**Research article**
08 May 2018

**Research article** | 08 May 2018

# Instantaneous variance scaling of AIRS thermodynamic profiles using a circular area Monte Carlo approach

Jesse Dorrestijn, Brian H. Kahn, João Teixeira, and Fredrick W. Irion

**Jesse Dorrestijn et al.**Jesse Dorrestijn, Brian H. Kahn, João Teixeira, and Fredrick W. Irion

- Jet Propulsion Laboratory, California Institute of Technology, 4800 Oak Grove Drive, Pasadena, CA 91109, USA

- Jet Propulsion Laboratory, California Institute of Technology, 4800 Oak Grove Drive, Pasadena, CA 91109, USA

**Correspondence**: Brian H. Kahn (brian.h.kahn@jpl.nasa.gov)

**Correspondence**: Brian H. Kahn (brian.h.kahn@jpl.nasa.gov)

Received: 17 Dec 2017 – Discussion started: 03 Jan 2018 – Revised: 31 Mar 2018 – Accepted: 18 Apr 2018 – Published: 08 May 2018

Satellite observations are used to obtain vertical profiles of variance
scaling of temperature (*T*) and specific humidity (*q*) in the atmosphere. A
higher spatial resolution nadir retrieval at 13.5 km complements previous
Atmospheric Infrared Sounder (AIRS) investigations with 45 km resolution
retrievals and enables the derivation of power law scaling exponents to
length scales as small as 55 km. We introduce a variable-sized circular-area
Monte Carlo methodology to compute exponents instantaneously within the swath
of AIRS that yields additional insight into scaling behavior. While this
method is approximate and some biases are likely to exist within non-Gaussian
portions of the satellite observational swaths of *T* and *q*, this method
enables the estimation of scale-dependent behavior within instantaneous
swaths for individual tropical and extratropical systems of interest. Scaling
exponents are shown to fluctuate between $\mathit{\beta}=-\mathrm{1}$ and −3 at scales
≥500 km, while at scales ≤500 km they are typically near $\mathit{\beta}\approx -\mathrm{2}$, with *q* slightly lower than *T* at the smallest scales observed. In
the extratropics, the large-scale *β* is near −3. Within the tropics,
however, the large-scale *β* for *T* is closer to −1 as small-scale moist
convective processes dominate. In the tropics, *q* exhibits large-scale *β*
between −2 and −3. The values of *β* are generally consistent with
previous works of either time-averaged spatial variance estimates, or
aircraft observations that require averaging over numerous flight
observational segments. The instantaneous variance scaling methodology is
relevant for cloud parameterization development and the assessment of time
variability of scaling exponents.

The author's copyright for this publication is transferred to California Institute of Technology.

In the atmosphere, energy that is present at larger scales tends to cascade towards the smaller scales where kinetic energy is turned into heat by dissipation on the Kolmogorov length scale (Hunt and Vassilicos, 1991; Kolmogorov, 1991). In two-dimensional turbulence, or quasi-geostrophic turbulence, energy that is injected at smaller scales can also be transferred to larger scales (Lindborg, 1999; Charney, 1971; Fjørtoft, 1953). Schertzer et al. (2012) give an alternative theory of energy transfer using fractal dimension turbulence. A review on upscale energy propagation is found in Tuck (2010). Numerous processes affect the atmosphere at different length scales (e.g., the large-scale planetary circulation, synoptic-scale systems, organized and isolated deep convection, shallow convection, turbulence, and molecular diffusion). As a result, the rate at which the variance of atmospheric properties changes as a function of length scale, the “variance scaling”, is not uniform over the entire range of scales within Earth's atmosphere.

Observations have been frequently used to demonstrate that atmospheric
variables satisfy specific scaling laws. Julian et al. (1970) showed that on
larger scales (> 1500 km) the kinetic energy spectra follow a *k*^{−3}
law. At smaller scales (< 500–700 km) the spectra are shallower and
follow a ${k}^{-\mathrm{5}/\mathrm{3}}$ law more closely. Transitions in between these regimes
have been clearly demonstrated with aircraft observations of wind and
temperature by Nastrom and Gage (1985). The Nastrom and Gage (1985) variance power
spectra diagram (their Fig. 3) is often cited and reproduced (e.g.,
Lindborg, 1999; Tung and Orlando, 2003; Palmer, 2012). The precise variance scaling
exponents of these atmospheric variables are, however, more complicated and
subtle. For instance, exponents that transition from $-\mathrm{5}/\mathrm{3}$ to −2.4 between
100 and 500 km were observed in aircraft wind measurements by
Pinel et al. (2012).

Kahn and Teixeira (2009) (KT09 hereafter) used satellite observations of temperature
(*T*) and the specific humidity of water vapor (*q*) to derive sensitivities of
scaling exponents to multiple factors such as the location on Earth (e.g., land,
ocean, latitude), the season, and the existence of clouds or clear sky. The
underlying causes of these variations and more complex phenomena, such as
scale breaks and reverse scale breaks (demonstrated to exist by KT09), are
not yet fully understood. One of the reasons may be the paucity of extensive
observational data sets that correspond to well-defined atmospheric
conditions over several orders of length scales. Clear patterns of scaling
exponents only appear after averaging over a sufficient time period on the
order of a season (KT09).

