Discussion of and correction to: Deformation of the NE Basin and Range Province: the response of the lithosphere to the Yellowstone plume?

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1 Geophys. 1. nf. (1991) 104, RESEARCH NOTE Discussion of and correction to: Deformation of the NE Basin and Range Province: the response of the lithosphere to the Yellowstone plume? Rob Westaway Department of Geological Sciences, University of Durham, Soufh Road, Durham DH1 3LE, UK Accepted 1990 July 16. Received 1990 July 5; in original form 1990 February 27 SUMMARY This article corrects and further develops theory that is necessary in order to relate slip sense on faults in the upper crustal brittle layer to continuous plastic deformation of the underlying uppermost mantle. A reasonable starting point in order to do this, but which some previous theoretical models for continental deformation fail to satisfy, requires that (in the absence of external forces acting on the brittle layer or the underlying plastically deforming material) velocity in both layers must match overall as closely as possible. n a region extending by pure shear only, fault slip vector azimuth gives the extension direction beneath the brittle layer, which equals the azimuth of the extensional eigenvector of the strain rate tensor beneath the brittle layer. n a region that takes up distributed simple shear on parallel faults that do not rotate around vertical axes, and extends by pure shear in the direction perpendicular to the simple shear, neither the eigenvectors of the strain rate tensor nor the slip vector azimuth on the faults gives the true extension direction. However, individual faults in these two types of region may appear the same, and unless one knows or infers independently the overall deformation sense in a region one cannot unambiguously make use of fault slip sense information from it. Earlier conclusions that tectonic deformation in the NE Basin and Range province of the western United States is driven by flow in the Yellowstone upwelling plume are confirmed by these additional investigations. Key words: Basin and Range Province, continental deformation, faults, Yellowstone. 1 NTRODUCTON Westaway (1989) has suggested that tectonic deformation of the upper crustal brittle layer to the south and west of the Yellowstone region in the western USA is driven by Row beneath the brittle layer caused by the Yellowstone upwelling plume being sheared southwestward by motion of the North American plate in this direction. He devised an empirical horizontal velocity field v(x, y) beneath this deforming region that he considered to resemble the likely velocity field in a sheared plume, and calculated values for observable parameters of the regional deformation pattern, which depend on horizontal gradients of this field, for comparison with observations. He suggested that extension beneath two seismically active zones in the region (the central daho seismic zone west of Yellowstone, and the ntermountain seismic belt south of Yellowstone in western Wyoming, eastern daho and northern Utah) is uniaxial and is associated with non-zero vertical vorticity. He estimated observed principal extensional strain rate El from slip rates on the spacings of active oblique normal faults, which he compared with a prediction calculated as the extensional eigenvalue of strain rate tensor for the plastic deformation beneath the brittle layer, derived from his empirical velocity field. He also estimated observed extension direction in the brittle layer in both zones by assuming it is equivalent to local slip vector azimuth on active normal faults, and compared this direction with a prediction calculated as the azimuth of the eigenvector corresponding to this extensional eigenvalue of the strain rate tensor. He showed that 647

2 648 R. Westaway observed extensional strain rate and extension direction in the brittle layer match reasonably well those predicted from his velocity field. He also showed that vertical vorticity derived from his horizontal velocity field matches that predicted from slip sense y on each fault in a parallel set, given local extensional strain rate and given a theoretical relationship: XZ cot (y) = - 2E, [equations (2.23) and (2.25) of Westaway This relationship was derived by considering uniaxial extension of the brittle layer being taken up on a single set of parallel faults that bound blocks which rotate around vertical axes at rate x,/2 in response to vertical vorticity x, in the underlying plastic deformation. The theoretical ideas developed by Westaway (1989) raise a number of questions that need to be answered before these ideas can be applied to investigate tectonic deformation in the western USA or any other region. First, what is the relationship between direction and magnitude of horizontal velocity in the brittle layer and in,the underlying plastic deformation? Second, under what circumstances will elongated blocks bounded by parallel faults rotate around vertical axes at rate equal to half the local vertical vorticity in the underlying plastic deformation? Third, under what circumstances does the existence of a single set of active oblique normal faults in a locality constrain the local strain rate tensor in the underlying plastic deformation to be uniaxial? Finally, under what circumstances is the relationship in equation (l), between slip sense on faults and the ratio of extensional strain rate to vertical vorticity, valid? 2 RELATONSHP BETWEEN HORZONTAL VELOCTY N THE BRTTLE LAYER AND N THE UNDERLYNG PLASTC DEFORMATON Application of results of rheological studies of rock samples with likely crustal and mantle mineralogy (e.g., Brace & Kohlstedt 1980) to the conditions within the outermost -100 km of the earth suggest that much of the strength of the deforming continental lithosphere is in the uppermost mantle. The uppermost km of the crust is too cold to deform by plastic processes, but may deform in a brittle manner by slip on faults. The near-total absence of earthquakes at upper mantle depths suggests that the upper mantle does not deform in a brittle manner. t is thus likely to deform plastically, with horizontal velocity a continuous function of position. The lower crust, which also deforms plastically, is weaker than both the overlying brittle upper crust and the underlying upper mantle. Some studies (e.g.. Sonder & England 1989) have modelled lithosphere deformation using rheology representative of the average properties of its mantle part, thus making predictions of patterns of plastic deformation, including extension direction, extensional strain and rotation angle around vertical axes, in the deforming upper mantle lithosphere. Because deformation in the upper mantle is not directly observable, these predictions can only be compared with observations of extension direction. extensional strain and rotation at the earth s surface. Clearly such comparisons only have meaning if strain and rotation are the same at any locality in both the brittle upper crust and the underlying upper mantle. Few attempts have, however, previously been made to address the implications for properties of the velocity field beneath the brittle layer of oblique slip on sets of parallel faults-with parallel slip vectors-in the brittle layer. Even the significance of such oblique slip for the kinematics of the brittle layer is a priori unclear. To investigate what will happen to blocks in the upper crustal brittle layer that are in a locality beneath which the upper mantle lithosphere is deforming in any given sense, it is necessary to make some assumptions concerning the nature of upper crustal blocks. One may, for example, choose to argue that blocks in the brittle layer may move relative to one another in some direction that is randomly oriented relative to the local deformation sense in the underlying plastically deforming upper mantle lithosphere. n this case, external forces must act on the blocks to maintain this motion against the viscous forces on the base of the brittle layer caused by viscous coupling through the lower crust to the plastically deforming upper mantle. Relative motion between the brittle layer and the underlying plastic deformation may occur in some regions (see, e.g., Westaway 1990a). However, it seems reasonable, in the absence of evidence to the contrary, to assume in general that blocks in the brittle layer are coupled to the underlying plastic deformation in the sense that the average horizontal velocity of a block equals the average horizontal velocity in the plastic deformation in the upper mantle beneath it, and the local horizontal velocity beneath any point in a block matches as well as possible the local horizontal velocity in the plastically deforming upper mantle beneath that point. These assumptions are followed throughout this article. Section 2.1 considers relationships between slip sense on faults and regional average strain rate tensors. t demonstrates that extensional eigenvector azimuth of the strain rate tensor determined using the well-known method by Kostrov (1974) for summing seismic moment tensors will not necessarily coincide with the slip vector azimuth on oblique normal faults that take up extension. This raises the question as to which, if either, of these estimates of regional extension direction (extensional eigenvector azimuth or slip vector azimuth) is correct. Section 2.2 considers a region where a set of parallel faults that do not rotate around vertical axes takes up distributed simple shear parallel to fault strike and extensional pure shear perpendicular to fault strike. t shows that neither estimate of extension direction determined in Section 2.1 coincides with the true extension direction in this case, which is perpendicular to fault strike. Section 2.3 examines a modification to Kostrov s (1974) method to enable it to address the kinematics of regions where deformation includes a component of simple shear taken up by sets of parallel faults that do not rotate around vertical axes. Section 2.4 considers the conditions under which the brittle layer will undergo uniaxial extension-the deformation style that Westaway (1989) suggested is occurring in each actively deforming zone in the NE Basin and Range province.

