observational, radiative and dynamical implications of ...susan hovde & adrian tuck crystal-face...
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Observational, Radiative and Dynamical Implications of Scale
Invariance Near the Tropical Tropopause
Susan Hovde & Adrian Tuck
CRYSTAL-FACE Science MeetingSalt Lake City, February 2003
Scaling Exponent H1
• Start with signal f(t).• The first order structure function associated with f is
where the independent variable r is called the “lag”.• Plot log(S1) vs. log(r). If the points describe a relatively
straight line, then f scales, and H1 is given by the slope of the line.
• H1 ranges from 0 to 1.– Values near 1: “persistence”, positive neighbor-to-
neighbor correlation.– Values near 0: “anti-persistence”, negative neighbor-to-
neighbor correlation.
S1 r; f( )= f t + r( )− f t( )
H1 for Air Along Aircraft Flight Tracks
• H1 is related to spectral exponent β by β = 2H1+1.• Kolmogorov theory predicts β = 5/3 for the atmosphere in
the horizontal direction, corresponding to Hh = 1/3. • Bolgiano’s arguments predict β = 11/5 in the vertical
direction, corresponding to Hv = 3/5.• Even in “level” flight, an aircraft samples a mix of the
horizontal and vertical structure of the atmosphere.• The scaling exponent along an aircraft flight track is given
by Hz = Hh/Hv = 5/9 ≈ 0.56.• Empirically, we see a slightly lower value, possibly due to
the inertia of the aircraft. (Under investigation.)
WAM 19980511 WB57
700
600
500
400
300
200
100
O3
(ppb
v)
70x103656055Time (seconds UTC)
420
400
380
360
340
320
300
Potential Tem
perature (K)
374 K ± 5 K
418 K ± 6 K
WAM WB57F 19980511
1.6
1.2
0.8
0.4
log 1
0(M
omen
t)
3210log10(Time Interval Length)
220
200
180
160
140
Ozo
ne (
ppbv
)
62x103
60585654Time (UTC Seconds)
Tropospheric Segment
2.0
1.6
1.2
0.8
log 1
0(M
omen
t)
3210log10(Time Interval Length)
H1 = 0.39±0.05
700
600
500
400
300
Ozo
ne (
ppbv
)
70x103
686664Time (UTC Seconds)
Stratospheric Segment
Synthetic Signal
-4.0-3.5-3.0-2.5-2.0-1.5-1.0-0.5
Log(
K(q
,η))
-1.0 -0.5 0.0 0.5 1.0Log(η)
α = 1.66 ± 0.12
0.20
0.15
0.10
0.05
0.00
K(q
)
3.53.02.52.01.51.00.50.0q
C1 = 0.025 ± 0.002
1.5
1.0
0.5
0.0
log 1
0(M
omen
t)
3.53.02.52.01.51.00.50.0log10(Time Interval Length)
H1 = 0.50±0.03
-60
-40
-20
0
20
40
f(t)
40003000200010000t
Synthetic Signal Plus 10% Gaussian Noise
