Understanding Earth Rotation Part 2: Physical Foundations and Interpretation

Understanding Earth Rotation Part 2: Physical Foundations and Interpretation Prof. Dr. Florian Seitz Technische Universität München (TUM) Deutsches Geodätisches Forschungsinstitut (DGFI) Munich, Germany International Summer School on Space Geodesy and Earth System August 21-25, 2012 Shanghai, China 1

Physical Foundations of Earth rotation Earth rotation is a rotational motion of many individual and mutually linked mass elements about one common axis This rotational motion is comparable to that of a physical gyroscope The physical description of Earth rotation is based on equations describing the motion of a gyro 2

Rotational motion of one individual mass element Viewed with respect to a non-rotating reference system: Angular momentum of rotating mass element M: with the relation between angular velocity and track speed v = ω r: with the moment of inertia J = Mr 2 : H = M (r v) H = (Mr 2 ) ω H = J ω Angular momentum is directly proportional to the angular velocity The proportionality constant is the moment of inertia J 4

Conservation of angular momentum Without external forces: Angular momentum is a conserved quantity: H = (Mr 2 ) ω = const. If the distance of the mass element to the axis of rotation decreases, the angular velocity has to increase accordingly. 5

Rigid system of point masses Total angular momentum of a rigid system of N point masses: with v i = ω r i : The moment of inertia is replaced by the tensor of inertia I: H = Σ M i (r i v i ) H = Σ (M i r i2 ) ω H = I ω The tensor of inertia describes the distribution of the mass elements in the system. N i=1 N i=1 6

Conservation of angular momentum & balance of angular momentum In a closed system the total angular momentum is conserved: N H = Σ M i (r i v i ) = H 1 + H 2 + H 3 +... + H N = const. i=1 7

Conservation of angular momentum & balance of angular momentum 8 In a closed system the total angular momentum is conserved: H = Σ M i (r i v i ) = H 1 + H 2 + H 3 +... + H N = const. i=1 If external forces are acting (e.g. gravitation): N N i=1 N i=1 N i=1 balance of total angular momentum H and total external torque L:

Non-rotating reference system For a rotary motion in a non-rotating reference system we have for every arbitrary vector x: (example: v = ω r ) Coordinate axes do not participate in the rotation. We have for the unity vectors e j (j = 1,2,3) : e 1 e 3 e 2 x x ω x = ω x e j = 0 9

Rotating reference system For a rotary motion in a rotating reference system we have also for the coordinate vectors: e j e j ω e j = ω e j 10

Rotating reference system For a rotary motion in a rotating reference system we have also for the coordinate vectors: So we have for every arbitrary vector x j e j + x j e j x j e j + x j ω e j 3 x = Σ x j e j j=1 e j = ω e j x x j e j + ω (x j e j ) = + ω x t 11

Rotating reference system For a rotary motion in a rotating reference system we have also for the coordinate vectors: So we have for every arbitrary vector x j e j + x j e j x j e j + x j ω e j Balance of angular momentum in a rotating reference system: 3 x = Σ x j e j j=1 dh = dt H t e j = ω e j x x j e j + ω (x j e j ) = + ω x t + ω H = L 12

Balance of angular momentum of a rotating solid body Balance of angular momentum in a rotating reference system: Angular momentum of a rotating solid body: + ω(t) H(t) = L(t) H(t) = I ω(t) 13

Balance of angular momentum of a rotating solid body Balance of angular momentum in a rotating reference system: Angular momentum of a rotating solid body: Assumption: Coordinate axes are principal axes: Then I has diagonal structure: (A < B < C: principal moments of inertia of the Earth) + ω(t) H(t) = L(t) H(t) = I ω(t) 14

Balance of angular momentum of a rotating solid body Balance of angular momentum in a rotating reference system: Angular momentum of a rotating solid body: Assumption: Coordinate axes are principal axes: Then I has diagonal structure: ω 1 ω 2 + ω(t) H(t) = L(t) H(t) = I ω(t) with ω = and L = expansion delivers L 1 L 2 ω 3 L 3 15