3 NE Basin and Range Province Relationships between slip sense on faults and regional average strain rate tensors Slip sense ys, or the angle between fault strike direction and slip vector azimuth, was shown by Westaway (1989) to satisfy tan (y,) = tan (A) cos (6) (2) [the first equation (2.26) of Westaway (1989)], where 6 and 1 are fault dip and rake angles. Aki & Richards (1980, p. 117) list all elements of the seismic moment tensor for coseismic slip on any fault as a function of its strike cf~ and 6 and A. Given this information, the eigenvalues and eigenvector orientations of this seismic moment tensor can be determined using standard methods. f ye is the angle between fault strike direction and eigenvector azimuth, then: tan ( ~d = sin (26) cos' (A)- 2 ~ cos, (26) sin (A) cos (6) sin (2A) sin (26) + 2Ei cos (6) cos (A)- sin (6) cos (26) sin (2A) (3) where Ej, the normalized eigenvalue of the tensor, is +1, 0 or -1 for i = 1 to 3. E, = +1 corresponds to the eigenvector along the T axis of the moment tensor. Kostrov (1974) showed that the strain rate tensor for the brittle layer in any deforming region may be proportional to the sum of seismic moment tensors for earthquakes in that region. f so, both tensors will have the same eigenvectors, which indicate the principal axes of the tensor that can be identified as indicating the directions of extension or shortening in the region. Molnar (1983) noted, however, that Kostrov's (1974) method gives a tensor that fails to correctly describe the deformation of a region if any of the seismically active faults in the region intersects any of its boundaries. The method may thus only be valid if all seismogenic faults are small compared with the region of interest, and all displacement on them or deformation around them is contained within the region. For an individual earthquake with a double-couple focal mechanism, the principal axes of the focal mechanism-the P, T and null axewoincide with the compressional, extensional and null eigenvectors of the seismic moment tensor. f the tensor determined by Kostrov's (1974) method validly describes the deformation of a region, then the azimuth of the eigenvector along the T axis of this tensor would be expected to indicate the extension direction. However, the azimuth of the slip vector also provides an indication of the direction in which coseismic extension occurred. One thus has two potential ways of deducing extension direction in the brittle layer in an actively extending region, from eigenvector azimuth of the strain rate tensor, which may itself be estimated from seismic moment tensors using Kostrov's (1974) method, and from fault slip vector azimuths. However, it is unclear a priori which of these two is the correct estimate of the extension direction in cases where they differ. 2.2 Comparison of estimates of regional extension direction Westaway (1990a) has compared the two methods for estimating extension direction-using slip vector azimuth of normal-faulting earthquakes and using the extensional eigenvector azimuth of the regional strain rate tensor-in SW Turkey, an actively extending region in the eastern Mediterranean. Over much of this region, slip vector azimuths of earthquake focal mechanisms for which the nodal plane ambiguity can be resolved and striations observed in the field on exposed fault planes (which indicate slip vector azimuths of older earthquakes in the recent geological past) average to S18" f SOW. The extensional eigenvector of the strain rate tensor calculated using Kostrov's (1974) method has similar azimuth, S22"W, indicating that both methods give consistent results for this region. Geodetic studies (Westaway 1990b) should in future provide another independent estimate of extension direction in western Turkey. However, western Turkey does not comprise a single set of exactly parallel faults with indefinite horizontal extent; it comprises numerous active faults with finite length and various orientations (Westaway 1990a), most of which are small compared with the overall extent of the region and thus do not affect its boundaries. The observed earthquake focal mechanisms indicate a range of fault orientations, as expected given this structure, but with uniform slip vector azimuth. n contrast, if the brittle layer in a region contains a single set of parallel faults with parallel slip vectors, all earthquakes occurring will have the same radiation pattern, and following Kostrov's (1974) method, the regional strain rate tensor will be proportional to the seismic moment tensor for this radiation pattern. Table 1 compares the two estimates for extension direction, y, and ye, in these circumstances for 6 = 45". With this dip equation (3) Table 1. Relationship between rake, slip vector azimuth and extensional eigenvector azimuth for left-lateral normal faults with 45" dip. A YE A is rake; by convention (Alu & Richards, ; p. 114) a pure left-lateral fault has rake 0" and a pure normal fault has rake -90". Assumed fault strike is 0'. us is slip vector azimuth, calculated using equation 2. fi is extensional eigenvector azimuth, calculated using equation 4. YE

4 650 R. Westaway Y l r l l ro 'L x A s Figure 1. Lower hemisphere equal area projection indicating the relationship between slip vector azimuth and orientation of the positive (i.e., extensional) eigenvalue of the strain rate tensor for a focal mechanism with strike O", dip 45" and rake -65". N, S and D denote the north, south and down directions. T. P and B denote the T-, P- and null axes of the moment tensor for this focal mechanism, which correspond to the eigenvectors of the strain rate tensor if Kostrov's (1974) method for estimating the strain rate tensor is used. E, denotes the horizontal projection of the T-axis. S, denotes the slip vector and R, its horizontal projection. S2 and R, would be the slip vector and its horizontal projection if the other nodal plane were the fault plane. simplifies to for E, = E, = 1. ys and ye differ for all senses of oblique slip; the larger the proportion of strike slip the greater the difference (Fig. 1). Although both directions are equivalent during extension by pure shear perpendicular to fault strike alone, it is thus important to sort out which, if either of them, is the correct extension direction during oblique extension on a set of parallel oblique faults with parallel slip vectors. Because each seismic moment tensor is symmetric, any sum of seismic moment tensors must also be a symmetric tensor. f the cumulative coseismic deformation includes a component of simple shear, this deformation cannot be described using a symmetric tensor (Molnar 1983). To illustrate this, consider a region where a set of parallel faults is arranged parallel to the x-axis and slip in the normal and right-lateral sense without rotating around vertical axes (Fig. 2). Suppose av,lax = a = 0, av,,/ax = c = 0, av,/ay = b and 3vYl3y =d. Slip vector azimuth makes angle GS with the x-axis where tan (&) = d/b. A notional extensional strain rate E, in this direction can be estimated as the (4) Figure 2. Schematic diagram showing a set of parallel oblique active normal faults that take up right-lateral distributed simple shear and extensional pure shear in the underlying plastic indicates the direction of relative motion of adjacent fault-bounded blocks relative to the x-axis that is parallel to fault strike. Tick marks indicate hanging walls of normal faults, although fault polarity is arbitrary; the same horizontal relative motion could be taken up with fault strike 180" different. Point A. situated on the centre-line of one block, can be defined as the origin of the Cartesian coordinate system; point B is situated on the centre-line of the adjacent block in the direction at distance r from A. Velocity of B relative to A is u. with x component u, and y component u,. See text for discussion. relative velocity of points in this direction v divided by their separation r: E, = v/r = V(vf + ii;)/r = d (9, given that y = r sin (QJ and (6) sin (9,) = d/v(d2 + b'). (7) However. the region is clearly not extending in the direction of $J*; it is extending perpendicular to the strike of the faults. The extensional strain rate for pure shear perpendicular to the strike of the faults will, however, equal the relative velocity of points in this direction divided by their separation, and hence will also equal d. The horizontal velocity gradient tensor L for the region (where L,, _= &,/ax,, with i. j = 1, 2 corresponding to the x and y directions) is =E+R =( 0 b/2 )+( O b/2), b/2 d -b/2 0 where E is the horizontal strain rate tensor, the symmetric part of L, and P is the rotation tensor, its antisymmetric part. E will have eigenvalues E, and E, oriented towards azimuths and &, where El = d[l + V(1 + b2/d2)]/2, E,=d[l - V(1 + b2/d2)]/2. Eigenvector azimuths will be such that tan $2 = 9, + 90". = [1+ V(1 + b2/d2)]d/b, (9) (10)

