Aoss 321 winter 2009 earth system dynamics lecture 11 2 12 2009
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AOSS 321, Winter 2009 Earth System Dynamics Lecture 11 2/12/2009. Christiane Jablonowski Eric Hetland [email protected] [email protected] 734-763-6238 734-615-3177. Today’s lecture. Derivation of the potential temperature equation (Poisson equation)

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Aoss 321 winter 2009 earth system dynamics lecture 11 2 12 2009 l.jpg

AOSS 321, Winter 2009Earth System DynamicsLecture 112/12/2009

Christiane Jablonowski Eric Hetland

[email protected]@umich.edu

734-763-6238 734-615-3177


Today s lecture l.jpg
Today’s lecture

  • Derivation of the potential temperature equation (Poisson equation)

  • Dry adiabatic lapse rate

  • Static stability, buoyancy oscillations

  • Derivation of the Brunt-Väisälä frequency


Thermodynamic equation l.jpg
Thermodynamic equation

(Divide by T)

Use equation of state(ideal gas law)


Thermodynamic equation4 l.jpg
Thermodynamic equation

For conservative motions (no heating, dry adiabatic: J = 0):


Derivation of poisson s equation 1 l.jpg
Derivation of Poisson’s Equation (1)

(integrateover Dt)

(integrate)


Slide6 l.jpg

Derivation of Poisson’s Equation (2)

Poisson’s Equation

 is calledpotential temperature!


Definition of the potential temperature l.jpg
Definition of the potential temperature 

with p0 usually taken to be constant withp0 = 1000 hPa

The potential temperature is the temperature a parcel would have if it was moved from some pressure level and temperature down to the surface.


Definition of potential temperature l.jpg
Definition of potential temperature

Does it makes sense that the temperature T would change in this problem? We did it adiabatically. There was no source and sink of energy.


Annual mean zonal mean temperature t l.jpg
Annual mean zonal mean temperature T

(hPa)

Pressure

Kelvin

1

260

230

10

200

100

220

210

1000

260

300

260

North Pole

South Pole

Equator

Source: ECMWF, ERA40


Annual mean zonal mean potential temperature l.jpg
Annual mean zonal mean potential temperature 

(hPa)

Kelvin

100

Pressure

350

330

285

285

300

1000

Equator

North Pole

South Pole

How does the temperature field look?

Source: ECMWF, ERA40


Dry adiabatic lapse rate l.jpg
Dry adiabatic lapse rate

  • For a dry adiabatic, hydrostatic atmosphere the potential temperature  does not vary in the vertical direction:

  • In a dry adiabatic, hydrostatic atmosphere the temperature T must therefore decrease with height.


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Class exercise

T?, ?

500 hPa

850 hPa

T = 17 ºC, ?

  • An air parcel that has a temperature of 17 ºC at the 850 hPa pressure level is lifted dry adiabatically. What is the temperature and density of the parcel when it reaches the 500 hPa level?


Dry adiabatic lapse rate derivation l.jpg
Dry adiabatic lapse rate: Derivation

Start with Poisson equation:

Take the logarithm of :

Differentiate with respect to height

= 0 (p0 constant)


Dry adiabatic lapse rate14 l.jpg
Dry adiabatic lapse rate

Use hydrostatic equationPlug in ideal gas law for p, then multiply by T:

For dry adiabatic, hydrostatic atmosphere with

d: dry adiabatic lapse rate (approx. 9.8 K/km)


Static stability l.jpg
Static stability

  • We will now assess the static stability characteristics of the atmosphere.

  • Static stability of the environment can be measured with the buoyancy frequency N.

  • N is also called Brunt-Väisälä frequency. The square of this buoyancy frequency is defined as

We will derive this equation momentarily, but first let’sdiscuss some static stability/instability conditions.


Static stability16 l.jpg
Static stability

We will now assess the static stability characteristics of the atmosphere.


Stable and unstable situations l.jpg
Stable and unstable situations

Check out this marble in a bowl:

http://eo.ucar.edu/webweather/stablebowl.html


Stable and unstable air masses l.jpg
Stable and unstable air masses

Stable air:A rising parcel that is cooler than the surrounding atmosphere will tend to sink back to its original position (why?).

Unstable air: A rising parcel that is warmer than the surrounding atmosphere will continue to rise (why?).

Neutral air:The parcel remains at the new location after being displaced, its temperature varies exactly as the temperature of the surrounding atmosphere.


Unstable air l.jpg
Unstable air

Unstable air: makes thunderstorms possible. Here: visible since clouds rise to high elevations!