A myriad of investigations using atmospheric variability generated by
numerical models have been performed. Jonker et al. (1999) used a large-eddy
simulation (LES) model to show that passive scalars in a turbulent field
exhibit different power spectra than the thermodynamic variables themselves.
Cusack et al. (1999) used the horizontal variance of moisture with global
weather model analysis data and constructed a cloud parameterization from it.
Hamilton et al. (2008) showed a transition from a steep *k*^{−3} law to a
shallower ${k}^{-\mathrm{5}/\mathrm{3}}$ law in the kinetic energy spectrum of a general
circulation model (GCM).

As with observations, numerical simulations have their own limitations considering the range of scales that are represented. Due to computational restrictions, LES models are not able to accurately simulate synoptic systems. GCMs and cloud resolving models (CRMs) are not able to accurately resolve smaller-scale processes (e.g., turbulence, shallow convection) that affect variance scaling exponents at all scales. Parameterizations of unresolved processes are based on assumptions about variance scaling exponents derived from larger scales (Bogenschutz and Krueger, 2013; Tompkins, 2002; Teixeira and Hogan, 2002; Larson et al., 2002), and therefore, cannot be used to infer independent estimates of variance scaling exponents near the subgrid-scales. In short, there remains a need for numerical and observational investigations that report the statistics of scaling exponents over a larger range of length scales, in particular near the GCM subgrid-scale (Kahn et al., 2011). A review of scaling properties in numerical models is found in Lovejoy and Schertzer (2013).

In a follow-up to the methodology described by KT09, this work presents a new
variance scaling method that is applied to vertically resolved, satellite
derived *T* and *q* with higher horizontal resolution than previously reported.
The new variance scaling method enables the calculation of instantaneous
variance scaling exponents along the swath of Earth observing satellites. For
a particular horizontal two-dimensional atmospheric field (e.g., *T* or *q*) at a
particular pressure level or altitude in the atmosphere, the standard
deviations are calculated over spatial areas for a range of length scales
from which variance scaling exponents are derived. Areas are chosen to be of
circular shape and are placed along the track of a satellite.
Variance spectra are estimated by varying the diameter of the circular areas.
Then exponents are derived by fitting power law exponents to the data. To
obtain robust estimates, a Monte Carlo method is employed that randomly
places smaller circles within the largest diameter circle.

The paper is organized as follows. Section 2 describes the temperature and specific humidity datasets, which is followed by the introduction of the new variance scaling method (Sect. 3). The variance scaling results are presented in Sect. 4. Lastly, Sect. 5 discusses the implications and conclusions of the main findings, and suggests future research that is enabled with this novel approach.

The *T* and *q* profiles are derived from high-spectral-resolution infrared (IR)
observations made by the Atmospheric Infrared Sounder (AIRS)
(Aumann et al., 2003; Chahine et al., 2006) onboard the Aqua spacecraft
(Parkinson, 2003). The Aqua satellite is one of the Earth Observing
System (EOS) satellites and shares its near-polar (98^{∘} inclination)
orbit with other satellites that form the afternoon A-Train constellation
(Stephens et al., 2008). Aqua orbits the Earth at ∼ 705 km altitude in a
sun-synchronous orbit with an equatorial crossing of 13:30
(01:30)
local time for ascending (descending) orbital segments. With a swath width of
1650 km, the AIRS instrument is able to provide a near-global daily coverage.

## 2.1 Three types of AIRS standard retrievals

The AIRS instrument is a cross-track scanning spectrometer with 90 AIRS–IR
ground footprints per swath and results in a horizontal resolution of 13.5 km at nadir view. The self-calibrating instrument enables the estimation of
vertical profiles of several atmospheric variables (e.g., temperature,
humidity) and minor gases (e.g., ozone, carbon dioxide) from the surface up
to an altitude of 40 km with a quality approaching conventional radiosonde
soundings and a vertical resolution of one kilometer (Chahine et al., 2006). The
*T* and *q* bias and root-mean-square estimates based on radiosonde matchups
generally affirm pre-launch requirements of AIRS soundings
(Divakarla et al., 2006; Wong et al., 2015).

AIRS is accompanied by two synchronized and aligned microwave instruments. The Advanced Microwave Sounding Unit (AMSU) is a two-unit microwave radiometer with 15 channels that observe frequencies between 23 and 89 GHz including the 60 GHz oxygen band, and a horizontal resolution of 45 km at nadir view. The Humidity Sounder for Brazil (HSB) is a four channel radiometer that observes frequencies between 150 and 190 GHz, centering on the 183 GHz water vapor line, and has a horizontal resolution of 13.5 km at nadir (Lambrigtsen and Calheiros, 2003).

The microwave instruments are used together with IR spectra by applying a
process called cloud clearing (Susskind et al., 2003). During the process, the
horizontal resolution is coarsened from 13.5 to 45 km because all of the
variability in the AMSU footprint that contains nine co-aligned AIRS
footprints is assumed to arise from cloud variations. The cloud-cleared
spectra are then used to retrieve *T* and *q* profiles for three different
instrument combinations: AIRS–AMSU–HSB, AIRS–AMSU, and AIRS–IR (also termed
AIRS-only), the last of which does not use microwave channels but is still
obtained at the same spatial resolution as AIRS–AMSU and AIRS–AMSU–HSB
(Chahine et al., 2006).

