AOSS 401, Fall 2007 Lecture 23 November 05, 2007 Richard B. Rood (Room 2525, SRB) Derek Posselt (Room 2517D, SRB)
Class News November 05, 2007 Homework 6 (Posted this evening) –Due Next Monday Important Dates: –November 16: Next Exam (Review on 14 th ) –November 21: No Class –December 10: Final Exam
Weather National Weather Service – –Model forecasts: 7loop.html 7loop.html Weather Underground – bin/findweather/getForecast?query=ann+arborhttp:// bin/findweather/getForecast?query=ann+arbor –Model forecasts: ?model=NAM&domain=US ?model=NAM&domain=US
Couple of Links you should know about –Library electronic journals it=Yes&SID=4Ajed7dbJbeGB3KcpBhhttp://portal.isiknowledge.com/portal.cgi?In it=Yes&SID=4Ajed7dbJbeGB3KcpBh –Web o’ Science
Material from Chapter 6 Quasi-geostrophic theory Quasi-geostrophic vorticity –Relation between vorticity and geopotential Geopotential prognostic equation Relationship to mid-latitude cyclones
One interesting way to rewrite this equation Advection of vorticity
Let’s take this to the atmosphere
Advection of planetary vorticity Φ 0 - Δ Φ Φ 0 + Δ Φ Φ0Φ0 Δ Φ > 0 A B C ٠ ٠ ٠ x, east y, north LL H ζ < 0; anticyclonic ζ > 0; cyclonic v g > 0 ; β > 0 v g 0
Advection of planetary vorticity Φ 0 - Δ Φ Φ 0 + Δ Φ Φ0Φ0 Δ Φ > 0 A B C ٠ ٠ ٠ x, east y, north LL H ζ < 0; anticyclonic ζ > 0; cyclonic -v g β < 0 -v g β > 0
Advection of relative vorticity Φ 0 - Δ Φ Φ 0 + Δ Φ Φ0Φ0 Δ Φ > 0 A B C ٠ ٠ ٠ x, east y, north LL H ζ < 0; anticyclonic ζ > 0; cyclonic Advection of ζ > 0 Advection of ζ < 0
Advection of vorticity Φ 0 - Δ Φ Φ 0 + Δ Φ Φ0Φ0 Δ Φ > 0 A B C ٠ ٠ ٠ x, east y, north LL H ζ < 0; anticyclonic ζ > 0; cyclonic Advection of ζ > 0 Advection of f < 0 Advection of ζ < 0 Advection of f > 0
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.
Advection of vorticity Φ 0 - Δ Φ Φ 0 + Δ Φ Φ0Φ0 Δ Φ > 0 A B C ٠ ٠ ٠ x, east y, north LL H ζ < 0; anticyclonic ζ > 0; cyclonic Advection of ζ tries to propagate the wave this way Advection of f tries to propagate the wave this way
Geopotential Nuanced
Assume that the geopotential is a wave
Remember the relation to geopotential
Advection of relative vorticity
Advection of planetary vorticity
Compare advection of planetary and relative vorticity
Advection of vorticity Φ 0 - Δ Φ Φ 0 + Δ Φ Φ0Φ0 Δ Φ > 0 A B C ٠ ٠ ٠ x, east y, north LL H ζ < 0; anticyclonic ζ > 0; cyclonic Advection of ζ tries to propagate the wave this way Advection of f tries to propagate the wave this way
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 Φ 0 - Δ Φ Φ 0 + Δ Φ Φ0Φ0 Δ Φ > 0 A B C ٠ ٠ ٠ x, east y, north LL H ζ < 0; anticyclonic ζ > 0; cyclonic Short waves Long waves
Go to the real atmosphere
An estimate of the January mean zonal wind north winter south summer -- u
Advection of relative vorticity for our idealized wave
An estimate of the January mean zonal wind north winter south summer What is the difference in the advection of vorticity at the two levels?
An estimate of the January mean zonal wind
Vertical Structure The waves propagate at different speeds at different altitudes. The waves do not align perfectly in the vertical. (This example shows that there is vertical structure, but it is only a (small) part of the story.)
A more general equation for geopotential
An equation for geopotential tendency
Another interesting way to rewrite vorticity equation (Flirting with) An equation for geopotential tendency An equation in geopotential and omega. (2 unknowns, 1 equation)
Quasi-geostrophic Geostrophic ageostrophic
Previous analysis In our discussion of the advection of vorticity, we completely ignored the term that had the vertical velocity. Go back to our original vorticity equation –Tilting –Divergence –Thermodynamic... (solenoidal, baroclinic) Which still exist after our scaling and assumptions?
