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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 length of day (LOD) varies significantly, and the observed fluctuations in the 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). 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]

Exchange of Angular Momentum

Observational evidence shows that there is no significant time delay between the change of AAM and its corresponding change of LOD for periods longer than about 10 days. This implies a strong coupling between atmosphere and solid Earth due to surface friction, with a time constant of not more than about 7 days, the spindown time of the Ekman layer. This spindown time is the characteristic time for the transfer of atmospheric axial angular momentum to the Earth's surface and vice versa.

The zonal wind component on the ground which is most effective for the transfer of axial angular momentum between Earth and atmosphere is the component describing rigid rotation of the atmosphere ^[7]. The zonal wind of this component has the amplitude u at the equator relative to the ground. u > 0 means superrotation; u < 0 means retrograd rotation with respect to the Earth. All other wind terms merely redistribute the AAM meridionally and become zero if averaged over the globe.

Surface friction allows the atmosphere to 'pick up' angular momentum from the Earth in the case of retrograd rotation or release it to the Earth in the case of superrotation. In the final stage, no exchange of axial angular momentum takes place. This implies that the climatic mean zonal wind component responsible for rigid rotation must be zero on the ground. Indeed, the observed meridional structure of the climatic mean zonal wind on the ground shows westerly winds (from the west) in middle and higher latitudes beyond about ± 30^o latitude and easterly winds (from the east) - the trade winds - in lower latitudes. The atmosphere picks up angular momentum from the Earth at lower latitudes and transfers exactly the same amount to the Earth at higher latitudes.

Any short term fluctuation of the rigidly rotating zonal wind on the ground is therefore accompanied by a corresponding change in the length of day. If one considers the total atmosphere to rotate rigidly with velocity u (in m/s), this value is related to the corresponding change of the length of day Δ (in msec) as

u ≃ 2.7 Δτ

The annual component of the change of the length of day of Δτ ≃ 0.34 ms corresponds then to a superrotation of u ≃ 0.9 m/s, and the semiannual component of Δτ ≃ 0.29 ms to u ≃ 0.8 m/s.

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.

