Modelling

This set of notebooks breaks the problem of modelling the Deterministic Boussinesq equations

$$ \begin{align} \frac{D \boldsymbol{U} }{Dt} &= -\nabla P + Ri_B B \boldsymbol{\hat{z}} + Re^{-1} \Delta \boldsymbol{U} + \boldsymbol{U}/\langle |\boldsymbol{U}|^2 \rangle, \ \frac{D B }{Dt} &= -W + Pe^{-1} \Delta B, \
\nabla \cdot \boldsymbol{U} &= 0, \end{align} $$

in terms of the time evolution of the PDF $f_{\boldsymbol{Y}}$ for the random variables $\boldsymbol{Y} = (W,B,Z)$ using the probabilistic equation

$$ \begin{equation} \frac{\partial f{\boldsymbol{Y}} }{\partial t} + \frac{\partial }{\partial z} \left( w f{\boldsymbol{Y}} \right) =
- \frac{\partial }{\partial b} \left( \left[ -w + Pe^{-1} E{\boldsymbol{Y}}[ \Delta B ] \right] f{\boldsymbol{Y}} \right) - \frac{\partial }{\partial w} \left( \left[ E{\boldsymbol{Y}}[ RiB B -\nablaZ P] + E{\boldsymbol{Y}}[ \frac{W}{\langle |\boldsymbol{U}|^2 \rangle}] + Re^{-1} E{\boldsymbol{Y}}[ \Delta W ] \right] f{\boldsymbol{Y}} \right). \end{equation} $$

into a hierarchial set of notebooks.

Directory Structure

1D Models

The 1D_Models directory contains Jupyter notebooks that simulate one-dimensional systems presumed to model $fB$. These models focus on simpler scenarios and are divided into two categories: - IEM: - `Part1IEMfB.ipynb: Implements the interaction by exchange with the mean (IEM) model for a one-dimensional system. For more details on the IEM model, refer to [Pope (1985)](https://doi.org/10.1017/S0022112085000942). -Part1IEMParticlefB.ipynb: Extends the IEM model to include particle-based simulations. - **Mapping**: -Part1MappingfB.ipynb: Implements the mapping closure for a one-dimensional system. For more information on the mapping closure, see [Klimenko (2005)](https://doi.org/10.1016/j.ces.2005.01.035). -Part1MappingParticle_fB.ipynb`: Combines mapping closure with particle-based simulations.

2D Models

The 2D_Models directory includes Jupyter notebooks for two-dimensional systems, categorized by the random variables modelled: - BW: contains Jupyter notebooks that simulate two-dimensional systems presumed to model $f{BW}$ - IEM: - `Part2IEM2dfBW.ipynb: Simulates the IEM model for a 2D system with buoyancy and width variables. For more details on the IEM model, refer to [Pope (1985)](https://doi.org/10.1017/S0022112085000942). -Part2IEMparticlefBW.ipynb: Extends the IEM model to include particle-based simulations. - **Mapping**: -Part2Mapping2dfBW.ipynb: Implements the mapping closure for a 2D system. For more information on the mapping closure, see [Klimenko (2005)](https://doi.org/10.1016/j.ces.2005.01.035). -Part2MappingfBW.ipynb: Focuses on mapping closure for buoyancy and width variables. - **BZ**: contains Jupyter notebooks that simulate two-dimensional systems presumed to model $f_{BZ}$ where $Z$ must be treated differently to $W$ and $B$ - **IEM**: -Part3IEM2dfBZLayered.ipynb: Implements the IEM model for a 2D system with buoyancy and height variables, focusing on layered structures. For more details on the IEM model, refer to [Pope (1985)](https://doi.org/10.1017/S0022112085000942). - **Mapping**: -Part3Mapping2dfBZLayered.ipynb`: Implements the mapping closure for a 2D system with buoyancy and height variables, using a layered approach. For more information on the mapping closure, see Klimenko (2005).

3D Models

The 3D_Models directory contains Jupyter notebooks for full three-dimensional system $f{WBZ}$: - IEM: - `Part4IEM3dfWBZ.ipynb`: Implements the IEM model for a 3D system, focusing on buoyancy $B$, vertical velocity $W$, and the vertical coordinate $Z$ variables. For more details on the particle implementation of the IEM model for this case, refer to Pope (1985).

Each directory is structured to include relevant datasets, scripts, and documentation to support the models.