by Leah H. Joseph
GS 662, 12/13/96
II. Isostatic Rebound
III. Effect and "Use" of Isostatic Rebound
"...I suggested that the enormous weight of ice laid upon the surface of the country might have caused a depression, while the melting of the ice would also account for the rising again of the land..." T. F Jamieson (1882)
Concepts concerning isostatic rebound have been discussed in scientific literature for over a century. Through geologic observations (mostly of old raised shorelines), it had become obvious to some of the geologists of the nineteenth century that parts of the land had experienced vertical uplift without mountain building. Not only did they suggest rebound processes, but took one step further and used these observations to make insightful statements about the structure of the earth as we are still trying to do today.
Isostasy forms the basis of the theory of isostatic rebound. Isostasy itself is based on the opposing influences of two main forces: buoyancy and gravity. For the earth, it is the reason that the relatively rigid lithospheric plates float at certain levels in the underlying ductile asthenosphere. Blocks, or plates, will adjust themselves vertically until the forces of buoyancy and gravity are balanced. When the forces are balanced, the blocks are considered to be in isostatic equilibrium and there will be no vertical movement.
The level at which both continental and oceanic blocks begin to float in the mantle is called the level of compensation and generally corresponds to the top of the asthenosphere. At this location, the mantle will flow in response to stress (e.g. a surficial load); therefore the principal stresses at this level are all equal to the lithostatic pressure (P).
Figure 1 is an spatial (rather than temporal) isostatic model of a continental block. If this system is in isostatic equilibrium, the lithostatic pressure under both the continental and oceanic blocks must be equal at the level of compensation. This pressure is equal to the weight of a column of material that extends from the level of compensation to the surface. This implies that:
(·tiri)ocean = (·tiri)cont = P/g
where t is the thickness of a certain layer and r is the density of that layer. One can calculate unknowns in this relationship by writing out the sums and assigning values to the "known" variables (This section from Middleton and Wilcock, 1994).
Isostatic rebound occurs when a load is imposed on or removed from the lithosphere. The surface tends to rise or sink as the lithosphere rises or sinks in the asthenosphere. Loads may consist of large lakes, oceans (on continental shelves during eustatic sea level rise), ice, sediment, thrust sheets, and volcanoes. The rising or sinking of the lithosphere will continue until isostatic equilibrium is reached.
Calculation: A load (e.g. ice sheet) imposed on the continental block in Figure 1 will cause the continent to sink. One can calculate the thickness of the ice sheet given the value of total isostatic rebound (total implies that the rebounded system has reached isostatic equilibrium) using the same mathematical relationship listed previously. This is an example of a temporal isostatic model where conditions change with time rather than space; crustal structure can be neglected in this calculation because it is the same before and after glaciation.
Assuming that the system had reached isostatic equilibrium when loaded with ice, the buoyancy pressure on the continent would equal the pressure that had been imposed by the ice sheet (equal but opposite forces in order to be in equilibrium). The buoyancy pressure is therefore equal to the weight of the column of asthenosphere that was displaced (density around 3300 kg/m3). The pressure imposed by the ice sheet equals the ice density (about 1000 kg/m3) multiplied by the thickness of the ice sheet.
(ice thickness)(density of ice) = (displacement of asthen.)(density of asthen.)
t x 1000 = 1000 x 3300
t = 3.3 km
While this model is fairly simple, it results in a fairly realistic ice sheet thickness (This section from Middleton and Wilcock, 1994).
This paper is based primarily on the effect of loading on surface topography. It is worth mentioning, however, that there may be other reasons that subsidence or rebound may occur. Dynamic surface topography is the theory that processes in the mantle (i.e. mantle convection) cause the topography that we see on earth today (subtracting the effect of height variations due to density differences). For example, this theory predicts the topographic highs at mid-ocean ridges as well as the sinking of the crust with distance (and age) from the ridge (e.g. Forte et al., 1993).
Flexure of the Lithosphere
Isostatic depression itself is only possible if the asthenosphere can flow away from the depressed area and if the lithosphere is able to move vertically, either along fractures or by elastic bending. Geological evidence indicates that lithospheric bending dominates over fracturing and we can therefore use flexural models to predict the shape of the depression and surrounding areas (or vice versa). Instead of all areas around a load becoming depressed, the flexural rigidity of the lithosphere (among other things) causes some areas to actually be elevated during this same time. This is known as the peripheral bulge which forms a ring around the outside of the load (Figure 2; Middleton and Wilcock, 1994).
Effect and "Use" of Isostatic Rebound
Now that the "basics" of isostatic rebound have been established, the rest of this paper will focus on one effect of rebound and insight gained about the earth through the modeling of rebound. I will confine my view to the rebound processes associated with glacial isostasy, although other types of loads may have similar effects.
