AOSS 401, Fall 2007 Lecture 24 November 07, 2007 Richard B. Rood (Room 2525, SRB) Derek Posselt (Room 2517D, SRB)

Class News November 07, 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

Rest of Course Wrap up quasi-geostrophic theory (Chapter 6) –Potential vorticity –Vertical velocity –Will NOT do Q vectors We will have a lecture on the Eckman layer (Chapter 5) –Boundary layer, mix friction with rotation We will have a lecture on Kelvin waves (Chapter 11) –A long wave in the tropics There will be a joint lecture with 451 on hurricanes (Chapter 11) Computer homework (perhaps lecture) on modeling Special topics?

Material from Chapter 6 Quasi-geostrophic theory Quasi-geostrophic vorticity –Relation between vorticity and geopotential Geopotential prognostic equation Quasi-geostrophic potential vorticity

Scaled equations in pressure coordinates (The quasi-geostrophic (QG) equations) momentum equation continuity equation thermodynamic equation geostrophic wind

Approximations in the quasi-geostrophic (QG) theory

Quasi-geostrophic equations cast in terms of geopotential and omega. THERMODYNAMIC EQUATION VORTICITY EQUATION

Add these equations to eliminate omega and we have a partial differential equation for geopotential tendency (assume J=0) GEOPOTENTIAL TENDENCY EQUATION

Add these equations to eliminate omega and we have a partial differential equation for geopotential tendency (assume J=0) f 0 * Vorticity Advection

Add these equations to eliminate omega and we have a partial differential equation for geopotential tendency (assume J=0) Thickness Advection

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 

Relationship between upper troposphere and surface vorticity advection thickness advection

To think about this Read and re-read pages in the text.

Idealized vertical cross section

Great web page with current maps: Real baroclinic disturbances Personalize your maps (create a login):

Real baroclinic disturbances: 850 hPa temperature and geopot. thickness warm air advection east of the surface low, enhances the ridge cold air advection, enhances trough

Real baroclinic disturbances: 500 hPa rel. vorticity and mean SLP sea level pressure Positive vorticity, pos. vorticity advection, increase in cyclonic vorticity

Real baroclinic disturbances: 500 hPa geopot. height and mean SLP Upper level systems lags behind (to the west): system still develops

With the benefit of hindsight and foresight let’s look back.

QG vorticity equation Advection of relative vorticity Advection of planetary vorticity Stretching term Competing THINKING ABOUT THESE TERMS

QG vorticity equation Advection of relative vorticity Advection of planetary vorticity Stretching term Competing WHAT ABOUT THIS TERM?

Consider our simple form of potential vorticity From scaled equation, with assumption of constant density and temperature. There was the assumption that the layer of fluid was shallow.

Fluid of changing depth What if we have something like this, but the fluid is an ideal gas?

Add these equations to eliminate omega and we have a partial differential equation for geopotential tendency (assume J=0) Still looks a lot like time rate of change of vorticity

Quasi-Geostrophic potential vorticity (PV) equation Simplify the last term of the geopotential tendency equation by applying the chain rule: = 0Why?

Quasi-Geostrophic potential vorticity (PV) equation Simplify the last term of the geopotential tendency equation by applying the chain rule: = 0Why? THERMAL WIND RELATION

Quasi-Geostrophic potential vorticity (PV) equation Simplify the last term of the geopotential tendency equation by applying the chain rule: = 0 Leads to the conservation law: Quasi-geostrophic potential vorticity: Conserved following the geostrophic motion Why?

Imagine at the point flow decomposed into two “components” A “component” that flows around the point.

Vorticity Related to shear of the velocity field. ∂v/∂x-∂u/∂y

Imagine at the point flow decomposed into two “components” A “component” that flows into or away from the point.

Divergence Related to stretching of the velocity field. ∂u/∂x+∂v/∂y

Potential vorticity (PV): Comparison Quasi- geostrophic PV: THESE ARE LIKE STRETCHING IN THE VERTICAL Barotropic PV: s -1 Ertel’s PV: m -1 s -1 Units: K kg -1 m 2 s -1

QG vorticity equation Advection of relative vorticity Advection of planetary vorticity Stretching term Competing WHAT ABOUT THIS TERM?

Fluid of changing depth What if we have something like this, but the fluid is an ideal gas? Conversion of thermodynamic energy to vorticity, kinetic energy. Again the link between the thermal field and the motion field.

