Aspects of Moat Formation Aspects of Moat Formation in Tropical Cyclone Eyewall in Tropical Cyclone Eyewall
Replacement CyclesReplacement Cycles
Christopher Rozoff 3 April 2005 2006 2007
Timeline of world history during Chris Rozoff’s time at CSU
Time (scale = many, many years)
2000 20022001 2005 2006 2007
The Clinton era endsThe Clinton era ends
A bunch of bad stuff happensA bunch of bad stuff happens
1 Average Lifespan of a Crow
2 Lifespans of House Sparrows
61 Lifespans of Honey Bees
Aspects of Moat Formation Aspects of Moat Formation in Tropical Cyclone Eyewall in Tropical Cyclone Eyewall
Replacement CyclesReplacement Cycles
Christopher Rozoff3 April 2007
AcknowledgementsAcknowledgements• My advisor – Prof. Wayne Schubert / My committee – Profs. William Cotton,
Richard Johnson, Iuliana Oprea (CSU mathematics)• Prof. Michael Montgomery (Naval Postgraduate School)• Other Collaborators: Paul Ciesielski, Prof. Scott Fulton (Clarkson U.), Dr. Jim
Kossin (UW-Wisc), Brian McNoldy, Rick Taft, Wes Terwey, and Jonathan Vigh
• Drs. Will Cheng, Louie Grasso, Sue van den Heever, and Mel Nicholls (U. Colorado) for help with RAMS throughout my CSU tenure.
• Drs. Michael Black (HRD), Neal Dorst (HRD), and Hugh Willoughby (FIU), and Michael Bell (NCAR) and Kevin Mallen for help with real hurricane data.
• Prof. Matthew Parker (NCSU) and Russ Schumacher for useful discussion on dynamic pressure perturbation analysis.
• Gail Cordova and department staff for making life easy for research and learning.
• Schubert group members and many others for an invigorating learning environment at CSU.
• Your tax dollars• My family for dedicated support and for attending my defense.• My wife Jill for unearthly patience, support, and encouragement.
OutlineOutline
1. Introduction
2. Rapid filamentation zones
3. Observations
4. Idealized cloud model results
5. Concluding Remarks
1. Introduction:1. Introduction:Eyewall replacement cycles and rapid intensity fluctuationsEyewall replacement cycles and rapid intensity fluctuations
10/19 0014 UTC130 kts/946 hPa
10/19 1214 UTC160 kts/882 hPa
10/19 1358 UTC157 kts/885 hPa
10/20 0000 UTC135 kts/892 hPa
10/20 1234 UTC130 kts/910 hPa
10/20 2347 UTC130 kts/923 hPa
10/21 1219 UTC125 kts/929 hPa
10/22 0220 UTC117 kts/932 hPa
Hurricane Wilma (2005)
SSM 85 GHz Composites
1. Introduction:1. Introduction:Formation of a secondary eyewallFormation of a secondary eyewall
• Axisymmetric (circularly symmetric) hurricane models– Forcing mechanism needed to initiate secondary eyewall:
• Symmetric instability (Willoughby et al.,1984; Zeng, 1996)• Other sources of low-level convergence (Hausman, 2001; Nong and
Emanuel, 2003)
– To sustain, wind-induced surface heat exchange (WISHE) (Willoughby et al., 1984; Nong and Emanuel, 2003)
z
rCenter of eye
z
rCenter of eye
Earlier Later
Subsidence Inversion Strong forcing
1. Introduction:1. Introduction:Formation of a secondary eyewallFormation of a secondary eyewall
2D, nondivergent barotropic models– Multiple vortex interactions (e.g., Kuo et al., 2004) in a
horizontal plane. (Asymmetric processes are important here!)
y
x
Extensive weaker vorticity (e.g., Convective rainbands)
Stronger vorticity (eyewall)
t = 0 hr t = 3 hr t = 12 hr
• Other perhaps crucial asymmetric processes:– Vortex Rossby waves and wave-mean flow interactions
accelerate mean flow at a radius determined by the mean vortex structure (e.g., Montgomery and Kallenbach, 1997)
– Convective rainbands generate potential vorticity (PV).
