A model for the formation of wrinkle ridges in volcanic plains on Venus

Physics of the Earth and Planetary Interiors 135 (2003) 161–171

A model for the formation of wrinkle ridges in volcanic plains on Venus Michele Dragoni∗ , Antonello Piombo Dipartimento di Fisica, Università di Bologna, Viale Carlo Berti Pichat 8, 40127 Bologna, Italy Received 15 May 2002; received in revised form 30 September 2002; accepted 02 October 2002

Abstract Ridged plains, the most abundant geologic terrain on Venus, are volcanic plains deformed by wrinkle ridges after their emplacement. It has been suggested that the ridges are the product of thermal stresses induced in the lithosphere by the increase of surface temperature due to the greenhouse effect of gases released during the emplacement of the volcanic plains. A model for the formation of wrinkle ridges is proposed where the ridges are assumed to be the surface effect of dislocations produced by compressive stress in the Venusian lithosphere. The lithosphere is modeled as a thermoelastic half-space, the surface of which is subject to a linear temperature increase during 100 Ma. The state of stress is calculated and the maximum distance is obtained at which dislocations can be placed in order that the compressive stress at the Venusian surface is everywhere lower than the rock strength. The model predictions in terms of dislocation density and crustal shortening appear to be consistent with observations. © 2002 Elsevier Science B.V. All rights reserved. Keywords: Venus; Volcanic plains; Wrinkle ridges; Thermal stress; Dislocations

1. Introduction Most of the surface of Venus is made of volcanic plains which originated in a period of extensive volcanic activity between 300 and 500 million years ago (e.g. Phillips and Hansen, 1998). As on the Earth, magma on Venus is originated in the mantle, an inner region where thermodynamic conditions produce the partial melting of rocks. A distinctive character of Venus is the close connection between the evolution of its atmosphere and its internal activity (Phillips et al., 2001). Global volcanic events, such as those required from the formation of plains, had a remarkable influence on the atmosphere of Venus. The degassing ∗ Corresponding author. Tel.: +39-51-2095020; fax: +39-51-2095058. E-mail address: [email protected] (M. Dragoni).

of the erupted magma released large quantities of volatiles, such as water and sulfur gases, contributing to the greenhouse effect (Bullock and Grinspoon, 1996, 2001). The external temperature variations affected the temperature and state of stress inside the planet, i.e. the factors controlling the surface tectonic activity, including volcanism (Anderson and Smrekar, 1999; Solomon et al., 1999). The most abundant geologic terrain on the surface of Venus is made by the ridged plains. These are formed by volcanic material deformed by wrinkle ridges subsequent to emplacement (Basilevsky and Head, 1996, 1998; Head and Basilevsky, 1998). The ridges in a plain are nearly parallel, having a consistent orientation over areas of thousands of kilometers, and their average distance is in the order of few tens of kilometers. Ridged plains are common in lowlands and make more than 60% of the surface of Venus. They should

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Nomenclature d D D∗ eij e¯ ij e¯ ∗ij g K L n p t ˆt t1

T

Tˆ u U w y0 z0

maximum depth extent of dislocations distance between neighboring dislocations maximum value of D strain tensor average strain tensor minimum average strain acceleration of gravity elastic bulk modulus horizontal length of a plain number of dislocations lithostatic pressure time upper value of the time interval time at which σ = τ at z = 0 temperature increase surface temperature increase at t = ˆt displacement field fault slip downdip width of dislocation surface position of fault trace depth at which σ = τ

Greek letters α thermal expansion coefficient β crustal shortening γ rate of temperature increase δ dip angle of dislocations κ strength coefficient λ, µ Lam´e constants ν Poisson modulus ρ mass density

σ differential stress σij stress tensor σijT

thermal stress

T σˆ yy

surface thermal stress at t = ˆt

σ¯ σ0 σ1 , σ2 , σ3 τ τ0 χ

a constant tectonic stress principal stresses rock strength rock strength at the Venus surface thermal diffusivity

