AOSS 401 Geophysical Fluid Dynamics: Atmospheric Dynamics Prepared: 20130926 Scale Analysis

1. AOSS 401Geophysical Fluid Dynamics:Atmospheric DynamicsPrepared: 20130926Scale Analysis Richard B. Rood (Room 2525, SRB) rbrood@umich.edu 734-647-3530 Cell: 301-526-8572

2. Class News • Ctools site (AOSS 401 001 F13) • Disruption of schedule • Lectures for missed classes posted. • Homework posted on 18th due 26th

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

4. Outline • Scale analysis • Vertical momentum equation • Continuity equation(check out online lecture) • Vertical velocity / horizontal divergence • A predictive equation • Discussion of Francis and Vavrus

5. Vertical Momentum Equation

6. What are the scales for the vertical momentum equation g W*U/L Uf U*U/a

7. What are the scales for the vertical momentum equation 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.

8. From this first scale analysis • For “large-scale” middle latitude, dynamics • There are two balances that dominate the observed flow. • Geostrophic balance • Hydrostatic balance • What is the importance of “middle latitude?”

9. Can we say w, vertical velocity, can be ignored?

10. How to compute vertical motion? • Vertical motion is important: rising motion leads to clouds and precipitation… • 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. • Hence: • w must be “diagnosed” from some balance • exception: small scales: thunderstorms, tornadoes • We will return to this in a moment…

11. Scale Analysis of theContinuity Equation

12. The continuity equation • Remember, we can write the continuity equation • Use the product rule (e.g., ) to go from the above equation to

13. 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)

14. Use the continuity equationEulerian form 0 Perturbation 0 0 Base State Advection Divergence

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

16. Look at the velocity divergence (ρ’U)/(Lρ0) W/H 10-7 s-1 10-6 s-1 These scale as U/L, but with opposite sign ~ 10-6 s-1

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

18. What is the point? Large-scale (synoptic) vertical motions are proportional to the vertically integrated divergence

19. What is the point? • We can look for where there is divergence of the horizontal wind in the middle to upper troposphere and diagnose vertical motion in the underlying atmosphere • We will return to this in detail later…

20. Scale for different problem

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

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

23. Cyclostrophic Balance:

24. What else can we do with scale analysis? • Our goals • Analysis of how the atmosphere works • Prediction of how the atmosphere will evolve in time

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

26. 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

27. Geostrophic & observed wind 300 mb

28. From the scale analysis of the horizontal momentum equation Assume that you look at spatial scale that is small enough that f is constant and density is a function of height only. What is the divergence of the velocity in this case?

29. We are faced with a difficult situation. • The middle latitude atmosphere is in a state of near balance – geostrophic and hydrostatic. • There is no vertical motion associated with geostrophic flow • There is no acceleration associated with geostrophic flow • How does this impact our goals?

30. We are faced with a difficult situation. • We know that the acceleration scales an order of magnitude less than the geostrophic balance. • We know that the vertical velocity scales MANY order of magnitudes less than the hydrostatic balance. • It is the DIFFERENCE from these balances that are important for forecasting. • A small difference from a strong balance.

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

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

33. Our prediction equation for large scale midlatitudes

34. Our prediction equation for large scale midlatitudes We used the definition of the geostrophic component of the wind, Which is within 10-15% of the real wind in middle latitudes. (large-scale)

35. 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.

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

37. Geostrophic & observed wind 300 hPa

38. Geostrophic & Full wind 500 mb

39. Ageostrophic Wind and Vertical Motion • Remember the scaled continuity equation • Vertical motion is related to divergence, but geostrophic wind is nearly nondivergent (is exactly nondivergent on a pressure surface—we will show this later…) • Divergence of ageostrophic wind leads to vertical motion on large scales These scale as U/L, but with opposite sign ~ 10-6 s-1

40. Ageostrophic Wind and Vertical Motion • Conservation of mass relates our prediction equation to the vertical motion. • How to diagnose where there is divergence of the ageostrophic wind? • More on this in the next two weeks…

41. Thinky problem 1 • What does this figure tell us about our scale analysis? • What does this picture tell us about vertical velocity?

42. Boundary Layer Observations Effect on winds near the surface Geostrophic Observed CONVERGENCE

43. Summary Points • We have used scale analysis to extract information about “large-scale” rotational flow. • Revealed two balances from the momentum equation: • Geostrophic balance • Hydrostatic balance • These balances are “diagnostic.” They are approximations. • We are interested in differences from these balances for prediction – time evolution. • Vertical velocity is small and related to the divergence of the horizontal wind – which is the ageostrophic wind.

44. Discussion of Francis and Vavrus • From Morning Edition September 25, 2013 • Visualization from SVS Goddard • Easier to run on Youtube