# AOSS 401, Fall 2006 Lecture 8 September 24, 2007 Richard B. Rood (Room 2525, SRB) 734-647-3530 Derek Posselt (Room 2517D, SRB)

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AOSS 401, Fall 2006 Lecture 8 September 24, 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 Contract with class. –First exam October 10. Homework 3 is posted. –Due Friday Solution sets for Homework 1 and 2 are posted.

Weather National Weather Service –http://www.nws.noaa.gov/http://www.nws.noaa.gov/ –Model forecasts: http://www.hpc.ncep.noaa.gov/basicwx/day0- 7loop.html http://www.hpc.ncep.noaa.gov/basicwx/day0- 7loop.html Weather Underground –http://www.wunderground.com/cgi- bin/findweather/getForecast?query=ann+arborhttp://www.wunderground.com/cgi- bin/findweather/getForecast?query=ann+arbor –Model forecasts: http://www.wunderground.com/modelmaps/maps.asp ?model=NAM&domain=US http://www.wunderground.com/modelmaps/maps.asp ?model=NAM&domain=US

Outline Vertical Structure Reset Stability and Instability –Wave motion Balances Thermal Wind Maps

Full equations of motion We saw that the first two equations were dominated by the geostrophic balance. What do we do for the vertical motion?

Thermodynamic equation (Use the equation of state)

Definition of potential temperature This is the temperature a parcel would have if it was moved from some pressure and temperature to the surface. This is Poisson’s equation.

This is a very important point. Even in adiabatic motion, with no external source of heating, if a parcel moves up or down its temperature changes. What if a parcel moves about a surface of constant pressure?

Adiabatic lapse rate. For an adiabatic, hydrostatic atmosphere the temperature decreases with height.

Another important point If the atmosphere is in adiabatic balance, the temperature still changes with height. Adiabatic does not mean isothermal. It means that there is no external heating or cooling.

The parcel method We are going displace this parcel – move it up and down. –We are going to assume that the pressure adjusts instantaneously; that is, the parcel assumes the pressure of altitude to which it is displaced. –As the parcel is moved its temperature will change according to the adiabatic lapse rate. That is, the motion is without the addition or subtraction of energy. J is zero in the thermodynamic equation.

Parcel cooler than environment z Warmer Cooler If the parcel moves up and finds itself cooler than the environment then it will sink. (What is its density? larger or smaller?)

Parcel cooler than environment z Warmer Cooler If the parcel moves up and finds itself cooler than the environment, then it will sink. (What is its density? larger or smaller?)

Parcel warmer than environment z Warmer Cooler If the parcel moves up and finds itself warmer than the environment then it will go up some more. (What is its density? larger or smaller?)

Parcel cooler than environment z Warmer Cooler If the parcel moves up and finds itself cooler than the environment, then it will sink. (What is its density? larger or smaller?) This is our first example of “instability” – a perturbation that grows.

Let’s quantify this. Under consideration of T changing with a constant linear slope (or lapse rate).

Let’s quantify this. Under consideration of T of parcel changing with the dry adiabatic lapse rate

Stable: temperature of parcel cooler than environment.

Unstable: temperature of parcel greater than environment.

Stability criteria from physical argument

What are the scales of the terms? W*U/L U*U/a Uf g 10 -7 10 -5 10 10 -3 10 10 -15

What are the scales of the terms? W*U/L U*U/a Uf g 10 -7 10 -5 10 10 -3 10 10 -15

Vertical momentum equation  Hydrostatic balance

Hydrostatic balance

But our parcel experiences an acceleration Assumption of adjustment of pressure.

But our parcel experiences an acceleration

Again, our pressure of parcel and environment are the same so

So go back to our definitions of temperature and temperature change above

Use binomial expansion

So go back to our definitions of temperature and temperature change above

Ignore terms in z 2

For stable situation Seek solution of the form

For stable situation Seek solution of the form

Parcel cooler than environment z Warmer Cooler If the parcel moves up and finds itself cooler than the environment then it will sink. (What is its density? larger or smaller?)

Example of such an oscillation

For unstable situation Seek solution of the form

Parcel cooler than environment z Warmer Cooler If the parcel moves up and finds itself cooler than the environment, then it will sink. (What is its density? larger or smaller?) This is our first example of “instability” – a perturbation that grows.

