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365 Days of Climate Awareness 322 – General circulation models 3: boundary conditions [1]
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Date: 2022-07-01
This is because a “differential” in mathematics is computed from a rate of change. How much does one variable (y, for example) change when another (x) changes? This rate of change itself can vary: an increase in (x) of 1 might under different circumstances produce different changes in (y). Look at the sine wave, a smoothly varying curve, in the illustration. With each identical forward step in the horizontal axis (x), the height (y) changes by a slightly different amount from the previous, and following, amounts. That’s what creates the curve, as opposed to a straight line. In a curve, the rate of change—the slope of the graph—changes as you move along it. That’s bad enough. How about more variables? What if (y) depends not only on (x), but also on (z), (a) and (b)? Hence the sorcery!
In nature, where just about nothing is constant (though we often pretend they are, to simplify the math), rates of change are the simplest means to approach almost any topic. Predicting the water level and current velocity of tides—extremely important for shipping—are an ideal application of diffEQ’s. Disease control, where microbes grow exponentially—with increasing speed as their population grows, until the system becomes full—is another. For a climate example, the temperature of ocean water will vary in space—east (x), north (y), depth (z), and in time (t). Those four fundamental variables determine quite a lot, but not everything!
But nature also has limits. COVID did not continue to spread unchecked, because earth’s population is finite, and the spread rate decreased, and (much) later the actual case load began to decrease. A cycle like the tides has inherent limits within itself—the regular motion of earth and moon—but their height and flow rates are also constrained by things such as the local bathymetry, which limit how much water can pass through. These natural limits, whether based on population, physical barriers like mountains or coastlines, or physical properties like wind speed or incoming solar radiation, are called boundary conditions.
A more complex species of diffEQ’s contains these built-in boundaries, so the system and its variables don’t become unrealistic. This is because climate models are relentlessly tuned to known, measured aspects of global climate. In climate models, most boundary conditions are empirical: that is, based on measurements, not theoretical derivations. Topography (mountains, valleys, plains.) and bathymetry (coastlines, channels and basins) are two obvious, measured boundary conditions. Other empirical boundary conditions include the amount of solar energy entering earth’s climate system (not constant! Because earth is in motion, and its orbit is elliptical), changing land use patterns (a very important aspect), and many more. These models are complicated.
Tomorrow: some of the main boundary conditions used in IPCC climate models.
Be brave, be steadfast, and be well.
Sources:
NOAA: climate modeling
More on diffEQ's
Boundary conditions
Boundary conditions in climate modeling
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