How Small-Scale Turbulence Sets the Amplitude and

Structure of Tropical Cyclones

Kerry Emanuel

PAOC

Some Critical Questions

What sets upper limit on TC intensity?

What determines TC structure?

Once formed, how do TCs intensify?

Remarkably, outflow turbulence has a strong influence on all of these

Axisymmetric Theory

Hydrostatic and gradient balance

Slantwise neutral vortex

Slab boundary layer

* *ˆePV fk ×V

Saturation Potential Vorticity

0

Slantwise convective neutrality

Thermal Wind

p

2 2

23

1

4

V MfV f r

r r r

21

2M rV fr

2

3

1

pr

M

r p r

Hydrostatic:

Gradient Wind:

Angular Momentum:

Thermal Wind:

2

3

1 *

*p p pr

M s

r p r s r

** p s

T

s p

2

3*

1 *

psr

M T s

r p p r

* *

p p

s ds M

r dM r

3

*

*

2M s

r r ds T

p M dM p

(Maxwell)

3

*

*

2M s

r r ds T

p M dM p

2 |

**

M

o

Mr

dsT T s

dM

Evaluate at top of boundary layer: b b bM rV

**

bb

b o

Vr

dsT T s

dM

To(s*) = temperature along M surfaces where Vb = 0

*( ) ,bb o

b

V dsT T

r dM

Gradient wind at top of PBL

Radius at top of PBL

T at top of PBL

Outflow T

Saturation entropy

Absolute angular momentum per unit mass

Set by Boundary Layer Processes

Slab Boundary Layer Entropy Balance (neglect dissipative heating):

s s s FM g

M P P

M grP

s Fgr g

P M P

Assume steady state and ignore vertical advection in boundary layer:

s

s

ds F

dM r

Integrate over depth of boundary layer:

| |s DC V V

*0| |s kF C k k V

*0

( )kb b o

b D s

k kr CVV T Tr C T

*02 ( )k

b b oD s

k kCV T T

C T

Not a closed expression: Need equation for k. Nothing about this derivation is peculiar to radius of maximum winds; previous work assumed constant To and tried to derive structure from k equation. This turns out to be wrong!

Basic Idea

Tropical cyclone outflow surfaces, rather than asymptoting to unperturbed environment, space themselves in vertical so as to achieve a critical Richardson Number

2

c

g d dRi

dz dz

U

Integration of Rotunno-Emanuel (1987) model, revised to ensure energy conservation

Streamfunction (black contours), absolute temperature (shading) and V=0 contour(white)

Outflow at V=0 is clearly T-stratified

Angular momentum surfaces plotted in the V-T plane. Red curve shows shape of balanced M surface originating at radius of maximum winds. Dashed red line is ambient

tropopause temperature.

Richardson Number (capped at 3). Box shows area used for scatter plot.

Ri=1

Dropsondes in TCs

Global Hawk-deployed sounding through outflow of Atlantic Hurricane Leslie of 2012 (left; courtesy Michael Black) and smoothed estimate of the inverse of the square root of the Richardson Number (right). Richardson Number criticality is indicated in the 150-300 hPa layer.

Implications for Outflow Criticality for Tropical Cyclone Structure and Intensity

2 *o c

t

T Ri dM

M r ds

Assumption of constant Richardson Number leads to equation for the dependence of outflow temperature on M:

* * * **0 0

212

k k

D b b Db

s s s sds C C

dM C rV C M r

2 1 *

2b b o

dsM r f T T

dM

Combine boundary layer equation:

And thermal wind equation:

System can be integrated inward from some outer radius ro, defined such that

0 oV at r r

Must choose either ro or rt.

In general, integrating this system will not yield To=Tt at r=rmax. Iterate value of rt until this condition is met.

If V >> fr, we ignore dissipative heating, and we neglect pressure dependence of s0*, then we can derive an approximate closed-form solution.

