Postglacial Rebound Modeling with Power-Law Rheology

Transcription

1 Postglacial Rebound Modeling with Power-Law Rheology Patrick Wu Dept. of Geology & Geophysics, University of Calgary, Calgary, Alberta T2N-1N4, Canada, Keywords: rheology, power law creep, relative sea levels, ambient tectonic stress Abstract. Recent progress in postglacial rebound modeling with power-law creep is reviewed. Results of earlier 2D finite element models with simple nonlinear rheology are extended to 3D models with realistic ice histories. In the absence of ambient tectonic stress, mantle with power law rheology is shown to be characterized by a localized flow with distinct uplift patterns inside and outside the former ice margin. An important implication of this is that relative sea-level observations in and around the centre of rebound in Laurentia cannot be simultaneously reconciled unless nonlinear rheology is limited to lie within a thin zone ( 2 km) in the upper mantle below the lithosphere. Results of 2D modeling shows that ambient tectonic stress have an important effect on the postglacial rebound process if the tectonic stress level exceeds 1 MPa. In the presence of large ambient stress, power law mantle loses its uplift characteristics inside the ice margin and thus the flow appears to be linear. However, the uplift characteristic of a power law medium outside the ice margin is still retained even with ambient stress as high as 1 MPa. Thus, in the presence of ambient stress, nonlinear mantles continue to have difficulty explaining the relative sea level data outside former Laurentia ice margin. 1. Introduction An important question in the study of the dynamics of the earth, is whether the flow law of the mantle is linear (Newtonian) or nonlinear (power-law). Creep experiments on rocks suggest that mantle deformation follows a linear creep law (diffusion creep) when the stress level is low or when the grain size is small. However, at high stress level or large grain size, the dominant creep mechanism is dislocation creep (i.e. power-law creep). It has been suggested that nonlinear rheology prevails in the upper part of the mantle [4]. However, due to the uncertainty of the transition conditions between diffusion and dislocation creep, it is not clear if the rest of the mantle is nonlinear, and if there is a transition in the dominant creep mechanism, where that may occur. Recently, Karato & Li [5] suggested that the lower mantle may not be nonlinear. Karato & Wu [6] showed that rheological transition may occur in the upper mantle and suggested that this question of rheological transition may be resolved with seismic anisotropy observations and postglacial rebound modeling. The purpose of this chapter is to review recent progress in postglacial rebound modeling with

2 power-law creep. In the next section, the method of calculation is briefly reviewed. In section 3 & 4, simple disc load calculations are presented. The characteristics of power-law halfspaces are discussed in section 3, and in the section 4, relative sea level data is used to infer whether the whole mantle is nonlinear. The depth of the rheological transition zone is also addressed. In section 5, results for a realistic 3D ice deglaciation model is considered. In section 6, the effect of ambient tectonic stress is investigated and the results are summarized the section The Model In the early studies of postglacial rebound with nonlinear rheology, simplifying assumptions were made to keep the calculation tractable: Post and Griggs [9], and later Crough [2], assumed that the surface depression kept a constant shape during the rebound process; Brennen [1] assumed that the strain rate within the earth decreases exponentially with depth; Yokokura & Saito [2] assumed that the relaxation time only depends on time and not on space; Nakada [8] assumed a shallow channel flow. Unfortunately, all these simplifying assumptions have been shown to be invalid [16], calling into question the results of these earlier studies. In this paper, 2D and 3D Finite Element Models will be used to calculate the deformation of the earth due to glacial loading and unloading. The advantage of the FE method is that none of the above simplifying assumptions have to be made and the solution is much more rigorous. The earth models considered are isotropic, incompressible viscoelastic flat-earths with power-law rheology and no self-gravitation. The gravitational acceleration, g, and the density of the halfspace, ρ o, are taken to be constants. The flat-earth approximation has been demonstrated to be adequate in describing the rebound process for loads even as large as the Laurentian Ice sheet [19]. In this paper, it is assumed that the rebound process sees the steady state creep and not the transient creep. For a viscoelastic material, the deformation can be expressed as a sum of the elastic response and the creep response. The elastic strain is related to the stress by Hooke s law while the steady state creep law is given by: C ε ij =A * n-1 σ E σ ij (1) where A* is the creep parameter, σ' ij are the deviatoric stress components and σ' E is the equivalent deviatoric stress with σ' E = 1 σ' ij σ' ij (2) 2 For linear rheology, n equals 1. For nonlinear rheology, n normally lies between 2 and 6 for mantle rocks. For later discussion and comparison of results with other papers, it is useful to define the Table 1. Elastic parameters of halfspace earth models to be used with a load with size comparable to the Fennoscandian ice sheet. Density (kg/m 3 ) 338 Young's Moulus (Pa).35x Poisson's ratio. 5