Balance of angular momentum of a rotating solid body Balance of angular momentum in a rotating reference system: Angular momentum of a rotating solid body: Equations of motion for the rotation of a rigid body in the principal coordinate system: + ω(t) H(t) = L(t) H(t) = I ω(t) A ω 1 + (C-B) ω 2 ω 3 = L 1 B ω 2 + (A-C) ω 3 ω 1 = L 2 C ω 3 + (B-A) ω 1 ω 2 = L 3 16

Euler s Equation of Motion 17 (Leonhard Euler: Painting by Emanuel Handmann, Switzerland) A ω 1 + (C-B) ω 2 ω 3 = L 1 B ω 2 + (A-C) ω 3 ω 1 = L 2 C ω 3 + (B-A) ω 1 ω 2 = L 3 Euler s equation of motion

From the rotating rigid body to a rotating deformable body Balance of angular momentum in the rotating reference system: Angular momentum of a rotating rigid body: Angular momentum of a rotating deformable body: with h(t) as relative angular momentum: N h(t) = Σ M i (r i v rel i ) i=1 + ω(t) H(t) = L(t) H(t) = I ω(t) H(t) = I(t)ω(t) + h(t) The relative angular momentum describes the effect of the motion of mass elements w.r.t. the reference system. 18

From the rotating rigid body to a rotating deformable body Balance of angular momentum in the rotating reference system: Angular momentum of a rotating rigid body: Angular momentum of a rotating deformable body: Euler-Liouville - Equation: + ω(t) H(t) = L(t) H(t) = I ω(t) H(t) = I(t)ω(t) + h(t) (I(t)ω(t) + h(t)) + ω(t) (I(t)ω(t) + h(t)) = L(t) t 19

Balance of angular momentum in the Earth system Model approaches for Earth rotation are based on the balance of angular momentum in an Earth-fixed coordinate system (Euler-Liouville Equation): with angular momentum L(t): I(t): h(t): ω(t): d dt external gravitational torques (Sun, Moon, planets) Earth s tensor of inertia relative angular momenta Earth rotation vector H() t + ω() t H() t = L() t H() t = I() t ω() t + h() t 20

Balance of angular momentum in the Earth system Model approaches for Earth rotation are based on the balance of angular momentum in an Earth-fixed coordinate system (Euler-Liouville Equation): with angular momentum L(t): I(t): h(t): ω(t): d dt external gravitational torques (Sun, Moon, planets) Earth s tensor of inertia relative angular momenta Earth rotation vector y + z x y x z 2 2 N i i i i i i 2 2 i i i i i i= 1 2 2 symm. xi + yi H() t + ω() t H() t = L() t H() t = I() t ω() t + h() t I() t = M x + z y z 21

Balance of angular momentum in the Earth system Model approaches for Earth rotation are based on the balance of angular momentum in an Earth-fixed coordinate system (Euler-Liouville Equation): with angular momentum L(t): I(t): h(t): ω(t): d dt external gravitational torques (Sun, Moon, planets) Earth s tensor of inertia relative angular momenta Earth rotation vector I() t = I + Δ I() t 0 H() t + ω() t H() t = L() t H() t = I() t ω() t + h() t approximate tensor of the Earth (time-invariant) temporal variations due to dynamical processes 22

Balance of angular momentum in the Earth system Model approaches for Earth rotation are based on the balance of angular momentum in an Earth-fixed coordinate system (Euler-Liouville Equation): with angular momentum L(t): I(t): h(t): ω(t): d dt external gravitational torques (Sun, Moon, planets) Earth s tensor of inertia relative angular momenta Earth rotation vector m() 1 t ω() t = Ω m() 2 t 1+ m 3( t) H() t + ω() t H() t = L() t H() t = I() t ω() t + h() t polar motion angular velocity, ΔUT / LOD 26

