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

Random flow

Haslam & Ronney (1995)

Cellular vortex flow

Pocheau & Harambat (2008)

Fronts in Flows The large-t limit Regimes I & II Regime III Summary FKPP vs G Conclusions

Reactive fronts in experimental flows

Random flow

Haslam & Ronney (1995)

Cellular vortex flow

Pocheau & Harambat (2008)

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 ∂tθ(x, t) = κ∆θ(x, t) + 1 τ θ(1 − θ), θ(x, 0) =

• 1

if x ≥ 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 ∂tθ(x, t) = κ∆θ(x, t) + 1 τ θ(1 − θ), θ(x, 0) =

• 1

if x ≥ 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.

κ=1

θ(x, t) = 0 θ(x, t) = 1 κ 1 τ 10 1 0.1

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 ∂tθ(x, t) + u(x) · ∇θ(x, t) = Pe−1∆θ(x, t) + Da θ(1 − θ), where Pe = Uℓ/κ and Da = ℓ/Uτ 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 ∂tθ(x, t) + u(x) · ∇θ(x, t) = Pe−1∆θ(x, t) + Da θ(1 − θ), where Pe = Uℓ/κ and Da = ℓ/Uτ 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 2π-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 2π-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 2π-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 2π-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 2π-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 I O(Pe−1) Pe− 3

4 C1(PeDa)

boundary-layer analysis II O((log Pe)−1) (log Pe)−1C2(Da log Pe) boundary-layer analysis III O(Pe) C3(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 I O(Pe−1) Pe− 3

4 C1(PeDa)⋆

boundary-layer analysis II O((log Pe)−1) (log Pe)−1C2(Da log Pe) boundary-layer analysis III O(Pe) C3(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 I O(Pe−1) Pe− 3

4 C1(PeDa)⋆

boundary-layer analysis∗ II O((log Pe)−1) (log Pe)−1C2(Da log Pe) boundary-layer analysis∗ III O(Pe) C3(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 I O(Pe−1) Pe− 3

4 C1(PeDa)⋆

boundary-layer analysis∗ II O((log Pe)−1) (log Pe)−1C2(Da log Pe) boundary-layer analysis∗ III O(Pe)† C3(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 ∂tθ(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 θ(x, t) ≈ e−t(g(c)−Da ) where c = x t = O(1) =

• ∞,

for c < g−1(Da) 0, for c > g−1(Da) 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 ∂tθ(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 θ(x, t) ≈ e−t(g(c)−Da ) where c = x t = O(1) =

• ∞,

for c < g−1(Da) 0, for c > g−1(Da) 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 ∂tθ(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 θ(x, t) ≈ e−t(g(c)−Da ) where c = x t = O(1) =

• ∞,

for c < g−1(Da) 0, for c > g−1(Da) 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 − (u1, u2) · ∇φ − 2Pe−1q∂xφ + (u1q + Pe−1q2)φ, where f is the Legendre transform of g g(c) = sup

q>0

(qc − f (q)), and φ(x, y) is 2π-periodic in x with ∂yφ = 0 at y = 0, 1.

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 − (u1, u2) · ∇φ − 2Pe−1q∂xφ + (u1q + Pe−1q2)φ, where f is the Legendre transform of g g(c) = sup

q>0

(qc − f (q)), and φ(x, y) is 2π-periodic in x with ∂yφ = 0 at y = 0, 1.

• r, look for solutions of the form

exp(−qx + (f (q) + Da)t)φ with φ > 0 from where c = inf

q>0

f (q) + Da q = g−1(Da).

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 − (u1, u2) · ∇φ − 2Pe−1q∂xφ + (u1q + Pe−1q2)φ, where f is the Legendre transform of g g(c) = sup

q>0

(qc − f (q)), and φ(x, y) is 2π-periodic in x with ∂yφ = 0 at y = 0, 1. For u = 0, f (q) = Pe−1q2 which recovers classic result c0 = 2

• Da/Pe = 2
• κ/τ.