have covered an even larger area at the time of their formation, since younger plains are in part superimposed with them today. The observation that very few of the impact craters, which are present in the plains, were deformed by the ridges shows that in the time interval between the formation of basaltic plains and the formation of ridges only few meteorites fell on Venus, suggesting that the ridge formation occurred in a geologically short time interval. An interval not longer than 100 Ma was proposed by Basilevsky and Head (1998), and Campbell (1999) suggests that crater counts for specific landforms offer little constraint on the duration of their development. Solomon et al. (1999) proposed a model for the evolution of Venus climate and made an approximate calculation of the thermal stress induced in the lithosphere by the temperature increase at the surface. They showed that climate driven variations in thermal stresses are consistent with the formation of many of the wrinkle ridges in the volcanic plains within 100 Ma of plain emplacement. During this period, the surface temperature of the planet increased linearly with time as a consequence of atmospheric warming, producing thermal stresses in the order of several tens of megapascals, able to fracture the lithosphere. The hypothesis that thermal stresses are responsible for the deformation at the Venus surface was also investigated by Anderson and Smrekar (1999), who employed a fixed plate and a strength envelope model to predict the depth of failure and the amount of strain. The formation of fold and thrust belts on Venus was explained by Zuber and Parmentier (1995) as the horizontal shortening of a laterally heterogeneous lithosphere. In the present paper, we consider a 100 Ma time interval and assume that the surface temperature of a thermoelastic half-space, representing the Venusian lithosphere, increases linearly with time. We calculate the thermal stress as a function of time and depth and suggest that the wrinkle ridges may be the effect of dip–slip dislocations on reverse faults formed by the fracturing of the lithosphere under compressive stresses. We shall assume that the stress release due to the dislocations controls the distance between the dislocations themselves and will look for relationships between the ridges density and the crustal shortening of volcanic plains. Thermoelastic models have often been considered in connection with the cooling of the terrestrial litho-

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sphere moving away from mid-ocean ridges. A thermoelastic half-space was used by Bratt et al. (1985), while Parmentier and Haxby (1986) employed the thin plate approximation. 2. The model We model the lithosphere of Venus as homogeneous and isotropic, thermoelastic half-space, with density ρ, Lamé parameters λ and µ, thermal diffusivity χ and thermal expansion coefficient α. Let the half-space be z > 0 in a Cartesian coordinate system (Fig. 1). We assume that the state of stress in the lithosphere is such that the vertical principal stress σ3 is equal to the lithostatic pressure σ3 = −p

(1)

and the lithosphere is subject to a constant horizontal compressive stress −σ0 in the y-direction, so that the principal stress in this direction is σ2 = −p − σ0

(2)

where σ0 > 0 (compressive stresses are negative). In plane strain conditions, the principal stress in the

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x-direction is σ1 = −p − νσ0

(3)

where ν is Poisson’s modulus (e.g. Jaeger and Cook, 1976). It is assumed that the stress σ0 (henceforth called tectonic stress) arises from lateral variations in topography or in lithospheric density which are not included in the model (e.g. Sandwell et al., 1997). The lithostatic pressure can be written as p(z) = ρgz

(4)

where g is the acceleration of gravity at the surface of Venus. Studies of the climate evolution on Venus usually span over several hundred Ma and make linear interpolations to plot the surface temperatures at time between those calculated by the model. The linear approximation employed here, regarding a time interval of 100 Ma or less, is therefore consistent with the results of Solomon et al. (1999), and Bullock and Grinspoon (1996). We suppose that for t > 0 the surface z = 0 of the half-space is subject to a linear temperature increase up to t = ˆt

T(0, t) = γt,

0 ≤ t ≤ ˆt

(5)

Fig. 1. Model of wrinkle ridges in the Venus lithosphere: (a) parallel reverse faults in the thermoelastic half-space; (b) geometry of a single fault.

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where γ is a constant. The temperature change in the half-space as a function of depth and time is then (Carslaw and Jaeger, 1959)

T(z, t)
   z2 z z 2 = γt 1 + erfc √ − √ e−z /4χt 2χt 2 χt πχt (6) We shall consider moderately shallow depths (z < 10 km), time t is in the order of tens of million years and thermal diffusivity χ = 10−6 m2 s−1 . With these √ values, z/(2 χt) 1 and for practical applications we can approximate (6) as   2z

T(z, t)  γt 1 − √ (7) πχt The thermal stress induced in a thermoelastic halfspace by the temperature increase T(z, t) can be easily obtained from the solution given by Boley and Weiner (1960) for a thermoelastic plate with finite thickness. In the limit of infinite thickness, the solution for the half-space is T T σxx (z, t) = σyy (z, t) = − T σzz (z, t)

2µαK

T(z, t) λ + 2µ

=0

(8) (9)

where K = λ + (2/3)µ is the elastic bulk modulus (Fig. 2). The thermal stress is compressive and adds to the pre-existing stresses given by (1)–(3). The maximum differential stress is then

