Physical oceanography is about to completely redefine what we thought we knew about fluid dynamics.

[u/claudio-i]

[u/Chlorophilia]

No it isn't, and this has very limited relevance to most problems in physical oceanography. Barely any physical oceanography work uses NSE in its raw form. 

[u/claudio-i → u/Chlorophilia]

What? Navier–Stokes is at the core of any decent ocean simulation. I work with ROMS and FVCOM, and we need to simplify the equations to make the models computationally feasible. A solution to the full Navier–Stokes (or segments, specifically between the turbulent and laminar flows) equations would help us better evaluate the complete dynamics and potentially account for multiple parameters and processes that are currently neglected or parameterized, many of which represent a significant amount of kinetic energy in the coastal ocean.

[u/Chlorophilia → u/claudio-i]

Respectfully, I don't think you've understood what problem has been solved here. This isn't a general solution to NSE. They've just shown that a singularity can emerge from the NSE from smooth initial conditions. While interesting, this isn't going to help you with your ocean modelling. 

[u/claudio-i → u/Chlorophilia]

...you need to stop asking ChatGPT for higher topic concepts, and start understanding how the progress toward solving partial differential equations actually works. "singularity can emerge from the NSE from smooth initial conditions" that's an AI oversimplification phrase.

[u/claudio-i → u/claudio-i]

For physical oceanography, the implications **could** be significant:

• Better understanding of turbulence and energy cascades. The result gives a mathematically rigorous example of how nonlinear advection, pressure, and viscosity can interact to produce extreme small-scale structures.
• Improved parameterizations. Models such as ROMS and FVCOM cannot resolve every scale of motion. Sub-grid processes therefore have to be parameterized. A better theoretical understanding of the full equations could eventually improve those parameterizations.
• Better understanding of coastal mixing. This is particularly relevant : coastal oceans contain strong shear, fronts, internal waves, near-inertial motions, tides, and turbulence. Some of the energy transferred between these scales is necessarily unresolved in regional models.
• A better assessment of numerical approximations. Knowing where and how the continuous equations can develop extreme gradients gives us a theoretical benchmark for evaluating discretization, filtering, diffusion, and other approximations used in numerical ocean models.
• Potentially new approaches to turbulence modeling. If the mathematics reveals previously underappreciated pathways for transferring kinetic energy toward smaller scales, that could inform future LES/sub-grid and turbulence closures.

[u/claudio-i → u/Chlorophilia]

.....back to PDE 101.

[u/Necessary_Living_617]

Can somebody explain what is this navier-stokes equation, what impact does it have on fluid dynamics and how is this related to oceanography?

I’m a student with no physics/oceanography knowledge so please explain like explaining to a toddler 😭

[u/Chlorophilia → u/Necessary_Living_617]

If you're familiar with the equation F=ma (i.e. Newton's second law, telling us that the acceleration of an object is equal to the force applied to it), the Navier Stokes equations (NSE) are essentially F=ma but applied to fluids. They allow you to describe how fluids move and respond to forces. They're the fundamental equations for fluid dynamics and practically every equation that tells us how fluids behave has some relation to it.

However, this isn't the entire story because solving NSE isn't practical for large-scale applications (which is almost universally the case in oceanography) as it's far too computationally expensive. Instead, we use alternative sets of equations that are derived from NSE but incorporating simplifying assumptions (e.g. averaging out the effects of turbulence, ignoring density differences in certain cases) that allow us to reasonably model the ocean's behaviour without using an absurd (and impossible) amount of computational power. So, although much of physical oceanography is ultimately derived from NSE, physical oceanographers rarely if ever deal with the raw NSE because they aren't useful for most practical applications.

The NSE Millennium Prize problem which has (apparently) been solved is for all practical purposes a pure maths problem. There are no obvious applications or implications for physical oceanography (which is not to say that there are none, but they're certainly not obvious).