the horizontal cascade structure of atmospheric fields
TRANSCRIPT
6/3/09 1
The Horizontal cascade structure of atmospheric 1
fields determined from aircraft data 2
S. Lovejoy1, A. F. Tuck2, D. Schertzer3,4 3
1 Physics, McGill University, 3600 University st., Montreal, Que. H3A 2T8, 4
2 NOAA Earth System Research Laboratory, Chemical Sciences Division, 325 6
Broadway, Boulder CO 80305-3337, USA, 7
3 CEREVE, Université Paris Est, France, 8
4Météo France, 1 Quai Branly, Paris 75005, France 9
10
6/3/09 2
11
Abstract: 12
Aircraft measurements of the power spectra of the horizontal wind field typically 13
find a transition from ≈ k-5/3 to ≈ k-2.4 at scales somewhere around 40 km (k is a 14
wavenumber). In the usual interpretation this represents a transition between an isotropic 15
3-D and an isotropic 2-D turbulence; we have recently argued that the turbulence is so 16
highly anisotropic that it has different exponents in the horizontal and vertical. When, 17
coupled with non-flat aircraft trajectories this can explain the break as a transition from a 18
roughly horizontal spectrum at small scales to the spurious appearance of the vertical 19
spectrum at large scales. If the atmosphere indeed has wide range horizontal scaling, then 20
it is important to test out the multiplicative cascade models that predict its statistical 21
behaviour. In this paper, we do this by analyzing wind, temperature, pressure and 22
humidity data from the Winter Storm 2004 experiment using 24 aircraft legs each 1120 23
km long and at 280 m resolution. We analyse both the turbulent fluxes and the 24
fluctuations showing that in spite of the non-flat trajectories that there is good evidence of 25
roughly planetary scale multiplicative cascades. By carefully determining the scale-by-26
scale effects of intermittency on the aircraft altitude and measurements we estimate the 27
corresponding scaling exponents. We argue that our results should finally permit the 28
emergence of a long needed consensus about the basic scale-by-scale statistical properties 29
of the atmosphere. They also point to the urgent need to development anisotropic scaling 30
models of turbulence. 31
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32
1. Introduction: 33
1.1 The biases in the wind statistics: 34
Data analysis requires both a theoretical framework in which the physical 35
quantities can be defined and understood as well as a more technical sampling/measuring 36
framework which assesses their degree of correspondence with the data. In the case of 37
aircraft measurements of atmospheric turbulence, it is usual to interpret the measurements 38
in an isotropic or quasi isotropic framework in which basic exponents are the same in the 39
horizontal and vertical directions. However, starting in the 1980’s studies of the vertical 40
structure of the atmosphere have almost invariably concluded that the horizontal wind is 41
scaling in the vertical, but with a different exponent than in the horizontal. This 42
empirically observed scaling stratification has thus been interpreted either with quasi-43
linear gravity wave models (Gardner, 1994), (Gardner et al., 1993), (Dewan, 1997), 44
(Koch et al., 2005) or the 23/9D strongly turbulent model, (Schertzer and Lovejoy, 45
1985), (Lovejoy et al., 2008b). 46
In the last few years, these results have prompted the examination of the 47
consequences of such anisotropic but scaling turbulence for the interpretation of aircraft 48
data; both for stratospheric aircraft following isomachs (Lovejoy et al., 2004) and for 49
tropospheric aircraft following isobars (Lovejoy et al., 2009c). The perhaps surprising 50
conclusions were that they are best explained by single wide range anisotropic scaling 51
regime. A corollary is that the usual interpretation - that the atmosphere has two (or 52
more) isotropic scaling regimes (e.g. small scale, 3D isotropic and large scale, 2-D 53
6/3/09 4
isotropic) is untenable. The key finding was that when measuring the wind field, rather 54
than detecting successively two different isotropic turbulence exponents, the aircraft first 55
detects at small scales the correct horizontal exponent, and then - at larger scales where 56
the departure of the aircraft from a flat trajectory becomes important - it spuriously 57
measures the different vertical exponent. In terms of traditional spectral exponents β, the 58
horizontal has β ≈ 5/3 (near the Kolmogorov value) while the vertical has β ≈ 2.4 (as 59
accurately estimated by drop sondes (Lovejoy et al., 2007)). As pointed out in (Lovejoy 60
et al., 2009c), the latter is almost exactly the same as value of the large scale exponents 61
found in all the major aircraft campaigns including GASP (Nastrom and Gage, 1985), 62
MOZAIC (Cho and Lindborg, 2001) and others (Gao and Meriwether, 1998), 63
(Bacmeister et al., 1996). 64
If the atmosphere is anisotropic and scaling over wide ranges, then it is 65
fundamentally important to characterize the anisotropic scaling regimes for as many of 66
the fields as possible, i.e. to determine the type of scaling (including the determination of 67
the basic turbulent parameters), as well as the limits to the scaling – the inner and outer 68
scales. The goal of this paper is therefore to go a step beyond the reinterpretation to see 69
how the data can be quantitatively used to characterize atmospheric turbulence while 70
attempting to avoid spurious aircraft induced effects. 71
The basic model accounting for the wide range anisotropic scaling of the fields is 72
anisotropic multiplicative cascades i.e. anisotropic extensions of the explicit 73
phenomenological cascade models that were developed through the 1960’s, 70’s and 80’s 74
((Novikov and Stewart, 1964), (Yaglom, 1966), (Mandelbrot, 1974), (Schertzer and 75
Lovejoy, 1987)). In recent papers, we have investigated the vertical cascade structure 76
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using drop sondes (Lovejoy et al., 2009b), the horizontal cascade structure using satellite 77
data (Lovejoy et al., 2009a) and vertical cross-sections using lidar backscatter data 78
((Lilley et al., 2004), (Lilley et al., 2008)). In addition, meteorological reanalyses as well 79
as numerical meteorological models (Stolle et al., 2009) have recently been shown with 80
high accuracy to have multiplicative cascade structures over almost their entire spatial 81
ranges and up to about 10 days in the time domain. What is missing from a fairly 82
complete spatial characterization of atmospheric cascades is the direct verification of the 83
multiplicative cascade structure in the horizontal on the standard meteorological fields 84
including the horizontal wind, the temperature, humidity, potential temperature; this is 85
