Sea Level Rise and Small Glaciers

1
Sea Level Rise and Small Glaciers

• The math and physics behind the science

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The Math is There

• See many claims about climate change.
• Rarely get to see the derivations leading to
  those claims.
• This lecture will show the math that lets us
  predict sea-level rise due to melting mountain
  glaciers.
• Primarily based on
• Bahr, Meier and Peckham, The physical basis of
  glacier volume-area scaling, 1997.
• Bahr, Global distributions of glacier properties
  A stochastic scaling paradigm, 1997.

3
Cant Show Everything

• My goal is to convince you that climate change
  science is based on real math and physics.
• No arm waving necessary!
• So Ill derive the technique but wont finish the
  actual application.
• Actual application requires analyzing oodles of
  satellite images and other data.
• Well derive the math that shows how to analyze
  those images.
• Alas, only have one hour.

4
Background

• People pump greenhouse gasses into atmosphere.
• Sun shines on Earth.
• Greenhouse gasses trap the resulting heat.
• Earth heats up.
• Glaciers melt.
• Melting water flows into oceans.
• Oceans rise.
• Entire island nations disappear underwater.
• Maldives, etc.
• Also Venice, US Gulf Coast, Bangladesh,
  Indonesia, etc.

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How much?

• Approx 1.7mm/year.
• Seems small, but adds up.
• From 1900 to 2100, thats approximately 0.5 m.
• Church and White (2006).
• 80 of the 1200 Maldives Islands are less than 1m
  above current sea level.
• No more beaches, no more islands.
• Kiss paradise goodbye. ?

Photo from National Geographic
6
Sea Level Rise Contributors

• Its not just melting glaciers.
• Sea level changes due to
• Thermal expansion of the ocean.
• Melting ice caps and ice sheets.
• Melting mountain glaciers.
• And host of other processes post-glacial
  rebound, groundwater pumping, ENSO (short term
  and localized), etc.

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Sea Ice Not a Contributor

• Polar sea ice is melting at a distressing rate.
• Important harbinger of things to come.
• Polar bears endangered.
• But doesnt change sea level.
• Sea ice is floating ice on the ocean surface.
• Its like ice cubes in your glass of water.
• When the ice cubes melt, the glass doesnt
  overflow.

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Land Ice is the Culprit

• Thermal expansion and glacier water (flowing from
  land into the oceans) causes most of the rise.
• On decade and century time scales.
• Well focus on glacier component.

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Greenland and Antarctica

• Greenland and Antarctic ice sheets are huge
  reservoirs of land ice.
• But takes a long time to transport their water to
  the ocean.
• Water melts at the surface.
• Percolates down into the ice sheet.
• Much of it refreezes in the firn (old snow)
  before reaching ocean.
• Some of it lubricates the bottom of the glacier
  and makes it flow faster.
• But can only flow out through a limited number of
  outlet glaciers. Restricted nozzles.

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Skiing Across Greenland in the Name of Science
Neil Humphrey (left), Tad Pfeffer (right), and me
(photographer).
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Measuring Percolation in Greenland
We cored a transect of Greenland to look for
meltwater that refreezes in the firn before
reaching the ocean.
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Icebergs From Illilusat (Jakobshavn) Outlet
Glacier
Ice flows out of the Greenland Ice Sheet and into
the Atlantic, breaking off as icebergs.
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But Mountain Glaciers are the Canaries

• Small mountain glaciers are very susceptible to
  warming.
• Small is a bit of a misnomer. Some are bigger
  than Rhode Island.
• They melt rapidly (decadal time scales).
• Of all the melting ice, 60 comes from these
  small mountain glaciers.
• Meier et al. Science, 2007.

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Need Volume of Mountain Glaciers

• How much can mountain glaciers contribute to sea
  level?
• 160,000 mountain glaciers.
• Meier and Bahr (1996)
• Each ones volume has to be measured.
• Aurgh! Measuring even one glacier takes a lot of
  money, time, and people.

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Measuring Glacier Volume

• Can only see the surface of a glacier.
• So have to drill holes everywhere through the
  glacier to measure volume.
• Or have to use ground penetrating radar.

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Weve Done That On Friendly Glaciers
Here we are drilling through the Worthington
Glacier, AK.
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But Most Glaciers Look Like This
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And This
Lets see you pull a drill or radar across
160,000 of these!
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Need Remote Sensing!

