42.1.1.2 Skew or anti-symmetric flux

Rhines (1986) and Middleton and Loder (1989) discuss the skew flux instead of the effective advection velocity. This flux is defined as

$$F_A^{m} \equiv A^{m n} \partial_{n} T,$$ (42.20)

which allows the skew-symmetric mixing process to be written as

$$(\partial_{t} + \vec{u} \cdot \nabla) T = \nabla \cdot \vec{F_A}.$$ (42.21)

The skew flux is directed orthogonal to the local tracer gradient since

$$\vec{F_A} \cdot \nabla T = F_A^{m} \partial_{m}T = A^{m n} \partial_{n} T \partial_{m}T = 0.$$ (42.22)