Mathematics Mathematics in the Oceanin 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

( )( ) 0

0

0

0

),(

),(

yty

xtx

yxvdt

dy

yxudt

dx

==

==

=

=

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

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

Coherent Structures: Eddies, Meddies, Rings & Jets

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

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’

Modeling Ocean Currents:Simplest Models

• Jets: Narrow, fast currents

• Meandering Jets: Oscillate in time

• Eddies: Strong circular currents

{ }( )( ) ( ){ }( )

( ) ( )x

tyxvy

tyxu

yyxxAyx

LctkxyKtyx

eddyjeteddyjet

eddyeddyeddy

jet

∂

Ψ+Ψ∂=

∂

Ψ+Ψ∂−=

−+−−=Ψ

−−=Ψ

),,(,),,(

exp),( :Eddy

)sin(tanh),,( :Jet22α

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: 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:Shallow Water Equations

( ) ( )

( ) ( ) ( )y

vx

uDt

D

y

v

x

uhhhh

Dt

D

tyxWyy

v

xx

v

y

hgfuv

Dt

D

tyxWyy

u

xx

u

x

hgfvu

Dt

D

bb

ye

xe

∂

∂+

∂

∂=

=⎟⎟⎠

⎞⎜⎜⎝

⎛∂

∂+

∂

∂−+−

+⎟⎟⎠

⎞⎜⎜⎝

⎛

∂∂

∂+

∂∂

∂+

∂

∂−=+

+⎟⎟⎠

⎞⎜⎜⎝

⎛

∂∂

∂+

∂∂

∂+

∂

∂−=−

0

),,('

),,('

22

22

υ

υ

ma = F:

Mass Conserved:

Non-Linear:

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 (?)

Modeling Ocean Currents:Shallow Water Equations

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

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

( )( ) 0

0

0

0

),(

),(

yty

xtx

yxvdt

dy

yxudt

dx

==

==

=

=

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! xdt

dy

ydt

dx

=

=

:Example Simplest

Dynamical Systems Theory:Hyperbolic Saddle Points

)exp(1

1)exp(

1

1)(

01

10

)(

)()(

21 tctctX

Xdt

dX

ty

txtX

−⎟⎟⎠

⎞⎜⎜⎝

⎛−

+⎟⎟⎠

⎞⎜⎜⎝

⎛=

⎟⎟⎠

⎞⎜⎜⎝

⎛=

⎟⎟⎠

⎞⎜⎜⎝

⎛=

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:don’t move too fastdon’t change shape too much 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 15ALONG THE EIGENVECTOR ASSOCIATED WITH THEPOSITIVE EIGENVALUE AND INTEGRATED FORWARD IN TIME

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

Dynamical Systems Theory:Making Manifolds

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 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 withoutside 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!