• Want a satellite to take a picture of a glacier
  and say That glacier is 100 km3.
• And That glacier over there is 1643 km3.
• And
• How?
• Measure surface area (easy with satellite) and
  convert to volume (cant measure from satellite).
• Use fancy scaling (math) analysis.

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Scaling Example

• Suppose I told you glaciers look like square
  boxes.
• All you can see is the surface of the box.

10 km

• Whats the volume of the glacier (box)?

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Right, And Satellites Could Do the Same Thing

• Use satellite to measure surface area, then
  convert that to a volume.
• Area width length
• Volume width length height
• Volume Area height
• And if a glacier looks like a box, then
• width length height
• height (Area)1/2
• So Volume (Area)3/2

Called a scaling relationship
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Problem Glaciers Dont Look Like Boxes

• Glaciers look like misshapen rectangles on a good
  day.
• Many look like networks.
• Like rivers or trees.
• Glaciers have fractal topologies!
• Bahr and Peckham (1996).
• Thats a very complex shape where a small part of
  the shape looks like a miniature representation
  of the whole glacier.

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Whats a Fractal?

• Visually, a part looks like the whole.
• A classic fractal the Sierpinski gadget.

This part looks like the whole picture when
expanded!
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Fractal Topology
Break off one branch of the glacier network and
that branch looks like the whole glacier
(mathematically speaking).
Columbia Glacier. Photo courtesy Tad Pfeffer.
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Can We Scale Fractal Glaciers?

• Yes! They have complex shapes, but they do have
  a characteristic shape.
• Consider human body.
• Everyone is built slightly differently.
• But humans have a characteristic
• shape.
• E.g., Arm span roughly equals height
• width ? (height)1.0
• Called a scaling relationship.
• 1.0 is called the scaling exponent.
• In general, when solving other problems,
• scaling exponents can be any fractional
• number like 1.3 or 2.4.

Consider a spherical cow
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Glacier Scaling

• Want volume from (fractal) area.
• Area width length
• Volume width length thickness
• Assume width is proportional to length.
• There is no reason for a glacier to prefer
  growing left or right versus forward or back.
• Then
• Area ? length2
• Volume ? length length thickness

Close, but not entirely true width ? length0.6
(Bahr, 1997), but that detail isnt important
here and doesnt change our later results.
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Length-Thickness Scaling

• Suppose we can find a scaling relationship
  between thickness and length.
• thickness ? lengthp (for some p).
• Volume ? length length lengthp lengthp2
  (length2)(p2)/2
• Volume ? (Area)(p2)/2
• Aha! If we can find a length-thickness
  relationship, then can calculate the glacier
  volume.
• So we use physics and calculus to find the
  correct scaling exponent p for thickness ?
  lengthp.

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Need Glaciers Characteristic Length-Height
Scaling

• Start with physics forces acting on glaciers.
• Then represent with math.
• Then derive scaling relationships from the math.
• Then use satellites to measure area.
• Use scaling relationships to convert to volume.
• That gives estimate of amount of water that will
  flow into oceans and cause sea level rise.

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Some Glacier Physics

• Glaciers flow downhill under the influence of
  gravity.
• Just like water in a river but much slower.
• Think of honey oozing off of a tilted plate.

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Glacier Flow
Surface of glacier
Mountain under the glacier
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Hows It Really Flow?

• Gravity creates forces.
• To calculate these forces we use
• Conservation of mass
• Conservation of momentum (force balance)
• Well start with conservation of mass.

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Hang Onto Your Hats!

• Ok, here comes the math that I promised!
• Its easy.
• Some of it may look scary.
• But Ill never use anything more than simple
  algebra, derivatives, and integrals.

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Conservation of Mass

• Imagine a small box cut out of the glacier.
• The amount of ice flowing into the box has to
  equal the amount of ice flowing out of the box.
• Ice is incompressible, so no extra ice can be
  shoved into the box and/or stored in the box.
• i.e., no mass disappears.

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Small Box Cut Out of Glacier
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Mass In and Out of Box

• mass in mass out
• How much mass flows in per second?
• Its the velocity (v) times the mass.
• The mass is density (r) times volume.

r vy(yDy)
y-axis
yDy
r vx(xDx)
r vx(x)
y
r vy(y)
x-axis
x Dx
x
And theres also a z component, r vz.
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Mass Balance

• Sum of all the mass in and out must equal zero.
• mass conservation

Why DxDy? Because the mass is flowing in on that
whole side of the box. And thats the area of
the side. (Also ensures that each term has units
of mass.)
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Divide By the Volume

• If divide by the volume, should look familiar

Why, those are just derivatives! (Remember the
definition of a derivative?)
38
Make the Box Really Small

• In the limit of a very small box
• i.e., take limit as DxDyDz approaches 0.
• Called the Continuity Equation.
• Its just mass conservation.