Excursion: Earth rotation parameters The rotation of the Earth and its temporal variation are monitored by geodetic and astrometric observation systems since decades with very high accuracy. Observations of the orientation of the Earth axis and the angular velocity of the rotation are transformed into time series of Earth rotation parameters (ERP). Polar motion: Location of the axis of the CIP w.r.t. an Earth-fixed reference system IERS C04 CTP Conventional Terrestrial Pole 28

Excursion: Earth rotation parameters 29 The rotation of the Earth and its temporal variation are monitored by geodetic and astrometric observation systems since decades with very high accuracy. Observations of the orientation of the Earth axis and the angular velocity of the rotation are transformed into time series of Earth rotation parameters (ERP). Polar motion: Location of the axis of the CIP w.r.t. an Earth-fixed reference system IERS C04 IERS C04 Length-of-day (LOD): Related to the Earth s angular velocity. LOD = length of a solar day with respect to a nominal day of 86400 s (= 24 hours)

Observation of Earth rotation and its importance The rotation of the Earth and its temporal variation are monitored by geodetic and astrometric observation systems since decades with very high accuracy. Observations of the orientation of the Earth axis and the angular velocity of the rotation are transformed into time series of Earth rotation parameters (ERP). Precise knowledge of temporal variations of ERP is fundamental, a.o. for the realisation of time systems for the highly precise computation of geodetic reference frames in order to relate Earth-fixed and space-fixed coordinate systems for precise navigationon Earth and in space interesting for various disciplines of geosciences, since dynamic processes in the Earth system are reflected in the temporal variations of the ERP Analysis of ERP time series allows for conclusions with respect to processes and changes in the Earth system on various temporal scales 30

Polar motion: Signal characteristics Mainly beat of two oscillations: 365 days (annual oscillation) 434 days (Chandler oscillation) 31

32 Polar motion: Signal characteristics x Component

Polar motion: Chandler oscillation Figure Axis ω Figure axis Rotation axis Free oscillation of rotation axis with respect to an Earth-fixed reference frame Period: Rigid Earth body: Euler-period (304 days) Non-rigid Earth body: Chandler-period (434 days) Free polar motion is superposed by forced polar motion 33

ERP integral quantities of the Earth system Variations are caused by a multitude of superposed effects (e.g. gravitational effects (torques, tides), dynamic processes in the Earth system involving the transport of masses in the Earth system) ERP provide an important information to the balance of angular momentum in the Earth system But: The time series do not allow for conclusions with respect to specific underlying processes without further knowledge The interpretation / separation of observed variations requires independent information from numerical model approaches 35

Effect of dynamic processes in the Earth system Displacement of mass elements temporal changes I(t) of the tensor of inertia mass term Motions of mass elements with respect to the reference system relative angular momentum h(t) motion term [Fig.: S. Böhm] 36

37 Expansion of the Euler-Liouville equation Ω

Expansion of the Euler-Liouville equation Assumptions: rotational symmetry of the Earth around its z-axis (A=B) products of small quantities (c ij, h i, m i ) and their derivatives are negligible A 1 2 m 1 + m2 = L 2 1 + Ω c23 Ωc 13 + Ωh2 h 1 Ω( C A) Ω ( C A) A 1 2 m 2 m1 = L 2 2 Ω c13 Ωc 23 + Ωh1 h 2 Ω( C A) Ω ( C A) 1 m 3 = Ωc33 h3 ΩC Components of the Earth rotation vector as functions of external torques, and mass redistributions and motions that are due to dynamical processes in the Earth system. 38

Expansion of the Euler-Liouville equation Assumptions: rotational symmetry of the Earth around its z-axis (A=B) products of small quantities (c ij, h i, m i ) and their derivatives are negligible A 1 2 m 1 + m2 = L 2 1 + Ω c23 Ωc 13 + Ωh2 h 1 Ω( C A) Ω ( C A) A 1 2 m 2 m1 = L 2 2 Ω c13 Ωc 23 + Ωh1 h 2 Ω( C A) Ω ( C A) 1 m 3 = Ωc33 h3 ΩC =Ψ 2 = Ψ 1 = Ψ 3 excitation functions 39 [Lambeck, 1980]