our task here. 86
87
2. The data, its Intermittency: 88
The data analysed were from the Winter Storm 2004 experiment and involved 10 89
aircraft flights over a roughly a 2 week period over the northern Pacific. Each flight 90
dropped 20 - 30 drop sondes which were analyzed to determine the vertical cascade 91
structure (Lovejoy et al., 2009b). For air traffic reasons, the Gulfstream 4 plane flew 92
along either the 162, 178, or 196 mb isobars, (to ±0.11mb i.e. the pressure level was ≈ 93
constant to within ±0.068%), it has a radar altimeter which is reliable over the ocean, but 94
which is only used to anchor the GPS estimates, these geometric altitudes and were used 95
here. Each had one or more roughly constant straight and constant altitude legs more 96
than four hundred kilometers long between 11.9, 13.7 km altitude. The data were 97
sampled every 1 s and the mean horizontal aircraft speed with respect to the ground was 98
280 m/s. In addition, we checked that the distance covered on the ground between 99
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measurements was constant to within ±2% so that the horizontal velocity was nearly 100
constant (in addition, using interpolation, we repeated the key analyses using the actual 101
ground distance rather than the elapsed time and found only very small differences). This 102
is the same data set analyzed in (Lovejoy et al., 2009c) where it is described in more 103
detail. 104
The basic scale by scale relations between the trajectory and the fields was 105
investigated using both spectral and cross-spectral analysis in (Lovejoy et al., 2009c). In 106
particular, it was shown that while there were strong phase relations and strong 107
coherencies between the altitude and the wind and between the pressure and the wind - 108
indicating a strong influence of the trajectory on the wind statistics at large scales (and 109
visa versa at small scales) - on the contrary, the corresponding phases and coherencies of 110
the altitude and pressure with the temperature and with the humidity were very low. 111
This is consistent with the systematic finding that the latter show no scale breaks, that 112
they have wide range scaling e.g. (Hovde et al., 2009; Tuck et al., 2004); see also (Lilley 113
et al., 2004) for aerosol lidar backscatter and Tuck et al., 2004 for ozone and nitrous 114
oxide concentrations. 115
The coherency and phase analysis suggests that the main effect of the trajectory 116
fluctuations is on the wind field and that its influence will be smaller for the other 117
atmospheric fields. Going beyond phases and coherencies, let’s consider the detailed 118
statistics of the trajectories. For scales less than about 4 km, the aircraft inertia smoothes 119
them out; the main variations are smooth and are associated with various roll modes and 120
the pilot/autopilot response. However, at scales from about 4 to 100 km (and from ≈3 to 121
about 300 km for the stratospheric ER-2 trajectories along isomachs), the trajectory is 122
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fractal with the altitude lagging behind the wind fluctuations. Finally at scales >≈ 40 km 123
the aircraft follows the isobars quite closely so that the wind lags behind the pressure. 124
These conclusions were reached by considering the average for 24 aircraft legs flying 125
between roughly 11 and 13 km, each 1120 km long as well as through a leg by leg 126
analysis which showed considerable variability in the scale at which these transition 127
occurred. According to this analysis we may anticipate that there will be a strong effect of 128
the variability (indeed, intermittency) of the aircraft altitude on the measurements – at 129
least for scales less than about 40 km where the wind leads the altitude, imposing its 130
strong intermittency on the aircraft. 131
In order to demonstrate this and to quantify the intermittency of the trajectory we 132
performed a standard multifractal analysis. Since we will perform similar analyses on the 133
meteorological fields we give a general explanation of the method, using the example of 134
the familiar Kolmogorov law of turbulence. 135
Consider wind fluctuations Δv over distances Δx with underlying turbulent energy 136
flux φ; we therefore have: 137
Δv = φΔxH (1) 138
The form 1, is a generic relation between a turbulent fluctuation Δv and turbulent 139
flux φ, the Kolmogorov law is the special case where φ = ε1/3 and H = 1/3. If over a 140
range of scales, φ is the result of a multiplicative cascade process, then it obeys the 141
following statistics: 142
M q =λ
λeff
⎛
⎝⎜
⎞
⎠⎟
K q( )
; λ =Lref
L; λeff =
Lref
Leff
(2) 143
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Where M q = φλq / φλ
q is the normalized qth moment at resolution L (scale ratio λ), Lref 144
is a convenient reference scale (taken below as the largest great circle distance on the 145
earth, Lref = 20,000 km), Leff is the “effective” scale at which the cascade begins, and K(q) 146
is the scaling exponent characterizing the intermittency. In order to test eq. 2 on the data, 147
it is sufficient to use the (absolute) fluctuations at the smallest available scales l; 148
(corresponding to the large scale ratio Λ); and then to estimate the normalized flux as: 149
φΛ
φΛ
=Δv l( )Δv l( ) ; Λ =
Ll
(3) 150
i.e. it is not necessary to know H or even the physical nature of the flux φ so that our 151
results are independent of any specific theory of turbulence. The fluctuations can be 152
estimated either by differences (when 0< H <1) or (equivalently) by wavelets. The 153
normalized flux at lower resolution λ < Λ is then obtained by straightforward spatial 154
averaging of the fine scale (Λ) resolution normalized fluxes. Note that we do not 155
determine the direction (i.e. from large to small or small to large) of the cascade. 156
When this method is applied to the aircraft altitude, we obtain figure 1a which 157
shows that eq. 2 is well verified with lines converging to log10λeff ≈ 2.9 corresponding to 158
Leff ≈ 30 km. This shows that for scales <≈ 30 km, the altitude has very strong 159
intermittency which is of the type theoretically predicted by turbulent cascades. This 160
result is compatible with the phase and coherence analysis summarized above to the 161
effect that for scales < ≈ 40 km that the phase of the wind leads the phase of the altitude. 162
The simplest explanation is that the altitude is intermittent because it varies in response to 163
the strong wind intermittency. At the larger scales the phase changes sign and the 164
pressure leads the wind so that (due to the autopilot) the aircraft starts to closely follow 165
