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

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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. 2. Strength stratification of strength jump (J) and continu- where ~ is the viscosity evaluated at the stress or ous strength (C) models. The former is described in terms of strain rate of the basic state. For a single Fourier the ratio of layer/substrate strength R = #~)/#~2), while the harmonic, perturbing velocities in the vertical and latter is described by the ratio a = ~/h of the viscosity decay horizontal directions: depth in the substrate to the strong layer thickness. The layer and substrate are described by uniform densities (O> P2) and = Wcos kx (10) power law exponents (nl, n2). =-k 1DWsin kx (c) is proportional to stress (o) to a power n I = 3. where D= d/dz and k (= 27r/)~) is the wave Deformation by distributed faulting can be ideal- number, satisfy the incompressibility condition. ized using a perfectly plastic material with a stress Within any layer in which the viscosity varies exponent of n I = m. The substrate is assumed to exponentially with depth, the equations of equi- deform by creep with n 2 = 3. In the strength jump librium are satisfied if: (J) model, the layer and substrate both have a D4W + 2~-lD3W+ [~-2 _ 2k2(Z/n _ 1)] D2W uniform strength with Oxx~'(1)> Uxx~'(2),where the super- scripts 1 and 2 refer to the layer and substrate, -2k2~-'(2/,- 1)DW+ k2[k 2 + ~,-2] W= 0 respectively. In the continuous strength (C) model, (11) the strength is uniform in the layer, continuous across the layer-substrate interface, and decays Solutions of (11) are given in Fletcher and Hallet exponentially in the substrate with an e-folding [6]. The flow in any layer with uniform strength or depth ~'. Since we are interested primarily in dy- viscosity can be obtained from (11) by taking the namic, stress-supported topography and numerous limit as ~" + ~. previous studies have considered the effect of iso- The velocities u and w and the stress compo- statically compensated crustal thickness variations, nents %, and ozz must be continuous across each the layer and substrate are each represented by a interface. At the surface, the vertical normal stress uniform density (p) with I)1 = 02. must equal the weight of the surface topography In a layered medium, any small perturbation and the shear stress must vanish. Expressed in along an interface will cause deviatoric stresses terms of the basic and perturbing stresses, the proportional to the amplitude of the perturbation stress continuity conditions at each interface give: resulting in the growth of an instability. The total 5x~(x, di)= [~x~/-') - ~i']d3,/dx (12) velocities (u, w), stresses (oo), and strain rates (q j) are written: G(x, 4) = (p,-, - 0,)ga, u=~+fi where d i is the depth of the interface. These w=~+~ equations have been linearized for small distor- % = o-,j + a,j (7) tions (3i) of the interface (cf. [7,8]). Vertical veloc- Eij = Eij + Eij ities arising from these perturbing stresses amplify initial deformation of the interfaces further en- where a bar represents the mean flow and a tilde hancing the perturbing stresses resulting in insta- the additional or perturbing flow. For an isotropic bility. At a given time, the shape of the ith inter- 377 face can be represented by the superposition of greater than unity and thus contributes to instabil- Fourier harmonics: ity. While both values of q are required to com- pletely describe the evolution of interface shapes, 6i(x, t)=~Ai(k, t) sin kx (13) solutions of (15) which retain only the positive k value of q yield a good approximation to the where i= 1, 2 refer to the free surface and the perturbed interface shape. Note the correspon- layer-substrate interface, respectively, and 8 and dence of (15) to the one-dimensional growth rate A are the spatial and wave number domain repre- equation (6) with the substitution of q for n. As sentations of the interface shapes. In terms of the subsequent results for the two-dimensional prob- perturbing vertical velocity, the time rate of change lem show, q varies directly with the stress expo- of amplitude A is: nent in the layer, but unlike the one-dimensional A, = W(k, d~) - ~x~A, (14) growth rate, q is a function wave number. The J and C models are each expressed in terms where A = dA/dt. The system of equations con- of two dimensionless parameters. In both models, sisting of (11), (14) and that obtained by