5 NE Basin and Range Province 651 With b non-zero both eigenvalues of the strain rate tensor are thus non-zero. Note also that when b is non-zero the fault-bounded blocks in this example undergo translation parallel to the x-axis and will not rotate around vertical axes, even though vertical vorticity is non-zero: x, = (V X v), = -6 #O [Westaway (1989) defined x, = (-V X v), instead. However, this sign convention is non-standard and its further use should be discouraged]. For example, in right-handed Cartesian coordinates and in cylindrical polars with r the distance from the axis of the coordinate system and Q the azimuth measured anticlockwise. Unless b = 0, in which case no strike-slip is occurring, extension direction estimated using Kostrov's (1974) method will differ both from the direction C#J~ calculated from the relative velocities of the fault-bounded blocks, and will also differ from the true extension direction that is perpendicular to fault strike. Kostrov's (1974) method also gives strain rate tensor eigenvalues that provide an estimate of the extensional strain rate that differs from the correct rate E, [obtained from equation (5)] that equals the velocity gradient 6. Kostrov's (1974) method thus does not validly describe the kinematics of regions that deform with a component of simple shear, where the simple shear is taken up on sets of parallel faults that do not rotate around vertical axes, even if this deformation is taken up in the brittle layer on faults that are small compared with the dimensions of the deforming region. This second category of deforming region for which Kostrov's (1974) method fails to work exists in addition to the first category mentioned above that was noted by Molnar (1983). t is interesting to consider why Kostrov's (1974) method gives the correct extension direction in SW Turkey, even though this region appears to be taking up distributed simple shear. This distributed simple shear is left-lateral, and occurs because southward velocity increases westward for reasons discussed by Westaway (1990a). However, instead of being taken up by left-lateral strike-dip on faults that strike parallel to the direction of simple shear and do not rotate, this distributed simple shear appears to be taken up by anticlockwise rotation. Palaeomagnetic studies (Kissel et al. 1987) indeed indicate that much of SW Turkey has rotated -40" anticlockwise since the present phase of tectonic deformation began. Application of Kostrov's (1974) method to this region was shown in Section 2.1 to yield a strain rate tensor involving uniaxial extension towards the SSW. The spatial averaging inherent in Kostrov's (1974) method means it is unable to resolve the westward increase in extension rate, and the method also cannot-in principle-resolve the local anticlockwise rotation. Excluding a minor component of eastsoutheastward extension (Westaway 1990a), the kinematics of SW Turkey do indeed comprise, to a good approximation, uniaxial extension towards the SSW (but at a rate that increases westward) together with anticlockwise rotation. n this case Kostrov's (1974) method thus correctly resolves the one feature of the region's deformation pattern that it is capable of resolving, the extension direction, as well as providing an estimate of the average extension rate. 2.3 A modification o Kostrov's method to cope with distributed simple shear Suppose that one is investigating a region where distributed simple shear is taken up on a set of parallel faults that do not rotate around vertical axes and have parallel slip vectors. Suppose that one has obtained using the method of Kostrov (1974) a symmetric strain rate tensor E by summation of seismic moment tensors from earthquakes in such a region, which is extending perpendicular to the direction of simple shear. One may adjust this tensor E to obtain a modified tensor F that reveals valid information about the deforming region by the following procedure. First, rotate the tensor E to give a new tensor E' in a coordinate system with x-axis parallel to the direction of simple shear, as in Fig. 2. Suppose t is the angle between the original and revised x-axes. The necessary rotation R is given by Tensor E' will also be symmetric, and should have the form of tensor E in equation (8), with E;, zero. E;2 will now be the strain rate for extensional pure shear perpendicular to the simple shear direction, equivalent to E, in equation (5). Next, form tensor F by adding an antisymmetric tensor of the form of S2 in equation (8), in sufficient proportion to make F2, zero: 6, = E:, + w, (i, j = 1, 2). Element F12 will now indicate the average velocity gradient for distributed simple shear, the vertical vorticity, in the deforming region. Tensor F is indeed an estimate for the regional velocity gradient tensor L assuming the region is deforming by distributed simple shear and pure shear perpendicular to the simple shear direction. To illustrate this method, consider the results of Eyidogan's (1988) investigation of coseismic deformation in the western part of the North Anatolian fault zone in NW Turkey, between Lat. 39.5" and 41.5"N and Lon. 26" and 31"E. This fault zone appears to be taking up distributed right-lateral simple shear and extension (Westaway 1990a). Strike of the North Anatolian fault zone varies from -N80"W near Lon. 31"E to -W20"S near Lon. 26"E. Most of the seismic moment release investigated by Eyidogan (1988) occurrred near the eastern end of this zone, where its average strike is -N84"W. The seismicity considered by Eyidogan (1988) thus occurred predominantly in order to take up distributed right-lateral simple shear sub-parallel to -NWW, as well as extension. Eyidogan (1988) obtained using Kostrov's (1974) method a strain rate tensor equivalent to E = ( 1.2 3'6 ) s-l

6 652 R. Westaway using a right-handed coordinate system with principal axes oriented northward and westward. Eyidogan (1988) interpreted the large El, element as indicating right-lateral simple shear, the small positive El, as north-south extension, and the small negative E,, as east-west shortening. However, this interpretation of shortening makes no sense given that the region is dominated by oblique normal faults and earthquakes with oblique normal-faulting focal mechanisms. This tensor has extensional eigenvalue El 3.7 x s-l oriented towards S37"W, with the other eigenvalue -3.4 x s- oriented orthogonally (Westaway 1990a). At first sight, it thus appears that the region is undergoing deformation involving biaxial pure shear with extension towards S37"E and shortening in the perpendicular direction. However, given that the presence of the North Anatolian fault zone suggests that the region is undergoing distributed simple shear (Westaway 1990a), the reasoning above indicates that this interpretation is also incorrect: neither eigenvector azimuth will correspond to the true extension or shortening direction. n order to correctly describe the region's kinematics, first, tensor E in equation (18) should be rearranged to match the coordinate system in Fig. 2. This involves transforming to a new coordinate system rotated 96" clockwise relative to the original coordinate system, with new principal axes x' and y' oriented towards E6"S and N6"E. n this coordinate system the strain rate tensor is E', where E,=(-o.ol 3.73)x*0-15s-l, Addition of an asymmetric tensor R of the form to E' gives tensor F equivalent to the velocity gradient tensor L where f all the coseismic deformation in the region is thus regarded as taking up distributed simple shear and extension perpendicular to the simple shear direction, then given the -100 km extent of this zone perpendicular to the direction of simple shear, this tensor implies -20mmyr-' of distributed simple shear is occurring by right-lateral slip towards N84"E, together with -1 mm yr-l of extension towards S6"W. This inferred extension direction is sub-parallel to the S21"W extension direction deduced by Westaway (19YOa) for SW Turkey, as described in Section 2.2. When extension direction is estimated using the correct method appropriate for the local style of deformation occurring in each part of western Turkey, it is thus observed to remain roughly constant throughout the region. 