σ = σ2 − σ3

(10)

or, for t > 0, T (z, t)

σ(z, t) = −σ0 + σyy

(11)

We assume that the half-space is brittle, i.e. it fractures when the limit of elasticity (or strength) of rocks is reached. This is consistent with our interest in shallow depths and with models suggesting that the thickness of the Venus lithosphere is several tens of kilometers (e.g. Phillips and Hansen, 1998). The rock strength is a function of both confining pressure and temperature (Jaeger and Cook, 1976). At constant temperature, the dependence of strength on

confining pressure at low values of p can be approximated with the linear relation (e.g. Paterson, 1978) τ = τ0 + κp

(12)

where τ0 is the value of strength at zero pressure and κ a coefficient with a value usually greater than 1. The dependence on temperature is less pronounced, entailing a slow decrease of strength under increasing temperature. In order to take into account the effect of temperature increasing with depth, we shall also admit values of κ smaller than 1. Accordingly, the strength depends on depth. The depth z0 at which the differential stress σ is equal to τ is given by | σ(z0 , t)| = τ(z0 )

(13)

which yields, thanks to (4), (7), (8), (11) and (12) z0 (t) =

2µαγKt − (λ + 2µ)(τ0 − σ0 ) √ πχ √ √ , 4µαγK t + κρg(λ + 2µ) πχ t ≥ t1

(14)

where t1 is the time when the differential stress equals τ at z = 0, where σ is maximum. Taking z0 (t1 ) = 0 in (14), we get t1 =

(λ + 2µ)(τ0 − σ0 ) 2µαγK

(15)

In this reasoning we assume σ0 ≤ τ0 , so that a time t1 must elapse before σ equals τ0 . For t > t1 , the depth interval where σ is greater than τ increases with time as the thermal stress increases. As noted by Solomon et al. (1999), the thermal stress can become large enough to rupture the lithoT  σ ). spheric rocks (for sufficiently large times, σyy 0 As a consequence of the stress distribution, ruptures will be parallel to the x-axis. They can be described as compressive faults having a dip angle δ with respect to the planet surface. According to Anderson theory of faulting (e.g. Turcotte and Schubert, 1982), the dip angle must be smaller than 45◦ , depending on the value of the coefficient of friction. The first rupture and the associated dislocation take place at time t = t1 , when the magnitude of differential stress first equals τ at the Venus surface (we assume that τ is uniform on the plane z = 0, but a small inhomogeneity is assumed at least at one place, where the first rupture occurs). We describe fractures as uniform slip

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dislocations, extending to a depth z = d. If w is the downdip dislocation width, we have d = w sin δ

(16)

produces a strain component eyy , which is obtained by differentiation of displacement uy eyy (y) =

and the fault trace is the right line y = y0 ,

z=0

(17)

where y0 = w cos δ = d cot δ

(18)

The value of d is controlled by the stress magnitude at depth; the value z0 given by (14) can be considered as the lower boundary of the actual value of d, which can be substantially greater due to the stress concentration at the lower dislocation edge (e.g. Dragoni et al., 1986). The displacement u produced by a dislocation at the free surface can be obtained from expressions given by Okada (1985) for rectangular dislocations, where we assume a very great fault length in the x-direction u(y) = Uf(y) (0, cos δ, sin δ)

(19)

where U is fault slip and f(y) = −

1 y cos δ + d sin δ arctan , π y sin δ − d cos δ

y = y0

(20)

f(y0 ±) =

∓ 21

(21)

The function f(y) is discontinuous at y = y0 and has an inflection point at y = 0 (Fig. 3). The dislocation

U cos δ d 2 π y + d2

(22)

while the strain component ezz is obtained from the free surface boundary condition. From σzz = 0, taking into account that exx = 0, we get ezz = −

λ eyy λ + 2µ

(23)

The stress component σyy produced by the dislocation is σyy = λ(eyy + ezz ) + 2µeyy

(24)

since exx = 0, or due to (23) σyy = 4µ

λ+µ eyy λ + 2µ

(25)

Thanks to (22), we can write σyy (y) = σ¯

y2

Ud + d2

(26)

where σ¯ is a function of the elastic constants and the dip angle σ¯ =

while at the fault trace

165

4µ cos δ λ + µ π λ + 2µ

(27)

The stress produced by the dislocation is tensile and has its maximum at y = 0, above the lower edge of the fracture (Fig. 4). As d increases, the stress at y = 0

Fig. 2. Temperature variation (left scale) and thermal stress (right scale) as functions of depth z in the Venus lithosphere, for different T /σ T, times. Temperatures and stresses are normalized to values corresponding to t = ˆt at the Venus surface: T  = T/ Tˆ , σ Tyy = σyy ˆ yy z = z/(πχˆt )1/2 , t  = t/ˆt .

