Tropical Cyclones: Steady StateTropical Cyclones: Steady ......rz 8 Variables conserved (or else...
Transcript of Tropical Cyclones: Steady StateTropical Cyclones: Steady ......rz 8 Variables conserved (or else...
![Page 1: Tropical Cyclones: Steady StateTropical Cyclones: Steady ......rz 8 Variables conserved (or else constant along streamlines) above PBL where flow is consideredstreamlines) above PBL,](https://reader036.fdocuments.net/reader036/viewer/2022071218/605363baf40c7e202d3a47ab/html5/thumbnails/1.jpg)
Tropical Cyclones: Steady StateTropical Cyclones: Steady State PhysicsPhysics
1
![Page 2: Tropical Cyclones: Steady StateTropical Cyclones: Steady ......rz 8 Variables conserved (or else constant along streamlines) above PBL where flow is consideredstreamlines) above PBL,](https://reader036.fdocuments.net/reader036/viewer/2022071218/605363baf40c7e202d3a47ab/html5/thumbnails/2.jpg)
EnerEnergy Productiongy Production
2
![Page 3: Tropical Cyclones: Steady StateTropical Cyclones: Steady ......rz 8 Variables conserved (or else constant along streamlines) above PBL where flow is consideredstreamlines) above PBL,](https://reader036.fdocuments.net/reader036/viewer/2022071218/605363baf40c7e202d3a47ab/html5/thumbnails/3.jpg)
a a t cu a e e e
Carnot Theorem: Maximum efficiency results from a pap rticular energy cycle:gy cyc
• Isothermal expansionIsothermal expansion • Adiabatic expansion • I th Isothermall compressiion • 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 coolingg is the same as if air were saturated and descendingg moist adiabatically.
Maximum rate of working: T T W s oo Qs W Q
Ts
3
![Page 4: Tropical Cyclones: Steady StateTropical Cyclones: Steady ......rz 8 Variables conserved (or else constant along streamlines) above PBL where flow is consideredstreamlines) above PBL,](https://reader036.fdocuments.net/reader036/viewer/2022071218/605363baf40c7e202d3a47ab/html5/thumbnails/4.jpg)
Total rate of heat input to hurricane:
* 3Q 2 r0 C | V | k k C | V | rdr k 0 D0
DissipativeSurface enthalpy flux heating
In steady state, Work is used to balance frictional dissipation:
W 2 0
r0 CCD || VV ||3 rdr rdr W 2
4
![Page 5: Tropical Cyclones: Steady StateTropical Cyclones: Steady ......rz 8 Variables conserved (or else constant along streamlines) above PBL where flow is consideredstreamlines) above PBL,](https://reader036.fdocuments.net/reader036/viewer/2022071218/605363baf40c7e202d3a47ab/html5/thumbnails/5.jpg)
T
Plug into Carnot equation:
r r| | rdr
T T C | | k k rdr 00 CD V 3
s
T o 0
0
k V 0*
o
If integgrals dominated byy values of integgrands near radius of maximum winds,
C T T2 k s o * ||V max || k 0 k max 0 C TD o
Not a closed expression since k at radius of maximum winds is unknown
5
![Page 6: Tropical Cyclones: Steady StateTropical Cyclones: Steady ......rz 8 Variables conserved (or else constant along streamlines) above PBL where flow is consideredstreamlines) above PBL,](https://reader036.fdocuments.net/reader036/viewer/2022071218/605363baf40c7e202d3a47ab/html5/thumbnails/6.jpg)
ProblemsProblems wwithith EnergyEnergy Bound:Bound:ProblemsProblems withwith EnergyEnergy Bound: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 thereproduction there
• Approximation of integrals dominated by high wind region is cruderegion is crude
6
![Page 7: Tropical Cyclones: Steady StateTropical Cyclones: Steady ......rz 8 Variables conserved (or else constant along streamlines) above PBL where flow is consideredstreamlines) above PBL,](https://reader036.fdocuments.net/reader036/viewer/2022071218/605363baf40c7e202d3a47ab/html5/thumbnails/7.jpg)
Local energy balance in eyewallLocal energy balance in eyewall regregiiiion:on:
7
Hei
ght
Radius
h
T = Tb
Ψ1Ψ2
T = To
Image by MIT OpenCourseWare.
