# LES Subgrid Coupling

When the fluid simulation runs in LES mode, the unresolved scalar flux contributes a subgrid diffusivity that must be added to the molecular value:

$$D_{\text{total}} = D_0 + D_{\text{SGS}}, \qquad D_{\text{SGS}} = \frac{\nu_{\text{SGS}}}{Sc_t}.$$

The turbulent Schmidt number $Sc_t$ is a configurable constant; the default value is $Sc_t = 0.7$, the standard choice in atmospheric LES.
The same SGS constant plays the role of the turbulent Prandtl number $Pr_t$ when the transported scalar is temperature: the subgrid thermal diffusivity is $\kappa_{\text{SGS}} = \nu_{\text{SGS}} / Pr_t$, with $Pr_t = Sc_t$ entering the identical closure (see {doc}`thermal`).

```{admonition} Key idea: the scalar inherits its subgrid diffusivity from the fluid
---
class: note
---
The scalar does not run its own subgrid model. It takes the fluid's Smagorinsky eddy viscosity $\nu_{\text{SGS}}$ and converts it with a single constant, $D_{\text{SGS}} = \nu_{\text{SGS}} / Sc_t$. One closure, one tunable constant: the unresolved scalar mixing is tied directly to the unresolved momentum mixing the fluid already computes.
```

The node-local scalar relaxation rate is then

$$\tau_g = \frac{D_{\text{total}}}{c_{s,\phi}^2} + \frac{1}{2}, \qquad \omega_\phi = \tau_g^{-1}.$$

The subgrid viscosity $\nu_{\text{SGS}}$ is taken directly from the fluid Smagorinsky model (see {doc}`/theory/LES/index`): the scalar reads the fluid's stored node-local relaxation $\omega_{\text{LES}}$ and reconstructs $\nu_{\text{SGS}} = \nu(\omega_{\text{LES}}) - \nu_0$, so fluid and scalar see identical subgrid values without storing $\nu_{\text{SGS}}$ as a separate macroscopic. The same node-local $\tau_g$ enters the multiblock level transfer of $q^{neq}_\alpha$ through the flux conversion $J = q^{neq} (1 - \omega_\phi/2)$ described in {doc}`coupling`, mirroring the fluid's per-node viscosity treatment in the stress transfer.

In lattice units both $\nu_{\text{SGS}}$ and $D_{\text{SGS}}$ have the same dimension $[\Delta x^2 / \Delta t]$, so the closure $D_{\text{SGS}} = \nu_{\text{SGS}} / Sc_t$ applies directly with no conversion factor between the fluid lattice and the scalar lattice.

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