Added mom6 instructions to Eddy-Mean_Kinetic_Energy_Decomposition.ipynb

[julia-neme]

Closes #580

Additional improvements: loading of variables with intake.

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navidcy commented on 2025-12-19T21:51:53Z
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refer to [INSERT UPCOMING TUTORIAL ON GRIDS].

We can't have this here. We can add it when the upcoming tutorial is a reality.... But we can't have a promise for the future because often promises never materialise and then it looks bad and it also makes us feel an internal pressure for achieving something we "promised"?

[navidcy]

I don't understand why you state that in a fundamental way, $w$ does not appear in the mechanical energy budget. Can you elaborate a bit?

[navidcy]

Consider the hydrostatic equations of motion (without surface fluxes and stresses etc...). Below $\boldsymbol{u}_h = u\hat{\boldsymbol{x}}+v\hat{\boldsymbol{y}}$ is the horizontal component of velocity and $\boldsymbol{v}$ is the 3D velocity $\boldsymbol{v} = \boldsymbol{u}_h + w\hat{\boldsymbol{z}}$.

$$ \begin{align} \partial_t u & = - u \partial_x u - v \partial_y u - w \partial_z u - \partial_x p + f v - g \partial_x η\\ \partial_t v & = - u \partial_x v - v \partial_y v - w \partial_z v - \partial_y p - f u - g \partial_y η\\ 0 & = - \partial_z p + b \\ 0 & = \boldsymbol{\nabla}_h \boldsymbol{\cdot}\boldsymbol{u}_h + \partial_z w \\ \partial_t η & = w \Big|_{z=0} = - \boldsymbol{\nabla \cdot } \int \boldsymbol{u}_h \mathrm{d}z \end{align} $$

So if we consider the horizontal kinetic energy $K = (u^2+v^2)/2$ we get:

$$ \begin{align} \partial_t K &= u \partial_t u + v \partial_t v \\ & = - u^2 \partial_x u - uv \partial_y u - u w \partial_z u - u \partial_x p - g u \partial_x η - u v \partial_x v - v^2 \partial_y v - v w \partial_z v - v \partial_y p - g v \partial_y η \\ & = - u \partial_x (u^2/2) - v \partial_y (u^2/2) - w \partial_z (u^2/2) - u \partial_x (v^2/2) - v \partial_y (v^2/2) - w \partial_z (v^2/2) - u \partial_x p - v \partial_y p - g (u \partial_x η + v \partial_y η) \\ & = - (u \partial_x + v \partial_y + w \partial_z) K - (\underbrace{u \partial_x p + v \partial_y p}_{=\boldsymbol{\nabla}_h \boldsymbol{\cdot} (p \boldsymbol{u}_h) - p \boldsymbol{\nabla}_h \boldsymbol{\cdot}\boldsymbol{u}_h}) - g (u \partial_x η + v \partial_y η) \\ & = - \boldsymbol{v} \boldsymbol{\cdot} \boldsymbol{\nabla} K - \boldsymbol{\nabla} \boldsymbol{\cdot} (\boldsymbol{v} p) + w b - g (u \partial_x η + v \partial_y η) \end{align} $$

I don't see why $w$ fundamentally disappears...

[navidcy]

In the recipe you wrote "$w$ does not appear in the mechanical energy budget". But even if you consider both the kinetic and potential energy, the $w b$ energy conversion term will vanish but still there is $w$ e.g. in the advection of energy.

[julia-neme]

Heyaa! I just added MOM6 instructions and loaded data with intake. I haven't made this recipe or written those comments. I'll remove the reference to the upcoming #602 tutorial.