Fronts in Flows The large-t limit Regimes I & II Regime III Summary FKPP vs G Conclusions

Example: Pe = 250

−1 −0.5 0.5 1 0.5 1 1.5 2 2.5 3 3.5 4 4.5

c g

Da Near the origin, homogenization theory applies: g(c) ≈

√ Pe 4 c2

(since κeff ≈ 2νPe1/2 and ν ≈ 0.58)

Childress (1979), Shraiman (1987), Soward (1987)

Fronts in Flows The large-t limit Regimes I & II Regime III Summary FKPP vs G Conclusions

Example: Pe = 250

−1 −0.5 0.5 1 0.5 1 1.5 2 2.5 3 3.5 4 4.5

c g

Da

0.5 1 1 2 3 4

c Da

Near the origin, homogenization theory applies: g(c) ≈

√ Pe 4 c2

(since κeff ≈ 2νPe1/2 and ν ≈ 0.58)

Childress (1979), Shraiman (1987), Soward (1987)

Fronts in Flows The large-t limit Regimes I & II Regime III Summary FKPP vs G Conclusions

Regime I: Da = Pe−1, c = Pe− 3

4C1(PeDa)

Cell interior solve eigenvalue equation along streamlines, d dψ

• a(ψ) dφ

dψ

• ∼ f b(ψ)φ,

a(ψ) =

• |∇ψ|dℓ, b(ψ) =
• dℓ

|∇ψ| with initial condition at cell center and obtain lim

ψ→0± φ−1 dφ

dψ ∼ ∓F(f )

• n the boundaries.

Childress (1979), Sowers (1987), Haynes & Vanneste (2014)

Fronts in Flows The large-t limit Regimes I & II Regime III Summary FKPP vs G Conclusions

Regime I: Da = Pe−1, c = Pe− 3

4C1(PeDa)

Cell interior solve eigenvalue equation along streamlines, d dψ

• a(ψ) dφ

dψ

• ∼ f b(ψ)φ,

a(ψ) =

• |∇ψ|dℓ, b(ψ) =
• dℓ

|∇ψ| with initial condition at cell center and obtain lim

ψ→0± φ−1 dφ

dψ ∼ ∓F(f )

• n the boundaries.

Matching with the boundary layer: f ∼ Pe−1F −1

π2ν 4 q2Pe1/2

.

Childress (1979), Sowers (1987), Haynes & Vanneste (2014)

Fronts in Flows The large-t limit Regimes I & II Regime III Summary FKPP vs G Conclusions

Regime I: Da = O(Pe−1), c = Pe− 3

4C1(PeDa)

20 40 60 80 10 20 30 40 50

PeDa Pe3/4 c

Pe=500 Pe=250 Pe=125 Pe=50

Fronts in Flows The large-t limit Regimes I & II Regime III Summary FKPP vs G Conclusions

Regime I: Da = O(Pe−1), c = Pe− 3

4C1(PeDa)

20 40 60 80 10 20 30 40 50

PeDa Pe3/4 c

Pe=500 Pe=250 Pe=125 Pe=50

2.5 5 7.5 10 2.5 5 7.5 10

PeDa Pe3/4 c

⇐ Homogenization theory is only valid for Da ≪ Pe−1 Zoom near the origin

Fronts in Flows The large-t limit Regimes I & II Regime III Summary FKPP vs G Conclusions

Regime I: Da = O(Pe−1), c = Pe− 3

4C1(PeDa)

20 40 60 80 10 20 30 40 50

PeDa Pe3/4 c

Pe=500 Pe=250 Pe=125 Pe=50

(same as for the 2D setup)

2.5 5 7.5 10 2.5 5 7.5 10

PeDa Pe3/4 c

⇐ Homogenization theory is only valid for Da ≪ Pe−1 Zoom near the origin

Fronts in Flows The large-t limit Regimes I & II Regime III Summary FKPP vs G Conclusions

Regime II: Da = (log Pe)−1, c = (log Pe)−1C2(Da log Pe)

◮ Away from the corners Φ is controlled by diffusion across

streamlines.

◮ Around each corner, Φ jumps by (16Pe)f /2.

Matching eval pb that involves 4 corners (16Pe)

f 2 ˆ

Φ(ζ) = (Kˆ Φ)(ζ), where K is a 4 × 4 matrix composed of linear integral

• perators

⇒ f = 2 log λ(q,f )

log(16Pe) .