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the isobars which are much smoother (the response time of the autopilot is much shorter 166
than the 2- 3 minutes needed to cover ≈ 40 km but apparently, the turbulence is too 167
strong at smaller scales). 168
We now seek to quantify the accuracy with which the altitude intermittency 169
follows the cascade form. First, we may use the logarithmic slopes of Mq to estimate 170
K(q), and then use a parametric form of K(q) in order to reduce it to a manageable (finite) 171
number of parameters. This is most conveniently done by exploiting the existence of 172
stable, attractive cascade processes - the result of a kind of multiplicative central limit 173
theorem -which gives rise to “universal multifractals” (Schertzer and Lovejoy, 1987), 174
(Schertzer and Lovejoy, 1997). For universal multifractals K(q) has of the form: 175
K q( )=C1
α − 1qα − q( ); q ≥ 0; 0 ≤ α ≤ 2 (4) 176
The parameter C1 (0< C1 < D) characterizes the intermittency of the mean: C1 = K’(1), it 177
thus characterizes the intermittency of the mean field (D is the dimension of the 178
observing space, here D = 1). The second parameter is the multifractal index α which 179
characterises the degree of multifractality, it quantifies how rapidly (with intensity, with 180
q), the statistics deviate from the monofractal case α = 0). In table 1 we give the 181
estimates of the parameters for the altitude and various other fields. 182
To quantify the accuracy, we may characterize the deviations by the mean 183
absolute residuals for the statistical moments Mq of order q from 0 to 2 for all points 184
between the scale of the grid and the scales in the figure (about 100 times larger): 185
( ) ( )10 10log ( ) log /q effM K q λ λΔ = − (5) 186
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To convert Δ to a percent deviation, use δ = 100(10Δ-1), which we find for the altitude 187
(fitting between 1 km and 20 km) is ≈ ±2.6% (see table 1 for the mean δ averaged over 188
the q ≤ 2). 189
190
191
192
193
Fig. 1 a: The normalized moments q = 0.2, 0.4, … 3, for aircraft altitude, z; log10λ 194
corresponds to 20,000 km (= Lref). 195
196
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T Logθ h vlong vtrans p z
C1 0.052 0.052 0.040 0.033 0.046 0.031 0.23
α 2.15 2.20 2.10 2.10 2.10 2.2 2.15
Leff(km) 5000. 10000. 10000. 105 25000. 1600. 30.
δ(%) 0.5 2.0 0.5 0.4 0.8 0.5 2.6
Table 1: Horizontal cascade parameter estimates. These are estimated over the 197
range 100 km down to 2 km except for z which is over the range 20 km to 0.5 km. The 198
regressions force the lines to pass through a common external scale. We have not given 199
error estimates on the parameters due to the significant systematic (trajectory) effects 200
discussed below. Note that the aircraft α estimates are a bit too big since the theoretical 201
maximum is α = 2. They were estimated with the double trace moment technique which 202
depends largely on the statistics of the weaker events and these could be affected by 203
aircraft turbulence. In our experience, values of α > 2 are always accompanied by error 204
bars that encompass values < 2. 205
206
207
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3. Scaling analyses: 208
3.1 Trace moments and the cascade structure of the fields: 209
Based on the analysis of the aircraft altitude and cross-spectra, we have argued 210
that the aircraft trajectory fluctuations and their coupling with the fields must be taken 211
into account for quantitative analysis of the statistics. However, before we attempt a 212
more refined analysis, we present the basic cascade (flux) and then fluctuation analyses. 213
As indicated in section 2 (c.f. eq. 3), the dimensionless, normalized flux is estimated 214
using absolute fluctuations at the smallest scales. In the case of the horizontal wind in the 215
≈k-5/3 regime, it is presumably the energy flux to the one third power, but knowledge of 216
the exact physical nature of the flux is not necessary to test the prediction of cascade 217
processes, eq. 2. First we decompose the wind into longitudinal and transverse 218
components anticipating that the altitude fluctuations will affect each somewhat 219
differently. Fig. 1b shows the results along with those of the pressure. Although the 220
basic structure is reasonably close to (near) planetary scale cascades, we can see evidence 221
of the three regimes discussed above; at scales smaller than about 4 km, the variability is 222
a little too large, logMq is quite linear from 4 – 40 km, and then flattens out a bit at scales 223
> 40 km. Qualitatively this can be understood by the action of inertially smoothed 224
trajectories at small but nonzero slopes (scales < ≈ 4 km), by turbulent wind induced 225
altitude fluctuations for scales between 4 and 40 km and then at larger scales an excess of 226
variability since the aircraft starts to follow the isobaric slopes and responds to the 227
(larger) vertical intermittency. For the pressure, on the contrary the variability at the 228
largest scales is low since the aircraft doesn’t stray far from the isobars. We may also 229
note that for the wind, the outer scales are somewhat larger than the planetary scales 230
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(table 1), especially for the longitudinal component. In (Lovejoy et al., 2009c), we have 231
already argued on the basis of cross-spectral analysis that the longitudinal component was 232
the most affected by the trajectory; it is presumably partially responsible for the large 233
value of Leff. In comparison, the value of Leff for the transverse component – although 234
perhaps still too large - is about the same as that reported for satellite radar reflectivities 235
from precipitation (Lovejoy et al., 2008a); the fact that it is larger than the largest great 236
circle distance (20,000 km) simply means that even at planetary scales there is residual 237
variability due to the interactions of the wind with other atmospheric fields. In 238
comparison the outer scale of the pressure is somewhat smaller than planetary scales; this 239
is not surprising since the aircraft was attempting to fly along isobars. 240
241
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Fig. 1b: Same as fig. 1a but for the fields strongly affected by the trajectories: the 242
longitudinal wind (top left a), the transverse wind (top right, b), pressure (c, lower left 243
right). 244
245
Turning our attention to the analyses of the relatively unbiased temperature, 246
humidity and log potential temperature fields (fig. 1c), we again see evidence for the 247
three regimes and we note that the cascade structure is somewhat more closely followed 248
with external scales somewhat smaller than 20,000 km. Overall, we conclude that the 249