3 effective viscosity as : η eff = 1 3A * (n - 1) σ E (3) Note that this differs from the usual definition [11, ] by a factor of 3/2, and the reason is discussed in Ranalli [1] (p.75-8). 3. Characteristic Deformation Pattern of Nonlinear Halfspaces The deformation pattern of a nonlinear mantle is quite distinct from that for a linear one. In this section, we shall review these differences in the deformation pattern as the stress exponent n increases from 1 to 4. We shall also investigate how the creep parameter A* affect this characteristic deformation pattern. The implication of the characteristic deformation pattern for a power-law medium on relative sea level (RSL) data is also discussed. For simplicity, the earth is considered to be a viscoelastic uniform halfspace with no ambient tectonic stress. The parameters of the earth model is listed in Table 1. The ice history is taken to be the Heaviside loading of a disc load with 9 km radius and a uniform magnitude of 2 MPa inside the load. Displ. (m) Vert t =. ka ka A = 3.33E n = 1 (a) t =. ka ka A=1.66E-28 n=2 (b) Displ. (m) Vert t =. ka ka A=8.33E-35 n=3 (c) t =. ka ka A=4.16E-41 n=4 (d) Radius (km) Radius (km) Figure 1. Vertical displacements at different times after the Heaviside loading of a uniform disc load on viscoelastic halfspaces with different stress exponent n. Elastic parameters of the models are given in Table 1.

4 The effects of stress exponent n on the vertical displacements is shown in Fig. 1. Here, the creep parameters, A*, are chosen so that the effective viscosity is 1 21 Pa-s. Comparing the deformation for the linear halfspace (Fig. 1a) with the nonlinear halfspaces, we see that power-law rheological models (n>1) are characterized by: i) Higher initial rate of deformation followed by much slower rate under the load [8]. Wu [16, 17, 18] have shown that this is due to the stress dependence of the effective viscosity for power-law medium (see equation 3). Thus, the initial flow rate is fast when the rebound stress is large, but slows down as the rebound stress relaxes. ii) Sharper cornered trough near the edge of the load. This is due to the high stress concentration near the edge of the load which results in low effective viscosity and fast relaxation of the shorter wavelengths of deformation. iii) Narrower peripheral trough and a larger amplitude, sharper cornered, peripheral bulge. This behavior is very similar to that for a linear channel and again can be explained by the ridge of high stress that extends beneath the load and peaks near the edge of the load (see Fig. 3 in Wu [17]). iv) Existence of a narrow viscously stationary zone (VSZ) or point. Within this narrow zone or point, the viscous displacement (i.e. displacement other than the elastic one at t=), almost remain stationary with time. From the centre of the load to this VSZ, land sinks continuously after the application of the load. Outside this VSZ, land rises continuously (see also Figs. 2b, c & d). On the other hand, a linear halfspace is characterized by a broad RSL transition zone outside the edge of the load, where land submergence is followed by emergence (see also Fig. 2a). The existence of a VSZ Displ. (m) Vert t= (a) A=3E n= ka t=.2.5 (b) A=1.66E n= ka Displ. (m) Vert t=.5 2. VSZ (c) ka Radius (km) A=3.33E-35 n= VSZ (d) A=1.66E-35 n= ka Radius (km) Figure 2. Vertical displacements at different times after the Heaviside loading of a uniform disc load on viscoelastic halfspaces with different combinations of A* and n. Elastic parameters of the models are given in Table 1.

