Satellite observations are used to obtain vertical profiles of variance
scaling of temperature (

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

Observations have been frequently used to demonstrate that atmospheric
variables satisfy specific scaling laws.

A myriad of investigations using atmospheric variability generated by
numerical models have been performed.

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

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

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

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

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

The standard deviation of the temperature at 500 hPa as a function
of length scale

The microwave instruments are used together with IR spectra by applying a
process called cloud clearing

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

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

Same as Fig.

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

When plotting the standard deviation as a function of length scale, while
using logarithmically scaled horizontal and vertical axes, the scaling
exponent

Following KT09,

The well-known

Same as Fig. 5 using the water vapor mass mixing ratio.

Variance scaling exponents

Same as Fig. 7 using the water vapor mass mixing ratio.

Probability density functions of AIRS–AMSUv6

Scatterplots, linear fits, and correlations between physical quantities and the slope

Two variance-scaling plots for AIRS–AMSU–HSB derived temperature at
500 hPa with detected scale breaks:

Probability density functions of the scale break length scale using (left) temperature and (right) water vapor fields at three pressure levels.

In these double scale break examples it can be seen that the slope can change at scales below 1.5

Standard deviations are computed within circular areas of diameter

To obtain standard deviations corresponding to length scales smaller than

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

The intent of this method is to move the 15.4

The diameters of the circles can vary with arbitrary increments; we select
0.5

For

We note that

To quantify the length scale

To demonstrate the methodology, we aim to construct variance scaling diagrams
that are analogous to Fig.

The standard deviation typically increases as a function of

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.

The corresponding

We now focus on the scaling exponents

The three scaling exponents derived from AIRS–OE retrievals are shown in
Fig.

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

Local maxima of the standard deviation at

Increased separation between the three values of

The corresponding

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.

The three coarser-resolution AIRS products are similar when the yield is
high. Therefore, we show only AIRS–AMSU derived exponents in Figs.

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

The values of

Histograms of

The asymmetry of

These types of statistical distributions are valuable for the development and
evaluation of cloud parameterizations based on PDF schemes. This is
especially true for

To relate

The

Figure

We show PDFs in Fig.

Lastly, an example of a double scale break detection is applied to the
AIRS–OE dataset

The scale-dependent variability of temperature (

The precise value of

The small-scale variance scaling exponents

Deviations from typical values of

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

A major advantage of the “poor man's spectral analysis” method

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

The results show that there is a preference for scale break length scales
(

The exponent

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

The AIRS version 6 data sets were processed by and obtained
from the Goddard Earth Services Data and Information Services Center
(

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