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 =>