![Page 8: Tropical Cyclones: Steady StateTropical Cyclones: Steady ......rz 8 Variables conserved (or else constant along streamlines) above PBL where flow is consideredstreamlines) above PBL,](https://reader036.fdocuments.net/reader036/viewer/2022071218/605363baf40c7e202d3a47ab/html5/thumbnails/8.jpg)
Definition of streamfunction, ψ:
rw , ru rr z z
Flow parallel to surfaces of constant ψ, satisfies mass continuity:
1 1 ru rw 0 r r r zr zr r
8
![Page 9: Tropical Cyclones: Steady StateTropical Cyclones: Steady ......rz 8 Variables conserved (or else constant along streamlines) above PBL where flow is consideredstreamlines) above PBL,](https://reader036.fdocuments.net/reader036/viewer/2022071218/605363baf40c7e202d3a47ab/html5/thumbnails/9.jpg)
Variables conserved (or else constant along streamlines) above PBL where flow is consideredstreamlines) above PBL, where flow is considered reversible, adiabatic and axisymmetric:
1 22Energy: E T L gz | || |VEnergy: E c T L q V p v 2 L q *L qvEntropy: s* cp ln T Rd ln p
T
Angular 1 2M rV frMomentum: 2
9
![Page 10: Tropical Cyclones: Steady StateTropical Cyclones: Steady ......rz 8 Variables conserved (or else constant along streamlines) above PBL where flow is consideredstreamlines) above PBL,](https://reader036.fdocuments.net/reader036/viewer/2022071218/605363baf40c7e202d3a47ab/html5/thumbnails/10.jpg)
f
First definition of s*:
Tds* pd vd * dp (1)d * c dT L dq * d
Steadyy flow:
p pdp dr dz rr z z
Substitute from momentum equations:
V 2 p g (2)dp dz g V w dr fV V u ( )
r
10
![Page 11: Tropical Cyclones: Steady StateTropical Cyclones: Steady ......rz 8 Variables conserved (or else constant along streamlines) above PBL where flow is consideredstreamlines) above PBL,](https://reader036.fdocuments.net/reader036/viewer/2022071218/605363baf40c7e202d3a47ab/html5/thumbnails/11.jpg)
Identity:
11 2 2 11Vw dz Vu dr d u w d , (3)2 r
u w where azimuthal vorticity
z r
Substituting (3) into (2) and the result into (1) gives:
V 2 1 Tds* VdV fV dr d (4)Tds * dE dE VdV fV dr d (4) r r
11
![Page 12: Tropical Cyclones: Steady StateTropical Cyclones: Steady ......rz 8 Variables conserved (or else constant along streamlines) above PBL where flow is consideredstreamlines) above PBL,](https://reader036.fdocuments.net/reader036/viewer/2022071218/605363baf40c7e202d3a47ab/html5/thumbnails/12.jpg)
One more identity:
V 2 M 1 VdV fV dr f dM 2 r r 22 r r
Substitute into (4):
M 1 1 *Tds Tds dM d EE fMdM dd d fM , (5) (5)2 r r 2
Note that third term on left is very small: Ignore
12
![Page 13: Tropical Cyclones: Steady StateTropical Cyclones: Steady ......rz 8 Variables conserved (or else constant along streamlines) above PBL where flow is consideredstreamlines) above PBL,](https://reader036.fdocuments.net/reader036/viewer/2022071218/605363baf40c7e202d3a47ab/html5/thumbnails/13.jpg)
Integrate (5) around closed circuit:
Right side vanishes; contribution to left only from end points
13
Hei
ght
Radius
h
T = Tb
Ψ1Ψ2
T = To
Image by MIT OpenCourseWare.
![Page 14: Tropical Cyclones: Steady StateTropical Cyclones: Steady ......rz 8 Variables conserved (or else constant along streamlines) above PBL where flow is consideredstreamlines) above PBL,](https://reader036.fdocuments.net/reader036/viewer/2022071218/605363baf40c7e202d3a47ab/html5/thumbnails/14.jpg)
1 1 T T ds * 2 2 MdM 0T Tb o ds * MdM 0 r r b o
1 1 ds * ( ) T T (6)T T2 2 bb orb ro MdM
Mat re storm r r :Mature storm: ro rb :
M ds *M ds T T 2 b o (7)
rb dM
14
![Page 15: Tropical Cyclones: Steady StateTropical Cyclones: Steady ......rz 8 Variables conserved (or else constant along streamlines) above PBL where flow is consideredstreamlines) above PBL,](https://reader036.fdocuments.net/reader036/viewer/2022071218/605363baf40c7e202d3a47ab/html5/thumbnails/15.jpg)
In inner core, V fr
M rV
V r T T ds * (8)b b b o dMdM
Convective criticality: s* sbb
V r T dsbb (9) V T TT b b b o dM
15
![Page 16: Tropical Cyclones: Steady StateTropical Cyclones: Steady ......rz 8 Variables conserved (or else constant along streamlines) above PBL where flow is consideredstreamlines) above PBL,](https://reader036.fdocuments.net/reader036/viewer/2022071218/605363baf40c7e202d3a47ab/html5/thumbnails/16.jpg)
dsb /dM determined by boundary layer processes:
16
Hei
ght
Radius
h
T = Tb
Ψ1Ψ2
T = To
Image by MIT OpenCourseWare.