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Fig. 3. The function f(y), proportional to the displacement components uy and uz at the surface of Venus (δ = π/6).

decreases and the stress distribution as a function of y becomes more and more flat. As a consequence, the distance at which the tensile stress has a given value first increases and then decreases as a function of d.

The tensile stress produced by the dislocation lowers the compressive stress σ. We may presume that once the first dislocation takes place at y = y0 , the second dislocation is produced at y = y0 + D, such

Fig. 4. Tensile stress produced by a dislocation at the Venus surface, for different values of the maximum dislocation depth d (λ = µ = 6 × 105 MPa, δ = π/6).

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that the total stress is everywhere smaller than τ in the interval 0 ≤ y ≤ D. Assuming that the value of stress at z = 0 controls the occurrence of dislocations, the spacing D between the two dislocations is such that, at y = D/2 and z = 0 | σ + 2σyy | = τ

(28)

We may extend the reasoning to a whole array of parallel dislocations, since the stress σyy decreases very rapidly with increasing |y|, so that at any point only the contributions of the closest dislocations to the stress field can be considered. In this context, we consider the maximum value of thermal stress, which is reached at t = ˆt , when the increase of surface temperature is

Tˆ = γ ˆt . Accordingly, (28) becomes σ0 +

2µαK Tˆ 8 σ¯ Ud − 2 = τ0 λ + 2µ D + 4d 2

whence D = 2 (D∗ d − d 2 )1/2 ,

d < D∗

(30)

where D∗ =

2σU(λ ¯ + 2µ) 2µαK Tˆ − (λ + 2µ)(τ0 − σ0 )

We conclude that, as a consequence of tectonic plus thermal stresses, an ideal volcanic plain having a length L in the y-direction will be fractured at n equally spaced positions, where n ≈ L/D. The distance D (or the dislocation density 1/D) is a function of the thermal and mechanical parameters of the lithosphere (α, λ, µ, K, τ0 ), the increase Tˆ of surface temperature, the tectonic stress σ0 and the dislocation parameters (U, d, δ). The crustal shortening produced by n dislocations with parallel traces can be calculated as  L eyy (y) dy (32) β=n 0

For very large L, the integration of (22) gives β = nU cos δ

(29)

(31)

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(33)

The average strain due to n dislocations is then e¯ yy =

β β U = = cos δ L nD D

(34)

and is proportional to the dislocation density. We can write the average strain (or the dislocation density) as a function of d, introducing (30) into (34) and using

Fig. 5. Average horizontal strain at the surface of the lithosphere as a function of the depth extent d of dislocations, for different values of the difference τ0 − σ0 between rock strength and tectonic stress at the Venus surface (λ = µ = 6 × 105 MPa, α = 3 × 10−5 K−1 , γ = 0.5 K Ma−1 , ˆt = 100 Ma, δ = π/6).

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Fig. 6. Favorite distance D∗ between neighboring dislocations as a function of the difference τ0 − σ0 between rock strength and tectonic stress at the Venus surface. Curves for different values of slip U are shown (λ = µ = 6 × 105 MPa, α = 3 × 10−5 K−1 , γ = 0.5 K Ma−1 , ˆt = 100 Ma).

Fig. 7. Distance D between neighboring dislocations as a function of the ratio d/U. Curves for different values of the difference τ0 − σ0 are shown (λ = µ = 6 × 105 MPa, α = 3 × 10−5 K−1 , γ = 0.5 K Ma−1 , ˆt = 100 Ma).