substitut- ing (13) into (12) has solutions of the form Acc S = (Pa- Po)gh/[a~lx)/2] where P0 is the density above the layer, determines the relative importance exp[(q - 1)~xt ], where q is the growth rate factor of surface topography which stabilizes the defor- which measures the degree of instability of the disturbance at a particular wave number. The mation and stresses due to surface distortion which help drive the instability. In the J model, R = value of unity subtracted from q represents the ~1)/7,(2) the layer/substrate strength ratio, con- kinematic interface distortion from the second x/Uxx, trols the relative magnitude of stresses due to term on the right-hand side of (14). Complete distortion of the surface and the layer-substrate ~olutions of this system of equations can be writ- interface. In the C model, the layer-substrate inter- ten: face does not help to drive the instability because Al(k, t) = A,1 exp[(q,- 1)exx ] the strength is continuous across it. The amount of viscous resistance to deformation in the substrate +A12 exp[(q2 - 1)exx] is determined by the ratio of the viscosity decay (15) A2(k, t) =A21 exp[(q 1 - 1)exx ] depth to the layer thickness a = ~/h. +A22 exp[(q2 - 1)exx] 3. Results In the limit of large layer/substrate viscosity where ql and q2 are eigenvalues, (An, A12 ) and contrast, the one-dimensional growth rate agrees (A21 , A22 ) are the eigenvectors corresponding to with the growth rate of the dominant wavelength ql and q2, respectively, and exx (=ixx t) is the for the two-dimensional formulation, as noted by total horizontal extension at a time t. The length Emerman and Turcotte [9]. However, Fig. 3a shows of the eigenvectors are determined from the pre- that the growth rate spectrum for one-dimensional scribed initial conditions. The surface topography extension is independent of wave number so this is assumed to be initially flat [81(x, 0)= 0], and model fails to predict a dominant wavelength. the initial shape of the layer-substrate interface is Thus, to explain the regular development of geo- given by a prescribed function. The Fourier trans- logic structures using one-dimensional theory, ini- form of the initial interface shapes is substituted tial thickness perturbations along layer interfaces into (15), and the perturbed surface topography is must be distributed periodically. In the absence of found from the inverse Fourier transform of a process that defines a natural length scale, the Al(k, t). Patterns of deformation in the layer and occurrence of periodic initial disturbances seems substrate can be determined by calculating the unlikely. displacements due to the perturbing flow at points Fig. 3 also shows growth rate spectra for two- on a coordinate grid. dimensional models with limiting plastic (n 1 = 10 4) The number of growth rate factors at each wave and power law viscous (nl = 3) surface layers number (in this case two) equals the number of plotted as a function of dimensionless wave num- interfaces. In general, only one value of q is ber k' (=2~rh/)Q. The wave number (k~) at 378

a) b} a) -- nl= r, ~..-3 / 1.5 L one-d xO.5 I0 d-Model "I -- nt=3 /~=m ...... ~. | ,~ ...... xlh ql.O ~10_ 2 -I0 b) 0.5 -2C r~:~/[ o ! d model \ k' -3o [ l C model I Fig. 4. (a) Rift morphology for limiting plastic (n 1 = 104, solid Fig. 3. (a) Growth rate spectra as a function of wave number line) and power law viscous (nl = 3, dashed line) surface layers non-dirnensionalized by the strong layer thickness k' ( = kh). for the strength jump (J) model. For both cases S = 1.2, R = 50 The growth rate for one-dimensional flow in a layer with stress and n 2 = 3. For comparison, the surface topography in each exponent n = 3 is simply equal to n over the entire range of k'. case is normalized by the depth of the central depression. (b) Thus the one-dimensional model fails to predict a dominant Rift morphology for the C model with plastic surface layer and wave number. For two-dimensional flow, the solid and dashed with S=1.2, ct=5 and n2=3. lines refer to cases with limiting plastic (n] = 104) and power law viscous (n 1 =3) surface layers, respectively, for the J and arise in response to the perturbing flow in- model. For both cases S = 1.2, R = 50 and n 2 = 3. The peaks in the growth rate spectra define the dominant wave numbers. duced as the layer necks. For the J model, a plastic (b) Two-dimensional growth rate spectra for the C model for surface layer (n] = 10 4) results in deformation plastic