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Mathematics in the Ocean

Andrew Poje Mathematics Department College of Staten Island. Mathematics in the Ocean. M. Toner A. D. Kirwan, Jr. G. Haller C. K. R. T. Jones L. Kuznetsov … and many more!. U. Delaware. Brown U. April is Math Awareness Month. Why Study the Ocean?. Fascinating !

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Mathematics in the Ocean

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  1. Andrew PojeMathematics Department College of Staten Island Mathematics in the Ocean • M. Toner • A. D. Kirwan, Jr. • G. Haller • C. K. R. T. Jones • L. Kuznetsov • … and many more! U. Delaware Brown U. April is Math Awareness Month

  2. Why Study the Ocean? • Fascinating! • 70 % of the planet is ocean • Ocean currents control climate • Dumping ground - Where does waste go?

  3. Ocean Currents: The Big Picture • HUGE Flow Rates (Football Fields/second!) • Narrow and North in West • Broad and South in East • Gulf Stream warms Europe • Kuroshio warms Seattle image from Unisys Inc. (weather.unisys.com)

  4. Drifters and Floats:Measuring Ocean Currents

  5. Particle (Sneaker) Motion in the Ocean

  6. Particle Motion in the Ocean:Mathematically • Particle locations: (x,y) • Change in location is given by velocity of water: (u,v) • Velocity depends on position: (x,y) • Particles start at some initial spot

  7. Ocean Currents: Time Dependence • Global Ocean Models: • Math Modeling • Numerical Analysis • Scientific Programing • Results: • Highly Variable Currents • Complex Flow Structures • How do these effect transport properties? image from Southhampton Ocean Centre:. http://www.soc.soton.ac.uk/JRD/OCCAM

  8. Coherent Structures: Eddies, Meddies, Rings & Jets • Flow Structures responsible for Transport • Exchange: • Water • Heat • Pollution • Nutrients • Sea Life • How Much? • Which Parcels? image from Southhampton Ocean Centre:. http://www.soc.soton.ac.uk/JRD/OCCAM

  9. Coherent Structures: Eddies, Meddies, Rings & Jets

  10. Mathematical Modeling: Simple, Kinematic Models (Functions or Math 130) Simple, Dynamic Models (Partial Differential Equations or Math 331) ‘Full Blown’, Global Circulation Models Numerical Analysis: (a.k.a. Math 335) Dynamical Systems: (a.k.a. Math 330/340/435) Ordinary Differential Equations Where do particles (Nikes?) go in the ocean Mathematics in the Ocean:Overview

  11. Abstract reality: Look at real ocean currents Extract important features Dream up functions to mimic ocean Kinematic Model: No dynamics, no forces No ‘why’, just ‘what’ Modeling Ocean Currents:Simplest Models

  12. Jets: Narrow, fast currents Meandering Jets: Oscillate in time Eddies: Strong circular currents Modeling Ocean Currents:Simplest Models

  13. Modeling Ocean Currents:Simplest Models Dutkiewicz & Paldor : JPO ‘94 Haller & Poje: NLPG ‘97

  14. Particle Dynamics in a Simple Model

  15. Add Physics: Wind blows on surface F = ma Earth is spinning Ocean is Thin Sheet (Shallow Water Equations) Partial Differential Equations for: (u,v): Velocity in x and y directions (h): Depth of the water layer Modeling Ocean Currents:Dynamic Models

  16. Modeling Ocean Currents:Shallow Water Equations ma = F: Mass Conserved: Non-Linear:

  17. Modeling Ocean Currents:Shallow Water Equations • Channel with Bump • Nonlinear PDE’s: • Solve Numerically • Discretize • Linear Algebra • (Math 335/338) • Input Velocity: Jet • More Realistic (?)

  18. Modeling Ocean Currents:Shallow Water Equations

  19. Modeling Ocean Currents:Complex/Global Models • Add More Physics: • Depth Dependence (many shallow layers) • Account for Salinity and Temperature • Ice formation/melting; Evaporation • Add More Realism: • Realistic Geometry • Outflow from Rivers • ‘Real’ Wind Forcing • 100’s of coupled Partial Differential Equations • 1,000’s of Hours of Super Computer Time

  20. Shallow Water Model b-plane (approx. Sphere) Forced by Trade Winds and Westerlies Complex Models:North Atlantic in a Box

  21. Particle Motion in the Ocean:Mathematically • Particle locations: (x,y) • Change in location is given by velocity of water: (u,v) • Velocity depends on position: (x,y) • Particles start at some initial spot

  22. Particle Motion in the Ocean:Some Blobs S t r e t c h

  23. Dynamical Systems Theory:Geometry of Particle Paths • Currents: Characteristic Structures • Particles:Squeezed in one direction Stretched in another • Answer in Math 330 text!

  24. Dynamical Systems Theory:Hyperbolic Saddle Points Simplest Example:

  25. Dynamical Systems Theory:Hyperbolic Saddle Points

  26. Saddle points appear Saddle points disappear Saddle points move … but they still affect particle behavior North Atlantic in a Box:Saddles Move!

  27. Dynamical Systems Theory:The Theorem • As long as saddles: • don’t move too fast • don’t change shape too much • are STRONG enough • Then there are MANIFOLDS in the flow • Manifolds dictate which particles go where

  28. Main Theorem

  29. Dynamical Systems Theory:Making Manifolds UNSTABLE MANIFOLD: A LINE SEGMENT IS INITIALIZED ON DAY 15 ALONG THE EIGENVECTOR ASSOCIATED WITH THE POSITIVE EIGENVALUE AND INTEGRATED FORWARD IN TIME STABLEMANIFOLD: A LINE SEGMENT IS INITIALIZED ON DAY 60 ALONG THE EIGENVECTOR ASSOCIATED WITH THE NEGATIVEEIGENVALUE AND INTEGRATED BACKWARDIN TIME

  30. Dynamical Systems Theory:Mixing via Manifolds

  31. Dynamical Systems Theory:Mixing via Manifolds

  32. Each saddle has pair of Manifolds Particle flow: IN on Stable Out on Unstable All one needs to know about particle paths (?) North Atlantic in a Box:Manifold Geometry

  33. BLOB HOP-SCOTCH BLOB TRAVELS FROM HIGH MIXING REGION IN THE EAST TO HIGH MIXING REGION IN THE WEST

  34. BLOB HOP-SCOTCH:Manifold Explanation

  35. RING FORMATION • A saddle region appears around day 159.5 • Eddy is formed mostly from the meander water • No direct interaction with outside the jet structures

  36. ABSOLUTELY! Modeling + Numerical Analysis = ‘Ocean’ on Anyone’s Desktop Modeling + Analysis = Predictive Capability (Just when is that Ice Age coming?) Simple Analysis = Implications for Understanding Transport of Ocean Stuff …. and that’s not the half of it …. Summary:Mathematics in the Ocean? April is Math Awareness Month!

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