Atmospheric Dynamics Studies: Scale Analysis and Energy Conservation Equations

1. AOSS 401, Fall 2007Lecture 6September 19, 2007 Richard B. Rood (Room 2525, SRB) rbrood@umich.edu 734-647-3530 Derek Posselt (Room 2517D, SRB) dposselt@umich.edu 734-936-0502

2. Class News • Homework 1 graded • Homework 2 due today • Homework 3 posted on ctools between now and Friday

3. Weather • NCAR Research Applications Program • http://www.rap.ucar.edu/weather/ • National Weather Service • http://www.nws.noaa.gov/dtx/ • Weather Underground • http://www.wunderground.com/modelmaps/maps.asp?model=NAM&domain=US

4. Outline • Review from Monday • Continuity Equation • Scale Analysis • Conservation of Energy • Thermodynamic energy equation and the first law of thermodynamics • Potential temperature and adiabatic motions • Adiabatic lapse rate and static stability

5. From last time

6. Conservation of Mass • Conservation of mass leads to another equation; the continuity equation • Continuity  Continuous • No holes in a fluid • Another fundamental property of the atmosphere • Need an equation that describes the time rate of change of mass (density)

7. Dz Dy Dx Eulerian Form of the Continuity Equation In the Eulerian point of view, our parcel is a fixed volume and the fluid flows through it.

8. The change in mass (density) following the motion is equal to the divergence Convergence = increase in density (compression) Divergence = decrease in density (expansion) Lagrangian Form of the Continuity Equation

9. Scale Analysis of theContinuity Equation • Define a backgroundpressure field • “Average” pressure and density at each level in the atmosphere • No variation in x, y, or time • Hydrostatic balance applies to the background pressure and density

10. Scale Analysis of theContinuity Equation Total pressure and density = sum of background + perturbations (perturbations vary in x, y, z, t) Start with the Eulerian form of the continuity equation, do the scale analysis, and arrive at

11. Scale Analysis of theContinuity Equation • Expand this equation Remember, ρ0does not depend on x or y

12. Scale Analysis of theContinuity Equation • The vertical motion on large (synoptic) scales is closely related to the divergence of the horizontal wind

13. Scale Analysis of the Horizontal Momentum Equations Largest Terms

14. Geostrophic Balance • There is no D( )/Dt term (no acceleration) • No change in direction of the wind (no rotation) • No change in speed of the wind along the direction of the flow (no divergence)

15. What are the scales of the terms? For “large-scale” mid-latitude Analysis (Diagnosis) Geostrophic Prediction (Prognosis) Ageostrophic

16. Remember the definition of geostrophic wind • Our prediction equation for large scale midlatitudes

17. Ageostrophic Wind and Vertical Motion • Remember the scaled continuity equation • Vertical motion related to divergence, but geostrophic wind is nondivergent. • Divergence of ageostrophic wind leads to vertical motion on large scales.

18. Closing Our System of Equations • We have formed equations to predict changes in motion (conservation of momentum) and density (conservation of mass) • We need one more equation to describe either the time rate of change of pressure or temperature (they are linked through the ideal gas law) • Conservation of energy is the basic principle

19. Conservation of Energy: The thermodynamic equation • First law of thermodynamics: • Change in internal energy is equal to the difference between the heat added to the system and the work done by the system. • Internal energy is due to the kinetic energy of the molecules (temperature) • Total thermodynamic energy is the internal energy plus the energy due to the parcel moving

20. Thermodynamic Equation For a Moving Parcel • J represents sources or sinks of energy. • radiation • latent heat release (condensation/evaporation, etc) • thermal conductivity • frictional heating. • cv = 717 J K-1 kg-1, cvT = a measure of internal energy • specific heat of dry air at constant volume • amount of energy needed to raise one kg air one degree Kelvin if the volume stays constant.

21. Thermodynamic Equation • Involves specific heat at constant volume • Remember the material derivative form of the continuity equation • Following the motion, divergence leads to a change in volume • Reformulate the energy equation in terms of specific heat at constant pressure

22. Another form of the Thermodynamic Equation • Short derivation • Take the material derivative of the equation of state • Use the chain rule and the fact that R=cp-cv • Substitute in from the thermodynamic energy equation in Holton • Leads to a prognostic equation for the material change in temperature at constant pressure

23. (Ideal gas law) (Material derivative) (Chain Rule) (Use R=cp-cv) Substitute in from the thermodynamic energy equation (Holton, pp. 47-49) (Cancel terms)

24. Thermodynamic equation • Prognostic equation that describes the change in temperature with time • In combination with the ideal gas law (equation of state) the set of predictive equations is complete

25. Atmospheric Predictive Equations

26. Motions in a Dry (Cloud-Free) Atmosphere • For most large-scale motions, the amount of latent heating in clouds and precipitation is relatively small • In absence of sources and sinks of energy in a parcel, entropy is conserved following the motion • Why is this important? • Large scale vertical motion • Atmospheric stability (convection)

27. Motions in a Dry (Cloud-Free) Atmosphere • Goal: find a variable that • Is conserved following the motion if there are no sources and sinks of energy (J) • Describes the change in temperature as a parcel rises or sinks in the atmosphere • Adiabatic process: “A reversible thermodynamic process in which no heat is exchanged with the surroundings” • Situations in which J=0 referred to as • Dry adiabatic • Isentropic • Why is this useful?

28. Synoptic Motions

29. Cooling Warming Forced Ascent/Descent

30. Derivation of Potential Temperature (Energy Equation divided by temperature) (No sources or sinks of energy) (Adiabatic process) (Integrate between two levels) (Use the properties of the natural logarithm) (Take exponential ofboth sides)

31. Definition of the Potential Temperature  Note: p0 is defined to be a constant reference level p0 = 1000 hPa Interpretation: the potential temperature is the temperature a parcel has when it is moved from a (higher or lower )pressure level down to the surface.

32. 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

33. 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?

34. (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)

35. 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)

36. Atmospheric Static Stabilityand Potential Temperature • Static: considering an atmosphere at rest (no u, v, w) • Consider what will happen if an air parcel is forced to rise (or sink) • Stable: parcel returns to the initial position • Neutral: parcel only rises/sinks if forcing continues, otherwise remains at current level • Unstable: parcel accelerates away from its current position

37. Static Stability • Displace an air parcel up or down • Assume the pressure adjusts instantaneously; the parcel immediately assumes the pressure of the altitude to which it is displaced. • Temperature changes according to the adiabatic lapse rate

38. Static Stability • Adiabatic: parcel potential temperature constant with height • For instability, the temperature of the atmosphere has to decrease at greater than 9.8 K/km • This is extremely rare… • Convection (deep and shallow) is common • How to reconcile lack of instability with presence of convection?

39. 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

41. Consider the Upper Atmosphere

42. Atmosphere in Balance • Hydrostatic balance (no vertical acceleration) • Geostrophic balance (no rotation or divergence) • Adiabatic lapse rate (no clouds or precipitation) • What we are really interested in is the difference from balance. • 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.

43. Next time • Ricky will be lecturing Friday, Monday, and Wednesday • We have essentially completed chapters 1-2 in Holton • We have derived a set of governing equations for the atmosphere • Chapter 3 will introduce simple applications of these equations • First exam covers chapters 1-3—three weeks from today!