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Estimating the Period and Q of the Chandler Wobble
from Observations and Models of its Excitation
Richard S. Gross
Jet Propulsion Laboratory, California Institute of Technology, Pasadena, USA

Jolanta Nastula
Space Research Centre, Polish Academy of Sciences
Abstract. Any irregularly shaped solid body rotating about some axis that is not aligned with its figure axis will freely wobble as it rotates. For the Earth, this free wobble is

known as the Chandler wobble in honor of S. C. Chandler, Jr. who first observed it in 1891. Unlike the forced wobbles of the Earth, such as the annual wobble, whose periods are the same as the periods of the forcing mechanisms, the period of the free Chandler wobble is a function of the internal structure and rheology of the Earth, and its decay time constant, or quality factor Q, is a function of the dissipation mechanism(s), like mantle anelasticity, that are acting to dampen it. Improved estimates of the period and Q of the Chandler wobble can therefore be used to improve our understanding of these properties of the Earth. Here, estimates of the period and Q of the Chandler wobble are obtained by finding those values that minimize the power within the Chandler band of the difference between observed and modeled polar motion excitation spanning 1962-2010.


Introduction
· Polar motion
· Earth not rotating about figure axis
· So, Earth wobbles as it rotates

Data Sets
· Observed polar motion variations
· COMB2010 combined EOP series
· Combination of optical astrometric, LLR, SLR, VLBI, & GPS observations · Polar motion rate observations not used (contaminated by tidal artifacts) · Daily values at midnight spanning January 20, 1962 to July 15, 2011
where:

Estimation Strategy
· Polar motion equation:
p (t ) + i #p = $( t ) "cw #t
polar motion p(t) ! xp(t) ­ i yp(t ) excitation function (t) !
cw 1

· Forced wobbles
· Forced by changes in relative motion (winds and currents), surficial loading processes, glacial isostatic adjustment, etc. · Frequency of wobble same as frequency of forcing mechanism

(t) + i

2

(t)

· Helmholtz Centre Potsdam ­ GFZ
· Consistent estimates of AAM, OAM, & HAM computed at GFZ
· AAM computed from European Centre for Medium-Range Weather Forecasts · OAM computed from Ocean Model for Circulation and Tides (OMCT) · HAM computed from Land-Surface Discharge Model (LSDM) · Ocean and hydrology models driven by ECMWF fields · Global atmosphere/oceans/hydrology mass conservation imposed

is complex-valued frequency of Chandler wobble

· Free wobbles
· Chandler wobble of 14-month period · Nearly Diurnal Free Wobble (Free Core Nutation) of retrograde diurnal period · Unlike forced wobbles, frequencies of free wobbles depend on Earth's interior structure and dissipation processes

· Recover
·

cw

using

· Observed values of polar motion p(t) and polar motion rate d p/dt
Polar motion rate d p/dt determined using Kalman filter

· Models of excitation functions
· Atmospheric, oceanic a nd hydrologic angular momentum

· Estimate period and Q of Chandler wobble
· Observed polar motion and polar motion rate estimates · Modeled atmospheric, oceanic, and hydrologic excitation · During 1962­2010

· Frequency domain
· Isolate Chandler band · Minimizes effects of polar motion measurement errors · Find that value of cw which minimizes the difference in power betwe en observed and modeled excitation functions (Furuya and Chao, 1996)
· Observed excitation functions computed using polar motion equation specifying different values for Chandler period and Q

· Merge AAM, OAM, & HAM from ERA-40 / ERA-Interim
· Adjust bias of ERA-40 series to agree with that of ERA-Interim series · Merged series spans January 1, 1958 to December 31, 2010 at daily intervals

Sensitivity to Data Length
GFZ ECMWF AAM OMCT OAM LSDM HAM!

Sensitivity to Excitation Model
GFZ ECMWF AAM OMCT OAM LSDM HAM

1962­2010 1971­2010 1981­2010 1991­2010 2001­2010!

AAM AAM+OAM AAM+OAM+HAM

Preferred estimate: T = 431.3 days Q = 117

Independent Excitation Model
· Oceanic angular momentum
· ECCO/JPL 50-year simulation
· 10-day values at midnight spanning 1949-2002 · Sum of current and bottom pressure terms

Sensitivity to Data Length
NCEP Rean AAM ECCO/JPL OAM

Sensitivity to Excitation Model
NCEP Rean AAM ECCO/JPL OAM

· ECCO/JPL data assimilating model (kf080)
· Hourly values spanning January 1, 1993 to present · Averaged to 10-day values at midnight · Sum of current and bottom pressure terms
1962­2010 1971­2010 1981­2010 1991­2010 2001­2010! AAM AAM+OAM

· Merge 50-year and kf080 series
· Adjust bias of 50-year series to agree with that of kf080 series · Merged series spans 1949 to present at 10-day intervals

· Atmospheric angular momentum
· NCEP/NCAR Reanalysis
· 6-hour values spanning January 1, 1948 to present · Averaged to 10-day values at midnight · Sum of wind and inverted barometer pressure terms
Preferred estimate: T = 431.9 days Q = 111

Estimates of Chandler Frequency!

Summary
· Estimate period and Q of Chandler Wobble
· From polar motion and polar motion rate observations
· COMB2010 combination of astrometric, LLR, SLR, VLBI, & GPS observations

· And models of atmospheric, oceanic, & hydrologic excitation
GFZ ECMWF Preferred estimate: T = 431.3 days Q = 117 NCEP & ECCO/JPL Preferred estimate: T = 431.9 days Q = 111

· Consistent ECMWF AAM, OMCT OAM, and LSDM HAM from GFZ · NCEP/NCAR Reanalysis AAM and ECCO/JPL OAM

· Estimated values
· Sensitive to accuracy of modeled excitation
· For longest data span, best fit to observations in Chandler band with AAM+OAM

· Sensitive to length of data analyzed
· Most consistent estimates obtained using longest data spans
GFZ ECMWF Preferred estimate: T = 431.3 days Q = 117 NCEP & ECCO/JPL Preferred estimate: T = 431.9 days Q = 111

Gross (2014)

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