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

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

4. 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)

5. Drifters and Floats:Measuring Ocean Currents

6. Particle (Sneaker) Motion in the Ocean

7. 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

8. 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

9. 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

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

11. 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

12. 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

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

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

15. Particle Dynamics in a Simple Model

16. 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

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

18. 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 (?)

19. Modeling Ocean Currents:Shallow Water Equations

20. 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

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

22. 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

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

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

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

26. Dynamical Systems Theory:Hyperbolic Saddle Points

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

28. 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

29. Main Theorem

30. 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

31. Dynamical Systems Theory:Mixing via Manifolds

32. Dynamical Systems Theory:Mixing via Manifolds

33. 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

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

35. BLOB HOP-SCOTCH:Manifold Explanation

36. 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

37. 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!