basic predictions of the cascade theories are well respected; in particular there is no 250
evidence for a break anywhere near the meso-scale (≈ 10 km). The outer scales of these 251
fields are in fact very close to those of visible, infra red and passive microwave radiances 252
as determined by satellite (Lovejoy et al., 2009a). Note that the log equivalent potential 253
temperature is not shown here or below because the humidity is sufficiently low at the 254
aircraft altitude that the graph is nearly indistinguishable from that of the log potential 255
temperature. 256
257
258
259
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260
Fig. 1c: Same as 1b but for the fields that are relatively unaffected by the trajectory: 261
temperature (top left), relative humidity (top right), log potential temperature (lower left 262
right). 263
3.2 Fluctuation analysis using structure functions: 264
In section 3.1 we discussed the statistics of the turbulent flux, we now turn our 265
attention to the fluctuations, here estimated using differences: e.g. 266
v Δx( )= v x + Δx( )− v x( ) . Taking the ensemble average of eq. 1 and defining the qth 267
order structure function (Sq): 268
Sq Δx( )= Δv Δx( )q ∝ Δxξ q( ); ξ q( )= qH − K q( ) (6) 269
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The results for altitude are given in fig. 2a. We again see evidence for three 270
regimes; the large scale regime very nearly corresponds to linear variations (H = 1), i.e. to 271
a constant slope with the aircraft nearly exactly following the isobars. As we shall see in 272
more detail in the next section - where we estimate H as a function of scale from the 273
logarithmic slopes of fig. 2a - H decreases at the smallest scale from a value ≈1 274
corresponding to inertially smoothed trajectories, to a minimum value ≈ 0.5 at around 4 275
km, and then increases to the maximum value 1 for scales > 40 km where the trajectories 276
are close to isobars and are again, relatively smooth. Note that in fig. 2a, b, c we plot the 277
nondimensional structure functions obtained by dividing the fluctuations by the mean at 278
the smallest scale. 279
280
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Fig. 2 a: Structure functions of order q = 0.2, 0.4, ..2.0 for the aircraft altitude. 281
Distances Δx are in km. Separate sets of regression lines are shown for both small and 282
large scales in order to emphasize the change with scale. 283
284
In fig. 2b we show the corresponding results for the longitudinal and transverse 285
wind and pressure. While the behaviour for the wind has essentially two scaling regimes 286
with a 4 - 40 km transition regime – unsurprisingly - the pressure has poor scaling; we 287
reserve a detailed analysis for the next section. In fig. 2c we show the corresponding plot 288
for those fields less affected by the trajectory fluctuations, the temperature, humidity and 289
log potential temperature; we see that the scaling is indeed very good. 290
291
292
293
294
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295
Fig. 2 b: Same as fig. 2a but for the fields strongly affected by the aircraft trajectory: the 296
longitudinal wind (upper left), the transverse wind (upper right) and the pressure (lower 297
left). 298
299
300
301
302
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303
304
Fig. 2 c: Same as for 2a except for those fields relatively unaffected by the aircraft 305
trajectories: the temperature (upper left), the humidity (upper right) and the log potential 306
temperature (lower left). 307
308
4. Refined analysis: 309
4.1 horizontal exponent estimates 310
In section 3 we considered the raw flux statistics; no attempt was made to 311
quantitatively correct or take into account the effects of the trajectory, we turn to this task 312
now. From the “raw” flux and fluctuation analyses, we saw that the aircraft trajectories 313
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are highly intermittent for scales < 40 km (fig. 1a) after which they begin to accurately 314
follow isobars and have nearly constant slopes (fig. 2a). We also saw that for the fields 315
not strongly affected by the trajectory (temperature, humidity, log potential temperature), 316
that the overall scaling was very good. However, if we want to quantitatively 317
characterize the corresponding horizontal scaling and estimate K(q) and hence the 318
parameters H, C1, α, it is not clear what is the optimum range of scales that should be 319
used. In order to obtain a more exact picture of the scale by scale variations in the 320
statistics, we first consider scale by scale estimates of the basic flux and intermittency 321
parameters using the logarithmic derivatives of the second flux exponent (K(2)) and first 322
order (q = 1) structure function exponent H. K(2) was chosen to characterize the 323
amplitude of the intermittency rather than the more fundamental C1 because it is directly 324
related to the spectral exponent β = 1+2H - K(2) and so can be directly used to determine 325
the contribution of intermittency corrections to the spectrum (note that since α ≈ 2 we 326
have C1 ≈ K(2)/2, eq. 4). 327
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328
Fig. 3a: Second order trace moments (right column) and their logarithmic derivatives 329
(estimates of K(2), an intermittency index), left column. Top row: wind, longitudinal 330
(blue) transverse (red). The region within the dashed lines has particularly low K(2) (≈ 331
0.05). Middle row: pressure (blue), altitude (red). Bottom, temperature (blue), humidity 332
(red), log potential temperature (gold), reference lines K(2) = 0.12. Log10λ = 0 333
corresponds to 20,000 km (= Lref). 334
335
Fig. 3a shows the results for the second moment of the flux M2; the right column 336
for the moments themselves, the left for the corresponding logarithmic derivatives 337
estimated from the corresponding right hand graphs by performing regressions over an 338
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octave of scales centred on the scale indicated. Focusing on the top row for the wind, we 339
see that the logarithmic derivatives bring out the three regions quite clearly. At small 340
scales < ≈ 4 km (large λ), the intermittency is large corresponding to the strong 341
intermittency of the trajectory while at the largest scales >≈ 40 km (small λ) it is again 342
large due to the fact that the near constant slope means that the aircraft is picking up the 343
(large) vertical intermittency indeed, if we use the vertical value K(2) ≈ 0.17 estimated 344
from the drop sondes we find that it is not far from the value of the upper left (velocity) 345
graph in fig. 3a. Note that the estimate of the logarithmic slope is itself highly fluctuating 346
