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AOSS 401, Fall 2007 Lecture 22 November 02 , 2007

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## AOSS 401, Fall 2007 Lecture 22 November 02 , 2007

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**AOSS 401, Fall 2007Lecture 22November 02, 2007**Richard B. Rood (Room 2525, SRB) rbrood@umich.edu 734-647-3530 Derek Posselt (Room 2517D, SRB) dposselt@umich.edu 734-936-0502**Class News November 02, 2007**• Homework 5 (Due Monday) • Posted to web • Computing assignment posted to ctools under the Homework section of Resources • Next Test: November 16**Seminars Today**• Professor Cecilia Bitz is giving the Dept. of Geological Sciences’ Smith Lecture on Friday (tomorrow, Nov. 2) from 4-5pm. The lecture is held in room 1528 in C.C. Little and is followed by a reception. Cecilia is an expert in high-latitude climate, climate change and variability. The title of her lecture will be: • “Future thermohaline collapse and its impact are unlike the past”**Seminars Today**• Dr. Guy Brasseur - Professor and Associate Director, National Center for Atmospheric Research and Director of the Earth and Sun Systems Laboratory will visit U of M on Thursday/Friday. • Impact of solar variability and anthropogenic forcing on the whole atmosphere: Simulations with the HAMMONIA Model • Friday, November 2, 3:30 pm -- refreshments at 3 pm North Campus AOSS Auditorium, Room #2246**Weather**• National Weather Service • http://www.nws.noaa.gov/ • Model forecasts: http://www.hpc.ncep.noaa.gov/basicwx/day0-7loop.html • Weather Underground • http://www.wunderground.com/cgi-bin/findweather/getForecast?query=ann+arbor • Model forecasts: http://www.wunderground.com/modelmaps/maps.asp?model=NAM&domain=US**Material from Chapter 6**• Quasi-geostrophic theory • Quasi-geostrophic vorticity • Relation between vorticity and geopotential**Mathematics**• Remember the idea that mathematics is a language to use to help us explore a complex system. • Verb: equal, • Qualification: not equal, greater than, less than, approximately • We own the equations and can do to them what we want, as long as we remember equal and not equal.**Tangential coordinate system**Place a coordinate system on the surface. x = east – west (longitude) y = north – south (latitude) z = local vertical or p = local vertical Ω R=acos(f) R a Φ Earth**Tangential coordinate system**Relation between latitude, longitude and x and y dx = acos(f)dl l is longitude dy = adf f is latitude dz = dr r is distance from center of a “spherical earth” f=2Ωsin(f) Ω b=2Ωcos(f)/a R a Φ Earth**Equations of motion in pressure coordinates(using Holton’s**notation)**Scaled equations of motion in pressure coordinates**Definition of geostrophic wind Momentum equation Continuity equation ThermodynamicEnergy equation**Approximate horizontal momentum equation**• This equation states that the time rate of change of the geostrophic wind is related to • the coriolis force due to the ageostrophic wind and • the part of the coriolis force due to the variability of the coriolis force with latitude and the geostrophic wind. • Both of these terms are smaller than the geostrophic wind itself.**Derived a vorticity equation**• Provides a “suitable” prognostic equation because need to include div(ageostrophic wind) in the prognostics. • Remember the importance of divergence in vorticity equations.