Passive scalar

A passive scalar is an advected and diffusing scalar field whose value has no dynamical effect on the transporting velocity field. Concentration of a dilute nonreacting dye is a common example. Its transport can depend on the flow while the flow equations remain independent of it.

The horizontally integrated plume concentration is the integral of a local concentration $C$ across the plume. It measures scalar amount per unit vertical length and unit span, and its vertical integral is total scalar amount per unit span. Unlike a local concentration, its dimensions include one horizontal length. It must be distinguished from horizontally averaged plume concentration when a plume's width varies with height.

For a plume of full width $w(z)$, its horizontally averaged concentration is $\overline{C}(z,t)=c(z,t)/w(z)$, where $c$ is horizontally integrated plume concentration. In a line plume with $w=\mu z$, division by $z$ generally changes the location of maximum concentration. A gradient closure for $c$ differs from one for $\overline{C}$: $w{\partial }_z\overline{C}=c_z-c/z$, so integrated diffusive flux $-Kw{\partial }_z\overline{C}$ equals $-K(c_z-c/z)$.