20 40 60 80 100 120 1 2 3 4 5 6

Dalog16Pe log16Pec

Pe=500 Pe=250 Pe=125 Pe=50 1 2 3 4 5 1 2 3 4 5

Haynes & Vanneste (2014)

Fronts in Flows The large-t limit Regimes I & II Regime III Summary FKPP vs G Conclusions

Regime III: Da = O(Pe), c = C3(Da/Pe)

Look for solns of the linearized θ-eqn of the form θ(x, y, t) ≈ e−PeI(x,y,t)+Dat =

• ∞,

for I(x, y, t) > Dat/Pe 0, for I(x, y, t) < Dat/Pe, The front speed then satisfies G3(c) = lim

t→∞ I(ct, y, t)t−1 = Da/Pe.

At leading order, ∂tI + H(∇I, x, y) = 0 where H = ∇I2 + u · ∇I.

Fredlin & Wentzell (1984), Majda & Souganidis (1994)

Fronts in Flows The large-t limit Regimes I & II Regime III Summary FKPP vs G Conclusions

Regime III: Da = O(Pe), c = C3(Da/Pe)

This leads to g(c) = Pe G3(c) where G3(c) = lim

t→∞ I(ct, y, t)t−1 = lim t→∞

1 4t inf

ϕ(·)

t ˙ ϕ(s)−u(ϕ(s))2 ds, s.t. ϕ(0) = (0, ·), ϕ(t) = (ct, ·).

π −π π

x y

As c → ∞, instanton → straight line

We simplify by considering smooth periodic trajectories (2π, τ) ϕ(· + τ) = ϕ(·) + (2π, 0) where τ = 2π/c.

Fronts in Flows The large-t limit Regimes I & II Regime III Summary FKPP vs G Conclusions

Regime III: Da = O(Pe), c = C3(Da/Pe)

Slow instantons

For c ≪ 1, seek for an instanton ϕ∗(t) = (x(t), y(t)) satisfying cy′ ≈ − cos x sin y with x(0) = 0, y(0) = x(π/2c−1) = π/2, ˙ y(π/2c−1) = 0.

π

π π/2 π/2

Ι II

Region I: ¨ x ≈ x, ˙ y ≈ − sin y Region II: ˙ x ≈ sin x, ˙ y ≈ −y cos x. Matching: G3(c) ∼ (8/π)ce−π/c

Fronts in Flows The large-t limit Regimes I & II Regime III Summary FKPP vs G Conclusions

Regime III: Da = O(Pe), c = C3(Da/Pe)

Fast instantons

For c ≫ 1, seek for an instanton in the form ϕ(t) = (ct, y0)+c−1(x1(t), y1(t)) + . . . , where x1(t), x2(t) satisfy xi(2π/c) = xi(0) for i = 1, 2. Substituting + minimizing gives ϕ∗(t) =

• ct, π

2

• + c−1(0, −2 sin(ct)) + . . . ,

G3(c) = c2/4 − 3/8 + . . .

Fronts in Flows The large-t limit Regimes I & II Regime III Summary FKPP vs G Conclusions

Regime III: Da = O(Pe), c = C3(Da/Pe)

Speed

c = G −1

3

(γ) where γ = Da/Pe.

2 4 6 8 10 2 4 6

γ c

0.001 0.01 0.1 1 1 2 3 4

γ ≫ 1 c ∼ 2√γ

• c0:bare speed
• 1 +

3 16γ + . . .

• γ ≪ 1

c ∼ π/Wp(8γ−1)

Fronts in Flows The large-t limit Regimes I & II Regime III Summary FKPP vs G Conclusions

Regime III: Da = O(Pe), c = C3(Da/Pe)

Speed 2 4 6 8 10 0.5 1 1.5

Da c

Pe=50 Pe=125 Pe=250 Pe=500

Fronts in Flows The large-t limit Regimes I & II Regime III Summary FKPP vs G Conclusions

The three regimes together

5 10 15 20 0.5 1 1.5 2

Da c

Pe=125 Pe=250 Pe=500 Pe=50

Fronts in Flows The large-t limit Regimes I & II Regime III Summary FKPP vs G Conclusions

The three regimes together

5 10 15 20 0.5 1 1.5 2

Da c

Pe=125 Pe=250 Pe=500 Pe=50 0.5 1 0.2 0.4 0.6 0.8 1

Da c

Fronts in Flows The large-t limit Regimes I & II Regime III Summary FKPP vs G Conclusions

FKPP vs G equation

Burned Fuel Unburned Fuel Flame Front V

L

s n ( , ) G x t = G < G >

Xin & Yu (2014)

The sharp front when Da = O(Pe) is often approximated by a curve of a function G(x, t) satisfying ∂tG + u · ∇G = c0|∇G|.

