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/

QG theory (for primitive equation)

Primitive equation in 𝛽 plane ( )

f = f₀ + βy

Du

Dt − ( f₀ + βy) v = − ∂Φ∂x +X Dv

Dt + (f₀ + βy) u = − ∂Φ

∂y +Y

∂Φ

∂z* = − R H T

∂u

∂x + ∂v

∂y + 1 ρ₀

∂ (ρ₀ w*)

∂z* = 0 Dθ

Dt = Q

where

,

w* ≡ Dz*

D Dt

Dt ≡ ∂

∂t + u ∂

∂x + v ∂

∂y + w* ∂

∂z*

T = θe^−κz/H e^−κz/H = (p_s p )

−κ

Q ≡ J c_p

θ T

Scale of synoptic phenomena

(from web.kma.go.kr)

L -3 -12

1470 1410

Typical synoptic scale in mid-latitude U ~ 10 m/s W ~ 10^-2 m/s

L ~ 10⁶ m t ~ 10⁵ s (~1 day) f ~ 10^-4 /s 𝜷 ~ 10^-11 /sm

𝜃 ~ 10² K 𝛿𝜃 ~ 10 K 𝛿𝚽 ~ 10³ m²/s²

QG theory (primitive, log-P)

Geostrophic balance

Using hydrostatic balance

(u_g , v_g) = f₀⁻¹(−Φ_y , Φ_x )

(Φ − Φ₀(z))_z = R

H (T − T₀(z)) f₀ ψ_z = R

H (T − T₀(z)) Stream function (u , v) = (−ψ_y , ψ_x )

Where

f₀ ψ = Φ′(x, y, x, t)

= Φ − Φ₀(z)

|∂θ_e/∂z| ≪ |∂θ₀/∂z|

QG theory (primitive, log-P)

Note: some useful manipulation

f₀ ψ_z = R

H (T − T₀(z)) = R

H (θ − θ₀(z)) e^−κz/H = g

⟨θ⟩ θ_e f₀ ψ_z = g

⟨θ⟩

dθ₀

dz θ_e/ dθ₀ dz

f₀

N² ψ_z = θ_e (θ₀)_z

QG theory (primitive, log-P)

u_t + uu_x + vu_y + wu_z − ( f₀ + βy) v = − Φ_x

∂u

∂x + ∂v

∂y + 1 ρ₀

∂ (ρ₀ w*)

∂z* = 0

QG theory (primitive, log-P)

u_t + uu_x + vu_y + wu_z − ( f₀ + βy) v = − Φ_x

QG theory (primitive, log-P)

* Define reference thermal structure (vertical only)

And we know

θ_t + uθ_x + vθ_y + wθ_z = Q

θ (x, y, z, t) = θ_e(x, y, z, t) + θ₀(z)

|∂θ_e/∂z| ≪ |∂θ₀/∂z|

QG equations (log-P)

D_g u_g − f₀ v_a − βyv_g = X D_g v_g + f₀ u_a + βyu_g = Y D_g θ_e + w_a (θ₀)_z = Q

(u_a)_x + (v_a)_y + ρ₀⁻¹(ρ₀w_a)_z = 0

where

,

D_g ≡ ∂

∂t + u_g ∂

∂x + v_g ∂

∂y

T = θe^−κz/H e^−κz/H = ( p_s p )

−κ

Q ≡ J c_p

θ T

QGPV equation (log-P)

Vorticity equation

Thermodynamic energy equation

D_g (f₀ + βy + ψ_xx + ψ_yy) = f₀ 1

ρ₀ (ρ₀w)_z −X_y + Y_x

D_g (θ_e/θ_0z) + w = Q/θ_0z

= f₀ N² ψ_z

QGPV equation (log-P)

Vorticity equation

Thermodynamic energy equation

Quasi-Geostrophic Potential Vorticity (QGPV) equation

D_g (f₀ + βy + ψ_xx + ψ_yy) = f₀ 1

ρ₀ (ρ₀w)_z −X_y + Y_x D_g (θ_e/θ_0z) + w = Q/θ_0z

D_g

[f₀ + βy + ψ_xx + ψ_yy + 1

ρ₀ (ρ₀ f₀²

N² ψ_z)_z] = f₀

ρ₀ (ρ₀ Q

θ_0z )

z

− X_y + Y_x

= f₀ N² ψ_z