[mathjax]The two primary oscillating factors that we have identified in the CSALT model of global temperature are the Southern Oscillation Index (SOI) and the Length of Day (LOD).  The distinguishing factor in terms of impact  is that the SOI is characterized by intradecadal oscillations while the LOD fluctuates across decades [1].

If we can model the SOI deterministically, as demonstrated here, the hope is we may be able to model the LOD as well.  But first, we need to understand the significance of the LOD and its possible origin.

Fig 1:  As a premise for Length of Day (LOD) variations we consider that the rotational moment of inertia changes along the planetary surface.  If a band of water positioned along the equator shifts to higher latitude, the rotational moment of inertia decreases and the rotational velocity increases, thus shortening the length of day.

The LOD measure has a close correspondence to the Atmospheric Circulation Index (ACI) [2][3], which characterizes the proportion of meridional transport of air mass (i.e. North/South) versus the zonal transport (i.e. West/East). This index is essentially a count of the number of days that one mode of transport occurs versus the other. Figure 2 shows the correlation between -LOD (faster angular rotation) against the zonal index of ACI.  In this case, a progressively shorter LOD anomaly corresponds to greater zonal transport of the air masses.

Fig 2: (TOP) The compliment of the LOD corresponds to faster angular rotation of the earth.  (BOTTOM) ACI index with respect to zonal transport. From reference [2].

“The first type, zonal circulation, is characterised by increasing intensity of the zonal circulation at all latitudes and pole-ward shift of the wind intensity maximums. The circulation is accompanied somewhat by a decrease in the overall range of surface-air temperature between the equator and poles and by an overall increase in the mean global surface-air temperatures. Ocean-surface temperatures tend to increase in high latitudes. The second type, meridional circulation, is characterised by weakening in zonal circulation, shift of the main atmospheric streams toward lower latitudes, and overall decrease in global temperature (Lamb 1972). Both easterly and westerly winds increase during the zonal type of circulation and both decrease in the periods of the meridional type of the circulation.”

The rationale for associating the change in LOD with differences in transport direction has to do with the potential for surface water to displace from the equator to higher latitudes, thus reducing the rotational moment of inertia and causing the earth to speed up and reduce the length of day slightly.   Compare Figure 1 to the analogy of Figure 3, where the figure skater pulls in her arms and spins faster.

The tightening radius of the band towards higher latitudes corresponds to a reduction of the rotational moment of inertia as water is displaced away from the maximum radius at the equator. Since angular momentum must be conserved, the rotational velocity must be increased.

$$ r_1 > r_2 $$

$$ omega_1 < omega_2 $$

In terms of angular momentum, the conservation law is established as :

$$ L = I omega $$

$$ dL = 0 = d(I omega) = I domega + omega dI $$

since a delta increase in angular frequency corresponds to a drop in the LOD time

$$ frac{dI}{I} = – frac{domega}{omega} = frac{dLOD}{LOD} $$

The rotational moment of inertia of a solid sphere such as the earth is

$$ I = frac{2}{5} m_e r^2 $$

while a thin band at the same radius would be

$$ dI = m_b r^2 $$

The sanity check is to determine whether a band of this water volume mass can impact LOD if it is shifted from a large radius at the equator to a minimal radius near the pole.

The earth’s mass m_e is known, while the band mass m_b can be geometrically constructed given the density of water and volume of that band.

$$ m_b = 2 pi r cdot Width cdot Depth cdot rho $$

so that the relation is

$$ frac{dI}{I} = frac{5}{2} frac{m_b}{m_e} = frac{dLOD}{LOD} $$

The radius of the earth at the equator  6380 km and the mass is 6e24 kg.

For a dLOD of approximately 0.3 msec out of 86,400 seconds per year (an average change in Figure 2) , the Width x Depth product has to be about 200,000 square meters.

This product could be a width of water 2000 km straddling the equator with a depth of 0.1 meters shifting toward the poles, or a width twice this and a depth half of 0.1 meter. It is not clear what the value is exactly, only that this is a physically plausible possibility, although only remotely feasible due to the large volumes of water involved. The outcome is that the movement of water leading to greater buildup pole-ward brings along with it subtle changes in global sea-surface temperature (yet not so much on land) of +/- 0.1C, according to CSALT model fits.