![Page 17: Tropical Cyclones: Steady StateTropical Cyclones: Steady ......rz 8 Variables conserved (or else constant along streamlines) above PBL where flow is consideredstreamlines) above PBL,](https://reader036.fdocuments.net/reader036/viewer/2022071218/605363baf40c7e202d3a47ab/html5/thumbnails/17.jpg)
,
Put (7) in differential form:
ds M dM T T 0, (10)b o 2dtdt rr dt dt Integrate entropy equation through depth of boundary layer:depth of boundary layer:
ds 1 CC V k * kk C | |3 FF , (11) h | || | k C | | (11) h V VV k s D b dt Ts
where Fb is the enthalpy flux through PBL top. Integrate angular momentum equation through depth of boundary layer:
dM V V (12)h dM
C r | |dt D
17
![Page 18: Tropical Cyclones: Steady StateTropical Cyclones: Steady ......rz 8 Variables conserved (or else constant along streamlines) above PBL where flow is consideredstreamlines) above PBL,](https://reader036.fdocuments.net/reader036/viewer/2022071218/605363baf40c7e202d3a47ab/html5/thumbnails/18.jpg)
Substitute (11) and (12) into (10) and set Fb to 0:
C T T2 k s o * | | k (13)V k0CC TTD o
Same answer as from Carnot cycle This is still not a closed Same answer as from Carnot cycle. This is still not a closed expression, since we have not determined the boundary layer enthalpy, k
18
![Page 19: Tropical Cyclones: Steady StateTropical Cyclones: Steady ......rz 8 Variables conserved (or else constant along streamlines) above PBL where flow is consideredstreamlines) above PBL,](https://reader036.fdocuments.net/reader036/viewer/2022071218/605363baf40c7e202d3a47ab/html5/thumbnails/19.jpg)
ttWhat Determines OutflowWhat Determines Outflow
TT ??TTemperaemperattureure??
19
![Page 20: Tropical Cyclones: Steady StateTropical Cyclones: Steady ......rz 8 Variables conserved (or else constant along streamlines) above PBL where flow is consideredstreamlines) above PBL,](https://reader036.fdocuments.net/reader036/viewer/2022071218/605363baf40c7e202d3a47ab/html5/thumbnails/20.jpg)
ttSimulations with Cloud Permitting,Simulations with Cloud--Permitting,
A iA iAAxxiisymmesymmett iirriicc MM ddooddeellMM ll
20
![Page 21: Tropical Cyclones: Steady StateTropical Cyclones: Steady ......rz 8 Variables conserved (or else constant along streamlines) above PBL where flow is consideredstreamlines) above PBL,](https://reader036.fdocuments.net/reader036/viewer/2022071218/605363baf40c7e202d3a47ab/html5/thumbnails/21.jpg)
Saturation entropy (contoured) and V=0 line (yellow)
21
![Page 22: Tropical Cyclones: Steady StateTropical Cyclones: Steady ......rz 8 Variables conserved (or else constant along streamlines) above PBL where flow is consideredstreamlines) above PBL,](https://reader036.fdocuments.net/reader036/viewer/2022071218/605363baf40c7e202d3a47ab/html5/thumbnails/22.jpg)
Streamfunction (black contours), absolute temperature (shading) and V=0 contour(white)
22
![Page 23: Tropical Cyclones: Steady StateTropical Cyclones: Steady ......rz 8 Variables conserved (or else constant along streamlines) above PBL where flow is consideredstreamlines) above PBL,](https://reader036.fdocuments.net/reader036/viewer/2022071218/605363baf40c7e202d3a47ab/html5/thumbnails/23.jpg)
Angular momentum surfaces plotted in the V-T plane. Red curve shows shape of balanced M surface originating atcurve shows shape of balanced M surface originating at radius of maximum winds. Dashed red line is ambient
tropopause temperature. 23
![Page 24: Tropical Cyclones: Steady StateTropical Cyclones: Steady ......rz 8 Variables conserved (or else constant along streamlines) above PBL where flow is consideredstreamlines) above PBL,](https://reader036.fdocuments.net/reader036/viewer/2022071218/605363baf40c7e202d3a47ab/html5/thumbnails/24.jpg)
RiRichharddson NNumbber ((cappedd at 3) t 3). BBox shhows area usedd for scatter plot.
