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Navier stokes stacked shallow water #29

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Update print statement from 'Hello' to 'Goodbye'
chrisb13 May 14, 2026
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Merge remote-tracking branch 'origin/Fortran101-part2' into NS-stacke…
aekiss May 22, 2026
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latex test
aekiss May 22, 2026
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35 changes: 33 additions & 2 deletions documentation/docs/pages/season2.md
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Expand Up @@ -737,8 +737,39 @@ you to go deep into the codebase!

Date: 28/05/2026.

Presenter: Andrew Kiss (@aekiss).

Presenter: Andrew Kiss (@aekiss).

Navier-Stokes equation: fully 3D

$
% shorthand for \partial
\newcommand {\PD}{\partial}
% 1st partial derivative
\newcommand {\pd}[2]{\frac{\PD #1}{\PD #2}}
% 2nd partial derivative
\newcommand {\pdTwo}[2]{\frac{\PD^{2} #1}{\PD #2^{2}}}
% 2nd partial derivative, allows more general denominator (eg dxdy)
\newcommand {\pdTwoX}[2]{\frac{\PD^{2} #1}{ #2}}
% 3rd partial derivative
\newcommand {\pdThree}[2]{\frac{\PD^{3} #1}{\PD #2^{3}}}
% 3rd partial derivative, allows more general denominator (eg dx^{2}dy)
\newcommand {\pdThreeX}[2]{\frac{\PD^{3} #1}{ #2}}
% 4th partial derivative
\newcommand {\pdFour}[2]{\frac{\PD^{4} #1}{\PD #2^{4}}}
% 4th partial derivative, allows more general denominator (eg dx^{3}dy)
\newcommand {\pdFourX}[2]{\frac{\PD^{4} #1}{ #2}}
% small 1st partial derivative
\newcommand {\pds}[2]{\PD #1/\PD #2}
% small 2nd partial derivative
\newcommand {\pdTwos}[2]{\PD^{2} #1/\PD #2^{2}}
\underbrace{
\underbrace{\underbrace{\pd{\mathbf{u}}{t}}_{\text{Eulerian acceleration}}
\underbrace{+(\mathbf{u}\cdot\nabla)\mathbf{u}}_{\text{advection}}}_{\frac{D\mathbf{u}}{Dt}\text{: material (Lagrangian) acceleration relative to Earth}} \quad
\underbrace{+2\bm{\Omega}\times\mathbf{u}}_{\text{Coriolis}}}_\text{Lagrangian acceleration relative to non-rotating frame} \quad
=\underbrace{\underbrace{-\frac{1}{\rho}\nabla p}_{\text{pressure gradient}}
\underbrace{+A\nabla^2\mathbf{u}}_{\text{eddy viscosity}}\quad
\underbrace{+\mathbf{g}}_{\text{gravity}}}_{\frac{\mathbf{F}}{\rho}\text{: Force per unit density}}
$


## Generalised vertical coordinates
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