AOSS 401, Fall 2007 Lecture 21 October 31, 2007 Richard B. Rood (Room 2525, SRB) Derek Posselt (Room 2517D, SRB)

Class News October 31, 2007 Homework 5 (Due Friday) –Posted to web –Computing assignment posted to ctools under the Homework section of Resources Next Test November 16 Thanks for the comments on the Midterms evaluations

A Diversion Santa Ana Winds

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

Material from Chapter 6 Quasi-geostrophic theory Quasi-geostrophic vorticity –Relation between vorticity and geopotential

Quasi-Geostrophic Derive a system of equations that is close to geostrophic and hydrostatic balance, but includes the effects of ageostrophic wind Comes from scale analysis of equations of motion in pressure coordinates Scale analysis = make assumptions (where do these assumptions break down?)

Equations of motion in pressure coordinates (using Holton’s notation)

Scale factors for “large-scale” mid-latitude

From the scale analysis we introduced the non- dimensional Rossby number A measure of planetary vorticity compared to relative vorticity. A measure of the importance of rotation.

Scale analysis of equations in pressure coordinates Start: –horizontal flow is approximately geostrophic –vertical velocity much smaller than horizontal velocity

We will scale the material derivative Ignore // small This is for use in the advection of temperature and momentum. ω comes from div(ageostrophic wind)

Variation of Coriolis parameter L, length scale, is small compared to the radius of the Earth In the calculation of geostrophic wind, assume f is constant; f = f 0 We cannot assume f is constant in the Coriolis terms…

Variation of Coriolis parameter

Scale of first two terms.

Continuity equation becomes

Thermodynamic equation geostrophic wind can be used here. static stability, S p, is large; ω cannot be ignored

Thermodynamic equation (use the fact that atmosphere is near hydrostatic balance) split temperature into basic state plus deviation

Thermodynamic equation (and with the hydrostatic equation) note the inverse relation of heating with pressure

The Momentum Equation

horizontal flow is approximately geostrophic L, length scale, is small compared to the radius of the Earth In the calculation of geostrophic wind, assume f is constant; f = f 0

horizontal flow is approximately geostrophic

Forcing terms in momentum equation approx of coriolis parameter Use definition of geostrophic wind in the pressure gradient force def’n of the full wind

Forcing terms in momentum equation

Approximate horizontal momentum equation This equation states that the time rate of change of the geostrophic wind is related to 1.the coriolis force due to the ageostrophic wind and 2.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.

A Point All of the terms in the equation for the CHANGE in the geostrophic wind, which is really a measure of the difference from geostrophic balance, are order Ro (Rossby number). –Again, reflects the importance of rotation to the dynamics of the atmosphere and ocean

Scaled equations of motion in pressure coordinates Definition of geostrophic wind Momentum equation Continuity equation Thermodynamic Energy equation

What is the point? Set of equations that describes synoptic- scale motions and includes the effects of ageostrophic wind (vertical motion)

Scale Analysis = Make Assumptions Quasi-geostrophic system is good for: Synoptic scales Middle latitudes Situations in which V a is important Flows in approximate geostrophic and hydrostatic balance Mid-latitude cyclones

Scale Analysis = Make Assumptions Quasi-geostrophic system is not good for: Very small or very large scales Flows with large vertical velocities Situations in which V a ≈ V g Flows not in approximate geostrophic and hydrostatic balance Thunderstorms/convection, boundary layer, tropics, etc…

What will we do next? Derive a vorticity equation for these scaled equations. –Actually provides a “suitable” prognostic equation because need to include div(ageostrophic wind) in the prognostics. –Remember the importance of divergence in vorticity equations.

Derive a vorticity equation Going to spend some time with this.

Vorticity relative vorticity velocity in (x,y) plane shear of velocity suggests rotation

Vorticity view this as the definition of relative vorticity

If we want an equation for the conservation of vorticity, then We want an equation that represents the time rate of change of vorticity in terms of sources and sinks of vorticity.

Conservation (continuity) principle dM/dt = Production – Loss

Newton’s Law of Motion Which is the vector form of the momentum equation. (Conservation of momentum) Where F is the sum of forces acting on a parcel, m mass, v velocity

What are the forces? Total Force is the sum of all of these forces –Pressure gradient force –Gravitational force –Viscous force –Apparent forces Derived these forces from first principles

If we want an equation for the conservation of vorticity, then We could approach it the same way as momentum, define the sources and sinks of vorticity from first principles. –But that is hard to do. What are the first principle sources of vorticity? Or we could use the conservation of momentum, and the definition of vorticity to derive the equation.

Newton’s Law of Motion (components)

Combine definition and conservation principle

Operate on momentum equation

Subtract so a time rate of change of vorticity will come from here.

Subtract so details will depend on d( )/dt. For an Eulerian fluid d( )/dt = D( )Dt, material derivative. For a Lagrangian description could write immediately.

Expand derivative

Dζ/Dt comes from here.

Expand derivative Other things comes from here.

Collect terms

Let’s return to our quasi- geostrophic formulation

Scaled horizontal momentum in pressure coordinates

Use definition of vorticity  vorticity equation

An equation for geopotential tendency

Barotropic fluid

Perturbation equation

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.

Study Questions

Study question 1 Thermodynamic equation Why can we pull this p outside of the derivative operators?

Study Question 2 Forcing terms in momentum equation Do the derivation of this approximation.

Study Question 3 An estimate of the July mean zonal wind north summer south winter Compare this figure to the similar figure for January. What is similar and different between the tropospheric jets? (magnitude, position). Without referring to the plots of temperature, what can you say about the temperature structure?

Study question 4

Study question 5 What is the scale of the horizontal divergence of the wind to the total vorticity in middle latitudes. Would you expect the same in the tropics? What are the units of potential vorticity?

Study question 6 Use definition of vorticity  vorticity equation Carry out this derivation.

Scaled equations of motion in pressure coordinates Set up and start the derivation for the time rate of change of divergence of the horizontal divergence. Show how to extract D(div(u horizontal )/Dt= ….