Expansion of the Euler-Liouville equation Assumptions: rotational symmetry of the Earth around its z-axis (A=B) products of small quantities (c ij, h i, m i ) and their derivatives are negligible A 1 2 m 1 + m2 = L 2 1 + Ω c23 Ωc 13 + Ωh2 h 1 Ω( C A) Ω ( C A) A 1 2 m 2 m1 = L 2 2 Ω c13 Ωc 23 + Ωh1 h 2 Ω( C A) Ω ( C A) A 1 = Ω( C A) σ r 1 m 3 = Ωc33 h3 ΩC (Euler-period = 304 days) 40 [Lambeck, 1980]

Expansion of the Euler-Liouville equation Assumptions: rotational symmetry of the Earth around its z-axis (A=B) products of small quantities (c ij, h i, m i ) and their derivatives are negligible A 1 2 m 1 + m2 = L 2 1 + Ω c23 Ωc 13 + Ωh2 h 1 Ω( C A) Ω ( C A) A 1 2 m 2 m1 = L 2 2 Ω c13 Ωc 23 + Ωh1 h 2 Ω( C A) Ω ( C A) 1 m 3 = Ωc33 h3 ΩC Solution of these equations for m i delivers the Earth rotation vector 41 [Lambeck, 1980]

Simple case: Rigid Earth body and no external forces 42 no variations of the tensor (c ij = 0) and no relative angular momenta (h i = 0) no external torques (L i = 0) m m 1 + m = 1 2 σ r 1 m = 2 1 σ r 0 0 m 3 = 0 A 1 = Ω( C A) σ r Ψ 1, Ψ 2, Ψ 3 = 0 (Euler-period = 304 days) [Lambeck, 1980]

Simple case: Rigid Earth body and no external forces no variations of the tensor (c ij = 0) and no relative angular momenta (h i = 0) no external torques (L i = 0) m m 1 + m = 1 2 σ r 1 m = 2 1 σ r 0 0 m 3 = 0 Ψ 1, Ψ 2, Ψ 3 = 0 no variation of the angular velocity (LOD = constant) 43

Simple case: Rigid Earth body and no external forces no variations of the tensor (c ij = 0) and no relative angular momenta (h i = 0) no external torques (L i = 0) m m 1 + m = 1 2 σ r 1 m = 2 1 σ r 0 0 Ψ 1, Ψ 2, Ψ 3 = 0 Complex formulation: m = m1+ im2 Im(m) x-component m 1 m 2 y-component 44 Re(m)

Simple case: Rigid Earth body and no external forces no variations of the tensor (c ij = 0) and no relative angular momenta (h i = 0) no external torques (L i = 0) m m 1 + m = 1 2 σ r 1 m = 2 1 σ r Solution: 0 0 0 i t m e σ r mt () = Ψ 1, Ψ 2, Ψ 3 = 0 1 i m + m = σ r 0 Undamped circular motion at frequency σ r (1 rev / 304 d) 46

Real case: Various dynamic processes in the Earth system influence Earth rotation Model approach: Development of physically consistent and comprehensive dynamic Earth models for the simulation of the Earth s rotational dynamics 47

Required information for modelling ERP Distribution and motion of mass elements in various components of the Earth system (atmosphere, ocean, hydrosphere, cryosphere, solid Earth, ): variations of the Earth s tensor of inertia relative angular momentum pressure related loading effects Ephemerides of Sun, Moon, external gravitational torques tidal deformations Geometrical, physical, and rheological parameters of the Earth (Earth ellipsoid, mean tensor of inertia, Love numbers, ) Model result is highly dependent on completeness of the effects considered in the model quality and consistency of the applied forcing model parameters 48

