The high predictability of the atmosphere and ocean depends on the existence of a 'slow manifold', which contains the solutions of equations describing only large-scale motions. This unique compendium succinctly describes major recent advances in showing that these equations can be solved independently.The book is a new edition of a similar book published 15 years ago. The explanation of the mathematical techniques has been expanded. Many new theoretical results are included. Illustrations derived from production atmosphere and ocean models are also incorporated to cover the full range between rigorous mathematics and state-of-the-art numerical modelling.The author is a dynamical meteorologist with long experience and international standing. The mathematical results in the book were proved by many of the world's leading analysts. The results come from the Met Office Unified Model, which is one of the world's leading weather and climate models.Related Link(s)
Author(s): Mike Cullen
Publisher: World Scientific Publishing
Year: 2021
Language: English
Pages: 410
City: Singapore
Contents
Dedication
Preface
1. Introduction
2. The Governing Equations and Asymptotic Approximations to Them
2.1 Basic equations
2.2 Scale analysis, key dimensionless parameters, and reduced equations
2.2.1 Dimensionless equations
2.2.2 Asymptotic regimes and reduced equations
2.2.3 The shallow atmosphere hydrostatic approximation
2.2.4 Properties of the shallow atmosphere hydrostatic equations
2.2.5 Classification of shallow atmosphere hydrostatic regimes
2.3 Various approximations to the shallow water equations
2.3.1 The shallow water equations
2.3.2 Key parameters
2.3.3 General equations for slow solutions
2.3.4 Slow solutions on small scales
2.3.5 Quasi-geostrophic solutions
2.3.6 Slow solutions on large scales
2.3.7 Large scale slow solutions in the tropics
2.4 Various approximations to the three-dimensional hydrostatic Boussinesq equations
2.4.1 The hydrostatic Boussinesq equations
2.4.2 Key parameters
2.4.3 General equations for slow solutions
2.4.4 Slow solutions with large aspect ratio
2.4.5 Quasi-geostrophic solutions
2.4.6 Slow solutions with small aspect ratio
2.4.7 Large scale slow solutions in the tropics
2.5 Large scale solutions of the fully compressible equations in spherical geometry
2.5.1 Choice of semi-geostrophic scaling
2.5.2 Derivation and properties of the equations
2.6 Illustrations of asymptotic behaviour from observations
2.6.1 Atmosphere and ocean spectra
2.6.2 Identification of different regimes in the atmosphere
2.7 The appropriateness of the semi-geostrophic approximation for large-scale ows in the atmosphere
2.7.1 Implications of the atmospheric basic state
2.7.2 Curvature of trajectories
2.7.3 Validation of the inertial stability condition
2.7.4 Demonstration of the accuracy of semi-geostrophic theory for large-scale flows
2.7.5 The transition from the semi-geostrophic regime to the tropical long wave regime
3. Solution of the Semi-geostrophic Equations in Plane Geometry
3.1 The solution as a sequence of minimum energy states
3.1.1 The evolution equation for the geopotential
3.1.2 Solutions as minimum energy states
3.1.3 Physical meaning of the energy minimisation
3.2 Solution as a mass transport problem
3.2.1 Solution by change of variables
3.2.2 The equations in dual variables
3.2.3 Solution using optimal transport
3.3 The shallow water semi-geostrophic equations
3.3.1 Solutions as minimum energy states
3.3.2 Solution by change of variables
3.3.3 The equations in dual variables
3.3.4 Solution using optimal transport
3.3.5 The Boussinesq semi-geostrophic equations with a free surface
3.3.6 The fully compressible semi-geostrophic equations in pressure coordinates with a free surface
3.4 Discrete solutions of the semi-geostrophic equations
