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STATISTICAL COMPARISON OF WINDS FROM ERS-1 SCATTEROMETER AND ECMWF MODEL IN TIME AND WAVENUMBER DOMAIN
| E. Bauer | Potsdam Institute for Climate Impact
Research, P.O.Box 60 12 03, 14412 Potsdam, Germany; phone: +49-331-2781-165; fax: +49-331-288-2600 |
ABSTRACT
The statistical properties of global homogeneous wind vector
fields from the scatterometer of the ERS-1 satellite are analysed
using the one-year data set of 1993. The mean, the total variance
and the spectral energy density clearly show seasonal
variability. But the spectral energy density reveals a constant
power-law dependence on wavenumber for meso-scales throughout the
year. The power-law is 
as
retrieved from along-swath series of 10,000 km length. This
power-law lies between the power-law predicted by the theory of
quasi-geostrophic turbulence with 
and by the
theory of 2d isotropic turbulence with 
.
The wavenumber spectra from the scatterometer wind vectors
reveal that the kinetic energy of the near-surface winds of the
ECMWF model is significantly underestimated on scales downward
1000 km. This underestimation causes an underestimation of the
spectral variance of the modeled ocean waves which is significant
on scales downward 1200 km. This is evident from comparison of
wavenumber spectra of significant wave height 
from the wave model WAM and from the measured of the ERS-1 altimeter.
INTRODUCTION
Global wind measurements with the scatterometer on board the polar orbiting satellites ERS-1/2 provide valuable information for earth monitoring, weather prediction and for geophysical modeling, in general. The geophysical modeling of, for instance, currents, waves and exchange of heat, water vapour, gases and aerosols at the interface between the ocean and the atmosphere gains importance for the understanding and prediction of the earth climate. The prediction of weather has already improved through the assimilation of scatterometer wind measurements which is reflected in an increased forecast skill.
Figure 1: Sketch
of kinetic energy E(k) of atmospheric motions as function of
horizontal wavenumber k based on theory of quasi-geostrophic
turbulence and 2d isotropic turbulence derived from data studies.
The quasi-geostrophic regime is characterized by 
. This regime receives enstrophy 
through baroclinic instabilities at Q1.
Enstrophy is transferred at a constant rate to lower wavenumbers.
The 2d isotropic turbulence regime is characterized by 
and by a constant flux of energy 
toward lower wavenumbers. Input of energy may occur at Q2 and/or
Q3 and may lead to a variable level of energy density which is
indicated by the dashed line. The transition between the two
regimes is marked by S which corresponds to the sink of enstrophy
and energy.
Furthermore, the scatterometer wind vectors are useful to
study the distribution and the flux of the kinetic energy in the
atmosphere within a wide range of length scales. A first attempt
to determine kinetic energy spectra from polar orbiting satellite
measurements has been performed by Freilich and Chelton (1986).
Freilich and Chelton (1986) analysed scatterometer winds from
Seasat taking 15 days of data from 6-20 September, 1978 from the
Pacific. Meridional energy spectra obtained from sea surface
winds for wavelength between 250 and 2200 km were consistent with
a power-law dependence. The power-law dependence on wavenumber k
was 
for midlatitude regions and 
for
tropical regions. Although the data set was rather small these
power-law dependencies indicated a deviation from the power-law
predicted by the theory of isotropic turbulent motions.
Figure 2:
Meridional wavenumber spectra of 10 m wind component u (bold
line) and v (thin line) obtained from the ERS-1 scatterometer
measurements of November 1992 for the northern ( 
N- 
N) (top panel), the tropical ( N- S) (middle panel) and the southern regions
( S- S) (bottom panel). Slopes of wavenumber
spectra are given in table 1. Slopes inferred from theoretical
power-laws and from isotropy conditions are shown by straight
lines.
This study is concerned in temporal changes of wavenumber spectra analysing the global scatterometer wind data from ERS-1 of the year 1993. The next section 2 gives a brief introduction to the theory of isotropic turbulence. Section 3 presents monthly wavenumber spectra from extra-tropical and tropical latitudes. The differences of the energy density spectra obtained from the scatterometer winds and from the ECMWF model winds are described in section 4. Section 5 demonstrates that the underestimation of the meso-scale energy density by the atmospheric model is passed on to the spectral density of significant wave height. This results in an underestimation of the meso-scale variance of modeled ocean waves. This findings are discussed in the concluding section 6.
