Turbulent lithosphere deformation in the Tibetan Plateau

In this work, we show that the Tibetan Plateau deformation demonstrates turbulence-like statistics, e.g., spatial invariance across continuous scales. A dual-power-law behavior is evident to show the existence of two possible conservation laws for the enstrophy-like cascade in the range $500\lesssim r\lesssim 2000\mathrm{km}$ and kinetic-energy-like cascade in the range $50\lesssim r\lesssim 500\mathrm{km}$. The measured second-order structure-function scaling exponents $\zeta \left(2\right)$ are similar to their counterparts in the Fourier scaling exponents observed in the atmosphere, where in the latter case the earth's rotation is relevant. The turbulent statistics observed here for nearly zero-Reynolds-number flow can be interpreted by the geostrophic turbulence theory. Moreover, the intermittency correction is recognized with an intensity close to that of the hydrodynamic turbulence of high-Reynolds-number turbulent flows, implying a universal scaling feature of very different turbulent flows. Our results not only shed new light on the debate regarding the mechanism of the Tibetan Plateau deformation but also lead to new challenges for the geodynamic modeling using Newton or non-Newtonian models because the observed turbulence-like features have to be taken into account.

Figure 1

(a) Spatial distribution of the 553 GPS monitoring stations at the Tibetan Plateau, where the velocity amplitude in cm is encoded by symbol size. The velocity unit vector is indicated by an arrow. Large clockwise rotation of crustal material around the eastern Himalayan syntaxis are illustrated by the big arrow. The GPS velocity data are taken from Ref. [9]. The color map is the elevation provided by ETOPO1 [35]. (b) The experimental number distribution of the neighbor distance, where $r$ is the great circle distance and $h=70\mathrm{km}$ is the average depth of the lithosphere. The solid line indicates a power-law relation with a scaling exponent 2 for reference.

Figure 2

(a) Measured second-order structure functions. The solid line is the fitting on the range $50\lesssim r\lesssim 500\mathrm{km}$ (resp., $0.7\lesssim r/h\lesssim 7$) and $500\lesssim r\lesssim 2000\mathrm{km}$ (resp., $7\lesssim r/h\lesssim 28$) with respectively scaling exponent $0.72\pm 0.07$ and $2.50\pm 0.07$. (b) The corresponding compensated curve, e.g., $S_2\left(r\right)r^{-\zeta \left(2\right)}{C}^{-1}$ using a fitted parameter, to highlight the power-law behavior.

Figure 3

Experimental probability density function $p\left(x\right)$ and the corresponding integral kernel $p\left(\mathrm{\Delta }{u}_r\right){\left|\mathrm{\Delta }{u}_r\right|}^q$ for various separation scales $r$: (a) $50\lesssim r\lesssim 500\mathrm{km}$, and (b) $500\lesssim r\lesssim 2000\mathrm{km}$, where the thin line with different color indicates different separation scales and the thick solid line is the normal distribution for reference. Because of the finite sample size, the fourth-order structure function slightly deviates from statistical convergence.

Figure 4

(a) Measured high-order structure functions $S_q\left(r\right)$ for $q=1,3,4$. The solid line illustrates the power-law fitting. (b) The corresponding compensated curve to highlight the power-law behavior.

Figure 5

(a) Measured high-order scaling exponent $\zeta \left(q\right)$ of small-scale ($◯$) and large-scale ($\square$) motions for $-1\le q\le 4$, where $\zeta \left(q\right)=q/3$ (dashed line) and $\zeta \left(q\right)=q$ (solid line) are illustrated for reference. (b) The extended self-similarity plot of the measured scaling exponent ${\zeta }_{\text{E,1}}\left(q\right)=\zeta \left(q\right)/\zeta \left(1\right)$. For comparison, the scaling exponents compiled for velocity (thick solid line) [41] and passive scalar (thin solid line) [42] are also shown. The error bar indicates a 95% fitting confidence level. The scaling exponent for topology is illustrated as $▵$.

Figure 6

(a) Experimental second-order structure function for the topography using the ETOPO1 data. Power-law behavior is observed on the range $50\lesssim r\lesssim 500\mathrm{km}$ with a scaling exponent $0.77\pm 0.07$. (b) The corresponding compensated curve using fitted parameters to highlight the power-law behavior.