5 VSZ and the lack of a broad RSL transition zone for power-law medium implies that nonlinear halfspaces cannot explain the sea-level data in the transition zone outside the former ice margin. As we shall see later, this existence of a VSZ or the lack of a broad RSL transition zone is characteristic of power-law medium even in the presence of ambient tectonic stress. Due to the importance of this VSZ on the interpretation of RSL data, we shall now investigate whether the creep parameter A* can affect the existence of the VSZ. In Fig. 2, the deformation near the edge of the load is plotted for nonlinear halfspaces (Figs. 2b-d) with n=3 and values of A* ranging from 1.66x1-35 to 1.66x1. For comparison, the deformation pattern for the linear halfspace is also shown (Fig. 2a). Fig. 2d shows that for small values of A*, the VSZ is located around 95 km. With the value of A* increased to 1.66x1, a VSZ is also found around 95 km for t < 2 ka, but as time progresses, this zone no longer remains stationary. Since the main difference between A*=1.66x1-35 and A*=1.66x1 is that the latter has a smaller effective viscosity than the former, this means that the VSZ in the former model is only stationary for the time under consideration. This zone is actually quasi-stationary - given enough time, VSZ does not actually exist. In order to see how the quasi stationary zone affect relative sea levels, vertical displacements relative to that at t= ka have been computed for the Heaviside unloading of the same uniform disc load, and plotted in Fig. 3 for four sites. Fig. 3a confirms that, at the centre of rebound (r= km), the nonlinear halfspaces give faster initial rate of uplift than the linear halfspace, but the present rates of uplift for the nonlinear halfspaces are much lower. The model with A*=1.66x1 has the lowest rate of uplift at the present because it has the smallest effective viscosity. Outside the load at r=92 km, the linear halfspace and the nonlinear model with A*= 1.66x1-35 show continuos land emergence(also shown in Fig. 2d), while the other nonlinear models show land emergence followed by submergence. Displ. (m) (a) R = km A=1.66E-35 n=3 A=8.33E-35 n=3 A=1.66E n=3 A=3.33E n=1 (b) R = 92 km Rel Displ. (m) (c) R = 948 km (d) R=165 km 25 Rel Time (ka BP) Time (ka BP) -5 Figure 3. Relative vertical displacement curves at different radius from the centre of a Fennoscandian size ice load.

6 However, the transition from land emergence to submergence takes place soon after the Heaviside unloading event around ka BP. At the VSZ around r=948 km, the model with A*= 1.66x1-35 show small land emergence followed by zero relative displacement after about 1 ka BP. Again, the other two nonlinear models show land emergence followed by submergence with very early time of transition. On the other hand, the linear halfspace predict a transition from land emergence to submergence around 6 ka BP. For r > 95 km, land submergence is predicted by all the nonlinear models except immediately after deglaciation at ka BP. However, the amount of submergence are very small in the last 8 ka. In contrast, the linear halfspace continue to predict land emergence followed by submergence around 8 ka BP, and the amount of submergence is about twice as much as that predicted by the nonlinear models. In summary, linear halfspace predicts a broad RSL transition zone, whereas nonlinear halfspace predicts a viscously quasi-stationary zone which result in a very narrow or nonexistent RSL transition zone. Even for those nonlinear models with a narrow RSL transition zone, the time of transition from emergence to submergence is much earlier than that predicted by the linear model and the amplitude of submergence is much smaller than that of the linear halfspace. Thus, RSL data near the former ice margin can be used to distinguish between linear and nonlinear halfspaces. 4. More complex earth models with parabolic disc load In this section, we are interested in finding earth models with nonlinear rheology and zero ambient tectonic stress that have the potential in explaining relative sea level data in Hudson Bay and along the east coast of Canada and the United States. For a load with size comparable to the Laurentia ice sheet, the deformation is sensitive to earth structure much deeper than the 67 km transition. Thus, the earth models considered here are stratified viscoelastic earths with a 22 km thick lithosphere (and no ambient tectonic stress). The elastic parameters of the earth models are listed in Table 2. Here, the Laurentian ice sheet is taken to be axially symmetric and the ice thickness at radius r is given by H ice (r) = H max α r where α =19 km is the radius of the disc load and H max is the maximum ice height at glacial maximum. Again, the load is assumed to be at isostatic equilibrium before being removed instantaneously from the surface of the earth at time T D. Note that H max and T D are not well determined, so the values for a linear uniform 1 22 Pa-s mantle ( H max =35 meter and T D = ka BP) will be used initially. However, in order to fit the sea level data with the nonlinear earth models, the timing of deglaciation T D is allowed to shift by 2 ka and H max can also be modified. Table 2. Elastic Parameters of stratified earth models Depth Density (kg/ m 3 ) Young's Modulus (Pa) Poisson Ratio Lithosphere Layer Layer Lower 1 2 Mantle - 22 km 34.87x km 34.87x km x1 below 67 km x

7 Model Name U22 U34 U35 U34B Lithospheric thickness (km) n in the mantle A* in the mantle Table 3. Uniform rheological earth models and ice models (Pa s - 1 ) 3.33x1 3.33x x1 3.33x1 Maximum ice thickness (m) Time of deglaciation (ka BP) 1 4.a Linear & Nonlinear Uniform Mantle Let us start with models that consists of a 22 km thick elastic lithospheres overlying stratified mantle with uniform rheology. The parameters for earth rheology and ice history are listed in Table 3. The letter U in the name of the models denote that rheology in the mantle is uniform, the numbers that follow are the negative of the exponent in the creep parameter. Fig. 4 shows the comparison of the observed sea level data and the predicted vertical displacements relative to the present. As shown in Wu & Johnson [19], relative vertical displacement curves give good approximations to relative sea level curves in the last 7 ka. Inspection of Fig. 4 shows that Model U22 (with linear rheology) is able to fit the sea level data simultaneously at the four sites - although a slightly thicker ice sheet will give a better fit to the data in Ottawa Island. On the other hand, all the nonlinear models predict land emergence instead of the observed submergence for sites outside the former ice margin (e.g. Boston Rel. Displ. (m) Ottawa Is Model U22 Model U35 Model U34 Model U34B NW NewFoundland Rel. Displ. (m) Boston Delaware Age (ka BP) Age (ka BP) -2 Figure 4. Comparing the predicted relative displacement curves with the observed sea level data (error bars) at 4 sites in Canada and along the east coast of the U.S. Axisymmetric (2D) parabolic ice models are used. Model parameters are given in Table 3.