at low λ’s since the statistics get progressively worse at large scales due to the smaller 347
and smaller number of large scale structures. Detailed consideration of the scale by scale 348
estimates of K(2) for the altitude show that it is actually not so constant in spite of the 349
relatively straight appearance of the lines in fig. 1a. For the velocity, the region 4 - 40 km 350
is the least intermittent (in the sense of the lowest exponent K(2)); this is presumably the 351
optimum region for estimating the scaling exponents: at smaller scales, the aircraft 352
trajectory is too intermittent whereas at the larger scales, the intermittency increases 353
again since – at least for the wind - the aircraft is moving significantly in the vertical 354
along a near constant isobaric slope (in any case at the large scales, we obtain the vertical 355
rather than the horizontal exponents and the vertical intermittency is larger, see below). 356
We therefore used this regime to estimate the exponents (table 2 estimated by taking 357
regression over the range 4 – 40 km). We see that in conformity with the cross-spectral 358
analyses and our previous discussion, that the scale by scale intermittency (i.e. K(2) = 359
dlogMq/dlogλ) for T, logθ and h is nearly constant for scales >≈ 4 km confirming that the 360
main deviation from scaling for these fields is only at the smallest scales. We also 361
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observe that the exponents are nearly the same for all three fields. In table 2 we also 362
give the corresponding C1 estimates obtained by numerically estimating K’(1) using 363
regressions over the range 4 – 40 km. 364
365
366
Table 2: Parameter estimates over the “optimum” range 4 to 40 km. β is the spectral 367
exponent, Hst is from the first order structure function, Hβ = (β + K(2)-1)/2, and Hmean is 368
the average of the two. C1 = K’(1) from the trace moments and C1,st = ξ(1) - ξ’(1), C1mean 369
is the average. 370
371
Turning our attention to the fluctuation exponent H, (fig. 3b) we see (top row, 372
right, see the reference lines) that they reasonably follow successively the values H = 1/3, 373
0.75 (i.e. the standard Kolmogorov value and the observed vertical value respectively). 374
T logθ h
α 1.78 1.82 1.81
β 1.89 1.91 1.99
K(2) 0.12 0.12 0.11
C1 0.064 0.063 0.051
C1,st 0.040 0.042 0.028
C1mean 0.052±0.012 0.052±0.010 0.040±0.012
Hst 0.49 0.50 0.52
Hβ 0.51 0.52 0.50
Hmean 0.50±0.01 0.51±0.01 0.51±0.01
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However the scale by scale estimates (top, left) show that this mean behaviour hides an 375
increase in H from ≈ 1/3 at the smallest scales to H ≈ 0.75 at scales 40 -100 km. 376
According to the leg by leg analyses in (Lovejoy et al., 2009c), this fairly systematic 377
increase of ensemble mean H hides highly variable transition scales in individual legs; 378
the observed fairly continuous change in H is in fact the result of the transition scale 379
varying greatly from leg to leg. On the structure function (upper right) we can also see 380
that fitting a single line through the entire range is not so bad, and yields roughly the 381
mean of 1/3 and 0.75; this may explain the wind value H ≈ 0.56±0.02 from (Hovde et al., 382
2009) (using nearly the same data set as here). This can be compared to the stratospheric 383
estimates over (different) fractal isomach trajectories of 0.50±0.02, 0.52±0.03 384
(transverse, longitudinal wind respectively, (Lovejoy et al., 2004)) and 0.53±0.01 (Tuck 385
et al., 2004). Note that the latter finds significant differences in H for the wind for 386
stratospheric trajectories across or along the polar jet (0.45±0.14 and 0.55±0.12 387
respectively) but this could be a reflection of a systematic shift in the transition scale 388
from an H ≈ 1/3 to an H ≈ 0.75 regime (see however the interpretation in Tuck et al., 389
2004 and Tuck 2008, which is consistent with the thermal wind equation). 390
Moving down to the second row in fig. 3b, we see that at small enough scales, the 391
altitude has Htr ≈ 1 presumably as a consequence of the inertial smoothing leading to near 392
linear behaviour (nearly constant slopes; the subscript “tr” is for “trajectory”). At larger 393
scales, the inertia effect is gradually overcome by turbulence so that at 4 km it reaches a 394
minimum near Htr ≈ 0.48 after which it systematically rises again to Htr ≈ 1 (for scales > 395
≈ 40 km) as the aircraft flies increasingly close to a constant isobaric slope. Up to this 396
scale, the altitude statistics are thus close to those of the ER-2 stratospheric trajectories 397
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except that the latter maintain a value Htr ≈ 0.58 along isomachs (over the range ≈ 3 km 398
until ≈ 300 km when the systematic vertical rise of ≈ 1 m/km (due to fuel consumption) 399
eventually implies Htr ≈ 1 (on average, for scales >≈ 300 km; (Lovejoy et al., 2004)). As 400
noted above, along this fairly wide fractal range, the ER-2 had an “anomalous” wind 401
exponent Hwind ≈ 0.5 which is nearly the mean of the horizontal and vertical values (1/3, 402
0.75) so that the effect of following a fractal trajectory with fairly well defined fractal 403
dimension (= 1+Htr) apparently leads the ER2 to sample both horizontal and vertical 404
exponents yielding overall an average of the two. 405
Consider now the pressure exponent. At the smallest scales, it starts off with a 406
value near 2/3 decreasing systematically to zero at scales >≈ 40 km indicating very low 407
fluctuations consistent with near isobaric trajectories (there are some large fluctuations in 408
the scale by scale estimates presumably due to poor statistics). The small scale value ≈ 409
2/3 is presumably a consequence of the dynamic pressure relation Δp ≈ ρΔv2/2 so that 410
Hpress = 2Hwind with Hwind ≈ 1/3. 411
Finally, on the bottom row, we show H estimates for T, logθ, h, again finding 412
relatively good scaling (constant exponents) for scales >≈ 4 km with H ≈ 0.50 for all three 413
(see table 2). For temperature, this is also close to the value 0.52±0.02 obtained from the 414
similar tropospheric data (Hovde et al., 2009) and also not far from stratospheric ER-2 415
analyses with Htemp ≈ 0.45±0.02, (Lovejoy et al., 2004), and Htemp ≈ 0.54±0.01 (Tuck et 416
al., 2004). This value is so close to a ratio of small integers (1/2) that one would expect 417
there to be a straightforward theoretical explanation (or at least dimensional analysis) 418
leading to this value but we are not aware of any adequate theory. The usual argument is 419
to consider temperature to be a passive scalar in which case Corrsin-Obukhov theory 420
6/3/09 26
