1. Climate variability simulation predictions don't work
2. Fluid dynamics contains difficult math
3. When the math does work, can't understand fully why
*phenomenological* : "A phenomenological model is a scientific model that describes the empirical relationship of phenomena to each other, in a way which is consistent with fundamental theory, but is not directly derived from theory."
This blog post by Tao provides an indication on how difficult the math is to solve in a generalized fully 3D Navier-Stokes formulation (as defined per the Clay prize). Interesting that these points that Tao makes are all elements to the LTE solution for ENSO
Even if one gets close, you may not be that close, unless you choose wisely:
> "... so there is basically no chance of a reduction of the non-perturbative case to the perturbative one unless one comes up with a highly nonlinear transform to achieve this (e.g. a naive scaling argument cannot possibly work)."
> "... we are thus left with Strategy 2 – discovering new bounds, stronger than those provided by the (supercritical) energy. This is not a priori impossible, but there is a huge gap between simply wishing for a new bound and actually discovering and then rigorously establishing one. Simply sticking in the existing energy bounds into the Navier-Stokes equation and seeing what comes out will provide a few more bounds, but they will all be supercritical, as a scaling argument quickly reveals. The only other way we know of to create global non-perturbative deterministic bounds is to **discover a new** conserved or monotone quantity. In the past, when such quantities have been discovered, they have always been connected either to geometry (symplectic, Riemmanian, complex, etc.), to physics, or to some consistently favourable (defocusing) sign in the nonlinearity (or in various “curvatures” in the system). "
> "Strategy 2 would require either some exceptionally good intuition from physics, or else an incredible stroke of luck. "
So these “blue-sky” long shot approaches (elaborated more in Tao's blog post):
1. Work with ensembles of data, rather than a single initial datum.
2. Work with a much simpler (but still supercritical) toy model.
3. Develop non-perturbative tools to control deterministic non-integrable dynamical systems.
4. Establish really good bounds for critical or nearly-critical problems.
5. Try a topological method.
> "while it is good to occasionally have a crack at impossible problems, just to try one’s luck, I would personally spend much more of my time on other, more tractable PDE problems than the Clay prize problem, though one should certainly keep that problem in mind if, in the course on working on other problems, one indeed does stumble upon something that smells like a breakthrough in Strategy 1, 2, or 3 above. (In particular, there are many other serious and interesting questions in fluid equations that are not anywhere near as difficult as global regularity for Navier-Stokes, but still highly worthwhile to resolve.)"
then, in response to a question in the comments section
> "From a physical viewpoint, it may well be that one of these modified equations is in fact a more realistic model for fluids than Navier-Stokes. But for the narrow purposes of solving the Clay Prize Problem, we’re stuck with the original Navier-Stokes equation :-) ."
> "Self-promotion of one’s own papers is against the stated comment policy of this blog. If you wish to discuss your own research papers, please do so using another venue, such as your own personal web pages."
When it comes down to it, a not-well-understood model that obeys Navier-Stokes and that will phenomenologically match the data in a parsimonious and plausible manner may be just what's needed to proceed.