Richard B. Rood (Room 2525, SRB) University of Michigan rbrood@umich

1. AOSS 401Geophysical Fluid Dynamics:Atmospheric DynamicsPrepared: 20130918Potential TemperatureLapse Rate Richard B. Rood (Room 2525, SRB) University of Michigan rbrood@umich.edu

2. Weather Links • National Weather Service • Model forecasts: • Weather Underground • Model forecasts: • NCAR Research Applications Program

3. Outline • Potential Temperature • Lapse Rate

4. DefinitionFrom AMS Glossary • Potential temperature is the temperature that an unsaturated parcel of dry air would have if brought adiabatically and reversibly from its initial state to a standard pressure, ps, typically 1000 hPa. • In oceanography it is the temperature that a water parcel would have if brought adiabatically to the sea surface.

5. Mathematical Form The mathematical representation of potential temperature follows from the first law of thermodynamics which is stated as: The increase in internal energy of a closed system is equal to the difference of the heat supplied to the system and the work done by it. Many meteorological texts start with the enthalpy form of the first law of thermodynamics, where enthalpy is the total energy of a thermodynamic system. For a closed system this is written as Where h is enthalpy, s is entropy and other variables as previously defined.

6. Mathematical Form For adiabatic motions ds is zero. Then for air using the ideal gas law

7. Return to meteorological form of the thermodynamic equation

8. Thermodynamic equation(Use the equation of state to reform)

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

10. Thermodynamic equation (conservative, no heating, adiabatic)

11. What skills are needed? • Note that this is a perfect differential • Then substitute in for the ln • We are “moving” the parcel by using integration

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

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

14. The temperature at the top of the continental divide is -10 degrees celsius (about 263 K) • The pressure is 600 hPa, R=287 J/kg/K, cp=1004 J/kg/K • Compute • potential temperature at the continental divide • The temperature the air would have if it sinks to the plains (pressure level of 850 hPa) with no change in potential temperature 304 K

15. Average vertical temperature structure This cooling with height is related to dynamics – vertical mixing. The change of temperature with height is called the lapse rate.

16. Dry Adiabatic Lapse RateChange in Temperature with Height • 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 decrease with height. How quickly does the temperature decrease?

17. Derive the dry adiabatic lapse rate • Remember that you own the equation, but you have to treat the equation according to the rules. • You must respect the “equal sign.” • Think of the derivations that we do as exploration, not as prescription.

18. (logarithm of potential temperature) (take the vertical derivative) (Definition of d lnx and derivative of a constant) (Multiply through by T) (Hydrostatic balance) (Equation of State)

19. Dry adiabatic lapse rate The adiabatic change in temperature with height is For dry adiabatic, hydrostatic atmosphere d: dry adiabatic lapse rate (approx. 9.8 K/km)

20. Average vertical temperature structure This profile should be very close to the adiabatic lapse rate in a dry atmosphere.

21. Fundamental to remember • 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?

22. Fundamental to remember • 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.

23. 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.

24. Average vertical temperature structure In fact, the temperature decline is not the adiabatic lapse rate.

25. Instantaneous T and H2O profile NCAR Research Applications Program

26. Static Stability and Moisture • The atmosphere is not dry—motion is not dry adiabatic • If air reaches saturation (and the conditions are right for cloud formation), vapor will condense to liquid or solid and release energy (J≠0) • Average lapse rate in the troposphere: -6.5 oC/km • Moist (saturated) adiabatic lapse rate: -5 oC/km

27. Cooling Warming Forced Ascent/Descent

28. Summary • Potential temperature is an important variable because for adiabatic motion it does not change. • Useful for theoretical development • Often true in, say, stratosphere • Potential temperature can be modified to account for moisture. • Lapse rate, derived from consideration of vertical structure of potential temperature, explains the vertical temperature of the troposphere.