Mathematics in the Ocean Andrew Poje Mathematics Department

Mathematics in the Ocean • Andrew Poje Mathematics Department College of Staten Island • • • April is Math Awareness Month M. Toner U. Delaware A. D. Kirwan, Jr. G. Haller C. K. R. T. Jones L. Kuznetsov … and many more! 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 • • • image from Unisys Inc. (weather. unisys. com) West Broad and South in East Gulf Stream warms Europe Kuroshio warms Seattle

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 image from Southhampton Ocean Centre: . http: //www. soc. soton. ac. uk/JRD/OCCAM transport properties?

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

Coherent Structures: Eddies, Meddies, Rings & Jets

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

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 è è 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 è (h): Depth of the water layer è

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

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

Modeling Ocean Currents: Shallow Water Equations

Modeling Ocean Currents: Complex/Global Models • Add More Physics: è è è Depth Dependence (many shallow layers) Account for Salinity and Temperature Ice formation/melting; Evaporation • Add More Realism: è è è Realistic Geometry Outflow from Rivers ‘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 b-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: Theorem • As long as saddles: don’t move too fast è don’t change shape too much è are STRONG enough è • Then there are MANIFOLDS in the flow • Manifolds dictate which particles go where

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 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: Mixing via 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 159. 5 • 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!