and power law viscous surface layers with S = 1.2, a = 5 localized into a region overlying the initial thick- and n 2 = 3. In this model the growth rate for a viscous surface ness perturbation. With a power law viscous layer is always less than unity and therefore the layer is stable surface layer (hi = 3) deformation is more broadly with respect to necking. distributed. For the C model, the relative amount of uplift of the flanks is much greater than in the J which the growth rate factor is maximized (qd) model, and small surface depressions occur outside defines the dominant wavelength )k d (= 2'n'/kd). the flanking highs. A variety of initial perturbation For the J model (Fig. 3a), the magnitude of the shapes including Gaussian, boxcar, and triangular growth rate for a lithosphere with a plastic surface functions have been examined. The topography is layer is approximately two orders of magnitude independent of the shape of the initial perturba- greater than that for a power law viscous layer tion. with the same value of R. This indicates that a The amplitude of the deformation is dependent lithosphere in which the near-surface deforms on the amplitude of the initial disturbance, the plastically is much more unstable in extension amount of horizontal extension exx, and the mag- than a lithosphere which deforms viscously nitude of the growth rate factor. In the linearized throughout. The growth rate spectrum for the C theory, the amplitude of deformation for a given model is shown in Fig. 3b for both plastic and horizontal extension is exactly proportional to the viscous surface layers. For n I = 3 the growth rate amplitude of the initial disturbance. Since for ac- is everywhere less than unity, which indicates that tual rift zones the amplitude of this disturbance is the lithosphere is stable with respect to necking. unknown, only relative amplitudes, which depend An initial disturbance in strong layer thickness on the growth rate spectrum and horizontal exten- will decay with time, and deformation of the sion, are shown in Fig. 4. Fig. 4a shows that the medium will be manifest as uniform thinning. flanking uplifts are narrower and have a greater This holds for any reasonable range of lithosphere amplitude relative to the depth of the central mechanical properties. depression for the J model with large n t. For the Topographic profiles for the two-dimensional same initial disturbance amplitude, the absolute growth rate spectra in Fig. 3 and an initial thick- magnitude of the topography for the plastic surface ness perturbation of the form ~2(X, O)(X e (~/d)2 layer is greater than that for the viscous layer, where d << h are shown in Fig. 4. The topography reflecting the larger growth rate for the former in each case consists of a central depression and case as shown in Fig. 3a. flanking uplifts which are dynamically supported As for the topography discussed above, the 379

IIllllllllllll~ .. ~llllllllllll z']']l']'~'l I',li IIIIIIlIIl]l Illlllllllllllllll~ _/x "~4.UIIIIIIIIIIIIIIIII VE-Zx

Fig. 5. Deformation of an extending plastic surface layer with an initial thickness perturbation at its base. The width of the disturbance is determined by the growth rate spectrum in Fig. 3a. The shape of the initial perturbation of width d (<< h) is shown schematically by the dashed line. The central depression I0° K~ 102 R i0~ I0 V3 I/6 V9 VI2 I/f5 and uplifted flanks produced due to the unstable growth of the Va initial perturbation are characteristic topographic features of Fig. 7. (a) Dominant growth rate factor qd vs• R for the J rift zones (cf. Fig. 1). model• (b) qd VS. 1/a for the C model• For r/1 = 3, q is always less than unity and the lithosphere extends uniformly. S = 0 pattern of strong layer deformation shown in Fig. and n 2 = 3 for both cases. 5 reflects the shape of the growth rate spectrum. While the absolute amplitude is arbitrary, the rela- model with a plastic surface layer. However, the tive amplitudes are everywhere properly repre- flank-to-flank width of the rift zone for the C sented. Hence if the topography at the surface or model is about twice the dominant wavelength. In another interface is known, the amplitude of de- this case the width of the central depression is formation at any point in the layer or substrate is approximately equal to the dominant wavelength. determined. For example, Fig. 5 shows that the The relationships between rift zone width and the amount of upwarping at the base of the