Sea Level Rise and Small Glaciers. The math and physics behind the science. 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.
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Sea Level Rise and Small Glaciers The math and physics behind the science
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.
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.
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.
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. Photo from National Geographic
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.
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.
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.
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.
Skiing Across Greenland in the Name of Science Neil Humphrey (left), Tad Pfeffer (right), and me (photographer).
Measuring Percolation in Greenland We cored a transect of Greenland to look for meltwater that refreezes in the firn before reaching the ocean.
Icebergs From Illilusat (Jakobshavn) Outlet Glacier Ice flows out of the Greenland Ice Sheet and into the Atlantic, breaking off as icebergs.
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.
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.
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. $
We’ve Done That On “Friendly” Glaciers Here we are drilling through the Worthington Glacier, AK.
And This Let’s see you pull a drill or radar across 160,000 of these!
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.
Scaling Example • Suppose I told you glaciers look like square boxes. • All you can see is the surface of the box. 10 km • What’s the volume of the glacier (box)?
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
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.
What’s a Fractal? • Visually, a part looks like the whole. • A classic fractal – the “Sierpinski gadget”. This part looks like the whole picture when expanded!
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.
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…
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 isn’t important here and doesn’t change our later results.
Length-Thickness Scaling • Suppose we can find a scaling relationship between thickness and length. • thickness ∝ lengthp (for some p). • Volume ∝ length * length * lengthp = lengthp+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 ∝ lengthp.
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.
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.
Glacier Flow Surface of glacier Mountain under the glacier
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.
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.
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.
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. r vy(y+Dy) y-axis y+Dy r vx(x) r vx(x+Dx) y r vy(y) x-axis x + Dx x And there’s also a z component, r vz.
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 that’s the area of the side. (Also ensures that each term has units of mass.)
Divide By the Volume • If divide by the volume, should look familiar… Why, those are just derivatives! (Remember the definition of a derivative?)
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.
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.
Adding Up a Column of Boxes b Snow melts at the surface at rate b.
Adding Up the Boxes • Remember that a sum and an integral are the same thing! • So the sum of the boxes is
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? It’s now b! It’s 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.
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.
Ugh, Make it Simpler! • For simplicity, we’ll assume there’s a nice well-defined average velocity so that the integral goes away.
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.
Stretching length • Let’s 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. • It’s 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
Rescale and Factor • The original equation: • Rescale each variable: • Factor out the constants:
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
Scaling the Variables • Now we can really simplify! • lf = le+c-ais the same as • And separating, Values for the original smaller glacier Values for the bigger glacier
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