The tropical cyclone boundary layer The tropical cyclone boundary layer

The tropical cyclone boundary layer The tropical cyclone boundary layer

Roger K. Smith

L d i M i ili U i i f M i h Ludwig-Maximilians University of Munich

Frictional effects on the secondary circulation

Pressure Secondary

i l i

primary circulation

gradient force circulation

F i ti l

0 5 1 k r Friction layer

v 0.5-1 km

v

Centrifugal force and Coriolis force are reduced by friction

“Tea cup” Experiment

Boundary-layer scaling Boundary layer scaling

Continuity equation

S = U/V A = Z/R Re = VZ/K

S = U/V A = Z/R Re = VZ/K

Full BL equations

Nonlinear terms

U2

R

U2

R

U2

R

VV R

 V ²

R fV 

2

KU

R R R R R ZZ

UV R

 UV

R

 VU

R ²

KV Z UV 

R

 UV

R

 VU

R fU

Assume U V

N li U

Nonlinear terms Nonlinear terms U 0.3

Linear terms  V 

Linearized equations q

Ekman equations q

Justification?

Scale analysis

Representation of frictional stress

Linear solution Linear solution

Can solve analytically

V(r)

ζ(r)

 < o

 < o

Linear theory

Linear theory

Linear theory

10

5

0 -1

Linear theory versus Ekman theory

ntpergradien

1 p 0

 r

  

 w

sup inflo

Dynamics of vortex spindown

Vertical cross-section

V



w  z 0

V t







w  0

V

 

  z 0

Boundary layer Level of nondivergence

Conventional view of tropical cyclone spin up 15

10 kmz

M 1

v fr

r 2

 

5 M conserved

0 50 r km 100

0

M not conserved, boundary layer supplies clouds with moisture

Emanuel’s 1986 steady-state TC model

T ( ) 

*e

M,

out e

T ( ) 

eight

e

H ^Assumes

gradient-wind balance in the

BL

h Region I Region II Region III

BL

r_m Radius

In J. Atmos Sci.

Emanuel’s 1997 model for TC spin up

out *

T ( ) 

*e

M,

out

( )

HeightH

h

BL model based on thermodynamics – still assumes gradient balance!

Radius

Model sets: v_g(h) = v_b

In J. Atmos Sci.

Questions Questions

• Does the boundary layer determine the tangential windDoes the boundary layer determine the tangential wind speed of air that ascends into the updraught?

• Consistent only if vy _gg(h) = v( ) _bb!

• Considerations: steady boundary layer equations are parabolic => information travels inwards. Region of ascent out of the boundary layer requires an open boundary condition.

• Can one improve the Emanuel model by relaxing the assumption of gradient wind balance in the boundary layer?y

Smith et al. (2008), Smith and Montgomery (2010)

ht

Smaller N²

Heigh

Smaller N

Momentum source Radius

Momentum source Radius

Zhang et al (2001) found that spin up occurred in the BL in Zhang et al (2001) found that spin up occurred in the BL in Hurricane Andrew.

Revised view of tropical cyclone spin up

15

10m

M 1

5

z km

M 1

v fr

r 2

 

5

M conserved

0 50 r km 100

M reduced by friction, but strong convergence  small r

From Montgomery, Nguyen & Smith (2009): QJRMS