• 3D modeling with sufficiently small grid spacing (Houze et al., 2007; Terwey and Montgomery, 2006; Wang, 2006; Yau et al., 2006; Zhang et al., 2005) produces concentric eyewalls in intense hurricanes.
• Where are secondary eyewalls unlikely to form?
1. Introduction:1. Introduction:Formation of a secondary eyewallFormation of a secondary eyewall
• Formation of a moat – Region of subsidence as a secondary eyewall matures (Dodge
et al., 1999; Houze et al., 2007)– Region of intense horizontal strain before and after secondary
eyewall formation (Shapiro and Montgomery, 1993; Kossin et al., 2000; R. et al., 2006)
– Which processes dominate in the moat region before and after secondary eyewall formation?
1. Introduction:1. Introduction:
The “moat”
2. Rapid filamentation zones2. Rapid filamentation zones
( ) 022 =∇•+∂≡ q
Dt
qDt u
⎟⎟⎠
⎞⎜⎜⎝
⎛−+−
=ns
sn
SS
SS
ζ
ζ
2
12V
uv yx ∂−∂=ζvuS yxn ∂−∂=uvS yxs ∂+∂=
022 =⎟
⎟⎠
⎞⎜⎜⎝
⎛
∂∂
+⎟⎟⎠
⎞⎜⎜⎝
⎛
∂∂
DtD
y
xT
y
xV
222
2
1 ζλ −+±= sno SS
From a materially conserved tracer q in a horizontal, 2D plane, we can form a tracer gradient equation:
where V2 is the velocity gradient tensor:
and where
Assuming V2 is constant, we obtain the Okubo-Weiss criterion (which is thefrequency associated with the solution of the tracer gradient equation):
2. Rapid filamentation zones2. Rapid filamentation zones
022 =⎟
⎟⎠
⎞⎜⎜⎝
⎛
∂∂
+⎟⎟⎠
⎞⎜⎜⎝
⎛
∂∂
DtD
y
xT
y
xV 02222
2
22 =⎟⎟⎠
⎞⎜⎜⎝
⎛∂∂
⎟⎟⎠
⎞⎜⎜⎝
⎛−+⎟⎟
⎠
⎞⎜⎜⎝
⎛∂∂
Dt
Dqq
Dt
D
y
xTTT
y
x VVV
( ) 2/1222222
2 22
1⎥⎦⎤
⎢⎣⎡ −+±−+±= ζζλ &&&
snsn SSSS
Rather than assuming a constant velocity gradient tensor, we obtain a second order equation describing tracer gradient growth, which yields more accurate solutions (Hua and Klein, 1998):
Which has the following eigenvalues:
2. Rapid filamentation zones2. Rapid filamentation zones
}max{/1fil iλτ =
min30convfil =<ττ
Okubo-Weiss and Hua-Klein eigenvalues are frequencies associated with either oscillatory or exponential decay/growth. An e-folding type timescale can be defined – the filamentation time – for the real part λi
(i.e., where there is exponential growth rates):
Given typical convective overturning timescales of about 30 min, we definea rapid filamentation zone as a region where:
We hypothesize that deep convection is strongly deformed and susceptible to enhanced entrainment and subsequent suppression in such regions.
2. Rapid filamentation zones2. Rapid filamentation zones
Hua-Klein τfilOkubo-Weiss τfil
Gaussian vortices
2. Rapid filamentation zones2. Rapid filamentation zones
ζυζψζ 2
),(
),(∇=
∂
∂+
∂
∂
yxt < 2.5 min
2.5 -7.5 min
7.5 - 15 min
15 - 30 min
> 30 min
Infinity min
Hua-Klein τfilRel Vorticity
ψζ 2∇=
Pseudo-spectral numerical integration of:
Initial Conditions:-Random vorticity elements between 20 – 40 km.-Random vorticity has 1/10 magnitude of central vortex.-Positive bias to random vorticity field.
