Semi-geostrophic equations A large-scale model in meteorology - PowerPoint PPT Presentation

Semi-geostrophic equations A large-scale model in meteorology Stefania Lisai Supervised by B. Pelloni (HWU) and J. Vanneste (UoE) Evolution Equations and Friends (6th October 2017)

Historical overview ❼ 1948 - SG are introduced by Eliassen, in [Eli48]; ❼ 1971 - rediscovered by Sir Hoskins, in [Hos71]; ❼ 1987 - Cullen et al introduce the Stability principle , in [CS87]; ❼ 1998 - Benamou and Brenier construct a weak-type solution to SG in a dual space, in [BB98].

Plan of the presentation 1. Existence of solutions: dual space ❼ Derivation of the dual system ❼ Existence of dual weak solutions ❼ A space-preserving transformation 2. Ongoing and future work ❼ Surface semi-geostrophic equations ❼ Stability: energy minimisation ❼ Explicit solutions of SG

Benamou and Brenier’s transformation [BB98] Incompressible SG on a rigid domain Ω ⊂ R 3 : ∂ t u g 1 + u · ∇ u g 1 − u 2 + ∂ 1 p = 0 ∂ t u g 2 + u · ∇ u g 2 + u 1 + ∂ 2 p = 0 ∇ · u = 0 ( ∂ t + u · ∇ ) ρ = 0 u g 1 := − ∂ 2 p u g 2 := ∂ 1 p ∂ 3 p = − ρ u · n = 0 on ∂ Ω p ( · , 0) = p 0 , derived as an asymptotic limit of Euler equations.

Benamou and Brenier’s transformation [BB98] Incompressible SG on a rigid Introduce generalised pressure P domain Ω ⊂ R 3 : P ( x , t ) = p ( x , t ) + 1 2( x 2 1 + x 2 ∂ t u g 1 + u · ∇ u g 2 ) 1 − u 2 + ∂ 1 p = 0 ∂ t u g 2 + u · ∇ u g 2 + u 1 + ∂ 2 p = 0   x 1 + ∂ 1 p ( x , t )  . ∇ · u = 0 x 2 + ∂ 2 p ( x , t ) T ( x , t ) = ∇ P ( x , t ) =  ∂ 3 p ( x , t ) ( ∂ t + u · ∇ ) ρ = 0 u g 1 := − ∂ 2 p Equations for u and T u g 2 := ∂ 1 p D t T = J ( T − Id Ω ) ∂ 3 p = − ρ ∇ · u = 0 u · n = 0 on ∂ Ω u · n = 0 on ∂ Ω p ( · , 0) = p 0 , T 0 ( x ) = ( x 1 , x 2 , 0) + ∇ p 0 ( x ) derived as an asymptotic limit of T t = ∇ P t Euler equations.

Working on the dual space Eulerian physical space D t T = J ( T − Id Ω ) ∇ · u = 0 Lagrangian physical space T t = ∇ P t P ( · , 0) = P 0 Ω if u x locally Lipschitz ∂ t ˜ T = J ( ˜ T − X ) ∂ t X ( x , t ) = u ( X ( x , t ) , t ) u · n = 0 on ∂ Ω ∇ · u = 0 ˜ T t = ∇ P t ◦ X t ˜ T ( · , 0) = P 0 Dual space X ( x , 0) = x Ω ∂ t α + U · ∇ α = 0 u · n = 0 on ∂ Ω t ∈ C 2 ( R 3 ), if ∇ P t diffeo + P ∗ U ( y , t ) = J ( y − ∇ P ∗ ( y , t )) using weak formulation α = det D 2 P ∗ +change of variables y = T t ( x ) α 0 = ∇ P 0# L Ω R 3