◮ For u = 0, cG = cFK = c0 = 2

• Da/Pe.

◮ For u = 0?

Fronts in Flows The large-t limit Regimes I & II Regime III Summary FKPP vs G Conclusions

FKPP vs G equation

Burned Fuel Unburned Fuel Flame Front V

L

s n ( , ) G x t = G < G >

Xin & Yu (2014)

The sharp front when Da = O(Pe) is often approximated by a curve of a function G(x, t) satisfying ∂tG + u · ∇G = c0|∇G|.

◮ For u = 0, cG = cFK = c0 = 2

• Da/Pe.

◮ For u = 0?

Fronts in Flows The large-t limit Regimes I & II Regime III Summary FKPP vs G Conclusions

Variational Principles

The front speed is given by cG = 2π TG , where TG = inf

x(·)

• dx

˙ x(t) with | ˙ x(t) − u(x(t))|2 = c2

0.

The FKPP equation is analogously given by cFK = 2π TFK , where TFK = inf

x(·)

• dx

˙ x(t) with 1 T T | ˙ x(t) − u(x(t))|2dt = c2

0.

(Here x(T) = x(0) + (2π, 0)).

◮ The pointwise constraint on the relative velocity is replaced by

a slacker, time-averaged constraint. Thus, cFK ≥ cG.

Fronts in Flows The large-t limit Regimes I & II Regime III Summary FKPP vs G Conclusions

Variational Principles

The front speed is given by cG = 2π TG , where TG = inf

x(·)

• dx

˙ x(t) with | ˙ x(t) − u(x(t))|2 = c2

0.

The FKPP equation is analogously given by cFK = 2π TFK , where TFK = inf

x(·)

• dx

˙ x(t) with 1 T T | ˙ x(t) − u(x(t))|2dt = c2

0.

(Here x(T) = x(0) + (2π, 0)).

◮ The pointwise constraint on the relative velocity is replaced by

a slacker, time-averaged constraint. Thus, cFK ≥ cG.

Fronts in Flows The large-t limit Regimes I & II Regime III Summary FKPP vs G Conclusions

Numerical evaluation

◮ For shear flows u(x) = (u(y), 0) it is easy to show that

cFK = cG = c0 + maxyu(y).

◮ For cellular flows, the two variational problems are

approximated numerically.

2 4 6 8 10 0.01 0.02

Fronts in Flows The large-t limit Regimes I & II Regime III Summary FKPP vs G Conclusions

Asymptotic limits

0.01 0.02 0.05 0.1 0.2 0.3 0.5 0.2 0.4 0.6 0.8 1 1 2 3 4 5 10

• 100
• 10-1
• 10-2
• 10-3

◮ For c0 ≪ 1, cG ∼ − π 2 ln−1 c0π 8

• and cFK ∼

π Wp(32c−2

0 )

∗. ◮ For c0 ≫ 1, cG ∼ c0

• 1 + 3

4c−2

− 109

64 c−4

• and

cFK ∼ c0

• 1 + 3

4c−2

− 429

256c−4

• .

∗ For c0 ≪ 1, cFK ∼ −π log−1 c0 ∼ π

2 log−1 Pe.

Fronts in Flows The large-t limit Regimes I & II Regime III Summary FKPP vs G Conclusions

Asymptotic limits

0.01 0.02 0.05 0.1 0.2 0.3 0.5 0.2 0.4 0.6 0.8 1 1 2 3 4 5 10

• 100
• 10-1
• 10-2
• 10-3

◮ For c0 ≪ 1, cG ∼ − π 2 ln−1 c0π 8

• and cFK ∼

π Wp(32c−2

0 )

∗. ◮ For c0 ≫ 1, cG ∼ c0

• 1 + 3

4c−2

− 109

64 c−4

• and

cFK ∼ c0

• 1 + 3

4c−2

− 429

256c−4

• .

∗ For c0 ≪ 1, cFK ∼ −π log−1 c0 ∼ π

2 log−1 Pe.

Fronts in Flows The large-t limit Regimes I & II Regime III Summary FKPP vs G Conclusions

Conclusions

◮ Large deviation theory is a neat way to obtain the front speed:

Letting θ ≍ exp[−t(g(x/t) − Da)] gives: c = g−1(Da), where the rate function g is calculated by solving an eigenvalue problem.