-3.5-3.0-2.5-2.0-1.5-1.0-0.5
Log(
K(q
,η))
-1.0 -0.5 0.0 0.5 1.0Log(η)
α = 1.64 ± 0.17
0.25
0.20
0.15
0.10
0.05
0.00
K(q
)
3.53.02.52.01.51.00.50.0q
C1 = 0.027 ± 0.002
1.6
1.5
1.4
1.3
1.2
1.1
log 1
0(M
omen
t)
3.53.02.52.01.51.00.50.0log10(Time Interval Length)
H1 = 0.29±0.02
H1 = 0.02±0.02-80
-40
0
40
f(t)+
nois
e
40003000200010000t
WAM WB57F 19980501 MTP Isentrope Curtain
WAM WB57F 19980501 minlat-begindescent
-1.0-0.8-0.6-0.4-0.20.00.2
log 1
0(M
omen
t)
3210log10(Time Interval Length)
H1 = 0.30±0.07 0.40.30.20.10.0
98765Water (ppmv)
98765W
ater
(pp
mv)
68x103
6664Time (UTC Seconds)
20x10-3
1510
50
400300200100Ozone (ppmv)
2.01.61.20.8
log 1
0(M
omen
t)
3210log10(Time Interval Length)
H1 = 0.35±0.07400
300
200
100Ozo
ne (
ppbv
)
68x103
6664Time (UTC Seconds)
WAM WB57F 19980501 minlat-begindescent
1.51.00.50.0
-0.5-1.0lo
g 10(
Mom
ent)
3210log10(Time Interval Length)
H1 = 0.65±0.06
0.20
0.15
0.10
0.05
0.00
210205200195Temperature (K)
205
200
195
Tem
pera
ture
(K
)
68x103
6664Time (UTC Seconds)
-1.5-1.0-0.50.00.51.0
log 1
0(M
omen
t)
3210log10(Time Interval Length)
H1 = 0.70±0.02
80x10-3
60
40
20
5040302010Wind Speed (m/s)
5040
3020
10Win
d Sp
eed
(m/s
)
68x103
6664Time (UTC Seconds)
ACCENT 19990921 WB57
40
30
20
10
0
Nav
Rec
orde
r Win
d S
peed
(m
/s)
76x10374727068666462Time (seconds UTC)
360
350
340
330
320
310
300
Potential Tem
perature (K)
360 K ± 3 K
19990921 Scaling Exponents
-1.5
-1.0
-0.5
0.0
0.5
log(
<|f(
x+r)
- f(x
)|> )
43210log( r )
TemperatureH1 = 0.55 ± 0.041.0
0.5
0.0
-0.5
log(
<|f(
x+r)
- f(x
)|> )
43210log( r )
Wind SpeedH1 = 0.48 ± 0.04
0.8
0.4
0.0
-0.4
log(
<|f(
x+r)
- f(x
)|> )
43210log( r )
Total H2OH1 = 0.31 ± 0.05
1.6
1.2
0.8
0.4log(
<|f(
x+r)
- f(x
)|> )
43210log( r )
OzoneH1 = 0.34 ± 0.04
Flight Segments Included In AnalysisAircraft Date
yyyymmddSegmentduration(s)
Coordinates (ºlat, ºlon) Segmentcenter time(UTCs)
Approximate mean θ (K)
ER-2 19870203 6971 (14ºS,129ºE) (18ºS, 117ºE) 91635 412ER-2 19870814 6845 (7ºN, 79ºW) (5ºS, 78ºW) 48470 391ER-2 19941026 8207 (15ºN, 158ºW)(1ºN,159ºW) 76878 410WB57F 19980409 7017 (28ºN, 93ºW) (17ºN, 84ºW) 59534 368WB57F 19980409 5943 (19ºN, 85ºW) (27ºN, 91ºW) 66911 413WB57F 19980501 8209 (28ºN, 94ºW) (14ºN, 95ºW) 57962 362WB57F 19980501 7242 (13ºN, 95ºW) (26ºN, 95ºW) 66259 370WB57F 19980504 7926 (14ºN, 95ºW) (28ºN, 95ºW) 68442 420WB57F 19980507 5144 (30ºN, 93ºW) (33ºN, 83ºW) 56910 373WB57F 19980511 8872 (28ºN, 95ºW) (11ºN, 95ºW) 58357 374WB57F 19980511 8052 (13ºN, 95ºW) (28ºN, 95ºW) 67906 418WB57F 19990920 13525 (28ºN, 87ºW) (5ºN, 94ºW) 46669 362WB57F 19990921 10992 (13ºN, 82ºW) (28ºN, 94ºW) 68936 360WB57F 20020726 3125 (22ºN, 86ºW) (17ºN, 84ºW) 63738 369
