Fronts in Flows The large- t limit Regimes I & II Regime III Summary FKPP vs G Conclusions The eigenvalue problem for the speed Linearising around the tip of the front, θ ≈ 0 ∂ t θ ( x , t ) + u ( x ) · ∇ θ ( x , t ) = Pe − 1 ∆ θ ( x , t ) + Da θ ////// (1 − θ ) . For t ≫ 1 we employ the long-time large-deviation form which at leading order is where c = x θ ( x , t ) ≈ e − t ( g ( c ) − Da ) t = O(1) � for c < g − 1 (Da) ∞ , = for c > g − 1 (Da) 0 , The front speed is given by c = g − 1 (Da). G¨ artner & Freidlin (1979)

Fronts in Flows The large- t limit Regimes I & II Regime III Summary FKPP vs G Conclusions The eigenvalue problem for the speed We solve for g ( c ) via an eigenvalue equation f ( q ) φ = Pe − 1 ∆ φ − ( u 1 , u 2 ) · ∇ φ − 2Pe − 1 q ∂ x φ + ( u 1 q + Pe − 1 q 2 ) φ, where f is the Legendre transform of g g ( c ) = sup ( qc − f ( q )) , q > 0 and φ ( x , y ) is 2 π -periodic in x with ∂ y φ = 0 at y = 0 , 1.