THEORY OF ISOTROPIC TURBULENCE
Atmospheric models approximate the large-scale circulation
according to the theory for quasi-geostrophic turbulence
(Charney, 1971). The regime of quasi-geostrophic turbulent
motions represents an inertial subrange in which neither energy
nor enstrophy (= mean squared vorticity) is generated or
dissipated. Within this inertial subrange a constant flux of
enstrophy from large to small scales takes place, but no flux of
energy (Batchelor, 1969; Kraichnan, 1971). The wavenumber
spectrum has a characteristic power-law of .
Energy and enstrophy are fed to the atmospheric system through baroclinic instabilities which are most effective at scales of about 5000 km. This length scale determines the upper boundary of the inertial subrange. The lower boundary is connected with a sink of enstrophy. The existence of such an inertial subrange is supported by various analyses using measurements of the upper atmosphere (e.g., Julian et al., 1970; Chen and Wiin-Nielsen, 1978; Nastron and Gage, 1985).
However, atmospheric turbulent motions are not only excited
through baroclinic instabilities, but also at smaller scales
through, for instance, convection, shear flow and breaking waves.
Another inertial subrange might evolve which is associated with a
constant flux of energy from lower toward larger scales. The
turbulent motions in this inertial subrange are horizontally
isotropic. This 2d isotropic motions are characterized by a
wavenumber spectrum with slope (Kraichnan,
1971). This power-law dependence follows from the similarity
theory of Kolmogoroff (1941) which founds the power-law dependence of the small-scale 3d turbulent
and local isotropic motions. Kraichnan (1971) showed that if
energy and enstrophy are conserved also 2d isotropic motions obey
the power-law. This theoretical considerations have been
verified with measurements (e.g., Gage, 1979; Balsley and Carter,
1982; Lilly and Peterson, 1983; Nastron and Gage, 1985).
The results of the historical studies are combined in a sketch for the wavenumber density spectrum of atmospheric motions covering four length scales (Figure 1). For the macro-scale regime the same considerations might apply as for the meso-scale regime with 2d isotropic motions. Large uncertainties exist on the transition wavenumbers between the different regimes, and on the level of the energy density. The level of the energy density depends largely on the height in the atmosphere and on the seasonal variations.
The wind measurements of an orbiting satellite yields 1d wavenumber spectra only. But 1d wavenumber spectra of the wind vector components are sufficient to test whether the atmospheric motions are conform with the property of isotropy connected with the inertial subranges. If the kinetic energy density is isotropically distributed in the horizontal then the spectrum of each wind component has the same power-law dependence in both directions. This follows from the definition of the total kinetic energy density
where 
is the Fourier transform
of the horizontal field of the wind component j, and
where 
denote the horizontal
wavenumbers, 
are the u and v
wind components, and x,y are the horizontal space
coordinates.
The meridional spectrum 
adds up by the meridional spectra 
and 
of the wind components u and v,
respectively
Assuming that the energy density is horizontally isotropic and free of divergence, and that the meridional spectrum has the power-law with slope -b
then generalizing the calculations of Leith (1971) the isotropy condition
is derived. According to the isotropy condition atmospheric
motions may be called isotropic if the meridional spectral
density of the zonal wind
component u is enhanced by the factor b with
respect to the spectral density of the meridional wind component v.
Table 1:
Meridional wavenumber spectra of u and v wind component of the
data [D], from the region [R] and from the period [P] based on N
meridional series have slope -b determined from linear regression
with 99% confidence interval. The data are from the ERS-1
scatterometer [sc] and from the collocated ECMWF [E] model winds.
The regions are the northern ( N- N) [N], the tropical ( N- S) [T], the southern latitudes ( S- S) [S], and the global area ( N- S) [G].
WAVENUMBER SPECTRA OF SCATTEROMETER WINDS
Wavenumber spectra are computed from the ERS-1 scatterometer
winds provided by IFREMER. These winds are 
winds which refer to winds 10 m above the sea
surface. Along the scatterometer swath of 500 km width only three
series of wind vectors are taken. The series taken are separated
by 150 km across the swath to obtain about independent series.
Only series without data gaps are used for the spectral analysis
to avoid any influence from interpolation. For the same reason
the ERS-1 path with inclination angle 
is assumed
to be oriented in meridional direction. (Freilich and Chelton
(1986) interpolated the wind to a x,y coordinate
system by an optimum interpolation scheme.)
Figure: Meridional wavenumber spectra of 10 m wind component u (bold line) and v (thin line) obtained from ERS-1 scatterometer (top panel) and ECMWF model (bottom panel) from series 10,000 km long from November 1992 until October 1993. For slopes of wavenumber spectra see table 1, and for further explanations see Figure 2.