predicts H = 1/3 (see Tuck 2008 for arguments about the nature of atmospheric 421
temperature that imply it is not a passive scalar). Alternatively, consider the prediction of 422
Bolgiano-Obkuhov theory for isotropic buoyancy driven turbulence, which for the wind 423
predicts H = 3/5 which is near the correct vertical value. However the same theory 424
predicts Htemp = 1/5 which is even further from the empirical value (see (Monin and 425
Yaglom, 1975)). Further support for the non standard value H ≈ ½ for temperature 426
(corresponding to β ≈ 1.9) comes from aircraft temperature spectra in the lower 427
troposphere (Chigirinskaya et al., 1994) where β ≈ 1.9 fits very well over the range > 1 428
km in scale, and stratospheric potential temperature spectra where β ≈ 1.9 is also very 429
close to data for scales >≈3 km in scale (Bacmeister et al., 1996) (recall that K(2)≈ 0.1 430
and β =1+2H - K(2)). 431
432
433
434
6/3/09 27
435
Fig. 3b: First order structure functions (right column) and their logarithmic derivatives 436
(estimates of H), left column. Top row: wind, longitudinal (blue) transverse (red), the 437
reference lines correspond to the exponents 1/3, 0.75. Middle row: pressure (blue), 438
altitude (red), the reference lines correspond to H = 1 i.e. constant mean slopes. Bottom, 439
temperature (blue), humidity (red), log potential temperature (gold), reference lines H = 440
0.5. 441
442
443
444
6/3/09 28
Finally, we consider the spectra (fig. 3c). As indicated above, since the spectrum 445
is a second order statistic, we expect that the scale by scale analysis of the latter (again by 446
logarithmic derivatives) should at least roughly satisfy the equation β = 1+2H- K(2) 447
(actually, this equation is strictly only valid when the scaling is satisfied over wide ranges 448
so hence some deviations are to be expected). Starting with the wind (the top row), we 449
see the spectral version of the transition discussed earlier: at low wavenumbers (k <(40 450
km)-1), β ≈ 2.4 (= 2x0.75- 0.1) to β ≈ 1.6 (= 2x1/3- 0.05 for (40 km)-1 > k > (4 km)-1). In 451
the high wavenumber regime, we see a new feature which is a slight bump near k ≈ (1 452
km)-1, (magnified in the derivative on the left) which may be a residual signature of 453
various aircraft roll modes and autopilot feedbacks. Turning our attention to the middle 454
row (altitude, pressure), we see that the small scales are indeed particularly smooth with 455
the larger scales following near Kolmogorov (β ≈ 5/3) behaviours (probably the 456
maximum near (200 km)-1 is only a statistical fluctuation). Interestingly, the regime with 457
reasonable β ≈ 5/3 scaling seems to be a consequence of the partial cancellation of a 458
continuously varying H and K(2) (compare the corresponding plots of 3a, 3b). Finally, 459
the temperature, humidity and log potential temperature (bottom row) is seen to have 460
significant fluctuations (left), but the overall scaling (far right) is nevertheless fairly good. 461
We now consider the parameter estimates (table 2), taken from the “optimum” 462
regime 4 – 40 km. For both the C1’s and H’s there are two slightly different ways to 463
estimate them. H can be estimated from both structure functions: Hst = ξ(1) and also 464
from the spectral exponent β via: Hβ = (β-1+K(2))/2. The C1 can be estimated from the 465
flux moments: C1 = K’(1) and also from the structure function exponent: C1 = ξ(1) - ξ’(1). 466
In the table we have given all these estimates as well as the mean of the two as the best 467
6/3/09 29
guess, and with the error as half the difference. From the table we see that the three 468
fields have nearly identical exponents, and that the important parameter H - which should 469
be determinable by dimensional analysis - is near the value 1/2. 470
In fig. 4 we show the moment scaling exponents K(q) estimated over the 4 - 40 471
km range. We see that T, logθ, h remain almost indistinguishable out to very large q 472
values (corresponding to rare extreme fluxes) whereas the longitudinal and transverse 473
components separate for q >≈3 (already a large value of q). The asymptotic linearity is 474
predicted as a consequence of a “multifractal phase transition”, (Szépfalusy et al., 1987), 475
(Schertzer et al., 1993) and depends on the finite size of the sample (the asymptotic slope 476
is simply the largest singularity present in the sample). 477
478
479
480
6/3/09 30
481
Fig. 3c: Spectra (right column) and their logarithmic derivatives (estimates of -β), left 482
column. Top row: wind, longitudinal (blue) transverse (red). The thin reference lines 483
correspond to β = 5/3, 2.4. Middle row: pressure (blue), altitude (red) with reference 484
lines corresponding to β = 5/3. Bottom, temperature (blue), humidity (red), log potential 485
temperature (gold), reference lines β = 2. Vertical dashed lines indicate the region with 486
scales 4 – 40 km where the velocity intermittency is low. 487
488
6/3/09 31
489
Fig. 4: The K(q) functions for altitude and pressure (upper, lower blue), temperature, 490
humidity, log potential temperature (lower, middle, upper green), longitudinal, transverse 491
wind (upper, lower, red). 492
493
4.2 Horizontal - Vertical comparison, vertical stratification, 494
estimating Hz, Del: 495
We have argued that atmospheric fields are compatible with wide range horizontal 496
scaling and have estimated the corresponding exponents. This work complements that of 497
(Lovejoy et al., 2009b) where high resolution drop sondes were used to estimate the 498
corresponding vertical cascades and exponents; the overall conclusion is thus that the 499
dynamics are scaling and turbulent but anisotropic. The simplest anisotropic turbulence 500
6/3/09 32
model involves a unique scale function for all the fields. This would imply that the ratio 501
of horizontal and vertical components is Hhor/Hver = Hz = constant, so that for universal 502
multifractals αhor = αver and C1hor/C1ver= Hz. In (Lilley et al., 2008) there was an 503
extensive analysis of this for the lidar backscatter data (B) from passive pollutants. 504
Recall that the significance of Hz is that it controls the scaling of the aspect ratio of 505
structures in vertical sections. In particular, if we assume horizontal isotropy, then 506
volumes of structures vary as L2+Hz
where L is their horizontal extent, hence the “elliptical 507
dimension” Del controlling the rate of change of volumes of nonintermittent structures is 508
Del = 2+Hz. The 23/9 D model derives its name from Hz = 5/9, the result of Kolmogorov 509
scaling (H = 1/3) in the horizontal and Bolgiano-Obukhov scaling in the vertical (H = 510
3/5). In comparison, the popular quasi-linear gravity wave theories ((Gardner, 1994), 511