8 and Delaware). This first order effect is due to the presence of VSZ in nonlinear mantle models and no modification of ice thickness nor time of deglaciation can change the predicted land emergence to submergence. Thus, nonlinear mantles are rejected by relative sea level data outside the former ice margin. Fig. 4 also illustrates the problems nonlinear mantles have in fitting the sea level data inside the ice margin. For H max =35 meter and T D = ka BP, Model U34 (short dashed line) is able to fit the RSL data in NW Newfoundland but under-predicts the height of the ancient beaches and the present-day rate of uplift in Ottawa Island. The RSL data in Ottawa Is. can be better explained by Model U34 if one delays the deglaciation time T D by 2 ka (Model U34B). However, Model U34B ( long dashed lines) now over predicts the height of the ancient beaches in NW Newfoundland, and for sites outside the former ice margin, land emergence is still predicted. Similarly, Model U35 can either explain the data in Ottawa Is. or in NW Newfoundland, but not both simultaneously. Thus, unlike the linear case, a simple ice profile cannot be used by nonlinear models to fit the RSL data in Ottawa Is. and Newfoundland simultaneously. 4.b Nonlinear Zone in the Upper Mantle with different thickness From the above, we saw that nonlinear rheology in the whole mantle is rejected by relative sea level data outside the former ice margin. In this subsection, nonlinear rheology is confined to a zone beneath the elastic lithosphere. This scenario may occur in the upper mantle because the viscosity for dislocation creep is generally smaller than the viscosity for diffusion near the base of the lithosphere, but the viscosity for diffusion becomes lower than that for dislocation for deeper depth [6]. Thus, the dominant creep mechanism may be dislocation near the base of the lithosphere, but may change to diffusion at deeper depth. The creep parameter in the NLZ is kept fixed at A*=3.33x1 (Pa -3 s -1 ), but the thickness of this nonlinear zone (NLZ) is varied (see Table 4). The purpose here is to find the thickest NLZ that can explain the RSL data in and around the centre of rebound. In the names of these models, the numbers following the label NLZ denote the thickness of the nonlinear zone in km. The predictions of these models are plotted in Fig. 5. For reference, the model U34 is also plotted. Table 4. Models with Nonlinear Zones of various thickness Model Name NLZ45 NLZ2 NLZ1 Lithospheric thickness (km) Thickness of NLZ in upper mantle (km) n in NLZ A* of NLZ (Pa s - 1 ) 3.33x1 3.33x1 3.33x1 n below NLZ A* in the mantle below the NLZ (Pa s - 1 ) 3.33x1 3.33x1 3.33x1 Maximum ice thickness (m) Time of deglaciation (ka BP)

9 Displ. (m) Rel Ottawa Is Model U34 Model NLZ45 Model NLZ2 Model NLZ1 NW NewFoundland Displ. (m) 2 1 Boston Delaware 2 1 Rel Age (ka BP) Age (ka BP) -2 Figure 5. Same as Fig. 4 except for the models in Table 4. Fig. 5 shows that with linear rheology in the lower mantle, land submergence is now predicted outside the former ice margin in Boston and Delaware. However, a nonlinear upper mantle above 67 km depth (Model NLZ45) under predicts the amount of land emergence in Ottawa Is. and also under predicts the amount of submergence outside the former ice margin. With the NLZ thickness reduced to 2 km or less, the RSL data both inside and immediately outside the former ice margin can be better explained, although a slightly thicker ice sheet would give even better fit to the data in Ottawa Is. Table 5. Models with Nonlinear Zones and various lower mantle viscosities Name NLZ45 Model NLZ45A NLZ45B Lithospheric thickness (km) Thickness of NLZ in upper mantle (km) n in NLZ A* of NLZ (Pa s - 1 ) 3.33x1 3.33x1 3.33x1 n in the lower mantle A* in the lower mantle (Pa s - 1 ) 3.33x1 1.11x x1 Maximum ice thickness (m) Time of deglaciation (ka BP)

