For a scalar tracer $$ \frac{\partial C}{\partial t} = C_t = - \mathbf{u} \cdot \nabla C \; (\text{advection}) + k \nabla^2 C \; (\text{diffusion}) + S \; (\text{source}) $$

$$ T = \overline{T} + T', \quad \overline{T} = \frac{1}{\Delta t} \int_0^{\Delta t} T \, dt, \quad \overline{T'} = 0 $$

For a 1D tracer with source $H$ $$ T_t = - u T_x + k T_{xx} + H $$ Insert $T = \overline{T} + T'$, $u = \overline{u} + u'$, $H = \overline{H} + H'$ $$ (\overline{T} + T')_t = - (\overline{u} + u') (\overline{T} + T')_x + k (\overline{T} + T')_{xx} + (\overline{H} + H') $$

Taking the Reynolds average yields $$ \overline{T}_t = - \big( \overline{u} \, \overline{T}_x + \overline{u'T'_x} \big) + k \, \overline{T}_{xx} + \overline{H}. $$ Here the new correlation term $\overline{u'T'_x}$ arises. The unresolved correlation $\overline{u'T'_x}$ represents the effect of turbulent fluctuations. This is the closure problem: the mean equation now contains an unknown term