Model config:
- 600 x 600 km - 1024 x 1024 collocation points => 1.76 km res.- = 20 m2 s-1
3. Moat observations3. Moat observations
• Dropsondes and aircraft data from Frances (2004) and Rita (2005).• NOAA P3s give 1 s T, Td, p, u, v. • T, Td corrected for instrument wetting (Zipser et al., 1981).• GPS dropsondes – p, T, R.H., u, and v at 5 m intervals (2 Hz.)
(QC’d on ASPEN or Editsonde (HRD)).• Data tranformed into cylindrical coordinates – Willoughby and
Chelmow (1982) center-finding technique (~3 km error).• Data composites defined as:
∫
∫+
−
+
−= xx
xx
xx
xxo o
o
o
o
dxxK
dxxKx
x δ
δ
δ
δ
η
η
)(
)()(
)(
xxx
xxxx
xxxx
xxx
xxxx
xxxxxK
o
oo
oo
o
o
o
δδ
δδ
δδδδ
+≥+<≤<≤−
−<
⎪⎪⎩
⎪⎪⎨
⎧
+−+−
=
if
if if if
0/)(/)(
0
)(
3. Moat observations3. Moat observations
Hurricane Frances (2004)
Figure taken from Beven (2004/NHC)
Best track data (NHC)
3. Moat observations3. Moat observations
Atlantic Hurricane Frances on30 August 2004.
NOAA P3 data collected inthis storm.
(a) & (b) 1804 – 1822 UTC(c) & (d) 1924 – 1943 UTC(e) & (f) 2108 – 2126 UTC
v
T
Td
3. Moat observations3. Moat observationsAtlantic Hurricane Frances on30 August 2004.
Composite profile: - 2 δr = 6 km on a r = 250 m grid. - 700 hPa flight-level data only (1804 – 1822 UTC; 2108 – 2126 UTC).
TOP:Blue (Individual Flight-level Tangential Wind)Red (Filamentation Time (min))Black Composite)
BOTTOM:Red (Temperature)Green (Dew Point)Black (Composites)
T
Td
v
τfil
3. Moat observations3. Moat observations
Moat
r = 24 km r = 29 km r = 32 km
Dropsonde data points shown tothe right.
The moat of Frances had eye-like dropsondes in the moat.Low-level instability was marginal.
TTd
Eye
wall
Eye
wall
T Td Td TTparcel
3. Moat observations3. Moat observations
Hurricane Rita (2005)
Figure taken from Knapp et al. (2005/NHC)
Best track data (NHC)
3. Moat observations3. Moat observationsRita 21 September 2005 (N43)
Rita 22 September 2005 (N43)
Radar imagery from HRD/RAINEX
1459 UTC
1752 UTC1612 UTC1457 UTC
1936 UTC1517 UTC1510 UTC
1911 UTC
525047
4542
dBZ
403735
323027
252220
216
km21
6 km
3. Moat observations3. Moat observations
Rita 21 September 2005 (N43)
640 hPa1855 – 1956 UTC
700 hPa1507 – 1616 UTC
T
Td
T
Td
v
v
τfil
τfil
3. Moat observations3. Moat observationsRita 21 September 2005 Composite Dropsondes
Composite profile: - 2 δp = 10 hPa on a p = 0.5 hPa grid. - N43/NRL drops - (a) 25 km < r < 55 km - (b) 55 km < r < 85 km - Std Dev ~ 0.9oC
TTd
Eye
wall
TTd Tparcel
3. Moat observations3. Moat observations
700 hPa1437 – 2057 UTC
2.1 km1705 – 1735 UTC
1.5 km1754 – 2213 UTC
Rita 22 September 2005 Flight-level Composites
v
v
v
τfil
τfil
τfil
T
T
T
Td
Td
Td
3. Moat observations3. Moat observations
Composite profile: - 2 δp = 10 hPa on a p = 0.5 hPa grid. - N43/N42/NRL drops - 25 km < r < 40 km
16 – 19 UTC 19 - 22 UTC
Eye-like soundingsconsistent with Houze et al. (2007;Science)
Eye
wall
Eye
wall
Moat
Td TdT TTparcel
3. Moat observations:3. Moat observations:Balanced vortex suggestionsBalanced vortex suggestions
• 5-region approximation to the Sawyer-Eliassen equation (Similar approaches are used in Schubert et al., 2007; Shapiro and Willoughby, 1982; Schubert and Hack, 1982). This model diagnoses the secondary circulation for a given tangential wind profile and prescribed diabatic heating.