10 Displ. (m) Rel Ottawa Is Model NLZ45 Model NLZ45A Model NLZ45B NW NewFoundland Displ. (m) Boston Delaware Rel Age (ka BP) Age (ka BP) Figure 6. Same as Fig. 4 except for the models in Table c Nonlinear Upper Mantle and Linear High Viscosity Lower Mantle For the previous models with nonlinear zones, the rheology of the lower mantle is linear with viscosity kept at 1x1 21 Pa-s. In this set of computation, we investigate the effect of increasing lower mantle viscosity to 3x1 21 Pa-s and 1x1 22 Pa-s (see table 5). The results, which are plotted in Fig. 6, show that increased lower mantle viscosity give land emergence instead of the observed submergence outside the former ice sheet. This is because increased lower mantle viscosity promotes channel flow which causes the forebulge to migrate outwards and result in land emergence outside the ice margin. Within the ice margin, the high viscosity in the lower mantle result in higher present-day rate of uplift, and thus the height of the ancient beaches are over-predicted. Thus, nonlinear upper mantle with high viscosity lower mantle is not compatible with sea level observations outside the former ice margin. Table 6. Earth models with Nonlinear Lower Mantle Model Name NLM21 NLM22 NLM23 Lithospheric thickness (km) n in upper mantle A* of upper mantle (Pa s - 1 ) 3.33x x x1 n in the lower mantle below 67 km A* in the lower mantle (Pa s - 1 ) 3.33x1 3.33x1 3.33x1 Maximum ice thickness (m) Time of deglaciation (ka BP)

11 4.d Nonlinear Lower Mantle and Linear Upper Mantle with Various Viscosities For the sake of completeness, let us consider the less likely case where nonlinear rheology is restricted to lie in the lower mantle and the viscosity in the upper mantle is varied from 1 2 Pa-s to 1 22 Pa-s (see table 6). In the name of the models, NLM stands for nonlinear lower mantle and the numbers that follow are the exponent of the creep parameter. The predicted relative displacement curves are compared with the observational data in Fig. 7. Inspection of Fig. 7 shows that a low viscosity (1 2 Pa-s) upper mantle on top of a nonlinear lower mantle (Model NLM21) cannot explain the RSL data outside the edge of the former ice margin because land emergence instead of the observed submergence is predicted. When the viscosity of the upper mantle is increased to 1 21 Pa-s (NLM22), a small amount of submergence is predicted outside the former ice margin, but the present-day rate of uplift outside the former ice margin is close to zero and the transition between land submergence and emergence is predicted to be as early as 11 ka BP. Furthermore, the height of the ancient beaches in Ottawa Is. are under-predicted by Model NLM22. When the viscosity of the upper mantle is further increased to 1 22 Pa-s (NLM23), the height of the 4-7 ka BP beaches in Ottawa Is. and the 8-1 ka BP beaches in NW Newfoundland can be explained, however, the amount of submergence outside the former ice margin is under predicted. To explain the amount of submergence outside the former ice margin, the thickness of the ice sheet has to be drastically increased, however, that would significantly over predict the height of the ancient beaches in Ottawa Is. and Newfoundland. Thus, a nonlinear lower mantle cannot satisfactorily explain all the sea level data in and around the centre of rebound simultaneously. In summary, of all the earth models investigated, the only type of earth model with nonlinear rheology and zero ambient tectonic stress that is consistent with sea level data inside and outside the former ice margin of the Laurentia is one that has a thin ( 2 km ) nonlinear zone. Rel. Displ. (m) a) Ottawa Is Model NLM21 Model NLM22 Model NLM23 b) NW NewFoundland Rel. Displ. (m) 4 c) Boston 2 d) Delaware Age (ka BP) Age (ka BP) Figure 7. Same as Fig. 4 except for the models in Table 6.