(Dewan, 1997)) have H = 1/3 in the horizontal and H = 1 in the vertical so that Hz = 1/3 512
and Del = 7/3. 513
Combining the results from the aircraft and the drop sondes and taking into account a 514
small apparent altitude dependence of the sonde intermittency exponents C1 (so as 515
estimate them at the 200 mb aircraft level), we obtain table 3. Note that the value for the 516
horizontal wind is given as the theoretical value 1/3 whereas the detailed leg by leg 517
analysis in (Lovejoy et al., 2009c) which fits two power laws (one for small, the other 518
for large scales) gives 0.26±0.07, 0.27±0.13 for the dominant small scale exponent 519
(transverse and longitudinal respectively). Given the strong coupling between the 520
aircraft trajectory and the wind, it seemed best to assume that these estimates support the 521
Kolmogorov value and it was used without an error bar. Similarly, the value of Hz for 522
6/3/09 33
velocity is from the analysis of (Lovejoy et al., 2009c) which fits two power laws one 523
fixed at Hh = 1/3, the other (large scale one corresponding to Hv) varying. 524
It should be noted that although in table 3 we give the ratio of the C1 values, since 525
their values are small, their relative errors are large and consequently their ratios have 526
large uncertainties. Since the H’s are larger, the ratio Hz = Hh/Hv is more reliable than 527
C1h/C1v, indeed in the latter case the error is very hard to reliably estimate and is not 528
indicated except in the lidar case. The main conclusions are a) for T, log θ and v , the 529
exponent ratio Hz is close to 0.47, b) for T, log θ, B, Hz is within one standard deviation 530
of the 23/9 D model value Hz = (1/3)/(3/5) = 5/9, c) the value for the humidity is 531
somewhat larger. Note that finding virtually identical exponents for T and logθ is not 532
surprising since if the pressure is exactly constant, then the two have a one to one (albeit 533
nonlinear) relation. Finally, also using the Gulfstream 4 data, (Hovde et al., 2009) find H 534
= 0.45±0.03 for the humidity (see also Tuck 2008 for a possible explanation for the low 535
value). 536
If the ratios in table 3 are taken at face value then we are lead to the conclusion that 537
two or possibly three scale functions are required to specify the scale of atmospheric 538
structures. While this is certainly possible, let us for the moment underline the various 539
difficulties in obtaining the in situ estimates: the nontrivial vertical dropsonde outages, 540
the nontrivial aircraft trajectory fluctuations. In addition, detailed analysis of the altitude 541
dependence of the horizontal velocity exponent in (Lovejoy et al., 2007) indicates that 542
starting with the theoretical Boligano - Obukhov value 3/5 near the surface, that the 543
exponent increases somewhat with altitude to the value ≈ 0.75 at 10 - 12 km. Similarly, 544
6/3/09 34
the humidity (and hence logθE) exponents may have both horizontal and vertical 545
variations which may account for their high Hz values (recall however that the values of 546
H are expected to be determined by dimensional analysis on fluxes; they are not expected 547
to have truly continuous variations). We should therefore regard these studies as only 548
early attempts to quantify the stratification. 549
550
551
T Logθ h v B
αh 1.78 1.82 1.81 1.94 1.83
αv 1.70 1.90 1.85 1.90 1.82
Hh 0.50±0.01 0.51±0.01 0.51±0.01 1/3 0.33±0.02
Hv 1.07±0.18 1.07±0.18 0.78±0.07 0.75±0.05 0.60±0.02
Hz=Hh/Hv 0.47±0.09 0.47±0.09 0.65±0.06 0.46±0.05 0.55±0.02
C1h 0.052±0.012 0.052±0.0100.040±0.012 0.04 0.076
C1v 0.072 0.071 0.091 0.088 0.11
Hz=C1h/C1v 0.72 0.71 0.44 0.45 0.69±0.2
Table 3: The above uses the estimate of the vertical Hv, C1v from sondes (table 2 in 552
(Lovejoy et al., 2009b)). The (horizontal) values for Hh for T, logθ, h are from table 2 553
(from 4 – 40 km, see above), for the lidar reflectivity B it is from (Lilley et al., 2008). 554
Finally, the C1 for v (velocity) is for the range 4- 40 km using trace moments. The 555
horizontal α values were for nonlinear fits of the K(q) for 0<q<3 (same range of scales). 556
557
6/3/09 35
558
5. Conclusions: 559
Satellites and other remote methods make the determination of the scale by scale 560
statistical properties of atmospheric radiances relatively easy to establish. In comparison 561
our knowledge of the corresponding variability of the standard atmospheric fields (wind, 562
temperature humidity etc.) is quite meagre and is based almost entirely on problematic in 563
situ measurements. The fact that today’s in situ atmospheric data is often orders of 564
magnitude better than those available even twenty years ago - and is often easily 565
accessible - has unfortunately encouraged naïve applications. For example, in a series of 566
papers, we have argued that the extreme (and highly clustered) nature of data outages in 567
drop sonde profiles must be carefully taken into consideration when they are analysed 568
(Lovejoy et al., 2009b). Recently in a companion paper, we have argued that due to 569
strong coupling over wide ranges of scale between the wind and the trajectories, that 570
aircraft measurements require a theory of anisotropic scaling turbulence for their 571
interpretation (Lovejoy et al., 2009c). Whereas the classical interpretation of aircraft 572
wind statistics assumes that the turbulence is isotropic and interprets the observed break 573
at around 40 km as a transition between two isotropic regimes, in the reinterpretation, it is 574
spurious and is simply a transition from horizontally to vertically dominated parts of the 575
trajectory. 576
In this paper, we have attempted to exploit this new understanding in order to 577
both demonstrate and to quantify the wide range horizontal scaling of the key 578
atmospheric fields: wind, temperature, humidity, potential temperature. The basic 579
theoretical framework for such wide range scaling is anisotropic cascades; we 580
6/3/09 36
demonstrate the horizontal cascade structure by directly analyzing the raw turbulent 581
fluxes finding that they all follow a multiplicative cascade structure with external scales 582
of the order of 10,000 km. Although this basic structure was present in all the analyzed 583
fields, we noted the presence of the three different regimes predicted on the basis of their 584
cross-spectra with the aircraft altitude and with the pressure. The deviations were fairly 585
strong for the wind and pressure which were strongly coupled with the trajectory, but 586
relatively small for the temperature, humidity and potential temperature which were only 587
weakly coupled. A rough summary is that at scales <≈ 4 km the trajectories vary 588
smoothly but are affected by aircraft roll and pilot/autopilot controls, for scales ≈ 4 - 40 589
