Eddy diffusion (${\partial }_t\sigma \supset D{\nabla }^2\sigma$)

Mixing by turbulence is represented approximately by a diffusive flux down a mean concentration gradient. Its coefficient is the eddy diffusivity. A constant-coefficient model gives a $D{\nabla }^2\sigma$ term and a smoothing rate $D{k}^2$ for a Fourier mode. Effective diffusion can suppress slow gravitational concentration even when microscopic diffusion is negligible.

In a narrow tube with unstable mean density gradient $q={\partial }_z\bar{\rho }>0$, width-limited turbulent eddies have velocity scale $u_{\ast }\sim W\sqrt{gq/{\rho }_0}$. A mixing-length diffusivity gives downward density flux $F_{\rho }=K{W}^2\sqrt{g/{\rho }_0}{q}^{3/2}$ per unit area. With upward coordinate $z$, conservation is ${\bar{\rho }}_t={\partial }_z{F}_{\rho }$ and $q_t=K{W}^2\sqrt{g/{\rho }_0}{\partial }_z^2(q^{3/2})$. The closure presupposes established high-Reynolds number turbulence and requires boundary fluxes and finite available potential energy to be respected.

Ignoring end boundaries, the buoyancy-driven turbulent density diffusion model admits $\bar{\rho }={\rho }_0[1+A(t)C{z}^5]$ for $C>0$. Since $({\partial }_z\bar{\rho }{)}^{3/2}\propto A^{3/2}{z}^6$, the amplitude obeys $A^{\prime }=65^{3/2}K{W}^2\sqrt{gC}{A}^{3/2}$. With $A(0)=1$, $A=(1-t/t_{\ast }{)}^{-2}$ and $t_{\ast }=[35^{3/2}K{W}^2\sqrt{gC}{]}^{-1}$. Its formal growth is not a physical finite-time singularity of a closed mixing tube: ignored end fluxes, loss of the Boussinesq approximation and the finite energy budget invalidate the continuation.

Molecular scalar diffusion removes density-fluctuation variance at the positive rate displayed here. With this convention the budget for one half of the variance has dissipation term $-\chi$; the full variance budget has $-2\chi$. In a stationary locally homogeneous shear flow the half-variance budget gives $\overline{{\rho }^{\prime }{w}^{\prime }}{\overline{\rho }}_z+\chi =0$. An integral-scale model writes $\chi =\overline{{\rho }^{\prime 2}}q/L_{\rho }$, absorbing any normalization constant into $L_{\rho }$.

The ratio of turbulent momentum diffusivity to buoyancy or heat diffusivity. Their ratio determines how the gradient Richardson number is related to the energy-budget Flux Richardson number. It is generally a closure-dependent function of the flow, not necessarily one.

An ideal correspondence between turbulent momentum and scalar transfer. In the local gradient-diffusion form used for a surface layer, the ideal analogy sets ${\mathrm{Pr}}_t=1$. It is an additional modeling approximation; measured buoyancy and momentum similarity functions need not be identical.

The covariance of vertical velocity and buoyancy fluctuations has dimensions length squared per time cubed. It supplies buoyancy production in the local turbulent kinetic energy balance. This local flux density differs dimensionally from the section-integrated buoyancy flux of a plume, which includes an area or width integration.

For positive dense-fluid reduced gravity $G$ and upward coordinate $z$, an unstable gradient $G_z>0$ supplies buoyant work of order $d^2{G}_z$ over mixing length $d$. velocity $u\sim d\sqrt{G_z}$ and eddy diffusivity $K\sim ud$ follow. Order-one constants depend on the turbulence model. This closure is for an unstable gradient; its square root must not be applied to a stable negative gradient.

An upward volume flux $Q$ and a downward turbulent buoyancy transport $d^4(G_z{)}^{3/2}$ balance in a steady reactor. Matching $G=G_z=0$ below $-H$ gives $QG=d^4(G_z{)}^{3/2}$. Integrating yields the cubic profile above the front. If downward source buoyancy flux is $B_s$, then $H=3d^{8/3}{B}_s^{1/3}/Q$. The corresponding eddy diffusivity is $K=Q(z+H)/(3d^2)$. All these prefactors use the same area and mixing-length normalization; an actual cross-sectional area $S$ replaces the denominator in $K$ by $3S$.

In the area normalization $d^2$, the buoyancy-gradient mixing-length closure transports dense-fluid buoyancy downward with magnitude $\mathcal{F}=d^{2}K{G}_z=d^4(G_z{)}^{3/2}$. Its convergence in an upward coordinate is ${\mathcal{F}}_z$. Transport has units $L^4{T}^{-3}$ and convergence units $L^3{T}^{-3}$. Constant positive $G_z$ has nonzero transport and zero convergence; the two quantities cannot be identified.