12 5. Nonlinear Mantle Rheology with Realistic ICE3G Model In the last two sections, only simple disc load with Heaviside loading/unloading history has been considered. In reality, the ice profile at glacial maximum was more complicated than a parabolic profile. In addition, the rate of deglaciation was not instantaneous but varied with time and location. Furthermore, the important effect of migration of the ice margin is not captured in those simple ice models. In this section, the realistic ICE3G model of Tushingham and Peltier [13, 14] with several saw-tooth glacial cycles that have slow buildup time of 9 ka but rapid deglacial time of 1 ka are used. The details of the ice model and the elastic parameters of the earth model can be found in Wu & Johnston [19]. In this section, only four uniform rheological models are considered - L22, L34, L35 & L36 (see Table 7). The letter L in their names denote that they all have a 15 km thick elastic lithosphere. The rheology of the underlying mantle is uniform and the numbers in their name represent the negative of the exponent in the creep parameter of the mantle. Except for Model L22, all the other models are nonlinear. The ambient tectonic stress is assumed to be zero here. In Fig. 8, the relative displacement curves for these models are compared with the observed RSL data in six sites in North America. Taking into account the problem of the earth s curvature for far-away sites (i.e. Brigantine and Southport) [19], Fig. 8 still show that the linear earth model has the best chance to explain all the RSL data simultaneously. However, in Churchill (Manitoba), neither the linear model nor the nonlinear models can predict the observed height of the ancient beaches - apparently, a thicker local ice or a delayed deglaciation is required by any of the models to give a better fit to the observations. In Ottawa Is., Model L34 significantly under predicts the height of the ancient beaches whereas in NW Newfoundland, Model L36 under predicts their height. In Boston, Model L36 predicts emergence rather than the observed submergence; Model L35 predicts zero rate of uplift but Model L34 is able to predict a small amount of submergence. In Brigantine (New Jersey), all three nonlinear models are able to predict submergence, but again the amount of submergence are too small. In Southport (North Carolina), all models, except Model L34, have no problem explaining the height of the submerged beaches. Comparing Figs. 4 & 8 shows that the effect of a realistic ice sheet is mainly in sites close to the ice margin at the last glacial maxima: With the inclusion of shrinking ice margin, land submergence are now predicted by nonlinear mantles with ICE3G. However, the amount of submergence is still not large enough to explain the observed RSL data. In summary, 3D models with realistic ice history show that nonlinear mantles are still rejected by RSL data just outside the ice margin at the last glacial maxima. However, more studies with realistic ice histories are needed to see if the conclusions for the other earth models in section 3 remain valid. Table 7. Uniform rheological earth models used with ICE3G Model Name L22 L34 L35 L36 Lithospheric thickness (km) n in the mantle A* of the mantle (Pa s - 1 ) 3.33x1 3.33x x x1

13 RSL (m) Ottawa Island Model L22 Model L36 Model L35 Model L34 Churchill RSL (m) NW Newfoundland Boston RSL (m) Brigantine, NJ Southport, NC Time (ka BP) Time (ka BP) -2 Figure 8. Same as Fig. 4 except for the earth models in Table 7 and realistic ice model ICE3G. 6. Effect of Ambient Tectonic Stress So far, all the rebound calculations in this paper have assumed that the ambient tectonic stress is zero. In the presence of ambient tectonic stress, σ T ij, the state of stress can be written as σ ij = σ ij T +σ ij R (4) where R σ ij is the deviatoric rebound stress. Substituting equation (4) into equations (1) to (2), and following the procedure outlined in Schmeling [11], one obtains: σ E = σ TE Σ i,j σ ij T σ ij R σ (5) where σ TE = 1 2 Σσ T T ij ij σ ij is the equivalent deviatoric tectonic stress. Also the creep law can

14 be written as: C T ε ij = εij +A n-1 σ TE σ R ij +A* n-1 σ 2 n-3 TE σ ijt Σ σ T R i,j ij σ ij (6) where the first term on the right side is the strain rate due to tectonic stress alone and is defined as: T ε ij =A * n-1 T σ TE σ ij The strain rate seen by the rebound process is thus: R ε n-1 ij =A*σ TE σ R ij +A* n-1 2 σ n-3 TE σ ijt Σσ T R ij σ ij i,j (8) Furthermore, equation (3) shows that the effective viscosity experienced by the rebound process in the presence of tectonic stress is: (7) η eff = 1+ n ηt eff Σ i,j σ ij T σ ij R σ TE 2 (9) where, η T eff = 1 n-1 3A*σ TE (1) is the effective viscosity seen by the tectonic process in the absence of rebound stress. Thus, equation (9) shows that the viscosity seen by postglacial rebound is different from that seen by mantle convection [] when the rheology is nonlinear. Some insight on how the interaction between rebound stress and tectonic stress affect creep in the mantle can be gained if we consider the following special case: Assume that the tectonic stress and the rebound stress components are orthogonal except for the kl-th component, then Σσ T i,j ij σ R ij = σ T R kl σ kl (11) Thus, from equation (8), the kl-th component of the strain rate is: R ε kl = 1 A* σ n-1 2 TE T 2 n-1 σ 2 R kl σ TE +2 σ kl () and the rebound stress component R σ kl sees a linear creep law but the viscosity is dependent on the tectonic stress distribution [11, 15]. If one further assumes that only the kl-th component of the tectonic stress contribute to the equivalent deviatoric tectonic stress, i.e. then equation () can be written as, σ TE = 1 2 σ kl T T σ kl (13) R ε n-1 kl =A*σ TE n-1 σ 4 TE +1 σr kl (14) The creep law for the other components (where rebound stress and tectonic stress are orthogonal) is: ε ij R =A*σ TE n-1 n-1 σ T ij T σ σ kl R R + σ ij kl (15)