2

2

2

2

,

2

k

D

C

Cm

m k k

D D m

rrM

M C C rC C r

Assuming that Ri is critical in the outflow leads to an equation for To that, coupled to the interior balance equation and the slab boundary layer leads (surprisingly!) to a closed form analytic solution for the gradient wind (as represented by angular momentum, M, at the top of the boundary layer:

20 * *k

p b t eD

CV T T s s

C

22 2 1

2

k

D

k

D

C

CC

k Cm p

D

CV V

C

Defining

322

0

1

2 * *o

m

b t e

frr

T T s s

Also,

2 2 Dt m c

k

Cr r Ri

C

and

Predicted dependence on Ck/CD is weaker than square-root dependence

Explains effect of capping the wind speed in the surface enthalpy fluxes

Comparison of analytic model with numerical simulations

Actual and normalized evolution of maximum wind speed in RE numerical simulations

3*0

2

2

| || | | |

1 *2

*

1

2

b bD k b D

s

b o

o c

t

s sh C r V C s s C

M T

Mr

dsf T T

dMT Ri dM

M r ds

MV frr

VV V

Time-Dependent System

, , * .o t m bAlso T T at r r s s except in eye

Approximate System

Neglect pressure dependence of s0*

V~M/r (inner core)

Neglect dissipative heating

|V| ~ V

h=constant

2

1

2

0

*( ) ,

*,

* ** .

b o

o c

t

D k

sV T T M

M

T Ri s

M r M

s sh C VM C V s s

M

Combined system:

2

20

2 22

3 *

.

b oD k b o

b o

cD k

t

T T h V M VC V C T T s s

V T T V M

RiC M C V

r

Suppose that maximum winds always occur on the same M surface. Then, using

2 2 1,kmax max max max t c

D

CM r V and r r Ri

C

2 2

2m k

max m

V CV V

h

(14)

(15)

with

220

1*

2

k

D

k

D

C

CC

k k Cmax b t e

D D

C CV T T s s

C C

( ) tanh2k max

m max

C VV V

h

If V = 0 at t = 0, the integration of time-dependent equation gives

Comparison with numerical solution of (7) – (10)

(17)

Comparison with Rotunno-Emanuel 1987 Model

( ) tanh .2k max

m max

C VV V

h

Summary

Previous assumption that outflow asymptotes to environmental entropy surfaces appears to be wrong

Instead, outflow stratification appears to be set by the requirement that the Richardson Number remain at or above critical value

Implementing a critical Ri criterion in balance analytical model leads to closed form solution that brings theory into better agreement with numerical simulations

No longer need to make crude assumption about boundary layer entropy distribution outside of eyewall

Weaker dependence of Vmax on Ck/CD, as in numerical simulations

Time-dependent quasi-analytic model no longer needs to assume jump in entropy budget equation

TC intensity, structure, and development all depend on action of turbulence in TC outflow as well as in the boundary layer

Saturation entropy (contoured) and V=0 line (yellow)

2 1 *

2b b b o

dsM r f T T

dM

We can also re-write the thermal wind equation as

(1)

3

*0

| || |b

k b db

dsh C s s Cdt T

V

VBoundary layer entropy (with dissipative heating):

(2)

| |dMh r Vdt

VBoundary layer angular momentum

(3)

Combine (2) and (3):

* 20 | |bb k

D b

s sds C

dM C rV T rV

V

(4)

* 20 | |bb k

D b

s sds C

dM C rV T rV

V

Let *, | | ,b b bs s V V r rV

*0 ** k b

D b b b b

s sds C V

dM C rV T r

(5)

Thermal wind balance:

*bb o

b

V dsT T

r dM (6)

2 1 *

2b b b o

dsM r f T T

dM

Eliminate Vb between (5) and (6):

*20

2

** b k

o D b b o

s sds T C

dM T C r T T

(7)

Eliminate rb2 between (20) and (25):

2* *

2 0,b o

ds ds f

dM dM T T

where 0 * *

2b k

o D

T C s s

T C M

(8)

Remember that

2 *o c

t

T Ri dM

M r ds

(9)