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Pete BoslerPete Bosler
Modeling Geophysical Fluid
Flows
Modeling Geophysical Fluid
Flows
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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
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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
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Data InputData Input
Over LandSatellitesAirports and automated stations
Maritime: very sparse dataSatellitesShip observationsIslands
Over LandSatellitesAirports and automated stations
Maritime: very sparse dataSatellitesShip observationsIslands
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600 nm
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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
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North Wall
Warm Eddy
Cold Eddies
Jump ExampleJump Example
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= 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)
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Lorenz AttractorLorenz Attractor
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Shock ExampleShock Example
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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 =
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Burgers EquationBurgers Equation
€
∂u∂t
+ u∂u
∂x= 0
u = u(x, t)
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Singularity ExampleSingularity Example
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Where to go next?Where to go next?
Level Set Methodshttp://physbam.stanford.edu/
~fedkiw/
Level Set Methodshttp://physbam.stanford.edu/
~fedkiw/
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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”