**One interesting way to rewrite this equation**Expand material derivative**One interesting way to rewrite this equation**Equation of continuity Understand how this is equivalent**One interesting way to rewrite this equation**Advection of vorticity**One interesting way to rewrite this equation**Advection of vorticity Advection of relative vorticity Advection of planetary vorticity**Geopotential Map (Northern Hemisphere)**Where is geostrophic approximation valid? What other force balance is important? What is the sign of the geostrophic wind? ٠ ΔΦ > 0 B Φ0 - ΔΦ L L H Φ0 ٠ ٠ y, north Φ0 + ΔΦ A C x, east**Geostrophic wind**٠ ΔΦ > 0 vg < 0 vg > 0 B Φ0 - ΔΦ L L H Φ0 ٠ ٠ y, north Φ0 + ΔΦ A C x, east**Geopotential Map (Northern Hemisphere)**What is the sign of planetary vorticity? What is the sign of the relative vorticity? ٠ ΔΦ > 0 B Φ0 - ΔΦ L L H Φ0 ٠ ٠ y, north Φ0 + ΔΦ A C x, east**Vorticity**ζ < 0; anticyclonic ٠ ΔΦ > 0 β > 0 β > 0 B Φ0 - ΔΦ L L H Φ0 ٠ ٠ y, north Φ0 + ΔΦ A C x, east ζ > 0; cyclonic ζ > 0; cyclonic**Advection of planetary vorticity**ζ < 0; anticyclonic ٠ ΔΦ > 0 vg < 0 ; β > 0 vg > 0 ; β > 0 B Φ0 - ΔΦ L L H Φ0 ٠ ٠ y, north Φ0 + ΔΦ A C x, east ζ > 0; cyclonic ζ > 0; cyclonic**Advection of planetary vorticity**ζ < 0; anticyclonic ٠ ΔΦ > 0 -vgβ > 0 -vgβ < 0 B Φ0 - ΔΦ L L H Φ0 ٠ ٠ y, north Φ0 + ΔΦ A C x, east ζ > 0; cyclonic ζ > 0; cyclonic**Advection of relative vorticity**ζ < 0; anticyclonic ٠ ζ from >0 to <0 vg > 0; ug > 0 ΔΦ > 0 ζ from <0 to >0 vg < 0; ug > 0 B Φ0 - ΔΦ L L H Φ0 ٠ ٠ y, north Φ0 + ΔΦ A C x, east ζ > 0; cyclonic ζ > 0; cyclonic**Advection of relative vorticity**ζ < 0; anticyclonic ٠ Advection of ζ > 0 ΔΦ > 0 Advection of ζ < 0 B Φ0 - ΔΦ L L H Φ0 ٠ ٠ y, north Φ0 + ΔΦ A C x, east ζ > 0; cyclonic ζ > 0; cyclonic**Advection of vorticity**ζ < 0; anticyclonic Advection of ζ > 0 Advection of f < 0 Advection of ζ < 0 Advection of f > 0 ٠ ΔΦ > 0 B Φ0 - ΔΦ L L H Φ0 ٠ ٠ y, north Φ0 + ΔΦ A C x, east ζ > 0; cyclonic ζ > 0; cyclonic**Summary: Vorticity Advection in Wave**• Planetary and relative vorticity advection in a wave oppose each other. • This is consistent with the balance that we intuitively derived from the conservation of absolute vorticity over the mountain.**Summary: Vorticity Advection in Wave**• What does this do to the wave.**Advection of vorticity**ζ < 0; anticyclonic Advection of ζ tries to propagate the wave this way ٠ ΔΦ > 0 B Φ0 - ΔΦ L L H Φ0 Advection of f tries to propagate the wave this way ٠ ٠ y, north Φ0 + ΔΦ A C x, east ζ > 0; cyclonic ζ > 0; cyclonic**Wave like solutions**Dispersion relation. Relates frequency and wave number to flow. Must be true for waves.**Stationary wave**Wind must be positive, from the west, for a wave.**Advection of vorticity**ζ < 0; anticyclonic Advection of ζ tries to propagate the wave this way ٠ ΔΦ > 0 B Φ0 - ΔΦ L L H Φ0 Advection of f tries to propagate the wave this way ٠ ٠ y, north Φ0 + ΔΦ A C x, east ζ > 0; cyclonic ζ > 0; cyclonic**Compare advection of planetary and relative vorticity** Short waves, advection of relative vorticity is larger Long waves, advection of planetary vorticity is larger **Advection of vorticity**ζ < 0; anticyclonic Short waves ٠ ΔΦ > 0 B Φ0 - ΔΦ L L H Φ0 • Long waves ٠ ٠ y, north Φ0 + ΔΦ A C x, east ζ > 0; cyclonic ζ > 0; cyclonic