Working on the dual space Eulerian physical space D t T = J ( T − Id Ω ) ∇ · u = 0 Lagrangian physical space T t = ∇ P t P ( · , 0) = P 0 Ω if u x locally Lipschitz ∂ t ˜ T = J ( ˜ T − X ) ∂ t X ( x , t ) = u ( X ( x , t ) , t ) u · n = 0 on ∂ Ω ∇ · u = 0 ˜ T t = ∇ P t ◦ X t ˜ T ( · , 0) = P 0 Dual space X ( x , 0) = x Ω ∂ t α + U · ∇ α = 0 u · n = 0 on ∂ Ω t ∈ C 2 ( R 3 ), if ∇ P t diffeo + P ∗ U ( y , t ) = J ( y − ∇ P ∗ ( y , t )) using weak formulation α = det D 2 P ∗ +change of variables y = T t ( x ) α 0 = ∇ P 0# L Ω R 3

Existence of weak solution in dual space Theorem ([BB98]) Let Ω ⊂ R 3 be open bounded Lipschitz set and 0 ≤ α 0 ∈ L p ( R 3 ) compactly supported and � α 0 � L 1 ( R 3 ) = L 3 (Ω) . For any τ > 0 and p > 3 there exist [0 , τ ); L p ( R 3 ) α ∈ L ∞ � � ≥ 0 , α t compactly supported and α ( · , 0) = α 0 , [0 , τ ); W 1 , ∞ (Ω) ψ ∈ L ∞ � � convex with ψ ∗ ∈ L ∞ � [0 , τ ); W 1 , ∞ ( R 3 ) � convex in space, � 3 , loc ( R 3 ) ∩ BV loc ( R 3 )) � U ∈ L ∞ ([0 , τ ); L ∞ solutions of � τ � R 3 ( ∂ t ξ ( y , t ) + U ( y , t ) · ∇ ξ ( y , t )) α ( y , t ) dy dt ❼ 0 0 ( R 3 × [0 , τ )) � = − R 3 ξ ( y , 0) α ( y , 0) dy ∀ ξ ∈ C ∞ ∀ ( y , t ) ∈ R 3 × [0 , τ ) ❼ U ( y , t ) = J ( y − ∇ ψ ∗ ( y , t )) ∀ f ∈ C c ( R 3 ) . � � Ω f ( ∇ ψ ( x , t )) dx = R 3 f ( y ) α ( y , t ) dy ❼

Back to physical space? Given ( α, ψ ) weak solution of the dual problem, the couple ( u , P ) with P = ψ u ( x , t ) := ( ∂ t ∇ P ∗ t ◦ ∇ P t )( x ) + ( D 2 P ∗ t ◦ ∇ P t )( J ( ∇ P t ( x ) − x )) is formally a solution in Eulerian space. It is not clear what this means, because D 2 P ∗ t is a distribution (a measure).

Hoskins’ transfomation [Hos75] On the domain Ω = R 2 × [0 , 1] , Physical space the transformation   x 1 + ∂ 1 p H t T t y = H t ( x ) = x 2 + ∂ 2 p   x 3 Dual spaces brings to the system in dual space ∂ t β + U · ∇ β + ∂ 3 ( β u 3 ) = 0 ( ∂ t + U · ∇ + u 3 ∂ 3 )( 1 β ∂ 2 3 Φ) = 0 β = 1 + ( ∂ 2 1 Φ ∂ 2 2 Φ − ( ∂ 1 , 2 Φ) 2 ) − ( ∂ 2 1 Φ + ∂ 2 2 Φ) ( ∂ t + U · ∇ ) ∂ 3 Φ = 0 on ∂ Ω U = ( − ∂ 2 Φ , ∂ 1 Φ , 0) , 2 ( ∂ 1 p ( x , t ) 2 + 1 where Φ( y , t ) = Φ( H t ( x ) , t ) = p ( x , t ) + 1 2 ( ∂ 2 p ( x , t )) 2 .