• Consider axisymmetric, quasi-static, stratified, compressible, and inviscid motions on an f-plane.
• Assume a barotropic vortex.
)(ˆ rQ Heating:
rr1 r2 r3 r4
Q1 Q2
rr1 r2 r3 r4
Vorticity:
3. Moat observations:3. Moat observations:Balanced vortex suggestionsBalanced vortex suggestions
2-123
243
21
221 km) 50(dayK 125))(/())(/( =−+− rrcQrrcQ pp
1-5 s 10 x 5 −=f
T obs.
Td obs.
dT/dt (analytical)
w (analytical)
v
-12 s 10 x 1 −=N
Assume the following: Results:
3. Moat observations:3. Moat observations:Balanced vortex suggestionsBalanced vortex suggestions
Flux Mass Downward Total
Eye in theFlux Mass Downward=eyeσ
Flux Mass Downward Total
Moat in theFlux Mass Downward=moatσ
• A look at mass subsidence in the moat during an idealized eyewall replacement cycle
r4r3
2-123
243
21
221 km) 50(dayK 125))(/())(/( =−+− rrcQrrcQ pp
Frances
4. Idealized cloud model results4. Idealized cloud model results
• RAMS – 3D, compressible, nonhydrostatic, one-moment microphysics.
• f-plane, x = y = 500 m over 125 x 125 km. z = 160 m near surface, stretching to a maximum spacing of 500 m aloft. 25 km depth.
• Radiation neglected • Lower boundary is free slip.• Rayleigh friction layer at rigid lid and Klemp-
Wilhelmson (1978) lateral boundary conditions.• Smagorinsky (1963) diffusion.• Convection initiated with a 2 K bubble.
4. Idealized cloud model results4. Idealized cloud model results
• Sounding constructed using several outer-core dropsondes from Hurricane Isabel (2003) and carefully blended with a proximity sounding (13 Sep 2003) (courtesy W. Terwey and M. Bell)
• CAPE = 2067 J/kg and CIN = 1 J/kg.
• Background wind:
• vz = 0, 5, 10, and 20 m s-1 per 15 km and vx = 0, -2, -4, and -6 x 10-4 s-1. All cases are initialized in geostrophic and hydrostatic balance.
• The initial absolute vorticity, vx + f, is always equal to 1 x 10-4 s-1.
