Gross, R. S., Earth Roation Variations – Long Period, in Physical Geodesy, edited by T. A. Herring, Treatise on Geophysics, Vol. 11, Elsevier, Amsterdam, in press, 2007. 3.11 Earth Rotation Variations – Long Period Richard S. Gross Jet Propulsion Laboratory, California Institute of Technology, Pasadena, CA, USA ____________________________________________________________________________________________________________________________ 3.11.1 INTRODUCTION 1 3.11.2 THEORY OF EARTH ROTATION VARIATIONS AT LONG PERIODS 2 3.11.2.1 Instantaneous Rotation Vector 2 3.11.2.2 Celestial Intermediate Pole 7 3.11.3 EARTH ROTATION MEASUREMENT TECHNIQUES 10 3.11.3.1 Lunar Occultation 11 3.11.3.2 Optical Astrometric 12 3.11.3.3 Space-Geodetic 13 3.11.3.3.1 Very long baseline interferometry 13 3.11.3.3.2 Global navigation satellite system 13 3.11.3.3.3 Satellite and lunar laser ranging 14 3.11.3.3.4 Doppler orbitography and radio positioning integrated by satellite 15 3.11.3.4 Ring Laser Gyroscope 15 3.11.3.5 Intertechnique Combinations 16 3.11.4 OBSERVED AND MODELED EARTH ROTATION VARIATIONS 17 3.11.4.1 UT1 and Length-of-Day Variations 17 3.11.4.1.1 Secular trend, tidal dissipation, and glacial isostatic adjustment 18 3.11.4.1.2 Decadal variations and core-mantle interactions 20 3.11.4.1.3 Tidal variations and solid Earth, oceanic, and atmospheric tides 22 3.11.4.1.4 Seasonal variations 26 3.11.4.1.5 Interannual variations and the El Niño/Southern Oscillation 27 3.11.4.1.6 Intraseasonal variations and the Madden-Julian Oscillation 28 3.11.4.2 Polar Motion 29 3.11.4.2.1 True polar wander and glacial isostatic adjustment 30 3.11.4.2.2 Decadal variations, the Markowitz wobble, and core-mantle interactions 31 3.11.4.2.3 Tidal wobbles and oceanic and atmospheric tides 33 3.11.4.2.4 Chandler wobble and its excitation 34 3.11.4.2.5 Seasonal wobbles 35 3.11.4.2.6 Nonseasonal wobbles 37 REFERENCES 39 ____________________________________________________________________________________________________________________________ 3.11.1 INTRODUCTION of hydromagnetic motion, a mantle both thermally convecting and rebounding from the The Earth is a dynamic system—it has a fluid, glacial loading of the last ice age, and mobile mobile atmosphere and oceans, a continually tectonic plates. In addition, external forces due to changing global distribution of ice, snow, and the gravitational attraction of the Sun, Moon, and water, a fluid core that is undergoing some type planets also act upon the Earth. These internal Gross, R. S., Earth Roation Variations – Long Period, in Physical Geodesy, edited by T. A. Herring, Treatise 2 on Geophysics, Vol. 11, Elsevier, Amsterdam, in press, 2007. dynamical processes and external gravitational because the distribution of the Earth’s mass is forces exert torques on the solid Earth, or changing: displace its mass, thereby causing the Earth’s rotation to change. L(t) = h(t) + I(t)⋅ω(t) (2) Changes in the rotation of the solid Earth are studied by applying the principle of conservation Combining equations (1) and (2) yields the of angular momentum to the Earth system. Under Liouville equation: this principle, the rotation of the solid Earth€ changes as a result of: (1) applied external ∂ h(t) + I(t)⋅ω(t) + torques, (2) internal mass redistribution, and (3) ∂t [ ] the transfer of angular momentum between the solid Earth and the fluid regions with which it is ω(t)×[h(t) + I(t)⋅ω(t)] = τ (t) (3) in contact; concomitant torques are due to hydrodynamic or magneto-hydrodynamic stresses€ The external torques acting on the Earth due to acting at the fluid/solid Earth interfaces. the gravitational attraction of the Sun, Moon, and Here, changes in the Earth’s rotation that occur planets€ cause the Earth to nutate and precess. on time scales greater than a day are discussed. Since the nutations and precession of the Earth Using the principle of conservation of angular are discussed in chapter 12 of this volume, the momentum, the equations governing small external torques τ(t) in equation (3) will be set to variations in both the rate of rotation and in the zero. Note, however, that tidal effects on the position of the rotation vector with respect to the Earth’s rotation, which are also caused by the Earth’s crust are first derived. These equations gravitational attraction of the Sun, Moon, and are then rewritten in terms of the Earth rotation planets, is discussed here in Sections 3.11.4.1.3 parameters that are actually