Vector Identities and Derivations
To derive the prognostic equations in a GCM usually requires the use of vector identities. Here is a list of some links which give vector identities:
http://en.wikipedia.org/wiki/List_of_vector_identities
http://en.wikipedia.org/wiki/Product_rule
http://en.wikipedia.org/wiki/Vector_calculus_identities
http://mathworld.wolfram.com/VectorDerivative.html
http://planetphysics.org/encyclopedia/VectorIdentities.html
http://geo.phys.spbu.ru/~runov/SPIntro/SPIntro_15.pdf
I’ve been searching for the derivations of these identities and have found an excellent source:
Vector algebra is a powerful and needful tool for Physics but unfortunately, due to lack of mathematical skills, it becomes misleading for first undergraduate courses of science and engineering studies. Standard vector identities are usually proved using Cartesian components or geometrical arguments, accordingly. Instead, this work presents a new teaching strategy in order to derive symbolically vector identities without analytical expansions in components, either explicitly or using indicial notation. This strategy is mainly based on the correspondence between threedimensional vectors and skewsymmetric secondrank tensors. Hence, the derivations are performed from skew tensors and dyadic products, rather than cross products. Some examples of skewsymmetric tensors in Physics are illustrated.
http://www.citeulike.org/user/pak/article/4524046
http://arxiv.org/PS_cache/arxiv/pdf/0904/0904.1814v1.pdf
This will require some understanding of some concepts of tensor algebra. Here are some helpfull links of concepts that will be needed:
http://www.foamcfd.org/Nabla/guides/ProgrammersGuidese3.html
http://en.wikipedia.org/wiki/LeviCivita_symbol
http://en.wikipedia.org/wiki/Dyadics
http://en.wikipedia.org/wiki/Pseudovector
As a side note, the prognostic equations are typically derived from the vorticy equation. See:
http://en.wikipedia.org/wiki/Vorticity_equation
http://en.wikipedia.org/wiki/Barotropic_vorticity_equation
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