The central result about fast rotating-flow structures is the Taylor-Proudman theorem (TPT) which connects various aspects of the dynamics. Taylors geometrical proof of TPT is reproduced and extended substantially, with Lies theory for general frozen-in laws and the consequent generalized invariant circulation theorems, to compressible flows and to $d$-dimensional Euclidean space ($mathbb{E}^{d}$) with $dge 3$. The TPT relatives, the reduced models (with particular interests on passive-scalar problems), the inertial (resonant) waves and the higher-order corrections, are discussed coherently for a comprehensive bird view of rotating flows in high spatial dimensions.
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