tropical cyclones: steady state...
TRANSCRIPT
Tropical Cyclones: Steady State Physics
Image on page 27 is copyrighted by Oxford, NY: Oxford University Press, 2005. Book title is Divine wind: the history and science of hurricanes. ISBN: 0195149416. Used with permission.
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&
Dissipative heatingSurface enthalpy flux
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:
Figure by MIT OCW.
Hei
ght
Radius
h
T = Tb
Ψ1Ψ2
T = To
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:
21 | |2p vE c T L q gz= + + + VEnergy:
** ln ln vp d
L qs c T R pT
= − +Entropy:
212
M rV fr= +Angular Momentum:
First definition of s*:
* *p vTds c dT L dq dpα= + − (1)
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 (2)
Identity:
whereu wz r
ς ∂ ∂≡ −∂ ∂
azimuthal vorticity
Substituting (3) into (2) and the result into (1) gives:
2 1* VTds dE VdV fV dr dr r
ς ψρ
⎛ ⎞= − − + +⎜ ⎟
⎝ ⎠(4)
( ) ( ) ( )2 21 1 ,2
w dz u dr d u w drς ψ
ρ∇ + ∇ = + +V Vi i (3)
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
⎡ ⎤+ − = +⎢ ⎥⎣ ⎦* (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 pointsFigure by MIT OCW.
Hei
ght
Radius
h
T = Tb
Ψ1Ψ2
T = To
( ) 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)
:o br r>>Mature storm:
( )2
*b o
b
M dsT Tr dM
→ ≅ − − (7)
V fr>>In inner core,
M rV→ ≅
( ) *b b b o
dsV r T TdM
≅ − − (8)
* bs s=Convective criticality:
( ) bb b b o
dsV r T TdM
→ ≅ − − (9)
dsb /dM determined by boundary layer processes:
Figure by MIT OCW.
Hei
ght
Radius
h
T = Tb
Ψ1Ψ2
T = To
Put (7) in differential form:
( ) 2 0,b ods M dMT Tdt r dt
− + = (10)
Integrate entropy equation through depth of boundary layer:
( )* 31 | | | | ,k s D bs
dsh C k k C Fdt T
⎡ ⎤= − + +⎣ ⎦V V (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≡ + = + +
*bh h=Convective neutrality:
First law of thermo:
* * ln ,b b d bdh T ds R T d p= + (15)
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
∂= +
∂
( )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 hC T
*h−≅ − (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