Slab s mmetric and a i s mmetric Slab-symmetric and axi-symmetric
models
B l d l ti f t i l
Balanced evolution of tropical
cyclones y
f-plane
Slab-symmetric Hadley circulation
z
y
v
x
Axi-symmetric Tropical cyclone
Axi symmetric z Tropical cyclone
v
r
Slab-symmetric Axi-symmetric
2
v v v v 0
u w fu
t r z r
∂ + ∂ + ∂ + + =
∂ t ∂ r ∂ z r
∂ ∂ ∂
v ζ = ∂ x
∂
v v r r ζ = ∂ +
∂
Thermal wind
0
ru rz
r ρ r ρ
∂ + ∂ =
∂ ∂
Potential vorticity
Ertel PV
Sl b t i f
Slab-symmetric form
Sawyer-Eliassen Equation Slab-symmetric
Transform
Special case q = constant
Special case q constant
The membrane analogy
2 2
2 2
h h
F(x, y)
x y
∂ ∂
+ = −
∂ ∂
F(x,y) h(x,y)
F(x,y)
Equilibrium displacement of a stretched membrane over a
square under the force distribution F(x,y).
slippery glass walls
2 2
2 2
(x, y)
x y
∂ ψ ∂ ψ
+ = ζ
∂ ∂
y ψ = constant
x y
∂ ∂
ψ
ζ = ζ
cδ(x)δ(y)
x
2 2
∂
2∂
22 2
2 2
N f F(x,z)
x z
N
∂ ψ + ∂ ψ =
∂ ∂
2 2
Put z N z f
1 F(x z)
= ⇒
∂ ψ ∂ ψ
+ =
2 2 2
F(x, z) x + z = N
∂ ∂
R
L NH
= f
f
The Sawyer-Eliassen equation
Sawyer-Eliassen Equation
Axi-symmetric
Discriminant
SE equation is elliptic if D > 0
Parameters in SE Equation q
2v
2v f
ξ = r +
Potential vorticity Ertel PV
Slab symmetric Slab-symmetric
Axi-symmetric Axi symmetric Discriminant
Can show that
Larger I
2Smaller I
2H eight H
Heat source Radius
Larger N
2Larger N
H eight
Smaller N
2H
Momentum source Radius
Thermally-forced secondary circulation leads to spin up 15
10 km z
M conserved M 1
v fr
r 2
= −
5
z w v f)
t u(ζ v
∂
− ∂ +
−
∂ =
∂
0 50 r km 100
0
Prediction method
Initial condition: v(r z 0) given Initial condition: v(r,z,0) given
Heating and friction distributions given Solve for χ Solve SE-equation for ψ
Solve for u and w
Solve for u and w
Integrate for v(r,z,Δt)
Integrate for v(r,z,Δt)
Calculations by Dr. Hai Bui (Hanoi Uni)
Calculations by Dr. Hai Bui (Hanoi Uni)
Calculations by Dr. Hai Bui (Hanoi Uni)
Calculations by Dr. Hai Bui (Hanoi Uni)