strong dominant wavelength demonstrated by the above layer is approximately four times the depth of the examples hold for the complete range of cases we central depression. Likewise, the amount of rela- have examined for the J and C models. tive uplift of the flanks is about half the depth of Fig. 6 shows the variation of k~ with R for the the central depression. J model. For power law viscous near-surface be- The width of the rift zone is controlled by the havior k d varies as R -]/3. This relationship was dominant wavelength. For the J model, the domi- first determined for flexural buckling of a layered nant wave numbers shown in Fig. 3 correspond to medium [10] and was later shown to be valid for wavelength to layer thickness ratios of Xd/h = 4.0 boudinage and folding of a layered viscous medium and 10.6, respectively, for plastic and viscous in the limit of large viscosity contrast [7,8,11]. In surface layers. In the J model, the width of the rift the plastic limit k~ is essentially independent of from flank-to-flank is almost exactly equal to the R. For this case the width of the rift will be dominant wavelength. In the C model for n t = 10 4, the dominant wave number corresponds to ~d/h = 3.7, which is very similar to the result in the J 1.8 '~'- -~v...... -' -' -" -' ~,6o'8° 15 ~0 ~k' -- n,'lO'~ nl=~o 1.2t q~'X,~ - - n,-3 120 % io o %[ ~ ,oo 0.9[ 80

0.6 SO 40

• ~ - --Tkd...... 20 . . . 0 2 4 6 8 I0 12 14 S IOI 102 R IOa Fig. 8. Dominant wave number k~l and dominant growth rate Fig. 6. Dominant wave number k~ as a function of strength factor qd as a function of S for n] =104 (solid) and n 1 = 3 contrast R for the J model with S = 0 and n 2 = 3. (dashed) with R = 50 and n 2 = 3. 380 primarily a function of the strong layer thickness. as a mechanical response to localized extension. In The dominant wavelength decreases with increas- the present model the flanks are solely a conse- ing stress exponent in the layer (n 1), which is quence of viscous or plastic flow. consistent with results on folding and boudinage A thermal anomaly that is broader than the of a non-Newtonian layer embedded in a viscous region of crustal thinning may also explain flank- medium [11]. ing uplifts. Buck [15], Keen [16], and Steckler [17] The variation of the growth rate at the domi- suggest that thermally-produced uplifted flanks nant wave number with R for the J model and form as a result of horizontal heat transfer due to with 1/a for the C model is shown in Fig. 7. Note small-scale convection induced by the rift temper- that qd increases with R and l/a, indicating ature structure. Steckler [17] interpreted the enhanced instability for large strength contrasts amount of uplift in the Gulf of Suez to be indica- and small e-folding depths. Increased instability is tive of lithospheric heating greatly in excess of a result of the relative decrease in viscous resis- that expected due to uniform lithosphere exten- tance in the substrate for both cases. In the plastic sion. Buck [15] showed that the magnitude and layer case for the J model the magnitude of qa is distribution of uplift associated with continental generally proportional to R, indicating that the rifts may be explained by small-scale convection. strength contrast at the layer-substrate interface The relative contributions of thermal and me- provides a significant contribution to driving the chanical mechanisms should be reflected by the instability. timing of uplift on flanks. If uplift is dynamic, Fig. 8 summarizes the effects of the buoyancy/ then deepening of the central trough and uplift of strength parameter S for the J model and shows the flanks should occur simultaneously. If it is a that k~ and qa decrease with increasing S for thermal effect, then uplift should lag the forma- both large and small n~. As would be expected, an tion of the central depression. Flanking uplifts increase in density with depth across an interface along the Gulf of Suez formed during the main stabilizes the perturbing flow. The decrease in phase of rift development [17], but the relative dominant wave number with increasing S is con- timing of the formation of the rift valley and the sistent with results for folding of a layered visco- flanks is 'not yet clearly defined. Thermal uplift, elastic medium under the influence of gravity [12]. unless it is frozen in by a thickening elastic litho- Increasing S results in deepening of the rift and sphere, should decay due to cooling after exten- damping of the bounding flanks. sion ceases. Uplift due to viscous or plastic flow, produced while the extending lithosphere is at 4. Discussion yield, must also be frozen-in as stresses fall below the yield stress at the cessation of extension. 