km the aircraft follow a rough, intermittent fractal trajectory; finally for scales ≥ 40 km 590
the aircraft very closely follow the isobars which have significant vertical slopes. These 591
regimes are for the ensemble statistics; the transition points vary considerably from 592
trajectory to trajectory. 593
Refined analyses of the statistics was performed by considering the scale by scale 594
intermittency (characterized by the second order moment of the flux) as well as the scale 595
by scale smoothness of the fluctuations (characterized by the first order structure 596
function). It was argued that the optimum scale range for parameter estimation was ≈ 4 -597
40 km where the intermittency in the altitude was not strong, the effect of the 598
pilot/autopilot, rolls was weak and the constraint of flying on an isobar was not yet so 599
strong as to impose the verticals statistics on the wind field. We therefore used this range 600
to give refined parameter estimates. Perhaps one of the most surprising results was the 601
near identity of the scaling parameters H, C1, α for temperature, log potential temperature 602
and humidity, and in particular the non standard result H = 0.51±0.01 for all three fields 603
6/3/09 37
which is extremely close to the fraction ½ but which nevertheless (apparently) still belies 604
theoretical explanation (see however Tuck 2008). 605
Finally, we compared the estimates of the horizontal exponents with those from 606
drop sondes of the corresponding vertical exponents. Overall, the ratios Hz for v, T, logθ, 607
h were not far from the 23/9 D model prediction 5/9 (based on energy flux domination of 608
horizontal statistics and buoyancy variance flux domination of vertical statistics), yet 609
there were apparently systematic deviations: v, T, logθ had Hz ≈ 0.47 and for h, Hz ≈0.65. 610
For the moment there is no satisfactory theoretical explanation for these results, although 611
some clarification may come from a comparison with the structure of atmospheric 612
reanalyses and (hydrostatic) numerical models which not only have nearly perfect 613
cascade structures in the horizontal, but also at least some similar (and non-standard) 614
parameters. For example they notably show that along isobars, the wind spectral 615
exponent is almost exactly β = 2.4 i.e. the same as that observed here (Stolle et al., 616
2009). The fact that we find qualitatively similar wide range cascade structures in the 617
simulations and in the atmosphere opens up new avenues for statistically verifying the 618
models: their scale by scale statistics should be the quantitatively the same as those of the 619
measurements. 620
It is remarkable that in spite of the current golden age of meteorological 621
observations that there is still no scientific consensus about the atmosphere’s basic scale 622
by scale statistical properties, in particular those of the dynamical (wind) field. However, 623
if we reinterpret the (systematically observed) transition from ≈ 5/3 to ≈ 2.4 spectral 624
scaling in terms of a single anisotropic scaling turbulence then there are no longer 625
seriously obstacles to the emergence of a consensus. To complete this emerging “new 626
6/3/09 38
synthesis” (Lovejoy and Schertzer, 2009) the analyses presented here must be extended 627
to the time domain, to different altitudes, should be rechecked with different sensors and 628
in particular to aircraft flying on isoheights. Perhaps most importantly, completion of the 629
synthesis requires that we abandon the dogma of isotropy (or quasi isotropy) and embark 630
upon the systematic development of anisotropic but scaling theories of turbulence. 631
632
6. Acknowledgements: 633
We thank the crew of the Gulfstream 4 who gathered the data. This work was 634
performed for scientific purposes only; it did not enjoy any specific funding. 635
636
7. References: 637
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723
List of Figure Captions:
Fig. 1 a: The normalized moments q = 0.2, 0.4, … 3, for aircraft altitude, z; log10λ
corresponds to 20,000 km (= Lref).
Fig. 1b: Same as fig. 1a but for the fields strongly affected by the trajectories: the
longitudinal wind (top left a), the transverse wind (top right, b), pressure (c, lower left
right).
Fig. 1c: Same as 1b but for the fields that are relatively unaffected by the trajectory:
temperature (top left), relative humidity (top right), log potential temperature (lower left
right).
Fig. 2 a: Structure functions of order q = 0.2, 0.4, ..2.0 for the aircraft altitude. Distances
Δx are in km. Separate sets of regression lines are shown for both small and large scales
in order to emphasize the change with scale.
Fig. 2 b: Same as fig. 2a but for the fields strongly affected by the aircraft trajectory: the
longitudinal wind (upper left), the transverse wind (upper right) and the pressure (lower
left).
Fig. 2 c: Same as for 2a except for those fields relatively unaffected by the aircraft
trajectories: the temperature (upper left), the humidity (upper right) and the log potential
temperature (lower left).
Fig. 3a: Second order trace moments (right column) and their logarithmic derivatives
(estimates of K(2), an intermittency index), left column. Top row: wind, longitudinal
(blue) transverse (red). The region within the dashed lines has particularly low K(2) (≈
0.05). Middle row: pressure (blue), altitude (red). Bottom, temperature (blue), humidity
(red), log potential temperature (gold), reference lines K(2) = 0.12. Log10λ = 0
corresponds to 20,000 km (= Lref).
Fig. 3b: First order structure functions (right column) and their logarithmic derivatives
(estimates of H), left column. Top row: wind, longitudinal (blue) transverse (red), the
reference lines correspond to the exponents 1/3, 0.75. Middle row: pressure (blue),
altitude (red), the reference lines correspond to H = 1 i.e. constant mean slopes. Bottom,
temperature (blue), humidity (red), log potential temperature (gold), reference lines H =
0.5.
Fig. 3c: Spectra (right column) and their logarithmic derivatives (estimates of -β), left
column. Top row: wind, longitudinal (blue) transverse (red). The thin reference lines
correspond to β = 5/3, 2.4. Middle row: pressure (blue), altitude (red) with reference
lines corresponding to β = 5/3. Bottom, temperature (blue), humidity (red), log potential
temperature (gold), reference lines β = 2. Vertical dashed lines indicate the region with
scales 4 – 40 km where the velocity intermittency is low.
Fig. 4: The K(q) functions for altitude and pressure (upper, lower blue), temperature,
humidity, log potential temperature (lower, middle, upper green), longitudinal, transverse
wind (upper, lower, red).