Mathematics in the Ocean

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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!

U. Delaware
Brown U.
April is Math Awareness Month
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Why Study the Ocean?

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

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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)
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Drifters and FloatsMeasuring Ocean Currents
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Particle (Sneaker) Motion in the Ocean
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Particle Motion in the OceanMathematically

• 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

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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
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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
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Coherent Structures Eddies, Meddies, Rings
Jets
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Mathematics in the OceanOverview

• 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

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Modeling Ocean CurrentsSimplest Models

• 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

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Modeling Ocean CurrentsSimplest Models

• Jets Narrow, fast currents
• Meandering Jets Oscillate in time
• Eddies Strong circular currents

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Modeling Ocean CurrentsSimplest Models
Dutkiewicz Paldor JPO 94 Haller Poje
NLPG 97
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Particle Dynamics in a Simple Model
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Modeling Ocean CurrentsDynamic Models

• 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

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Modeling Ocean CurrentsShallow Water Equations
ma F
Mass Conserved
Non-Linear
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Modeling Ocean CurrentsShallow Water Equations

• Channel with Bump
• Nonlinear PDEs
• Solve Numerically
• Discretize
• Linear Algebra
• (Math 335/338)
• Input Velocity Jet
• More Realistic (?)

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Modeling Ocean CurrentsShallow Water Equations
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Modeling Ocean CurrentsComplex/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
• 100s of coupled Partial Differential Equations
• 1,000s of Hours of Super Computer Time

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Complex ModelsNorth Atlantic in a Box

• Shallow Water Model
• b-plane (approx. Sphere)
• Forced by Trade Winds and Westerlies

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Particle Motion in the OceanMathematically

• 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

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Particle Motion in the OceanSome Blobs S t r
e t c h
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Dynamical Systems TheoryGeometry of Particle
Paths

• Currents Characteristic Structures
• Particles Squeezed in one
  direction Stretched in another
• Answer in Math 330 text!

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Dynamical Systems TheoryHyperbolic Saddle Points
Simplest Example
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Dynamical Systems TheoryHyperbolic Saddle Points
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North Atlantic in a BoxSaddles Move!

• Saddle points appear
• Saddle points disappear
• Saddle points move
• but they still affect particle behavior

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Dynamical Systems TheoryThe Theorem

• As long as saddles
• dont move too fast
• dont change shape too much
• are STRONG enough
• Then there are MANIFOLDS in the flow
• Manifolds dictate which particles go where

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Main Theorem
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Dynamical Systems TheoryMaking 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
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Dynamical Systems TheoryMixing via Manifolds
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Dynamical Systems TheoryMixing via Manifolds
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North Atlantic in a BoxManifold Geometry

• Each saddle has pair of Manifolds
• Particle flow IN on Stable
  Out on Unstable
• All one needs to know about particle paths (?)

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BLOB HOP-SCOTCH
BLOB TRAVELS FROM HIGH MIXING REGION IN THE EAST
TO HIGH MIXING REGION IN THE WEST
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BLOB HOP-SCOTCHManifold Explanation
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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
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SummaryMathematics in the Ocean?

• ABSOLUTELY!
• Modeling Numerical Analysis Ocean on
  Anyones Desktop
• Modeling Analysis Predictive Capability
  (Just when is that Ice Age coming?)
• Simple Analysis Implications for Understanding
  Transport of Ocean Stuff
• . and thats not the half of it .

April is Math Awareness Month!