4.1. Flanking uplifts 4.2. Continental rifts Uniform stretching of the lithosphere resulting in crustal thinning and subsequent conductive The width of a rift zone formed by necking is a cooling may account for the subsidence of rift function of the thickness of the strong layer of the basins (e.g. [13]) but does not explain the flanking lithosphere. The strength stratification of the litho- highs characteristic of rift zones. Flanking uplifts sphere is determined by brittle behavior at shallow in our necking model form in response to dynamic depths and creep at greater depths (cf. [18]). The upwarming produced by the unstable flow. Since brittle strength increases linearly with depth to a there are no density contrasts at depth, the flanks maximum value determined by a creep strength are supported entirely by stresses in the layer. As which decreases with depth as temperature in- in the models of Vening Meinesz [1] and Bott [3] creases. Neither the J nor C model is an exact which treat the near-surface as an elastic layer, the representation of this strength stratification since flanking uplifts regionally compensate the rift de- in both models the linear increase of brittle strength pression. Finite element solutions [14] incorpo- with depth is approximated by a layer of uniform rating elastic as well as plastic and viscous behav- strength. In addition, in the C model the strength ior have also shown that flanking uplifts can occur falls to zero at depth while in the J model the 381 strength is discontinuous at the base of the strong turbation much wider than the spacing of individ- layer. For the purposes of the present study, the ual basins and ranges ( -~ 30 km). The depth distri- strong layer thickness will be taken to correspond bution of seismicity [21], high heat flow [22], and to the depth of the brittle-ductile transition. In a low Pn velocities [23] in this area suggest that an model lithosphere with a plastic surface layer, the upper mantle thermal anomaly of large horizontal layer-substrate interface corresponds to a change extent could be the cause of the initial perturba- in power law exponent, indicating the change in tion. the mode of deformation from faulting to ductile flow. Strength decreases rapidly with depth below 4.3. Rifting on Venus the brittle-ductile transition. If estimated from this strength stratification, the strong layer thickness A number of features on Venus with the mor- would not be significantly greater than the depth phology of rift zones have been recognized from to the brittle-ductile transition. Pioneer Venus radar altimetry [24]. High-resolu- As shown in Fig. 4, a rift formed in a litho- tion earth-based radar images have recently been sphere with a plastic surface layer has a flank-to- obtained for a major rift in Beta Regio [25]. As flank width = 4h. If the strong upper crustal shown in the across-strike profiles in Fig. lb, the region of the continental lithosphere is best de- topography consists of a central depression and scribed by a plastic theology, then a continental bounding highs which may represent uplifted rift with a typically observed width in the range flanks. The width of the rift varies along strike but 35-60 km [19] requires a strong layer thickness of averages about 150 km from flank to flank. Two 9-15 km. An extending continental lithosphere in volcanic shields occur along the rift, Rhea Mons in which a quartz rheology approximates flow in the the north and Theia Mons in the south. While crust undergoes the transition from brittle to volcanic construction could contribute to the ductile behavior at a depth of about 15-20 km flanking highs, they are best developed where the depending on the geothermal gradient and strain central depression is deepest and in areas not rate (cf. [18]). This is comparable to the depth to associated with the volcanoes (profiles G and H in which earthquakes are observed in the continental Fig. lb). lithosphere and to which faulting penetrates in The high surface temperature of Venus, ap- continental rifts as shown in seismic reflection proximately 700 K, suggests that ductile deforma- profiles. tion will occur at shallower depths than on the For a strong viscous surface layer with nl = 