43.0.2 Functional formalism

A fundamental property of diffusion is that it dissipates tracer variance. Unfortunately, it is not guaranteed that every discretization of a diffusion operator will satisfy this desired dissipative property. Fortunatly, there is a means to ensure variance reduction by employing a straightforward formalism. The mathematical property that is exploited with this formalism is to note that the diffusion operator, when acting on a passive tracer, is a linear self-adjoint operator. As such, it has an associated negative semi-definite functional (e.g., Courant and Hilbert, 1953). For example, Laplacian diffusion in an isotropic media, $R(T) = \nabla^{2}T$, is identified with the functional derivative $R(T) = \delta {\cal F} / \delta T$, where ${\cal F} \equiv -(1/2)\int \vert\nabla T\vert^{2} d\vec{x} \le 0$ is the associated functional. For a general diffusion tensor, the functional is given by ${\cal F} \equiv (1/2) \int d\vec{x} \; \vec{F} \cdot \nabla T$ $= (1/2) \int d\vec{x} \; T_{m} \, K^{mn} \, T_{n}$, where $T_{m} = \partial_{m}T$ and repeated indices are summed. The negative semi-definiteness of the functional ${\cal F}$ is related to the dissipative property of the diffusion operator; i.e., one implies the other. It is also directly related to the symmetric positive semi-definiteness of the diffusion tensor K^mn = K^nm and the associated downgradient orientation of the diffusive flux $\vec{F} \cdot \nabla T \le 0$.

On the lattice, not every consistent^43.1 discrete diffusion operator corresponds to a negative semi-definite discrete functional. Therefore, a consistent numerical diffusion operator does not necessarily possess the dissipative properties of the continuum operator. For the Laplacian in an isotropic media, it is trivial to produce a dissipative numerical operator. In the anisotropic case, such as isoneutral diffusion, it is nontrivial. Indeed, the original discretization of the isoneutral diffusion operator in the GFDL model (Cox, 1987) is numerically consistent but not dissipative. The approach taken in the derivation of the new discretization of this operator is to focus on discretizing the functional first and then to take the discrete version of the functional derivative in order to derive the discrete diffusion operator. This approach ensures that the discretized operator is dissipative, no matter how the functional is discretized.