◮ For the particular case of a cellular flow, we have identified

three regimes. For Pe ≫ 1, we obtain that:

• The homogenization result is a limiting case of Regime I.
• Regime II reveals a slow growth with Da and a logarithmic

dependence on Pe.

• Periodic instantons control Regime III which are

straightforward to calculate.

◮ The difference between the front speed obtained from the G

equation and that obtained from the FKPP equation is very small: the G equation only slightly underpredicts the front speed.

Fronts in Flows The large-t limit Regimes I & II Regime III Summary FKPP vs G Conclusions

Conclusions

◮ Large deviation theory is a neat way to obtain the front speed:

Letting θ ≍ exp[−t(g(x/t) − Da)] gives: c = g−1(Da), where the rate function g is calculated by solving an eigenvalue problem.

◮ For the particular case of a cellular flow, we have identified

three regimes. For Pe ≫ 1, we obtain that:

• The homogenization result is a limiting case of Regime I.
• Regime II reveals a slow growth with Da and a logarithmic

dependence on Pe.

• Periodic instantons control Regime III which are

straightforward to calculate.

◮ The difference between the front speed obtained from the G

equation and that obtained from the FKPP equation is very small: the G equation only slightly underpredicts the front speed.

Fronts in Flows The large-t limit Regimes I & II Regime III Summary FKPP vs G Conclusions

Conclusions

◮ Large deviation theory is a neat way to obtain the front speed:

Letting θ ≍ exp[−t(g(x/t) − Da)] gives: c = g−1(Da), where the rate function g is calculated by solving an eigenvalue problem.

◮ For the particular case of a cellular flow, we have identified

three regimes. For Pe ≫ 1, we obtain that:

• The homogenization result is a limiting case of Regime I.
• Regime II reveals a slow growth with Da and a logarithmic

dependence on Pe.

• Periodic instantons control Regime III which are

straightforward to calculate.

◮ The difference between the front speed obtained from the G

equation and that obtained from the FKPP equation is very small: the G equation only slightly underpredicts the front speed.

Fronts in Flows The large-t limit Regimes I & II Regime III Summary FKPP vs G Conclusions

Conclusions

◮ Large deviation theory is a neat way to obtain the front speed:

Letting θ ≍ exp[−t(g(x/t) − Da)] gives: c = g−1(Da), where the rate function g is calculated by solving an eigenvalue problem.

◮ For the particular case of a cellular flow, we have identified

three regimes. For Pe ≫ 1, we obtain that:

• The homogenization result is a limiting case of Regime I.
• Regime II reveals a slow growth with Da and a logarithmic

dependence on Pe.

• Periodic instantons control Regime III which are

straightforward to calculate.

◮ The difference between the front speed obtained from the G

equation and that obtained from the FKPP equation is very small: the G equation only slightly underpredicts the front speed.

Fronts in Flows The large-t limit Regimes I & II Regime III Summary FKPP vs G Conclusions

Conclusions

◮ Large deviation theory is a neat way to obtain the front speed:

Letting θ ≍ exp[−t(g(x/t) − Da)] gives: c = g−1(Da), where the rate function g is calculated by solving an eigenvalue problem.

◮ For the particular case of a cellular flow, we have identified

three regimes. For Pe ≫ 1, we obtain that:

• The homogenization result is a limiting case of Regime I.
• Regime II reveals a slow growth with Da and a logarithmic

dependence on Pe.

• Periodic instantons control Regime III which are

straightforward to calculate.

◮ The difference between the front speed obtained from the G

equation and that obtained from the FKPP equation is very small: the G equation only slightly underpredicts the front speed.

Fronts in Flows The large-t limit Regimes I & II Regime III Summary FKPP vs G Conclusions

Conclusions

◮ Large deviation theory is a neat way to obtain the front speed:

Letting θ ≍ exp[−t(g(x/t) − Da)] gives: c = g−1(Da), where the rate function g is calculated by solving an eigenvalue problem.

◮ For the particular case of a cellular flow, we have identified

three regimes. For Pe ≫ 1, we obtain that:

• The homogenization result is a limiting case of Regime I.
• Regime II reveals a slow growth with Da and a logarithmic

dependence on Pe.

• Periodic instantons control Regime III which are

straightforward to calculate.

◮ The difference between the front speed obtained from the G

equation and that obtained from the FKPP equation is very small: the G equation only slightly underpredicts the front speed.