This is our first explicit solution of the wave equation These are called buoyancy waves or gravity gaves. The restoring force is gravity, imbalance of density in the fluid. We extracted an equation through scaling and use of balances. –This is but one type of wave that is supported by the equations of atmospheric dynamics. Are gravity waves important in the atmosphere?

Near adiabatic lapse rate in the troposphere Troposphere: depth ~ 1.0 x 10 4 m Troposphere ------------------ ~ 2 Mountain Troposphere ------------------ ~ 1.6 x 10 -3 Earth radius GTQ: What if we assumed that the atmosphere was constant density? Is there a depth the atmosphere cannot exceed?

Looking at the atmosphere What does the following map tell you?

Forced Ascent/Descent Warming Cooling

An Eulerian Map

Some meteorologist speak Zonal = east-west Meridional = north-south Vertical = up and down

What are the scales of the terms? U*U/L U*U/a U*W/a Uf Wf 10 -4 10 -5 10 -8 10 -3 10 -6 10 -12

What are the scales of the terms? U*U/L U*U/a U*W/a Uf Wf 10 -4 10 -5 10 -8 10 -3 10 -6 10 -12 Largest Terms

Geostrophic balance High Pressure Low Pressure

Atmosphere in balance Hydrostatic balance Geostrophic balance Adiabatic lapse rate We can use this as a paradigm for thinking about many problems, other atmospheres. Suggests a set of questions for thinking about observations. What is the rotation? How does it compare to acceleration, represented by the spatial and temporal scales?

Atmosphere in balance Hydrostatic balance Geostrophic balance Adiabatic lapse rate But what we are really interested in is the difference from this balance. And this balance is like a strong spring, always pulling back. It is easy to know the approximate state. Difficult to know and predict the actual state.

Let’s think about another possible balance

Thermodynamic balance (velocity and acceleration = 0) Compare with geostrophic balance.

Specify something for J

Where we ignore for latent heat release for convenience (e.g. dry atmosphere). We know frictional heating is zero for no velocity.

We can show Horizontal gradients of both pressure and density must equal zero. –Hence horizontal temperature gradient must be zero. T=T(z) If there is a horizontal temperature gradient then there is motion. If differential heating in the horizontal then temperature gradient. Hence motion.

Transfer of heat north and south is an important element of the climate at the Earth’s surface. Redistribution by atmosphere, ocean, etc. SURFACE Top of Atmosphere / Edge of Space ATMOSPHERE CLOUD heat is moved to poles cool air moved towards equator This is a transfer. Both ocean and atmosphere are important!

Hurricanes and heat

Middle latitude cyclones

Thermodynamic Balance The atmosphere and ocean are NOT in thermodynamic balance. If there is a temperature gradient, then there is motion. Temperature gradients are always being forced.

The geostrophic balance U*U/L U*U/a U*W/a Uf Wf 10 -4 10 -5 10 -8 10 -3 10 -6 10 -12 Largest Terms

The geostrophic balance

How do we link the horizontal and vertical balances?

The geostrophic balance Take a vertical derivative of the equation.

The geostrophic balance Use equation of state to eliminate density. Thermal wind relationship in height (z) coordinates

moving block Shear? (1) stationary surface There is force due to fact that there is a velocity and when the moving blocks are in contact the interfaces experience a force – say, friction, the surfaces can distort. One form of distortion is shearing.

moving fluid Shear? (2) Shear is a word used to describe that velocity varies in space. more slowly moving fluid There is force due to fact that there is a velocity gradient, and because our fluid is a fluid, the fluid surface responds to this gradient, which is called the shear.

moving fluid Shear? (3) Shear is a word used to describe that velocity varies in space. more slowly moving fluid z

The geostrophic balance What does this equation tell us? Thermal wind relationship in height (z) coordinates

Can we start to relate vertical structure and wind? Troposphere: depth ~ 1.0 x 10 4 m Troposphere ------------------ ~ 2 Mountain Troposphere ------------------ ~ 1.6 x 10 -3 Earth radius

An estimate of the January mean temperature north winter south summer tropopause stratopause mesosphere stratosphere troposphere note where the horizontal temperature gradients are large

An estimate of the January mean zonal wind north winter south summer note the jet streams

An estimate of the July mean zonal wind north summer south winter note the jet streams

Gosh, that’s a lot Think about it! Do your homework? This is new material now? From that July wind field, what are the differences between January and July temperatures.

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