One of the challenges of doing interdisciplinary work is the constant back and forth between mathematical understanding and, for lack of a better word, “physical” understanding.
What I mean by mathematical understanding is knowledge about how mathematical objects of a particular type behave, such as the fact that differential equations of the form $$frac{d^2x}{dt^2}+afrac{dx}{dt} + b = 0$$ have solutions that tend to 0 as time goes to infinity, if the quadratic equation r^2+ar+b = 0 has negative roots (and the proper initial or boundary conditions). This kind of knowledge is independent of the system that the equation is modeling.
On the other hand, we could look at spring-mass systems and see how they behave, and the observations we make will be explainable using concepts like acceleration, friction, and mass, and we can describe the spring’s motion conceptually (rather than quantitatively), and use this conceptual understanding to make predictions about the general behavior of the spring if we change parameters of the system, such as the size of the mass at the end of the spring, the stiffness of the spring, etc.
One important criterion for trusting a model is how well the model predicts well understood conceptual relationships. For example, increasing a spring’s stiffness should decrease the size of the oscillations for a fixed mass. The spring constant parameter in the differential equation exhibits a similar control on the solution of the differential equation. Thus, in some sense, the two types of reasoning agree with each other, taking the spring constant in the equation as the analog for the spring’s stiffness.
For more complicated models, like weather prediction models, these ideas translate well. What is different is the state of the mathematical understanding of these models. For the equation above, the mathematical understanding is complete. The relationship between the model solution and the initial conditions, boundary conditions, and parameters has been thoroughly explored using theoretical tools (at least for the simplest deterministic setting). The equations that make up an atmospheric model are more complex because they include dependence on both space and time, and because they have other mathematical features that make them less tractable.
The conceptual physical understanding that accompanies different weather phenomena is also incomplete, in particular weather events that are especially dynamic in nature, like severe thunderstorms, tornados, hurricanes, and the like. One good reason for this is that observations of these phenomena are less common than of the larger scale conditions. Another is that they’re simply more complicated objects than larger, slower moving things, and so require more sophisticated physical ideas to explain them.
The reason I’m thinking about this is that I’m currently trying to evaluate the effectiveness of including different types of observations on predicting precipitation. In the particular case that I’m looking at, the different methods all have different results, and I’d really like to understand why, and by that I mean I want to know what the model and assimilations are doing differently from a physics perspective. Is one result due to more heating in a certain area, or different wind patterns, or something else? These are relationships that the model is thought to capture well, and so the arguments I construct from conceptual reasoning should be verifiable with model results, once I know where to look. As a mathematician, the conceptual understanding is the place where I need help. Thankfully, I’m in a research lab where everyone is more than willing to fill my ear full of meteorology.
In the next post I’ll show some preliminary results of my research, and hopefully any readers out there will have things to say that will lessen my ignorance.
Sean M. R. Crowell 