Atmospheric Dynamics Overview: Scales, Analysis, Equations

1. AOSS 401, Fall 2007Lecture 5September 17, 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 • Posselt office hours: Tues/Thurs AM and after class • Homework 2 due date postponed to this Wednesday • Homework 3 will be assigned this week

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: Material Derivative • Conservation of Mass and Scale analysis • Lagrangian and Eulerian derivations of continuity equation • Scale analysis of the momentum equations • Scale analysis of the continuity equation

5. From last time

6. Lagrangian Eulerian Material Derivative Calculate the change in a property of the atmosphere (for example, temperature) over time Δt, following the parcel. Expand the change in temperature in a Taylor series around the temperature at the initial position, ignore second order and higher terms, and take the limit as Δt0

7. 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)

8. Remember our particle of atmosphere, our parcel r≡ density = mass per unit volume (DV) DV = DxDyDz m = rDxDyDz ------------------------------------- p ≡ pressure = force per unit area acting on the particle of atmosphere Dz Dy Dx

9. The Eulerian point of view our parcel is a fixed volume and the fluid flows through it. Dz Dy Dx

10. (x, y, z) Dz . Dy Dx x Introduce mass flux, ρu • ρu = mass flux at (x, y, z) in the x direction. • Flux is mass per unit time per area • Mass flux out =

11. Introduce mass flux, ρu (x, y, z) • ρu = mass flux at (x, y, z) in the x direction. • Flux is mass per unit time per area • Mass flux in = Dz . Dy Dx x

12. Mass out right (downstream) face Mass in left (upstream) face What is the change of mass inside the fixed volume? The change of mass in the box is equal to the mass that flows into the box minus the mass that flows out of the box = (flux) x (area)

13. Note: this is change in mass per unit volume. Recognizing the definition of divergence Extend to 3-Dimensions The change of mass in the box is equal to the mass that flows into the box minus the mass that flows out of the box = (flux) x (area)

14. 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.

15. Dz Dy Dx The Lagrangian point of view is that the parcel is moving. And it changes shape…

16. In-Class Exercise:Derive the Lagrangian Form • Remember, we can write the continuity equation • Use the chain rule (e.g., ) to go from the above equation to

17. 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

18. Our System of Equations We have u, v, w, ρ, p which depend on (x, y, z, t). We need one more equation for the time rate of change of pressure… (Wednesday)

19. Scale Analysis • Remember, we want to solve the system of equations that describes the atmosphere so that we can • understand how the atmosphere works • predict the motion and state of the atmosphere • Scale analysis  simplify the equations • Identify which processes are most important

20. Our System of Equations(so far…)

21. Consider x and y components of the momentum equations Remember the units—each term must have units of acceleration m/sec2 or L/t2

22. Would like to define scales in terms of wind, pressure, and density distance = rate x time Estimate time as distance/(some average rate) L ≡ some characteristic distance U ≡ some characteristic speed Characteristic time ≡ L/U

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

24. acceleration Let us define:

25. U*W/a U*U/L Uf Wf U*U/a What are the scales of the terms?Horizontal momentum equations:

26. What are the scales of the terms? For “large-scale” mid-latitude

27. What are the scales of the terms? For “large-scale” mid-latitude Largest Terms

28. Low Pressure High Pressure Consider only the largest termsNo D( )/Dt term = balance In this case: geostrophic balance

29. What are the scales of the terms? For a tornado (In-class exercise)

30. What are the scales of the terms? For a tornado Largest Terms

31. Cyclostrophic Balance:Pressure gradient vs. Centrifugal Acceleration here is the centrifugal acceleration

32. What are the scales of the terms? For the vertical motion g W*U/L Uf U*U/a

33. What are the scales of the terms? For the vertical motion Hydrostatic relation The vertical acceleration Dw/Dt is 8 orders of magnitude smaller than this balance. The ability to use the vertical momentum equation to estimate w is essentially nonexistent.

34. How to compute vertical motions? • The vertical acceleration Dw/Dt is 8 orders of magnitude smaller than hydrostatic balance. • The ability to use the vertical momentum equation to estimate w is essentially nonexistent. • Vertical motion is important: rising motion leads to clouds and precipitation… • Hence: • w must be “diagnosed” from some balance • exception: small scales: thunderstorms, tornadoes • We will return to this in a moment…

35. Scale Analysis of theContinuity Equation

36. Define a background pressure 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 • Total pressure and density = sum of background + perturbations (perturbations vary in x, y, z, t)

37. Use the continuity equationEulerian form Perturbation Base State Advection Divergence

38. Divide by ρ0 and assume ρ’/ρ0 ~ 10-2 <<1 (ρ’U)/(Lρ0) W/H 10-7 s-1 10-6 s-1

39. Look at the velocity divergence (ρ’U)/(Lρ0) W/H 10-7 s-1 These are U/L, but they balance 10-6 s-1 ~ 10-6 s-1

40. Cancel relatively small terms Remember, horizontal variations of ρ0 are zero and we can add them back in Multiply through by ρ0 and we can write

41. What can we do with scale analysis? • Remember our goals • Analysis of how the atmosphere works • Prediction of how the atmosphere will evolve in time

42. What are the scales of the terms? For “large-scale” mid-latitude Largest Terms

43. Geostrophic Balance There is no D( )/Dt term. Hence, no acceleration, no change with time. This is a DIAGNOSTIC equation that can be used to analyze how the atmosphere works

44. Geostrophic & observed wind 300 mb

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

46. Our prediction equation for large scale midlatitudes

47. Our prediction equation for large scale midlatitudes We used the definition of the geostrophic component of the wind. Within 10-15% of real wind in middle latitudes, large-scale.

48. Our prediction equation for large scale midlatitudes For middle latitudes and large scales, the acceleration can be computed directly as the difference from geostrophic balance. Remember: pressure and density are buried inside the definition of the geostrophic wind. The mass field and velocity field are linked.

49. Ageostrophic Wind • Acceleration can be computed from the difference between the real wind and the geostrophic wind • Acceleration defined as: • Change in direction (curvature/rotation) • Along-flow change in speed (convergence/divergence)

50. Geostrophic & Full wind 500 mb