Secular vs. Eustatic Sea Level
Sea level changes have left numerous signs to their occurrence such as abandoned shorelines, drowned rivers, marine extensions into continents, biological (e.g. corals) organism changes, seismic reflector truncations, etc. (Plag, et. al., 1996). In fact, it was these signs of sea level change that led early geologists to think about isostatic rebound processes (e.g. Jamieson, 1882 and Shaler, 1974). Today, the knowledge of eustatic sea level variations is very important in climatic studies. However, since apparent sea level is affected by two different, but linked, processes, deciphering purely eustatic sea level can be complicated. The processes affecting sea level are: 1) formation of continental ice and 2) glacial compression and rebound. One area where this complication is now obvious is the east coast of the United States. The North American continent was partially covered with the Laurentide ice sheet until about 6000 years ago (Farrand, 1988). The east coast was the peripheral bulge of this ice sheet and the land there is now experiencing subsidence. This sinking makes it appear as though sea level were rising more quickly than it truly is.
One solution to this problem of differentiating between eustatic sea level and apparent sea level rise caused by glacial rebound processes is to study an area that would not be affected by glaciation. A sea level curve from Barbados has been derived for this reason (Peltier, 1995). However, there may be other problems associated with using this area (e.g. tectonic elevation or subsidence, etc.).
Another solution is one presented by Peltier (1996). In this study, Peltier compares the variations of sea level derived from tide gauge data (short term) with radiocarbon data (long term) which would record the variations of sea level over the past several millennia on the east coast of the United States (Figure 3). Peltier felt that since the Laurentide ice sheet (which had partially covered North America) had disappeared by 6000 years ago, there should be no eustatic sea level change due to the melting of ice (it should have already happened by this point) since this time. The tide gauge data should reflect both apparent sea level change as well as any eustatic sea level change that we are presently experiencing. Subtracting the long term, isostatically controlled rates of sea level change from the total change determined by the tide gauge data should result in the present eustatic sea level change. Using this method, Peltier obtained a eustatic sea level rise of about two millimeters per year. This is supported, generally, by other sources (Plag, 1996).
Modeling of Conditions Influencing Postglacial Rebound to Gain Insight into the Earth
Much of the focus on isostatic rebound today is on the information that can be obtained about the earth's structure and processes through modeling. Studies on the viscosity of the mantle abound, as well as studies on general mantle and lithospheric properties. Most of the studies consist of mathematical and computer modeling; many assumptions and simplifications are necessary. Most models try to see the effect that certain conditions would have on rebound processes, and then check to see if the results are present geologically.
Various properties of the mantle have been explored via modeling of the effects these properties have on postglacial rebound. These include stratification of the mantle both vertically and horizontally (Fang and Hager, 1994), mantle rheology (Newtonian vs. Power law behavior; Gasperini et al., 1992; Wu, 1995), mantle phase changes (Christensen, 1985; O'Connell, 1976), gravity anomalies (James, 1992), and, of course, mantle viscosity (example in Middleton and Wilcock, 1994).
Calculation : Mantle Viscosity (Middleton and Wilcock, 1994). One way to decipher the viscosity of the mantle is to calculate how fast the surface of the earth rises after the load produced by an ice sheet is removed (melts). Figure 4 shows a simple model of a two dimensional cross-section through the ice sheet (the diamond shape) and the asthenosphere (neglecting lithospheric effects, apparently!). Assuming that the ice sheet had reached isostatic equilibrium before melting, the pressure exerted by the ice sheet would be balanced by the buoyancy pressure exerted by the asthenosphere.
Definition of variables
W = total width of ice sheet, H = height of ice sheet at x = 0
h = height of ice sheet at some x, L = thickness of asthenosphere
r = density, u = horizontal flow rate of asthenosphere averaged over L
v = rate of vertical movement averaged over W
The origin of the coordinate system is below the center of the ice sheet.
Ice pressure (P) is given by rgh and h = H - (Hx)/(W/2) where (Hx)/(W/2) is the correction for the thickness of the ice as you move away from the origin.
