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 ⊂ R3: ∂tug

1 + u · ∇ug 1 − u2 + ∂1p = 0

∂tug

2 + u · ∇ug 2 + u1 + ∂2p = 0

∇ · u = 0 (∂t + u · ∇)ρ = 0 ug

1 := −∂2p

ug

2 := ∂1p

∂3p = −ρ u · n = 0 on ∂Ω p(·, 0) = p0, derived as an asymptotic limit of Euler equations.

Benamou and Brenier’s transformation [BB98]

Incompressible SG on a rigid domain Ω ⊂ R3: ∂tug

1 + u · ∇ug 1 − u2 + ∂1p = 0

∂tug

2 + u · ∇ug 2 + u1 + ∂2p = 0

∇ · u = 0 (∂t + u · ∇)ρ = 0 ug

1 := −∂2p

ug

2 := ∂1p

∂3p = −ρ u · n = 0 on ∂Ω p(·, 0) = p0, derived as an asymptotic limit of Euler equations. Introduce generalised pressure P P(x, t) = p(x, t) + 1 2(x2

1 + x2 2)

T(x, t) = ∇P(x, t) =   x1 + ∂1p(x, t) x2 + ∂2p(x, t) ∂3p(x, t)   . Equations for u and T DtT = J(T − IdΩ) ∇ · u = 0 u · n = 0 on ∂Ω T0(x) = (x1, x2, 0) + ∇p0(x) Tt = ∇Pt

Working on the dual space

DtT = J(T − IdΩ) ∇ · u = 0 Tt = ∇Pt P(·, 0) = P0 ∂t ˜ T = J( ˜ T − X) ∂tX(x, t) = u(X(x, t), t) ∇ · u = 0 ˜ Tt = ∇Pt ◦ Xt ˜ T(·, 0) = P0 X(x, 0) = x ∂tα + U · ∇α = 0 U(y, t) = J(y − ∇P∗(y, t)) α = det D2P∗ α0 = ∇P0#LΩ u · n = 0 on ∂Ω u · n = 0 on ∂Ω Ω Ω R3 if ∇Pt diffeo + P∗

t ∈ C 2(R3),

using weak formulation +change of variables y = Tt(x) if ux locally Lipschitz Eulerian physical space Lagrangian physical space Dual space

Working on the dual space

DtT = J(T − IdΩ) ∇ · u = 0 Tt = ∇Pt P(·, 0) = P0 ∂t ˜ T = J( ˜ T − X) ∂tX(x, t) = u(X(x, t), t) ∇ · u = 0 ˜ Tt = ∇Pt ◦ Xt ˜ T(·, 0) = P0 X(x, 0) = x ∂tα + U · ∇α = 0 U(y, t) = J(y − ∇P∗(y, t)) α = det D2P∗ α0 = ∇P0#LΩ u · n = 0 on ∂Ω u · n = 0 on ∂Ω Ω Ω R3 if ∇Pt diffeo + P∗

t ∈ C 2(R3),

using weak formulation +change of variables y = Tt(x) if ux locally Lipschitz Eulerian physical space Lagrangian physical space Dual space

Existence of weak solution in dual space

Theorem ([BB98])

Let Ω ⊂ R3 be open bounded Lipschitz set and 0 ≤ α0 ∈ Lp(R3) compactly supported and α0L1(R3) = L3(Ω). For any τ > 0 and p > 3 there exist α ∈ L∞ [0, τ); Lp(R3)

• ≥ 0, αt compactly supported and α(·, 0) = α0,

ψ ∈ L∞ [0, τ); W 1,∞(Ω)

• convex with

ψ∗ ∈ L∞ [0, τ); W 1,∞(R3)

• convex in space,

U ∈

• L∞([0, τ); L∞

loc(R3) ∩ BVloc(R3))

3, solutions of ❼ τ

• R3 (∂tξ(y, t) + U(y, t) · ∇ξ(y, t)) α(y, t) dy dt

= −

• R3 ξ(y, 0)α(y, 0) dy

∀ξ ∈ C ∞

0 (R3 × [0, τ))

❼ U(y, t) = J(y − ∇ψ∗(y, t)) ∀(y, t) ∈ R3 × [0, τ) ❼

• Ω f (∇ψ(x, t)) dx =
• R3 f (y)α(y, t) dy

∀f ∈ Cc(R3).

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 ◦ ∇Pt)(x) + (D2P∗ t ◦ ∇Pt)(J(∇Pt(x) − x))

is formally a solution in Eulerian space. It is not clear what this means, because D2P∗

t is a distribution (a measure).

Hoskins’ transfomation [Hos75]

On the domain Ω = R2 × [0, 1], the transformation y = Ht(x) =   x1 + ∂1p x2 + ∂2p x3   brings to the system in dual space Physical space Dual spaces

Ht Tt

∂tβ + U · ∇β + ∂3(βu3) = 0 (∂t + U · ∇ + u3∂3)( 1 β ∂2

3Φ) = 0

β = 1 + (∂2

1Φ∂2 2Φ − (∂1,2Φ)2) − (∂2 1Φ + ∂2 2Φ)

(∂t + U · ∇)∂3Φ = 0 on ∂Ω U = (−∂2Φ, ∂1Φ, 0), where Φ(y, t) = Φ(Ht(x), t) = p(x, t) + 1

2(∂1p(x, t)2 + 1 2(∂2p(x, t))2.

Surface semi-geostrophic equations (SSG)

If the potential vorticity is uniform, then β = ∂2

3Φ and the dual system is

simplified to 1 + (∂2

1Φ∂2 2Φ − (∂1,2Φ)2) − (∆Φ) = 0

• n Ω

(∂t − ∂2Φ∂1 + ∂1Φ∂2)∂3Φ = 0

• n {y3 = 0, 1}

∂3Φ(·, 0) = g

• n {y3 = 1}

∂3Φ(·, t) = 0

• n {y3 = 0}∀t ∈ [0, τ).

Surface semi-geostrophic equations (SSG)

If the potential vorticity is uniform, then β = ∂2

3Φ and the dual system is

simplified to 1 + (∂2

1Φ∂2 2Φ − (∂1,2Φ)2) − (∆Φ) = 0

• n Ω

(∂t − ∂2Φ∂1 + ∂1Φ∂2)∂3Φ = 0

• n {y3 = 0, 1}

∂3Φ(·, 0) = g

• n {y3 = 1}

∂3Φ(·, t) = 0

• n {y3 = 0}∀t ∈ [0, τ).

Introducing a Dirichlet-to-Neumann operator T we write the system above in R2 (∂t + w · ∇)f = 0 w = ∇⊥ψ f = T(ψ) f (·, 0) = g. Physical space Dual spaces

Ht Tt

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 E =

• Ω

1 2|ug(x, t)|2 + ρx3

• dx

with respect to the rearrangements of particles that conserve the absolute momentum (ug

1 − x2, ug 2 + x1) 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 − IdR3)

∇ · u = 0 is stable if, for any fixed time t, ∇P(·, t) is the optimal transport map that solves the following Monge problem min

T∈Tt

• Ω

|x − T(x)|2 2 dx with Tt := {T : Ω → R3|T#LΩ = ∇Pt#LΩ.}

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.