@@ Line 12: / Line 12: @@
 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). LOD can presently be measured to a high accuracy with integration times of only a few hours,<ref>{{cite journal|last=Robertson|first=Douglas|title=Geophysical applications of very-long-baseline interferometry|journal=Reviews of Modern Physics|year=1991|volume=63|issue=4|pages=899–918|doi=10.1103/RevModPhys.63.899|bibcode = 1991RvMP...63..899R }}</ref> and [[general circulation model]]s of the atmosphere allow high precision determination of changes in AAM.<ref>{{cite journal|last=Eubanks|first=T. M.|coauthors=Steppe, J. A.; Dickey, J. O.; Callahan, P. S.|title=A Spectral Analysis of the Earth's Angular Momentum Budget|journal=Journal of Geophysical Research|year=1985|volume=90|issue=B7|pages=5385|doi=10.1029/JB090iB07p05385|bibcode = 1985JGR....90.5385E }}</ref> 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&nbsp;milliseconds, maximizing on February 3, and a semiannual period with an amplitude of 0.29&nbsp;milliseconds, maximizing on May 8,<ref>{{cite journal|last=Rosen|first=Richard D.|title=The axial momentum balance of Earth and its fluid envelope|journal=Surveys in Geophysics|year=1993|volume=14|issue=1|pages=1–29|doi=10.1007/BF01044076|bibcode = 1993SGeo...14....1R }}</ref> as well as 10&nbsp;day fluctuations of the order of 0.1&nbsp;milliseconds. Interseasonal fluctuations reflecting [[El Nino]] events and quasi-biennial oscillations have also been observed.<ref>{{cite journal |author=Carter, W.E. |coauthors=D.S. Robinson |title=Studying the earth by very-long-baseline interferometry |journal=Scientific American |volume=255 |pages=46&ndash;54 |year=1986 |doi=10.1038/scientificamerican1186-46 |issue=5}}</ref> 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.<ref>{{cite journal|last=Hide|first=R.|coauthors=Dickey, J. O.|title=Earth's Variable Rotation|journal=Science|year=1991|volume=253|issue=5020|pages=629–637|doi=10.1126/science.253.5020.629|pmid=17772366|bibcode = 1991Sci...253..629H }}</ref>
+==Exchange of Angular Momentum==
-==Atmospheric coupling==
+Observational evidence shows that there is no significant time delay between the change of AAM and its corresponding change of LOD for periods longer than about 10&nbsp;days. This implies a strong coupling between atmosphere and solid Earth due to surface [[friction]],  with a time constant of not more than about 7&nbsp; days, the spindown time of the [[Ekman layer]]. This spindown time is the characteristic time for the transfer of atmospheric axial angular momentum to the Earth's surface and vice versa.
-{{Section OR|date=June 2012}}
-Observational evidence shows that there is no significant time delay between AAM and its corresponding LOD for periods longer than about 10&nbsp;days. This implies a strong coupling between atmosphere and solid Earth with a time constant of not more than about 7&nbsp;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.
+The zonal wind component on the ground which is most effective for the transfer of axial angular momentum between Earth and atmosphere is the component describing rigid rotation of the  atmosphere  <ref name=Volland>{{cite journal|last=Volland|first=H.|title=Atmosphere and Earth's rotation|journal=Surveys in Geophysics|year=1996|volume=17|issue=1|pages=101–144|doi=10.1007/BF01904476|bibcode = 1996SGeo...17..101V }}</ref>. The zonal wind of this component has the  amplitude u at the equator relative to the ground. u > 0 means superrotation; u < 0 means retrograd rotation with respect to the Earth.  All other wind terms merely redistribute the AAM meridionally and become zero if averaged over the globe.
-In [[Euler]]s 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 function]]s:
+Surface friction allows the atmosphere to 'pick up' angular momentum from the Earth in the case of retrograd rotation  or release it to the Earth in the case of superrotation. In the final stage, no exchange of axial angular momentum takes place. This implies that the climatic mean zonal wind component responsible for rigid rotation must be zero on the ground. Indeed, the observed meridional structure of the climatic mean zonal wind on the ground shows westerly winds (from the west) in middle and higher latitudes beyond about ± 30<sup>o</sup> latitude and easterly winds (from the east) - the [[trade wind]]s - in lower latitudes.  The atmosphere picks up angular momentum from the Earth at lower latitudes and transfers exactly the same amount to the Earth at higher latitudes.
-(1) {{pad|4em}} u  = ∑ u<sub>2s-1</sub> P<sub>2s-1</sub><sup>1</sup>(φ){{pad|3em}} (for s = 1 to ∞)
+Any short term fluctuation of the rigidly rotating zonal wind on the ground is therefore accompanied by a corresponding  change in the [[length of day]]. If one considers the total atmosphere to rotate  rigidly with velocity u (in m/s),  this value is related to the corresponding change of the length of day Δ (in msec) as
-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) {{pad|4em}}  u<sub>1</sub> P<sub>1</sub><sup>1</sup>(φ) = u<sub>1</sub> cosφ
+{{pad|4em}}   u ≃ 2.7 Δτ
+The annual component of the change of the length of day of Δτ ≃ 0.34&nbsp;ms corresponds then to a superrotation of u ≃ 0.9&nbsp;m/s, and the semiannual component of Δτ ≃ 0.29&nbsp;ms to
-remains.<ref name=Volland>{{cite journal|last=Volland|first=H.|title=Atmosphere and Earth's rotation|journal=Surveys in Geophysics|year=1996|volume=17|issue=1|pages=101–144|doi=10.1007/BF01904476|bibcode = 1996SGeo...17..101V }}</ref> 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.
+u ≃ 0.8&nbsp;m/s.
-If one does an equivalent development for pressure loading on the ground (z = 0), only the two pressure terms remain:
-(3) {{pad|4em}}   p<sub>0</sub>(0) P<sub>0</sub><sup>0</sup>(φ) = p<sub>0</sub>(0)
-and
-(4) {{pad|4em}}  p<sub>2</sub>(0) P<sub>2</sub><sup>0</sup>(φ) = p<sub>2</sub>(0) 0.5 (3 sin<sup>2</sup>φ - 1)
-The term in (3) represents the quiet (rigid) atmosphere. Its moment of inertia is
-C<sub>A</sub> ≃ 1.39 x 10<sup>32</sup> kg&nbsp;m<sup>2</sup>. Together with the moment of inertia of crust and mantle of
-C<sub>m</sub> ≃ 7.12 x 10<sup>37</sup>&nbsp;kg&nbsp;m<sup>2</sup> 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 u<sub>1</sub> and the change of the length of the day Δτ as
-(5)  {{pad|4em}}   u<sub>1</sub> ≃ 2.7 Δτ
-with u<sub>1</sub> 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&nbsp;ms corresponds then to a superrotation of
-u<sub>1</sub> ≃ 0.9&nbsp;m/s, and the semiannual component of Δτ ≃ 0.29&nbsp;ms to  u<sub>1</sub> ≃ 0.8&nbsp;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)  {{pad|4em}}  u<sub>1</sub> = du<sub>1</sub>/dz = 0 {{pad|2em}} (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)  {{pad|4em}}  P<sub>3</sub><sup>1</sup>(φ) = 1.5 cosφ (5 sin<sup>2</sup>φ - 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 = ±&nbsp;30<sup>o</sup> and easterly winds - the trade winds - in lower latitudes (the zero of the spherical function in equation (7) is at φ = ±&nbsp;27<sup>o</sup>).
-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==

Revision as of 13:55, 22 November 2012

Introduction

Observations

Exchange of Angular Momentum

References

Further reading