Basics of Numerical Weather Prediction (NWP):

1. THE HORIZONTAL MOMENTUM EQUATION:
\[
\frac{d\mathbf{V}}{dt} + f\hat{k} \times \mathbf{V} = -\nabla \phi + \frac{\sigma}{p_s} \frac{\partial \phi}{\partial \sigma} \nabla p_s + \mathbf{F}
\]

2. THE CONTINUITY EQUATION:
\[
\frac{\partial p_s}{\partial t} + \nabla \cdot (p_s \mathbf{V}) + \frac{\partial}{\partial \sigma}(p_s \dot{\sigma}) = 0
\]

3. THE THERMODYNAMIC ENERGY EQUATION:
\[
\frac{1}{R} \frac{d}{dt} \left[ \sigma \frac{\partial \phi}{\partial \sigma} \right] + \frac{RT}{C_p p} \left[ p_s \dot{\sigma} + \sigma\dot{p_s} \right] = -Q
\]

4. HYDROSTATIC EQUATION:
\[
\frac{\partial \phi}{\partial \sigma} = -\frac{RT_v}{\sigma}
\]

5. SURFACE PRESSURE TENDENCY EQUATION:
\[\displaystyle
\frac{\partial p_s}{\partial t} = -\int_{0}^{1} \nabla\cdot (p_s \mathbf{V}) \, d\sigma
\]

6. MOISTURE EQUATION:
\[\displaystyle
\frac{\partial}{\partial t} (p_s q) + \nabla\cdot (p_s q \mathbf{V}) + \frac{\partial}{\partial \sigma} (p_s q \dot{\sigma}) = p_s S
\]

The six primary unknowns are: \(\mathbf{V}\) (horizontal wind velocity), \(p_s\) (surface pressure), \(T\) (temperature), \(q\) (specific humidity or moisture), \(\phi\) (geopotential), and \(\dot{\sigma}\) (sigma velocity or vertical velocity in \(\sigma\)-coordinates).

#NWP #Weather #NumericalWeatherPrediction #Meteorology #Climate #ClimateScience #Earth #EarthScience #ClimateChange #ClimateSciences #Science #WeatherPrediction #Humidity #Moisture #Pressure #Velocity #SurfacePressure #HydrostaticEquation #WeatherPrediction #Ocean #Atmosphere #AOS #ClimateDynamics #WeatherDynamics #Geopotential #SigmaVelocity #VerticalVelocity #MoistureEquation #Thermodynamics #Dynamics #NavierStokes

Imagine being a brilliant physicist/mathematician and still avoiding the most important problems because your career depends on publishing frequent papers, not solving the biggest mysteries in the world.

That's why you can't do things like this in academia.

https://english.elpais.com/science-tech/2025-06-24/spanish-mathematician-javier-gomez-serrano-and-google-deepmind-team-up-to-solve-the-navier-stokes-million-dollar-problem.html

#NavierStokes #GoogleDeepMind #DeepMind #MillenniumProblems #Existence #Smoothness #Fluid #FluidDynamics #Turbulence #Dynamics #TurbulentFlows #Research #Engineering #Physics #Math #Maths #Mathematics #UnsolvedProblems #BiggestMystery #Flows #MillionDollarProblem

Blow-Up or Not? - Terence Tao on Lex Fridman

#navierstokes #blowup #millenniumprize

🌊 Scientists have advanced in understanding turbulence, a long-standing puzzle for physicists. This progress aids in solving the Navier–Stokes equations, a major challenge in math and physics, and a Millennium Prize Problem by the Clay Mathematics Institute. Recent developments are crucial for fluid dynamics, impacting engineering, meteorology, and more.

@goodnews

#GoodNews #Physics #Turbulence #NavierStokes #ScienceBreakthrough
https://edition.cnn.com/2025/02/06/science/turbulence-physics-oldest-unsolved-problem/index.html

Researchers claim to have solved Hilbert’s sixth problem by unifying three theories of #FluidDynamics at different levels of granularity:

+ Newton’s laws of motion at the microscopic level where fluids are composed of particles - little billiard balls bopping around and occasionally colliding

+ The Boltzmann equation at the mesoscopic level where the equation considers the likely behavior of a typical particle

+ Euler and #NavierStokes equations at the macroscopic level where the fluids are a single continuous substance

https://www.scientificamerican.com/article/lofty-math-problem-called-hilberts-sixth-closer-to-being-solved

Preprint https://arxiv.org/abs/2503.01800

An Article in the Annual Review of Condensed Matter Physics on Turbulence by KR Sreenivasan and J Schumacher
https://www.annualreviews.org/content/journals/10.1146/annurev-conmatphys-031620-095842

What is the turbulence problem, and when can we say it’s solved? 🌪️ This deep dive by Sreenivasan & Schumacher explores the math, physics, and engineering challenges of turbulence—from Navier-Stokes equations to intermittency and beyond. A must-read for anyone fascinated by chaos, complexity, and the unsolved mysteries of fluid dynamics! 🌀

A summary of the talk presented by KR Sreenivasan in December 2023 at the International Center for Theoretical Sciences (ICTS-TIFR) in Bengaluru, as part of a program on field theory and turbulence.
https://www.youtube.com/watch?v=fwVSBYh-KC4

"Field Theory and Turbulence" program link: https://www.icts.res.in/discussion-meeting/ftt

#FluidDynamics #Physics #NavierStokes #UnsolvedMystery #Mechanics #Dynamics #FluidMechanics #Science #Chaos #TurbulentMotion #Randomness #Chaotic #Fluid #ClassicalMechanics
#Turbulence