15 If one further assumes that the ij-components of tectonic stress is zero, then: ε ij R =A*σ TE n-1 σ ij R (16) Again we have an effectively linear creep law for the ij-th component. Comparing equations (14) and (15) or (16), we see that the creep law for the ij-th component is different from that for the kl-th component - this means that the rebound process sees a linear but anisotropic creep law [11]. If the conclusions of this special case is applicable in general, then the rebound process will basically see a linear rheology even though the creep law is actually non-linear. This seems to imply that the RSL data inside and outside the former ice margin can be explained simultaneously by nonlinear mantle provided the right amount of ambient stress is applied. But are the conclusions of this special case applicable in general? Gasperini et al. [3] used the finite element method to study the vertical displacement history of a nonlinear upper mantle when the ambient tectonic stress is included in the equivalent stress (σ' E ) term only. With this simplified nonlinear creep law, they were able to show that the uplift history of linear models at the centre of the load can be approximated by nonlinear upper mantle models. So this does lend some support to the conclusions of the special case. Unfortunately, the uplift history in the RSL transition zone was not investigated by Gasperini et al [3]. Instantaneous loading/removal constant g of ice load o ICE Elastic Lithosphere ViscoElastic Power-law Mantle Fluid core Figure 9. Schematic diagram of the model that includes the interaction between ambient tectonic stress and rebound stress. The length of the arrows indicate the magnitude of the tectonic shear stress. After Wu [18]. Model Name A22 A36 A38 AUM36 Max. Ambient Stress (MPa) Lithospheric thickness (km) n in the upper mantle A* of upper mantle n in the lower mantle below 67km depth Table 8. Earth models with Ambient Tectonic Stress (Pa s - 1 ) 3.33x1 A* in the lower mantle below 67 km (Pa s x1 ) x x x x x1 3.33x1 Maximum ice thickness (m) Time of deglaciation (ka BP)

16 Vertical Displacement (m) A) Max. Shear =.1 MPa A=4E-35, n=3 t= B) Max. Shear = 1 MPa A=4E-35, n=3 t= kyr 16 kyr C) Max. Shear = 1 MPa A=4E-35, n=3 Ambient Stress (lines) vs No Ambient Stress (symbols) Ambient Stress (lines) vs No Ambient Stress (symbols) t = ka t = 1 ka t = 4 ka t =16 ka Distance from the centre of the load (km) Figure 1. Effect of ambient stress magnitude (.1, 1 and 1 MPa) on the deformation pattern at four time steps. Deformation calculated with the inclusion of ambient stress (lines) are compared with that calculated without ambient stress (symbols). After Wu [18]. Wu [18] also investigated this problem with the finite element method. In his study, the full interaction between tectonic stress and rebound stress in a power-law (n=3) medium was taken into account and no approximation was made in the creep law. However, to keep the problem simple, the plane strain assumption was employed, i.e. the Fennoscandian sized ice load was assumed to be a boxcar load with infinite extent in the y-direction perpendicular to the plane (see Fig. 9). The load was applied/removed instantaneously above the centre of a large convection cell, so that the tectonic stress seen were effectively horizontal shear stresses that varies quasi-linearly with depth and reached maxima at the top and bottom of the cell. The magnitude of the maximum shear was not fixed but was varied from.1 to 1 MPa in order to study the effect of ambient stress magnitude. His results showed that: (1) Ambient stress can be neglected in the calculation if its magnitude is much less than 1 MPa (Fig. 1a); (2) For ambient stress level of the order of 1 MPa, the effect of ambient stress is still small within the ice margin, but becomes important outside the former ice margin (Fig. 1b); (3) When the ambient stress level is much larger than 1 MPa, the large ambient stress results in a smaller but more time-independent effective viscosity (equation 3), thus the rate of defor-

