= Scalar variance
{title2=$V=|\Omega|^{-1}\int_\Omega(\Theta-\overline\Theta)^2\,dV$}
<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.
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