1/29 Thermosolutal Convection at Infinite Prandtl Number with or without rotation: Bifurcation and Stability in Physical Space Jungho Park Department of Mathematics New York Institute of Technology SIAM Conference on Analysis of PDE Nov. 15, 2011
2/29 1.Motivation This is a part of a research program to study - low frequency variability of the atmosphere and oceanic flows Primary goal in climate dynamics is - to document the presence of climate low frequency variability - to verify the robustness of this variability s characteristics to changes in model parameters - to help explain its physical mechanisms
3/29 Governing Equations for the atmosphere or ocean or coupled atmosphere-ocean models - general equations of hydrodynamic equations with (i) conservation laws for quantities such as the energy, humidity and salinity (ii) proper boundary and interface conditions Typical sources of climate low frequency variability: - Thermosolutal(vertical) circulation. - Wind-driven(horizontal) circulation. - El Niño Southern oscillation.
2. Methods Bifurcation theory enables us to determine how qualitatively different flow regimes appear and disappear as a control parameter varies, and to explore the theoretical limits of predicting these flow regimes. - Lyapunov-Schmidt Method - Krasnoselskii-Rabinowitz Theorem - Hopf Bifurcation Numerical Studies - Pseudo-arclength continuation methods [2, 6] applied to atmospheric and oceanic models Rigorous Analysis for the original PDE models associated with typical phenomena 4/29
5/29 3. Thermosolutal Convection The main objective is to study the stability and transitions of the Thermosolutal convection in both phase and physical space. Non-dimensional Boussinesq equations: [ ] 1 u t +(u )u = u p+ Ta k u+(r T T R S S)k, (1) P r T t = T +u 3 (u )T, (2) S t = τ S +u 3 (u )S, (3) u = 0, (4)
6/29 Parameters: R T = αg(t 0 T 1 )h 3 κ T ν R S = γg(s 0 S 1 )h 3 κ T ν, thermal Rayleigh number,, salinity Rayleigh number, P r = ν κ T, Prandtl number, Ta = 4Ω2 h 2 ν 2 Taylor number, τ = κ S κ T, Lewis number,
7/29 Infinite Prandtl number Convection : u p+ Ta k u+( λt ηs)k = 0, (5) T t = T + λu 3 (u )T, (6) S t = τ S + ηu 3 (u )S, (7) u = 0. (8) where, λ = R T and η = R S are system parameters. Domain: D = R 2 (0,1) R 3
8/29 Initial and Boundary Conditions (u,t,s)(x,t) = (u,t,s)(x 1 +k 1 L 1,x 2 +k 2 L 2,x 3,t) for any k 1,k 2 Z, u 1 T = S = 0, u 3 = 0, = u 2 = 0 at x 3 = 0,1, (10) x 3 x 3 (u,t,s) t=0 = (u 0,T 0,S 0 ). (11) (9) Function Spaces: H = L 2 (D) 2, V = {ψ = (T,S) H 1 (D) 2 H ψ is periodic in x 1,x 2 direction, ψ = 0 at x 3 = 0,1}, H 1 = V H 2 (D) 2.
9/29 Operator Equation: Define L λη = A+B λη : H 1 H and G : H 1 H by A(T,S) = ( T, τ S), B λη (T,S) = ( λu 3 (T,S), ηu 3 (T,S)), G(T,S) = ( (u(t,s) )T, (u(t,s) )S). (12) Then we have an operator equation, which is equivalent to the (5) (7), in terms of only T and S: where ψ = (T,S). dψ dt = L ληψ +G(ψ) (13)
10/29 4. Physical Phenomena Thermal structure of the Earth s mantle or the evolution of the Earth, Pr 10 24 Fluids with Pr > 1 convects in a similar fashion Convection cells have almost uniform size for the lower Rayleigh number, in particular, near the critical number R c but become irregular for the higher Rayleigh number. Free-Free boundary condition(first considered by Rayleigh) : can be realized only under very artificial conditions but theoretically interested. The results can be applied for other physically sound boundary conditions.
11/29 5. 2D Non-Rotating Convection Eigenvalue problem of linear equation: L λη ψ = β(λ,η)ψ, which is equivalent to u+ p ( λt ηs)k = 0, T + λu 2 = β(λ,η)t, τ S + ηu 2 = β(λ,η)s, u = 0. (14)
12/29 Separation of variables: For Ψ = (u 1,u 2,T,S), Ψ(x 1,x 2 ) = p(x 1,x 2 ) = where, a k = 2kπ L. (u 1 k (x 2),u 2 k (x 2),T k (x 2 ),S k (x 2 ))e ia kx 1, (15) k=0 p k (x 2 )e ia kx 1, (16) k=0
13/29 Lemma 1 Let λ c = η +(1+τ) 27π4 4 and λ = λ c. If η > η, then the linear eigenvalue problem (14) of the non-rotating thermosolutal convection has two pure imaginary eigenvalues for all k 0 and j 1. However, the problem has two real eigenvalues if η < η. Here, η = τ2 1 τ 27π 4 4.
