geophysical fluid dynamics of the earth€¦ · geophysical fluid dynamics of the earth. the earth...
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![Page 1: Geophysical Fluid Dynamics of the Earth€¦ · Geophysical Fluid Dynamics of the Earth. The Earth is a spinning sphere •Coriolis force depends on latitude •solar flux depends](https://reader034.vdocuments.net/reader034/viewer/2022052611/5f08a0767e708231d422f0b3/html5/thumbnails/1.jpg)
Jeffrey B. WeissUniversity of Colorado, Boulder
Geophysical Fluid Dynamics of the Earth
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The Earth is a spinning sphere• Coriolis force depends on latitude
• solar flux depends on latitude
Michael Ritter, http://www.uwsp.edu/geo/faculty/ritter/geog101/textbook/title_page.html
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Atmosphere is turbulent• global scale forcing:
L ~ 107 m• dissipation:
νair ~ 10-5 m2/s
• jet stream velocity: U ~ 102 m/s
• Ratm= 1014
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Direct simulation impossible• Kolmogorov scale
η = 1 mm
• current global models: ~ 1o resolution ~ 100 km
• improvement needed: 108
• computer speed-up: 1032 ~ 2100
• Moore’s Law: 200 years• need to be smarter (or very patient)
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Climate system self-organizes• layers:
troposphere• circulations
Hadley cell• spatially localized organization
ocean vortices• spatio-temporal organization
El-Niño
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Radiative forcing
IPCC
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Atmospheric Layers• troposphere is weather layer• height governed by
radiation convection dynamics
• tropopause dynamics interesting• stratosphere stably stratified
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planetary boundary layer• directly feels surface• 3d turbulence• strong diurnal cycle• surface layer
lower 10% log wind profile
• free atmosphere affected by rotation/stratification
Arya, S.P. Air Pollution Meteorology and Dispersion
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ocean mixed layer
http://hpl.umces.edu/ocean/sml_main.htm
• directly feels atmosphere• mixed by wind• stabilized by solar heating
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deep ocean• thermocline
50 - 1000 m• below stably stratified• slow currents
http://www.windows.ucar.edu/tour
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Large scale circulations• 3-cell model
trades westerlies ITCZ deserts frontal zone
• shifts seasonallyNASA JPL
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wind-driven ocean circulation• winds + basins = gyres
• horizontal• large geostrophic component• timescales: months
geographyalltheway.com
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serendipitous tracers• 1990: 20,000 Nike shoes washed into
Pacific Gyre• 1992: 29,000 bathtub toys• locations from
beachcombers
http://en.wikipedia.org/wiki/Image:Friendly_Floatees.png
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thermohaline circulation• density driven vertical overturning• forcing:
small scale intermittent convection uncertain vertical mixing
• timescale: 1000 yrs• large climate impact
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spatially localized features• Coherent structures
vortices, jets, fronts• rotation and stratification suppress
vortex stretching• inverse cascade to large scales
http://eosweb.larc.nasa.gov/HPDOCS/misr/misr_images/arctic_vortex.jpg
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Spatio-temporal phenomena• El Niño, Monsoons, NAO, MJO, PDO, …• As records lengthen, expect to see more• can have extreme multi-scale components
NOAA
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Monsoon• seasonal circulation associated with land-
sea contrast• complex intermittency• experience the North American Monsoon
burst breakhttp://www.wrh.noaa.gov/fgz/science/monsoon.php?wfo=fgz
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Balances• often find balances between subset of
forces• provide insight into dynamics• departures important even if small• asymptotic analysis gives reduced
models
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Hydrostatic Balance• aspect ratio α = H/L• Froude number Fr = U/NH• non-rotating Boussinesq equations for
departures about basic state b(z)• vertical momentum eqn
• α2 Fr2 << 1 gives hydrostatic balance U2/L2 N2 << 1 tbuoy
2 << tadv2
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• large scale troposphere N ~ 10-2/s, H ~ 10 km, L ~ 1000 km, U ~ 10 m/s Fr ~ 0.1 α ~ 0.01 α2 Fr2 ~ 10-6
fails at fronts and convection• large scale ocean
N ~ 10-2/s, H ~ 1 km, L ~ 1000 km, U ~ 0.1m/s Fr ~ 0.01 α ~ 0.001 α2 Fr2 ~ 10-10
hydrostatic down to small horizontal scales fails at deep convection sites
• preconditioning: Fr near unity• plumes: H/L large
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Geostrophic balance• Rossby number: Ro = U/ f L• horizontal momentum eqn
scaling:
• Ro << 1: geostrophy• flow along isobars• diagnostic relation
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upper air map
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thermal wind• combine geostrophic and hydrostatic• obtain wind shear
• midlatitudes heating decreases with latitude eastward wind increases with height jet intensification
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turbulent cascades• compare homogeneous isotropic 3d and 2d• energy injection at large scale• dissipation at small scale• intermediate: inertial scales, only depend on
energy flux ε• focus on spectra:• assume local in k and universal
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• dimensional analysis
• T requires α = 2/3 L requires β = −5/3
Vallis, Atmospheric and Oceanic Fluid Dynamics, 2006
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2d cascades• In 2d, two inviscid invariants
energy E and enstrophy Z in 3d vortex stretching creates enstrophy Z(k) = k2 E(k)
• centroid wavenumber• width• initial narrow E(k) spreads
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• energy cascades to large scales: inverse
• similar argument: enstrophy to smallscales: direct
Vallis, Atmospheric and Oceanic Fluid Dynamics, 2006
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• energy inertial range to large scales• enstrophy inertial range to small scales• energy inertial ranges same scaling as 3d• need mechanism to remove energy at
large scales, e.g. Ekman drag• enstrophy flux
η ~ Z/T ~1/T3
E(k) ~ η2/3 k-3
Vallis, Atmospheric and Oceanic Fluid Dynamics, 2006
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cascades on 2d β-plane• vorticity equation on 2d β-plane
f = β y
• Rossby waves
• dispersion relation
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• scaling
• small scales advection and turbulence dominate
• crossover: Rhines scale
• large scales, β dominates waves inhibit cascades
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• anisotropy of Rossby waves advection freq Uk ~ Rossby wave freq ω
- 1.5 -1 . 0 -0 . 5 0.0 0.5 1.0 1.5
- 1.5
- 1.0
- 0.5
0.0
0.5
1.0
1.5
waves
turbulence
Vallis, Atmospheric and Oceanic Fluid Dynamics, 2006
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ocean jets• zonal jets observed to populate ocean
Maximenko, et al 2005
instantaneous
18 week avg
200 week avg