The turbulence of solids: a multifractal plate tectonic model with Guttenberg-Richter plate “quakes”

Over thirty years ago, Y. Kagan speculated that seismicity could fruitfully be considered as “the turbulence of solids”.  Indeed, fluid turbulence and seismicity have many common features: they are both highly nonlinear with huge numbers of degrees of freedom.  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 – we could add – multifractal statistics.

Kagan was referring to seismicity as usually conceived, as a sudden rupture process  occurring over very short time periods.  We argue that even at one million year time scales, that the movement of tectonic plates is “quake-like” and is quantitatively close to seismicity, in spite of being caused by relatively smooth mantle convection. 

To demonstrate this, we develop a multifractal model grounded in convection theory and the analysis of the GPlates data-base of 1000 point trajectories over the last 200 Myrs.  We analyzed the statistics of the dynamically important vector velocity differences where Dr is the great circle distance between two points and Dt is the corresponding time lag.  The longitudinal and transverse velocity components were analysed separately.  The longitudinal scaling of the mean longitudinal difference follows the scaling law <Δv(Δr)> ≈ Δr^H with empirical H close to the mantle convection theory value  H = 1.  This high value implies that  mean fluctuations vary smoothly with distance.  Yet at the same time,  the intermittency exponent C₁ is extremely high (C₁ ≈ 0.55) implying that from time to time there are enormous “jumps” in velocity: “Plate quakes”.  For comparison, laminar (nonturbulent) flow has H = 1 but is not intermittent (C₁ = 0), whereas fully developed isotropic fluid turbulence has the (less smooth) value H = 1/3 (Kolmolgorov) but with non-negligible intermittency C₁ ≈ 0.07 and seismicity has very large C₁ ≈ 1.3.  Our study thus quantitatively shows how smooth fluid-like behaviour for the longitudinal velocity component can co-exist with highly intermittent quake-like behaviour.

Whereas the longitudinal component is well modelled by (highly intermittent) convection, the transverse velocity is well modelled by Brownian motion.  In the temporal domain both components (including their strong correlations) display such diffusion behaviour (i.e. with classical exponent H = ½), but are highly intermittent (C_1time = C_1space/2 ≈ 0.27).  Finally, the extreme velocity differences (that appear as occasional spikes in the velocities) have power law probability tails; the “Guttenberg-Richter” exponents in the seismology literature.

The advection - diffusion model is based on an underlying multifractal space-time cascade process.  Using mantle convection theory, we show how the driving multifractal flux (ψ) is related to vertical heat fluxes, expansion coefficients, densities, viscosities and specific heats. Taking typical values predict driving fluxes very close to the observed mean <ψ> ≈ 1/(400 Myrs).  Trace moment analysis shows that the outer space-time scales of the cascade process are ≈17000 km in space and ≈ 50Myrs in time.   Whereas the former corresponds to half the Earth’s circumference, the latter is the typical time required for a plate to randomly “walk” the same distance.