## Abstract

Using estimated strain rates across the actively deforming region of Asia to infer variations in viscous stress, it is shown that gradients of stress point uphill toward the center of Tibet, where gravitational potential energy per unit surface area reaches a maximum. Thus, the dynamics of the deformation seem to obey the equation of creeping flow, which expresses a balance between gradients in stress and the gravitational body force. This balance, in the region of the Tibetan plateau, yields an estimate of 10^{22} pascal second for the average viscosity of the Tibetan lithosphere, which is only about 10 to 100 times greater than the viscosity of the convecting upper mantle.

Plate tectonics gained acceptance 25 to 30 years ago because it passed a simple quantitative test of its central tenet, that the kinematics of surface motions can be described by the rigid-body motions of a small number of plates. A single angular velocity prescribes the relative velocities of all points on any pair of rigid bodies on the surface of a sphere; relative motions measured across the narrow (tens of kilometers wide) bands of intense seismicity and deformation that separate broad regions with little or no seismic activity agreed with the motions expected from the relative rotation of rigid plates, and demonstrated the validity of plate tectonics, at least for the oceanic parts of Earth (1).

No comparably simple description of the tectonics of the continents has been accepted. The deformation of continental regions is not confined to narrow bands, but is spread over regions hundreds to thousands of kilometers in horizontal extent. The actively deforming part of Asia, which we consider here, is larger than several of the smaller oceanic plates. The diffuse nature of the deformation led to the inference that the continents, like Earth's mantle, behave as continuously deforming solids, rather than as rigid plates [for example, (2-4)].

The hypothesis of rigidity of plates was tested by measurements at widely spaced points around the edges of plates (1), without the need for measuring velocities in the plate's interior. In contrast, testing the hypothesis that continental deformation is continuous requires knowledge of spatial derivatives of stress, quantities much more difficult to determine than velocities. Furthermore, measurements must be made not only at the boundaries of deforming continents, but throughout entire deforming regions. Making some simplifications to the equation of stress balance in a continuous medium, we carried out a test of its applicability to the tectonics of Asia, estimating the stress field from observations of Quaternary faulting.

Inertial terms are negligible in tectonic processes, so the governing equation for the deformation of a continuous lithosphere is the stress balance equation, which states that gradients of stress are balanced by the force of gravity per unit volume(1)where σ_{ji} is the*ij*
^{th} component of the stress tensor,*x*
_{j} is the *j*
^{th} coordinate direction, ρ is the density, and *g*
_{i} is the*i*
^{th} component of acceleration due to gravity (5).

Because neither stress nor strain can be measured at depth within the lithosphere, it is impossible to determine the vertical gradients of stress in Eq. 1 (6). We therefore treat deformation of the lithosphere in terms of vertical averages of stress and strain rate in a thin sheet of fluid [for example, (3, 4, 7)]. This simplification is appropriate where the horizontal dimensions of a region of active deformation exceed by many times the thickness of the lithosphere (8).

Gravity anomalies show that, on length scales >100 to 200 km, density contrasts within the lithosphere are isostatically compensated. Thus, the weight per unit area of any column of rock is supported by the vertical traction, σ_{zz}, on its base(2)where *z* is depth, *z*′ is a variable of integration, and *g* is the acceleration due to gravity. With this simplification, σ_{zz} can be specified from a knowledge of the density structure of the lithosphere.

The horizontal equations in Eq. 1 may be expressed in terms of deviatoric stress (9), and simplified by taking vertical averages and neglecting shear tractions on the base of the lithosphere [for example, (10)]
(3)where *L* is the thickness of the lithosphere, τ_{ij} is the *ij*
^{th}component of the deviatoric stress tensor, averaged vertically through the lithosphere, and(4)For isostatically balanced columns of lithosphere, differences in Γ are equal to differences in the gravitational potential energy per unit area between columns (*4, 10*–*12*).

Equation 3 states that horizontal gradients of the deviatoric stress required to deform a thin viscous sheet are balanced by horizontal gradients of its gravitational potential energy (4,7, 10, 11). A simple test of the hypothesis that deformation of the continents is described by this balance would be to determine whether the left-hand side of Eq. 3, derived from observations of deformation, corresponds to gradients of lithospheric potential energy, derived from a knowledge of the density structure of the lithosphere. Because there are no observations of components of vertically averaged stress in the lithosphere and no direct observations of the gravitational potential energy of the lithosphere, we proceeded by using observable proxies for both quantities.

The stresses in Eq. 3 can be related to deformation by specifying a relation between the vertically averaged deviatoric stresses and the vertically averaged strain rates in the lithosphere. The vertical averages of a wide range of possible rheological profiles for the continental lithosphere probably obey a relation of the form(5)where ɛ˙_{ij} is the*ij*
^{th} component of the strain rate, assumed independent of depth(6)where *u*
_{i} is the*i*
^{th} component of velocity, and*E˙* =, with the convention of summation over repeated subscripts (13). Equation 5 expresses the rheology of a “power law” fluid, whose effective viscosity, 1/2*BE˙*
^{1/}
^{n}
^{−1}, decreases with increasing magnitude of the strain rate (13).

We make the simplifying assumption, which is doubtless wrong in detail, that the variations in components of the average deviatoric stress in the lithosphere,τ̄_{ij}, arise principally from variations in strain rate (that is, we assume *B* to be constant). With this assumption, we can define a dimensionless stress(7)and Eq. 3 becomes
(8)where(9)is the dimensionless potential energy of a column of lithosphere.