The three-instrument AIRS suite enables the estimation of three-dimensional
(3-D) atmospheric profiles along the orbit of Aqua, since 30 August 2002 until
present (except until 5 February 2003 for HSB). Swath measurements are
organized in files that contain six minutes of data (Level 2) and are termed
a “granule”. Each day 240 granules are produced, each consisting of 30 × 45 vertical profiles of *T* and *q*. Figure 1a displays an
AIRS–AMSU–HSB Version 6 (v6) temperature field at 500 hPa in the very first
granule that is available at NASA Goddard Earth Sciences (GES) Data and
Information Services Center (DISC). Further detail about the AIRSv6 datasets
are found in Susskind et al. (2014).

## 2.2 AIRS infrared-only optimal estimation (AIRS–OE)

Other alternative methods are undergoing development that treat clouds during
the retrieval process through a more sophisticated approach without reducing the
horizontal resolution of the *T* and *q* fields that are described in
Chahine et al. (2006). The optimal estimation (OE) retrieval system for AIRS
(AIRS–OE) is described by Irion et al. (2018) and is used in addition to the
three coarser-resolution AIRS data products described previously. The
methodology is based on the works of Bowman et al. (2006) and
Rodgers (2000). Cloud detection and cloud property estimation is enhanced
with coincident high spatial resolution imaging data from the Moderate
Resolution Imaging Spectroradiometer (MODIS) instrument with a horizontal
resolution of 0.25–1.0 km at nadir and a swath width of 2330 km and also
resides on EOS Aqua with AIRS (King et al., 2003; Parkinson, 2003; Platnick et al., 2003).

Our approach is to calculate standard deviations as a function of length scale, then scaling exponents are calculated that correspond to a particular range in length scales (as in KT09). The scaling exponents obtained using standard deviations are referred to as “variance scaling” exponents.

If a power-law relation exists between the standard deviation and the length
scale, then given two length scales *l*_{1}<*l*_{2} with standard deviations
*σ*_{1} and *σ*_{2}, the scaling exponent *α* is as follows:

When plotting the standard deviation as a function of length scale, while
using logarithmically scaled horizontal and vertical axes, the scaling
exponent *α* determines the slope of the line from (*l*_{1},*σ*_{1}) and
(*l*_{2},*σ*_{2}). This line is straight if a power-law relation exists and is
half as steep as for variances, which can equivalently be used instead of
standard deviations to calculate the variance scaling exponents
(Vogelzang et al., 2015).

Following KT09, *α*_{L} is defined as the “large-scale” exponent for
scales between 6 and 12^{∘}, and *α*_{S} is defined as the
“small-scale” exponent for scales between 1.5 and 4^{∘}. In
addition, we added a third exponent *α*_{T} that is defined as the
“tiny-scale” exponent for scales between 0.5 and 1.5^{∘}.
The length scale is expressed in degrees over great circles. To relate the
computed *α* values to the more commonly used power spectral exponents
*β*, *α* values are interchanged with *β* values by using the
following equation (KT09, Davis et al., 1996; Yu et al., 2017):

The well-known $\mathit{\beta}=-\mathrm{5}/\mathrm{3}$ and $\mathit{\beta}=-\mathrm{3}$ correspond to $\mathit{\alpha}=\mathrm{1}/\mathrm{3}$
and *α*=1, respectively. Below we describe the estimation of standard
deviations within the AIRS swath following the ground track of Aqua.

## 3.1 Circular geometry

Standard deviations are computed within circular areas of diameter *l*. The
maximum length scale is determined by the fixed swath width of AIRS, $L=\mathrm{15.4}{}^{\circ}$. In that limiting case, a circle with radius 7.7^{∘} is
positioned with its center on Aqua's ground track (at nadir), after which the
standard deviation of valid *T* and *q* values are calculated within the circle.
A depiction of the 500 hPa *q* using AIRS–OE retrievals that are inside a
15.4^{∘} diameter circle is found in Fig. 2a. The smallest
length scale is the other limiting case and is determined by the horizontal
resolution of the observations. Here, we require that the minimum number of
valid retrievals that are necessary to calculate a standard deviation from a
circle with a given diameter is five, as assumed in KT09. Taking this
requirement into consideration for AIRS–OE retrievals, the smallest length
scale and hence the smallest diameter of the circles is chosen to be
*l*=0.5^{∘}. For the three coarser-resolution AIRSv6 data products
(AIRS–IR, AIRS–AMSU, AIRS–AMSU–HSB) the smallest length scale is chosen to be
*l*=1.5^{∘}.

## 3.2 Monte Carlo calculations

To obtain standard deviations corresponding to length scales smaller than $L=\mathrm{15.4}{}^{\circ}$ (i.e., $\mathrm{0.5}{}^{\circ}\le l<\mathrm{15.4}{}^{\circ}$), smaller circles
are randomly placed inside the largest circle. Given that a smaller circle
with diameter $l<\mathrm{15.4}{}^{\circ}$ is randomly placed within the largest
circle, we further require that the smallest circle is entirely within the
larger circle. Then, the standard deviation of *T* and *q* values located within
the smaller circle are computed.

To obtain a more robust estimate, a Monte Carlo estimation procedure
is employed. The random placement is repeated 10 000 times for each of the
smaller circle diameters. The average standard deviation over all 10 000
values is used as the estimate of the standard deviation corresponding to
*l*. The random placement should be done such that the 10 000 smaller circles
cover the largest circle as uniformly as possible. This procedure is repeated
for all length scales $\mathrm{0.5}{}^{\circ}\le l<\mathrm{15.4}{}^{\circ}$ for AIRS–OE and
down to 1.5^{∘} for AIRSv6 products. Two out of the 10 000 smaller
circles at $l=\mathrm{6}{}^{\circ}$ are displayed in Fig. 2b.