refzx vzvxvtzyxv ++=),,,(
4. Idealized cloud model results4. Idealized cloud model results
vz = 20 m s-1 (15 km)-1
vx = 0 x 10-4 s-1
4. Idealized cloud model results4. Idealized cloud model results
vz = 0 m s-1 (15 km)-1
vx = -4 x 10-4 s-1
4. Idealized cloud model results4. Idealized cloud model results
vz = 0 m s-1 (15 km)-1
vx = -6 x 10-4 s-1
4. Idealized cloud model results4. Idealized cloud model results
τl (h) hmax (km) wmax wmin
v00h0 1.8 14.9 37.8 -9.2
v00h2 0.9 12.9 30.8 -9.1
v00h4 1.0 10.9 22.6 -6.2
v00h6 0.2 8.4 12.3 -4.0
• Practical rapid filamentation occurs for vx = -6 x 10-4 s-1 (exp. v00h6)⎟
⎟⎠
⎞⎜⎜⎝
⎛−−= )(
,
'
vtotalov
v rrgBθ
θ
z
v
x
w
z
u
y
w
tr
r
r
rtilt∂∂
∂∂
−∂∂
∂∂
=∂∂ 'ζ
z = 1.25 km at 0.6 h
z = 1.25 km at 0.6 h
z = 0.08 km at 0.6 h
z = 1.25 km at 0.6 h z = 1.25 km at 0.6 h
z = 1.25 km at 0.6 h z = 1.25 km at 0.6 h
z = 1.25 km at 0.6 h z = 1.25 km at 0.6 h
3 m s-1
Vertical Motion (m s-1)
Pert.RelativeVorticity(x 10-4 s-1)
Pert.RelativeVorticity(x 10-4 s-1)
Exp. v00h6
4. Idealized cloud model results4. Idealized cloud model results
SBzDt
Dwvo ++
∂∂
−='π
Exp. v00h4 x 10-4 s-1 x 10-2 m s-2 x 10-2 m s-2
First column -w: Vertical velocity (contoured)ζ: Pert. vert. vorticity (shaded)
Second column –Dynamic perturbation pressure gradient
Third column –Sum of buoyancy andbuoyant perturbationgradient
z=1.25 km
''''' cbdh πππππ +++=o
4. Idealized cloud model results4. Idealized cloud model results
vz = 20 m s-1 (15 km)-1
vx = -2 x 10-4 s-1
4. Idealized cloud model results4. Idealized cloud model results
x 10-2 m s-2x 10-2 m s-2x 10-4 s-1 x 10-4 s-1 x 10-2 m s-2 x 10-2 m s-2
Exp. v20h2 Exp. v20h4
z=1.25 km
Convergence of ζa>0
+
4. Idealized cloud model results4. Idealized cloud model resultsSummary of cloud dynamicsSummary of cloud dynamics
Vertical Shear Horizontal Shear
++ -
-
-
Dynamic pressure perturbations/buoyant forcing important in forcing
primary updrafts. Dynamicpressure perturbations also
force an upright updraft.
Buoyant forcing along edges of coldpool are important in forcing
primary updrafts.
x
y
z z
x
y
LL
v
v
Convergence of ζa>0
4. Idealized cloud model results4. Idealized cloud model results Sensitivity ExperimentsSensitivity Experiments
τl (h) hmax (km) wmax wmin
v00h0 + 0.3 + 1.0 + 3.5 - 7.4
v00h6 + 0.8 + 4.0 + 13.3 - 2.6
v20h0 + 0.2 - 2.5 + 7.2 - 4.9
v20h6 + 1.0 + 2.5 + 10.4 - 5.7
τl (h) hmax (km) wmax wmin
v00h0 - 1.3 + 0.5 + 5.9 - 4.4
v00h6 + 0.6 + 4.0 + 13.3 - 3.1
v20h0 + 0.4 + 0.5 + 0.5 - 1.2
v20h6 + 0.6 + 3.0 + 13.4 - 3.3
“Unstable” – “Control” “Moist” – “Control”
5. Conclusions5. Conclusions
• Rapid filamentation zones (RFZs), defined from local kinematics, are regions where the filamentation time is smaller than the typical timescale of convective overturning.
• Observations suggest moats coincide with RFZs. Moats contain marginal thermodynamic conditions for the existence of deep, moist convection.
• As a moat forms, balanced theory suggests eye-like downward mass fluxes can take place in the moat early in an eyewall replacement cycle.
• Rapid filamentation is most likely relevant prior to mature moat formation.
5. Conclusions5. Conclusions
• Cloud simulations suggest that, in relatively marginal thermodynamic conditions, adverse filamentation occurs for sufficiently strong horizontal shear.
• We’ve uncovered new dynamics of horizontally sheared convection. Future work should include low-level inflow.
• PV wakes left behind sheared convection could be important in the genesis of secondary eyewalls (e.g., Franklin et al., 2006).
• Slight changes in the thermo has profound impacts on sheared convection. A refined definition of rapid filamentation should include the instability.
Questions?Questions?
16 September 2006Montrose, SDRemnants of Ioke?