Stable air l.jpg
Stable air

Stable air: makes oscillations (waves) in the atmosphere possible, visible due to the clouds!


Stable air21 l.jpg
Stable air

Stable air: makes oscillations (waves) in the atmosphere possible, what is the wave length?


Stable air22 l.jpg
Stable air

Stable air: temperature inversions suppress rising motions. Here: stratiform clouds have formed.


Let s take a closer look temperature as function of height l.jpg

z

Let’s take a closer look: Temperature as function of height

Cooler

z

- ∂T/∂z is defined as lapse rate

T

Warmer


Let s take a closer look temperature as function of height24 l.jpg

z

Let’s take a closer look: Temperature as function of height

Cooler

z

- ∂T/∂z is defined as lapse rate

T

Warmer


Let s take a closer look temperature as function of height25 l.jpg

z

Let’s take a closer look: Temperature as function of height

Cooler

z

- ∂T/∂z is defined as lapse rate

T

Warmer


Let s take a closer look temperature as function of height26 l.jpg

z

Let’s take a closer look: Temperature as function of height

Cooler

z

- ∂T/∂z is defined as lapse rate

T

Warmer


The parcel method l.jpg
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 l.jpg

z

Parcel cooler than environment

Cooler

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

Warmer


Parcel cooler than environment29 l.jpg

z

Parcel cooler than environment

Cooler

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

Warmer


Parcel warmer than environment l.jpg

z

Parcel warmer than environment

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?)

Warmer


Parcel warmer than environment31 l.jpg

z

Parcel warmer than environment

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?)

Warmer

This is our first example of “instability” – a perturbation that grows.


Let s quantify this characteristics of the environment l.jpg
Let’s quantify this:Characteristics of the environment

We assume that the temperature Tenv of the environment changes with a constant linear slope (or lapse rate ) in the vertical direction.

Tsfc: temperature at the surface


Let s quantify this characteristics of the parcel l.jpg
Let’s quantify this:Characteristics of the parcel

We assume that the temperature Tparcel of the parcel changes with the dry adiabatic lapse rate d.


Stable temperature of parcel cooler than environment l.jpg
Stable:Temperature of parcel cooler than environment

parcel

environment

environment

parcel

compare the lapse rates


Unstable temperature of parcel greater than environment l.jpg
Unstable: Temperature of parcel greater than environment.

parcel

environment

environment

parcel

compare the lapse rates


Stability criteria from physical argument l.jpg
Stability criteria from physical argument

Lapse rate of the environment

Dry adiabatic lapse rate of the parcel



But our parcel experiences an acceleration small displacement z l.jpg
But our parcel experiences an acceleration, small displacement z

Assumption of immediate adjustment of pressure.



But our parcel experiences an acceleration l.jpg
But our parcel experiences an acceleration

use ideal gas law:

Recall:



Back to our definitions of temperature change l.jpg
Back to our definitions of temperature change

Second-order, ordinary differential equation:


Recall dry adiabatic lapse rate l.jpg
Recall: Dry adiabatic lapse rate

Taking the logarithm of , differentiating with respect to height, using the ideal gas law and hydrostatic equation gives:

dry adiabatic lapse rate (approx. 9.8 K/km)


Rearrange44 l.jpg
Rearrange:

With the Brunt-Väisälä frequency


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Buoyancy oscillations

stable, the solution to this equation describes a buoyancy oscillation with period 2/N

2) unstable, corresponds to growing perturbation, this is an instability

3) neutral


Solution to the differential equation l.jpg
Solution to the differential equation

The general solution can be expressed via

the exponential function with a complex argument:

with A: amplitude, N: buoyancy frequency,

If N2 > 0 the parcel will oscillate about its initial level with a period  = 2/N.Average N in the troposphere N ≈ 0.01 s-1


Remember euler s formula l.jpg
Remember Euler’s formula

with x: real number

Physical solution:

If N2 > 0 (real) the solution is a wave with period  = 2/N (more on waves in AOSS 401)


Stable solution parcel cooler than environment l.jpg

z

Stable solution:Parcel cooler than environment

Cooler

If the parcel moves up and finds itself cooler than the environment then it will sink (and rise again). This is a buoyancy oscillation.

Warmer


Stable and unstable air masses49 l.jpg
Stable and unstable air masses

Picture an invisible box of air (an air parcel). If we compare the temperature of this air parcel to the temperature of air surrounding it, we can tell if it is stable (likely to remain in place) or unstable (likely to move).

http://eo.ucar.edu/webweather/stable.html


Temperature soundings l.jpg
Temperature soundings

z

Sounding of the parcel

Sounding of the environment:T inversion


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