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(31) and (27). We obtain 2µαK Tˆ − (λ + 2µ)(τ0 − σ0 ) 16µ(λ + µ)   2 −1/2 d d − × ∗ D D∗

e¯ yy = π

(35)

It is straightforward to see that the average strain (or dislocation density) is minimum when d = D∗ /2, corresponding to the maximum value D = D∗ of the distance between dislocations (Fig. 5). Therefore, D∗ is the most favorable value of D, it corresponds to a dislocation array lowering the compressive stress below the rock strength with the minimum number of dislocations. We note that D∗ , given by (31), is proportional to the slip amplitude U and becomes larger as the difference τ0 − σ0 increases (Fig. 6). When d = D∗ /2, (35) yields e¯ ∗yy = π

2µαK Tˆ − (λ + 2µ)(τ0 − σ0 ) 8µ(λ + µ)

(36)

which is the minimum average strain which solves the problem. It is independent of the dislocation parameters U, d and δ and is a linearly decreasing function of τ0 − σ0 . The value e¯ ∗yy determines the upper boundary of the ratio d/U between the depth extent d of dislocations and the slip amplitude U. Since d < D∗ and D∗ depends on U, it results from (31) and (36) that d/U < cos δ/¯e∗yy . This value increases with increasing τ0 − σ0 and is in the order of 1000 (Fig. 7). A complete list of the variables used in the model is given in the nomenclature.

3. Discussion The model developed in the previous section is now applied to the formation of wrinkle ridges. According to the model by Solomon et al. (1999), we take γ = 0.5 K Ma−1 (or 1.6 × 10−14 K s−1 ) for the rate of temperature increase at the surface of Venus in the time interval 0 ≤ t ≤ ˆt . It is believed that the formation of ridges occurred within at most 100 Ma from the emplacement of volcanic plains (Basilevsky and Head, 1996). With the assumed value for γ, a temperature increase Tˆ = 50 K takes place at the surface of the planet in a time interval ˆt = 100 Ma

169

(Fig. 2). If Tˆ = 50 K, α = 3 × 10−5 K−1 and K = 1011 Pa, the magnitude of the thermal stress at the surface of Venus reaches a value of 100 MPa at t = ˆt . If ˆt = 100 Ma, the depth interval in the abscissa of Fig. 2 is about 10 km (with χ = 10−6 m2 s−1 ). Since the compressive strength of rocks under conditions appropriate to Venus may range from 10 to 60 MPa (Schultz, 1993), the thermal stress is able to rupture the lithospheric rocks. If the strength at the surface of Venus is τ0 = 10 MPa, the linear increase given by (12) may lead to values exceeding 100 MPa at depth z = 10 km, depending on the value of the coefficient κ. We use ρ = 3000 kg m−3 for the density and g = 8.9 m2 s−1 for the acceleration of gravity. With the same value of τ0 , (14) shows that the depth interval where the differential stress becomes greater than τ is in the order of several kilometers at t = ˆt , depending on the value of κ. Higher values of τ0 would further limit this interval. The time t1 , when τ first reaches the Venus surface, is proportional to the difference τ0 −σ0 and can be considered the lower boundary for the inception time of wrinkle ridge formation. For instance, if σ0 = 1 MPa and τ0 = 10 MPa, t1 is about 10 Ma. The assumed linear temperature increase may be of course an approximation of the actual time dependence of T . The model of the Venus climate proposed by Solomon et al. (1999) indicates that the surface temperature increased with an average rate of about 0.5 K Ma−1 during the first 100 Ma after the emplacement of volcanic plains and with a much lower rate afterwards. On the other hand an upper limit equal to 100 Ma has been set to the formation time of wrinkle ridges (Basilevsky and Head, 1998). If the temperature increase during this time interval was not linear, the formation time of wrinkle ridges may be shorter. An initially faster temperature increase would entail a faster growth of thermal stress, so that the rock strength τ would be reached earlier. As a consequence, the time t1 would be smaller and wrinkle ridges would require a shorter time than 100 Ma to be produced. This might be roughly modeled by a higher value of γ and a lower value of ˆt . However, for a given value of the temperature difference Tˆ , the space distribution of wrinkle ridges would be scarcely affected by the thermal history, since the favorite distance D∗ does not depend separately on γ and ˆt , but only on their product Tˆ .

