Numerical solutions

The thin-sheet approximation assumes that vertical tractions on vertical planes may be neglected, so that density variations within the lithosphere are in local isostatic balance. Under this condition, the stress balance equation for creeping flow in spherical coordinates is: 2 r ∂ τθθ ∂ θ + 1 r ∂ τφφ ∂ θ + 1 rsinθ ∂ τθφ ∂ φ + cotθ r τθθ−τφφ = 1 r L ∂Γ ∂ θ 1 rsinθ ∂ τθθ ∂ φ + 2 rsinθ ∂ τφφ ∂ φ + 1 r ∂ τθφ ∂ θ + 2 cotθ r τθφ = 1 rsinθL ∂Γ ∂ φ (A.14)

whereθandφare respectively the co-latitude and the longitude (in radians) andris the ra- dial distance from the center of the earth,Lis the thickness of the lithosphere, andΓis the gravitational potential energy per unit area of lithosphere (GPE) [Bird and Piper, 1980;Eng- land and Mc_{Kenzie, 1983;Gordon and Houseman, 2015;Houseman and England, 1986].}

The relevant components of the strain-rate tensor are ˙ εθθ = 1_r∂_∂θuθ (A.15) ˙ εφφ = 1 rsinθ ∂u_φ ∂φ + uθ r cotθ (A.16) ˙ ε_{θφ =} 1 2 1 rsinθ ∂uθ ∂φ + 1 r ∂uφ ∂θ − uφ r cotθ ! (A.17)

We non-dimensionalize these equations using the lithospheric thickness,L, as a scale length and using a scale velocity,U0, whence:

˙ ε_{i j}′ ₌ ε˙i j L U0 (A.18) τ_{i j}′ ₌ τi j gρcL(1−ρc/ρm) = ˙ ε′ i j ˙ ε′ klε˙′kl 1/n−1 Ar (A.19) Γ′ = Γ gρcL2(1−ρc/ρm) (A.20) whereAr is the Argand number [England and Mc_{Kenzie, 1982]}

Ar =gρcL(1−ρc/ρm)

B(U₀/L)1/n , (A.21)

and all variables are identified in Table 1.

The equations (A.14), in their discretised non-dimensional form, are solved foruθ,uφ subject to a prescribed set of velocities and forces on the external boundary to the sheet, using a finite-element method that was adapted from the method ofHouseman and Eng- land[1986];Houseman et al.[2008] by projecting the spherical problem onto the Carte- sian plane using the sinusoidal equal area projection and neglecting the small variation in r. A finite-element mesh of approximately equidimensional triangles is generated using the triangleprogram ofShewchuck[2002], with the size of the elements being reduced un- til well-resolved solutions are obtained. This mesh include nodes at the locations of all GPS measurements used to constrain the calculations (Section 2).

With this non-dimensionalisation, model velocities scale asArn_{. BecauseU}

0, the scale factor that relates model velocities to observed velocities, is unknowna priori, we choose an arbitrary value ofAr (that results in dimensionless velocities of order 1) and then deter- mine the scale factorU₀using Eq. (8). Bis then obtained from Eq. (9). BecauseU₀scales asAr−n_{, the resulting value ofBis independent of our choice of Ar. The Argand number is,}

therefore, indeterminate in these calculations. This indeterminacy arises because the bound- ary conditions contain no velocity scale (Section 3.2). Had we chosen to specify a non-zero velocity on any part of our solution domain, that choice would have set a velocity scale – henceB– and the Argand number would have become an additional parameter.

Acknowledgments

Lynn Evans has made numerous contributions to the development and maintenance of the 2D finite element codebasilused to make the calculations in this paper. We are grateful to many colleagues with whom we have collaborated during three decades of GPS work in this region. Principal among these are B. Aktuğ, A. Avallone, H. Billiris, P. Briole, A. Cingöz, P. Clarke, R. Davies, M. Floyd, A. Kılıçoğlu, A. Kurt, O. Lenk, K. Palamartchouk, D. Par- adissis, B. Parsons, C. Raptakis and G. Veis. This work has been supported by the Natural Environment Research Council through grant NER/A/S/2001/01607, by funding of the Na- tional Centre for Earth Observation (COMET+) and by the National Environmental Research Council and the Economic and Social Research Council through grant NE/J02001X/1. The figures were made using GMT [Wessel and Smith, 2013]. The data used in this analysis can be found in the supporting information. We thank Z-K Shen for providing the code used for calculating strain rates from GPS measurements. We are grateful to Peter Molnar and an anonymous reviewer for helpful reviews.

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