Monthly wavenumber spectra of the u and v wind
component are computed from series of 4000 km length from the
Extra-Tropics of both hemispheres between and latitude and from the tropical latitudes between
them. The horizontal resolution of the scatterometer wind
measurements is 50 km. Typical spectra for the three regions are
shown in Figure 2 for November 1992. The 99% confidence interval
of the spectral estimates are shown in the plots. The power-law
dependencies on wavenumber from linear regression are given in
table 1.
The difference between the slopes of the extra-tropical and the tropical spectra is small but significant on the 99% confidence level. The atmospheric motions are less energetic on the larger scales in the Tropics resulting in slightly smaller spectral slopes. This is in agreement with Freilich and Chelton (1986).
The power-law dependency is found to be same for every monthly data set of 1993. No variation with season is seen. The energy density levels of the wind components are found to be nowhere in agreement with the isotropy condition (5). It has to be added that the coherency spectra of the u and the v wind component show small values but the coherence is significantly larger than zero for the large scale motions. This implies that the motions are not entirely free of divergence.
The spectral energy density of the extra-tropical regions is larger in winter time than in summer time. The seasonal variability of the spectral density is more pronounced in the northern than in the southern regions. This is consistent with the seasonal variability of the (total) variance being larger in the northern hemisphere than in the southern hemisphere.
SPECTRA FROM MEASURED AND MODELED WINDS
Variance spectra from the scatterometer winds show a single
power-law dependence on a wide range of wavenumbers which lies
between the theoretical power-laws of and . No transition between different regimes is
recognized from the measurements. This is evident even more
clearly from a wave number spectrum calculated from series of
10,000 km length (Figure 3, top). This spectrum comprises
continuous series of scatterometer wind vectors of the entire
year 1993.
Figure 4: Meridional wavenumber spectra of significant wave height from ERS-1 altimeter (top panel) and wave model WAM (bottom panel) from series 7800 km long of 1993. Error bar shows 99% confidence interval.
In contrast, two different regimes are visible from the
corresponding spectrum of collocated winds of the ECMWF model (Figure 3, bottom). At
large scales a power-law of about is
recognized. At scales below 1000 km the spectral energy density
falls off steeply. This fall-off is presumably mainly induced by
numerical damping to assure numerical stability. The model winds
are from the T106 model version with a nominal resolution of 
in x and y direction. The influence of
interpolation of the ECMWF model winds from collocation distorts
the spectral estimates at scales smaller than 150 km and these
values should not be included in the comparison.
WAVENUMBER SPECTRA OF SIGNIFICANT WAVE HEIGHT
The wavenumber spectra of ERS-1 scatterometer winds are seen to contain increasingly higher spectral variance from scales downward 1000 km than the spectra of the collocated ECMWF model winds. The question arises whether the underestimation of the spectral variance by the ECMWF model is important for geophysical modeling.
Table 2:
Meridional wavenumber spectra of of the ERS-1 altimeter and the WAM model for the
global region from the year 1993 based on N meridional series
have slope -b determined from linear regression with 99%
confidence interval.
Here, the effect of the ECMWF model winds applied as forcing
fields to the wave model WAM is studied. Significant wave heights
( ) are a highly suitable
quantity to test the quality of winds. Although is nonlinear related to the wind (Komen
et al., 1994) the spectral variance of measured and of measured wind show about the same
power-law dependence. This is also found by Monaldo (1990) and by
Challenor (1993). As seen before for the winds, the wavenumber
spectra of from wave modeling are
underestimated compared to the spectra from measured (Figure 4).
Figure 4 represents wavenumber spectra from measured taken along the ERS-1 altimeter swath
(top panel) and from collocated modeled (bottom panel) of the year 1993. Table 2 contains
the slopes of the wavenumber spectra.
The modeled are derived with the WAM
model on 
grid and with ECMWF model
wind forcing. The spectra are calculated from continuous series
of about 7800 km length. The length of the series was chosen as a
compromise in order to obtain spectral estimates of acceptable
confidence intervals. The meridional separation of the data is 195 km due to an along-swath
averaging. The averaging of from the ERS-1 altimeter is performed to obtain
about the same spatial resolution as the of the WAM model. Figure 4 shows that the
underestimation of the spectral variance of modeled is significant on the 99% confidence
level for scales downward 1200 km.
CONCLUSIONS
Meso-scale wavenumber spectra of ocean surface winds are determined with a large statistical accuracy. From the spectral analysis some interesting conclusions may be drawn.