24
![Page 25: Tropical Cyclones: Steady StateTropical Cyclones: Steady ......rz 8 Variables conserved (or else constant along streamlines) above PBL where flow is consideredstreamlines) above PBL,](https://reader036.fdocuments.net/reader036/viewer/2022071218/605363baf40c7e202d3a47ab/html5/thumbnails/25.jpg)
Vertical Diffusivity (m2s-1)
25
![Page 26: Tropical Cyclones: Steady StateTropical Cyclones: Steady ......rz 8 Variables conserved (or else constant along streamlines) above PBL where flow is consideredstreamlines) above PBL,](https://reader036.fdocuments.net/reader036/viewer/2022071218/605363baf40c7e202d3a47ab/html5/thumbnails/26.jpg)
Ri=1
26
![Page 27: Tropical Cyclones: Steady StateTropical Cyclones: Steady ......rz 8 Variables conserved (or else constant along streamlines) above PBL where flow is consideredstreamlines) above PBL,](https://reader036.fdocuments.net/reader036/viewer/2022071218/605363baf40c7e202d3a47ab/html5/thumbnails/27.jpg)
cat o s o Out o e atu eImpplications for Outflow Temperaturepe
2 ds * r r m dMRi .M z
2 ds * MM rt m dM dM .z Ri c
s * ds * M ,
z dM z
27
![Page 28: Tropical Cyclones: Steady StateTropical Cyclones: Steady ......rz 8 Variables conserved (or else constant along streamlines) above PBL where flow is consideredstreamlines) above PBL,](https://reader036.fdocuments.net/reader036/viewer/2022071218/605363baf40c7e202d3a47ab/html5/thumbnails/28.jpg)
*
-
2 ds * 2
r t m (14)s ** t m dMdM (14)
.z Ri c
But the vertical gradient of saturation entropy is related to the vertical gradient of temperature:
T T T p s* , (15)
s ** s ** p ss * pUse definition of
Ts* and C -C :s and C. C.: T cp , 2s * L q * (16) pp L q (16) v1 2 R c T v p
28
![Page 29: Tropical Cyclones: Steady StateTropical Cyclones: Steady ......rz 8 Variables conserved (or else constant along streamlines) above PBL where flow is consideredstreamlines) above PBL,](https://reader036.fdocuments.net/reader036/viewer/2022071218/605363baf40c7e202d3a47ab/html5/thumbnails/29.jpg)
t
Substitute (16) into (15) and use hydrostatic equation:
TT
T c p m .
2 * (17)(17)s ** L q * s * v1 2 zR c T v pp
2 2T T r we can neglect first term 2 L q * 2 b o v bIf V c 1 Ri bb cc on left of (17) pp 22 22 on left of (17) Tb RRvcpT t T T r
Substitute (14) into (17):( ) ( ) T Ri dM
2 o c 2 . (18)s * rt ds *
Gives dependence of Outflow T on s*
29
![Page 30: Tropical Cyclones: Steady StateTropical Cyclones: Steady ......rz 8 Variables conserved (or else constant along streamlines) above PBL where flow is consideredstreamlines) above PBL,](https://reader036.fdocuments.net/reader036/viewer/2022071218/605363baf40c7e202d3a47ab/html5/thumbnails/30.jpg)
T T dM Using ,s * M ds *
T Ri dM We can re-write (18) as . (19)o 2 c M rt ds *
2 1 ds * We can also re-write (6) Mb rb f Tb To (20) as 2 dM
ds | |3 Boundary layer h b C V s * s C
V (21) | |kk 00 bb dd entropyentropy dtdt TTb
Boundary layer dMh rr | || |VV VVh (22)angular momentum (22) dt
30
![Page 31: Tropical Cyclones: Steady StateTropical Cyclones: Steady ......rz 8 Variables conserved (or else constant along streamlines) above PBL where flow is consideredstreamlines) above PBL,](https://reader036.fdocuments.net/reader036/viewer/2022071218/605363baf40c7e202d3a47ab/html5/thumbnails/31.jpg)
MIT OpenCourseWarehttp://ocw.mit.edu
12.811 Tropical Meteorology Spring 2011
For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms.
31