... components^42.1

The anti-symmetric component is sometimes called the skew-symmetric component in the literature.

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... variance^42.2

The variance of the tracer is given by $var(T) = V^{-1}[ \int d\vec{x} \; T^{\,2} - V^{-1} (\int d\vec{x} \; T)^{2} ] \ge 0$, with $V = \int d\vec{x}$ the domain volume. Reducing $\int d\vec{x} \; T^{\,2}$ is therefore equivalent to reducing var(T).

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... coefficient^42.3

Since this frame is flat (metric is the unit tensor), there is no distinction between lower and upper labels on the components of a tensor. In general, the convention for tensors such as $K^{\overline{m} \overline{n}}$, when written as a matrix, is that the first index indicates the row and the second index is for the column.

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... tensor^42.4

Note the rotation need not transform the $\hat{x} \leftrightarrow \hat{y}$ symmetry present in the (x,y,z) form of the small angle mixing tensor into a $\vec{e}_{\overline{1}} \leftrightarrow \vec{e}_{\overline{2}}$symmetry in the $(\vec{e}_{\overline{1}},\vec{e}_{\overline{2}},\vec{e}_{\overline{3}})$form. The (x,y) coordinate symmetry, however, is preserved.

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... consistent^43.1

Consistent in that the discretization reduces to the correct continuum operator as the grid size goes to zero.

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