Chandler wobble: Stochastic &

deterministic dynamicsAlejandro Jenkins

U. de Costa Rica & Academia Nacional de Ciencias

13th International Conference on Dynamical Systems Łódź, Poland

7 December 2015

Introduction• Chandler wobble: torqueless precession of Earth’s

rotational axis

• Main component of ‘latitude variation’ (or ‘polar motion’)

• Predicted by Euler in 1736; reported by Küstner in 1890

• Extensively studied by Chandler after 1891

• Amplitude: α = 0.1’’ - 0.2’’ (3-6 m)

• Period: 433 ± 1 days (~ 14 months)

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Latitude variation

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S R

Zx3pr

Ω

ω

x2x1

• x3 (symmetry axis) intersects Earth’s surface at S

• Ω (angular vel.) intersects at R

• R is instantaneous North pole

• Circle SR is ‘polhole’

Euler’s equations

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Mi = Iij⌦jAngular momentum:

See: Landau & Liftshitz, Mechanics, 3rd ed. (1976), ch. VI

Iij =

Zd3r ⇢(r)

�r2�ij � rirj

�Tensor of inertia:

Eq. of motion: K = M =d0M

dt+⌦⇥M

for d’/dt in body frame

Free precession

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⇢⌦1 = �!Eu⌦2

⌦2 = !Eu⌦1!Eu ⌘ I3 � I

I⌦3

) ⌦3 = const.

K = 0 ; I ⌘ I1 = I2Free symmetric top:

K1 = I1⌦1 + (I3 � I2)⌦3⌦2 ,

K2 = I2⌦2 + (I1 � I3)⌦1⌦3 ,

K3 = I3⌦3 + (I2 � I1)⌦2⌦1

In principal axes:

Chandler & Newcomb• Relative amplitudes of

astronomical (forced) precession & nutation give

• Implies wobble period

• Disagrees with Chandler’s finding of 433 days

• Discrepancy explained by Newcomb (1892)

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Seth Carlo Chandler, Jr.(1846 - 1913)

Simon Newcomb(1835 -1909)

2⇡

!Eu=

2⇡

⌦3

I

I3 � 1= 306 days

I/(I3 � I) = 306

Imperfect rigidity

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O

R

S

x

y

R = (x, y); S = (x0, y0)

O: fixed (average) North Pole

⇢x = �!Eu (y � y0)y = !Eu (x� x0)

semi-rigidity:

S = kw(x, y) ; 0 < kw < 1

With respect to O:

!Ch = !Eu (1� kw)

Imperfect rigidity, cont.

• Earth without ocean ~ 1.2 times as rigid as steel

• Consistent with relative magnitudes of oceanic & body tides (‘Love number’ k2)

• ωCh therefore agrees with free precession

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See: • Klein & Sommerfeld, Theory of the Top, vol. III

(Birkhäuser, 2012 [1903]); • Lambeck, Earth’s Variable Rotation (Cambridge, 1980)

kw = 1� !Ch

!Eu= 1� 306

433= 0.293

Dissipation

• For Earth: Ew = α2 × 1027 J

• Damping: α(t) = α0 e-t/τ

• Quality factor: Q = τ ωCh / 2

• Dissipated power:

• Maintaining α ~ 10-6 requires 108/Q watts

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Pw =Ew !Ch

Q' ↵2

Q⇥ 1020 W

Ew =I

2

✓1� I

I3

◆⌦2

? ' I

2

✓1� I

I3

◆⌦2↵2

Dissipation, cont.• Jeffreys (’40, ’68) estimated τ, supposing wobble re-

excited by stochastic perturbations to mass distribution

• Huge uncertainty: Q ~ 37 - 1000 (τ ~ 14 - 300 yrs)

• Tidal friction gives Q ~ 8,900 (τ ~ 3,400 yrs)

• Mantle inelasticities insufficient by at least ~ 102

• Similar problem if Q estimated as deterministically forced oscillation; see Mandelbrot & McCamy (1970)

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Irregularities

• Rare extinctions followed by phase jumps

• in 1850s, 1920s, 2000s

• Not associated with obvious geophysical events

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Malkin & Miller, Earth Planets Space 62, 943 (2010)

Singular spectrum analysis (SAS)

944 Z. MALKIN AND N. MILLER: CHANDLER WOBBLE: TWO MORE LARGE PHASE JUMPS REVEALED

Fig. 1. Original and filtered PM series used for our analysis, and corresponding spectra. One can see that both types of digital filtering allows us toeffectively suppress the annual signal. The CW signal looks similar in both filtered series. However, some differences can be seen near the ends ofthe interval.

for digital filtering of the PM series.

