Tropical Cyclones: Steady State Physics

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Tropical Cyclones: Steady State Physics

Energy 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: W T T s o = Q T s

Total rate of heat input to hurricane: r0 ( * ) 3 0 k 0 D Q = π ρ C V k k + C V rdr Surface enthalpy flux Dissipative heating In steady state, Work is used to balance frictional dissipation: r 0 0 3 W = π ρ C V rdr D

Plug into Carnot equation: T T C rdr C ( k ) 0 k = rdr r0 r 3 0 s o * ρ 0 D V ρ 0 k V To If integrals dominated by values of integrands near radius of maximum winds, C k T s T V o * max k 0 k C D T 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: ρ rw ψ =, ρru = r ψ z Flow parallel to surfaces of constant ψ, satisfies mass continuity: 1 ( ) 1 ρru + ( ρrw) = r r r z 0

Variables conserved (or else constant along streamlines) above PBL, where flow is considered reversible, adiabatic and axisymmetric: 1 E = cpt + Lvq+ gz+ V Lvq * s* = cplnt Rd ln p+ T Energy: Entropy: Angular Momentum: 1 M = rv + fr

First definition of s*: Tds* = c dt + L dq* α dp p v (1) Steady flow: p p αdp α = dr + α dz r z Substitute from momentum equations: V αdp = dz[ g + Vi w] + dr + fv Vi u r ()

Identity: 1 1 wdz+ udr= du + w + d, ρr ς ψ u w ς azimuthal vorticity z r ( ) ( ) ( Vi Vi ) where (3) Substituting (3) into () and the result into (1) gives: V 1 Tds* = de VdV + fv dr + d r ρr ς ψ (4)

One more identity: V M 1 VdV + + fv dr = f dm r r Substitute into (4): M 1 1 Tds* dm d d E fm r ρr ξ ψ + = +, (5) Note that third term on left is very small: Ignore

Integrate (5) around closed circuit: Right side vanishes; contribution to left only from end points

1 1 T T ds* + MdM = 0 r r b o ( ) b o 1 1 ds * = ( T ) b T (6) o r r MdM b o Mature storm: r >> r : o b b M ds * ( T ) b To r dm (7)

In inner core, V >> fr M rv ds* V r ( T T ) b b b o (8) dm Convective criticality: * b s = ds (9) ( ) b Vb rb Tb To dm s

ds b /dm determined by boundary layer processes:

Put (7) in differential form: ds M dm ( ) + = 0, Tb To dt r dt (10) Integrate entropy equation through depth of boundary layer: ds 1 h = C ( k k) C F dt T V + V + s * 3 k s D b, (11) where F b is the enthalpy flux through PBL top. Integrate angular momentum equation through depth of boundary layer: dm h = CDr V V, (1) dt

Substitute (11) and (1) into (10) and set F b to 0: C ( * ) k T s T V = o k 0 k (13) C D T o 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 quasiequilibrium as follows. First, use moist static energy, h instead of k: C ( * ) k T s T V = o h 0 h (14) C D T o h k+ gz = c T + L q+ gz p v

Convective neutrality: h = h* b First law of thermo: dh = Tds + RTdln p, * * b b d b (15) Go back to equation (10): ds M dm Tb To dt r dt ( ) + = 0, Use definition of M and gradient wind balance: R T d ln r p V = + r fv

( ) 1 1 1 V = + + + + r 1 1 1 = d V + frv + f r + RdTln p. 4 * Tb To ds d V frv f rdr fv dr Substitute into (15): T 1 1 1 dh = d V + frv + f r + R T ln p * b b d o Tb T o 4 Define an outer radius, r a, where V=0, p=p o. Take difference between this and radius of maximum winds: T 1 p 1 h = h V + fr V + R T ln f r ( ) * * b m b a max m max d o a p Tb T o a 4

Relate p m to V max using gradient wind equation. But simpler to use an empirical relation: RT ln p bv p m d s max a Also neglect fr m in comparison to V max. and neglect difference between T s and T b : Substitute into (14): V max T T ( ) 1 T h h f r C T T C D C k 1 T s 1 b CD To s o * * s s a a k o 4 o.

Absolute upper bound on storm size: r omax = 4 T s T T s o ( * h ) s ha f romax 1000 km For r o << r omax neglect last term in numerator. C T T V h h (16) ( ) k s o * * max s a CD To

h h = h h = c T T + L q q ( H ) ( ) ( ) * * * * s a s ab p s b v s ab Lq v * sa 1 e*( Ts ) = L ε v 1 p o ( H ) Since p depends on V, this makes (16) an implicit equation for V. But write expression for wind force instead: p C T T L ρv = V = e*( T ) 1 H o k s o v max max s RT d s CD To RT v s o ( )

Otherwise, use empirical relationship between V and p: RT d p= s p ln p e a b V p a bv RT d s C T T C T T e*( T ) V h h = L 1 ( ) ε ( H ) o k s o * * k s o s max s a v CD To CD To p C T T pe V L e*( T) 1 H V b max RT d s k s o a max vε s CD To ( ) o

Taking natural log of this, the result can be written in the form: ( ) max max AV ln V B = 0

Numerical simulations

Relationship between potential intensity (PI) and intensity of real tropical cyclones

Wind speed (m/s) Potential wind speed (m/s) 80 70 60 50 V (m/s) 40 30 0 10-80 -60-40 -0 0 0 40 60 80 100 Time after maximum intensity (hours)

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

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