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Balanced Dynamical Theoryy yThe primary and secondary
circulations
Chapter 3Chapter 3
Inviscid equations of motion
Absolute angular momentumg
Eq. (3.7) follows from r times Eq. (3.2)
Tropical cyclone intensification
Basic principle: conservation of absolute angular momentum:221
2M rv fr= +
M
r
vM
If r decreases, v increases!12
Mv frr
= −
Spin up requires radial convergencer
Conventional view of intensification: axisymmetric
15Thermally-forced secondary circulation leads to spin up
10M 1v frkm
5M d
v frr 2
= −
z k
M conserved
0 50 100r km
M not conserved, inflow feeds the clouds with moistureIs that it? See later for a surprise
Primary circulation
G di t i dGradient wind
iHydrostatic
Th l i d tiThermal wind equation
Physical interpretation
Thermal wind equation
2v 2v vg ln fv ln f 0⎛ ⎞∂ ∂ ∂⎛ ⎞ρ + + ρ+ + =⎜ ⎟ ⎜ ⎟
⎝ ⎠g
r r z r zρ ρ⎜ ⎟ ⎜ ⎟∂ ∂ ∂⎝ ⎠⎝ ⎠
Balance in toroidal circulation tendency
z z zlight heavylight
heavy
r r r
Thermal wind equation
Characteristics
On a characteristic
Generalized buoyancyy y
A typical vortex
AAM in a typical vortexyp
M
Barotropic stabilityp y
Net radial force on a displaced air parcel
Radial pressure gradient at BRadial pressure gradient at B
Net force on parcel at BNet force on parcel at B
Net radial force on a displaced air parcel
A scale analysis
A scale analysis
A scale analysis
A scale analysis
The secondary circulation
Thermal wind equationThermal wind equation
The secondary circulation
A balanced theory: The Sawyer-Eliassen equationA balanced theory: The Sawyer-Eliassen equation
Continuity y
Thermal wind
Prognostic equationsPrognostic equations
A balanced theory: The Sawyer-Eliassen equation
∂/∂t (thermal wind equation)
A balanced theory: The Sawyer-Eliassen equation
∂/∂t (thermal wind equation)
The Sawyer-Eliassen equation
Di i i tDiscriminant
Parameterse e s
The Sawyer-Eliassen equation
The Sawyer-Eliassen equation
Willoughby (1995)
The toroidal vorticity equation
With l b i i l ti thi bWith some algebraic manipulation this becomes
The Sawyer-Eliassen equation
(3.33).
More on buoyancy
See
See
End
Thermal wind equation
Gradient wind balance Hydrostatic balance
2p v fvr r
⎛ ⎞∂= ρ +⎜ ⎟∂ ⎝ ⎠
p g∂= −ρ
∂Write
r r∂ ⎝ ⎠ z∂
p p∂ ∂ ∂ ∂⎛ ⎞ ⎛ ⎞Eliminate p usingp p
r z z r∂ ∂ ∂ ∂⎛ ⎞ ⎛ ⎞=⎜ ⎟ ⎜ ⎟∂ ∂ ∂ ∂⎝ ⎠ ⎝ ⎠
21 v 1 2v vln fv ln f⎛ ⎞∂ ∂ ∂⎛ ⎞ρ + + ρ = − +⎜ ⎟⎜ ⎟ln fv ln fr g r z g r z
ρ + + ρ = − +⎜ ⎟⎜ ⎟∂ ∂ ∂⎝ ⎠⎝ ⎠
Thermal wind equation
Mathematical solution
21 1 2⎛ ⎞∂ ∂ ∂⎛ ⎞21 v 1 2v vln fv ln fr g r z g r z
⎛ ⎞∂ ∂ ∂⎛ ⎞ρ + + ρ = − +⎜ ⎟⎜ ⎟∂ ∂ ∂⎝ ⎠⎝ ⎠
Characteristics2dz 1 v fv
dr g r⎛ ⎞
= +⎜ ⎟⎝ ⎠
Characteristicsz
z(r) dr g r⎜ ⎟⎝ ⎠
r
z(r)
d 1 2v vln fdr g r
∂⎛ ⎞ρ = − +⎜ ⎟ ∂⎝ ⎠Governs the variation of ρ along characteristicsdr g r z⎜ ⎟ ∂⎝ ⎠ ρ along characteristics
Characteristics are isobaric surfaces
p = constantz p = constant
po ,ρo
z
z(r) d 1 2v vln fdr g r z
∂⎛ ⎞ρ = − +⎜ ⎟ ∂⎝ ⎠
r
dr g r z∂⎝ ⎠
r
2vgdz fv dr⎛ ⎞
= +⎜ ⎟Along a characteristic gdz fv drr
= +⎜ ⎟⎝ ⎠
o g c c e s c
2p p vd d d f d d 0⎛ ⎞∂ ∂⎜ ⎟
p pdp dr dz fv dr gdz 0r z r
= + = ρ + −ρ =⎜ ⎟∂ ∂ ⎝ ⎠
Inferences
p = constantzpo ,ρo
( )1 2v vl f∂ ∂⎛ ⎞⎜ ⎟z(r)
p
1 2v vln fr g r z∂ ∂⎛ ⎞ρ = − +⎜ ⎟∂ ∂⎝ ⎠
r
v∂ ∂ρ T∂v 0z∂
=∂
Barotropic vortexp
0r
∂ρ=
∂ p
T 0r
∂=
∂
v 0z∂
<∂
Baroclinic vortexp
0r
∂ρ>
∂ p
T 0r
∂<
∂
Equation of state T p / R= ρ
Summary
A barotropic vortex is cold cored if temperature contrastsb o op c vo e s co d co ed e pe u e co s sare measured at constant height.
A baroclinic vortex is warm cored if temperature contrastsA baroclinic vortex is warm cored if temperature contrastsare measured at constant height and if −∂v/∂z is largeenough.
A sample calculation =>