Calculations of H1
Segmentduration(s)
H1(H2O) H1(O3) H1([u2+v2]1/2) H1(T)
6971 - 0.34 ± 0.07 - 0.49 ± 0.036845 - - 0.35 ± 0.04 0.34 ± 0.048207 - 0.37 ± 0.04 0.49 ± 0.07 0.52 ± 0.027017 - - 0.56 ± 0.07 0.66 ± 0.025943 - 0.43 ± 0.05 0.36 ± 0.04 0.61 ± 0.038209 0.43 ± 0.05 0.51 ± 0.08 0.64 ± 0.06 0.65 ± 0.047242 0.30 ± 0.07 0.35 ± 0.07 0.65 ± 0.06 0.70 ± 0.027926 - 0.48 ± 0.10 0.36 ± 0.05 0.59 ± 0.035144 0.35 ± 0.05 0.51 ± 0.09 0.56 ± 0.03 -8872 - - 0.57 ± 0.06 0.65 ± 0.048052 - 0.39 ± 0.05 0.43 ± 0.04 0.62 ± 0.0313525 0.26 ± 0.04 0.37 ± 0.06 - 0.47 ± 0.0910992 0.31 ± 0.04 0.33 ± 0.04 0.48 ± 0.04 0.55 ± 0.043125 0.40 ± 0.04 0.43 ± 0.05 0.44 ± 0.08 0.47 ± 0.07
Mean 0.34 ± 0.06 0.41 ± 0.07 0.49 ± 0.11 0.56 ± 0.10
CRYSTAL-FACE 20020726 WB57
30
25
20
15
10
5
0
Hor
izon
tal W
ind
Spe
ed (
m/s
)
75x103706560Time (seconds UTC)
420
400
380
360
340
Potential Tem
perature (K)
370 K ± 5 K
20020726 Scaling Exponents
-0.8
-0.4
0.0
0.4
log(
<|f(
x+r)
- f(x
)|> )
3.02.52.01.51.00.50.0log( r )
Wind SpeedH1 = 0.44 ± 0.08
-1.0
-0.5
0.0
log(
<|f(
x+r)
- f(x
)|> )
3.02.52.01.51.00.50.0log( r )
TemperatureH1 = 0.47 ± 0.07
-0.8
-0.4
0.0
0.4
log(
<|f(
x+r)
- f(x
)|> )
3.02.52.01.51.00.50.0log( r )
H2O Vapor H1 = 0.40 ± 0.041.6
1.2
0.8
log(
<|f(
x+r)
- f(x
)|> )
3.02.52.01.51.00.50.0log( r )
OzoneH1 = 0.43 ± 0.05
Summary
• Wind speed and temperature scale with H1 = 5/9.• Scale breaks for ozone and water on some flights
at about 20 km appear to be instrumental.• Total water, water vapor and ozone scale in the
upper troposphere, as does carbon dioxide in the lower stratosphere, but not as passive scalars, i.e. H1 ≠ 5/9. Sources and sinks are operating faster than mixing.
Consequence
• Because the radiatively absorbing and emitting species scale, so must the energy input and output to the atmosphere: all scales play a role in forcing and dissipation.
• Better signal-to-noise ratio is required at high frequencies.
• Many more long, “constant level” flight tracks are needed in the troposphere, preferably with 10 Hz data.
