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CLIMATE/EARTH 401�Geophysical Fluid Dynamics�Dynamical Meteorology

Richard B. Rood

rbrood@umich.edu

https://clasp.engin.umich.edu/people/richard-b-rood/

https://openclimate.org/

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Buoyancy Wave

  • This material requires that you know the material presented on lapse rates.

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Average vertical temperature structure

  • In a dry atmosphere, this profile should be very close to the adiabatic lapse rate.

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Dry adiabatic lapse rate (AMS Glossary)

The adiabatic change in temperature with height is��

​

​

For dry adiabatic, hydrostatic atmosphere

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

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Consider a parcel in an atmosphere with a lapse rate

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Let’s return to our linear lapse rate.

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

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Temperature as function of height

z

Warmer

Cooler

z

T

∂T/∂z is defined as lapse rate

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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.

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Parcel cooler than environment

z

Warmer

Cooler

If the parcel moves up and finds itself cooler than the environment, then what will happen?

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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?)

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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?)

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Parcel cooler 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?)

​

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

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T changing with a constant linear slope (or lapse rate)

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T of parcel changing with the dry adiabatic lapse rate

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Stable: temperature of parcel cooler than environment

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Unstable: temperature of parcel greater than environment.

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Stability criteria from physical argument

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Use the vertical momentum equation

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We have an environment that looks like this

z

Warmer

Cooler

z

T

∂T/∂z is defined as lapse rate

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Hydrostatic balance

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Our parcel experiences an acceleration

Assumption of adjustment of pressure.

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Solve for pressure gradient

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Our parcel experiences an acceleration

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The pressure of parcel and environment are the same

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Use our definitions of temperature and temperature change

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Use binomial expansion

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Use our definitions of temperature and temperature change

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Ignore terms in z2

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For stable situation

Seek solution of the form

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For stable situation

Seek solution of the form

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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?)

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Example of such an oscillation

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For unstable situation

Seek solution of the form

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Parcel cooler 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?)

​

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

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End: Bouyancy Wave

Richard B. Rood

rbrood@umich.edu

https://clasp.engin.umich.edu/people/richard-b-rood/

https://openclimate.org/