AOSS 401, Fall 2006 Lecture 7 September 21, 2007 Richard B. Rood (Room 2525, SRB) Derek Posselt (Room 2517D, SRB)

Class News Contract with class. –First exam October 10. Homework 3 is posted. –Due next Friday –Seven problems First four should take about ½ hour. Last three a problem you have to “solve” –Exercises a whole suite of problem solving skills

Outline Scale analysis –In class problem –Exploring the atmosphere Vertical Structure –Isentropic and isothermal Stability and Instability –Wave motion Balances

Times scales Distance = velocity * time How long does it take you to.... ?

So for our equations D ( )/Dt can be characterized by 1/(L/U)

Let us define.

Geostrophic wind 300 mb

For “large-scale” mid-latitude

In class problem: Scaling Above are the horizontal components of the momentum equation. In class (and text) these equations were scaled with characteristic values. We saw that away from the surface that the viscosity term was very small. Near the Earth’s surface, however, viscosity is important. What is an appropriate vertical length scale near the surface? What is the approximate value of vertical shear of the horizontal wind? (assume  is constant = 1.46x10 -5 m 2 s -1 )

Geostrophic and observed wind 1000 mb (land)

Let’s spend some time with the atmosphere. Start with a typical upper tropospheric chart. What is a good estimate of the pressure at the surface? What is a good estimate of the pressure in the upper troposphere? How could you figure out the geometric height?

Geostrophic wind 300 mb How does this example relate to global scales?

300 mb

Some mid-latitude scales Mid-latitude cyclones about 1000 km and 1 day Jet stream. More on a planetary scale. 1000’s of kilometers

700 mb

500 mb

300 mb

50 mb

DJF 500 mb Average

JJA 500 mb Average

Anomaly 100 mb

What are the scales of the terms? (horizontal momentum equations, mid-latitudes, away from surface) U*U/L U*U/a U*W/a Uf Wf Geostrophic terms Acceleration

Some carry away points (Mid-latitudes) Motions are determined to a good approximation by the balance of the pressure gradient and the coriolis force –approximately in horizontal plane Multiple scales, scales embedded within larger scales Change of scale with altitude Change of scale from winter to summer Balance of forces with altitude Balance of forces with scale

What about the vertical?

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)

Thermodynamic equation (conservative, no heating, adiabatic) conservative, no heating, adiabatic all mean J=0

Thermodynamic equation conservative, no heating, adiabatic (solve, perfect differential) Know how to do this mathematical manipulation.

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.

Consider the vertical structure some more. Hydrostatic Eq. of State

Consider the vertical structure some more. T as constant - (Isothermal) Under special consideration of T as constant. (Isothermal)

Consider the vertical structure some more. Under special consideration of T as constant. (Isothermal)

Consider the vertical structure some more. Under special consideration of T as constant. (Isothermal) Units [R] = J/(kg*K)=kg*m*m/(s*s*kg*K), [T] = K [RT]=m*m/(s*s) [RT/g]=m (unit of length) GTQ: Given the magnitude of the scale of H what is the average temperature of the atmosphere?

Pressure altitude Exponential decrease with height

Consider a different vertical structure. Under special consideration of T changing with a constant linear slope (or lapse rate).

Consider a different vertical structure. Under special consideration of T changing with a constant lapse rate. Or a linear slope.

Consider a different vertical structure. Under special consideration of T changing with a constant lapse rate. Or a linear slope.  positive, T decreases with height, pressure decreases with height.  negative, T increases with height, pressure decreases with height.

Pressure altitude Under virtually all conditions pressure (and density) decreases with height. ∂p/∂z < 0. That’s why it is a good vertical coordinate. If ∂p/∂z = 0, then utility as a vertical coordinate falls apart.

Let’s return to our linear lapse rate. Under special consideration of T changing with a constant linear slope (or lapse rate).

Temperature as function of height z Warmer Cooler z T ∂T/∂z is defined as lapse rate

Temperature as function of height z Warmer Cooler z T ∂T/∂z is defined as lapse rate

Temperature as function of height z Warmer Cooler z T ∂T/∂z is defined as lapse rate

Temperature as function of height z Warmer Cooler z T ∂T/∂z is defined as lapse rate

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

Let’s return to the vertical momentum equation

What are the scales of the terms? W*U/L U*U/a Uf g

What are the scales of the terms? W*U/L U*U/a Uf g

Vertical momentum equation  Hydrostatic balance

Hydrostatic balance

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

Solve for pressure gradient

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.

What are the scales of the terms? W*U/L U*U/a Uf g

What are the scales of the terms? W*U/L U*U/a Uf g

Definition of potential temperature This is essentially the thermodynamic energy equation under the assumption of adiabatic motion. If the atmosphere were stirred adiabatically, then this is how the temperature would evolve.. This is Poisson’s equation.

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

Some basics of the atmosphere Troposphere: depth ~ 1.0 x 10 4 m Troposphere ~ 2 Mountain Troposphere ~ 1.6 x Earth radius This scale analysis tells us that the troposphere is thin relative to the size of the Earth and that mountains extend half way through the troposphere.

What are the scales of the terms? U*U/L U*U/a U*W/a Uf Wf

What are the scales of the terms? U*U/L U*U/a U*W/a Uf Wf Largest Terms

Geostrophic balance High Pressure Low Pressure

Looking at the atmosphere What does this map tell you?

Forced Ascent/Descent Warming Cooling

An Eulerian Map