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

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

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)

Drifters and Floats: Measuring Ocean Currents

Particle (Sneaker) Motion in the Ocean

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

Ocean Currents: Time Dependence Global Ocean Models: è Math Modeling è Numerical Analysis è Scientific Programing Results: è Highly Variable Currents è Complex Flow Structures How do these effect transport properties? image from Southhampton Ocean Centre:.

Coherent Structures: Eddies, Meddies, Rings & Jets Flow Structures responsible for Transport Exchange: è Water è Heat è Pollution è Nutrients è Sea Life How Much? Which Parcels? image from Southhampton Ocean Centre:.

Coherent Structures: Eddies, Meddies, Rings & Jets

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

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

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

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

Particle Dynamics in a Simple Model

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

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

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

Modeling Ocean Currents: Shallow Water Equations

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

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

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

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

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

Dynamical Systems Theory: Hyperbolic Saddle Points Simplest Example:

Dynamical Systems Theory: Hyperbolic Saddle Points

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

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

UNSTABLE MANIFOLD: A LINE SEGMENT IS INITIALIZED ON DAY 15 ALONG THE EIGENVECTOR ASSOCIATED WITH THE POSITIVE EIGENVALUE AND INTEGRATED FORWARD IN TIME STABLE MANIFOLD: A LINE SEGMENT IS INITIALIZED ON DAY 60 ALONG THE EIGENVECTOR ASSOCIATED WITH THE NEGATIVE EIGENVALUE AND INTEGRATED BACKWARD IN TIME Dynamical Systems Theory: Making Manifolds

Dynamical Systems Theory: Mixing via Manifolds

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

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

BLOB HOP-SCOTCH: Manifold Explanation

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

Summary: Mathematics in the Ocean? 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 …. April is Math Awareness Month!