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 · ∇) ug − fv + ∂xp = 0 (Euler-type equation) (∂t + u · ∇) v g + fu + ∂yp = 0 (Euler-type equation) div u = 0 (incompressibility condition) f ug := (−∂yp, ∂xp, 0) (geostrophic wind) g g0 θ = ∂zp (hydrostatic balance) (∂t + u · ∇)θ = 0 (continuity equation for θ) Boundary condition on a bounded domain Ω ⊂ R3: u · n = 0

• n ∂Ω

Semi-geostrophic equations

                     (∂t + u · ∇) ug − fv + ∂xp = 0 (∂t + u · ∇) v g + fu + ∂yp = 0 div u = 0 f ug := (−∂yp, ∂xp, 0) g g0 θ = ∂zp (∂t + u · ∇)θ = 0 ➒ u = ug + ua is the full velocity; ➒ θ is the buoyancy anomaly; ➒ f is the Coriolis frequency; ➒ p is the pressure. Boundary condition on a bounded domain Ω ⊂ R3: u · n = 0

• n ∂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 coordinates3

     X = x + vg

f

Y = y − ug

f

Z = z Φ = p + |ug|2 2

• 3B. J. Hoskins, Journal of the Atmospheric Sciences, 32 (1975), no. 2

Hoskins’ change of coordinates3

     X = x + vg

f

Y = y − ug

f

Z = z Φ = p + |ug|2 2 = ⇒ conservation laws for θ and PV + highly nonlinear equation for Φ

• 3B. 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) = θ0, where T[θ] = Φ|z=1 with          ∆Φ = ε(∂XXΦ ∂YY Φ − (∂XY Φ)2) ∂ZΦ|Z=0 = 0 ∂ZΦ|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) = θ0, 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 ’164: 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.

• 4F. 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(∂xxΦ ∂yyΦ − (∂xyΦ)2) ∂zΦ|z=0 = 0 ∂zΦ|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[θ0]; ➒ θ(1) solution on [0, h] of (∂t + w(0) · ∇)θ(1) = 0, θ(1)(·, 0) = θ0; ➒ w(1) = ∇⊥T[θ(1)(·, h)]; ➒ θ(2) solution on [h, 2h] of (∂t + w(1) · ∇)θ(2) = 0, θ(2)(·, h) = θ(1)(·, h); and so on...

Conclusion and future work

We showed local existence of classical solutions of SSG, given a small initial datum and a small Rossby number. 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. 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).