a deterministic view of modeling of the gulf of mexico guillaume vernieres (samsi/unc)

A Deterministic View of Modeling of the Gulf of Mexico

Guillaume Vernieres (SAMSI/UNC)

Outline

• Motivations• Some physical background• Mathematical formulation of the problem• Results• …That’s it …

Motivations

• Why do we care?

http://www.camex4.com/photos/Ivan.A2004258.1635.2km.jpg

Motivations

• Why do we care?

HURRICANE TRACK PREDICTION !!!!!!!!!

Motivations

• Why do we care?

Test bed for modeling methods

Physical background

• Ocean currents

http://www.waterencyclopedia.com/images/wsci_03_img0381.jpg

Physical background

• Global Wind

http://research.utep.edu/Portals/72/weather%20NOAA/global%20wind.gif

Physical background

• The Gulf Stream

Physical background

• The Gulf of Mexico: Shedding of eddies

Sea Surface Height in cm

Physical background

• The Gulf of Mexico: Shedding of eddies

Sea Surface Temperature

Mathematical formulation of the problem

Mathematical formulation of the problem

Mathematical formulation of the problem

Mathematical formulation of the problem

Simple conservation laws:

Mathematical formulation of the problem

Simple conservation laws:• Conservation of mass

Mathematical formulation of the problem

Simple conservation laws:• Conservation of mass

=

Mathematical formulation of the problem

Simple conservation laws:• Conservation of mass• Conservation of momentum

Mathematical formulation of the problem

Simple conservation laws:• Conservation of mass• Conservation of momentum• Rotating frame!! (yes the earth is turning!)

Mathematical formulation of the problem

Simple conservation laws:• Conservation of mass• Conservation of momentum• Rotating frame!! (yes the earth is turning!)• Hydrostatic pressure

Mathematical formulation of the problem

Simple conservation laws:• Conservation of mass• Conservation of momentum• Rotating frame!! (yes the earth is turning!)• Hydrostatic pressure• Neglect thermodynamics

Mathematical formulation of the problem

Simple conservation laws:• Conservation of mass• Conservation of momentum• Rotating frame!! (yes the earth is turning!)• Hydrostatic pressure• Neglect thermodynamics• L>>D

Mathematical formulation of the problem

Simple conservation laws:• Conservation of mass• Conservation of momentum• Rotating frame!! (yes the earth is turning!)• Hydrostatic pressure• Neglect thermodynamics• L>>D

Similar to the Navier-Sokes equations

Mathematical formulation of the problem

0

0

u

gz

M

uMukfz

uwuut

uHH

HHH

H

x & y momentum

Mathematical formulation of the problem

0

0

u

gz

M

uMukfz

uwuut

uHH

HHH

H

Hydrostatic assumption

Mathematical formulation of the problem

0

0

u

gz

M

uMukfz

uwuut

uHH

HHH

H

Continuity equation(conservation of mass)

Mathematical formulation of the problem

0

0

u

gz

M

uMukfz

uwuut

uHH

HHH

H

)),,,(),,,,(),,,,(()),,,(),,,,((

tzyxwtzyxvtzyxuutzyxvtzyxuuH

Mathematical formulation of the problem

0

0

u

gz

M

uMukfz

uwuut

uHH

HHH

H

Can be further simplified !!

Mathematical formulation of the problem

z

∞

Mathematical formulation of the problem

z

u1=u1(x,y,t) ρ1=cst

u2=u2(x,y,t) ρ2=cst>ρ1

∞

Mathematical formulation of the problem

),,(,...,1 )),,(),,,((

0)(

tyxhhNntyxvtyxuu

hut

h

uMukfdtud

nn

nnn

nnn

nnnn

Shallow water equations

),,(,...,1 )),,(),,,((

0)(

tyxhhNntyxvtyxuu

hut

h

uMukfdtud

nn

nnn

nnn

nnnn

•Discretized in space using FiniteDifference

ζ

η

x(ζ, η)=?y(ζ, η)=?

),,(,...,1 )),,(),,,((

0)(

tyxhhNntyxvtyxuu

hut

h

uMukfdtud

nn

nnn

nnn

nnnn

•Discretized in space using FiniteDifference•Discretized in time using Adams-Bashforth 2nd order

22500 grid points x

3 layersx

3 state variables (u,v,h)/layer=

202500 ODE’s

Some Results:

Eulerian and Lagrangian results

Can real drifter location be used to forecast the state of the GoM ?

How much information is contained in one single drifter ?

Higher Re

“Influence of a drifter on the state of the GoM”

“Influence of a drifter on the state of the GoM”