Uncertainty analysis of atmospheric friction torque
on the solid Earth
Haoming Yan
a,*, Yong Huang
a,b,caState Key Laboratory of Geodesy and Earth's Dynamics, Institute of Geodesy and Geophysics, Chinese Academy of Sciences, Wuhan 430077, China
b
Physics Department, School of Science, Wuhan University of Technology, Wuhan 430070, China cCollege of Earth Sciences, University of Chinese Academy of Sciences, Beijing, 100049, China
a r t i c l e i n f o
Article history:Received 11 December 2015 Accepted 1 February 2016 Available online 3 June 2016 Keywords: Uncertainty Wind stress Friction Torque Earth's rotation Annual amplitude Spectrum
a b s t r a c t
The wind stress acquired from European Centre for Medium-Range Weather Forecasts (ECMWF), National Centers for Environmental Prediction (NCEP) climate models and QSCAT satellite observations are analyzed by using frequency-wavenumber spectrum method. The spectrum of two climate models, i.e., ECMWF and NCEP, is similar for both 10 m wind data and model output wind stress data, which indicates that both the climate models capture the key feature of wind stress. While the QSCAT wind stress data shows the similar characteristics with the two climate models in both spectrum domain and the spatial distribution, but with a factor of approximately 1.25 times larger than that of climate models in energy. These differences show the uncertainty in the different wind stress products, which inevitably cause the atmospheric friction torque uncertainties on solid Earth with a 60% departure in annual amplitude, and furtherly affect the precise estimation of the Earth's rotation.
©2016, Institute of Seismology, China Earthquake Administration, etc. Production and hosting by Elsevier B.V. on behalf of KeAi Communications Co., Ltd. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).
How to cite this article:Yan H, Huang Y, Uncertainty analysis of atmospheric friction torque on the solid Earth, Geodesy and Geodynamics (2016), 7, 165e170, http://dx.doi.org/10.1016/j.geog.2016.05.003.
1.
Introduction
To better understanding of the angular momentum transform between solid Earth and atmosphere, ocean, or land hydrology, the torque method are usually involved in the
references[1e5]. Nowadays, the best studied geophysical fluid torque is the atmospheric torque on the solid Earth, while the oceanic torque is only touched by a few researchers (e.g., Fujita et al. [3]), and the land hydrology torque is still not systematically researched at present. For the atmospheric torque on the solid Earth, the studies focus mainly on the
*Corresponding author.
E-mail address:[email protected](H. Yan).
Peer review under responsibility of Institute of Seismology, China Earthquake Administration.
Production and Hosting by Elsevier on behalf of KeAi
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http://dx.doi.org/10.1016/j.geog.2016.05.003
1674-9847/©2016, Institute of Seismology, China Earthquake Administration, etc. Production and hosting by Elsevier B.V. on behalf of KeAi Communications Co., Ltd. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).
balance of global angular momentum in the atmospheric data set and Earth rotation, and the relations between meteorological oscillations such as El Nino-Southern Oscillation (ENSO) and the North Atlantic Oscillation (NAO) etc. and the regional atmospheric torques, and on the diurnal angular momentum budget changes, etc. [2,6e10]. The atmospheric torques can be dividing generally into mountain torque, friction torque, and gravity wave drag torque.
Mountain torque is a function of pressure and orography, which exerted the solid Earth through a difference in pressure across any raised Earth surface. The most significant moun-tain torque is locating in the mounmoun-tains or mounmoun-tain massifs regions. For example, if the pressure over the west slope of the mountain is stronger than that over the east side, it acts to push the Earth to rotate faster and slows the atmosphere rotation down, which imparting angular momentum from the atmosphere to the solid Earth[5,10]. The friction torque is the wind or oceanic current frictional force on the solid Earth surface, which will directly speed or slow down the rotation of solid Earth. If there is a net global eastward surface wind, the atmosphere wind friction force will speed the solid Earth's rotation up, transfer the atmospheric angular momentum to the solid Earth, and thus the atmosphere loses angular mo-mentum. The gravity wave drag torque is part of the mountain and friction torque that is too small to be resolved by present Global Circulation Models (GCMs), due to the nature of coarse resolution of climate models will not resolve the regional/local mountains and mountain-induced waves, and their contri-bution to the mountain torque (e.g., Palmer et al.[11]; Egger et al.[4]). For example, mountains usually have very jagged terrain, there can be turbulence and pressure (small spatial physical process of far below GCMs grid size) applied to the mountain that will not be picked up by the GCMs, thus the gravity wave drag torque was introduced as a way to remedy this[5].
