1	Sea Level Rise and Small Glaciers The math and physics behind the science
2	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	Can’t Show Everything My goal is to convince you that climate change science is based on real math and physics. No arm waving necessary! So I’ll derive the technique but won’t finish the actual application. Actual application requires analyzing oodles of satellite images and other data. We’ll 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.
5	How much? Approx 1.7mm/year. Seems small, but adds up. From 1900 to 2100, that’s 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. L
6	Sea Level Rise Contributors It’s 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.
7	Sea Ice Not a Contributor Polar sea ice is melting at a distressing rate. Important harbinger of things to come. Polar bears endangered. But doesn’t change sea level. Sea ice is floating ice on the ocean surface. It’s like ice cubes in your glass of water. When the ice cubes melt, the glass doesn’t overflow.
8	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. We’ll focus on glacier component.
9	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.
10	Skiing Across Greenland in the Name of Science
11	Measuring Percolation in Greenland
12	Icebergs From Illilusat (Jakobshavn) Outlet Glacier
13	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.
14	Need Volume of Mountain Glaciers How much can mountain glaciers contribute to sea level? 160,000+ mountain glaciers. Meier and Bahr (1996) Each one’s volume has to be measured. Aurgh! Measuring even one glacier takes a lot of money, time, and people.
15	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. $
16	We’ve Done That On “Friendly” Glaciers
17	But Most Glaciers Look Like This
18	And This
19	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 (can’t measure from satellite). Use fancy “scaling” (math) analysis.
20	Scaling Example Suppose I told you glaciers look like square boxes. All you can see is the surface of the box.
21	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
22	Problem: Glaciers Don’t 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). That’s a very complex shape where a small part of the shape looks like a miniature representation of the whole glacier.
23	What’s a Fractal? Visually, a part looks like the whole. A classic fractal – the “Sierpinski gadget”.
24	Fractal Topology
25	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.
26	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
27	Length-Thickness Scaling Suppose we can find a scaling relationship between thickness and length. thickness ∝ length^p (for some p). Volume ∝ length * length * length^p = length^p+2 = (length²)^(p+2)/2 Volume ∝ (Area)^(p+2)/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 ∝ length^p.
28	Need Glacier’s 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.
29	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.
30	Glacier Flow
31	How’s It Really Flow? Gravity creates forces. To calculate these forces we use Conservation of mass Conservation of momentum (force balance) We’ll start with conservation of mass.
32	Hang Onto Your Hats! Ok, here comes the math that I promised! It’s easy. Some of it may look scary. But I’ll never use anything more than simple algebra, derivatives, and integrals.
33	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.
34	Small Box Cut Out of Glacier
35	Mass In and Out of Box mass in = mass out How much mass flows in per second? It’s the velocity (v) times the mass. The mass is density (r) times volume.
36	Mass Balance Sum of all the mass in and out must equal zero. mass conservation
37	Divide By the Volume If divide by the volume, should look familiar…
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”. It’s just mass conservation.
39	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.
40	Adding Up a Column of Boxes
41	Adding Up the Boxes Remember that a sum and an integral are the same thing! So the sum of the boxes is
42	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
43	Ack! I Don’t Get It! Don’t panic! That’s ok. Trying to convince you that the climate change science isn’t 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, it’s online where you can peruse again. Or see me. I’d 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.
44	Ugh, Make it Simpler! For simplicity, we’ll assume there’s a nice well-defined average velocity so that the integral goes away.
45	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 let’s 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 Let’s stretch the variable x by a factor of l^a. i.e., x_stretched = l^a x the variable y by a factor of l^b. i.e., y_stretched = l^b y the variable v_x by a factor of l^c. the variable v_y by a factor of l^d. etc. In other words, we multiply each variable by that amount. It’s like we are saying, “make the glacier longer by a factor of l^a”, and “make the glacier wider by a factor of l^b”, etc.
47	Rescale and Factor The original equation: Rescale each variable: Factor out the constants:
48	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
49	Scaling the Variables Now we can really simplify! l^f = l^e+c-a is the same as And separating,
50	Scaling Relationship So for any two glaciers this ratio has to be the same! Big glaciers, small glaciers – it’s all the same. Must be some constant. So all of that work reduces to
51	But Wait, There’s 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.*
52	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. We’ll analyze the forces on the box to see how the ice flows.
53	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.
54	Shear Forces Shearing a book or deck of cards.
55	Notation Let s_xy be the forces on the x side of the cube acting in the y direction.
56	All the Forces
57	All the Forces in the x-Direction
58	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.
59	Recognize the Derivatives? Divide by the volume of the box DxDyDz.
60	Let the Box Get Really Small Now take the limit as the box gets really small.
61	Force Balance Equations So the net force is zero! And ditto in y- and z-directions.
62	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 Where’s that a come from? It’s the component of gravity in the x direction.
64	Huh? A Little Intuition Please! “g sin a” is the component of gravity acting along the glacier.
65	Intuition So is just the component of the gravitational force acting along the bottom of the glacier. The shear. Note sin a ≈ a
66	Strain Rates The shear force causes the glacier to move by deforming. Each unit of force causes a certain amount of deformation. Glaciers don’t accelerate, but they do have a velocity. The rate of deformation is measured as “strain rate”. It’s the change in velocity over a distance.
67	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).
68	Non-Linear Glacier Flow Law* Let be the rate of deformation caused by s_xz. Experiments (and theory) show that
69	Substitute Now we can substitute the strain rate for the stress. Remember that the angle a is just “rise over run”. So
70	Combine Mass and Momentum Finally, we can combine the derivations from mass and momentum conservation! Recall, And Together,
71	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).
72	Thickness Scales With Length Substituting gives This is like the person’s height scaling with their arm span. But the scaling exponent is different. (n+3)/(2n+2) instead of 1.
73	Volume and Area Finally we can get volume from surface area!
74	THAT’S IT! We’ve just derived the volume from the surface area.
75	And Data Backs It Up! Remember I said we did field work on a few “nice” glaciers? Here’s a log-log plot of volume versus area for 144 glaciers. The regression is 1.36, very close to the theory’s 1.375. From Bahr, Meier, and Peckham (1997).
76	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!
77	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.
78	We Need More People Doing This Kind of Climate Change Science!