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But Mass Is Lost!

• What if we consider a box at the surface?
• It snows and melts at the surface.
• Mass is lost at the surface!
• Let b be the mass added or lost at the surface.
• Now add up all the boxes from the bottom to the
  top of the glacier.

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Adding Up a Column of Boxes
b
Snow melts at the surface at rate b.
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Adding Up the Boxes

• Remember that a sum and an integral are the same
  thing!
• So the sum of the boxes is

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The Mass Conservation Equation

• Assume that the glacier has a thickness of h.
• In other words, our stack of boxes has a height
  h.
• Then integral becomes

What happen to the vz term? Its now b! Its
the amount of stuff leaving vertically through
the surface. And this also shrinks the height of
the glacier so we subtract dh/dt.
Technically, dh/dt comes from a dr/dt term in
the continuity equation. See me for details.
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Ack! I Dont Get It!

• Dont panic! Thats ok.
• Trying to convince you that the climate change
  science isnt pulled out of thin air.
• So my goal is to show you the math and leave out
  as little as possible!
• If this derivation seems mysterious, its online
  where you can peruse again. Or see me. Id love
  to explain it in gory detail!
• You can still follow the rest of this lecture if
  you accept that the equation has been properly
  derived.

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Ugh, Make it Simpler!

• For simplicity, well assume theres a nice
  well-defined average velocity so that the
  integral goes away.

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Still a Mess! Can we simplify?

• Yes, do a scaling analysis.
• Use a technique called nondimensionalization or
  stretching symmetries.
• If the equation is true for all glaciers, then it
  has to be true for glaciers of different sizes.
• So lets try stretching each variable as if we
  are trying to create a bigger glacier from the
  current glacier.
• Multiply each variable by a constant.
• This should give back the same equation, but for
  a bigger glacier.

46
Stretching
length

• Lets stretch
• the variable x by a factor of la. i.e.,
  xstretched la x
• the variable y by a factor of lb. i.e.,
  ystretched lb y
• the variable vx by a factor of lc.
• the variable vy by a factor of ld.
• etc.
• In other words, we multiply each variable by that
  amount.
• Its like we are saying, make the glacier longer
  by a factor of la , and make the glacier wider
  by a factor of lb , etc.

width
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Rescale and Factor

• The original equation
• Rescale each variable
• Factor out the constants

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Stretching Symmetries

• Note that the exact same original equation has to
  apply to the bigger glacier.
• In other words, the last equation (on previous
  slide) has to be the same as the first equation
  (on previous slide).
• That can only happen if
• So we have the requirements that
• f e c - a
• f e d - b

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Scaling the Variables

• Now we can really simplify!
• lf lec-a is the same as
• And separating,

Values for the bigger glacier
Values for the original smaller glacier
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Scaling Relationship

• So for any two glaciers this ratio has to be the
  same!
• Big glaciers, small glaciers its all the same.
• Must be some constant.
• So all of that work reduces to

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But Wait, Theres More

• That was just mass conservation.
• Now we have to do momentum conservation!
• These are two of the most important principles in
  physics.
• We need both to completely describe the flow of a
  glacier.

Also need energy conservation but on mountain
glaciers, the temperature is largely constant and
irrelevant.
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Forces on the Box

• Remember our little box cut out from the glacier?
• Ice in the box flows due to forces.
• Gravity
• Pressure from overlying ice.
• If I squish one side of a balloon, then the
  balloon bulges out on all the other sides.
• Same with the box. If I poke at one side, then
  ice oozes out the other sides.
• Well analyze the forces on the box to see how
  the ice flows.

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Normal and Shear Forces

• Two kinds of forces normal and shear.
• Normal is straight on.
• Like pressure.
• Shear is along the side.
• Like putting hand on a
• book and pushing sideways.

Technically we are dealing with stress, or force
per unit area. But Im going to gloss over that
for the moment.
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Shear Forces

• Shearing a book or deck of cards.

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Notation

• Let sxy be the forces on the x side of the cube
  acting in the y direction.

