1-4_geostrophic_adjustment copy

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′+ ¯u u′_x+ ¯vu_y′− fv = − gη_x v_t′+ ¯u v_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 = f₀η₀

H x > 0

− f₀η₀

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 = f₀η₀

H x > 0

− f₀η₀

H x < 0

PV equation becomes ψ_xx+ ψ_yy− f₀²

g Hψ = q₀

Consider uniformity in y ψ_xx− 1

L_d²ψ = f₀

Hη₀ ^sgn(x)

Rossby-Gill problem

q₀ = f₀η₀

H x > 0

− f₀η₀

H x < 0

Consider uniformity in y

Solution ψ_xx− 1

L_d²ψ = f₀

Hη₀ ^sgn(x)

ψ = − _f^g

0η₀(1− e^{− x/Ld}) x > 0

g

f₀η₀(1− e^x/Ld) x< 0

v = − _f^g₀^η_L⁰_de^{− x/Ld} x > 0

− _f^gη0

0L_de^x/Ld x< 0

Rossby-Gill problem

q₀ = f₀η₀

H x > 0

− f₀η₀

H x < 0