Earth and Planetary Science Letters, 77 (1986) 373-383 373 Elsevier Science Publishers B.V., Amsterdam - Printed in The Netherlands [61 Lithospheric necking: a dynamic model for rift morphology M.T. Zuber * and E.M. Parmentier Department of Geological Sciences, Brown Uniuersity, Providence, RI 02912 (U.S.A.) Received August 26, 1985; revised version received January 3, 1986 Rifting is examined in terms of the growth of a necking instability in a lithosphere consisting of a strong plastic or viscous surface layer of uniform strength overlying a weaker viscous substrate in which strength is either uniform or decreases exponentially with depth. As the lithosphere extends, deformation localizes about a small imposed initial perturbation in the strong layer thickness. For a narrow perturbation, the resulting surface topography consists of a central depression and uplifted flanks; the layer thins beneath the central depression. The width of the rift zone is related to the dominant wavelength of the necking instability, which in turn is controlled by the layer thickness and the mechanical properties of the lithosphere. For an initial thickness perturbation with a width less than the dominant wavelength, deformation concentrates into a zone comparable to the dominant wavelength. If the initial perturbation is wider than the dominant wavelength, then the width of the zone of deformation is controlled by the width of the initial perturbation; deformation concentrates in the region of enhanced thinning and develops periodically at the dominant wavelength. A surface layer with limiting plastic (stress exponent n = ~) behavior produces a rift-like structure with a width typical of continental rifts for a strong layer thickness consistent with various estimates of the maximum depth of brittle deformation in the continental lithosphere. The width of the rift is essentially independent of the layer/substrate strength ratio. For a power law viscous surface layer (n = 3), the dominant wavelength varies with layer/substrate strength ratio to the one-third power and is always larger than for a plastic surface layer of the same thickness. The great widths of rift zones on Venus may be explained by unstable extension of a strong viscous surface layer. 1. Introduction viscous lithosphere are evaluated. For simple rheo- logical stratifications we calculate the pattern of Rift zones are areas of localized lithospheric near-surface deformation which arises due to extension characterized by a central depression, horizontal extension and compare the results to uplifted flanks, and thinning of the underlying major morphological features of rifts, such as those crust. These features have been identified on many shown in Fig. 1. of the planets and their satellites; on the earth, The width and morphology of rift zones was rifts are found both on the continents and in first explained by Vening Meinesz [1], who sug- ocean basins, and represent the initial stage of gested that rifts form in an extending elastic-brittle continental breakup and seafloor spreading. Topo- layer which fails by normal faulting. Flexure of the graphic profiles illustrating the general mor- layer occurs in response to motion on a normal phology of rifts on continents and on the surface fault and a second normal fault forms where the of Venus are shown in Fig. 1. High heat flow, bending stresses in the layer are a maximum, thus broad regional uplift, and local magmatism are defining the width of the rift. Flanking highs form often associated with rifts, which suggests that by isostatic upbending of the elastic layer in re- thermal as well as mechanical effects play a role in sponse to graben subsidence. The elastic properties determining their morphology. In this study, we of the layer thus determine the width and mor- investigate the mechanical aspects of rifting by phology of the rift. Artemjev and Artyushkov [2] developing a simple model in which the dynamical qualitatively considered rifting due to necking in a consequences of flow in an extending plastic or ductile crust which is strong near the surface and weaker at depth. A perturbation in lower crustal * Now at: Geodynamics Branch, NASA/Goddard Space Flight thickness localizes during uniform extension and a Center. corresponding stress concentration in the brittle 0012-821X/86/$03.50 © 1986 Elsevier Science Publishers B.V. 374 (a) GULF OF SUEZ 2ooo -7 ~ 1',~o ~ RHINE GRABEN RIO GRANDE RIFT I I I 0 tO0 200 DISTANCE (km) 2"r~" 286". 37N .,o A~ ,Oo 'F" I _....~. i "3, B~ oE. .. o -/ .o CD,- 4 I I C' 0 1 D~ / / / , E~ -- BRIGHT FD,. ....... DARK o~"- I BRIGHT- LINEAR G~ EDGE OF BRIGHT ARFA :, I ,': Hm,- \ J~ dD,- ot::~- ""~"~°"" I "'1o :E' --q: Oe.