2.4 Conditions for uniaxial extension Suppose instead that a region is deforming with general velocity gradient tensor =E+P - a (c + + b)/2) ( 0 (b x c)/2 -((c+b)/2 d (c-b)/2 0 Strain rate tensor E has eigenvalues El,2 where El,' - a +d * J( (a ad + (b + c)' 2 4 with eigenvectors oriented at where > (23) to the x-axis, Suppose extension is uniaxial with E, zero. This will be satisfied provided 4ad = (b + c)'. (25) For this to be true either ad and 6 + c are non-zero, or b + c and either a or d is zero. n the latter case, the eigenvector for the non-zero eigenvalue El will be oriented parallel to the x-axis if a is non-zero and parallel to the y-axis if d is non-zero. n the former case the eigenvectors are oblique to the x- and y-axes. However, without loss of generality one may describe this deformation using a rotated coordinate system with x-axis parallel to the eigenvector corresponding to the non-zero strain rate tensor eigenvalue. n this coordinate system, the strain rate tensor is ') 0 0 ' where El, the non-zero eigenvalue of the strain rate tensor, equals a + d (equation 14). n these circumstances the plastic deformation beneath the brittle layer involves uniaxial extension by pure shear towards the x-axis with no shortening or extension in the y direction. Extension towards the x direction would thus be balanced by flattening in the vertical direction to conserve volume. f the rotation tensor R is zero for a region that is extending uniaxially with a strain rate tensor as in equation (26), the only horizontal velocity gradient in the region is au,/ax, which equals E,. For velocity in the brittle layer and the underlying plastic deformation to match as closely as possible, relative motion between points in the brittle layer may in these circumstances only occur in the x direction: the brittle layer may only extend in this direction, and slip vector azimuth on faults in the brittle layer must be parallel to this direction. Extension direction will thus be the same in both the brittle layer and the underlying plastic deformation. f the brittle layer in a region is extending uniaxially, slip vector azimuth on faults will thus be oriented parallel to the extension direction. This slip vector azimuth will thus equal the azimuth of the extensional eigenvector of the strain rate tensor in the plastic deformation beneath the brittle layer. 3 ROTATON RATE OF BLOCKS N THE BRTTLE LAYER DURNG UNAXAL EXTENSON Westaway (1989) showed (equations 2.14 to 2.16) that an object of arbitrary shape resting on or in an underlying medium that possesses uniform vertical vorticity x, = (VXv),, but is not extending or shortening, will rotate around a vertical axis at rate o, the same rate as the underlying medium. For vertical vorticity associated with coaxial circulation, this statement is clearly correct. For

7 NE Basin and Range Province 653 example, consider the kinematics of objects on a turntable that is rotating anticlockwise at angular velocity w equal to xj2, where v, = -wy (27) vy = wx (28) such that x, = (V x v)z = 2w. For example, a pencil dropped onto such a turntable will rotate at angular velocity w regardless of how elongated it is and regardless of its orientation relative to the local radial direction on the turntable. However, it is also clearly true that rotation rate around vertical axes is not necessarily equal to xz/2 during more general deformation in which the strain rate tensor is in general biaxial. Fig. 2 indicates a case where this is so; vertical vorticity is non-zero yet blocks will not rotate around vertical axes. Furthermore, if, in contrast, the blocks in such a zone of distributed simple shear were oriented with their long axes perpendicular to the distributed simple shear, their instantaneous rotation rate would equal x, (Lamb 1987). Westaway (1989) assumed, without offering any justification, that rotation rate around vertical axes equals xz/2 during uniaxial extension. t is therefore necessary to address here what the rotation rate around vertical axes will be during uniaxial extension. Consider plastic deformation in the upper mantle lithosphere that involves uniaxial extension parallel to the x-axis and anticlockwise rotation around vertical axes. The velocity gradient tensor for such deformation has L=( -b a 0 b, =E+R The velocity field in the upper mantle lithosphere beneath such a region has V, = M - by, (31) vy = bx. (32) Vertical vorticity within this velocity field is thus 2b. Suppose faults in the overlying brittle layer form a parallel set with strike oblique or perpendicular to the extension direction, and these faults take up all deformation of the brittle layer with the blocks between them not deforming. Suppose one block is centred on the origin of the x-y coordinate system. One can always choose a reference frame for v, and vy where this is so, and thus imposing this condition results in no loss of generality (Fig. 3). Suppose this block, A, is rotating around the origin 0 of the coordinate system at rate b. Velocity in the block will satisfy V, = -by, (33) vy = bx, (34) and will thus match the rotational component, but not the extensional component, of velocity in the underlying plastic deformation. Suppose the adjacent block B is rotating instantaneously around point P, where its centre-line intersects the extension direction in the underlying plastic deformation. To match the extension in this plastic deformation this point must have v, = 2aH, and thus the d' 0 2 Figure 3. Schematic diagram where uniaxial extension and anticlockwise rotation around vertical axes is described using a coordinate system with the x-axis along the extension direction. 0 is the origin of the coordinate system, positioned at the centre of block A. P is a point on the x-axis at the centre of adjacent block B. Faults are indicated with hanging wall ticks, although fault polarity is arbitrary. velocity in this block would be V, = 2aH - by, vy = b(x - 2H). (35) (36) With velocity in block B satisfying these equations, provided the fault separating blocks A and B has strike y relative to the extension direction, where y satisfies equation (l), the component of strike-slip on the fault joining the two blocks satisfies equation (2.19) of Westaway (1989), and Westaway's (1989) analysis that led to equation (1) is valid. However, to match this rotation and translation of block B, a suitable velocity field in the plastic deformation beneath it would have V, = M - by, (37) vy = b(x - 2H). (38) Comparison of this velocity field with that beneath adjacent block A indicates that, for these two velocity fields to exist simultaneously in the plastic deformation beneath the two blocks, a velocity discontinuity must exist in the plastic deformation beneath the fault that separates blocks A and B. Some people (e.g., Wernicke 1985) have proposed models for continental extension that include velocity discontinuities in the deforming upper mantle lithosphere beneath active extending regions. However, it is invalid to use, as Westaway (1989) did, equation (1) along with an assumed continuous velocity field in the plastic deformation beneath the brittle layer in order to calculate the behaviour of blocks in the brittle layer. Suppose instead that block B in Fig. 3 rotates, like block A, about point 0. The velocity in block B will then be V, = 2aH - by, (39) vy = bx. (40) Under these circumstances, the velocity may be continuous in the plastic deformation beneath both blocks, and the difference in v, at the adjoining edges of both blocks can be taken up by slip rate 2aH in the x direction on the fault that