Singular spectrum analysis (SSA). This method allows usto investigate the time series structure in more detailthan other digital filters. As shown in previous studies,it can be effectively used in investigations of the Earthrotation (see, e.g., Vorotkov et al., 2002; Miller, 2008).

Fourier filtering. We used the bandpass Fourier transform(FT) filter with the window 1.19±0.1 cpy. Such a widefilter band was used to preserve the complicated CW

structure. In the filtered PM series, the amplitude ofthe remaining annual signal is about 0.5 mas, i.e. 0.5%of the original value.

Hereafter we will refer to filtered PM time series as CWseries. Analyzed PM and CW time series and their spec-tra are shown in Fig. 1. We can see two main spectralpeaks of about equal amplitude near the central period ofabout 1.19 yr, and several less intensive peaks in the CWfrequency band. Discussion on its origin, and even reality,

Feedback• Displacement of Earth’s

symmetry axis affects displacement rotational axis (wobble)

• Wobble affects displacement of symmetry axis (non-rigidity)

• Can this produce positive feedback (i.e., self-oscillation)?

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�S ) �R

�R ) �S

O

R

S

x

y

Circulations• In body frame, geophysical fluids carry significant

energy, but no net M

• Path-rigidity of flows lets us write

• for εIij in Eulerian flow coordinates (body frame)

• obeys weighed superposition of eqs. of motion for precessions induced by Iij and εIij

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Itotij = Iij + "Iij

~

~

d0M

dt+⌦⇥M ; Mi =

⇣Iij + "Iij

⌘⌦j

Driving

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Superposing precessions for ε << 1, c << 2π / ωCh :x+ 2c"�(1� �)!2

Eux+ !

2Chx = 0

� ⌘ OS/OR

O

R S

x

y

S~

R = (x, y)

⇢x(t) = �!Eu [y(t)� �y(t� c)]y(t) = !Eu [x(t)� �x(t� c)]

!Eu ⌘ ⌦3(I3 � I)/I

Precession about S only:~S(t) = � (x(t� c), y(t� c))

Finite inertial delay c:

Driving, cont.• Wobble modulates circulations,

due to centrifugal deformation

• Modulation seen as precession of S

• For β > 1 and

resulting force on solid Earth leads wobble

• Heat engine: geophysical fluids as working substance, solid Earth as piston

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~

� ⌘ OS/OR

O

R S

x

y

S~

0 < !Chc < arccos (1/�)

Intermittence• Wobble anti-damped when

• R spirals away from O, until limited by non-linear damping,

• or until S falls inside OR (i.e. β <1)

• Stochastic β(t) has ‘Hopf bifurcation’ near β =1

• Random walk in β slow compared to ωCh

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~

� ⌘ OS/OR

O

R S

x

y

S~

� ⌘ 2c"�(� � 1)!2Eu > !Ch/Q

Outlook• Frède & Mazzega (2000):

“low dimensional unstable deterministic process”;

“strong fluctuations in the wobble stability can be seen from the time series of the local Lyapunov exponents”

• Peaks in power spectrum of atmospheric circulations at ωCh reported, with right phase to drive wobble: Plag (1997), Aoyama & Naito (2001), Aoyama et al. (2003)

• Stochastic intermittence of deterministic self-oscillation may give heavy-tailed distributions in other complex systems; see, e.g., Blanchard, Krueger & Volchenkov (2010)

• This work available at arXiv:1506.02810 [physics.geo-ph]

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