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Analysis of a Precambrian Resonance-Stabilized Day Length 10.1002/2016GL068912 Benjamin C View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Caltech Authors Geophysical Research Letters RESEARCH LETTER Analysis of a Precambrian resonance-stabilized day length 10.1002/2016GL068912 Benjamin C. Bartlett1 and David J. Stevenson1 Key Points: • Near resonance, atmospheric torque 1California Institute of Technology, Pasadena, California, USA cancels lunar torque, preserving day length • Snowball events 600 Ma likely caused During the Precambrian era, Earth’s decelerating rotation would have passed a 21 h period a break from resonant 21 h day length Abstract • Resonance is plausible given most that would have been resonant with the semidiurnal atmospheric thermal tide. Near this point, the variable choices and is consistent with atmospheric torque would have been maximized, being comparable in magnitude but opposite in direction geological data to the lunar torque, halting Earth’s rotational deceleration, maintaining a constant day length, as detailed by Zahnle and Walker (1987). We develop a computational model to determine necessary conditions for Correspondence to: formation and breakage of this resonant effect. Our simulations show the resonance to be resilient to B. C. Bartlett, atmospheric thermal noise but suggest a sudden atmospheric temperature increase like the deglaciation [email protected] period following a possible “snowball Earth” near the end of the Precambrian would break this resonance; the Marinoan and Sturtian glaciations seem the most likely candidates for this event. Our model provides Citation: a simulated day length over time that resembles existing paleorotational data, though further data are Bartlett, B. C., and D. J. Stevenson (2016), Analysis of a Precambrian needed to verify this hypothesis. resonance-stabilized day length, Geophys. Res. Lett., 43, 5716–5724, doi:10.1002/2016GL068912. 1. Introduction Received 30 MAR 2016 At some point during the Precambrian, the Earth would have decelerated to the point where it had a rotational Accepted 10 MAY 2016 Accepted article online 16 MAY 2016 period of 21 h, which would have been resonant with the semidiurnal atmospheric tide, with its fundamen- Published online 15 JUN 2016 tal period of 10.5 h. At this point, the atmospheric tidal torque would have been comparable in magnitude Corrected 12 JUL 2016 but opposite in sign to the lunar oceanic torque, which could create a stabilizing effect on the day length, preserving the 21 h day length until the resonance was broken, as first discussed in Zahnle and Walker [1987]. This article was corrected on 12 JUL The question then arises as to how the Earth broke out of its resonance-stabilized day length of 21 h to 2016. See the end of the full text for details. progress to its current day length of 24 h. In general, any sufficiently large sudden increase in temperature will shift the resonant period of the atmosphere by thermal expansion (resulting in a change of atmospheric column height) to a shorter period, as described in Figure 1, and could potentially break resonance, allowing for Earth to decelerate to longer day lengths. (Alternately, resonance could also be broken by increasing the lunar torque to surpass the peak atmospheric torque by the gradual change of the oceanic Q factor, defined for an arbitrary system as 2 ⋅ total energy , though the very low necessary atmospheric Q factor for energy dissipated per cycle resonance to form given the current oceanic torque makes this seem a less likely explanation and is not explored here.) This study develops a model of resonance formation and breakage that approximately outlines the necessary conditions for this constant day length phenomenon to occur for an extended period of time. In our model of atmospheric resonance, there are effectively three outcomes for this resonant phenomenon. First, the Earth could have entered a stable resonant state which lasted for some extended period of time before being interrupted by a global temperature increase, such as the deglaciation period following a pos- sible “snowball Earth” event. Specifically, the Sturtian or Marinoan glaciations make good candidates for this breakage event [Pierrehumbert et al., 2011; Rooney et al., 2014]. Second, the resonant stabilization could have never occurred, as the Q factor of the atmosphere could have been too low for the magnitude of the atmospheric torque to exceed that of the lunar torque, a necessary condition for a constant day length. Third, the resonance could have been of no interest, as atmospheric and temperature fluctuations could have been too high to allow a stable resonance to form for an extended period of time. ©2016. American Geophysical