## 3.3 Along-track instantaneous variance scaling estimates

The intent of this method is to move the 15.4^{∘} diameter circle along
with the orbit of the Aqua satellite to estimate standard deviations and
therefore variance scaling exponents at each successive scan line along the
satellite ground track. This requires stitching together successive granules.
Therefore, a novelty of this approach is that the variance scaling can be
derived instantaneously (i.e., no time averaging as in KT09).

The diameters of the circles can vary with arbitrary increments; we select
0.5^{∘}, which should yield sufficient resolution to resolve
scale-dependent breaks and other behavior introduced in Sect. 4.
After the standard deviations are calculated as a function of length scale,
the three exponents (i.e., the slopes) are estimated by a least squares fit
(Weisstein, 2017).

For *α*_{L}, the formula of Weisstein (2017) is:

where *x*_{i}=*l*_{i} and *y*_{i}=*σ*_{i}, with $\mathrm{6}{}^{\circ}\le {l}_{i}\le \mathrm{12}{}^{\circ}$ and *n*=13. For *α*_{S} and *α*_{T}, the
formulas are used with $\mathrm{1.5}{}^{\circ}\le {l}_{i}\le \mathrm{4}{}^{\circ}$,
$\mathrm{0.5}{}^{\circ}\le {l}_{i}\le \mathrm{1.5}{}^{\circ}$, and *n*=6 and *n*=3, respectively.

We note that *β*_{L}, *β*_{S}, and *β*_{T} are the analogs of
*α*_{L}, *α*_{S} and *α*_{T} and will be
interchangeably used with *α* values as *β* values are more commonly
used in the literature.

## 3.4 Scale break detection

To quantify the length scale *l*_{b} at which the exponents change (e.g., from
$\mathit{\beta}=-\mathrm{3}$ to $\mathit{\beta}=-\mathrm{5}/\mathrm{3}$) the standard deviation as a function of *l*
is approximated by two power laws, which is equivalent to fitting two
straight lines in a double-log scaled figure. When using the higher spatial
resolution AIRS–OE retrieval, the double scale break is examined by fitting
three straight lines in the variance scaling plots. To do this optimally, we
iterate over all possible (double) scale break positions $l\in \mathit{\{}\mathrm{1.5}{}^{\circ},\mathrm{\dots},\mathrm{15}{}^{\circ}\mathit{\}}$ to find the two (three) fitted lines that
minimize the sum of the squares of the vertical offsets from the data to the
lines. Since such a minimum always exists, albeit at times very subtly, we
find an optimal single scale break *l* in each variance scaling diagram for
AIRS–AMSU–HSB, AIRS–AMSU, and AIRS–IR, and two scale break values of *l* for
AIRS–OE. In future work, thresholds could be used to make a distinction
between variance scaling diagrams with and without scale breaks.

## 4.1 Variance scaling diagrams

To demonstrate the methodology, we aim to construct variance scaling diagrams
that are analogous to Fig. 3 of Nastrom and Gage (1985). In this work,
Fig. 3 shows the standard deviations of 500 hPa *T* at four selected
locations on the Aqua track. The four locations are shown because they
encompass typical behavior of the scaling. We note that the scaling behavior
drastically changes at these locations depending on the day. The four
available AIRS retrievals are shown in order to gain insight regarding the
uncertainty of the scaling that arises from sampling, retrieval algorithm,
and observation frequency differences.

The standard deviation typically increases as a function of *l*; only in
Fig. 3c do we find that the standard deviation decreases at larger
*l* for AIRS–OE. In Fig. 3a this increase is not constant: at larger
*l*, $\mathit{\beta}=-\mathrm{3}$, while at smaller *l*, $\mathit{\beta}=-\mathrm{5}/\mathrm{3}$ for AIRS–OE. In
Fig. 3a, the slope changes between $l=\mathrm{9}{}^{\circ}$ and $l=\mathrm{11}{}^{\circ}$ and is a clear example of a well-behaved scale break. Observe in
Fig. 3c that the slope at smaller *l* is steeper than at larger *l*,
and is an example of a reverse scale break previously reported in KT09 for
specific humidity. In Fig. 3d, there is no clear scale break at all
except for some subtle fluctuations in the spectra.

The differences in the standard deviations among the three coarser-resolution
AIRS data products are generally small and are partly attributed to sampling
differences from clouds. AIRS–OE generally yields higher standard deviations,
most notably in Fig. 3b, which is to be expected because of the
higher spatial resolution. Other contributions to discrepancies among the
retrievals may be attributed to spatial sampling differences that arise from
differences in the spatial distributions of unsuccessful retrievals. Given
that the variance of *T* and *q* is highly location dependent, the additional
sampling provided by the microwave frequencies will also lead to differences
in the four retrievals in Fig. 3. Observe that the relative
differences between the slopes of the four different retrievals appear to be
smaller than the magnitude differences themselves.

