The goal of this lab is to teach you about floating, buoyancy, and isostasy, an important geosphere phenomena. In addition you will gain some experience with simple computer modeling, and practice some basic math skills.

**Isostasy** is the
idea the the __crust floats on a very viscous fluid__, __and
responds to changes in load accordingly__. The critical parameter
here is **density**, which is the amount of **mass per unit
volume**. We will be using units of grams per cubic centimeter,
**gm/cc**, in this lab. The diagram below shows two blocks
floating in a fluid.

How can you tell the two blocks are of different densities, and which one is denser?

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List at least two different factors that determine the density of rocks.

1) _________________________________________________________________________

2) _________________________________________________________________________

British sailors used to be buried at sea. At the base of the corpse stones (from the ships ballast) or other weighty material would be attached. They were trying to increase the density of the burial package. The crime scene version of this is of course cement overshoes, where the gravity of the situation is high.

In fact, with only the information given above, and a ruler you can estimate what the densities are of the two blocks above in gm/cc (assuming the blocks are block shaped in 3-D and floating in water). Do so. This is easier than you might think.

density of block 1 ____________________________ density of block 2 _________________________

You have some Styrofoam, and two different types of wood blocks, and some containers that can hold water. Estimate the density of each of block.

density of Styrofoam ____________________

density of balsa wood ____________________

density of "hard" wood ____________________

If you add salt to the water, will it float in the same way? _________ Why or why not? ___________________

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A thought question: There is a way to tell
if your crust is in **isostatic equilibrium** or not. What
type of measurements are needed?

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One big difference between mountains and continental crust floating on mantle material at some depth is the time it takes to respond to a change in load, i.e. how long it takes to return to equilibrium if disturbed. Rebound often occurs after deglaciation because the weight of the ice has been removed from the crust. As the crust rebounds it leaves behind stranded beaches. On Kong Karls Land raised beaches about 10,000 years old are now at 130 m elevation above sea level. Another beach some 6500 years old in central Spitsbergen is at an elevation of 11 m above sea level. Calculate the rates of relative uplift in cms/yr for both situations. Info from: Hjelle, A., 1993, Geology of Svalbard, Norsk Polarinstitutt, Oslo, Norway, 163 p

long term average rate of uplift for Kong Karls Land ____________________________

long term average rate of uplift in central Spitsbergen ____________________________

Using Excel to model isostatic equilibrium and crustal roots.

You will need a computer to complete this exercise.

As discussed in lectures, computer models are extremely useful in modeling complex earth systems. Many of these models are very sophisticated, and would require substantial time to learn how to use. However, even simple modeling using Excel can be instructive. One reason computer models are so useful is that they allow us to easily explore how changing key variables changes system behavior. We can compare computer behavior with real world behavior, using the computer behavior to understand real world causes better, and using real world behavior to build a better computer model.

We will use Excel to a) model crustal isostasy and mountain roots (how the crust floats), and b) model what the history of rebound (the rise of the crust due to a removal of load) might look like in form.

First download the Excel model from Blackboard (or get a copy from your instructor). Open it up and follow the instructions and answer the questions below.

**a) Modeling crustal roots:**

First, familiarize your self with the columns of numbers and what real world phenomena they attempt to capture. Notice that the column with "average height of mountain topography in km above MSL" is populated by all zeros, except the middle row, which has a one in it. On the graph you see bars extended downwards from the zero level. They represent the depth to the bottom of the crust. Right next to them but offset are bars standing up - these represent the mountain topography. Except for the one bar associated with the 1 km cell, they all go down to 35 km. In this model this is taken as the depth of continental crust with its top at mean sea level (MSL) floating on mantle material in isostatic equilibrium.

Now populate the cells with new values representing increasing mountain topography, starting with 0 and going up to 5 km (the height of the Tibetan plateau).

What is the ratio between the mountain topography and the root? ________

What is the depth to the base of the crust underneath the 5 km high crust? _______

Print of a copy of the graph to hand in with your assignment and attach.

Increase the continental density cell to 2.75 gm/cc. What is the depth of crust underneath the 5 km high block now? _________

Decrease the mantle density by .1 gm cc. What is the depth of the crust underneath the 5 km high block now? ___________

Describe how the depth of the root beneath the mountain changes as you change the density contrast between the continental and mantle densities. ________________________________________________________________________________________

We know from seismic 'sounding' that the crust underneath the 5 km high Tibetan plateau is 70 km thick. Find a combination of mantle densities and continental densities that will produce such a crustal thickness by trial and error, and report the values below. As you play with the model to find an answer, consider the following. The mantle is more homogeneous and its density is better constrained. The continental crust is much more heterogeneous in composition.

continental density _______________________ mantle density _____________________________

Discuss in what ways this model is not realistic.

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**b) Modeling the history of isostatic rebound.**

When a load is removed from the crust it adjusts
by rising, i.e. it floats higher. ** What will be the history
of uplift or rise??** If you think about pushing a board
down into water and then letting go, it will push back up with
some velocity (don't try this at home, or without the guidance
of a physicist; i.e. let him or her push the board in the water
and let go). The greater the depth the board is pushed down the
greater the buoyancy force pushing it upward and the greater the
speed with which it "rebounds". For the relatively slow
movement of rebounding crust it then seems reasonable to propose
that the speed of uplift is controlled by the upward pushing force
at the base of the crust. Since with uplift the upward pushing
force diminishes one might expect the speed to diminish also.
In our particular case we are going to have the rebound rate be
one meter/year for the a case where 1 km of total rebound is possible
(removing some 200 m of load with a continental density). We will
consider changes in increments of 100 years. At each increment
we will recalculate the rebound rate as a function of the percentage
of remaining rebound possible. For example at the point where
500 m of rebound has been accomplished, then the rebound rate
will be .5 m/year, and if 750 m of total rebound has occurred
(of the possible 1 km) then the rate will .25 m/year.

Initially the model is frozen in time at 300 year. Change the first column so that you have an increase of 100 years for each cell below.

What form does the resulting history have? ___________________________________________________________

Print off a copy of the graph and attach it to your sheet.

The last Ice Age involved the relatively quick removal of ice caps about ten thousand years ago. The interior of Canada is still rebounding at a rate of over 1 cm year. Given this information, is the above model realistic for that rebound history?

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How might you change the model so that it would more realistically model the history of crustal rebound in this case?

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Finis!