For all three torques mentioned above, the most accurate one is the mountain torque, due to the accurate observation of the surface atmospheric pressure can be acquired from sur-face meteorological stations and GCMs. The last two torques encountered obvious problems. The wind friction stress can not be observed directly in practice, which must be converted from observed wind speed data by using experimental equa-tions with a parameter named as drag coefficient (e.g., Tren-berth et al. [12]; Rao et al.[13]). In general, the wind drag coefficient is the function of wind speed and roughness of the Earth's surface. The roughness definition is very difficult in both land and ocean regions. For example, in the land region, the very jagged terrain, unreachable regions with no observations, and time-variable of the true Earth surface due to rain, snow, runoff etc., will inevitably change the roughness of the friction surface, and make the wind drag coefficient change accordingly. In the ocean region, the status does not get better too. The sea surface state is the function of oceanic wave height and duration, atmospheric friction thickness, air-sea temperature difference, relative humidity at the air-sea interface, ocean surface current, oceanic turbulence, and atmospheric stability, etc. (e.g., Kara et al.[14,15]). All these factors will affect the value of wind drag coefficients to be determined accurately in space and
time domain, and inevitably make the wind stress conversion prone to be contaminated. For the gravity wave drag torque, due to gravity wave drag stress is the compensation of subgrid-scale orography effects of atmospheric gravity wave, it only can be estimated by parameterization scheme in the GCMs under some hypothesis theories (e.g., Lott and Miller [16]; Zhong and Chen[17]). Though the researches on the gravity wave drag have some progress and used in a lot of GCMs such as European Centre for Medium-Range Weather Forecasts (ECMWF), and National Centers for Environmental Prediction (NCEP) atmospheric models, it suffers unquestionable errors due to limited knowledge about the sub-scale atmospheric dynamics and limited comparison between the realistic investigation and the theory simulations. Furthermore, the non-orography gravity wave drag induced by atmospheric convention etc, is still mysterious for us at some degree and need to be clarified to improve the estimation accuracy of gravity wave drag stress.
In this paper, we will compare wind stress of GCMs and QSCAT satellite observations, and estimate the uncertainty of the wind stress field and draw a picture of these uncertainty effects on the calculation of friction torque on the solid Earth in global ocean region.
2.
Method and data
To estimate the uncertainty of atmospheric friction torque, five data sets are used. Three types of 10 m wind speed data acquired from: (1) ECMWF ERA-interim data with 1.5grid in both latitude and longitude, respectively (http://apps.ecmwf. int/datasets/data/interim_full_moda/); (2) NCEP reanalysis data with 1.875grid in both longitude and latitude, respec-tively (http://www.esrl.noaa.gov/psd/data/gridded/data.ncep. reanalysis.html); (3) QSCAT satellite observed wind speed with original 0.25 spatial grid (http://www.remss.com/ missions/qscat). Due to the friction torque calculation needs wind stress but not wind speed, we transfer the surface 10 m wind dataVto surface wind stress
s
by using experimental equation presented by Trenberth et al.[12], as follows
¼ ðse;s
nÞ ¼rCdjVjðu;vÞ (1) 103C d¼ 8 < : 0:49þ0:065jVj for jVj>10 m=s 1:14 for 3 jVj 10 m=s 0:62þ1:56jVj1 for jVj<3 m=s (2) where V is the wind vector with west-east component u (positive east) and north-south componentv(positive north), r¼1.3 kg/m3is the density of dry air,Cdis the drag coefficients which can be calculated from equation(2), ands
eands
nare the average east-west and north-south wind stress, respectively.Two types of direct model output wind stress data are also acquired from ECMWF interim data and NCEP reanalysis models with the same spatial resolution as the 10 m wind data. All these five wind stress data are then averaged to monthly interval from Jan. 2000 to Nov. 2009. The QSCAT data are available only in the most part of the ocean, thus we mask the NCEP and ECMWF climate model data to the same spatial
distribution as the QSCAT data. The total spatial area of QSCAT data occupies 46% of the whole Earth surface.
To examine the wind stress data in detail, all the above wind stress data are expand to time variable spherical har-monic coefficients (SHCs) with corresponding maximum de-gree/order (d/o), as,
Clm Slm ¼ 1 4p∬s
s
ð4;lÞPlmðsin4Þ cosml sinml ds (3)where,CnmðtÞandSnmðtÞare the expansion SHCs for the wind stress
s
(4,l,t);nandmare the harmonic degree and order;4 and l are latitude and longitude; Pnmðsin4Þ are the fully normalized associated Legendre functions.Furthermore, to analyze the different spatial scales char-acteristics of wind stress, the degree variance spectrumsn(or power per degreen) of a specific degreenover ordersm(m¼0, 1, 2,…,n), can be computed by sn¼ Xn m¼0 C2nmþS2nm (4)
The shortcoming of equation (4) is that it has no consideration of time variable of SHCs. We then use the global frequency-wavenumber spectrum method (equation (11) therein) presented by Wunsch and Stammer[18], to get the new degree variance spectra of wind stress which takes the time variable of SHCs into account.