Surface semi-geostrophic equations (SSG) If the potential vorticity is uniform, then β = ∂ 2 3 Φ and the dual system is simplified to 1 + ( ∂ 2 1 Φ ∂ 2 2 Φ − ( ∂ 1 , 2 Φ) 2 ) − (∆Φ) = 0 on Ω ( ∂ t − ∂ 2 Φ ∂ 1 + ∂ 1 Φ ∂ 2 ) ∂ 3 Φ = 0 on { y 3 = 0 , 1 } ∂ 3 Φ( · , 0) = g on { y 3 = 1 } ∂ 3 Φ( · , t ) = 0 on { y 3 = 0 }∀ t ∈ [0 , τ ) .

Surface semi-geostrophic equations (SSG) If the potential vorticity is uniform, then β = ∂ 2 3 Φ and the dual system is simplified to 1 + ( ∂ 2 1 Φ ∂ 2 2 Φ − ( ∂ 1 , 2 Φ) 2 ) − (∆Φ) = 0 on Ω ( ∂ t − ∂ 2 Φ ∂ 1 + ∂ 1 Φ ∂ 2 ) ∂ 3 Φ = 0 on { y 3 = 0 , 1 } ∂ 3 Φ( · , 0) = g on { y 3 = 1 } ∂ 3 Φ( · , t ) = 0 on { y 3 = 0 }∀ t ∈ [0 , τ ) . Introducing a Dirichlet-to-Neumann operator T we write the system above in R 2 Physical space ( ∂ t + w · ∇ ) f = 0 w = ∇ ⊥ ψ H t T t f = T ( ψ ) f ( · , 0) = g . Dual spaces

Cullen’s selection principle Question: What are the physically meaningful solutions of SG? Principle (Stability principle [CS87]) Stable solutions of SG correspond to those which, for any fixed time t , minimise the geostrophic energy � � 1 � 2 | u g ( x , t ) | 2 + ρ x 3 E = dx Ω with respect to the rearrangements of particles that conserve the absolute momentum ( u g 1 − x 2 , u g 2 + x 1 ) and the density ρ .

Cullen’s selection principle Principle (Stability principle in OT framework) A solution ( u , P ) of SG in Eulerian space � ( ∂ t + u · ∇ ) ∇ P = J ( ∇ P − Id R 3 ) ∇ · u = 0 is stable if, for any fixed time t , ∇ P ( · , t ) is the optimal transport map that solves the following Monge problem | x − T ( x ) | 2 � min dx 2 T ∈T t Ω with T t := { T : Ω → R 3 | T # L Ω = ∇ P t # L Ω . }

Other existence results ❼ Cullen and Gangbo (2001): Weak dual solutions for incompressible SG shallow water on a free surface; ❼ Cullen and Maroofi (2003): Weak dual solution for fully 3-D compressible SG; ❼ Cullen and Feldman (2005): Weak Lagrangian solution for incompressible 3-D rigid-boundary SG; ❼ Ambrosio et al. (2012): Eulerian weak solutions for 2-D SG on periodic domain; ❼ Ambrosio et al. (2012): Global-in-time Eulerian weak solution of 3-D SG on convex domain. Question: Can we build an explicit solution in Eulerian space?

References [BB98] Jean-David Benamou and Yann Brenier. Weak Existence for the Semigeostrophic Equations Formulated As a Coupled Monge-Amp` ere/Transport Problem. SIAM J. Appl. Math. , 58(5):1450–1461, October 1998. [CS87] M. J. P. Cullen and G. J. Shutts. Parcel Stability and its Relation to Semigeostrophic Theory. Journal of Atmospheric Sciences , 44:1318–1330, May 1987. [Eli48] Arnt Eliassen. The quasi-static equations of motion with pressure as independent variable. Geofis. Publ. , 17(3):5–44, 1948. [Hos71] Brian J. Hoskins. Atmospheric frontogenesis models: some solutions. Quarterly Journal of the Royal Meteorological Society , 97:139–151, 1971. [Hos75] Brian J. Hoskins. The Geostrophic Momentum Approximation and the Semi-Geostrophic Equations. Journal of Atmospheric Sciences , 32:233–242, February 1975.