8 654 R. Westaway bounds these blocks. n these circumstances, both blocks rotate at rate b, the same rate as elements in the underlying plastic deformation, but slip sense on faults at block margins reveals nothing about this sense of rotation. Furthermore, slip vector azimuth in these circumstances will be the same regardless of the strike of the faults that bound individual blocks in the brittle layer. One may also see readily that this is a valid solution for the kinematics of a set of blocks that are bounded by faults and are rotating around a vertical axis relative to some external reference frame by the following reasoning. Consider the problem transformed into a reference frame that is rotating around a vertical axis at angular velocity b. The velocity gradient tensor in the deforming upper mantle lithosphere thus has =E+R This velocity gradient tensor comprises only uniaxial extensional pure shear in this reference frame. As already discussed the brittle layer will extend towards the x direction in this coordinate system at strain rate a and will not rotate around a vertical axis relative to the underlying plastic deformation. Now transform back to the original coordinate system. This transformation cannot introduce relative rotation between blocks in the brittle layer and the underlying plastic deformation. Hence both the brittle layer and the underlying plastic deformation will rotate around vertical axes at the same rate when viewed in this reference frame also. Although equations (33)-(34) and (39)-(40) constitute a valid solution for the kinematics of blocks in the brittle layer that are coupled as much as is possible to underlying plastic deformation that has velocity gradient tensor as in equation (30), they do not necessarily comprise a unique solution. We have been unable to find any other solution that minimizes relative motion between the brittle layer and the underlying plastic deformation to any greater extent, and therefore suspect that none exists, but cannot rule out the possibility that others may exist. Assuming that in a given deforming region where the velocity gradient tensor comprises uniaxial extension and rotation around vertical axes and where velocity in the brittle layer follows equations (33)-(34) and (39)-(40) blocks in the brittle layer rotate around vertical axes at the same rate as elements in the underlying plastic deformation and slip vector azimuth on faults in the brittle layer reveals the local extension direction in the underlying plastic deformation. This deformation pattern corresponds to coaxial circulation beneath the brittle layer. Assuming the instantaneous deformation of a region follows equations (33)-(34) and (39)-(40) in the brittle layer and equation (30) in the underlying plastic deformation, one may begin to consider how this solution leads to finite deformation involving finite strain and rotation around vertical axes. f extension direction rotates around a vertical axis at the same rate as the faults in the brittle layer and elements in the underlying plastic deformation, then the slip sense y will remain constant. Unless one has palaeomagnetic information to indicate directly that rotation has occurred, observable features of this deformation are indistinguishable from those in which no rotation has occurred. n these circumstances, according to the terminology suggested by Lister & Williams (1983), the vertical vorticity in the underlying plastic deformation may be regarded as spin vorticity. Alternatively, the extension direction may remain fixed and blocks in the brittle layer may rotate relative to it. n these circumstances, the vertical vorticity in the underlying plastic deformation may be regarded as shear-induced vorticity. Alternatively, the extension direction may rotate around a vertical axis at a rate that is different from that of the blocks in the brittle layer, in which case the vertical vorticity beneath the brittle layer may be regarded as the sum of a component of spin vorticity and a component of shear-induced vorticity. f striations are preserved on exposed fault planes they may reveal slip sense at different stages in the evolution of the fault set, potentially enabling these alternatives to be distinguished. For example, extension has occurred on a set of sub-parallel active normal faults in central Greece since upper Miocene time. On the same time-scale these faults have rotated clockwise by " (Kissel, Laj & Mazaud 1986). Field studies by Mercier, Sore1 & Vergely (1989) that documented striations on these fault planes indicate that the angle y between slip vector azimuth and fault strike has remained roughly constant throughout. f central Greece is extending uniaxially, such that local fault slip vector azimuth coincides with local extension direction, this indicates that the local extension direction has rotated clockwise through the same angle as the faults. n terms of the terminology discussed above, the local vertical vorticity therefore counts as spin vorticity. 4 CONDTONS WHERE THE EXSTENCE OF A SNGLE SET OF PARALLEL FAULTS DOES NOT MPLY UNAXAL EXTENSON Figure 2, where a set of oblique normal faults takes up distributed simple shear parallel to fault strike, indicates circumstances where fault-bounded blocks respond to biaxial strain and vertical vorticity in the underlying plastic deformation by not rotating around vertical axes. This style of deformation appears to be occurring in some zones of distributed active extension, such as the western part of the North Anatolian fault zone in NW Turkey discussed in Section 2.3. However, given the slip sense on faults in the NE Basin and Range Province (see Fig. 2 and Table 2 of Westaway 1989), for this deformation style to be occurring in the region, western Wyoming east of the ntermountain seismic belt would be required to be translated southward relative to Yellowstone, and the region north of the Central daho seismic zone to be translated northnorthwestward. No one has ever asserted that this is a reasonable description for the active tectonics of the region. Moreover, it fails to account for the finite along-strike extent of substantial activity on faults in both seismic zones, which dies out both southward and northnorthwestward towards regions that are not deforming. A more feasible theoretical model that may a priori potentially described deformation in each of the seismically active zones in the NE Basin the Range Province has been

9 NE Basin and Range Province 655 i \ \ \ \ \ \ - \ \ Vy=Pdwt V.=Pbw d \\\\\\\ 110. Zrl D 10.0) 't Figure 4. Schematic diagram summarizing McKenzie 8~ Jackson's (1983) floating block model. A deforming zone bounded by rigid surroundings (indicated by diagonal shading) contains a set of parallel faults bounding blocks that are small compared with the width 2w of the zone. A and B denote the centres of two adjacent blocks along y = w. A unique orientation, with strike 4 relative to the x-axis, enables a single set of faults to take up biaxial extensional strain beneath the brittle layer (equation 42). f relative velocity across the deforming zone is u, = 2bw and uy = U w, and velocity gradients within the zone are uniform, vertical vorticity equals -b and blocks will rotate clockwise. Faults at block margins slip in the normal and left-lateral sense, with slip vector azimuth in the y direction. Faults are indicated with hanging wall ticks, although fault polarity is arbitrary. suggested by McKenzie & Jackson (1983). They considered deformation of a region with boundaries that are moving apart and in a transcurrent sense, and showed that, although the strain rate tensor within such a region is biaxial, a unique set of conditions exists whereby a single set of parallel faults may take up local deformation. Suppose the width of the deforming zone is 2w, the rate of transcurrent motion 2wb and the rate of separation 2wd (Fig. 4). f velocity gradients in the upper mantle lithosphere beneath this zone are uniform, then its velocity gradient tensor satisfies equation (8). McKenzie & Jackson (1983) showed that in these circumstances blocks that are small compared with the width of the deforming zone, are not pinned to the boundaries of the zone, and are bounded by faults with strike $J relative to the x-axis where b tan (4) = - 2d ' (42) will rotate around vertical axes at rate b/2. Under these circumstances a line joining the centres of blocks such as A and B will not rotate around a vertical axis, and the slip vector azimuth on faults at block margins will be. orthogonal to the zone boundaries. This model implicitly requires blocks to be equidimensional, because elongated blocks floating in a field of plastic deformation that includes a biaxial strain rate tensor will not necessarily rotate around vertical axes at a rate equal to half the local vertical vorticity (Lamb 1987). However, it possesses the property that slip sense