Tectonic &#8220;quakes&#8221;, scaling and the turbulence of solids
Authors: Lovejoy Shaun , Spiridonov Andrej , Balakauskas Lauras 
Published: March 2024

Over thirty years ago, Y. Kagan proposed that seismicity is &#8220;the turbulence of solids&#8221;.&#160; Indeed, fluid turbulence and seismicity have many common features: they are both highly nonlinear with huge numbers of degrees of freedom.&#160; Beyond that, Kagan recognized that they are both riddled with scaling laws in space and in time as well as displaying power law extreme variability and &#8211; we could add &#8211; multifractal statistics.Kagan was referring to seismicity as usually conceived, as a sudden rupture process &#160;occurring over very short time periods.&#160; We argue that even at million year time scales, that the movement of tectonic plates is &#8220;quake-like&#8221; and is quantitatively close to seismicity, yet caused by relatively smooth mantle convection fluid.&#160;To demonstrate this, we analyse the GPlates data base of 1000 point trajectories over the last 200 Myrs, analyzing the statistics of the dynamically important vector velocity differences&#160;where Dr is the great circle distance between two points and Dt is the corresponding time lag.&#160; The longitudinal and transverse velocity components are analysed separately.&#160; The longitudinal scaling of the mean longitudinal difference follows the scaling law &#8776; Dr^H with H close to the theoretically predicted value &#160;H = 1.&#160; This high value implies that&#160; mean fluctuations vary relatively smoothly with distance.&#160; Yet at the same time,&#160; the intermittency exponent C1 is extremely high (C1 &#8776; 0.5) implying that from time to time there are enormous &#8220;jumps&#8221; in velocity. For comparison, laminar (nonturbulent) flow has H = 1 and is not intermittent (C1 = 0), fully developed isotropic fluid turbulence has the (less smooth) value H = 1/3 (Kolmolgorov) but with non-negligible intermittency C1 &#8776; 0.07 and seismicity has very large C1 &#8776; 1.3.&#160; Our study thus quantitatively shows how smooth fluid-like behaviour can co-exist with highly intermittent quake-like behaviour.We find that the outer spatial scale is near the size of the Earth (&#8776;15000km) whereas the outer time scale is &#8776;60Myrs.&#160; We show that the statistics are multifractal with a very large intermittency parameter that is close to that of seismicity determined at sub-decadal time scales.&#160; The transverse scale function is the 2/3 power of the longitudinal scale function,&#160; the transverse intermittency exponent (C1 ) is reduced by this factor.&#160; The temporal scaling of the mean fluctuations of both the longitudinal and transverse components is close to a &#189; power of the time lag: Dr&#8776;Dt^(1/2).&#160; However since the spatial scaling of the longitudinal and transverse components are different, we obtain two somewhat different space-time diagrams.&#160; We link the parameter estimates to fundamental mantle convection parameters, and we make corresponding multifractal simulations.Finally, we discuss the implications for the megaclimate regime, including macro-evolution. Both megaclimate and macroevolution of global diversity are scaling processes with H>0 characterized by intermittent &#8212; climate &#8220;events&#8221;, such as P-Tr hyperthermal, in the case of former, and mass extinctions and originations in the case of latter. The tectonic scaling, and the extreme multifractal behavior grounds both&#8212;the long-term climate, and the biological evolution on the first principles of scaling in macroscopic physical systems.