Out At The Boundaries

Guest Post by Willis Eschenbach

I’ve heard many times that whereas weather prediction is an “initial-value” problem, climate prediction is a “boundary problem”. I’ve often wondered about this, questions like “what is the boundary?”. I woke up today thinking that I didn’t have an adequately clear understanding of the difference between the two types of problems.

For these kinds of questions I find it’s hard to beat Wolfram Reference, which is a reference to the various functions in the computer program Mathematica. Wolfram is a total genius in my opinion, and the Wolfram site reflects that. Here’s what Wolfram Reference says (emphasis mine):

Introduction to Initial and Boundary Value Problems

DSolve [a Mathematica function] can be used for finding the general solution to a differential equation or system of differential equations. The general solution gives information about the structure of the complete solution space for the problem. However, in practice, one is often interested only in particular solutions that satisfy some conditions related to the area of application. These conditions are usually of two types.

• The solution x(t) and/or its derivatives are required to have specific values at a single point, for example, x(0)=1 and x’(0)=2. Such problems are traditionally called initial value problems (IVPs) because the system is assumed to start evolving from the fixed initial point (in this case, 0).

• The solution x(t) is required to have specific values at a pair of points, for example, x(0)=1 and x(1)=5. These problems are known as boundary value problems (BVPs) because the points 0 and 1 are regarded as boundary points (or edges) of the domain of interest in the application.

The symbolic solution of both IVPs and BVPs requires knowledge of the general solution for the problem. The final step, in which the particular solution is obtained using the initial or boundary values, involves mostly algebraic operations, and is similar for IVPs and for BVPs.

IVPs and BVPs for linear differential equations are solved rather easily since the final algebraic step involves the solution of linear equations. However, if the underlying equations are nonlinear, the solution could have several branches, or the arbitrary constants from the general solution could occur in different arguments of transcendental functions. As a result, it is not always possible to complete the final algebraic step for nonlinear problems. Finally, if the underlying equations have piecewise (that is, discontinuous) coefficients, an IVP naturally breaks up into simpler IVPs over the regions in which the coefficients are continuous.

Now, as I read that, it says that for an initial value problem (IVP) we need to know the initial conditions at the starting time, and for a boundary value problem (BVP) we need to know the future conditions at a particular boundary. For example, suppose we are interested in the future thermal behavior of an iron rod with one end in a ice-water bath. The boundary condition is that the end of the iron rod in the ice-water bath is at 0°C.

So my question is two-fold. IF predicting weather is an IVP and predicting climate is a BVP, then

1) What is the “boundary” in question?, and

2) Once we determine what the boundary is, how do we know the future value of the boundary?

Some investigation finds that for US$48 I can read the following:

Existence and regularity theorems for a free boundary problem governing a simple climate model

Xiangsheng Xua

Abstract

From a class of mean annual, zonally averaged energy–balance climate models of the Budyko‐Sellers type, we arrive at a free boundary problem with the free boundary being the interface between ice‐covered and ice-free areas. Existence and regularity properties are proved for weak solutions of the problem. In particular, the regularity of the free boundary is investigated.

Fortunately, I don’t need to read it to see that the boundary in question is the ice-water interface. Now, that actually seems like it might work, because we know that at any time in the future, the boundary is always at 0°C. Since we know the future temperature values at that boundary, we can treat it as a boundary problem.

But then I continue reading, and I find Dr. Pielke’s excellent work , which says (emphasis mine):

One set of commonly used definitions of weather and climate distinguishes these terms in the context of prediction: weather is considered an initial value problem, while climate is assumed to be a boundary value problem. Another perspective holds that climate and weather prediction are both initial value problems (Palmer 1998). If climate prediction were a boundary value problem, then the simulations of future climate will “forget” the initial values assumed in a model. The assumption that climate prediction is a boundary value problem is used, for example, to justify predicting future climate based on anthropogenic doubling of greenhouse gases. This correspondence proposes that weather prediction is a subset of climate predictions and that both are, therefore, initial value problems in the context of nonlinear geophysical flow. The consequence of climate prediction being an initial value problem is summarized in this correspondence.

The boundaries in the context of climate prediction are the ocean surface and the land surface. If these boundaries are fixed in time, evolve independently of the atmosphere such that their time evolution could be prescribed, or have response times that are much longer than the time period of interest in the climate prediction, than one may conclude that climate prediction is a boundary problem.

So Dr. Pielke says that there is an entirely different boundary in play, the boundary between the atmosphere and the surface.

But then my question is, how would we know the future conditions of that boundary? If it’s a BVP, we have to know future conditions.

Dr. Pielke takes an interesting turn. IF I understand his method in another paper, Seasonal weather prediction as an initial value problem, he shows that the chosen boundary (the atmosphere/surface interface) doesn’t “evolve independently of the atmosphere such that their time evolution could be prescribed” and thus seasonal weather prediction is shown to be an IVP rather than a BVP.

However … he’s using an entirely different boundary than that used by Xiangsheng Xua above. Which one is right? One, both, or neither?

And the underlying problem, of course, is that IF climate is an initial value problem just like weather, given the chaotic nature of both we have little hope of modeling or predicting the future evolution of the climate.

My conclusion from all of this, which I think is shared by Dr. Pielke, is that climate prediction is an initial value problem. I say this in part because I see no difference in “climate” and “weather” in that both seem to be self-similar, non-linear, and chaotic.

This view is also shared by Mandelbrot, as was discussed about a decade ago over at Steve McIntyre’s excellent blog … have we really been at it that long? Mandelbrot analyzed a number of long-term records and found no change in the fractal nature of the records with timespan. In other words, there’s no break between the chaotic nature of the short, medium, and long-term looks at weather.

Now, it’s often argued that weather prediction has gotten much better over the decades … and this is true. But remember, weather prediction is an initial value problem. That means that the more accurately and specifically and finely we can measure the initial conditions, the better our prediction will be. Much of the improvement in our weather predictions is a result of satellites which give us our initial conditions in exquisite detail. And despite all our advances in predictive ability, lots of weekend barbecues still get rained on.

And at the end of the day, I’m left with my initial questions:

• If modeling the future evolution of the climate a boundary problem, what exactly is the boundary?, and

• Having specified the boundary, how can we know the future conditions of the boundary?

Egads … a post without a single graphic … curious.

w.

My Usual Request: If you think something is incorrect, please have the courtesy to quote the exact words that you disagree with so that everyone can understand your objections.