Figure 1: Instability regimes in λ η plane 14/29
15/29 Eigenvalues and Eigenvectors: For η < η = τ2 1 τ 27π 4 4, Ψ k2j 1 (x 1,x 2 ) = (u 1 k2j 1,u2 k2j 1,T k2j 1,S k2j 1 ) = ( (jπ)sina k x 1 cosjπx 2,a k cosa k x 1 sinjπx 2, c kjτ cosa k x 1 sinjπx 2,c kjτ cosa k x 1 sinjπx 2 ), Ψ k2j (x 1,x 2 ) = (u 1 k2j,u2 k2j,t k2j,s k2j ) = ((jπ)cosa k x 1 cosjπx 2,a k sina k x 1 sinjπx 2, c kjτ sina k x 1 sinjπx 2,c kjτ sina k x 1 sinjπx 2 ). (17) (18) where, c kjτ = a 2 k +(jπ) 2 1 τ = c kj 1 τ, c 2 kj = (2kπ/L)2 +(jπ) 2
β k,2j 1 (λ) = β k,2j (λ) = (λ η)a2 k 2[a 2 k +(jπ)2 ] 2 (1+τ) [a 2 k 2 +(jπ)2 ] = (λ η) a2 k 2c 4 kj (1+τ) 2 c 2 kj, The Principle of Exchange of Stabilities: < 0 if λ < λ c, β 11 (λ) = β 12 (λ) = 0 if λ = λ c > 0 if λ > λ c, β kj (λ c ) < 0 for (k,j) (1,1),(1,2), where, λ c = η +(1+τ) 27π4 4. 16/29
17/29 Theorem 1 Assume that η < η, we have 1. ψ = (T,S) = 0 is locally asymptotically stable for λ λ c. 2. The equations (5) (7) bifurcates from the basic solution (ψ,λ) = (0,λ c ) to an attractor Σ λ for λ > λ c. 3. Σ λ consists of exactly one cycle of steady state solutions. 4. Σ λ is homeomorphic to S 1.
18/29 Sketch of Proof Krasnoselskii Theorem on H odd = {ψ H ψ is an odd function in x 1 direction}, H odd 1 = H 1 H odd. implies that L λ,η +G bifurcates from the trivial solution to a steady state solution since β 11 (λ) is simple in H odd. Since the solutions are translation invariant if ψ 0 is a steady state solution, then represent S 1. ψ(x,t) ψ(x 1 +θ,x 2,t) Γ = {ψ 0 (x 1 +θ,x 2 ) θ R}
19/29 Center Manifold Reduction: For the Fourier expansion of ψ, ψ = (y k2j 1 ψ k2j 1 +y k2j ψ k2j ), k 0,j 1 the reduction equations on the center manifold: dy 11 dt dy 12 dt 1 = β 11 (λ)y 11 + < G 2 (ψ,ψ),ψ 11 > H, < ψ 11,ψ 11 > H 1 = β 11 (λ)y 12 + < G 2 (ψ,ψ),ψ 12 > H, < ψ 12,ψ 12 > H (19) where G 2 is the bilinear operator of G such that G 2 (ψ 1,ψ 2 ) = ( (u(t 1 ) )T 2, (u(s 1 ) )S 2 ), < G 2 (ψ 1,ψ 2 ),ψ 3 > H = [(u(t 1 ) )T 2 T 3 +(u(s 1 ) )S 2 S 3 ]dx. Ω
20/29 Let Φ be the center manifold function then ψ = y 11 ψ 11 +y 12 ψ 12 +Φ. By the approximation of the center manifold function, Φ(y) = Φ kj (y)ψ kj +O( β(λ) y 2 )+o( y 2 ), (k,j) (1,1),(1,2) where Φ kj (y) = 1 β kj (λ) 1 p,q 2 a kj pq =< G 2 (ψ 1p,ψ 1q ),ψ kj > H. a kj pqy 1p y 1q,
21/29 Computation of cubic coefficients of the reduced equation: dy 11 dt dy 12 dt = β 11 (λ)y 11 + = β 11 (λ)y 12 + 1 π 4 c 2 11τ c2 02τ < ψ 11,ψ 11 > H β 03 (λ) (y 2 11 +y 2 12)y 11 +O( y 11 3 + y 12 3 )+o( y 11 3 + y 12 3 ), 1 π 4 c 2 11τ c2 02τ < ψ 12,ψ 12 > H β 03 (λ) (y 2 11 +y 2 12)y 12 +O( y 11 3 + y 12 3 )+o( y 11 3 + y 12 3 ). (20) which implies that (i) ψ = 0 is locally asymptotically stable for λ λ c (ii) The attractor Σ λ is homeomorphic to S 1 for λ > λ c