The corresponding *q* spectra at 500 hPa are shown in Fig. 4. The
scaling of *q* in Fig. 4a is similar to scaling in *T* in
Fig. 3a; however, the slopes appear to be closer to $\mathit{\beta}=-\mathrm{5}/\mathrm{3}$
at smaller *l* in Fig. 4a. In Fig. 4b, a reverse scale
break is clearly visible near $l=\mathrm{9}{}^{\circ}$. In Fig. 4c, the
discrepancies between the four retrievals are more significant at larger *l*,
where AIRS–OE shows a decreasing standard deviation as a function of
increasing *l*. However, the AIRS–OE with a peak around 8^{∘} may be
a result of finer-scale fluctuations that are only captured by AIRS–OE. At
smaller scales, the slopes are similar and reside between $\mathit{\beta}=-\mathrm{3}$ and
$\mathit{\beta}=-\mathrm{5}/\mathrm{3}$. In Fig. 4d, the slopes are close to $\mathit{\beta}=-\mathrm{3}$ at
larger *l*, then are close to $\mathit{\beta}=-\mathrm{1}$ (i.e., *α*=0) around $l=\mathrm{7}{}^{\circ}$, then again increase (*α*) and decrease (*β*) at smaller length scales, in sharp contrast to *T* in
Fig. 3d.

## 4.2 Along-track variance scaling

We now focus on the scaling exponents *α*_{L}, *α*_{S}, and *α*_{T},
along 84 min of the Aqua track starting at the second granule available in
the AIRS archive. We estimate scaling exponents 500 hPa *T* and *q*. The 84 min
dataset is a very small subset of the ∼ 15 year AIRS dataset and
corresponds to just under one complete orbit. The centers of consecutive
15.4^{∘} diameter circles are 8 s apart from each other,
corresponding to the time it takes to make 30 AIRS–AMSU soundings along the
width of the swath. As consecutive 15.4^{∘} circular areas have high
overlap, there are large correlations in exponents along neighboring scan
lines.

The three scaling exponents derived from AIRS–OE retrievals are shown in
Fig. 5a. Observe that the exponent *α*_{L} fluctuates
between 0 and 1 (left vertical axis) that corresponds to *β* between −1
and −3 (right vertical axis). The exponent *α*_{S} has smaller
fluctuations around $\mathit{\alpha}=\mathrm{1}/\mathrm{3}$ ($\mathit{\beta}=-\mathrm{5}/\mathrm{3}$). The exponent
*α*_{T} exhibits even smaller fluctuations than
*α*_{S} and usually resides between 1∕2 and 1∕3, and
corresponds to *β* between −2 and −5/3.

The standard deviation estimates from which these scaling exponents are calculated are shown in Fig. 5b. The lowest line is the standard deviation that corresponds to $l=\mathrm{0.5}{}^{\circ}$, the line above corresponds to $l=\mathrm{1.0}{}^{\circ}$, and so forth. The standard deviations are usually, but not always, increasing as a function of length scale.

Local maxima of the standard deviation at 15.4^{∘} co-align with local
maxima of *α*_{L} (local minima of $\mathit{\beta}=-\mathrm{3}$). A large standard
deviation at synoptic scales is indicative of meridional temperature
gradients along the satellite track, and correlates well with *α*_{L}
(shown later). The latitude and longitude at nadir are depicted in Fig. 5c.

Increased separation between the three values of *α* (Fig. 5a)
suggest the existence of scale breaks at those latitudes. For example at
75 min (40^{∘} S), *α*_{L}=1 ($\mathit{\beta}=-\mathrm{3}$), while
*α*_{S}=0.6 and *α*_{T}=0.5. Nearer to the equator around 33 min,
the estimates are reversed: *α*_{L}=0, *α*_{T}=0.5
nearly unchanged, and *α*_{S} is in between the two other values
of *α*, indicating a double reverse scale break (steeper exponents at
smaller scales). Around 58 min near Antarctica, the three values of *α*
are nearly equal with no apparent scale breaks. Analogous variance scaling
diagrams would be similar to that shown in Fig. 3.

The corresponding *q* exponents are shown in Fig. 6a. The *α*_{L}
has a similar range as the *T* exponent. The *α*_{S} for *q* is
slightly larger than for *T*. However, the *α*_{T} for *q* is
significantly larger than for *T*. Figure 6b depicts the standard
deviations of *q*. In the tropics (location from Fig. 5c), the
standard deviation is much larger than in the extratropics (e.g., compare
granules 231 and 235). Maximum values of standard deviation are generally
co-aligned with maximum values of *α*_{L}.

## 4.3 Variance scaling at 850, 500, and 300 hPa

Previous studies have demonstrated that the magnitude of scaling exponents
depends on altitude, surface type, and cloud cover (e.g., KT09). Therefore,
we show variance scaling exponents along the same orbit segment at three
pressure levels (300, 500, and 850 hPa) in Fig. 7 for *T* and
Fig. 8 for *q*. The results are typically noisier at lower pressure
levels (e.g., compare Fig. 7c to a, b) and is
consistent with a reduction in the yield (percentage of successful
retrievals).

The three coarser-resolution AIRS products are similar when the yield is
high. Therefore, we show only AIRS–AMSU derived exponents in Figs. 7
and 8. The *α*_{L} (blue dash-dotted line) for *T*
fluctuates between 0 and 1, except for a portion of the 850 hPa pressure
level (Fig. 7c) around the South Pole (granule 235) where the yield
is exceptionally low. The small-scale exponent *α*_{S} (red solid
line) fluctuates within a smaller range except for a portion of 850 hPa
around 24 min in which *α*_{S}=1.

