Mathematical Principle and Fractal Analysis of Mesoscale Eddy

Preface
Contents
Abbreviations
Symbols
1 Introduction
2 Preliminaries
	2.1 Limit Cycle and Semi-stable Limit Cycle
	2.2 Criterion of Semi-stable Limit Cycle
		2.2.1 Limit Cycles of Oscillatory Approach and Monotone Approach
		2.2.2 Criterions
	2.3 Feature Scale, Scale-Free Domain, Fractal, Random Fractal, Dimension
	2.4 Iterative Function System and Fractal
	2.5 Dissipative System
	2.6 Attractor, Attracting Set, Basin of Attraction, Strange Attractor, and Semi-strange Attractor
	2.7 Relationship between Semi-stable Limit Cycles and Semi-strange Attractors
	2.8 Elementary Reaction and Reaction Rate
	2.9 Lagrangian Particle Dynamic System
3 Universal Mathematical Model of Mesoscale Eddy
	3.1 Mesoscale Eddy
	3.2 Mathematical Model of Mesoscale Eddy
		3.2.1 Bounded Motion
		3.2.2 Movement Asymptotic Unity and Uniform Tendency
	3.3 Universal Mathematical Model of Mesoscale Eddy
		3.3.1 Momentum of a Stochastic Ellipse
		3.3.2 Elementary Reaction Rate
		3.3.3 Basic Mathematical Model of Mesoscale Eddy
		3.3.4 Universal Mathematical Model of Mesoscale Eddy
4 Semi-stable Limit Cycle in Mathematical Model of Mesoscale Eddy
	4.1 Analysis of Parameter Distribution of Stable and Unstable Limit Cycles
		4.1.1 α and β are Positive and m is Odd
		4.1.2 α and β are Negative and m is Odd
		4.1.3 α is Positive, β is Negative and m is Odd
		4.1.4 α is Negative, β is Positive and m is Odd
		4.1.5 m is a Decimal
	4.2 Stable Limit Cycle
	4.3 Unstable Limit Cycle
	4.4 Parameter Distribution Analysis of Semi-stable Limit Cycle
		4.4.1 Special System
		4.4.2 General System
		4.4.3 Different Internal and External Stability
	4.5 Externally Unstable and Internally Stable Semi-stable Limit Cycle
	4.6 Externally Stable and Internally Unstable Semi-stable Limit Cycle
5 Semi-stable Limit Cycles and Mesoscale Eddies
	5.1 Semi-stable Limit Cycles and Mesoscale Cold Eddies
	5.2 Semi-stable Limit Cycles and Mesoscale Warm Eddies
6 Example Verification
	6.1 Basic Method
	6.2 Numerical Experiment
		6.2.1 Value in Special Circumstances
		6.2.2 Full Parameter Case
		6.2.3 Clockwise Model
		6.2.4 Anti-clockwise Model
		6.2.5 Algorithm Parallelization and Model Checking in Global Oceans
7 Spatiotemporal Structure of Mesoscale Eddies: Self-similar Fractal Behavior
	7.1 Spatiotemporal Fractal Structure of Mesoscale Warm Eddy
	7.2 Spatiotemporal Fractal Structure of Mesoscale Cold Eddy
	7.3 Self-similar Fractal Structure under Affine Transformation
		7.3.1 Transformation Relations of Spatial Coordinates
		7.3.2 Spatial Structure
8 Mesoscale Eddies: Disk and Columnar Shapes
	8.1 The Specific Implementation Process of Water Particle Motion …
		8.1.1 Specific Transformation
		8.1.2 Disk-Shaped Mesoscale Cold Eddy
	8.2 Specific Implementation Process of Water Particle Motion Transformation …
		8.2.1 Specific Transformation
		8.2.2 Disk-Shaped Mesoscale Warm Eddy
	8.3 Approximate Approximation of Mesoscale Disk-Shaped Mesoscale Eddy
9 Fractal Analysis and Prediction for Spatiotemporal Complexity of Mesoscale Eddy
	9.1 Spatiotemporal Structure of Mesoscale Eddies Based on Universal Model
		9.1.1 Mesoscale Cold Eddy
		9.1.2 Mesoscale Warm Eddy
	9.2 Mathematical Model and Complexity Analysis of Spatiotemporal Fractal Structure of Mesoscale Eddies
		9.2.1 Fractal Model of Snowflake
		9.2.2 Fractal Model of Random Snowflake
		9.2.3 Mesoscale Eddies and Spatiotemporal Fractal Structures of Cantor Self-Similar Fractal Sets
	9.3 Spatiotemporal Fractal Analysis and Prediction of the Complexity of Mesoscale Eddies
		9.3.1 Data
		9.3.2 Fractal Dimension of Mesoscale Eddy
		9.3.3 Fractal Processing of Mesoscale Eddies Profile of the Ocean
		9.3.4 Three-Dimensional Fractal Structure of Abnormal Salinity
		9.3.5 Comprehensive Analysis
10 Nonlinear Characteristics of Universal Mathematical Model of Mesoscale Eddy
	10.1 Dissipation of Nonlinear Systems
	10.2 Chaotic Behavior of Universal Nonlinear System  of Mesoscale Eddy
	10.3 Singularity of Mesoscale Eddy and its Physical Meaning
11 Same Solution Between Momentum Balance Equations and Mesoscale Eddies
	11.1 Navier-Stokes Equation
	11.2 Same Solution Between the Mathematical Model of Mesoscale Eddy …
	11.3 Necessary Conditions for Existence of Mesoscale Eddies in Special Model
	11.4 Sufficient and Necessary Conditions for the Existence of Mesoscale Eddies in the General Model
		11.4.1 No Stickiness
		11.4.2 Stickiness
		11.4.3 Perturbation Terms of Parameters with Pressure Change Rate
		11.4.4 Necessary Conditions
12 Momentum Balance Equation Based on Truncation Function and Mathematical Model of Mesoscale Eddies
	12.1 Sufficient Conditions of Mesoscale Eddies for the Two-Dimensional …
	12.2 Existence of Mesoscale Eddies in Two-Dimensional Momentum Balance …
		12.2.1 β-Plane Approximation and Viscosity
		12.2.2 β-Plane Approximation and Nonviscosity
	12.3 Mesoscale Cold and Warm Eddies Produced by Truncation Function and Circulation Factor
13 Interpolation Prediction of Mesoscale Eddies
14 Random Elliptic Curve and Brownian Motion Trajectory of Mesoscale Eddy
	14.1 Trajectory of Elliptical Arc
		14.1.1 Mesoscale Cold Eddy
		14.1.2 Mesoscale Warm Eddy
	14.2 Trajectory of Brownian Curve
15 Mathematical Model for Edge Waves of Mesoscale Eddies and Its Spatio-Temporal Fractal Structures
	15.1 Mathematical Model of Edge Waves
		15.1.1 Poincaré Cross-Section
		15.1.2 Edge Wave Motion and Its Duffing Dynamical System
	15.2 Mathematical Model of Edge Waves Based on Poincaré …
		15.2.1 Generators of Edge Waves
		15.2.2 A Mathematical Model for Random Fractal of Edge Waves
	15.3 Fractal Analysis of Internal Structure Complexity of Edge Waves
	15.4 New Problems Arising from Random Fractal Models of Edge Waves
Appendix  References