If P = rgh then P = rg (H-Hx)/(W/2) ---> (W/2)P = rgH - rgHx
The pressure gradient would therefore be dP/dx (where d represents the partial derivative). So if you take the partial derivatives of both sides of the equation with respect to both x and P, you obtain:
0dx + (W/2)dP = 0dP + 0dx - 0dP - rgHdx
---> dP/dx = -rgH/(W/2)
This value represents the pressure gradient that causes the asthenosphere to flow.
v = same as above which also equals the rate of isostatic rebound
u = same as above which also equals the average velocity between two parallel plates
m = viscosity of the asthenosphere
Since u equals the average velocity between two parallel plates, equations can be given by the theory of Couette flow (not given in this paper) and one can get an equation for u based on these equations as well as the pressure gradient calculated in Part I.
u = (rgHL2)/(6mW)
Since Wv = 2Lu (two horizontal flows contribute to one vertical flow) and therefore:
v = (2L/W)u = (rgHL3)/(3mW2)
Now you can solve for viscosity (m) by filling in the "knowns" to the final v equation. v itself, as the rate of rebound, can be estimated from geologic data, as can H and W. L can be estimated from seismological data. This leaves only the viscosity. Middleton and Wilcock obtained a viscosity of 4 x 1019 using this method; other more complex methods yield values near 1021 (James and Bent, 1994).
Using similar modeling principles, some of the properties of the lithosphere are also modeled for glacial rebound processes and compared to geologic evidence. These include studies of the thickness of the lithosphere (continental and oceanic; Peltier and Wu, 1983; Manga and OÕConnell, 1995), of strain rates resulting from glacial isostasy (James and Bent, 1994), and general lithospheric responses to loads, such as fracturing or bending (Middleton and Wilcock, 1994).
Thickness of the lithosphere
Many calculations about the earth need to know the thickness of the lithosphere. Some assume that it is 120 km, while others think it is closer to 250 km (e.g. Peltier and Wu, 1983). A study by Manga and OÕConnell (1995) shows that the lithosphere probably has a variable thickness- 250 km under continents (continental roots) and 100 km under the oceans (Figure 5).
This model of the effects of different lithospheric thicknesses on postglacial rebound was performed using an axisymmetric flow model where both the mantle and lithosphere were described as Newtonian fluids. The initial surface depression is assumed to be a gaussian curve and can be described as:
z = zoe-(r/R)2,
where zo is the original depth of the depression and is equal to -1 km, R is the radius of the depression and is equal to 1000 km, z is the current depth, and r is the distance from the center of the depression. These calculations were done for three different conditions: 1) a 250 km constant thickness lithosphere, 2) a 100 km constant thickness lithosphere, and 3) a variable thickness lithosphere were the continental root is 250 km thick but is reduced to 100 km thick at R = 1000 km.
The results of these calculations are shown in Figure 6. The relaxation rate of the variable thickness lithosphere is similar to that of the 250 km lithosphere when near the center of the depression. However, outside the depression the relaxation rate of the variable thickness lithosphere is similar to that of the 100 km thick lithosphere. This implies that there is a change of the thickness of the lithosphere at the edge of the continents. Near the edge of the depression, however, the relaxation rate of the variable thickness lithosphere is not similar to either of the other models, nor is it intermediate between the two. This may be a result of the shape that Manga and OÕConnell chose for the continental root (not realistic) or any of the other simplifications that were necessary. These results thus far imply that relaxation rates (and therefore relative sea level curves) near the edge of continents are not reliable; however, the relative sea level curves calculated for areas such as Hudson Bay, should not be affected by the lithospheric variations since they are far from plate boundaries.
Surface Strain Rates
This study by Manga and O'Connell (1995) continues by plotting surface strain rates for both constant and variable lithosphere thicknesses for increasing distances from the center of the depression (Figure 7). The type of deformation that results is written above each graph. The two top diagrams represent initial unloaded conditions and the two bottom diagrams are at present time (about 10 kyr later).
What is most interesting about the diagrams is the conclusions made by Manga and O'Connell. Neither strain pattern (constant or variable) is observed geologically today; this may be a result of the overpowering tectonic stresses that now exist. However, Manga and O'Connell feel that during the early stages of postglacial rebound these stresses were orders of magnitude larger and postglacial rebound effects may have dominated over the tectonic stresses (as well as the effect of the simplifications they used). The surface deformation that would be experienced at that time (top right diagram in Figure 7) would be that of concentric extension directly at the lithosphere thickness change and concentric compression beyond that. It has been suggested by Kemp (1994) that the initiation of subduction may occur in areas of extensional deformation. Combining both concepts implies that subduction may be initiated on the continental margin (where the lithosphere thickness changes) as a result of glacial unloading. This is important and interesting because it implies a fairly direct link between climate and mantle dynamics!
Isostatic rebound is the sinking or rising of the surface as the lithosphere responds to a surface load by sinking or rising in the asthenosphere. A load can be composed of many different materials; this paper primarily focused on ice sheets. Isostatic rebound has many effects, which include both the rising and sinking of the land surface, which complicates our calculations of sea level. Postglacial isostasy is often used in models to determine information about the earth.
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