This state of affairs, however, is not sufficient to assume that the lack of a well-posed problem can be dismissed. The properties obtained from such equation systems, relative to what we actually observe in nature, certainly leaves us with a few uncomfortable facts.

I agree.

The difficulty is that recognizing system A was illposed did not automatically mean well posed system B was more correct. Both could be wrong in some sense and the ways in which they were wrong could matter in different applications. So, whether a particular system was ill-posed or well-posed did not always “win the day” when it came to running codes to simulate actual systems.

While ill-posed equation systems might or might not violate the basic laws of physics, these systems produce material responses that are not consistent with the responses of materials as observed in nature. The most well-known is the fact that these systems predict that perturbations at the smallest scales grow, and grow the fastest, whenever stability of the physical and mathematical model are investigated. In nature, the smallest scales are the scales at which perturbations decay, they do not grow.

Some roots of the characteristic polynomial of these systems, those that represent propagation of pressure information, are complex; they have a real part and an imaginary part. This in itself is not good; at least it is not in agreement with all that is known about pressure-wave propagation in materials. More importantly, the presence of complex roots of the characteristic polynomial leads to all kinds of difficulties relative to analysis of the proper boundary conditions as determined by the eigenvectors associated with the eigenvalues.

If the imaginary part of the characteristics is somehow related to an elliptic-equation-system aspect, then there is an implication that information must be specified at future time whenever an IBVP is being considered.

The fundamental properties of the corresponding Euler-like systems for the single-phase and two-phase systems have almost nothing in common. And while the multi-phase model systems contain implicitly some rough approximations to all the classical instability situations which arise due to discontinuities in any flow-field property (velocity, temperature, density, pressure, and combinations of these, see D D Joseph), the equations do not calculate the correct growth rate of these instabilities.

It is true that these issues seldom arise in applications of these equation systems. The implicit and explicit regularization of the problem introduced at the discrete approximation stage, plus lack of sufficient spatial resolution, seems to make these non-issues. We simply crank up our computer code and happily grind away producing numbers by the gazallions.

This state of affairs, however, is not sufficient to assume that the lack of a well-posed problem can be dismissed. The properties obtained from such equation systems, relative to what we actually observe in nature, certainly leaves us with a few uncomfortable facts.

This is a good point. For models that can be thouroughly tested within a useful set of constraints and then used exclusively within those constraints, we can determine whether or not this problem matters. For example, we can use a model to simulate an aircraft wing that has a shape, size and wind speed that fall within tested capabilities of the model. With climate models, we have neither the tests nor the constraints and so the “may not matter” means that the model may or may not be useful and we are unable to determine whether it is in fact useful.

A perpetual motion machine is a system dominated by positive feedback.
The earths climate is being modeled as a system dominated by positive feedback.

The notion of a fixed sensitivity and feedback parameter pretty much have to be valid locally only. You can’t keep perturbing the system further (assuming the equilibrium even exists) and expect “all things to remain the same.”

Obviously we couldn’t get out of an ice age once we got in, if the system were uniformly dominated by positive feedback…

By the way Lucia have you looked at the correlation between cloud changes and temperature? Check out Ole Humlum’s site http://www.climate4you.com The clouds and climate page is particularly interesting, it identifies clouds/humidity as the rather obvious cause of the recent global warming

How close is the tip jar to the estimated publishing cost for a paper? I can contribute more if you need.

We prove that the Cauchy problem for the three-dimensional Navier–Stokes equations is ill-posed in *** source in the sense that a “norm inflation” happens in finite time. More precisely, we show that initial data in the Schwartz class *** source that are arbitrarily small in *** source can produce solutions arbitrarily large in *** source after an arbitrarily short time. Such a result implies that the solution map itself is discontinuous in *** source at the origin.

Anyway, there isn’t any real rule that says systems of equations that are ill-posed must be wrong. However, in multiphase flow, when some were proposing governing equations, there was a camp who wanted to insist the that systems of equations that were illposed must be unphysical. Others pointed out that lots of systems of equations are illposed and it doesn’t imply any violation of the laws of physics. So, there would be an impasse.

Still, if one is supposedly averaging some set of equations (like the navier stokes) one hopes the resulting set of averaged equations along with their paramerizations are well posed. Otherwise, they will be just as intractable as the original equations. The same held for multiphase flow, so the hope was that some set of averaged equations would be well-posed.

Here’s something from wikipedia:

The mathematical term well-posed problem stems from a definition given by Hadamard. He believed that mathematical models of physical phenomena should have the properties that

1. A solution exists
2. The solution is unique
3. The solution depends continuously on the data, in some reasonable topology.

Systems of equations can be ill posed without being wrong. Many inverse problems don’t have unique solutions.

Your post reminds me that there is debate to this day as to what the governing equations for a multiphase system even are.

Like…. is the pressure term α ∇P or ∇ (α P)?

Yep. No matter how you gin up multiphase flow equations, there are conceptual difficulties both when you are describing anything over a length scale small compared to some geometric feature of the two-phases (particle size, bubble size etc.) or as you get near wall. It’s not at all surprising some problems are ill-posed when you use approximations. Was it Don Drew who was concerned about the fact that a problem where the void fraction in a bubbly had a discontinuous step function description was ill-posed as you marched forward? ( Well… of course. If you really think about what averaged equations mean, void fraction cannot go from some finite value to 0 over a distance less than the diameter of a bubble.)

Let’s face it: The climatologist are, to a large extent, doing more or less what the bubbly flows guys all did. But, climate has even more bells and whistles to hang together.

Bingo! Welcome to the non-linear world of fluid dynamics :^)

Your post reminds me that there is debate to this day as to what the governing equations for a multiphase system even are. Phase variables (e.g. void fraction), for example, aren’t simple point functions, but in fact are usually described in the limit as you shrink a control volume to a point (just as the concept of fluid density doesn’t make any sense as you approach the molecular level). There was one paper I remember which cast the solutions in terms of “generalized functions” (i.e. functions which permit discontinuous step changes) as a way of dealing with the distributions of phase variables.

Also, you touch on the concept of a “well-posed” problem. It is my belief that the AOGCM formulations in general can not be proven to be well-posed, and in fact may not be. There is so much coupling and so many parameterizations (i.e. non-linear source terms) in the equations that it is anyone’s guess as to whether the equations are stable, consistent, and actually converge to a unique solution in the limit of decreasing mesh size (hence satisfying the Lax Equivalence Theorem for well-posed initial-value problems).

The design of these codes seems to follow these steps:

(1) create a reasonably stable Eulerian core
(2) hang a million source terms off these equations to model all of the actual physics (and this doesn’t include the solution of auxiliary equations for radiative transfer, moist processes, aerosol transport etc.).
(3) add a bunch of dissipation, “fixers” and filters to keep the solution from blowing up numerically.

A whole mental world that somehow fits on a spectrum between “robust” and “weather” …