Conclusions
WAM WB57F 19980506 endascent-beginchgalt
25
20
15
10
5
0
Wat
er (
ppm
v)
59x103
58575655Time (UTC sec)
800
700
600
500
400
300
200
Ozone (ppbv)
Water
Ozone
Tropopause Potential Temperature vs. Latitude
AAOE ER-2 19870814 endascent-beginchgalt1
-0.70
-0.60
-0.50
log 1
0(M
omen
t)
3210log10(Time Interval Length)
2.01.51.00.50.0
4.03.63.22.8Water (ppmv)
6
5
4
3
2
Wat
er (
ppm
v)
51x103
504948474645Time (UTC Seconds)
40x10-3
302010
0
160140120Ozone (ppmv)
1.0
0.8
0.6
log 1
0(M
omen
t)
3210log10(Time Interval Length)
H1 = 0.17±0.04160
140
120Ozo
ne (
ppbv
)
51x103
504948474645Time (UTC Seconds)
AAOE ER-2 19870814 endascent-beginchgalt1
-0.4
0.0
0.4
log 1
0(M
omen
t)
3210log10(Time Interval Length)
H1 = 0.35±0.04
0.6
0.4
0.2
0.0
199198197196Temperature (K)
199
198
197
196
Tem
pera
ture
(K
)
51x103
504948474645Time (UTC Seconds)
-0.8
-0.4
0.0
log 1
0(M
omen
t)
3210log10(Time Interval Length)
H1 = 0.34±0.04
0.250.200.150.100.050.00
1614121086Wind Speed (m/s)
16141210
86W
ind
Spee
d (m
/s)
51x103
504948474645Time (UTC Seconds)
ACCENT WB57F 19990921 63441-74432
0.4
0.0
-0.4log 1
0(M
omen
t)
43210log10(Time Interval Length)
H1 = 0.31±0.04 0.200.150.100.050.00
1614121086Water (ppmv)
16
12
8
Wat
er (
ppm
v)
74x103
7270686664Time (UTC Seconds)
30x10-3
25201510
5
10080604020Ozone (ppmv)
1.6
1.2
0.8
0.4log 1
0(M
omen
t)
43210log10(Time Interval Length)
H1 = 0.33±0.0410080604020
Ozo
ne (
ppbv
)
74x103
7270686664Time (UTC Seconds)
ACCENT WB57F 19990921 63441-74432
1.0
0.5
0.0
-0.5log 1
0(M
omen
t)
43210log10(Time Interval Length)
H1 = 0.48±0.04
0.30
0.20
0.10
204202200198196Temperature (K)
204202
200198196Te
mpe
ratu
re (
K)
74x103
7270686664Time (UTC Seconds)
-1.5-1.0-0.50.00.5
log 1
0(M
omen
t)
43210log10(Time Interval Length)
H1 = 0.55±0.04
0.100.080.060.040.020.00
252015105Wind Speed (m/s)
25201510
5
Win
d Sp
eed
(m/s
)
74x103
7270686664Time (UTC Seconds)
WAM WB57F 19980501 minlat-begindescent Ozone
-2.5
-2.0
-1.5
-1.0
-0.5
Log(
K(q
,η))
-1.0 -0.5 0.0 0.5 1.0Log(η)
α = 1.95 ± 0.20
0.5
0.4
0.3
0.2
0.1
0.0
K(q
)
3.53.02.52.01.51.00.50.0q
C1 = 0.047 ± 0.002
2.0
1.6
1.2
0.8log 1
0(M
omen
t)
3210log10(Time Interval Length)
H1 = 0.35±0.07
400
300
200
100
Ozo
ne (
ppbv
)
69x103
686766656463Time (UTC Seconds)
WAM WB57F 19980501 minlat-begindescent Wind Speed
-1.2
-1.0
-0.8
-0.6
-0.4
Log(
K(q
,η))
-1.0 -0.5 0.0 0.5 1.0Log(η)
α = 1.35 ± 0.63
0.6
0.4
0.2
0.0
-0.2
K(q
)
3.53.02.52.01.51.00.50.0q
C1 = 0.179 ± 0.010
1.5
1.0
0.5
0.0
-0.5
-1.0
log 1
0(M
omen
t)
3210log10(Time Interval Length)
H1 = 0.65±0.06
50
40
30
