NASA Technical Memorandum 86235 /• 1 • Intraplate Deformation Due To Continental Collisions: A Numerical Study of Deformation In A Thin Viscous Sheet tNasa-TM-86235) v INTfiaBLaTE DEFORHATIONr DOE ;. tN86-3>2012; TO COHTISENTaJDXCCIIISICHSr A RaflEBICAL > STCIDT ; -••'>:• ... ; .;;•;; i OF DEFOfiaaflONwlN a:,THIN! VISCOUS SHEET ;-.'. .. .• . :" , 27 p CSC^.08E Richard C. Morgan October 1985 NASA 47 ..a. tf tt1 fI NASA Technical Memorandum 86235 /v . f (/' Intraplate Deformation Due To Continental Collisions: A Numerical Study of Deformation In A Thin Viscous Sheet Steven C. Cohen Richard C. Morgan Geodynamics Branch Goddard Space Flight Center Greenbelt, Maryland NASA National Aeronautics and Space Administration Goddard Space Flight Center Greenbelt, Maryland 20771 INTRODUCTION The collision of tectonic plates involving large continental masses can produce significant crustal deformations, not only in the vicinity of the plate boundary, but also well into the interior of the plates. The best studied example of a contemporary continental collision is that involving the Indian and Eurasian plates, but other examples of both contemporary and ancient collisions are well known. The former category includes the collision between the Arabian and Eurasian Plates resulting in deformation in the Zagros Crush Zone, Anatolia, and elsewhere, and the incipient collision between Africa and the European portion of Eurasia. Ancient erogenic zones which display residual evidence of past continental collision include the Appalachian-Caledonian system, the Hercynian System in central Europe, and the Urals (Condie, 1982). Despite the attention given to both the general phenomena involved in continental collisions and to the geotectonics of specific collisions, the mechanical processes are still not well understood. In fact, controversy still exists about the correct kinematic description of the Indian-Eurasian collision. Nevertheless, the recent progress that has been made in understanding continental tectonics has sparked the emergence of several prototype models of collisional processes. The first of these models is the continental analog of the subduction of oceanic lithosphere under a continental plate near an oceanic trench. In the continental collision case crust from one plate overrides that of the other. The partial subduction of crust from one of the plates (which may have had a significant oceanic component as in the case of the Indian Plate) may be at a shallow angle with the possibility of considerable horizontal motion, or at a steep angle with implied penetration of the lithospheric plate into the mantle. In the case of the Indian-Eurasian colli- sion, there is little doubt that overthrusting of Asian continental crust onto lithospheric material from the oceanic portion of Indian plate occurred during the closing of the Tethys ocean. While localized thrusting occurs today in the vicinity of the Himalayas, it seems unlikely that this model can be used in the Tibet region, north of the Indus suture. One difficulty with the model is the fact that the buoyant Indian crust would tend to resist subduction. Both the absence of northly directed compres- sional seismic activity, and the general absence of intermediate and deep depth seismic activity around Tibet argue against using this model to explain deformation within the Eurasian plate. A second model assumes a viscous deformation of one plate as a consequence of the ram-like penetration into this plate by a more rigid collisional partner. This model is not suitable for studying the thrusting responsible for the Himalaya orogen, but it has been used with considerable success by Tapponier and Molnar (1977) and others to explain the eastward extrusion of Tibet along the Altyn Tagh and Kun-Lun faults, the plateau-like thickening of Tibetan crust, and a number of other Asiatic tectonic features. A third model involving the shearing of thin layers of crust and their overthrusting one on top of another to accommodate the convergence between the plates has come to be known as "flake tec- tonics" (Oxburgh, 1972) and has been used in the study of both Appalachian and Alpian geological features. A model for Tibet in which the uniform topography is explained by having the plateau float like a hydrostatic head on a fluid has been discussed by several authors. Recently Zhao and Morgan (1985) expressed the view that relatively cold and rigid Indian crust is penetrating into warm, low viscosity Asiatic lower crust with Tibet rising in response to the induced hydrostatic pressure. These various models are not entirely mutually exclusive and many variants of each model can be advanced. In this paper we propose to explore in some depth various features of the punch/viscous sheet model of continental deformation. Our aim here is to understand the details of the model and to test how sensitive the model predictions are to the choices of numerical values for the parameters and to some of the implicit and explicit