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Numerical-analytical modeling of the Earth's pole oscillations
Moscow Aviation Institute, Russia Authors: Markov Yu. G., Filippova A.S. Presented by: Filippova A.S


Achievements
· Refined model based on the celestial-mechanics methods is presented that is a generalized version of the previously developed basic model · Subtle effects are qualitatively explained based on the amplitude-frequency analysis and the computer numerical modelling of the oscillatory pole motion, where the key feature is the Chandler wobble

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Basic Model of the Polar Oscillations
Model of the polar oscillations is developed based on the gravitationaltidal mechanism:
c s c x p cx ( ) ax cos(2 N ) ax sin(2 N ) Nd x cos(2 ) d xs sin(2 ) s c s y p c y ( ) a c cos(2 N ) a y sin(2 N ) Nd x cos(2 ) d y sin(2 ) y

N 0.84 0.85
Model features: c,s s ,c c,s s ,c ·Coefficients should approximately match as ax ay , d x d y ·Parameters are prone to significant changes due to the inertia tensor perturbations. ·Factors like mechanical and physical planet parameters like large-scale natural events in the ocean and atmosphere are not taken into account ·During the anomaly periods the interpolation and forecast isn't sufficient.
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Generalized model requirements
· Consist of small amount of parameters · Agree qualitatively and quantitatively with previously developed basic model · Has the same structure features as the basic model · Averaging dynamic parameters correspond to the basic model parameters · Working with the geopotential in general
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Geopotential analysis
Earth figure ­dynamic geoid due to the inertia tensor variations. The additional perturbing potential W appears. Most significant component ­ perturbations caused by polar oblateness (second perturbing harmonic)
2 fmE RE W2 Y2 ( , ), 3 r fmE 3.98600442 1014 3 / c 2 ,

Normal spherical function

RE 6.38 106
Normal function variations:
Normal associated Legendre functions

Y2 ( , ) c20 P20 (cos ) c21 cos s21 sin P21 (cos ) c22 cos 2 s22 sin 2 P22 (cos )
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Polar coordinates differential equations: amplitude-phase form
Sometimes convenient to transform polar coordinates to amplitude-phase form:

x p cx a cos ,

y p cy a sin

From the dynamic Euler-Lliouville equations the differential equations for the amplitude and phase of the polar oscillations are derived taking into account the geopotential expansion
2 2mE RE * C * a r0 c22 1 * c22 a sin 2 p cos q sin , A* B N q cos 2 N p sin 2 a 1 q cos p sin .

Coefficients for the Chandler C * B* C B Np r0 , N q A* A External perturbations of the

wobble frequency: C * A* C A r0 B* B gravitational-tidal and geophysical nature:



p ,q



ch p ,q



h p ,q

Chandler and annual components

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Chandler wobble
Amplitude and phase of the Chandler wobble

0 va a ach ach ach r t , , ch c0h N *t ( C , c20 )dt cvh r N Chandler wobble for p, q components of the angular velocity
pch
0 ach cos ch

t, , N



r0 , qch

0 ach sin

ch

ch



r0 .

ch

For example, equations for p(t) of the perturbed motion of the Earth pole

p(t ) pch pch ph , pch ,h f
ch , h p

sin

ch



ch



cos

ch



f

ch , h p

(t )dt

cos

ch

ch



ch



sin

ch



f

ch , h p

(t )dt ,

ch

p N p 1 ch q ch p .

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Numerical modelling of perturbing factors
ch N ( C , c20 )
* var ch

t, N

Variations of the second zonal harmonic c20 of the geopotential: interpolation 1984-2008 yy., forecast 2009-2014 yy. comparing with SLR data
N *

N * frequency variation of
the perturbed Chandler wobble of the Earth pole in 1990-2014 yy.

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Generalized model
Chandler and annual wobble amplitude
acph,,qh r0 a
0 ch

pch f



ch , h p,q

(t )dt qch f

2

ch , h p,q

(t )dt

1 22



p ,q Equations for polar coordinates assuming ach,h ach,h

0 x p cx ach cos( c0h N *t ) ah cos( h ht ) 0 y p c y ach sin( c0h N *t ) ah sin( h ht )

( C , c20 )dt For the numerical modelling purposes transform to:
c s c x p cx ( ) ax ( ) cos 2 ch ( ) ax ( ) sin 2 ch ( ) Nd x cos 2 d xs sin 2 s c s y p c y ( ) a c ( ) cos 2 ch ( ) a y ( ) sin 2 ch ( ) Nd y cos 2 d y sin 2 y

ch ( ) N * ( ) ( ); N * 0.843 cycle/year
c,s s ,c c,s s ,c Apply structure features of the basic model: ax ( ) ay ( ), d x ( ) d y ( )

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Phase variations of the Earth Pole
Earth oscillations

Residuals

Polar phase variations (interpolation) without linear component and residuals. Dots ­ IERS observations, dashed curve­ basic model, solid line ­ generalized model.
var

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Numerical modelling of the polar coordinates variations
Interpolation for 1990-2012 yy. and forecast for 2013-2014 yy. Basic model ­ dashed curve, generalized model ­ solid curve. Dots for observation data from IERS Polar oscillations Residuals Mean square deviations for interpolation for the models: main and generalized.

x* 44.3067 x 24.1476, * 43.3290 y 20.2541, y x*y 61.9716, xy 31.5173.
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Forecast error analysis
Mean square deviation for the series of the annual forecasts of the Earth pole coordinates and polar trajectory. Gray line - for basic model, black line ­ for the generalized model. Dots for the iterations (fixing Chandler wobble value at the end of the interval). Basic model gives sufficient forecast during the stable times (Chandler frequency is a constant) - 2004-2005 yy. When anomaly occurs, generalized model gives much more accurate forecast ­ 20062009, 2012-2013

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Thank you for your attention!