Day length fluctuations
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A mean solar day contains 86,400 seconds. Exact measurements of time by atomic clocks and satellite laser ranging have revealed that this length of day (LOD) is not constant; i.e., the Earth's rotation varies. Short term fluctuations of the order of weeks to a few years have been observed. They are attributed to interactions between the dynamic atmosphere and the Earth.
Introduction
In the absence of external torques, the total angular momentum of the Earth as a whole system must be constant. Internal torques are due to relative movements and mass redistribution of the core of the earth, mantle, crust, oceans, atmosphere, and cryosphere. In order to keep the total angular momentum constant, a change of the angular momentum of one region must necessarily be balanced by angular momentum changes in the other regions.
Crustal movements (such as continental drift) or polar cap melting are slow secular events. The characteristic coupling time between core and mantle has been estimated to be on the order of ten years, and the so-called 'decade fluctuations' of the Earth's rotation rate are thought to result from fluctuations within the core, transferred to the mantle.[1] But even for time scales from a few years down to weeks, the LOD varies significantly, and the observed fluctuations in LOD - after eliminating the effects of external torques - are then a direct consequence of the action of internal torques. These short term fluctuations are very probably generated by the interaction between solid Earth and the atmosphere.
Observations
Any change of the axial component of the atmospheric angular momentum (AAM) must be accompanied by a corresponding change of the angular momentum of crust and mantle (due to conservation). Because the moment of inertia of the system mantle-crust is only slightly influenced by atmospheric pressure loading, this mainly requires a change of the angular velocity of the solid Earth (i.e. a change of the length of day). LOD can presently be measured to a high accuracy with integration times of only a few hours,[2] and general circulation models of the atmosphere allow high precision determination of changes in AAM.[3] A comparison between AAM and LOD shows that they are highly correlated. In particular, one recognizes an annual period of LOD with an amplitude of 0.34 milliseconds, maximizing on February 3, and a semiannual period with an amplitude of 0.29 milliseconds, maximizing on May 8,[4] as well as 10 day fluctuations of the order of 0.1 milliseconds. Interseasonal fluctuations reflecting El Nino events and quasi-biennial oscillations have also been observed.[5] There is now general agreement that most of the changes of LOD on time scales from weeks to a few years are excited by changes in AAM.[6]
Atmospheric coupling
 | This section possibly contains original research. (June 2012) |
Observational evidence shows that there is no significant time delay between AAM and its corresponding LOD for periods longer than about 10 days. This implies a strong coupling between atmosphere and solid Earth with a time constant of not more than about 7 days (surface friction), the spindown time of the Ekman layer. This spindown time is the characteristic time for the transfer of atmospheric angular momentum to the Earth's surface and vice versa.
In Eulers equation of a spinning body two excitation terms exist, including one for the zonal wind and one for pressure loading. The mean zonal wind can be developed in terms of symmetric spherical functions:
(1) u = ∑ u2s-1 P2s-11(φ) (for s = 1 to ∞)
If one introduces this series into the Euler equation of axial angular momentum, one finds from the orthogonal condition of spherical harmonics that only the term
(2) u1 P11(φ) = u1 cosφ
remains.[7] φ is the geographic latitude. All other wind terms merely redistribute the AAM meridionally, and become zero if averaged over the globe. The wind component in (2) corresponds to rigid superrotation (or retrograde rotation) of the whole atmosphere.
If one does an equivalent development for pressure loading on the ground (z = 0), only the two pressure terms remain:
(3) p0(0) P00(φ) = p0(0)
and
(4) p2(0) P20(φ) = p2(0) 0.5 (3 sin2φ - 1)
The term in (3) represents the quiet (rigid) atmosphere. Its moment of inertia is CA ≃ 1.39 x 1032 kg m2. Together with the moment of inertia of crust and mantle of Cm ≃ 7.12 x 1037 kg m2 and the condition of constant axial angular momentum of the whole system (the Earth's core is decoupled from the mantel for time constants much smaller than 10 years), one arrives at a relation between the equivalent amplitude of the value of the superrotation wind at the equator u1 and the change of the length of the day Δτ as
(5) u1 ≃ 2.7 Δτ
with u1 in m/s and Δτ in ms. 'Equivalent amplitude' means that the whole atmosphere is assumed to rotate rigidly. The annual component of the change of the length of the day of Δτ ≃ 0.34 ms corresponds then to a superrotation of u1 ≃ 0.9 m/s, and the semiannual component of Δτ ≃ 0.29 ms to u1 ≃ 0.8 m/s.
