365 Days of Climate Awareness 322 – General circulation models 3: boundary conditions

I’ll admit that the term “boundary conditions” used to make me twitch. Partial differential equations and 4-dimensional (three physical dimensions plus time) calculus were always difficult hurdles for me (and not a small part of the reason why I moved from physics over to geology!) But differential equations—diffEQ’s, as they’re known in the biz—are the real mathematic sorcery within climate, or almost any natural system model. In the end, you plug numbers into the many different variables and receive numerical output. You can look at all math as just increasingly fancy types of addition and subtraction. But what makes an equation “differential” is that it focuses on differences.

A sine wave, broken into a series of discrete measurements. 

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!

Ye olde diffEQ. The differential proper: dy ("change in y") over ("with respect to") dx ("change in x"). Not a constant! Change in the differential's value is the heart of mathematical modeling.

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!

Exponential increase in global COVID cases, early 2020.

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.

Measurements of tidal height (red) and current (blue) for the St. John's River in Jacksonville, FL Note that the curves are slightly out of phase. The cycle of tidal height lags slightly behind the cycle of current speed.

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.