49 Forward model for Earth rotation [Seitz et al., GJI, 2004]

Forward model for Earth rotation back-coupling effect of polar motion transition from rotating rigid body to rotating deformable body direct influence on the model s Chandler oscillation 50 [Seitz et al., GJI, 2004]

Rotational deformation Temporal variations of the Earth s centrifugal potential cause redistributions of masses in the solid Earth and in the oceans: 2 3 Ω a ΔC 21( t) = ( R ( k2)m 1( t) +I( k2)m 2( t)) 3GM 2 3 Ω a ΔS 21( t) = ( R( k2)m 2( t) I( k2)m 1( t)) 3GM k = k + Δk + Δk * O A 2 2 2 2 with the pole tide Love number. IERS-Conv. 2010: k 2 = 0.35 + 0.0036i response of the solid Earth and the oceans considering mantle anelasicity 51

Rotational deformation Temporal variations of the Earth s centrifugal potential cause redistributions of masses in the solid Earth and in the oceans: ΔC 21( ) 2 3 Ω a ΔC 21( t) = ( R ( k2)m 1( t) +I( k2)m 2( t)) 3GM 2 3 Ω a ΔS 21( t) = ( R( k2)m 2( t) I( k2)m 1( t)) 3GM with the pole tide Love number. t 21 ΔS ( t) IERS-Conv. 2010: k 2 = 0.35 + 0.0036i and are directly related to variations of the tensor of inertia ( ) c ()= t f ΔC ( t),δs ( t) ij k = k + Δk + Δk * O A 2 2 2 2 21 21 and thus enter into the Euler-Liouville-Equation. 52

Simulated free polar motion x-component Deformability of the Earth s body: Lengthening of the period of the free rotation from 304 to 432 days (Chandler-period) Consideration of mantle anelasticity: dissipation of energy due to friction Damping of the Chandler oscillation Without excitation the free polar motion would diminish within few decades (figure axis = rotation axis) 53

54 Forward model for Earth rotation [Seitz et al., GJI, 2004]

Model forcing Numerical values for ΔI(t) and h(t) are derived from model data of Earth system components, e.g. global atmospheric reanalyses of NCEP - assimilates meteorological observation data - atmospheric angular momentum from wind fields and surface pressure global ocean circulation model ECCO - forced by NCEP fields of wind stress, heat and freshwater fluxes - oceanic angular momentum from currents and water masses consistent representation of dynamics and mass transports in the subsystems atmosphere and ocean water, groundwater and snow fields from the global hydrological model LaD neglected: earthquakes, volcanoes, postglacial uplift, core/mantle, 55

Model results for polar motion x-component: corr.: 0,98; RMS-diff.: 29,5 mas y-component: corr.: 0,99; RMS-diff.: 23,3 mas 56

Model results for polar motion (1950-2010) Model forcing: NCEP + ECCO Full PM C01/C04 Chandler C01/C04 Corr-Coef.: 0.95 RMS-Diff.: 43.7 mas Corr-Coef.: 0.99 RMS-Diff.: 17.3 mas 57 [Seitz et al., JGR, 2012]

Dynamic processes and polar motion ftp://gemini.gsfc.nasa.gov

59 Variations of LOD

Dynamic processes and ΔLOD ftp://gemini.gsfc.nasa.gov

Summary Earth Rotation Parameters contain important information on geodynamic processes since the ERP are directly related to the balance of angular momentum in the Earth system. Angular momentum variations are linked to the motion and redistribution of masses in various components of the Earth system, and therefore a study of ERP allows for conclusions with respect to such processes. Interpretation of observed time series of ERP: Their integrative nature necessitates the development of advanced geophysical models in order to separate observations into individual effects. Current research topics: consideration of further system components in the simulations: can remaining gaps in the balance of angular momentum be closed? predictions on various time scales inverse modeling: Incorporation of ERP as constraints independent determination of mass transport processes from other observations (gravimetrical observations, altimetry, ) 61

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