Finally, we calculate the friction torque of wind stress on solid Earth expressed by polar motion (x,y) and LOD change (z) components as, Tx;y¼ R3 Z2p 0 Zp=2 p=2 ð
s
esin4þis
nÞcos4eildld4 (5) Tz¼ R3 Z2p 0 Zp=2 p=2s
ecos24dld4 (6)whereRis the average radius of the Earth from its center.
3.
Results
3.1. Uncertainties of five wind stress data
We first compare the 10 m wind speed data among two climate models, i.e., ECMWF and NCEP, and satellite observed QSCAT data. The two climate models monthly 10 m wind speed range from10 m/s to 10 m/s for year 2000, which is only half of the observed satellite QSCAT 10 m wind data. To see the spatial spectrum characteristics of these large different 10 m wind speed data, we calculate the frequency-wavenumber spectrum of five wind stress data (Fig. 1). The main energy of all the wind stress data is focus below SHCs degree 11, which means that the wind stress energy is mainly kept in the long-wave spatial scale. Other middle- and short-long-wave scale energy decrease as exponent rule. The ECMWF wind stress includinguandvcomponents have the similar wavenumber spectrum with that of corresponding NCEP terms, which indicate that both climate models capture the same characteristics of 10 m wind stress (Fig. 1). But if we compare the 10 m wind stress calculated by experiment equation of Trenberth et al.[12]with the direct model output wind stress results for the two climate models, we see about a ten times differences. This may mainly owing to the direct model output wind stress data are calculated with more complex
Fig. 1eWavenumber spectrum of wind stress from QSCAT, ECMWF and NCEP data (
se
ECMWF_10mandse
ECMWFmeans eastwardwind stress converted from 10 m wind speed and direct wind stress output from ECMWF model, respectively. Subscript n means northward wind stress.).
and accurate parameters, such as the consideration of seaeair interface status etc. (e.g., Kara et al.[14,15]), in the climate models running process.
Although the direct model output wind stress is ten times larger than 10 m wind stress converted by experiment equa-tions (1) and (2), it is still some smaller than our satellite observed QSCAT wind stress transferred from 10 m wind speed with the same experiment equation (Fig. 1). Meanwhile, we note that the spatial wavenumber spectrum characteristics are similar for both climate models and satellite observed data. This indicates that the wind stress data have large uncertainties in the climate models for both the original 10 m wind data itself and for the method used to transfer wind speed data to wind stress. Except that, Chelton and Freilich[19]point out that the uncertainties of QSCAT wind speed and direction are 1.7 m/s and less than 14, respectively. Thus, the direct model output wind stress may surfer errors from both model input driving data, such as ground and satellite based wind vector observations, and inner conversion process of wind speed to wind stress too. This large uncertainty in the wind stress data will inevitably transfer these uncertainties to the friction torque on the solid Earth which based on the wind stress data deeply. Furthermore, we here only check the wind stress uncertainties in most of the ocean area, but not consider the case in the land region. Due to the complex and roughness of land surface in mountain regions, pole regions, and lack of experimental coefficients to convert the wind speed to wind stress in some unreachable regions, etc., the wind stress data in these land regions are prone to be more seriously over- or under-estimated and will bring considerable uncertainties in friction torque on the solid Earth too.
To show the wind stress differences clearly between the climate models and the satellite observations in spatial re-gions, we calculate the standard deviation (std) of wind stress data in each spatial grid (120 d/o SHCs data, which corre-sponds to1.51.5 space grid) for year 2000e2009 (Fig. 2).
Considering the similar characteristics between ECMWF and NCEP climate models, we hereafter will only show ECMWF climate model results. For the QSCAT wind stress
s
e component, the largest wind stress std variations (std can be regarded as the annual variations approximately here because it is the key signal of wind stress) occur in the north-west Pacific, north-west Atlantic and North-West Indian Ocean regions. The second largest wind stress std variations are almost banded in the southern hemisphere at latitude band of 30S to 55S degree (Fig. 2a). Similar spatial distributions of ECMWF std ofs
e component are shown inFig. 2c, while the std value of ECMWF wind stress
s
e can generally explain 80% of QSCAT results in most ocean regions. For the QSCAT wind stresss
n component, the largest wind stress variations are located in the North Pacific, North Atlantic, Southern Pacific, and North-west Indian Ocean regions (Fig. 2b). Thes
nvalues of ECMWF can only capture some of the characteristics of QSCAT results and have less similarity to compare with thes
ecase (Fig. 2d). Although mosts
nof ECMWF can explain almost 80% of the std variations of QSCATs
n in global ocean region, these region is always not the largest wind stress occurred region as mentioned above (figure not shown). This indicates that for the wind stress data, the consistency ofs
e component between climate models and QSCAT result is better than that ofs
n component at some degree, especially in the spatial distributions.3.2. Comparisons of friction torques on solid Earth To estimate the wind stress uncertainties effect on the friction torques, we use direct model output wind stress data from ECMWF and NCEP climate models, and QSCAT wind stress data, to calculate the wind stress friction torques on solid Earth for polar motion and length-of-day (LOD) change terms from 2000e2009 (Fig. 3,Table 1). The two climate model results accord with each other perfectly for all three wind stress friction torque terms. But there are large differences
Fig. 2eStandard deviation (std) of QSCAT and ECMWF wind stress data (a and b are eastward and northward wind stress of QSCAT; c and d are eastward and northward wind stress of ECMWF.).