Two important definitions barotropic – density depends only on pressure. And by the ideal gas equation, surfaces of constant pressure, are surfaces of constant density, are surfaces of constant temperature (idealized assumption).  =  (p) baroclinic – density depends on pressure and temperature (as in the real world).  =  (p,T)

Barotropic/baroclinic atmosphere Barotropic: p p +  p p + 2  p p p +  p p + 2  p T+2  TT+  T T T T+2  T T+  T Baroclinic: ENERGY IN HERE THAT IS CONVERTED TO MOTION

Barotropic/baroclinic atmosphere Barotropic: p p +  p p + 2  p p p +  p p + 2  p T+2  TT+  T T T T+2  T T+  T Baroclinic: DIABATIC HEATING KEEPS BUILDING THIS UP

NOW WOULD BE A GOOD TIME FOR A SILLY STORY

VERTICAL VELOCITY

Vertical motions: The relationship between w and  = 0hydrostatic equation ≈ 10 hPa/d ≈ 1m/s 1Pa/km ≈ 1 hPa/d ≈ 100 hPa/d

Links the horizontal and vertical motions. Since geostrophy is such a good balance, the vertical motion is linked to the divergence of the ageostrophic wind (small). = 0 Link between  and the ageostrophic wind

Vertical pressure velocity  For synoptic-scale (large-scale) motions in midlatitudes the horizontal velocity is nearly in geostrophic balance. Recall: the geostrophic wind is nondivergent (for constant Coriolis parameter), that is Horizontal divergence is mainly due to small departures from geostrophic balance (ageostrophic wind). Therefore: small errors in evaluating the winds and lead to large errors in . The kinematic method is inaccurate.

Think about this... If I have errors in data, noise. What happens if you average that data? What happens if you take an integral over the data? What happens if you take derivatives of the data?

Estimating the vertical velocity: Adiabatic Method Start from thermodynamic equation in p-coordinates: - (Horizontal temperature advection term) S p :Stability parameter Assume that the diabatic heating term J is small (J=0), re-arrange the equation

Estimating the vertical velocity: Adiabatic Method If  T/  t = 0 (steady state), J=0 (adiabatic) and S p > 0 (stable): then warm air advection:  0 (ascending air) then cold air advection:  > 0, w ≈ -  /  g < 0 (descending air) Horizontal temperature advection term Stability parameter

Adiabatic Method Based on temperature advection, which is dominated by the geostrophic wind, which is large. Hence this is a reasonable way to estimate local vertical velocity when advection is strong.

Estimating the vertical velocity: Diabatic Method Start from thermodynamic equation in p-coordinates: Diabatic term If you take an average over space and time, then the advection and time derivatives tend to cancel out.

mean meridional circulation

Conceptual/Heuristic Model Plumb, R. A. J. Meteor. Soc. Japan, 80, 2002 Observed characteristic behavior Theoretical constructs “Conservation” Spatial Average or Scaling Temporal Average or Scaling Yields Relationship between parameters if observations and theory are correct

One more way for vertical velocity

Quasi-geostrophic equations cast in terms of geopotential and omega. THERMODYNAMIC EQUATION VORTICITY EQUATION ELIMINATE THE GEOPOTENTIAL AND GET AN EQUATION FOR OMEGA

Quasi-Geostrophic Omega Equation 1.) Apply the horizontal Laplacian operator to the QG thermodynamic equation 2.) Differentiate the geopotential height tendency equation with respect to p 3.) Combine 1) and 2) and employ the chain rule of differentiation (chapter in Holton, note factor ‘2’ is missing in Holton Eq. (6.36), typo) Advection of absolute vorticity by the thermal wind

Vertical Velocity Summary Though small, vertical velocity is in some ways the key to weather and climate. It’s important to waves growing and decaying. It is how far away from “balance” the atmosphere is. It is astoundingly difficult to calculate. If you use all of these methods, they should be equal. But using observations, they are NOT! In fact, if you are not careful, you will not even to get them to balance in models, because of errors in the numerical approximation.

One more summary of the mid- latitude wave

Idealized (QG) evolution of a baroclinic disturbance ( Read and re-read pages in the text.) L H + warm air advection - cold air advection - neg. vorticity advection + pos. vorticity advection 500 hPa geopotential p at the surface

Waves The equations of motion contain many forms of wave-like solutions, true for the atmosphere and ocean Some are of interest depending on the problem: Rossby waves, internal gravity (buoyancy) waves, inertial waves, inertial-gravity waves, topographic waves, shallow water gravity waves Some are not of interest to meteorologists, e.g. sound waves Waves transport energy, mix the air (especially when breaking)

Waves Large-scale mid-latitude waves, are critical for weather forecasting and transport. Large-scale waves in the tropics (Kelvin waves, mixed Rossby-gravity waves) are also important, but of very different character. This is true for both ocean and atmosphere. Waves can be unstable. That is they start to grow, rather than just bounce back and forth.

And, with that, Chapter 6, of Jim Holton’s book rested comfortably in the mind of the students.

Below Basic Background Material

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

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

Mid-latitude cyclones: Norwegian Cyclone Model ptic/cyclone.htmhttp:// ptic/cyclone.htm

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

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