Introduction to Mathematical Climatology

17 April 2012

In celebration of this blog opening for no-limits mathematics and science on climate, I’ll start on a book idea I’ve had — an introduction to mathematical climatology.  Maybe that should be a mathematical introduction to climatology.  At any rate, given that entire books and sets of books are written on small subsets of the climate system, I’m going to lean heavily on the math and rely on an adult audience to develop their physical intuition themselves.  In this, I hope to keep the introduction to something that I can do in blogging time.

My first step is to define our system.  Since I’m interested in ice ages this leads to a more extensive system than if I were only interested in 10-100 years.  Obviously the atmosphere and ocean are part of the climate system.  Plus the glaciers, ice shelves, ice sheets, and sea ice.  But we also need to consider the growth of deserts (they expand during glacial maxima) and changes of surface cover (changing types of forest changes the albedo).  And, on ice age time scales, the solid earth responds to ice sheet growth and decay — down to a depth of about 670 km.  The upper limit is … fuzzy.  It seems fairly clearly to be at least as high as the top of the homosphere (zone in which gasses are mixed uniformly without regard to molecular weight), or about 80 km.  For some purposes, like radios, we might also consider the ionosphere and up to the thermosphere (about 1000 km, which means many earth-observing satellites would be considered to be orbiting inside the climate system).  And a number of other things between 670 km below the surface and 80-1000 km above, such as clouds, permafrost, ozone layer (and its chemistry), ground hydrology, plant life (for its evapotranspiration and ground hydrology effects), ….

All those things are inside the system.  They interact with each other.

Outside the climate system, we have the sun, moon, and planets.  Obviously the sun is important to the climate system.  No sun = no interesting climate.  And it does vary in its output, both in total magnitude and in spectral distribution of the energy.  Mostly this is a matter of increasing UV output at high output times.  It is also important for tides.  The moon is even more important for tides.  Both sun and moon play a role in the precession of equinoxes, hence timing and amplitude of the seasons — certainly a climate matter.  Also, the planets (mainly Jupiter and Saturn) affect the earth’s orbit around the sun, and moon’s orbit around the earth on longer (ice age) time scales, leading to Milankovitch cycles in climate.

If we consider even longer time scales, millions of years and up, the moon’s orbit moves to being inside the climate system.  On these time scales, the moon is receding from the earth and the earth’s rotation is slowing.  But the rate of recession has much to do with the tides, which depend on the ocean, sea level, and distribution of continents.

On very long time scales, we also need to consider the evolution of the sun.  In early earth history, the sun was only 70% as bright as it is now.  Over the past 4.5 billion years, it has increased to present brightness.  And it will continue to brighten until it reaches red giant stage in another 5 billion years or so.

That’s a cursory summary of the climate system.  We’ll see more things to be considered as we examine individual components.  One arm of the mathematics will be statistical — for describing climate from observations.  The other may include statistics, but will be primarily analytical — for making predictions about the system.

Ocean current visualization

16 April 2012

A friend recently comments about how impressed he was by the degree of accord in ocean current features between the NASA ocean model with data assimilation — http://www.livescience.com/19664-nasa-animation-ocean-currents.html and the GFDL ocean model without data assimilation — http://www.youtube.com/watch?v=JMLy4jV7Xgo.  It’s a good news/bad news situation.  The bad news being that the ocean data assimilation isn’t doing as much as you might hope.  But the good news is, this means that the models must be doing something right to be representing the ocean so similarly in spite of their different histories.

Data assimilation is a great thing, but for ocean currents, what you see in a global ocean data assimilation is very largely what the model chose to do, regardless of the data.  ARGO, for instance, provides information at about 300 spots per day (3000 floats, reporting once every 10 days).  Given 300 million square km in the ocean, that’s 1 million square km per float.  If the ocean’s correlation scale were like the atmosphere’s, upwards of 1000 km Rossby radius, that would be ok sampling, and more or less comparable to the radiosonde network (1000 per day, give or take).  Unfortunately, the mid-latitude ocean’s Rossby radius is more like 100 km, making the ocean undersampled by a factor of 100 or so.

Satellite altimetry, if it was used for assimilation, improves the sampling problem.  But not by as much as satellite sounders do for the atmosphere.  The problem here is that the altimetry tracks are very narrow, so, again, little of the ocean is seen per day.

Where there are no data, the result of assimilation is the model’s output.  So ca. 99% of what you’re seeing in the video is still model.

The good news is that the two models, with different lineages, are producing such similar results.  Suggests that maybe we do know something about ocean behavior.

If you want to see current ocean model forecasts, to 6 days, see http://polar.ncep.noaa.gov/global/

Vickie is now blogging

10 September 2009

My wife is now blogging about her experiences volunteering at one of the few organizations that works with prostituted women.  Her blog is at vickiesprostitutionblog.blogspot.com. The writing is excellent. The lives are sad to tragic. But Vickie helps improve things.

Hello world!

5 August 2008

At some point I’ll start putting up posts.  They’ll be ‘no holds barred’ as far as the content goes — Partial Differential Equations will show up unabashedly.


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