on faults at block margins is diagnostic of the sense of local vertical vorticity: left-lateral slip at block margins, as in Fig. 4, is consistent with clockwise vertical vorticity. Blocks in the NE Basin and Range Province are not equidimensional; most are strongly elongated parallel to fault strike. Furthermore, some are of substantial size compared with the dimensions of the deforming zones in which they are situated. Moreover, slip vector azimuths are not even approximately orthogonal to the edges of the deforming zones; they are oriented relative to these edges at angles of between 30" and 65" (Westaway 1989, table 2). Thus these zones do not satisfy McKenzie & Jackson's (1983) conditions for a single set of parallel faults to take up biaxial extension in the underlying plastic deformation. Suppose instead that, on account of their elongated shape, blocks in the configuration in Fig. 4 rotate around vertical axes at rate b' relative to the surroundings of the deforming zone. The velocity of any point in any block in the brittle layer is the sum of the velocity of translation of the block centre and the velocity of rotation of the point around the block centre. For block A in Fig. 4, V, = bw + b'(y - W)/2, v,, = dw + b'x/2. (43) (44) n contrast, velocity in the plastic deformation beneath the brittle layer, will, if it satisfies equation (8), comprise v, = by, (45) vy = dy. (46) Substantial differences in the magnitude and direction of velocity will thus exist in these circumstances between blocks in the brittle layer and the underlying deforming upper mantle lithosphere. These differences in velocity will give rise to forces and torques in the lower crust that will act to eliminate them. Even supposing this pattern of relative velocities codld develop in the first place, it would become gradually eliminated over time by the action of these viscous forces and torques. f the brittle layer were weaker than the underlying plastically deforming upper mantle lithosphere this process would most likely result in the brittle layer becoming broken up by other active faults developing to take up the biaxial deformation in the upper mantle lithosphere. The most likely orientation of these faults would be with strike parallel to the regional simple shear direction, as in Fig. 2. Alternatively, if the upper mantle lithosphere were weaker than the brittle layer, this process would most likely result in the velocity field in the upper mantle lithosphere beneath sets of blocks bounded by parallel faults in the brittle layer adjusting to a configuration that more closely follows the local velocity in the brittle layer. For example, if velocity in adjacent blocks in the brittle layer were to follow equations (33)-(34) and (39)-(N), the velocity field in the underlying upper mantle lithosphere may adjust to that described by equations (23)-(24), which achieves a much better match between velocities of blocks in the brittle layer and the plastically deforming upper mantle lithosphere than equations (45)-(46) match (43)-(44). This revised velocity field satisfies the velocity gradient tensor in equation (30), which corresponds to uniaxial extension. Suppose average viscosity is q in the lower crust, and the difference in local horizontal velocity between the upper mantle lithosphere and the upper crustal brittle layer is u, with value uo at time = 0. Working in the reference frame of the upper mantle lithosphere, the shearing force df acting on horizontal area Cis of overlying upper crustal brittle

10 656 R. Westaway layer is (47) where HL is the thickness of the lower crust. Assuming no other forces act on the brittle layer, the equation of motion of this element of the brittle layer, which has thickness H, and density p, is du df= -ph, dsdt Equations (47) and (48) have the solution (49) Given estimates (e.g., Reilinger 1986) that suggest the lower crustal viscosity is -6 X 10" Pas, and with p kg m-3, HL-20 km and H, - 10 km, the time required for u to decrease to l/e of its initial value is - lo-'s. Even allowing for many orders of magnitude of uncertainty in q, it is most unlikely that substantial differences in relative velocity may persist between the brittle layer and the underlying upper mantle lithosphere over geological time-scales, unless other forces act. Given that the velocity field in the upper mantle lithosphere that matches best the relative velocities of blocks in the brittle layer that are bounded by parallel faults that rotate around vertical axes and have parallel slip vectors comprises uniaxial extension in the upper mantle lithosphere, one may thus reasonably have confidence that the local strain rate tensor in the upper mantle lithosphere is indeed uniaxial below sets of such faults in the brittle layer. One may regard this result as implying that the existence in the brittle layer of sets of parallel faults that rotate around vertical axes and have parallel slip vectors in a locality in general forces the local strain rate tensor in the underlying upper mantle lithosphere to be uniaxial. Of course, this constraint does not exist beneath localities where the brittle layer comprises more than one set of faults, or where faults in the brittle layer are irregular. Nor does it apply in the case in Fig. 2 where a single set of oblique normal faults with parallel slip vectors takes up distributed simple shear, is oriented parallel to the direction of simple shear, and does not rotate around vertical axes. This conclusion has important implications for the modelling of tectonic deformation in localities where the brittle layer comprises sets of parallel faults. First, continuum models for the velocity field in the deforming upper mantle lithosphere should include the constraint that the strain rate tensor for this velocity field is uniaxial beneath regions where the brittle layer comprises sets of parallel faults that rotate around vertical axes and have parallel slip vectors. Some published continuum models [e.g., Sonder & England's (1989) model for the Aegean region] do not include this constraint in localities where it is appropriate. Second, given that the strain rate tensor is likely to be uniaxial in the upper mantle lithosphere beneath sets of parallel faults in the brittle layer that rotate around vertical axes, McKenzie & Jackson's (1983) floating block model, which predicts parameters such as slip sense on faults and rotation sense around vertical axes assuming a single set of parallel faults takes up biaxial extension, is unlikely to be validly applicable to any real deforming region. This means that the simple relationship predicted by this model between sense of oblique slip on faults and sense of vertical vorticity beneath the brittle layer cannot be validly applied. McKenzie & Jackson's (1983) floating block model thus does not minimize velocity mismatch between the brittle layer and the underlying plastic deformation during deformation involving distributed simple shear and extension (or shortening) perpendicular to the simple shear direction. Anyone wishing to apply these models to any deforming region should give reasons why the brittle layer in the region follows this deformation style, given that the brittle layer could take up the underlying plastic deformation much more readily by breaking up instead into blocks bounded by faults that strike parallel to the regional simple shear. 