3, earth. On the basis of estimates of rock strength, the dominant wavelength, and therefore the rift surface temperature, and the spacing of features of zone width, depends on the ratio of layer to sub- presumed tectonic origin in the banded terrain, strate strength as shown in Fig. 6. For a rift zone Solomon and Head [26] suggest that the elastic 60 km wide, typical of the well-studied East Afri- lithosphere thickness on Venus is in the range can Rift or Rhinegraben, a strong layer thickness 1-10 km, and therefore too thin to explain the of 15 km requires an R only slightly greater than width of the Beta Regio rift by elastic flexure, unity. which would require an elastic-brittle layer thick- The width of the rift is controlled by the domi- ness in excess of 60 km. Schaber [24] estimated a nant wavelength as long as the initial perturbation brittle layer thickness in the range 47-69 km on width d < ?~d. However, if d > ?~a the width of the the basis of the widths of other presumed rift rift is governed by the width of the initial dis- valleys on Venus. Radar bright lineaments in Beta turbance. If an initial perturbation is much wider Regio, interpreted as faults [25], suggest the pres- than the dominant wavelength, corresponding per- ence of a brittle surface layer. On the basis of Fig. haps to broad-scale lithospheric thinning, exten- 4, the width of the Beta Regio rift would require a sion localizes in the thinned region and deforma- plastic layer thickness of about 40 km. If the tion within that region develops periodically at the near-surface of Venus is weaker than the earth, S dominant wavelength [6,20]. The vast regional ex- will be greater than that assumed in Fig. 4. The tent of the Basin and Range Province (> 103 km) limit of large S corresponds to a rift zone width may reflect an initial strong layer thickness per- = 6h and a plastic layer thickness of 25 km. This 382

is greater than the depth to the brittle-ductile strong layer corresponds to the maximum depth of transition on Venus estimated by Solomon and brittle deformation, determined either by experi- Head [26] and would require a geothermal gradi- mental flow laws or the observed depth of earth- ent less than that which they assume (20 K km-l). quakes, then the width predicted by this model is The 10-20 km spacing of bright lineaments where generally consistent with that of a typical con- they are most well-developed is comparable to the tinental rift zone. Dominant wavelength and there- width of radar bright bands in the banded terrain. fore rift zone width for a plastic surface layer is If the radar bright features are due to faulting essentially independent of the layer/substrate with a spacing comparable to the brittle-elastic strength contrast. For a strong viscous surface layer thickness, then this thickness in Beta Regio layer (n~ = 3), dominant wavelength varies as the should be similar to that in areas of banded ter- strength ratio to the one-third power. Extension of rain and too thin to explain the rift zone width. If a viscous surface layer always produces a rift zone the plastic region of near-surface faulting is thin, that is wider than that for a plastic surface layer. then lithospheric necking may be controlled by a The great width of rift zones on Venus, relative to thicker viscous layer deforming by creep. In this the spacing of other features of presumed tectonic case the width of the rift zone is determined by origin, may be explained by ductile necking of a the strong layer thickness and the layer/substrate strong viscous layer. strength contrast. On the basis of Fig. 6, 10 and 20 km thick strong layers require values of R = 100 Acknowledgements and 10, respectively, to explain the width of the Beta Regio Rift. This research was supported by NASA grant NSG-7605. We thank Ray Fletcher for helpful 5. Summary discussions and Philip England for a constructive review of the manuscript. Unstable extension of a lithosphere consisting of a strong surface layer overlying a weaker viscous References substrate results in a pattern of deformation that is consistent with the major morphological char- 1 F.A. Vening Meinesz, Les "graben" africains r6sultat de acteristics of rift zones. The surface topography compression ou de tension dans la croBte terrestre?, K. consists of a central depression, beneath which the Belg. Kol. Inst. Bull. 21, 539-552, 1950. layer thins by necking, and flanking uplifts. The 2 M.E. Artemjev and E.V. Artyushkov, Structure