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All the Forces
And ditto on the other faces of our box i.e.,
we will also have szx, syz, etc.
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All the Forces in the x-Direction
y-axis
syx (yDy)
y Dy
sxx (x)
sxx (xDx)
y
syx (y)
x-axis
x Dx
x
And ditto on the other faces of our box i.e.,
we will also have szx(x) and szx(xDx).
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Conservation of Momentum

• Take the sum of all the forces in the
  x-direction.
• Tells us how much the ice will move through the
  box in the x-direction.

Why DxDy? Because this force is acting on every
part of the xy face of the box. So have to add
the force for every point on that face.
g is gravity! And r is the density of ice. We
are considering the force due to gravity acting
in the x-direction. F ma. In this case, the
acceleration is gravity g, and the mass is given
by the density times the volume of ice, DxDyDz.
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Recognize the Derivatives?

• Divide by the volume of the box DxDyDz.

These are just derivatives! (Remember the
definition of derivatives?)
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Let the Box Get Really Small

• Now take the limit as the box gets really small.

• This is the net force, Fnet ma.
• But wait, glaciers dont accelerate! They move
  waaay too slowly. The change in velocity is
  miniscule, negligible, insignificant. In other
  words, a 0.

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Force Balance Equations

• So the net force is zero!
• And ditto in y- and z-directions.

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Ack! I Am Soooo Lost!

• Again, trying to be complete.
• Want to convince you that climate change science
  is backed by the real thing.
• No arm waving necessary.
• But if unhappy, feel free to accept the above
  equations and then move on!
• Can review this stuff in grad school we need
  people like you studying climate change science!

63
Again, What a Mess!

• Can we simplify?
• Yes, do stretching analysis again!
• We find a whole bunch of scaling relationships.
• Most important is
• Wheres that a come from? Its the component of
  gravity in the x direction.

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Huh? A Little Intuition Please!

• g sin a is the component of gravity acting
  along the glacier.

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Intuition

• So is just the component of the
  gravitational force acting along the bottom of
  the glacier.
• The shear.
• Note sin a a

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Strain Rates

• The shear force causes the glacier to move by
  deforming.
• Each unit of force causes a certain amount of
  deformation.
• Glaciers dont accelerate, but they do have a
  velocity.
• The rate of deformation is measured as strain
  rate. Its the change in velocity over a
  distance.

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Strain Rate Pictures

• For normal strain rates, deformation is only in
  one direction
• For shears, there is deformation in two
  directions, so
• Think of it as the rate at which the ice gets
  stretched (or compressed).

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Non-Linear Glacier Flow Law

• Let be the rate of deformation caused by sxz.
• Experiments (and theory) show that

(where n 3).
The full form of the flow law is more
complicated and contains deviatoric stresses,
but this is exact for shear terms.
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Substitute

• Now we can substitute the strain rate for the
  stress.
• Remember that the angle a is just rise over
  run. So

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Combine Mass and Momentum

• Finally, we can combine the derivations from mass
  and momentum conservation!
• Recall,
• And
• Together,

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Need to Use An Observation

• From data, we know that the amount of melting at
  the surface increases as a parabola with distance
  down a glacier.
• Bahr and Dyurgerov (1999).

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Thickness Scales With Length

• Substituting gives
• This is like the persons height scaling with
  their arm span. But the scaling exponent is
  different.
• (n3)/(2n2) instead of 1.

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Volume and Area

• Finally we can get volume from surface area!

(Recall that n 3.)
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THATS IT!

• Weve just derived the volume from the surface
  area.

That aint no standard box! Remember the box
was Volume Area1.5.
p.s. Scaling constant (0.034 in units of km to a
nasty power) comes from some observations.
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And Data Backs It Up!

• Remember I said we did field work on a few nice
  glaciers?
• Heres a log-log plot of volume versus area for
  144 glaciers.
• The regression is 1.36, very close to the
  theorys 1.375.
• From Bahr, Meier, and Peckham (1997).

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And Sea-Level Rise?

• So now, satellites can measure surface area of
  each glacier.
• All 160,000.
• Then the area of each glacier is converted to a
  volume with the volume-area scaling relationship.
• Then this tells us how much ice can melt into the
  oceans!
• 0.1 to 0.25 meters by 2100.
• Meier et al., Science, 2007.
• Phew!

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So Now You Know

• Ok, now you know the kind of math and physics
  that goes behind the climate change science.
• But this is only a very small part of the big
  picture!
• You can spend a long time learning all the
  details of the climate models.

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We Need More People Doing This Kind of Climate
Change Science!
Interested?