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## A Deterministic View of Modeling of the Gulf of Mexico

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**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**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**x & y momentum**Mathematical formulation of the problem**Hydrostatic assumption**Mathematical formulation of the problem**Continuity equation (conservation of mass)**Mathematical formulation of the problem**Can be further simplified !!**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**Shallow water equations**η**x(ζ, η)=? y(ζ, η)=? ζ**Discretized in space using FiniteDifference**• Discretized in time using Adams-Bashforth • 2nd order**22500 grid points**x 3 layers x 3 state variables (u,v,h)/layer = 202500 ODE’s**Can real drifter location be used to forecast the state of**the GoM ?