Tropical Cyclones: Steady State Physics

Energy ProductionEnergy Production

Carnot Theorem: Maximum efficiency results from a particular energy cycle:

• Isothermal expansion• Adiabatic expansion• Isothermal compression• Adiabatic compression

Note: Last leg is not adiabatic in hurricane: Air cools radiatively. But since environmental temperature profile is moist adiabatic, the amount of radiative cooling is the same as if air were saturated and descending moist adiabatically.

Maximum rate of working:s o

s

T TW QT−

=

Total rate of heat input to hurricane:

( )0 * 300

2 | | | |r

k DQ C k k C rdrπ ρ = − + ∫ V V

Surface enthalpy flux Dissipative heating

In steady state, Work is used to balance frictional dissipation:

0 3

02 | |

r

DW C rdrπ ρ = ∫ V

Plug into Carnot equation:

( )0 03 *00 0

| | | |r rs o

D ko

T TC rdr C k k rdrT

ρ ρ− = − ∫ ∫V V

If integrals dominated by values of integrands near radius of maximum winds,

( )2 *| |max 0C T Tk s oV k kC TD o

−→ ≅ −

Problems with Energy Bound:

• Implicit assumption that all irreversible entropy production is by dissipation of kinetic energy. But outside of eyewall, cumuli moisten environment....accounting for almost all entropy production there

• Approximation of integrals dominated by high wind region is crude

Local energy balance in eyewall region:

Definition of streamfunction, ψ:

,rw rur zψ ψρ ρ∂ ∂

= = −∂ ∂

Flow parallel to surfaces of constant ψ, satisfies mass continuity:

( ) ( )1 1 0ru rwr r r z

ρ ρ∂ ∂+ =

∂ ∂

Variables conserved (or else constant along streamlines) above PBL, where flow is considered reversible, adiabatic and axisymmetric:

Energy: 21 | |2p vE c T L q gz= + + + V

Entropy: ** ln ln vp d

L qs c T R pT

= − +

Angular Momentum:

212

M rV fr= +

First definition of s*:

* *p vTds c dT L dq dpα= + −Steady flow:

p pdp dr dzr z

α α α∂ ∂= +

∂ ∂

Substitute from momentum equations:

[ ]2Vdp dz g w dr fV ur

α

= − + ∇ + + − ∇

V Vi i

(1)

(2)

Identity:

( ) ( ) ( )2 21 1 ,2

w dz u dr d u w drς ψ

ρ∇ + ∇ = + +V Vi i

whereu wz r

ς ∂ ∂≡ −∂ ∂

azimuthal vorticity

(3)

Substituting (3) into (2) and the result into (1) gives:

2 1* VTds dE VdV fV dr dr r

ς ψρ

= − − + +

(4)

One more identity:

2

2

12

V MVdV fV dr f dMr r

+ + = −

Substitute into (4):

2

1 1 ,2

MTds dM d d E fMr r

ξ ψρ

+ − = +

Note that third term on left is very small: Ignore

(5)*

Integrate (5) around closed circuit:

Right side vanishes; contribution to left only from end points

( ) 2 2

1 1* 0b ob o

T T ds MdMr r

− + − =

( )2 2

1 1 *b o

b o

dsT Tr r MdM

→ = − − (6)

Mature storm: :o br r>>

( )2

*b o

b

M dsT Tr dM

→ ≅ − − (7)

In inner core, V fr>>

M rV→ ≅

( ) *b b b o

dsV r T TdM

≅ − − (8)

Convective criticality: * bs s=

( ) bb b b o

dsV r T TdM

→ ≅ − − (9)

dsb /dM determined by boundary layer processes:

Put (7) in differential form:

( ) 2 0,b ods M dMT Tdt r dt

− + =

Integrate entropy equation through depth of boundary layer:

( )* 31 | | | | ,k s D bs

dsh C k k C Fdt T

= − + + V V

(10)

(11)

where Fb is the enthalpy flux through PBL top. Integrate angular momentum equation through depth of boundary layer:

| | ,DdMh C r Vdt

= − V (12)

Substitute (11) and (12) into (10) and set Fb to 0:

( )2 *| | 0C T Tk s oV k kC TD o

−→ = − (13)

