When I was applying to grad school for applied math, one place I applied was the Division of Applied Math at Brown University. I had a business trip to the small jet engine shop of GE in Lynn, MA, so went down to Brown to meet some profs. Had lunch. Talked. I got one piece of advice: "Stay away from the Navier-Stokes equations."

Yup, good advice.

The Navier-Stokes equations are old stuff now. Basically they are just Newton's second law, the gas law, etc. -- just basic physics. If you want to calculate the flows of fluids -- liquids and/or gasses -- from first principles of the basic physics, then you are stuck with the Navier-Stokes equations. It's like calculating the trajectory of a home run baseball -- need Newton's second law, the law of gravity, and at least something first cut on air resistance (right, in fine detail, the air resistance would again be the Navier-Stokes equations but good enough for a baseball or artillery shell, there are some good enough, simple approximations).

Or for calculating what fluids do, from the basic physics, that is, from what we DO know about the basic physics, we just can't avoid the Navier-Stokes equations anymore than for the baseball we can avoid Newton's second law force = mass times acceleration. Yup, this is a case were we know too much: We DO know that, for calculating from a solid basis, that is, from first principles, for the math we need, we DO have the equations and they are just the Navier-Stokes equations. Sorry 'bout that.

You mentioned that you noticed that the Navier-Stokes equations are complicated math -- right!

As I mentioned, maybe we can do well with the Navier-Stokes equations for some boat hull or some airplane wing. Okay. Early in my career, I got started on the Navier-Stokes equations at the US Naval Ship R&D Center at Carderock, MD -- right, with the big towing tank used for designing ship hulls. That was before the guy at Brown told me "stay away from the" equations. He was right!

So, you are correct: Trying to solve the Navier-Stokes equations for all the oceans and all the atmosphere of the earth is a wild thigh slapper, absurd, out of the question. We don't have such a computer. Even if we did, we don't have even the required initial conditions -- that is, the current state of the flows in the oceans and the atmosphere. Or, we are stuck-o.

We will also want to know that the Navier-Stokes equations are nicely stable, that is, that a butterfly flapping its wings in NY will not cause rain instead of sunshine in Japan (to borrow from various movies). IIRC, Richard Bellman wrote his Princeton dissertation on the stability of ordinary differential equations -- IIRC the stability of the Navier-Stokes equations is a challenging topic, e.g., appears to be part of one of the Clay Math problems along with P versus NP, etc.

And the real problem is still worse: E.g., likely we would have to handle turbulence, known to be difficult. When I was at Carderock, there was a guy working on turblence; he had been for years; maybe he is still there still working on turbulence; maybe in another 100 years he will have some good progress! In principle, we are talking about handling winds blowing through trees (would need the details on all the leaves of all the trees!!!) and the resulting turbulence. Would need some good details on associated biology. Would need .... And after have all of that, as the climate started to change, we would need good details on how the biology would change, and we don't have any equations from first principles for that.

So, why do we need to do the fluid flow calculations? Well, we're talking about CO2. For that, we want to know where it goes, e.g., into the water, out of the water, into/out of seashells into the upper atmosphere, close to the ground, as it warms, as in the greenhouse effect, where it goes, sucked up by the plants, reacts with rocks, etc.

So, what people have done is use various assumptions, simplifications, and approximations. It's a little like in freshman physics where we assume a block slides down a plane, and the plane has no friction, or a ball rolls down a plane and we ignore the moment of inertia of the ball.

So, people tried such approximations, etc. We DO know the basic physics, and a big part of that is the Navier-Stokes equations. And we have more physics on the black body radiation that is the source of the infrared radiation that is the source of the warming of the CO2 that is the warming of the greenhouse effect. We have a lot of the basic physics and chemistry. And that basic science just does not tell us that there are some nice, easy approximations that will let us predict the climate.

E.g., suppose some day during the years we are predicting, it rains. It might! Then after the rains, where is the CO2? Do we have partially carbonated rain water? In principle, we will want to know where the CO2 goes. So, part of our calculation will be to predict when it rains. Hmm ....

Yes, as is often the case in physics, for some purposes we can just use the law of conservation of energy, calculate energy into the earth from the sun and energy out of the earth from radiation to space and get the balance and temperature change. Okay. But the details, if we want them, of the energy in and energy out will take us back to the Navier-Stokes equations. So, maybe we can make some simplifying assumptions. Apparently then ... we come up with the models that predicted much higher temperatures by now.

Then, in all of this, we have an assumption that is now looking like week old dead fish: It's all about the greenhouse effect and CO2 and not something else. What "something else"?

Apparently an argument can be made that, really, at anything like currently realistic levels of CO2, CO2 and the greenhouse effect are essentially irrelevant and the main cause is just clouds from water droplets from cosmic rays blocked or not by the solar wind from sun spots. In that case, we can calculate all we want with the Navier-Stokes equations, make more bad predictions, and accumulate some big computer bills. So, for accurate predictions, we'd be into predicting the sun spot activity of the sun. Hmm .... Is that at all promising?

The lecture on geology and CO2 was really interesting: It got into the orbit of the earth, the power (energy per unit time) to the earth from solar radiation, some geology, and some tricky chemistry but not sun spots!

Net, so far, it's tough to predict either the weather or the climate. Sorry 'bout that! It's also tough to cure cancer, explain dark matter and dark energy, ..., etc.