First, the power-law on wavenumber is independent on the
seasonal variability. The power-laws for the extra-tropical wind
components lies between and 
and for both tropical wind components is obtained. These power-laws and the slight
differences between extra-tropical and tropical winds are in
close agreement with Freilich and Chelton (1986). This result
motivated to study the kinetic energy density on meso-scales.
Second, the global distribution of the spectral variance of
meso-scales is found consistent with one single power-law on
wavenumber with about 
. The finding of a single power-law suggests that a
kind of self-stabilizing flux of kinetic energy is typical for
the atmospheric motions.
Third, the measured energy density of the meridional wind
component is more energetic than of the zonal wind component
which is in contradiction with the isotropic condition. The slope
of kinetic energy density spectrum of ocean near-surface winds
agrees not with the slope of spectrum determined from wind
measurements in the upper atmosphere in previous studies. Motions
in the upper atmosphere have been shown to be well approximated
by the theory of quasi-geostrophic turbulence with for larger scales and by the 2d isotropic
turbulence with for smaller scales.
Forth, wavenumber spectra obtained from winds of the atmospheric ECMWF model indicate a significant underestimation of the spectral energy density for scales downward 1000 km. The wavenumber spectra of the ECMWF have the same energy density for both wind components whereas the scatterometer winds show that the meridional wind component is more energetic than the zonal wind component.
Global atmospheric models exhibit a high degree of confidence in predicting the mean fields and the large-scale distributions. Regionalized model versions often show deficiencies which is manifested by too little variance. On the other hand it is known that turbulent eddies are involved in the transfer of kinetic energy to larger scales and to the main flow. Therefore, an underestimation of the spectral energy density at smaller scales might have impact also on the spectral energy density on larger scales.
The present study clearly supports that ocean wave modeling is a very useful procedure to evaluate winds of an atmospheric model. Conversely, we may expect that the prediction of waves will improve through improvements of the wind modeling.
REFERENCES
Balsley, B.B. & D.A. Carter, 1982, The spectrum of atmospheric velocity fluctuations at 8 km and 86 km. Geophys. Res. Lett., 9, 465-468.
Batchelor, G.K., 1969, Computation of the energy spectrum in homogeneous two-dimensional turbulence. Phys. Fluid, 12 (Suppl. II), 233-239.
Challenor, P.G., 1993, Spatial scales of wave height. Proc. First ERS-1 Symposium, ESA SP-359, 493-498.
Charney, J.G., 1971, Geostrophic turbulence. J. Atmos. Sci., 28, 1087-1095.
Chen, T.-C. & A. Wiin-Nielsen, 1978, On nonlinear cascades of atmospheric energy and enstrophy in a two-dimensional spectral index. Tellus, 30, 313-322.
Freilich, M.H. & D.B. Chelton, 1986, Wavenumber spectra of pacific winds measured by the Seasat scatterometer. Am. Met. Soc., 16, 741-757.
Julian, P.R., W.M. Washington, L. Hembree & C. Ridley, 1970, On the spectral distribution of large-scale atmospheric kinetic energy. J. Atmos. Sci., 27, 376-387.
Kolmogoroff, A.N., 1941, The local structure of turbulence in incompressible viscous fluids for very large Reynolds numbers. C. R. Acad. Sci. URSS, 30, 376-387.
Komen, G.J., L. Calvaleri, M. Donelan, K. Hasselmann, S. Hasselmann, & P. A. E. M. Janssen, 1994, Dynamics and modelling of ocean waves. Cambridge University Press, Cambridge, UK, 560 pp.
Kraichnan, R.H., 1971, Inertial-range transfer in a two- and a three-dimensional turbulence. J. Fluid Mech., 47, 525-535.
Leith, C.E., 1971, Atmospheric predictability and two-dimensional turbulence. J. Atmos. Sci., 28, 145-161.
Lilly, D.K., & E.L. Petersen, 1983, Aircraft measurements of atmospheric energy spectra. Tellus, 35A, 379-382.
Monaldo, F., 1990, Corrected spectra of wind speed and significant wave height. J. Geophys. Res., 95(C3), 3399-3402.
Nastrom, G.D. & K.S. Gage, 1985, A climatology of atmospheric wavenumber spectra of wind and temperature observed by commercial aircraft. J. Atmos. Sci., 42, 950-960.
Keywords: ESA European Space Agency - Agence spatiale europeenne, observation de la terre, earth observation, satellite remote sensing, teledetection, geophysique, altimetrie, radar, chimique atmospherique, geophysics, altimetry, radar, atmospheric chemistry