5 MPLCATONS FOR DEFORMATON OF THE NE BASN AND RANGE PROVNCE n this correction have demonstrated the following results. First, for a set of parallel faults to take up uniaxial extension in the underlying plastic deformation, slip vector azimuth will reveal the extension direction in the underlying plastic deformation, provided neither the brittle layer nor the underlying plastically deforming upper mantle lithosphere is rotating around a vertical axis relative to the reference frame in which the deformation is described (Section 2.4). Second, if the plastic deformation in the upper mantle lithosphere includes uniaxial extension and rotation around a vertical axis and involves coaxial circulation, then, regardless of the orientation of faults in the brittle layer relative to the extension direction in this underlying plastic deformation, blocks in the brittle layer will rotate around vertical axes relative to an external reference frame at the same rate as elements in the underlying plastic deformation; this rate is equal to half the vertical vorticity in the upper mantle lithosphere when this vorticity is measured in the same external reference frame (Section 3). n contrast, if the plastic deformation in the upper mantle lithosphere involves vertical vorticity associated with distributed simple shear, the rotation rate of elongated blocks is more complicated: in particular it will be zero if faults bounding these blocks are parallel to the distributed simple shear, and will equal the vertical vorticity in the distributed simple shear if faults bounding the blocks are instantaneously perpendicular to the distributed simple shear. Third, during uniaxial extension, slip vector azimuth on faults in the brittle layer indicates instantaneous extension direction in the upper mantle lithosphere (Section 3). Fourth, the strain rate tensor in the upper mantle lithosphere will be uniaxial in localities where the brittle layer comprises a single set of parallel faults that rotate around vertical axes and have parallel slip vectors (Section 4). Westaway (1989) was incorrect to conclude that slip sense on any fault in the brittle layer in the NE Basin and Range Province is diagnostic of any particular local sense of vertical vorticity in the underlying upper mantle lithosphere. Given that slip vector azimuths on faults in the two actively deforming zones in the NE Basin and Range Province are not orthogonal to the trend of the zones, neither zone can satisfy the unique set of conditions identified by McKenzie

11 NE Basin and Range Province 657 & Jackson (1983) whereby deformation involving a biaxial strain rate tensor, which includes distributed simple shear and extensional pure shear, can be taken up on a single set of parallel faults that strike obliquely to the direction of simple shear and rotate around vertical axes. Furthermore, given the reasoning already set out that the strain rate tensor in the upper mantle lithosphere will anyway be uniaxial beneath sets of parallel faults in the brittle layer that rotate around vertical axes, this unique set of conditions may never exist. Westaway's (1989) assumption that extension in the plastic deformation beneath the deforming zones in the NE Basin and Range Province is uniaxial is thus reasonable. Provided extension in the plastic deformation beneath these zones is uniaxial, Westaway's (1989) use of fault slip vector azimuth to estimate extension direction beneath the brittle layer is valid, regardless of whether or not blocks in the brittle layer are rotating around vertical axes. Westaway's (1989) empirical model for the velocity field u in the sheared Yellowstone plume covered only the x component of v, v,, where the x-axis was oriented southwestward. This was because Westaway (1989) believed that horizontal gradients of uy, auy/ax and auy/dy, were most likely small compared with gradients of V,, and could thus be reasonably neglected. Vertical vorticity (equation 13) and eigenvalues and eigenvector azimuths of the regional strain rate tensor would thus be controlled by the gradients of v,. t is likely, however, that &,/ax and possibly also i3vx/i3y will in reality be non-zero beneath this region, and, if each of the seismically active zones is extending uniaxially, the velocity gradient tensor L (equation 22) will satisfy the condition for uniaxial extension (equation 25). However, the velocity field that Westaway (1989) used to calculate observable quantities had only gradients of u, non-zero, such that =E+P b/2 0 -b/2 0 Strain rate tensor E has eigenvalues El and E, where E -!+ J(7). a'+ b2 '-2 and E,=i- J(+ a' + b2 with eigenvectors oriented at angles ql,, to the x-axis, where (53) Because b2>0 beneath both zones, eigenvalue El will be >a, and eigenvalue E, will be <O and thus correspond to shortening. Because b < 0 when y > 0 and b > 0 when y < 0 (see Westaway 1989), G < 0 when y > 0 and q1 > 0 when y < 0. The eigenvalue and eigenvector azimuth plotted in Fig. ll(c-d) of Westaway (1989), and which approximate to the observed extensional strain rate and extension direction in both zones, are thus El and ql. Suppose the region were extending uniaxially with du,/dx (=a) >> duy/dy (=d). With avy/ay small compared with du,/ax, the extensional strain rate El would approximate to au,/ax. Assuming Westaway's (1989) values of aux/3x and du,/dy are correct, the procedure that he used determined a value for E, that is somewhat larger than the correct value. Because laux/2y( is larger than av,/ax beneath both seismically active zones, the values of E, displayed by Westaway (1989) may exceed the values that he would have obtained had his empirical velocity field included non-zero terms for auy/ax and auy/ay with the appropriate values required to satisfy equation (25) and make the strain rate tensor uniaxial (with extensional eigenvalue approximately equal to av,/ax) at each locality. Westaway (1989) noted that his values for El exceed those observed in both seismically active zones. This mismatch between observed and predicted values of El can now be explained. The second horizontal eigenvalue E, that was not displayed by Westaway (1989), which is negative (implying shortening) and corresponds to an eigenvector with azimuth orthogonal to that for El, is an artefact of Westaway's (1989) horizontal velocity field having zero gradients of uy and thus not being consistent with uniaxial extension. There is no evidence for shortening orthogonal to the local extension direction in any part of the NE Basin and Range Province. f each of the seismically active zones in the NE Basin and Range Province is extending uniaxially, the extensional strain rate El equals a + d, rather than satisfying equation (51), and extension direction 1~ (equivalent to extensional eigenvector azimuth) satisfies 2(E1-a) 2d tan (111) = *- b+c b (54) (from equation 24). To evaluate El and + correctly thus requires velocity gradients a, b and d to be defined and to be non-zero beneath all points in both actively deforming zones. Velocity gradient c, or du,/dx, may be either zero or small compared with dux/3y. Westaway's (1989) reasoning that led to the deduction that the y component of velocity beneath these zones is inward, towards the axis of the Snake River plain, was based on the use of equation (1). As discussed above, for this equation to be valid discontinuities in velocity must exist in the deforming upper mantle lithosphere on the scale of individual blocks in the brittle layer, a condition that is most unlikely to be satisfied in this region. An alternative chain of reasoning can establish instead the likely direction and magnitude of the y component of velocity in these zones following the assumption that extension in these zones is uniaxial. Equation (25) must be satisfied for extension to be uniaxial, providing an interrelationship between horizontal velocity gradients beneath the brittle layer. Westaway's (1989) sheared plume model was based on the suggestion, which remains reasonable, that du,/dx is >O beneath both zones. n order to satisfy equation (25) this requires &,lay > 0 beneath both zones, indicating that vy is outward away from the axis of the Snake River plain beneath both zones. By rearrangement of equation (24), the observation that extension is oriented at -45" towards the Snake River plain in both zones, or in terms of equation (24) that q is --45" when y > 0 and - -45" when y < 0, leads to (55)