22/29 6. Structural Stability D r (D,R 2 ) = {v C r (D,R 2 ) v = 0,v 2 = 0 at x 2 = 0,1}, B1(D,R r 2 ) = {v D r (D,R 2 ) v 2 = v 1 = 0 at x 2 = 0,1}, x 2 B2(D,R r 2 ) = {v B1(D,R r 2 ) vdx = 0}. Definition 2 Two vector fields u, v C r (D,R 2 ) are called topologically equivalent if there exists a homeomorphism of ϕ : D D, which takes the orbits of u to that of v and preserves their orientations. Definition 3 Let X = D r (D,R 2 ). A vector field v X is called structurally stable in X if there exists a neighborhood U X of v such that for any u U, u and v are topologically equivalent. D
23/29 Theorem 4 For any ψ 0 = (T 0,S 0 ) H\Γ, there exists a time t 0 0 such that for any t t 0, the associated vector field u = u(t,ψ,ψ 0 ) is structurally stable and is topologically equivalent to the structure as shown in Figure 2, where Ψ = (u(t,ψ 0 ),ψ(t,ψ 0 )) is the solution of (5) (7), Ψ 0 = (u 0 (ψ 0 ),ψ 0 ) ( H H)\Γ and Γ is the stable manifold of ψ = 0 with co-dimension 2 in H. where, H = {u L 2 (D) 2 u = 0, D and u 2 = 0 at x 2 = 0,1}, udx = 0, u 1 is periodic in x 1 direction H 1 = {u H H 2 (D) 2 u satisfies (9) and (10)}.
24/29 Figure 2: The structure of solutions of Thermosolutal convection with infinite Prandtl number.
25/29 7. 3D Rotating Convection Lemma 2 Let λ c = η +(τ +1)c 4 111 + π2 (τ+1)(c 4 111 +Ta ) and λ = λ a 2 c. If 11 η > η, then the linear eigenvalue problem (14) of the rotating thermosolutal convection has two pure imaginary eigenvalues for all k 1,k 2 0 and j 1. However, the problem has two real eigenvalues if η < η. Here, η = τ2 1 τ a 2 11 c4 111 +π2 (c 4 111 + Ta) a 2 11
26/29 First critical thermal Rayleigh number λ c : - 2D non-rotating convection: - 3D rotating convection: λ c = η +(1+τ) 27π4 4 λ c = η +(τ +1)c 4 111 + π2 (τ +1) ( c 4 111 + Ta) a 2 11 For sufficient large Ta, the asymptotic behavior of λ c is ( π 2 ) 2/3 ) Ta π 2 2/3 λ c 3(τ +1) < 6( Ta 2/3 2 2 =17.3906142 Ta 2/3.
27/29 20 18 16 log Rc 14 12 10 8 6 0 5 10 15 20 25 30 log Ta Figure 3: The values of R c T = λ c determined by Ta.
28/29 8. Further Studies Transitions of solutions in physical space - The western boundary current separation Wind driven circulation Tropical atmospheric circulations
29/29 9. References References [1] B. Chen, M. Ghil, E. Simonnet, and S. Wang, Hopf bifurcation in quasi-geostrophic channel flow, SIAM J. Appl. Math., Vol.64, (2003), pp. 343 368. [2] B. Legras and M. Ghil, Persistent anomalies blocking and variations in atmospheric predictability, J. Atmos. Sci., vol. 42, (1985), pp. 433 471. [3] J. Park, Two-Dimensional Infinite Prandtl Number Convection : Structure of Bifurcated Solutions, J. Nonlinear Sci., Vol.17, (2007) pp.199-220. [4] J. Park, Bifurcation of Infinite Prandtl Number Rotating Convection, Nonlinear Anal., Vol.73, No.7, 2010-2021, 2010. [5] J. Park, Thermosolutal Convection at Infinite Prandtl Number with or without rotation: Bifurcation and Stability in Physical Space, J. Math. Phys., 52, 053701, 2011. [6] S. Speich, H. Dijkstra, and M. Ghil, Successive bifurcations in a shallow water model, applied to the wind driven ocean circulation, Nonlin. Proc. Geophys., vol.2, (1995), pp. 241 268.