This is similar and potentially related to the tidal mechanism that leads to LOD changes on a more predictable basis — which is correlated to global temperature changes via aligning the phase of the CSALT tidal factors to the LOD cyclical shifts.   The issue is that the interdecadal changes in LOD are much stronger than these rather subtle tidal changes.

In fact, an ocean transport process isn’t the preferred mechanism based on the research literature. So what other mechanism is driving the change in LOD and the simultaneous temperature correlations? J. Dickey of NASA JPL who has done much of the research on LOD [1] indicates that the inertial change occurs within the Earth’s core, not in the ocean or atmosphere. From a JPL press release:

“These longer fluctuations are too large to be explained by the motions of Earth’s atmosphere and ocean. Instead, they’re due to the flow of liquid iron within Earth’s outer core, where Earth’s magnetic field originates. This fluid interacts with Earth’s mantle to affect Earth’s rotation. While scientists cannot observe these flows directly, they can deduce their movements by observing Earth’s magnetic field at the surface. Previous studies have shown that this flow of liquid iron in Earth’s outer core oscillates, in waves of motion that last for decades with timescales that correspond closely to long-duration variations in Earth’s length of day.

Still other studies have observed a link between the long-duration variations in Earth’s length of day and fluctuations of up to 0.2 degrees Celsius (0.4 degree Fahrenheit) in Earth’s long-term global average surface air temperature.”

This moves the inertial band in Figure 1 to below the earth’s surface and Dickey offers this hypotheses for the correlation to temperature variations:

“Since scientists know air temperature can’t affect movements of Earth’s core or Earth’s length of day to the extent observed, one possibility is the movements of Earth’s core might disturb Earth’s magnetic shielding of charged-particle (i.e., cosmic ray) fluxes that have been hypothesized to affect the formation of clouds. This could affect how much of the sun’s energy is reflected back to space and how much is absorbed by our planet. Other possibilities are that some other core process could be having a more indirect effect on climate, or that an external (e.g. solar) process affects the core and climate simultaneously.”

Even though this is not as parsimonious explanation, the fact that the earth’s core is five times more dense than the ocean’s water lends it some weight. This consideration needs to be compared to the inertial lever-arm that goes as r^2 —  remember that the water exists on the surface of the planet and therefore has a much stronger impact on momentum conservation, even with its lower density.

If the origin is indeed subsurface, we may never extract a deterministic source for the LOD shifts, as volcanic geologic activity appears much more random than a potential luni-solar correlation at the surface. However, if the influence is gravitational, we may be able to discover a deterministic cyclic forcing factor that will allow us to predict the LOD into the future, and therefore the temperature anomaly that this feeds.  Whether that it grows much beyond it’s current +/- 0.1 C impact over the past 130+ years would also be an important piece of the puzzle.

References

[1] J. O. Dickey, S. L. Marcus, and O. de Viron, “Air Temperature and Anthropogenic Forcing: Insights from the Solid Earth,” Journal of Climate, vol. 24, no. 2, pp. 569–574, 2011.
[2] United Nations Food and Agricultural Organization (FAO) report, “2. DYNAMICS OF CLIMATIC AND GEOPHYSICAL INDICES.” [Online]. Available: http://www.fao.org/docrep/005/y2787e/y2787e03.htm. [Accessed: 04-Mar-2014].
[3] M. G. Wyatt and J. A. Curry, “Role for Eurasian Arctic shelf sea ice in a secularly varying hemispheric climate signal during the 20th century,” Climate Dynamics, pp. 1–20, 2013.
(This is the planetary “stadium wave” paper. Note the pole-ward origin for the stadium wave, where the LOD and ACI are included in the set of stadium wave indices.)
[4] F. P. Chavez, J. Ryan, S. E. Lluch-Cota, and M. Ñiquen, “From anchovies to sardines and back: multidecadal change in the Pacific Ocean,” Science, vol. 299, no. 5604, pp. 217–221, 2003.
[5] Klyashtorin et al, “Cyclic changes of climate and major commercial stocks of the Barents Sea”, Marine Biology Research, Vol.5, 2009