20
10
Hor
izon
tal W
ind
Spee
d (m
/s)
69x103
686766656463Time (UTC Seconds)
WAM WB57F 19980501 minlat-begindescent Temperature
-2.5
-2.0
-1.5
-1.0
-0.5
Log(
K(q
,η))
-1.0 -0.5 0.0 0.5 1.0Log(η)
α = 1.72 ± 0.25
0.5
0.4
0.3
0.2
0.1
0.0
K(q
)
3.53.02.52.01.51.00.50.0q
C1 = 0.061 ± 0.002
-1.5
-1.0
-0.5
0.0
0.5
1.0
log 1
0(M
omen
t)
3210log10(Time Interval Length)
H1 = 0.70±0.02205
200
195Tem
pera
ture
(K
)
69x103
686766656463Time (UTC Seconds)
Scaling Exponents as a Function of Potential Temperature
420
410
400
390
380
370
360
Pote
ntia
l Tem
pera
ture
0.70.60.50.40.3H1
420
410
400
390
380
370
360
Pote
ntia
l Tem
pera
ture
0.160.140.120.100.080.060.04C1
420
410
400
390
380
370
360
Pote
ntia
l Tem
pera
ture
2.52.01.51.0α
temperature wind speed water ozone
Water vapor and ozone along the Canberra flight track 55 N-68 N
AASE ER-2 19890207 takeoff-endascent
420
400
380
360
340
320
Pote
ntia
l Tem
pera
ture
(K
)
300025002000150010005000Ozone (ppbv)
806040200Water (ppmv)
Ozone
Water
19890207 Back Trajectories at 390 K Potential Temperature
19620131 Potential Vorticity, 12 UTC, 375 K
19620201 Potential Vorticity, 12 UTC, 375 K
19620206 Potential Vorticity, 12 UTC, 375 K
Ozone vs. Water, 1962 & 1989
WAM WB57F 19980507 endascent-beginchgalt, 370 K
12
10
8
6
4
Wat
er (
ppm
v)
59x103
58575655Time (UTC sec)
50
40
30
20
10
Wind Speed (m
/s)
Water
Wind Speed
ASHOE ER-2 19940327 Hawaii-Fiji
10 0 -10Latitude (°N)
6
5
4
3
2
Wat
er (
ppm
v)
90x103
858075Time (UTC Seconds)
450
400
350
300
Potential Temperature (K
)
1.70
1.65
1.60
1.55
1.50
Met
hane
(ppm
v)
Water
Methane
Pot Temp
ASHOE ER-2 19940327 Hawaii-Fiji
5.0
4.8
4.6
4.4
4.2
4.0
3.8
Wat
er (
ppm
v)
1.701.651.601.551.50Methane (ppmv)
Outer Tropics Inner Tropics
Scaling Analysisof Airborne Observations
Near the TropopauseAdrian Tuck, Susan Hovde
NOAA Aeronomy LaboratoryShaun Lovejoy
McGill UniversityDaniel Schertzer
U i ité d Pi t M i C i
H1 Example (Synthetic Data)
-60
-40
-20
0
20
40f(t
)
40003000200010000t
1.5
1.0
0.5
0.0log(
<|f(
x+r)
- f(x
)|> )
3.02.01.00.0log( r )
H1 = Slope = 0.50 ± 0.03
-80
-40
0
40f(t
)
40003000200010000t
Effect of Adding 10% Gaussian Noise
1.6
1.5
1.4
1.3
1.2
1.1log(
<|f(
x+r)
- f(x
)|> )
3.02.01.00.0log( r )
H1 = 0.29 ± 0.02
H1 = 0.02 ± 0.02
2.0
1.6
1.2
0.8log(
<|f(
x+r)
- f(x
)|> )
3210log( r )
H1 = 0.38 ± 0.05
700
600
500
400
300
O3
(ppb
v)
80006000400020000t
Stratospheric Segment
220200
180160
140
O3
(ppb
v)
80006000400020000t
Tropospheric Segment
1.6
1.2
0.8
0.4
log(
<|f(
x+r)
- f(x
)|> )
3210log( r )
H1 = 0.69 ± 0.12
H1 = 0.15 ± 0.03