reported by Earth and 3.11.4.2.3. rotation measurement services. The techniques The Earth’s rotation deviates only slightly from that are used to monitor the Earth’s rotation by a state of uniform rotation, the deviation being a the measurement services are then reviewed, a few parts in 108 in speed, corresponding to description of the variations that are observed by changes of a few milliseconds (ms) in the length these techniques is given, and possible causes of of the day, and about a part in 106 in the position the observed variations are discussed. of the rotation axis with respect to the crust of the Earth, corresponding to a variation of several hundred milliarcseconds (mas) in polar motion. Such small deviations in rotation can be studied 3.11.2 THEORY OF EARTH ROTATION by linearizing equation (3). Let the Earth initially VARIATIONS AT LONG PERIODS be uniformly rotating at the constant rate Ω about 3.11.2.1 Instantaneous Rotation Vector the z-coordinate axis of the body-fixed reference frame and orient the frame within the Earth in In a rotating reference frame that has been such a manner that the inertia tensor of the Earth attached in some manner to the solid body of the is diagonal in this frame: Earth, the Eulerian equation of motion that relates changes in the angular momentum L(t) of ω o = Ω zˆ (4) the Earth to the external torques τ(t) acting on it is (Munk and MacDonald, 1960; Lambeck, 1980, A 0 0 1988; Moritz and Mueller, 1988; Eubanks, 1993): € Io = 0 B 0 (5) ∂L(t) + ω(t)× L(t) = τ (t) (1) 0 0 C ∂t where the hat denotes a vector of unit length, Ω is where, strictly speaking, ω(t) is the angular the mean angular velocity of the Earth, and A, B, velocity of the rotating frame with respect€ to and C are the mean principal moments of inertia € inertial space. But since the rotating frame has of the Earth ordered such that A < B < C. In this been attached to the solid body of the Earth, it is initial state, in which all time-dependent also interpreted as being the angular velocity of quantities vanish, the Earth is rotating at a the Earth with respect to inertial space. In constant rate about its figure axis, there are no general, the angular momentum L(t) can be mass displacements, and there is no relative written as the sum of two terms: (1) that part h(t) angular momentum. So, for example, the due to motion relative to the rotating reference atmosphere, oceans, and core are at rest with frame, and (2) that part due to the changing inertia tensor I(t) of the Earth which is changing Gross, R. S., Earth Roation Variations – Long Period, in Physical Geodesy, edited by T. A. Herring, Treatise 3 on Geophysics, Vol. 11, Elsevier, Amsterdam, in press, 2007. respect to the solid Earth and merely co-rotate 1/ 2 1 ∂my (t) A (C − A) with it. − m (t) = σ ∂t B (C − B) x Now let this initial state be perturbed by the r appearance of mass displacements and relative 1/ 2 A 1 ∂φr,y (t) angular momentum. In general, since the crust − + φr,x (t) (9) and mantle of the Earth can deform, they can B Ω ∂t undergo motion relative to the rotating reference€ frame and hence can contribute to the relative 1 ∂m (t) 1 ∂φr,z (t) z = − (10) angular momentum. However, let the body-fixed Ω ∂t Ω ∂t reference frame in the perturbed state be oriented€ in such a manner that the relative angular where the external torques have been set to zero, momentum due to motion of the crust and mantle vanishes. In this frame, which is known as€ the C A C B σ 2 = − − Ω 2 (11) Tisserand mean-mantle frame of the Earth r A B (Tisserand, 1891), the motion of the atmosphere, oceans, and core have relative angular momentum, but the motion of the crust and and the φr,i(t), known as excitation functions, are: mantle does not. € In the Tisserand mean-mantle frame, the hx (t) + Ω ΔI xz (t) φr,x (t) = (12) perturbed instantaneous rotation vector and Ω (C − A) (C − B) inertia tensor of the Earth can be written without loss of generality as: hy (t) + Ω ΔI yz (t) φr,y (t) = (13) ω(t) = ωo + Δω(t) € Ω (C − A) (C − B) = Ω zˆ + Ω m (t) xˆ + m (t) yˆ + m (t) zˆ (6) 1 [ x y z ] φ (t) = h (t) + Ω ΔI (t) (14) r,z C Ω [ z zz ] € I(t) = Io + ΔI(t) Equations (8) and (9) are coupled, first-order € differential equations that describe the motion of A 0 0 ΔI xx (t) ΔI xy (t) ΔI xz (t) € the rotation pole in the rotating, body-fixed = 0 B 0 + ΔI xy (t) ΔI yy (t) ΔI yz (t) (7) reference frame as it responds to the applied 0 0 C ΔI xz (t) ΔI yz (t) ΔI zz (t) excitation.