3.4.1 The discrete problem
3.4.2 Example: frontogenesis
3.4.3 Example: outcropping
3.5 Rigorous results on existence of solutions
3.5.1 Summary of the solution procedure
3.5.2 Solutions of the mass transport problem
3.5.3 Existence of semi-geostrophic solutions in dual variables
3.5.4 Solutions in physical variables
3.6 Approximation of Euler solutions by semi-geostrophic solutions
3.6.1 General principles
3.6.2 Convergence of the two-dimensional semi-geostrophic equations to the Euler equations
3.6.3 Convergence for the Eady problem
3.6.4 Convergence to the solution of the three-dimensional Euler equations
4. Solution of the Semi-geostrophic and Related Equations in More General Cases
4.1 Solution of the semi-geostrophic equations for compressible flow
4.1.1 The compressible equations in Cartesian geometry
4.1.2 The solution as a sequence of minimum energy states
4.1.3 Solution by change of variables
4.1.4 The equations in dual variables
4.1.5 Rigorous weak existence results
4.2 Semi-geostrophic theory on a sphere
4.3 The shallow water spherical semi-geostrophic equations
4.3.1 Solutions as minimum energy states
4.3.2 Classical solutions for a short time interval
4.3.3 Formulation of the problem as a Monge-Kantorovich problem
4.4 The theory of axisymmetric flows
4.4.1 Forced axisymmetric flows
4.4.2 The vortex as a minimum energy state
4.4.3 Solution by change of variables
4.4.4 Generation of a mass conservation equation in the new variables
4.4.5 Solution using optimal transport
4.4.6 Properties of the vortex
4.4.7 Solutions of the evolution equation with forcing
4.4.8 Stability of the vortex
4.5 Zonal flows on the sphere
4.5.1 Governing equations
4.5.2 Zonal flow as a minimum energy state
4.5.3 Global solutions
4.5.4 Generation of a mass conservation equation in angular momentum and potential temperature coordinates
4.5.5 Solution procedure
4.5.6 Application
4.5.7 Almost axisymmetric flows
4.6 Stability theorems for semi-geostrophic flow
4.6.1 Extremising the energy by rearrangement of the potential density
4.6.2 Properties of rearrangements
4.6.3 Analysis of semi-geostrophic shear flows
5. Properties and Validation of Asymptotic Limit Solutions
5.1 Numerical methods for solving the semi-geostrophic equations
5.1.1 Solutions using the geostrophic coordinate transformation
5.1.2 The geometric method
5.1.3 Finite difference methods
5.1.4 Approximation of weak solutions given by Lagrangian conservation laws
5.2 Shallow water solutions
5.2.1 Nature of solutions
5.2.2 Examples of the asymptotic behaviour
5.2.3 Numerical demonstration of the asymptotic convergence
5.3 The Eady wave
5.3.1 Governing equations
5.3.2 Numerical experiments
5.3.3 Nature of the solution
5.3.4 Convergence of the Euler solution to the semi-geostrophic solution
5.3.5 Results from a fully Lagrangian scheme
5.3.6 Summary of Eady problem
5.4 Simulations of baroclinic waves
5.4.1 Diagnosis of vertical motion
5.4.2 Stability of frontal zones
5.4.3 Evolution of a baroclinic wave
5.5 Orographic flows
5.6 Tropical-extratropical interaction
6. Inclusion of Additional Physics
6.1 Inclusion of friction
6.1.1 Semi-geotriptic theory
6.1.2 The semi-geotriptic equations
6.1.3 Sea-breeze circulations
6.1.4 Modelling low-level jets
6.1.5 Accuracy of semi-geotriptic theory
6.1.6 Tropical performance
6.2 Inclusion of moisture
6.2.1 Moist semi-geostrophic equations
6.2.2 Solutions for stable data
6.2.3 Moist instability and moist rearrangements
6.2.4 The moist rearrangement problem
6.2.5 Discrete solutions of the moist rearrangement problem
6.2.6 Measure-valued solutions of the moist rearrangement problem
6.2.7 Examples of solutions of the moist rearrangement problem
6.2.8 Examples of large-scale flows including moist instability
7. Summary
Bibliography
Index