We used these equations to get previous equation for geopotential tendency
Now let’s use this equation
Rewrite the thermodynamic equation to get geopotential tendency
Rewrite this equation to relate to our first equation for geopotential tendency.
Scaled equations of motion in pressure coordinates Note this is, through continuity, related to the divergence of the ageostrophic wind Note that it is the divergence of the horizontal wind, which is related to the vertical wind, that links the momentum (vorticity equation) to the thermodynamic equation
Scaled equations of motion in pressure coordinates Note that this looks something like the time rate of change of static stability
Explore this a bit. So this is a measure of how far the atmosphere moves away from its background equilibrium state
Add these equations to eliminate omega and we have a partial differential equation for geopotential tendency (assume J=0)
Vorticity Advection
Add these equations to eliminate omega and we have a partial differential equation for geopotential tendency (assume J=0) Thickness Advection
How do you interpret this figure in terms of geopotential? Φ 0 - Δ Φ Φ 0 + Δ Φ Φ0Φ0 Δ Φ > 0 A B C ٠ ٠ ٠ x, east y, north LL H ζ < 0; anticyclonic ζ > 0; cyclonic Short waves Long waves
Add these equations to eliminate omega and we have a partial differential equation for geopotential tendency (assume J=0) This is, in fact, an equation that given a geopotential distribution at a given time, then it is a linear partial differential equation for geopotential tendency. Right hand side is like a forcing. You now have a real equation for forecasting the height (the pressure field), and we know that the pressure gradient force is really the key, the initiator, of motion.
Add these equations to eliminate omega and we have a partial differential equation for geopotential tendency (assume J=0) An equation like this was very important for weather forecasting before we had comprehensive numerical models. It is still important for field forecasting, and knowing how to adapt a forecast to a particular region given, for instance, local information.
Think about thickness advection Thickness Advection
Weather National Weather Service – –Model forecasts: 7loop.html 7loop.html Weather Underground – bin/findweather/getForecast?query=ann+arborhttp:// bin/findweather/getForecast?query=ann+arbor –Model forecasts: ?model=NAM&domain=US ?model=NAM&domain=US
Cold and warm advection cold warm
Question What happens when warm air is advected towards cool air? COOLWARM
Question What happens when warm air is advected towards cool air? COOL WARM
Question What happens the warm air? –Tell me at least two things. COOL WARM
Add these equations to eliminate omega and we have a partial differential equation for geopotential tendency (assume J=0) Thickness Advection
Lifting and sinking
Add these equations to eliminate omega and we have a partial differential equation for geopotential tendency (assume J=0) Thickness Advection
A nice schematic artments/geography/nottingham/atmosphe re/pages/depressionsalevel.htmlhttp://atschool.eduweb.co.uk/kingworc/dep artments/geography/nottingham/atmosphe re/pages/depressionsalevel.html
More in the atmosphere (northern hemisphere) South North Warm Cool Temperature What can you say about the wind?
Idealized vertical cross section
Increasing the pressure gradient force
Relationship between upper troposphere and surface divergence over low enhances surface low // increases vorticity
Relationship between upper troposphere and surface vertical stretching // increases vorticity
Relationship between upper troposphere and surface vorticity advection thickness advection
Relationship between upper troposphere and surface note tilt with height
Mid-latitude cyclones: Norwegian Cyclone Model
Fronts and Precipitation CloudSat Radar Norwegian Cyclone Model
What’s at work here?
Mid-latitude cyclone development
Mid-latitude cyclones: Norwegian Cyclone Model ptic/cyclone.htmhttp:// ptic/cyclone.htm
Below Basic Background Material
Tangential coordinate system Ω R Earth Place a coordinate system on the surface. x = east – west (longitude) y = north – south (latitude) z = local vertical or p = local vertical Φ a R=acos( )
Tangential coordinate system Ω R Earth Relation between latitude, longitude and x and y dx = acos( ) d is longitude dy = ad is latitude dz = dr r is distance from center of a “spherical earth” Φ a f=2Ωsin( ) =2Ωcos( )/a
Equations of motion in pressure coordinates (using Holton’s notation)
Scale factors for “large-scale” mid-latitude
Scaled equations of motion in pressure coordinates Definition of geostrophic wind Momentum equation Continuity equation Thermodynamic Energy equation