Union. We discuss the plausibility of each of these scenarios in greater detail below and ultimately conclude that the All Rights Reserved. first scenario is the most likely to have occurred. BARTLETT AND STEVENSON STABILIZED PRECAMBRIAN DAY LENGTH 5716 Geophysical Research Letters 10.1002/2016GL068912 2. Analysis of Atmospheric Resonance The details of the atmospheric tide are quite complex, but the essential features can be appreciated with the following toy model of the torque. (For interested readers, a more complete treatment of the atmospheric tidal problem is given in Lindzen and Chapman [1969], most specifically in section 3.5.C.) Given a fluid with column density 0 and equivalent column height h0 under√ gravitational acceleration g, with ≪ ≫ Lamb waves of amplitude h h0 and wavelength h0, wave speed of gh0, Cartesian spatial coordinates of x, a forced heating term h , and a damping factor Γ= 1 (with t defined as the total energy over power f t Q Q loss of the system, such that Q = 0tQ), we start with the forced wave equation without drag (we will add this later): 2h 2h 2h = gh + f . (1) t2 0 x2 t2 We are interested in a heating term of the form F = F0 cos(2 t + 2kx), with F0 as the average heating per 2 unit area, as the angular frequency, and k = at the equator, with R⊕ the Earth’s equatorial radius. Thus, 2R⊕ for C as the specific heat at constant pressure and T as mean surface temperature, we have C T dhf = p 0 0 p 0 dt F0 cos(2 t + 2kx) or F sin(2t + 2kx) 0 . hf = (2) 2 0CpT0 Expressing h = A sin(2t + 2kx) and defining the equivalent height (which is currently 7.852 km [Zahnle and 42R2 Walker, 1987], with resonant effects occurring at about 10 km) as h ≡ ⊕ , where is the relevant eigenvalue 0 g . to Laplace’s tidal equation. Using present values for h0, , g, and k, we obtain ≈ 0 089, which agrees with 2 42 Lamb [1932], p. 560. Near resonance gh = 4 ≈ 0 , so we obtain via equation (1) that 0 k2 k2 F A =− ( 0 ) . (3) 2 2 2 0CpT0 0 − < At present day, 0, making A negative, so the positive peak of A sin(2 t + 2kx) is at 2 t + 2kx =− .At 2 noon (t = 0), this occurs spatially at x =− =− R or −45∘. This result determines the sign of the torque, as 4k 4 the mass excess closer to the Sun exists such that it is being pulled in the prograde rotational direction. Note > that for period of time where the length of day is less than the resonant period of 21 h, that is, for 0, the resultant torque of A will exert a decelerating effect on Earth. However, at the point of resonance in question, < where the lunar torque is canceled by the atmospheric torque, 0 by a small factor, corresponding to a day length slightly above 21 h. Addressing drag in our model, we assume that any excess velocity formed from the tidal acceleration in the atmosphere is quickly dissipated into the Earth through surface interactions with a damping factor Γ, and that this surface motion is relatively quickly dissipated into the rotational motion of the entire Earth, as given by Hide et al. [1996], writing the dissipative Lamb wave forces, we have v h v h h =−g − vΓ h =− + f (4) t x 0 x t t from which we obtain ( ( ) ) F (2 − iΓ) 4 2 − 2 + 2iΓ A = 0 ⋅ ( ) 0 . (5) 2 0CpT0 2 2 2 2 16 − 0 + 4 Γ In this model, the imaginary component ℑ(A) represents amplitude which would create a force with an angle of with respect to the Sun, and thus does not exert any torque on Earth. We need only to concern ourselves 2 with the real part ℜ(A) then. Thus, we have ( ) F 4 2 − 2 + Γ2 ℜ(A)= 0 ⋅ ( )0 . (6) 2 2 0CpT0 2 2 2 2 4 − 0 + Γ Since we know the atmospheric torque to be directly proportional to the atmospheric displacement A,we can use the fact that the present accelerative atmospheric torque, 2.5 × 1019 Nm, is approximately 1 that 16 of the present decelerative lunar torque, 4 × 1020 Nm, as given in Lambeck [1980], to scale the atmospheric BARTLETT AND STEVENSON STABILIZED PRECAMBRIAN DAY LENGTH 5717 Geophysical Research Letters 10.1002/2016GL068912 Figure 1. Torque values for atmospheric torques assuming various Q factors compared to lunar torque. Torques are scaled at the 24 h endpoint such that they have a value 1 that of the lunar torque, while the contour of the curve is determined by the A term derived in section 2. Note that the minimum value of Q 16 required to form a resonance (the value such that its magnitude exceeds the lunar torque) varies linearly with the lunar torque. During the Precambrian, when the lunar torque was thought to be approximately a fourth of its current value [Zahnle and Walker, 1987], very low values of Q could have permitted resonance to form. ℜ 휏 휔 torque along the curve following (A), giving the total atmospheric torque atm( ) as a function of the Earth’s rotational frequency, as detailed in Figure 1.
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