A reverse scale break in the tropics (granule 231) is clearly visible at
500 hPa and to a lesser extent at 850 and 300 hPa, which is consistent with
KT09. The scaling exponent *α*_{T} (cyan dashed line) derived
from AIRS–OE fluctuates between *α*_{T}=0.2 to 0.6 for all time
segments at the three pressure levels. Variations with surface type and cloud
fraction (not shown) are less obvious in Figs. 7 and 8 and
become clearer only after averaging over long time series (e.g., KT09).

The values of *α* for *q* exhibit more rapid along-track fluctuations
compared to *T* at the three pressure levels. The *α*_{L} for *q*
fluctuates between 0 and 1 in Fig. 8. The *α*_{S} for
*q* is similar to *T*, but *α*_{T} for *q* is typically larger
than 0.5 ($\mathit{\beta}<-\mathrm{2}$) and has a larger dynamic range for *q* than for *T*
(compare Figs. 7 and 8). In granule 233 at 300 hPa,
*α*_{T} exceeds 1.0 ($\mathit{\beta}<-\mathrm{3}$).

## 4.4 Distributions of variance scaling exponents

Histograms of *α*_{L} and *α*_{S} for 500 hPa *T* and *q* obtained from
five days of Aqua orbits are shown in Fig. 9. To increase the
computational speed, only 100 circles are used in the Monte Carlo method
described earlier, which leads to a slight increase in the number of extreme
values of *α*_{L} and *α*_{S}. The histograms of *α*_{L} have larger
ranges than *α*_{S}, both for *T* and *q*. The *α*_{S} exhibit a more
symmetric distribution and the maximum number of values is near *α*_{S}=0.5.

The asymmetry of *α*_{L} for *T* is caused by different values in the
extratropics and tropics. In the tropics, *α*_{L} is closer to 0, while in
the extratropics it is closer to 1. The values of *α*_{S} do not have a
strong latitude dependency and thus the distribution is more symmetric. The
*α*_{L} for *q* is skewed in the opposite direction compared to *α*_{L}
for *T*, because *α*_{L} is closer to 1 in the tropics and closer to 0 in
the exratropics.

These types of statistical distributions are valuable for the development and
evaluation of cloud parameterizations based on PDF schemes. This is
especially true for *α*_{T} (not shown); the AIRS–OE retrieval methodology
is in development and the sample size from the limited set of granules is
unable to yield a robust histogram. Our intent is to instead demonstrate the
new scaling approach. A much larger and statistically robust dataset is
outside the scope of this work. Furthermore, the computational expense is
excessive using the Monte Carlo methodology with 10 000 circles rather than
100, and new ways of improving the speed of the calculations remains
necessary.

## 4.5 Correlation of scaling exponents to other quantities

To relate *β*_{L} and *β*_{S} to additional geophysical quantities,
correlation analysis is performed using the AIRS–AMSU retrieval and the
results are summarized in Fig. 10. A total number of 686 values (the number
of L2 retrieval swaths) cover a slightly longer extent of the orbital segment
than the 84 min portion used in Sect. 4.2 and
4.3. The largest correlation coefficient (*r*) is
found between *β*_{L} and the mean *T* gradient (slope) in the along-track
direction of Aqua at nadir view (Fig. 10a). The *T* change we consider
is the difference between the average 500 hPa *T* over consecutive
15.4^{∘} diameter circles. If the *T* gradient is large, *β*_{L} is
closer to −3. If the *T* gradient is small or near zero, *β*_{L} is closer to
−1.

The *β*_{L} also correlates strongly with the standard deviation
of *T* in the 15.4^{∘} diameter circle (Fig. 10b); this
confirms the co-alignment of peaks observed in Fig. 5a and b. For
*q*, *β*_{L} is moderately correlated with the standard deviation
in the 15.4^{∘} diameter circle (Fig. 10d) and the along-track
*q* gradient (Fig. 10e). The exponents *β*_{L} and
*β*_{S} are positively correlated for both *T* (Fig. 10c)
and *q* (Fig. 10f), but the correlations are notably larger for *T*.
The surface type (land vs. ocean) is not strongly correlated with
*β*_{L} or *β*_{S} at 850 hPa (not shown). Lastly, the
cloud fraction has a rather weak correlation with *β*_{L} and
*β*_{S} at 500 hPa (not shown); again, a larger sample size may yield
different results.

## 4.6 Scale break detection results

Figure 11 shows an example of the methodology to detect scale breaks.
This example makes it clear that the length scale of scale breaks *l*_{b}
varies substantially along the Aqua tracks. The *l*_{b} differ by a factor of
two (9 and 4.5^{∘}, respectively). *l*_{b} fluctuates by an
order of magnitude between 1.5 and 15^{∘} along the Aqua
track at 850, 500, and 300 hPa for all four AIRS retrieval products, and for
both *T* and *q*.

We show PDFs in Fig. 12 to gain additional insight in the
distributions of *l*_{b}. The resolution of AIRS–OE is fixed to
$\mathrm{1.5}{}^{\circ}\le l\le \mathrm{15.4}{}^{\circ}$ such that similar scale breaks will be
detected as the three standard AIRS data retrievals. The maximum frequency of
occurrence in the PDFs is between 7 and 10.5^{∘} for *T* and
between 5 and 8^{∘} for *q* at 850 hPa (Fig. 12e, f),
between 8 and 11^{∘} for *T* at 500 hPa (Fig. 12c) and
around 9^{∘} for *q* at 500 hPa (Fig. 12d), between 5.5 and 9^{∘} for *T* at 300 hPa (Fig. 12a)
and around 9^{∘} for *q* at 300 hPa (Fig. 12b). These results,
to some extent, are in agreement with the 500–700 and 450–750 km *l*_{b}
reported by Gage and Nastrom (1985) and Tung and Orlando (2003), respectively. The PDFs of *T*
(Fig. 12a, c) have less well-defined maxima compared to *q* (Fig. 12b, d, f), except for *T* at 850 hPa (Fig. 12e), in which a
clear peak is apparent at 7^{∘} (Fig. 12e) for three AIRS
standard retrieval products. A tentative physical and algorithmic explanation
for the large spread of *l*_{b} is provided in Sect. 5.