-~ .,I 0 T ',"°,~ . d.' 4,00 (b) . -?" Fig. 1. (a) Topographic profiles across several continental rift zones. From Buck [15]. (b) Map of radar bright and dark linear features, interpreted as faults, and topographic profiles across rift zone in Beta Regio, Venus. Note the presence of a central trough bounded by uplifted flanks. The arrows represent the approximate locations of major bounding faults shown on map. Vertical lines mark edge of the map. The locations of Rhea and Theia Mons, interpreted as volcanic shields, are shown for reference. From Campbell et al. [251. 375 upper crust results in a narrow zone of normal ness h(x) which has a viscosity much higher than faulting. Bott [3] developed a model for rifting by its surroundings. To satisfy the condition of equi- crustal stretching in response to external tension librium in the layer, the horizontal force F where: on the basis of the elastic model of Vening Meinesz. F=axx h (1) In this model, the elastic upper crust responds to tension by normal faulting and graben subsidence must be independent of x. To satisfy the condition while the ductile lower crust undergoes thinning of incompressibility: due to horizontal flow beneath the subsiding Cxx = -h-I Oh/Ot (2) graben. In this study, we examine rifting as the growth where the minus sign indicates that the layer thins of a necking instability. To nucleate a rift, a small as it extends. The constitutive relationship be- thickness perturbation is imposed at the base of a tween stress and strain rate is: strong layer which overlies a weaker substrate. For ,xx =AoL (3) a range of rheological parameters we evaluate the where n is the stress exponent of the layer and A conditions for which the initial disturbance will is a constant. By combining (1) to (3), the rate of amplify as the lithosphere extends, and we de- change of layer thickness can be written: termine the associated pattern of near-surface de- formation. As shown later (see Fig. 5), the pattern Oh/Ot = -AF"h 1-" (4) of deformation consists of a central depression For later comparison with our linearized two-di- beneath which the strong layer thins due to neck- mensional formulation, h can be expressed: ing, and flanking uplifts, all of which are char- acteristic of rift zones. Horizontal extension of the h = n(t) - 8(x, t) (5) lithosphere, which drives the instability, could oc- where H(t) is the layer thickness for uniform cur in response to remotely applied forces or thinning and 8 is a small (<< h) thickness varia- horizontal forces due to regional doming on a tion. By substituting (5) into (4), expanding, and scale much larger than the width of the rift zone. integrating with respect to time, the amplitude of The model results are thus applicable to both the layer thickness variation may be written: passive and active rifting (e.g. [4]). The initial ~(x, t) = 6(x, 0) exp[(n- 1),xxt ] (6) thickness perturbation could correspond to a pre- existing structural weakness such as a tectonic where 8(x, 0) describes the shape of the initial suture or a thermal anomaly due to an igneous perturbation and Cxx is independent of time. The intrusion, diapiric upwelling, or convective trans- stress exponent n defines the growth rate of a port of heat to the base of the lithosphere. All of thickness perturbation. If n > 1 an initial dis- these have been suggested in association with ter- turbance will grow with time, while if n < 1 a restrial rifts. A thickness perturbation due to di- disturbance will decay and the layer will thin apirism or convective upwelling may be much uniformly. Equation (6) demonstrates that n > 1 is wider than the rift, while that due to igneous necessary for extensional instability and that the intrusion or a pre-existing weakness may be nar- horizontal layer thickness variation retains the rower than the rift. As discussed later, differences same shape as the perturbation grows in ampli- in the width of the perturbation may have implica- tude. tions for the character of rifting. 2.2. Linearized two-dimensional extension 2. Model formulation In a two-dimensional formulation, the total flow can be expressed as the sum of a mean flow or 2.1. One-dimensional extension basic state of uniform horizontal extension and a perturbing flow which arises due to instability. The simplest formulation for the growth of an The lithosphere is represented as a strong layer of instability in an extending layer treats the flow as thickness h overlying a weaker substrate. Two locally one-dimensional with a uniaxial stress Oxx strength stratifications, illustrated in Fig. 2, are and strain rate c** [5]. Consider a layer of thick- considered. For creep deformation, the strain rate 376 d-Model C-Model viscous fluid: -f oij = 21acij - p3ij (8) h where # is the dynamic viscosity and p is the + pressure. Substituting the stresses and strain rates R=Ga,d from (7) into (8) and linearizing about the basic state gives: 6,, X = (2~t/n)gxx-~ ;/h G = (9) 6xz = 2~gxz Fig.