Quasi-Geostrophic (QG) theory

1. Perturbation theory (background) 2. QG theory (for shallow water)

3. QG theory (for primitive equations), QGPV equation

Quasi-Geostrophic (QG) theory

Estimated solutions of Schrödinger equation using Perturbation theory https://demonstrations.wolfram.com/

Story

1. Developed for celestial mechanics (motion of planet)
 e.g.) deviation of moon’s orbit from Keplerian ellipse 2. Extend and generalized in 18~19th century


by Lagrange and Laplace

3. Adopted for quantum mechanics in 20th century
 for atomic and subatomic processes

Concept (problem solving method)


A simplified (solvable, well-balanced) solution 


+ perturbed solutions that better match the original problem.


(where )

Note: Reviewed by Laplace, Poisson and Gauss


Contribute to the discovery of the planet Neptune!

A = A

₀

+ εA

₁

+ ε

²

A

₂

+ ε

³

A

₃

+ . . . ε ≪ 1

Perturbation theory

QG theory (for shallow water)

Shallow water system in 𝛽 plane ( )

f = f₀ + βy u_t + uu_x + vu_y − fv = − gη_x

v_t + uv_x + vv_y + fu = − gη_y

h_t + uh_x + vh_y = − h(u_x + v_y)

Fig. �.�

u, v h

QG theory (for shallow water)

Shallow water system in 𝛽 plane ( )

f = f₀ + βy u_t + uu_x + vu_y − ( f₀ + βy) v = − gη_x

v_t + uv_x + vv_y + ( f₀ + βy) u = − gη_y h_t + uh_x + vh_y = − h(u_x + v_y)

Fig. �.�

u, v h

QG theory (for shallow water)

Shallow water system in 𝛽 plane ( )


Scale analysis




f = f₀ + βy u_t + uu_x + vu_y − ( f₀ + βy) v = − gη_x

v_t + uv_x + vv_y + ( f₀ + βy) u = − gη_y h_t + uh_x + vh_y = − h(u_x + v_y)

u ∼ U, δx ∼ L, t ∼ T ( ∼ L/U), δη ∼ δH, h ∼ H

(u = Uu*, v = Uv*, t = Tt*, y = Ly*, η = δHη*, . . . )

u_t + uu_x + vu_y − f₀ v + βy v = − gη_x

QG theory (for shallow water)

Shallow water system in 𝛽 plane ( )


Scale analysis




f = f₀ + βy u_t + uu_x + vu_y − ( f₀ + βy) v = − gη_x

v_t + uv_x + vv_y + ( f₀ + βy) u = − gη_y h_t + uh_x + vh_y = − h(u_x + v_y)

u ∼ U, δx ∼ L, t ∼ T ( ∼ L/U), δη ∼ δH, h ∼ H

(u = Uu*, v = Uv*, t = Tt*, y = Ly*, η = δHη*, . . . )

u_t + uu_x + vu_y − f₀ v + βy v = − gη_x

εu*_t + εu*u*_x + εv*u*_y − v* + εy* v* = η*_x

QG theory (for shallow water)

Geostrophic balance (QG relies on “strong geostrophic balance”)

Rossby number,

Consider small terms that are not in geostrophic balance (ageostrophic terms), and their scale could be

−f₀ v₀ = − gη_x +f₀ u₀ = − gη_y

U

f₀ L = ε ≪ 1

∼ O(ε) u* = u₀ + εu₁ + ε²u₂ + ε³u₃ + . . .

v* = v₀ + εv₁ + ε²v₂ + ε³v₃ + . . . η* = η

QG theory (for shallow water)

u_t + uu_x + vu_y − ( f₀ + βy) v = − gη_x

QG theory (for shallow water)

(where )

h_t + uh_x + vh_y = − h(u_x + v_y) h = H + η −η_b

QG theory (for shallow water)

Order 1 balance: Geostrophic balance

Order 𝜀 balance: Quasi-Geostrophic equations

−f₀ v₀ = − gη_x +f₀ u₀ = − gη_y

u_0t + u₀u_0x + v₀u_0y − f₀ εv₁ − βyv₀ = 0 v_0t + u₀v_0x + v₀v_0y + f₀ εu₁ + βyu₀ = 0

h_t + u₀h_x + v₀h_y = − εH(u_1x + v_1y)

Assumptions (or conditions), MAD Andrew p.120










Ro ≡ U/f₀L ≪ 1

∂/∂t ≪ f₀ βL ≪ f₀

|X|, |Y| ≪ f₀U

ε ≪ 1

QG Potential Vorticity (for shallow water)

QG momentum equations

Vorticity equation (of shallow water system)

u_0t + u₀u_0x + v₀u_0y − f₀ εv₁ − βyv₀ = 0 v_0t + u₀v_0x + v₀v_0y + f₀ εu₁ + βyu₀ = 0

ζ_0t + u₀ζ_0x + v₀ζ_0y + εf₀ (u_1x + v_1y) + βv₀ = 0

QG Potential Vorticity (for shallow water)

QGPV equation

Recall full PV equation D₀

Dt [ ζ₀ + βy − f₀ η

H + f₀ η_b

H ] = 0

D

Dt [ f + ζ

h ] = 0

Benefit: QGPV can be expressed as 


a function of (or ) only

η ψ