Lastly, an example of a double scale break detection is applied to the
AIRS–OE dataset $\mathrm{0.5}{}^{\circ}\le l\le \mathrm{15.4}{}^{\circ}$ for *T* and *q* in
Fig. 13. This example once again demonstrates the variety of
variance scaling that is potentially observable in the atmosphere. The slope
in the spectrum is steeper at the smaller scales less than 1.5^{∘}.
This behavior is of importance for cloud parameterizations based on PDF
schemes. Less variance exists at the smaller scales than what one obtains
from a simple extrapolation of exponents at larger scales to smaller scales;
this was previously shown with in situ aircraft observations in
Kahn et al. (2011) in a limited region of the subtropical southeastern Pacific
Ocean.

The scale-dependent variability of temperature (*T*) and specific humidity (*q*)
is derived from Level 2 satellite swath data. The variance scaling of *T* and *q*
uses data from the Atmospheric Infrared Sounder (AIRS; Chahine et al., 2006)
instrument suite onboard the Aqua satellite. While these exponents are
frequently close to the canonical $\mathit{\beta}=-\mathrm{5}/\mathrm{3}$ and $\mathit{\beta}=-\mathrm{3}$ values,
deviations from these values are more the rule than the exception. The large
scale exponents *β*_{L} that correspond to length scales between
6 and 12^{∘} fluctuate between −3 and −1, respectively.

The precise value of *β*_{L} depends strongly on the standard deviation of
*T* and *q* within larger spatial areas. The synoptic-scale *T* and *q* gradient in
the along-track direction also impacts the magnitude of *β*_{L}. When the
large scale fluctuations or gradient are large, then ${\mathit{\beta}}_{\mathrm{L}}\approx -\mathrm{3}$.
When large scale *T* fluctuations are reduced such as in the tropics, and the
atmosphere is dominated by small scale fluctuations, ${\mathit{\beta}}_{\mathrm{L}}=-\mathrm{1}$. In the
tropics, small-scale *T* fluctuations dominate because of the preponderance of
deep convection. In contrast, large-scale *q* fluctuations are more dominant in
proximity to and within the tropics, which results in ${\mathit{\beta}}_{\mathrm{L}}\approx -\mathrm{3}$.

The small-scale variance scaling exponents *β*_{S} that correspond to
length scales between 1.5 to 4^{∘} are more often near to −2,
and less often close to −3. By using recently developed retrievals from
single-footprint AIRS data (Irion et al., 2018), we show that at the smaller
scales from 0.5 to 1.5^{∘}, the exponents *β*_{T} are
closer to −2 for *T* and slightly lower (between −2 and −3) for *q*. The PDFs of
small-scale *β*_{S} exhibit a maximum around −2 for both *T* and *q*. This is
somewhat surprising since previous studies have suggested values closer to
$-\mathrm{5}/\mathrm{3}$.

Deviations from typical values of *β* ($-\mathrm{5}/\mathrm{3}$ and −3) have been reported in
the literature previously (e.g., KT09). Lovejoy et al. (2008) show that exponents
derived from drop sondes over the northern Pacific Ocean reside in between
$-\mathrm{5}/\mathrm{3}$ and −3 and have strong height dependence. It was also shown that the
vertical exponents are not equal to the horizontal exponents, and suggests 3-D
anisotropy. Lovejoy et al. (2009) and Pinel et al. (2012) showed that scale breaks
detected by in situ aircraft observations may be the result of 3-D anisotropy
in atmospheric properties. In Pinel et al. (2012), scale breaks are observed in
the 100–500 km range with horizontal exponents that transition from $-\mathrm{5}/\mathrm{3}$ to
−2.4. In the vertical direction, an exponent of −2.4 is derived and suggests
that gently sloping isobaric aircraft trajectories are the source of the
transition to −2.4. Since the *T* and *q* exponents reside on isobaric surfaces
(e.g., 500 hPa) in this work, one may expect that the vertical exponents may
alias into the large-scale horizontal exponents. However, we do not find a
clear indication of ${\mathit{\beta}}_{\mathrm{L}}=-\mathrm{2.4}$, although a focused effort on obtaining
vertical scaling exponents with satellite soundings warrants further
investigation. Unfortunately, the relative coarse vertical resolution of
∼2 km from AIRS retrievals is not ideal for obtaining reliable
estimates of vertical scaling exponents; dropsondes and radiosondes remain
the standard and are much better suited to this observational challenge.

