Alphabeta MathRotating stratified and shallow-water mathematical models
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1 · Prerequisites
- Arc Length and Rectifiable Curves
- Areas of Elementary Plane Figures
- Binary Operations, Monoids, Groups and Subgroups
- Compactness
- Compactness in Metric Spaces
- Completeness, Completion, and Uniform Continuity
- Construction of the Natural Numbers
- Construction of the Real Numbers via Cauchy Sequences
- Construction of the Real Numbers via Dedekind Cuts
- Continuity, IVT, EVT, and Uniform Continuity
- Cosets, Index and Lagrange's Theorem
- Countability and Uncountability
- Determinants of Matrices over a Commutative Ring
- Divisibility, Euclidean Domains, Principal Ideal Domains and Unique Factorisation
- Dual Spaces, Bilinear and Quadratic Forms, and Sylvester's Law of Inertia
- Filters and Ultrafilters
- Finite Counting, Factorials and Binomial Coefficients
- Formal parameterized fluid systems and observables
- Foundations of the Real Numbers for Analysis
- Fubini and Change of Variables
- Group Actions, Orbits, Stabilisers and Cayley's Theorem
- Ideals, Quotient Rings and the Isomorphism Theorems for Rings
- Inner Product Spaces, Gram-Schmidt, Projections and Adjoints
- Lebesgue Measure on Euclidean Space
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- Rⁿ as a Normed Space; Vector-Valued Functions
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- The Derivative and the Mean Value Theorems
- The Determinant of a Linear Operator, Cofactors and Cramer's Rule
- The Divergence Theorem and Classical Stokes
- The Fundamental Theorems of Calculus
- The Inverse and Implicit Function Theorems
- The Inverse Function Theorem Completed
- The Lebesgue and Riemann Integrals Compared
- The Lebesgue Integral and the Convergence Theorems
- The Riemann Integral in Rᵐ and Jordan Content
- The Riemann Integral: Definition and Integrability
- The Topology of Euclidean Space
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- The ZFC Axioms and the Basic Set Constructions
- Topological Spaces and Continuity
- Topology of ℝ
- Vector Spaces, Linear Subspaces, Span and Direct Sums
2 · Logical flowchart
3 · Items
Definition
Rotating stratified and shallow-water equation families
Theorem
Positive linear stratified energy and diffusion
Theorem
Smooth rotating shallow-water energy balance
Sources
- Geoffrey K. Vallis, Fluid Dynamics of the Atmosphere and Ocean, University of Exeter lecture notes, Autumn 2018, §§1.6.2–1.6.3 and §2.4.2
- Darryl D. Holm, Ideal shallow water dynamics in a rotating frame, Imperial College London MPE CDT PDE lecture notes, Autumn 2016, §§1.1 and 1.4
- Paola Rizzoli, Wave Motion in the Ocean and the Atmosphere, MIT OpenCourseWare 12.802, Lecture 12: Shallow water equations with rotation — Poincaré waves, Spring 2008
- Geoffrey K. Vallis, Fluid Dynamics of the Atmosphere and Ocean, University of Exeter lecture notes, Autumn 2018, §1.6.3, equations (1.72)–(1.78)
- Geoffrey K. Vallis, Fluid Dynamics of the Atmosphere and Ocean, University of Exeter lecture notes, Autumn 2018, §2.4.2, equations (2.55)–(2.59)
- Darryl D. Holm, Ideal shallow water dynamics in a rotating frame, Imperial College London MPE CDT PDE lecture notes, Autumn 2016, §§1.1 and 1.4, equations (1.1) and (1.14)