12 658 R. Westaway and where k, and k - are both - +1, with k+ = (cot(v) (Y >0) (57) and k- =cot (w) (y < 0). (58) Westaway (1989) also suggested that typically in the NE Basin and Range Province lau,/dyl>> lau,/ax(. n the absence of any direct evidence that this is incorrect, this suggestion is followed here also. Given that au,/ay >O beneath both zones, and neglecting duyldx, du,ldy is <0 when y > 0, and au,/ay is >O when y <O. Given the definition of vertical vorticity in equation (13), and again neglecting du,jdx in comparison with au,/ay, this implies that predicted vertical vorticity is anticlockwise where y > 0 and clockwise where y < 0; the same senses predicted by Westaway (1989). Provided extension in both zones is uniaxial, Westaway's (1989) predictions of rotation sense are thus correct. Because the actively deforming zones in the NE Basin and Range Province appear to take up distributed simple shear on faults that are roughly perpendicular to the simple shear direction (see fig. 2 of Westaway 1989), rotation rates of these blocks can be estimated as all, not huy, the vertical vorticity beneath both zones, or approximately au,/ay (see Section 3). Westaway's (1989) model sheared plume velocity field, which has u, varying by - 10 mm yr- ' across -200 km distance in the y direction, thus predicts average rotation rate -3" Myr- ' in both zones. Given that no information is at present available to confirm or contradict these rotation rates, they are thus predictions that may potentially be verified in future when appropriate palaeomagnetic or geodetic studies have been carried out. The typical magnitude of av,/ay can be estimated, by neglecting dv,/dx and combining equation (13) with equation (55). as (59) Assuming xz has the same magnitude in both zones, this implies that lav,/ay( should be greater in the ntermountain seismic belt, where w is greater, than in the Central daho seismic zone. Figure 1 of Westaway (1989) shows a gravity anomaly associated with the Yellowstone upwelling plume in NW Wyoming and a similar feature -400 km SW beneath SE Wyoming and northern Colorado. Geochemical studies of local basalts (Thompson et af. 1990) indicate that this northern Colorado feature is another upwelling plume. One would expect that flow from both plumes would interfere, particularly beneath the ntermountain seismic belt that is situated between them. n particular, the presence of plumes on opposite sides of the ntermountain seismic belt, with Yellowstone at x = 0, y = 0 and the other plume at x = 0, y = +400 km, would be expected to make uy zero or small beneath this zone, and may well make u, beneath this zone larger (as the return flow from both plumes is channelled southwestward) than it would be with only the Yellowstone plume present. f this is so, u, may decrease less rapidly between the Snake River plain and the ntermountain seismic belt than Westaway (1989) suggested, and Jav,/dyl may be substantially smaller beneath the ntermountain seismic belt than beneath the central daho seismic zone. This implies that x,1 may be substantially smaller beneath the SB than the -3" Myr-' estimated beneath the central daho seismic zone, and typical rotation rates are smaller in proportion. Equation (59) can then be satisfied with both Jxzl and au,/ay smaller beneath the ntermountain seismic belt than the central daho sesimic zone, even though tan ( 1 ~ 1 ) is larger beneath the ntermountain seismic belt. 6 DSCUSSON AND CONCLUSONS Westaway's (1989) main objective was to investigate whether it is feasible that return flow from the Yellowstone upwelling plume is directly driving tectonic deformation in the NE Basin and Range Province, or whether this tectonic deformation must instead only be regarded as an indirect effect of the plume. Given the evidence presented by Westaway (1989), the u priori alternative possibility that the tectonic deformation in the NE Basin and Range Province is unrelated to the presence of this plume nearby can reasonably be disregarded. Westaway (1989) made several mistakes in his attempt to relate the kinematics of the brittle layer in the NE Basin and Range Province to the pattern of horizontal velocity beneath the brittle layer. However, with the (hopefully) correct theory necessary to investigate such a relationship now established, the main points in his reasoning remain unaffected. A case can be made, either in general (e.g., White 1989) or with regard to Yellowstone in particular (e.g., Anders et af. 1989) that extension of the surroundings to a plume may result from outward collapse of the lithosphere above a plume, which will be both heated-and hence weakenedand uplifted by the plume. The question whether plume-related volcanism is caused by extension, or whether (as we suggest for Yellowstone) both volcanism and extension are driven by flow in the plume, has indeed been widely debated in the past. Decompression melting accompanying extension of the uppermost mantle above a hot plume will undoubtedly generate a lot of magma (e.g., Mckenzie & Bickle 1988), and upwelling of a plume will also cause uplift of the Earth's surface above it. However, these processes will occur regardless of whether the plume drives the extension and volcanism or the volcanism is a consequence of the extension, and consequently cannot be used to distinguish these possibilities. t is, however, worth noting that volcanism above oceanic plumes such as Hawaii is unaccompanied by extension, suggesting that volcanism above upwelling plumes in general is not necessarily a consequence of local extension. Hooper (1990) has suggested that the volcanism -16 Myr ago in the Columbia plateau region of the NW USA, which was associated-like Yellowstone today-with motion of the North American plate across the Yellowstone plume, preceded local plume-related extension. Both this evidence and our own investigations support the conclusion that tectonic extension and volcanism associated with Yellowstone is driven directly by the plume, rather than the volcanism being only a consequence of the extension or the extension being only a consequence of the topographic uplift.

13 NE Basin and Range Province 659 ACKNOWLEDGMENTS This research was supported by Natural Environment Research Council grant GR3/6966. REFERENCES Aki, K. & Richards, P. G., Quantitative Seismology, Theory and Methodr, W. H. Freeman, San Francisco. Anders, M. H., Geissman, J. W., Piety, L. A. & Sullivan, J. T., Parabolic distribution of circumeastern Snake River Plain seismicity and latest Quaternary faulting: migratory pattern and association with the Yellowstone hotspot, 1. geophys. Res., 94, Brace, W. F. & Kohlstedt, D. L., Limits on lithospheric stress imposed by laboratory experiments, 1. geophys. Res., 85, Eyidogan, H., Rates of crustal deformation in western Turkey as deduced from major earthquakes, Tecronophysics, 148, Hooper, P. R., The timing of crustal extension and the eruption of continental flood basalts, Nature, 345, Kissel, C., Laj, C. & Mazaud, A., First paleomagnetic results from Neogene formations in Evia, Skyros and the Volos region, and the deformation of central Aegea, Geophys. Res. Lett., W, Kissel, C., Laj, C., Sengor, A. M. C. & Poisson, A,, Paleomagnetic evidence for rotation in opposite senses of adjacent blocks in northeastern Aegea and western Anatolia, Geophys. Res. Lett., 14, Kostrov, V. V., Seismic moment and energy of earthquakes and seismic flow of rock, zv. Acad. Sci. USSR, Phys. Solid Earth, Lamb, S. H., A model for tectonic rotations around a vertical axis, Earth. planet. Sci. Lett., 84, Lister, G. S. & Williams, P. F., The partitioning of deformation in flowing rock masses, Tectonophysics, 92, McKenzie, D. P. & Jackson, J. A,, The relationship between strain rates, crustal thickening, paleomagnetism, finite strain and fault movements within a deforming zone, Earth. planet. Sci. Lett., 65, (with 1984 correction: 70, 444). McKenzie, D. P. & Bickle, M. J., The volume and composition of melt generated by extension of the lithosphere, J. Pitroi., 29, Mercier, J. L., Sorel, D. & Vergely, P., Extensional tectonic regimes in the Aegean basins during the Cenozoic, Basin Rex, 2, Molnar, P., Average regional strain due to slip on numerous faults of different orientations, J. geophys. Res., 88, Reilinger, R., Evidence for postseismic viscoelastic relaxation following the 1959 M=7.5 Hebgen Lake, Montana, earthquake, J. geophys. Res., 91, Sonder, L. J. & England, P. C., Effects of a temperature-dependent rheology on large-scale continental extension, 1. geophys. Res., 94, Thompson, R. N., Leat, P. T., Dickin, A. P., Morrison, M. A., Hendry, G. L. & Gibson, S. A., Strongly potassic mafic magmas from lithospheric mantle sources during continental extension and heating: evidence from Miocene minettes of northwest Colorado, USA, Earth planet. Sci. Lett., 98, Wernicke, B., Uniform sense normal simple shear of the continental lithosphere, Can. J. Earth Sci., 22, Westaway, R., Deformation of the NE Basin and Range Province: the response of the lithosphere to the Yellowstone plume?, Geophys. J. Znt., 99, Westaway, R., 1990a. Block rotation in western Turkey: 1. Observational evidence, J. Reophys. Res., in press. Westaway, R., 1990b. Measurement of tectonic deformation in western Turkey using GPS satellite geodesy, Modern Geol., in press. White, R. S., gneous outbursts and mass extinctions, EOS, Trans. Am. geophys. Union, 70,