The methodology described uses circles to calculate standard deviations. The
optimal shape of an area used to calculate variance remains an open question.
Rectangles have been used previously (e.g., KT09) and are generally accepted,
because GCM grid columns are often (nearly) rectangular. The orientation of a
rectangle or square should not be of major importance when calculating
variance scaling exponents. One could argue that incremental rotation of the
rectangle about a central axis could be used to trace out the area of a
circle. Within each rectangle, the variance can be calculated, and the same
for each slight rotation of the rectangle about the center axis, until it is
rotated 360^{∘}. This procedure of incremental rotation could be
performed with any arbitrary shape. In all cases, the circle is the final
result, and therefore one may conclude that the circle is the “optimal
shape” to calculate variance scaling exponents. A circle is optimal in the
event that rotational symmetry is desired if, for example, the underlying
field is isotropic. This is consistent with Pressel and Collins (2012) who found that
variance scaling of *q* is approximately isotropic.

A major advantage of the “poor man's spectral analysis” method (Lorenz, 1979) is that relatively small datasets are sufficient to estimate variance scaling exponents. Reliable spectral power diagrams of observational data arise only after averaging over relatively large datasets. For instance, Nastrom and Gage (1985) obtained their spectral power diagrams by averaging over observations collected during 6000 commercial aircraft flights. The calculation of spatial variances is still possible in the event of missing or poor quality data, in which case conventional spectral analysis cannot be employed (Vogelzang et al., 2015).

The variance scaling exponents are computed nearly instantaneously without
using multiple satellite overpasses (no time averaging) in this work. The
exponents are derived from satellite observations within a 15.4^{∘}
diameter circle over a few minute time window, thus strictly speaking, the
method is “approximately” instantaneous. A result of the instantaneous
approach is that a much wider variety of scaling exponents is revealed. The
large variety of exponents is likely due to some extent from the turbulent
structure of *T* and *q* fields with long-tailed non-Gaussian distributions
(Tuck, 2010). These behaviors may inhibit the precise estimation of
variance scaling exponents from observations. Further research is necessary
to determine the impacts of the non-Gaussian distribution shapes of *T* and *q*
on derived exponents, and their scale-dependence of non-Gaussianity. This
effect likely contributes to spreading out the PDFs of exponents.

The results show that there is a preference for scale break length scales
(*l*_{b}) around 7^{∘} (at 850 hPa) and 9^{∘} (at 500 hPa and
300 hPa). This is slightly larger than the 500–700 and 450–750 km *l*_{b}
reported by Gage and Nastrom (1985) and Tung and Orlando (2003), respectively, and smaller
than 1000 km reported by Bacmeister et al. (1996). Pinel et al. (2012) report
*l*_{b} between 100 and 500 km. The spread around these values is large in our
results and a preferred length scale is only inferred from the maxima in the
PDFs. An explanation for the large spread in the PDFs is that convective
systems of different sizes exist. The existence of a reverse scale break
depends on the scale of convective systems: the larger the scale, the larger
*l*_{b}. Furthermore, as *l*_{b} is obtained in the along-track dimension, for
instance when the satellite observation transitions from a regime with ${l}_{\mathrm{b}}=\mathrm{2}{}^{\circ}$ to a regime with ${l}_{\mathrm{b}}=\mathrm{15}{}^{\circ}$, then the intermediate
length scales between 2 and 15^{∘} are also retrieved in
between the two regimes as the circular area advances along the swath. Due to
the overlapping nature of the circular areas, this additionally smooths out
the peaks in the PDFs, but further work is necessary to quantify the
magnitude of this effect compared to the spreading due to non-Gaussianity.

The exponent *β*_{T} (0.5 to 1.5^{∘}) for *q* was shown to
attain smaller values, i.e., closer to $\mathit{\beta}=-\mathrm{3}$ than the exponent *β*_{S}
(1.5 to 4^{∘}). This means that less variance is present at
length scales between 0.5 and 1.5^{∘} than if extrapolated
from exponents derived from length scales between 1.5 and
4^{∘}. Scale breaks at subgrid scales in GCMs are of significance for
cloud parameterizations in GCMs that extrapolate variability
(Tompkins, 2002; Teixeira and Hogan, 2002; Teixeira and Reynolds, 2008).

This novel instantaneous variance scaling methodology may enable detailed examination of the variance scaling of the time evolution of storm systems, such as extratropical cyclones at different stages in their life cycle as previously demonstrated with numerical simulations by Waite and Snyder (2013), or with deep convection along the Mei-Yu front by Peng et al. (2014). The changes in the kinetic energy spectra in Waite and Snyder (2013) and Peng et al. (2014) occur on time scales of hours to several days. We postulate that scaling exponents derived from instantaneous snapshots obtained from satellite swath data will be useful observational constraints for time-dependent spectra generated from numerical modeling experiments. To conclude, it is well known that the time scale of predictability is closely linked to the spatial scale of the phenomenon of interest (Lorenz, 1969). In the case of moist baroclinic waves, steeper (shallower) spectral slopes at small scales for individual baroclinic waves are inherently more (less) predictable as the slope portrays the relative importance of convection within any given disturbance (Zhang et al., 2007). As a result, the instantaneous scaling exponents are expected to potentially offer a new type of observational constraint with relevance to the predictability of individual tropical or extratropical disturbances.

The AIRS version 6 data sets were processed by and obtained from the Goddard Earth Services Data and Information Services Center (http://daac.gsfc.nasa.gov/; Teixeira, 2013). All rights reserved. Government sponsorship acknowledged.

The authors declare that they have no conflict of interest.

Part of this research was carried out at the Jet Propulsion Laboratory (JPL), California Institute of Technology, under a
contract with the National Aeronautics and Space Administration. All authors were partially supported by the AIRS project at JPL.

Edited by: Andrew Sayer

Reviewed by: two anonymous referees

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