1. 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

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:.

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:.

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

10. Mathematics in the Ocean: Overview
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

11. Modeling Ocean Currents: Simplest 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’

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

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

14. Particle Dynamics in a Simple Model

15. Modeling Ocean Currents: Dynamic 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

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

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

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
STABLE MANIFOLD:
A LINE SEGMENT
IS INITIALIZED ON DAY 60
ALONG THE EIGENVECTOR
ASSOCIATED WITH THE
NEGATIVE EIGENVALUE
AND INTEGRATED
BACKWARD IN TIME

30. Dynamical Systems Theory: Mixing via Manifolds

31. Dynamical Systems Theory: Mixing via Manifolds

32. 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 (?)

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. 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!