42.1.2 Tracer moments

Multiplying the tracer equation

$$(\partial_{t}+ \vec{u} \cdot \nabla)T =\partial_{m}( J^{m n}\partial_{n} T)$$ (42.23)

by T^N-1, where N is some positive integer, yields

$${1 \over N} \partial_{t} T^{N} + {1 \over N} \nabla \cdot (\vec{u}T^{N}) T^{N-1}\partial_{m}( J^{m n}\partial_{n} T)$$ = (42.24)

$$\partial_{m}(T^{N-1} J^{m n} \partial_{n}T) - \partial_{m}T^{N-1} \partial_{n}T J^{m n}$$ = (42.25)

$${1 \over N} \partial_{m} (J^{m n} \partial_{n} T^{N}) - (N-1)T^{N-2}\partial_{m}\partial_{n}J^{m n}.$$ = (42.26)

The time tendency is therefore given by

$$\partial_{t}T^{N} -\nabla \cdot (\vec{u}T^{N}) + \partial_{m}(J^{m n} \partial_{n}T^{N}) - N(N-1)T^{N-2} \partial_{m}T\partial_{n}T J^{m n}$$ = (42.27)

$$\nabla \cdot(\vec{F} T^{N-1} - \vec{u}T^{N}) - N(N-1)T^{N-2}\partial_{m}T\partial_{n}T K^{m n},$$ = (42.28)

where the flux

$$J^{m n} \partial_{n} T$$ F^m = (42.29)

$$K^{m n}\partial_{n} T + A^{m n}\partial_{n} T$$ = (42.30)

$$\equiv$$ F^m_K + F^m_A (42.31)

was introduced. Note that the above steps assumed a divergence-free velocity field ( $\nabla \cdot \vec{u} = 0$) and the identity (B.7) ( $A^{m n} \partial_{m}T \partial_{n} T = 0$) was employed.

Integrating the previous equation over some domain yields

$$\partial_{t} \int d\vec{x} \; T^{N} = \int d\vec{x} \; \lef... ... - N(N-1)T^{N-2} K^{m n} \partial_{m}T \partial_{n}T \right].$$ (42.32)

Stokes' Theorem brings this expression to

$$\partial_{t} \int d\vec{x} \; T^{N} = \int dA \; \hat{n} \c... ... \int d\vec{x} \; T^{N-2} K^{m n}\partial_{m}T \partial_{n}T,$$ (42.33)

where $\hat{n}$ is the outward normal to the boundary. Assuming the normal velocity vanishes at the domain boundary ( $\vec{u} \cdot \hat{n} = 0$) yields for the time tendency of the globally integrated N'th tracer moment

$$\partial_{t} \int d\vec{x} \; T^{N} = \int dA \; \hat{n} \cdo... ... \int d\vec{x} \; T^{N-2} K^{m n}\partial_{m}T \partial_{n}T.$$ (42.34)

It is of interest to consider some special cases. First, consider the case with no diffusive mixing, which means there is a zero symmetric component to the tracer mixing tensor ( K^{m n} = F^m_K = 0). Therefore, all moments of the tracer are conserved when there is no normal skew flux at the boundaries ( $\vec{F}_{A} \cdot \hat{n} = 0$), or equivalently there is zero normal induced velocity ( $\vec{u}_{A} \cdot \hat{n} = 0$) at the boundaries. For example, the skew flux associated with the Gent-McWilliams effective transport velocity satisfies these conditions and so preserves all tracer moments (Gent and McWilliams 1990).

For the case with zero anti-symmetric but nonzero symmetric tracer mixing tensor, the first moment, or the total tracer, is conserved in source free regions where there is no tracer flux normal to the boundary; i.e., if $\hat{n} \cdot \vec{F}_{K} \equiv \hat{n}_{m}K^{m n} \partial_{n}T = 0$ at the domain boundaries. In the absence of sources, the tracer's second moment satisfies the evolution equation

$$\partial_{t} \int d\vec{x} \; T^{2} = -2 \int d\vec{x} \; \partial_{m}T K^{m n}\partial_{n}T.$$ (42.35)

Therefore, this mixing tensor dissipates the tracer variance ^42.2 over the source-free domain if the tensor K^{m n} is a positive semi-definite tensor (i.e., the right hand side is negative semi-definite if K^{m n} is positive semi-definite). This property of diffusion is fundamental, and is taken as the foundation for the numerical discretization of isopycnal diffusion in MOM. All higher moments of the tracer are also dissipated by diffusion assuming the usual case of a non-negative tracer concentration.