Γ in Eq. 3 is related to the gravitational potential energy of the whole lithosphere (11). Assuming constant crustal density and ignoring lateral variations in the density of the mantle, the difference ΔΓ in gravitational potential energy per unit area between any two isostatically balanced columns of lithosphere is simply(10)where ρ_{c} and ρ_{m} are, respectively, the densities of crust and mantle, and *S*
_{1} and *S*
_{2}are the thicknesses of crust in the two columns. We use Eq. 10 to approximate gravitational potential-energy differences, estimating crustal thickness from surface elevation by assuming isostatic balance with a standard column of oceanic ridge (14). We recognize that possible variations in the thickness and temperature structure of the mantle portion of the lithosphere can also contribute to potential-energy contrasts within the lithosphere [see, for example, (11, 15, 16)], but these variations are poorly determined in Asia.

Because logistical and political difficulties preclude the direct measurement of strain rates over the whole of the actively deforming region of Asia, determination of the active strain of this region relies on incomplete information from earthquakes (17, 18) and from the rates of slip on faults (19, 20). Strain rates estimated from such observations are likely to be inconsistent between neighboring regions, thus forming an unreliable basis for estimating the gradients of strain that are required for the test proposed above.

We circumvented this deficiency [following (18, 21)] by imposing on the observations the constraint that the strain rates be self-consistent (or compatible in the sense of St. Venant) in that they should be derivatives of a velocity field (19, 22). We covered the region of interest with a mesh of triangles (Fig.1), in each of which we determined strain rates from estimated Quaternary slip rates of the major faults in Asia, using the method of Kostrov (23). We related strain-rate components within each triangle to the relative velocities of its vertices, forming a set of normal equations that could be solved to yield the velocity field (Fig. 1). Self-consistent strain rate components were then derived by differentiation of these velocities.

The most active deformation in eastern Asia occurs within the region 70° to 110°E, and 25° to 55°N, and we restricted our analysis to this region. If the deformation in Asia can be approximated by that of a thin viscous sheet, then the gradients of dimensionless stress in Eq. 8 should point in directions of increasing potential energy, and therefore, approximately uphill on a map of regional topography (Eq.10).

We converted the self-consistent strain rates into dimensionless stress, using Eq. 7 (see Fig. 2), then interpolated them onto a regular grid of spacing 2° in latitude and longitude, using a Gaussian smoothing function with a half-width of 200 km. From these gridded data, we constructed the spatial derivatives of dimensionless stresses by standard space-centered finite difference approximations. Estimating the left-hand side of Eq. 8 from these derivatives yields vectors (Fig. 3; arrows) that indeed point uphill around the Tibetan Plateau.

A second comparison can be made by integrating these gradients of dimensionless stresses to yield the dimensionless potential energy (Eqs. 8 and 9). This quantitity reaches its maximum over the Tibetan plateau (Fig. 3), as should be expected if this region deforms, at least approximately, as a viscous sheet (Eq. 3). There is, however, far from perfect agreement between estimated and calculated potential energy in other parts of Asia. For example, large variations of potential energy are calculated in the northeastern part of the region, around Lake Baikal, where the topographic slopes are small, and the calculated potential-energy decreases across the Tien Shan, in disagreement with the increase that is expected from the increase in surface height.

These disagreements may arise because of the simplifing assumptions made in our analysis. To calculate gradients in stress, and estimates of potential energy from these gradients, we assumed that the same rheological parameter *B* applies to the entire region. Variations in *B*, as well as influencing the distribution of deformation, would contribute significantly to mismatches in potential energy (24); such variations have been suggested among northwest Tibet, the Tarim basin, and the Tien Shan [for example, (25)]. The assumption of Airy isostatic compensation to estimate potential energy also introduces an error; variations in potential energy due to lateral variations in the temperature structure of the upper mantle can be as large as those due to changes in crustal thickness (11, 15); such variations may affect the region of Mongolia and Lake Baikal, which seems to be underlain by buoyant upper mantle (26). The assumption that differences in the potential energy can be determined from differences in surface height works best where crustal thickness variations are largest; in Asia, this is in the region of Tibet. Finally, the application of Eq. 3implies that basal shear tractions are negligible—a reasonable assumption for Tibet but perhaps not for other regions, such as the Himalaya, where Indian lithosphere is thrust beneath the southernmost Tibetan plateau. For these reasons, we consider the comparisons of potential energy to be meaningful only in the region of Tibet. The potential energy calculated from strain rates and the potential energy calculated from topography are completely independent quantities; agreement between the two in the region of Tibet provides support for the hypothesis that Eq. 3 describes the active deformation there.

If deformation of a homogeneous viscous sheet characterized the tectonics of the region, then estimates of dimensionless potential energy made from the strain-rate field would vary linearly with potential energy estimated from topography, with a slope equal to the value of *BL* (Eq. 9). Comparison of these two estimates of potential energy per unit area over Asia (Fig.4) reveals considerable scatter for low areas but a linear trend for the region around Tibet. Assuming*L* = 100 km, the slope of this trend yields an estimate for the spatially averaged rheological parameter, *B* = 1.3 × 10^{12} N m^{–2} s^{1/3}. For this value of *B*, the effective viscosity of the lithosphere is approximately 10^{22} N m^{–2} s^{–1}(Table 1). This viscosity, ∼10^{22} Pa·s, appropriate to a typical strain rate of 10^{–16} to 10^{–15} s^{–1}, is greater than the viscosity of the convecting upper mantle by only one, or at most two, orders of magnitude [for example, (27)], suggesting that the continental lithosphere of Asia is more properly regarded as belonging to the fluid portion of the solid earth than to the relatively small fraction of Earth that behaves as rigid plates.

↵* To whom correspondence should be addressed. E-mail: philip.england{at}earth.ox.ac.uk