and isostasy rift is nucleated by introducing a small amplitude of the Baikal Rift and the mechanism of rifting, J. Geo- phys. Res. 76, 1971. thickness perturbation at the base of the layer. For 3 M.H.P. Bott, Formation of sedimentary basins of graben an initial perturbation narrower than the domi- type by extension of continental crust, Tectonophysics 36, nant wavelength, deformation concentrates in a 77-86, 1976. zone of width comparable to the dominant wave- 4 A.H.C. SengSr and K. Burke, Relative timing of rifting and length. For an initial thickness perturbation wider volcanism on earth and its tectonic implications, Geophys. Res. Lett. 5, 419-421, 1978. than the dominant wavelength, deformation devel- 5 E.W. Hart, Theory of the tensile test, Act. Metall. 15, ops periodically at the dominant wavelength in the 351-355, 1967. region above the perturbation. 6 R.C. Fletcher and B. Hallet, Unstable extension of the The dominant wavelength is controlled by the lithosphere: a mechanical model for Basin and Range layer thickness and by the growth rate spectrum of structure, J. Geophys. Res. 88, 7457-7466, 1983. 7 R.C. Fletcher, Folding of a single viscous layer: exact extensional instability, which is a function of the infinitesimal amplitude solution, Tectonophysics 39, layer/substrate strength contrast, the density 593-606, 1977. stratification, and the stress exponents describing 8 R.B. Smith, Unified theory of the onset of folding, flow in the layer and substrate. Extension of a boudinage, and mullion structure, Geol. Soc. Am. Bull. 86, strong surface layer that deforms by distributed 1601-1609, 1975. 9 S.H. Emerman and D.L. Turcotte, A back-of-the-envelope faulting, idealized by plastic behavior (n 1 = oo), approach to boudinage mechanics, Tectonophysics 110, results in a rift zone width approximately four 333-338, 1984. times the plastic layer thickness. If the base of the 10 M.A. Biot, Folding instability of a layered viscoelastic 383

medium under compression, Proc. R. Soc. London, Ser. A evolutionary trends of continental rifts, Proc. 27th Int. 242, 444-454, 1957. Geol. Congr. 7, 165-216, 1984. 11 R.B. Smith, Formation of folds, boudinage, and mullions in 20 M.T. Zuber, E.M. Parmentier and R.C. Fletcher, Extension non-Newtonian materials, Geol. Soc. Am. Bull. 88, 312-320, of continental lithosphere: A model for two scales of Basin 1977. and Range deformation, J. Geophys. Res., in press, 1985. 12 M.A. Biot, The influence of gravity on the folding of a 21 G.P. Eaton, The Basin and Range Province: origin and layered viscoelastic medium under compression, J. Franklin tectonic significance, Annu. Rev. Earth Planet. Sci. 10, Inst. 267, 211-228, 1959. 409-440, 1982. 13 D. McKenzie, Some remarks on the development of sedi- 22 A.H. Lachenbruch and J.H. Sass, Models of an extending mentary basins, Earth Planet. Sci. Lett. 40, 25-32, 1978. lithosphere and heat flow in the Basin and Range Province, 14 H.D. Lynch, Numerical Models of the Formation of Con- in: R.B. Smith and G.P. Eaton, eds., Geol. Soc. Am. Mem. tinental Rifts by Processes of Lithospheric Necking, pp. 152, 209-250, 1978. 164-216, Ph.D. Thesis, New Mexico State University, Las 23 R.B. Smith, Seismicity, crustal structure, and intraplate Cruces, N.M., 1983. tectonics of the interior of the western cordillera, in: R.B. 15 W.R. Buck, Small-scale convection induced by passive rift- Smith and G.P. Eaton, eds., Geol. Soc. Am. Mem. 152, ing: the cause for uplift of rift shoulders, submitted to 111-144, 1978. Earth Planet. Sci. Lett., 1985. 24 G.G. Schaber, Venus: limited extension and volcanism 16 C.E. Keen, The dynamics of rifting: Deformation of the along zones of lithospheric weakness, Geophys. Res. Lett. lithosphere by active and passive driving mechanisms, Geo- 9, 499-502, 1982. phys. J. R. Astron. Soc. 80, 95-120, 1985. 25 D.B. Campbell, J.W. Head, J.K. Harmon and A.A. Hine, 17 M.S. Steckler, Uplift and extension at the Gulf of Suez: Venus: volcanism and rift formation in Beta Regio, Science Indications of induced mantle convection, Nature 317, 226, 167-170, 1984. 135-139, 1985. 26 S.C. Solomon and J.W. Head, Venus banded terrain: 18 W.F. Brace and D.L. Kohlstedt, Limits on lithospheric tectonic models for band formation and their relationship stress imposed from laboratory experiments, J. Geophys. to lithospheric thermal structure, J. Geophys. Res. 89, Res. 85, 6248-6252, 1980. 6885-6897, 1984. 19 I.B. Ramberg and P. Morgan, Physical characteristics and