Surface semi-geostrophic equations Stefania Lisai Supervised by B. - PowerPoint PPT Presentation

Surface semi-geostrophic equations Stefania Lisai Supervised by B. Pelloni (HWU), J. Vanneste (UoE) and M. Wilkinson (HWU) GFD workshop (19th June 2018)

Plan of the presentation 1. Semi-geostrophic equations ➒ History of SG ➒ Hoskins’ change of coordinates 2. Surface semi-geostrophic equations ➒ Comparison with SQG ➒ Local existence of classical solutions 3. Conclusion and future work

Semi-geostrophic equations ( ∂ t + u · ∇ ) u g − fv + ∂ x p = 0  (Euler-type equation)   ( ∂ t + u · ∇ ) v g + fu + ∂ y p = 0  (Euler-type equation)      div u = 0 (incompressibility condition)   f u g := ( − ∂ y p , ∂ x p , 0) (geostrophic wind)  g   θ = ∂ z p (hydrostatic balance)   g 0     ( ∂ t + u · ∇ ) θ = 0 (continuity equation for θ )  Boundary condition on a bounded domain Ω ⊂ R 3 : u · n = 0 on ∂ Ω

Semi-geostrophic equations ➒ u = u g + u a is the full ( ∂ t + u · ∇ ) u g − fv + ∂ x p = 0   velocity;  ( ∂ t + u · ∇ ) v g + fu + ∂ y p = 0     ➒ θ is the buoyancy   div u = 0   anomaly; f u g := ( − ∂ y p , ∂ x p , 0) ➒ f is the Coriolis g    θ = ∂ z p  frequency;  g 0     ➒ p is the pressure. ( ∂ t + u · ∇ ) θ = 0  Boundary condition on a bounded domain Ω ⊂ R 3 : u · n = 0 on ∂ Ω

Historical overview ❼ 1948 - introduction of SG (Eliassen); ❼ 1971 - SG for frontogenesis (Hoskins); ❼ 1975 - surface semi-geostrophic (SSG) (Hoskins); ❼ 1987 - Stability principle (Cullen and Shutts); ❼ 1998 - weak dual solution to SG (Benamou and Brenier); ❼ 2016 - SSG solved numerically (Badin and Ragone).

Hoskins’ change of coordinates 3  X = x + v g f Φ = p + | u g | 2   Y = y − u g f 2  Z = z  3 B. J. Hoskins, Journal of the Atmospheric Sciences, 32 (1975), no. 2

Hoskins’ change of coordinates 3  X = x + v g f Φ = p + | u g | 2   Y = y − u g f 2  Z = z  = ⇒ conservation laws for θ and PV + highly nonlinear equation for Φ 3 B. J. Hoskins, Journal of the Atmospheric Sciences, 32 (1975), no. 2

Surface semi-geostrophic (SSG) Regime of constant PV = ⇒ SSG.  ( ∂ t + w · ∇ ) θ = 0   w ( · , t ) = ∇ ⊥ T [ θ ( · , t )]  θ ( · , 0) = θ 0 ,  where T [ θ ] = Φ | z =1 with  ∆Φ = ε ( ∂ XX Φ ∂ YY Φ − ( ∂ XY Φ) 2 )    ∂ Z Φ | Z =0 = 0  ∂ Z Φ | Z =1 = θ − 1    � Ω Φ = 0 . 

SSG: comparison with SQG SSG and SQG are active scalar equations  ( ∂ t + w · ∇ ) θ = 0   w = ∇ ⊥ T [ θ ]  θ ( · , 0) = θ 0 ,  with different Neumann-to-Dirichlet operators: ➒ in SQG, T = ( − ∆) − 1 2 ; ➒ in SSG, T is not a pseudo-differential operator and it is associated to a highly non-linear boundary value problem.

SSG: comparison with SQG Figure: From Badin ’16 4 : Snapshots at T = 100 of θ at z = 1 for տ SQG, ր SQG under coordinate transformation with ε = 0 . 2, ւ SSG for ε = 0 . 2 in geostrophic coordinates, ց SSG for ε = 0 . 2 in physical coordinates. 4 F. Ragone and G. Badin, Journal of Fluid Mechanics, 792 (2016)

Existence of smooth solutions (L. and Wilkinson) Step 1: construction of the Neumann-to-Dirichlet operator. Fix time. Given small Rossby number ε > 0 and small θ = θ ( x , y ), there exists a smooth solution Φ of the boundary value problem  ∆Φ = ε ( ∂ xx Φ ∂ yy Φ − ( ∂ xy Φ) 2 )    ∂ z Φ | z =0 = 0  ∂ z Φ | z =1 = θ − 1    � Ω Φ = 0 .  The solution Φ is unique in a small ball. We define T [ θ ] := Φ | z =1 .

Existence of smooth solutions (L. and Wilkinson) Step 2: time-stepping. Let h = τ ∗ N , and define ➒ w (0) = ∇ ⊥ T [ θ 0 ]; ➒ θ (1) solution on [0 , h ] of ( ∂ t + w (0) · ∇ ) θ (1) = 0 , θ (1) ( · , 0) = θ 0 ; ➒ w (1) = ∇ ⊥ T [ θ (1) ( · , h )]; ➒ θ (2) solution on [ h , 2 h ] of ( ∂ t + w (1) · ∇ ) θ (2) = 0 , θ (2) ( · , h ) = θ (1) ( · , h ); and so on...

➒ ➒ ➒ Conclusion and future work We showed local existence of classical solutions of SSG, given a small initial datum and a small Rossby number.

Conclusion and future work We showed local existence of classical solutions of SSG, given a small initial datum and a small Rossby number. What next? ➒ Existence for any initial datum; ➒ Generalisation to weak solutions; ➒ Rigorous derivation of SG as an asymptotic limit of Boussinesq equations.

References [1] M. J. P. Cullen, A Mathematical Theory of Large-Scale Atmosphere/Ocean Flow . Imperial College Press, 2006. [2] B. J. Hoskins and F. P. Bretherton, Atmospheric Frontogenesis Models: Mathematical Formulation and Solution . Journal of Atmospheric Sciences, 29:11-37 (1972). [3] B. J. Hoskins, The Geostrophic Momentum Approximation and the Semi-Geostrophic Equations . Journal of Atmospheric Sciences, 32:233-242 (1975). [4] F. Ragone and G. Badin, A Study of Surface Semi-Geostrophic Turbulence: freely decaying dynamics , Journal of Fluid Mechanics, 729:740-774 (2016).