Same answer as from Carnot cycle. This is still not a closed expression, since we have not determined the boundary layer enthalpy, k. We can do this using boundary layer quasi-equilibrium as follows. First, use moist static energy, h instead of k:

( )2 *| | 0C T Tk s oV h hC TD o

−= − (14)

p vh k gz c T L q gz≡ + = + +

Convective neutrality: *bh h=First law of thermo:

* * ln ,b b d bdh T ds R T d p= +

Go back to equation (10):

( ) 2 0,b ods M dMT Tdt r dt

− + =

Use definition of M and gradient wind balance:

2lnd

p VR T fVr r

∂= +

∂

(15)

( )2

* 2 2

2 2 2

1 1 12 2 2

1 1 1 ln .2 2 4

b o

d

VT T ds d V frV f rdr fv drr

d V frV f r R T p

→ − = − + + + + = − + + +

Substitute into (15):

* 2 2 21 1 1 ln2 2 4

bb d o

b o

Tdh d V frV f r R T pT T

= − + + + −

Define an outer radius, ra, where V=0, p=po. Take difference between this and radius of maximum winds:

( )* * 2 21 1ln2 4

2b mb a max m max d o a

ab o

T ph h V fr V R T f rpT T = − + + − −

Relate pm to Vmax using gradient wind equation. But simpler to use an empirical relation:

2ln md s max

a

pR T bVp ≅ −

Also neglect frm in comparison to Vmax. and neglect difference between Ts and Tb:

Substitute into (14):

( )* * 2 2

2

14 .

112

s o ss a a

k o omax

D k s

D o

T T Th h f rC T TVC C T b

C T

−− −

≅

− −

Absolute upper bound on storm size:

( )*2

24 s as oomax

s

h hT TrT f

−−=

1000omaxr km≈

For ro << ro max neglect last term in numerator.

( )2 * *k s omax s a

D o

C T TV h hC T

−≅ − (16)

( ) ( )( )

( )

* * * *

* 1*( ) 1

s a s ab p s b v s ab

v sa

sv

h h h h c T T L q q

L qe TLp

ε

− = − = − + −

≅ −

= −

o

o

H

H

Since p depends on V, this makes (16) an implicit equation for V. But write expression for wind force instead:

( )2 2 *( ) 1k s o vmax max s

d s D o v s

C T T LpV V e TR T C T R T

ρ −= = − oH

Otherwise, use empirical relationship between V and p:

2

2ln

d s

d sa

Vb R Ta

pR T bVp

p p e−

≅ −

=

( ) ( )

( )

2 * *

2

*( ) 1

*( ) 12max

d s

k s o k s o smax s a v

D o D o

Vb R T k s oa max v s

D o

C T T C T T e TV h h LC T C T p

C T Tp e V L e TC T

ε

ε−

− −≅ − = −

−→ ≅ −

o

o

H

H

Taking natural log of this, the result can be written in the form:

( )2 2ln 0max maxAV V B− − =

Numerical simulations

Relationship between potential intensity (PI) and intensity of

real tropical cyclones

10

20

30

40

50

60

70

80

-80 -60 -40 -20 0 20 40 60 80 100

V (m/s)

Time after maximum intensity (hours)

Wind speed (m/s)

Potential wind speed (m/s)

Evolution with respect to time of maximum intensity

Evolution with respect to time of maximum intensity, normalized by peak wind

Evolution curve of Atlantic storms whose lifetime maximum intensity is limited by declining potential intensity, but not by landfall

Evolution curve of WPAC storms whose lifetime maximum intensity is limited by declining potential intensity, but not by landfall

CDF of normalized lifetime maximum wind speeds of North Atlantic tropical cyclones of tropical storm strength (18 m s−1) or greater, for those storms whose lifetime maximum intensity was limited by landfall.

CDF of normalized lifetime maximum wind speeds of Northwest Pacific tropical cyclones of tropical storm strength (18 m s−1) or greater, for those storms whose lifetime maximum intensity was limited by landfall.

Evolution of Atlantic storms whose lifetime maximum intensity was limited by landfall

Evolution of Pacific storms whose lifetime maximum intensity was limited by landfall

Composite evolution of landfalling storms