Pete BoslerPete Bosler

Modeling Geophysical Fluid

Flows

Modeling Geophysical Fluid

Flows

OverviewOverview

“Geophysical Fluid Flow”Ocean & Atmosphere

Physical oceanography and meteorologyAcross spatial scales of O(10 m) to O(1000

km) Modeling

Deriving & SimplifyingNumerical solutions

Application and use of modelingForecasts

“Geophysical Fluid Flow”Ocean & Atmosphere

Physical oceanography and meteorologyAcross spatial scales of O(10 m) to O(1000

km) Modeling

Deriving & SimplifyingNumerical solutions

Application and use of modelingForecasts

State of models todayState of models today

Global ModelsWorld Meteorological Association

Ex: NOGAPS, GFS

Regional ModelsBetter resolution

Can resolve smaller scale phenomenaMore realistic topographic interaction

Boundary conditions are an added issue

Global ModelsWorld Meteorological Association

Ex: NOGAPS, GFS

Regional ModelsBetter resolution

Can resolve smaller scale phenomenaMore realistic topographic interaction

Boundary conditions are an added issue

Data InputData Input

Over LandSatellitesAirports and automated stations

Maritime: very sparse dataSatellitesShip observationsIslands

Over LandSatellitesAirports and automated stations

Maritime: very sparse dataSatellitesShip observationsIslands

600 nm

MathematicsMathematics

Physics of these fluids can turn out to be “not nice.”Sensitive dependence on initial

conditionsChaotic dymanics

Discontinuities may ariseJumpsShocksSingularities

Physics of these fluids can turn out to be “not nice.”Sensitive dependence on initial

conditionsChaotic dymanics

Discontinuities may ariseJumpsShocksSingularities

North Wall

Warm Eddy

Cold Eddies

Jump ExampleJump Example

= Stream Function

=Temperature perturbation

Convection in a slabConvection in a slab

€

∂∂t∇ 2ψ −

∂ψ

∂z

∂

∂x∇ 2ψ − gε

∂θ

∂x−ν∇ 4ψ = 0

∂θ

∂t−∂ψ

∂z

∂θ

∂x+∂ψ

∂x

∂θ

∂z−

ΔT0

H

∂ψ

∂x−κ∇ 2θ = 0

€

ψ(x,z, t)

θ(x,z, t)

Lorenz AttractorLorenz Attractor

Shock ExampleShock Example

Updraft Velocity

Rainwater Mixing Ratio

Virtual temperature excess

“Generation Parameter”

Downward velocity of raindrops

Precipitation vs. UpdraftPrecipitation vs. Updraft

€

∂U∂t

+U∂U

∂z= g

ΔT

T− R

⎛

⎝ ⎜

⎞

⎠ ⎟

∂R

∂t+ U −Vc( )

∂R

∂z=UG + RVc

1

ρ

dp

dz

€

U(z, t) =

R(z, t) =

ΔT(z) =

G(z) =

Vc =

Burgers EquationBurgers Equation

€

∂u∂t

+ u∂u

∂x= 0

u = u(x, t)

Singularity ExampleSingularity Example

Where to go next?Where to go next?

Level Set Methodshttp://physbam.stanford.edu/

~fedkiw/

Level Set Methodshttp://physbam.stanford.edu/

~fedkiw/

References/Additional ReadingReferences/Additional Reading Davis, 1988, “Simplified second order Godunov-type

methods” Gottleib & Orszag, 1987, “Numerical Analysis of Spectral

Methods” Lorenz, 1963, “Deterministic Nonperiodic Flow” Leveque, 2005, “Numerical Methods for Conservation Laws” Malek-Madani, 1998, “Advanced Engineering Mathematics” Rogers & Yau, 1989,“A Short Course in Cloud Physics” Saltzman, 1962, “Finite amplitude free convection as an

initial value problem” Smoller, 1994, “Shock Waves and Reaction-Diffusion

Equations” Srivastava, 1967, “A study of the effect of precipitation on

cumulus dynamics”

Davis, 1988, “Simplified second order Godunov-type methods”

Gottleib & Orszag, 1987, “Numerical Analysis of Spectral Methods”

Lorenz, 1963, “Deterministic Nonperiodic Flow” Leveque, 2005, “Numerical Methods for Conservation Laws” Malek-Madani, 1998, “Advanced Engineering Mathematics” Rogers & Yau, 1989,“A Short Course in Cloud Physics” Saltzman, 1962, “Finite amplitude free convection as an

initial value problem” Smoller, 1994, “Shock Waves and Reaction-Diffusion

Equations” Srivastava, 1967, “A study of the effect of precipitation on

cumulus dynamics”