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In order to calculate the pattern of wrinkle ridges, we assume that after a time ˆt = 100 Ma, the differential stress in the lithosphere is everywhere lower than the rock strength τ. This can be accomplished by an array of dislocations situated at distance D from each other and having appropriate values of slip amplitude U and penetration depth d into the lithosphere. The dip angle δ has a limited effect on the results; in the following examples we shall assume δ = π/6. The tensile stress produced by each dislocation is a function of fault slip and fault width, for example, if U = 10 m and d = 5 km, σyy has a maximum equal to 100 MPa at y = 0, i.e. above the lower dislocation edge (Fig. 4). From the observed density of wrinkle ridges and the estimated shortening across a typical ridge feature, values of horizontal strain in volcanic plains on Venus are estimated to be 10−3 to 10−2 (Kreslavsky and Basilevsky, 1998; Bilotti and Suppe, 1999). The average strain produced by a dislocation array as a function of d can be read in Fig. 5, for different values of the stress difference τ0 − σ0 . Taking λ = µ = 6 × 1010 Pa (which implies K = 1011 Pa), strain values in the order of 10−3 are obtained for d around the most favorite value D∗ /2 and values of τ0 − σ0 up to several tens of megapascals. Observations show that in many cases the distance between ridges is in the order of tens of kilometers. The maximum distance D∗ between dislocations is mainly a function of the difference τ0 − σ0 and of slip amplitude U. Fig. 6 shows that, if τ0 ranges between 10 and 60 MPa and σ0 is in the order of megapascals, a value of D∗ between about 10 and 20 km is obtained with U = 10 m; between 20 and 40 km with U = 20 m. The relation between D and d is shown in Fig. 7, where the maxima of the different curves correspond to the most favorite condition d = D∗ /2. A typical ridged plain on Venus is Rusalka Planitia, situated in the proximity of the Venusian equator, east of Aphrodite Terra. Here, northeast–southwest compression produced wrinkle ridges spaced approximately 20–30 km (Solomon et al., 1999). On the basis of the present model, if we assume D ≈ D∗ , the observed ridge distribution is consistent with an array of dislocations with slip U ranging between 20 and 30 m, according to whether the stress difference τ0 − σ0 is near to the upper or to the lower boundary of its range (Fig. 6). Accordingly, the ratio D/U is between 1000 and 2000 from Fig. 7. As to the depth d, it can vary

within a relatively wide interval around d = D∗ /2, i.e. d = 10 or 15 km, at nearly constant strain (Fig. 5), being probably smaller than these values and closer to the values of z0 giving the depth above which the differential stress exceeds the rock strength. The tectonic stress σ0 and the rock strength τ are the less constrained parameters in the model. Tectonic stresses on Venus have been estimated by (Sandwell et al., 1997). However, the model predictions on the spacing of dislocations depend on the difference τ0 − σ0 . As regards the average strain, Fig. 5 shows that e¯ yy is not very sensitive to changes in τ0 − σ0 . Of course, the favorite distance D∗ between dislocations is more affected by the value of τ0 − σ0 (Fig. 6), particularly when this difference is large. The dependence of rock strength on depth mainly controls the depth extent of the dislocation. In turn, this controls the tensile stress produced by dislocations and hence the distance D. Fig. 7 shows that the ratio D∗ /U (given by the maximum of each D/U curve) changes by a factor 2 when τ0 − σ0 increases from 0 to 50 MPa. 4. Conclusion The model shows that wrinkle ridges may be the surface effect of dip–slip shear dislocations produced by compressive stresses on reverse faults in the Venusian lithosphere. Such stresses can be for a large part thermal stresses induced in the lithosphere by atmospheric warming, as suggested by Solomon et al. (1999). Assuming that the differential stress in the lithosphere is everywhere lower than rock strength after the ridge formation, we calculated the distance between ridges and the associated average strain. The density of ridges is controlled by the elastic and thermal properties of the lithosphere, by the thermal stress, by the difference between rock strength and tectonic stress and by fault slip. A larger compressive strength or a larger slip allow a greater distance between dislocations and a lower ridge density. The observed patterns of wrinkle ridges can be the surface effects of dislocation arrays, where each dislocation has a cumulative slip amplitude of several tens of meters and extends to several kilometers of depth. The model involves many physical quantities, which are the consequence of taking into account diverse physical processes such as thermal conduction,

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thermoelastic deformation and fracture. For some of these quantities the current values are relatively well constrained (e.g. density, elastic constants, thermal expansion coefficient, acceleration of gravity). Others derive from models of the geological history of Venus and the thermal evolution of the atmosphere. Values of rock strength and tectonic stress are probably more uncertain and appropriate ranges of values are considered. Of course, several simplifying assumptions have been made, also in view of the relatively uncertain knowledge of physical conditions in the Venusian lithosphere. However, the model predictions appear to be sufficiently consistent with available observations and current speculations on the history of Venus to suggest that dislocations induced by thermal stresses may be the mechanism of the formation of wrinkle ridges.

Acknowledgements The authors wish to thank Oded Aharonson and an anonymous referee for comments and suggestions which were helpful to improve the paper.

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