42.1.1.1 Effective advection velocity

Consider the tracer mixing operator constructed from an anti-symmetric tensor

$$R(T)_{A} = \partial_{m}(A^{m n}\partial_{n}T).$$ (42.10)

Property (B.8) implies

$$R(T)_{A} = (\partial_{m}A^{m n})\partial_{n}T,$$ (42.11)

which allows for the identification of a velocity

$$u_{A}^{n} \equiv -\partial_{m} A^{m n},$$ (42.12)

and which brings the anti-symmetric mixing process into the form of an advection

$$R(T)_{A} = -\vec{u}_{A} \cdot \nabla T.$$ (42.13)

It is important to note that the advection velocity $\vec{u}_{A}$ is divergence-free

 

$$\nabla \cdot \vec{u}_{A} \partial_{n} u_{A}^{n}$$ = (42.14)

$$-\partial_{n}\partial_{m}A^{m n}$$ = (42.15)

= 0, (42.16)

where the last identity used relation (B.9) above. Therefore, tracer mixing as parameterized with an anti-symmetric tensor

$${DT \over D t} \equiv (\partial_{t} + \vec{u} \cdot \nabla) \; T = R(T)_{A}$$ (42.17)

can be written

$$[\partial_{t} + (\vec{u} + \vec{u}_{A})\cdot \nabla]\; T = 0,$$ (42.18)

which allows for the identification of an effective advective transport velocity (Plumb and Mahlman 1987)

$$\vec{U} \equiv \vec{u} + \vec{u}_{A}.$$ (42.19)