The pressure term (4) causes a redistibution of the ground pressure that is a change in the moment of inertia of the atmosphere from its basic stage. This term, together with the higher-order terms, is connected with the meridional structure of winds and temperature.
If there were no surface friction, the climatic mean of the AAM would be zero because a zonal thermal wind cell has westerly winds in its upper branch and easterly winds in its lower branch, and the sum of the angular momenta of both branches is zero. Surface friction allows the atmosphere to 'pick up' angular momentum from the Earth (or vice versa). In the final stage, no exchange of angular momentum takes place. This implies that the climatic mean zonal wind component responsible for superroration, as well as its vertical change with height, must be zero on the ground:
(6) u1 = du1/dz = 0 (at z = 0)
In order to maintain mass continuity after the loss of the lower branch of the circulation cell, mass exchange within the boundary layer must occur via turbulence and diffusion. For the climatic mean, the zonal wind on the ground has now the form of the remaining spherical functions in equation (1), with its dominant term
(7) P31(φ) = 1.5 cosφ (5 sin2φ - 1)
The meridional structure of that wind simulates already, to a first approximation, the observed climatic zonal wind on the ground with westerly winds in middle latitudes beyond about φ = ± 30o and easterly winds - the trade winds - in lower latitudes (the zero of the spherical function in equation (7) is at φ = ± 27o).
In the case of the climatic mean, the atmosphere picks up angular momentum from the Earth at lower latitudes and transfers exactly the same amount to the Earth at higher latitudes. A shear force is therefore applied to the Earth. The force is, however, relatively small and can be compensated for by the almost-rigid crust.
Of course, the picture given above is highly simplified, and assumes the surface of the Earth is homogeneous. The real orographic surface certainly leads to corrections (which are probably second-order effects).
References
- ^ Hide, R. (1989). "Fluctuations in the Earth's Rotation and the Topography of the Core--Mantle Interface". Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences. 328 (1599): 351–363. Bibcode:1989RSPTA.328..351H. doi:10.1098/rsta.1989.0040.
- ^ Robertson, Douglas (1991). "Geophysical applications of very-long-baseline interferometry". Reviews of Modern Physics. 63 (4): 899–918. Bibcode:1991RvMP...63..899R. doi:10.1103/RevModPhys.63.899.
- ^ Eubanks, T. M. (1985). "A Spectral Analysis of the Earth's Angular Momentum Budget". Journal of Geophysical Research. 90 (B7): 5385. Bibcode:1985JGR....90.5385E. doi:10.1029/JB090iB07p05385.
{{cite journal}}: Unknown parameter|coauthors=ignored (|author=suggested) (help) - ^ Rosen, Richard D. (1993). "The axial momentum balance of Earth and its fluid envelope". Surveys in Geophysics. 14 (1): 1–29. Bibcode:1993SGeo...14....1R. doi:10.1007/BF01044076.
- ^ Carter, W.E. (1986). "Studying the earth by very-long-baseline interferometry". Scientific American. 255 (5): 46–54. doi:10.1038/scientificamerican1186-46.
{{cite journal}}: Unknown parameter|coauthors=ignored (|author=suggested) (help) - ^ Hide, R. (1991). "Earth's Variable Rotation". Science. 253 (5020): 629–637. Bibcode:1991Sci...253..629H. doi:10.1126/science.253.5020.629. PMID 17772366.
{{cite journal}}: Unknown parameter|coauthors=ignored (|author=suggested) (help) - ^ Volland, H. (1996). "Atmosphere and Earth's rotation". Surveys in Geophysics. 17 (1): 101–144. Bibcode:1996SGeo...17..101V. doi:10.1007/BF01904476.
Further reading
- Lambeck, Kurt (2005). The earth's variable rotation : geophysical causes and consequences (Digitally printed 1st pbk. ed.). Cambridge: Cambride University Press. ISBN 9780521673303.
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