between the QSCAT result and that of the two climate models, although all the wind stress friction torques show obvious annual terms. For polar motion X and Y component, the QSCAT annual amplitude are 6.5 and 5.3 Hadley unit (1Hadley unit¼ 1018kg$m2/s2), which are 86% and 160% of that of climate models, respectively. While for the LOD change component, the corresponding value is 8.6 Hadley, which is 160% of that of the climate models (Table 1). This shows that the observed QSCAT annual wind stress friction torque is about 1.6 times larger than the climate model results for polar motion Y component and LOD change component, which is a large uncertainties in the calculation of wind stress friction torques. If we take QSCAT results as a criterion, and view it from the annual amplitude of the polar motion Y component and LOD change term, both ECMWF and NCEP climate models largely underestimate the wind stress friction torques. While for polar motion X component, the climate models results overestimate the wind stress
torque by about 15%. Furthermore, all three results annual phases accord with each other perfectly (Table 1).
To further examine the wind stress friction torque un-certainties among QSCAT, ECMWF, and NCEP for a broad band (not only the annual frequency), we calculate the mean and standard deviation of absolute value of each two datasets in
Fig. 3, and show the results inTable 2. We find that the mean uncertainties between climate model results and that of QSCAT are about 2.0 and 3.8 Hadely for polar motion and LOD change terms, respectively, which are large enough in the excitation of friction torque and cannot be neglected. Meanwhile, the uncertainties between the two climate models are somewhat small, though the difference value itself are still noticeable.
4.
Conclusion
From the above comparisons, we find that ECMWF and NCEP climate models have limited ability to capture the large changes of wind stress in mid-latitude region (i.e. 30e60). Generally speaking, both the climate models underestimate the wind speed and stress when compare with the corre-sponding satellite observed QSCAT data, which is previously noted by Rao et al. [13] too. This reminds us that the uncertainties of wind stress are not negligible because not only there is no direct method to observe wind stress, but also the conversion from wind speed to wind stress can produce large uncertainties for using of different experimental equations and less accurate determined drag coefficients in equations (1) and (2). These uncertainties inevitably transfer to the wind friction torque on the solid Earth up to 60% differences, and make the friction torque precision less than the angular momentum results used in the Earth's rotation research traditionally. Thus further improvement of wind stress data quality and accuracy by using more precise observation data, and more robust methods, such as consideration of the time-variable airesea interaction and land surface dynamic process, may enhance
Fig. 3eWind stress friction torques for QSCAT, ECMWF
and NCEP data.
Table 1eAnnual amplitude and phase of wind stress friction torques for QSCAT, ECMWF and NCEP.
Term Polar motion X Polar motion Y LOD change
Amp. (Hadley) Ph. (deg.) Amp. (Hadley) Ph. (deg.) Amp. (Hadley) Ph. (deg.) QSCAT 6.5±0.5 168±4 5.3±0.5 186±5 8.6±0.7 6±4 ECMWF 7.6±0.4 177±3 3.1±0.4 194±7 5.1±0.6 14±6 NCEP 7.5±0.5 176±3 3.4±0.4 194±7 5.3±0.5 13±6
Table 2eWind stress friction torques uncertainties
among QSCAT(Q), ECMWF(E) and NCEP (N) dataset (unit: Hadely).
Uncertainty term Polar motion LOD change
X Y
Abs(Q-E) 1.8±1.5 2.2±1.7 3.3±2.5 Abs(Q-N) 2.0±1.5 2.2±1.7 4.3±2.6 Abs(E-N) 1.4±0.6 0.9±0.5 2.9±1.1
the precision of geophysical fluids torque on solid Earth and benefit the research on the Earth's rotation.
Acknowledgment
This research is supported by research projects of National Basic Research Program of China (2012CB957802), the National Natural Science Foundation of China (41321063, 41374087), and Open Fund of the State Key Laboratory of Geodesy and Earth's Dynamics (2014-2-1-E).
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Haoming Yan, Ph. D, Researcher at Institute of Geodesy and Geophysics, Chinese Acad-emy of Sciences. His interests include time-variable gravity theory and application, Earth's rotation, geophysical explanation of climate change, GPS time series analysis, climate oscillation, data processing.
