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That’s $2-million in prize money, and maybe a Fields medal or two, if you get them both. But it doesn’t stop there! There’s the closure problem of turbulence in fluids: Not only can’t we determine the flow in a pipe from first principles, we can’t even get the lowest order statistic, after 150 years of trying. There is no Clay prize, but instant fame awaits you. Good luck.

What about computers? Computers don’t have enough numbers (i.e. finite representation). The mathematics is too big to fit. Consequently computer arithmetic is different: garbage out even without garbage in. You must crack differential equations like eggs to put them onto computers. The shattered remains are an approximation, but with different physics.

They don’t usually conserve things like energy! Such differences tell when integrated over long times (i.e. climate). If you invent a computer scheme that conserves all the correct things, the computer’s solution amounts to an exact solution of the original equation. If you figure out how to do that for the infinite number of conserved quantities expected for Navier Stokes, you win $1-million!

Finite representation means that the smashed equations must be hung on a grid. Think of pixels on a computer screen. Between grid points there is nothing. Grid spacings must be smaller than anything you hope to see. Everything else is lost. Proper computation calls for spacings to be smaller than all of the wiggles in the equation’s solution. But the enormous scales and complexity in climate mean that the wiggles are much smaller than grid spacings.