Scalar variance ($V=|\Omega {|}^{-1}{\int }_{\Omega }(\Theta -\overline{\Theta }{)}^{2}dV$)

Scalar variance measures spatial departure from the conserved mean. For zero mean, incompressible flow bounded by impermeable walls and homogeneous scalar Neumann boundary conditions, integration by parts gives $d\int {\Theta }^2/dt=-2\kappa \int |\nabla \Theta {|}^2$. Advection preserves the instantaneous quadratic integral but can sharpen gradients, allowing the diffusion equation to remove scalar variance faster.