17 Displ. (m) Rel A) Centre of Load Model A22 Model A36 Model A38 Model AUM Time (ka BP) B) Transition zone Time (ka BP) Figure 11. Effect of ambient stress on relative vertical displacement curves for the models in Table 8. Modified after Wu [18]. mation is higher inside the ice margin (Fig. 1c) but there is no rapid change in the rate of uplift as seen in nonlinear halfspaces (Fig. 3a). In this way, the deformation inside the ice margin appears to be linear. This is illustrated in Fig. 11a, where the relative displacement curves for the four earth models of Table 8 are shown. The letter A in the name of the models indicates that ambient tectonic stress is superimposed. The letters UM indicate that the upper mantle is nonlinear but the lower mantle is linear. The numbers that follow are the exponent of the creep parameter. Fig. 11a shows that at the centre of the ice load, models with nonlinear mantle do deform like the linear model, that is without the strong time dependent rate of deformation. The model with nonlinear rheology restricted to the upper mantle gives even better match to the linear model. On the other hand, Fig. 11b shows that RSL data outside the former ice margin can still be used to distinguish between a linear mantle from nonlinear mantle or nonlinear upper mantle because the nonlinear models produce too small rates of uplift at the present and the amount of submergence for the past 8 ka is under-predicted. The reason is again due to the rebound stress concentration near the edge of the ice sheet. Thus, the results of simple models show that large ambient tectonic stress (>> 1 MPa) can affect the rebound process in a power-law medium. In such case, the creep law may appear linear for the postglacial rebound within the ice margin, confirming the predictions of Weertman [15] and Schmeling [11]. On the other hand, for sites outside the ice margin, the RSL signature of a power-law mantle is still distinguishable from that due to deep flow in a linear mantle. This however needs to be verified with more realistic rebound modeling. 7. Conclusions Assuming that the rebound process sees the steady state creep and not the transient creep, it is shown that : 1) If the ambient tectonic stress level is low (<<1 MPa), nonlinear mantle rheology are characterized by a localized flow with strong time dependent uplift rate inside the ice margin and a viscously quasi-stationary zone outside the ice margin - thus the relative sea level observations outside the Laurentia ice margin cannot be explained by power-law mantle. 2) In the presence of large ambient tectonic stress (>> 1 MPa), land uplift in a power law mantle loses its strong time dependency for uplift rate inside the ice margin but maintains its characteristics outside the ice margin. 3) If the ambient stress level is low, a nonlinear lower mantle or a high viscosity lower mantle beneath

18 a nonlinear upper mantle are rejected by the sea level data outside the Laurentia ice margin. (However one has to be cautioned that sea level data may not be able to resolve the rheology at depth greater than about 15 km depth [7], thus lower mantle here actually means the upper part of the lower mantle.) 4) If the ambient stress level is low, a thin nonlinear rheology zone, with thickness around 2 km or less, embedded between a 22 km thick elastic lithosphere and a linear mantle is permitted by the sea level data in North America. Thus, the transition in dominant creep mechanism may occur around 3-4 km depth. This is consistent with the observations in seismic anisotropy [6]. 5) Preliminary 3D models, with realistic deglaciation histories but without ambient stress, confirm the characteristics of power law mantles found with 2D simple ice models. However, more 3D modeling with realistic ice histories are needed to confirm these finding for layered nonlinear rheology in the presence of ambient stress. References [1] C. Brennen, J. Geophys. Res., 79 (1974), p [2] S.T. Crough, Geophys.J.R.astr.Soc. 5 (1977), p [3] P. Gasperini, D.A. Yuen & R. Sabadini, Geophys. Res. Lett. 19 (1992), p [4] C. Goetze & D.L. Kohlstedt, J. Geophys. Res. 78 (1974), p [5] S. Karato & P. Li, Science 255 (1992), p [6] S. Karato & P. Wu, Science 26 (1993), p [7] J.X. Mitrovica & W.R. Peltier, Geophys.J.Int. 2 (1995), p [8] Nakada, M., J.Phys.Earth 31 (1983), p [9] R.L. Post & D.T. Griggs, Science 181 (1973), p [1] G. Ranalli, Rheology of the Earth: Deformation and Flow processes in Geophysics and Geodynamics, Allen & Unwin Inc. (1987). [11] H. Schmeling, H., Earth plant.sci Lett. 84 (1987), p [] D.L. Turcotte & G.Schubert, Geodynamics: applications of continuum physics to geological problems, Wiley & Sons, New York (1982). [13] A.M. Tushingham & W.R. Peltier, J.Geophys.Res. 96 (1991), p.4,497-4,523. [14] A.M. Tushingham & W.R. Peltier, J.Geophys.Res. 97 (1992), p.3,285-4,523. [15] J. Weertman, Phil.Trans.R.Soc.Lond.A. 288 (1978), p [16] P. Wu, Geophys.J.Int. 18 (1992), p [17] P. Wu, Geophys.J.Int. 114 (1993), p [18] P. Wu, Geophys. Res. Lett. 22 (1995), p [19] P. Wu & P. Johnston, this volume. [2] T. Yokokura & M. Saito, J.Phys.Earth 26 (1978), p