# Scalar Transport

Nassu can solve the advection-diffusion equation (ADE) for one or more passive scalars $\phi$ on top of the fluid simulation.
Typical applications include temperature in atmospheric and urban-scale flows, pollutant or tracer concentration, and any quantity transported by the resolved velocity field with linear diffusion.

The ADE solved by nassu is

$$\partial_t \phi + u_\alpha \partial_\alpha \phi = D \nabla^2 \phi + S$$

where $u_\alpha$ is the fluid velocity (read from the LBM step), $D$ is the molecular diffusivity (with an LES subgrid component when active), and $S$ is an optional source term.

## Method

The scalar is solved with a **Double Distribution Function** (DDF) lattice Boltzmann scheme: a second set of populations $g_i$ on a velocity set independent from the fluid lattice evolves under streaming and a recursive regularised collision, recovering the ADE in the macroscopic limit.

```{figure} /_static/img/theory/scalar_transport/ddf_concept.svg
---
width: 65%
align: center
---
The double distribution function. The scalar rides its own populations $g_i$ on the same grid as the fluid $f_i$, with its own collision, streaming and transport coefficient. It reads the fluid velocity $u_\alpha$ as its advection field; in one-way coupling the fluid does not read the scalar back.
```

```{admonition} Key idea: the scalar rides its own distribution function
---
class: note
---
The scalar does not modify the fluid. It carries its own populations $g_i$ on a separate, smaller lattice and reads the fluid velocity $u_\alpha$ at each node. This decoupling lets the scalar reuse the fluid's velocity field while keeping its own diffusivity, sound speed, and boundary treatment independent.
```

The scalar is decoupled from the fluid (except optionally through Boussinesq buoyancy, see {doc}`thermal`) and reads the fluid velocity at every step:

$$g_i(\mathbf{x} + \mathbf{c}_i \Delta t,\, t + \Delta t) = g_i(\mathbf{x},t) + \Omega_i^{(\phi)}, \qquad \phi = \sum_i g_i.$$

The pages below cover the velocity set and unit system, the collision and macroscopic state, LES coupling, the Boussinesq path, boundary conditions, solid-body boundaries via voxelization, and integration with multiblock, initialisation, and IBM.

```{toctree}
---
maxdepth: 1
hidden:
---
Velocity sets and units <velocity_set.md>
Collision and state storage <collision.md>
LES subgrid coupling <les_coupling.md>
Thermal extension <thermal.md>
Boundary conditions <boundary_conditions.md>
Solid-body voxelization <voxelization.md>
Coupling with other modules <coupling.md>
```