assumptions. We will limit the analysis to cases having simple geometries where our intuitive insight is sharper than might be the case for geologically truer condi- tions. We refer the interested reader to the papers in the references for more detailed discussions of the model justifications and its applicability to specific contemporary environments. MODEL DESCRIPTION Figure 1 shows one version of the punch/deformable sheet model. By and large we will employ the mathematical development of this model that was derived by England and McKenzie (1982, 1983). Original calculations based on this model used finite difference techniques which required that the punch boundary condition be expressed as a flux of material rather than a rigid indentation. Houseman and England (1985) improved the model by using finite element techniques with prescrib- ed boundary velocity conditions representing the punch motion. We use our own formulation of the U,V - 0 y— s(x,y,t)_ 4 / ' N 1 'iY 1 pc AJivirncooiDLC, mini , E,X \CO\JS SHEET 1 1 1 \X = 0 \ \ i 4\ ! i i X PUNCH U,V = 0 Figure 1: Punch/viscous sheet model of continental collisions. The penetration of a rigid punch into a deformable sheet is represented by moving the boundary between points A and B north- ward at a uniform, constant velocity. Stress and deformation variables are averaged over the depth of the sheet. The sheet has a two layer density structure. The crustal density is gc; this layer has a thickness, s. The mantle density is Qm. The sheet is thin compared to its length and breadth, is subject to gravitational as well as collisional stresses, and deforms as an incompressible viscous fluid. finite element approach with differences from the earlier work that will be introduced in the discus- sions to follow. The viscous sheet is assumed to be incompressible and in local isostatic equilibrium. The isostacy condition is maintained by Airy type compensation against a crustless ocean plate. The sheet thickness is assumed to be small compared to its length (the X or eastward direction) and breadth (the Y or northward direction). The deformation of the sheet is governed by a power law flow of the form: T, = BE(H &y CD where Ty and Sy are components of the deviatoric stress and the strain rate tensor respectively, E is the second invariant of the strain rate tensor, and n is a power coefficient for linear (n = l) or nonlinear (n> 1) flow. The stress and strain rate variables are averaged over the thickness of the plate. The density structure of the plate is two layered. The crust has a density QC and a thickness, s (initial value s0); the underlying mantle has a density gm. The initial thickness of the sheet is L; after collision its thickness at a point x,y is L+h(x,y) where h can be interpreted as a topographic height and we require h< <L everywhere. The isostatic condition relates the crustal thickness, topographic height, and densities: h = s(i -ec/em) (2) The depth averaged equilibrium equations provide the quasi-static equations of motion for the model (England and McKenzie; 1982, 1983), viz, 1 _1_ dx dy/J ' fly I"" 2 \dy ' 3x, (3) = l-£.c (} -n./n_-» — £1—1 and d_ a I y L. \ / j |_ \ / J (4) = -^£\r^ (i -gc/em) -5oy— where u and v are the velocities in the x and y directions respectively. The left hand sides of Equa- tions 3 and 4 contain dynamical quantities relating to the velocity field; while the right hand sides contain thickness and density variables which are quantities related to the gravitational stresses. The equilibrium conditions can be solved for the velocity field whenever the crustal thicknesses are known. In the general case, the continuity or conservation of mass condition provides a third equa- tion for the three unknown quantities: u(x,y,t), v(x,y,t), and s(x,y,t). For finite difference calcula- tions the continuity condition is best expressed in an Eulerian frame of reference with fixed positions for all nodal points. The Eulerian frame can also be used with finite element calculations, but a Langranian frame in which nodal points move with the material flow, is more convenient. In this frame of reference the continuity condition expressing conservation of crustal mass is: 3s / 3u 3v The last form of the equality is a consequence of the incompressiblity condition, &XX +£yy + &zz = 0 (6) As we will now show the equilibrium equations have a useful and particularly simple form for a linear rheology. First we define a term, the gravitational strain rate, £g, which has strain rate units and can be used to express the contribution of gravity to the equilibrium conditions, / 2 • _ g ec(i-ec em)s (i) ss BL The parameter B=2 77 where r; is the viscosity. The linear equilibrium equations can now be written (8a) dy2 dxdy dx L BL or 8 2 3x and r -I 4"d72~+~dF"+ 3 dylR = "dy" \_ ~ BL° m J (9a) or = j 2 dy When the terms involving the gravitational strain are small, the equilibrium equations can be solved for the velocities; these velocities can then be used in the continuity equation.