Geostrophic adjustment

1. Linearized PV equation

2. Rossby-Gill problem (Vallis chapter 3.8)

Geostrophic adjustment

Holton (IDM, 2004)

Linearized PV equation

Shallow water equations (u, v, h)

( η = h, if bottom does not exist)

Linearized PV equation u′_t + ¯uu′_x + ¯vu′_y − fv = − gη_x v′_t + ¯uv′_x + ¯vv′_y + fu = − gη_y

η′_t + ¯uη′_x + ¯vη′_y = − H(u_x + v_y)

D

Dt (ζ − f₀ η

H ) = 0

Initial condition

Rossby-Gill problem

η

₀

− η

₀

q₀ = f₀ η₀

H x > 0

−f₀ η₀

H x < 0

Will fine geostrophic balance, while conserving PV

Introduce

f₀u = − g ∂η

∂y , f₀v = g ∂η

∂x ζ − f₀ η

H = q₀ ψ = gη/f₀

Initial condition

Rossby-Gill problem

η

₀

− η

₀

q₀ = f₀ η₀

H x > 0

−f₀ η₀

H x < 0

PV equation becomes ψ_xx + ψ_yy − f₀²

gH ψ = q₀

Consider uniformity in y

ψ_xx − 1

L_d² ψ = f₀

H η₀ ^sgn(x)

Rossby-Gill problem

q₀ = f₀ η₀

H x > 0

−f₀ η₀

H x < 0

Consider uniformity in y

Solution

ψ_xx − 1

L_d² ψ = f₀

H η₀ ^sgn(x)

ψ = − _f^g

0 η₀ (1 − e^−x/Ld) x > 0

g

f₀ η₀ (1 − e^x/Ld) x < 0

v = − _f^gη0

0L_d e^−x/Ld x > 0

− _f^gη0

0L_d e^x/Ld x < 0

Rossby-Gill problem

q₀ = f₀ η₀

H x > 0

−f₀ η₀

H x < 0