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AOSS 401, Fall 2007 Lecture 26 November 12 , 2007

AOSS 401, Fall 2007 Lecture 26 November 12 , 2007. Richard B. Rood (Room 2525, SRB) rbrood@umich.edu 734-647-3530 Derek Posselt (Room 2517D, SRB) dposselt@umich.edu 734-936-0502. Class News November 12 , 2007. Homework 6 due today (how did it go?)

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AOSS 401, Fall 2007 Lecture 26 November 12 , 2007

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  1. AOSS 401, Fall 2007Lecture 26November 12, 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 November 12, 2007 • Homework 6 due today (how did it go?) • (From the number of emails I got over the weekend…) • Computing assignment • Second component assigned this week • Due 5 December (or thereabouts) • I will introduce this today… • Important Dates: • November 16: Next Exam (Review on Wed. 14th) • November 21: No Class • December 10: Final Exam

  3. Today • Planetary boundary layer effects on midlatitude weather systems • A few last illustrations of cyclone development • (Introduce computing assignment)

  4. Atmospheric Boundary Layer • Until now, we have applied the governing equations to the free atmosphere • Left out the effects of friction (and turbulence) • Friction and turbulence are important • Wind direction near the surface • Decay of low pressure systems

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

  6. Boundary Layer ObservationsTurbulence (how to quantify this?) http://lidar.ssec.wisc.edu/experiments/wisc98/jan/011998/index.htm

  7. Atmospheric Boundary Layer Topics for today: • Flow in the boundary layer (Ekman layer) • (How to represent the effects of turbulence?) • (How much does the wind turn? How far above the surface will it be roughly geostrophic?) • Vertical motion near the surface • (Convergence  rising motion… how fast?) • Spin-down time for midlatitude cyclones • (How quickly will a storm decay and why?)

  8. Atmospheric Boundary Layer Topics for today: • Flow in the boundary layer (Ekman layer) • (How to represent the effects of turbulence?) • (How much does the wind turn? How far above the surface will it be roughly geostrophic?) • Vertical motion near the surface • (Convergence  rising motion… how fast?) • Spin-down time for midlatitude cyclones • (How quickly will a storm decay and why?)

  9. Need to average small scale turbulence to quantify the effects on the large scale flow Flow in the Boundary Layer • Turbulence • Small scale • Random • Fluctuates quickly • Flow in Midlatitude Storms • Large scale • Deterministic (can use equations of motion) • Changes slowly

  10. Large-Scale Effectsof Turbulence: Averaging • Divide flow into mean and perturbation • What if we have the mean of, say, uw? • Mean of perturbation = 0, mean of a mean is… the mean

  11. Large-Scale Effectsof Turbulence: Averaging • Why doesn’t ? • In the past, we assumed the product of perturbations was small—this is different! • Here we are trying to capture the accumulated effect of all of these perturbations over some time • What is ?

  12. Boundary Layer ObservationsTurbulence (how to quantify this?) http://lidar.ssec.wisc.edu/experiments/wisc98/jan/011998/index.htm

  13. Equations of Motion for the Large Scale Boundary Layer Make some assumptions… • Constant density (except for when we are looking at buoyancy… which we are not) • Constant coriolis parameter (f) (Consequence of constant density)

  14. Now, bear with me……we’re going to write the material derivative in flux form… Expand the material derivative

  15. Now, bear with me……we’re going to write the material derivative in flux form… Expand the material derivative Add zero

  16. Now, bear with me……we’re going to write the material derivative in flux form… Expand the material derivative Add zero Rearrange

  17. Now, bear with me… …we’re going to write the material derivative in flux form… Expand the material derivative Add zero Rearrange Use the chain rule…

  18. Flux form… why would we do this? • Remember our averaging? • We can now put together a set of averaged equations (u, v, and continuity)

  19. Material derivative with each wind component = average Averaged Equations:Material Derivative What does look like? Use flux form and the continuity equation and remember

  20. Averaged Equations We can now write equations of motion that include the mean effect of turbulence!

  21. assume horizontally homogeneous turbulence; only the terms involving w’ matter Averaged Equations We’re done, right? Nope. Sorry. What do we do with u’, v’, w’ ?

  22. Boundary Layer ObservationsTurbulence (how to quantify this?) http://lidar.ssec.wisc.edu/experiments/wisc98/jan/011998/index.htm

  23. Flux Gradient Theory • Assume that turbulent transport acts like molecular diffusion (friction) • Transport of u-momentum by w’ is proportional to the vertical gradient of u (vertical shear; du/dz) • Remember when we first talked about friction? (way back to lecture 2…)

  24. Viscous force velocity ≡ u m/sec Ocean Land Biosphere Velocity is zero at the surface; hence, there is some velocity profile.

  25. Viscous force (How do we think about this?) • Recognize the change in u with height can be represented by ∂u/∂z F = μA (∂u/∂z) u(h) = u0 Fzx/A = μ(∂u/∂z) h u(z) u(0) = 0 • = dynamic viscosity coefficient (depends on properties of the fluid)

  26. Viscous force (using definition of t) n ≡ m/r = kinematic viscosity coefficient

  27. Flux Gradient TheoryTurbulence acts like friction… • Write each of the turbulence terms involving w’ as (for example) • Km is the “eddy viscosity coefficient” and has units of m2/s (Typical value of Km is approximately 5 m2/s) Fzx/A = μ(∂u/∂z)

  28. Averaged Equations Plug in the assumption of horizontally homogeneous turbulence and Km

  29. Flux Gradient TheoryTurbulence acts like friction… • Does the turbulent friction look familiar? Turbulent friction Molecular viscosity

  30. Solving the averaged equations • A few more simplifications… • From scale analysis, the Du/Dt and Dv/Dt and friction terms are small compared to pressure gradient, coriolis, and turbulence • Use the definition of the mean geostrophic wind

  31. Averaged Equations We’re done! These are the Ekman Layer Equations A three-way balance between pressure gradient force, coriolis, and turbulent friction

  32. PGF FRIC COR three-way balance between pressure gradient force, coriolis, and friction Wind north Φ0 ΔΦ Φ0+ΔΦ Φ0+2ΔΦ Φ0+3ΔΦ east south west

  33. DONE Atmospheric Boundary Layer Topics for today: • Flow in the boundary layer (Ekman layer) • (How to represent the effects of turbulence?) • (How much does the wind turn? How far above the surface will it be roughly geostrophic?) • Vertical motion near the surface • (Convergence  rising motion… how fast?) • Spin-down time for midlatitude cyclones • (How quickly will a storm decay and why?)

  34. DONE Atmospheric Boundary Layer Topics for today: • Flow in the boundary layer (Ekman layer) • (How to represent the effects of turbulence?) • (How much does the wind turn? How far above the surface will it be roughly geostrophic?) • Vertical motion near the surface • (Convergence  rising motion… how fast?) • Spin-down time for midlatitude cyclones • (How quickly will a storm decay and why?)

  35. Ekman Layer Equations • What is the angle between real wind and geostrophic wind near the surface? • How far above the surface will the wind be roughly geostrophic?

  36. Solving the Ekman Layer Equations • Note: u = v = 0 at the surface, u ug, v  vg as z  infinity • Assume ug is constant with height and vg = 0 • (Skip a whole lot of details…)

  37. Ekman Layer Solution • At z =π/γ, the wind is geostrophic • v-component is due to friction • Depth of the Ekman Layer is De =π/γ • For typical f, Km, De≈ 1 km

  38. Ekman Layer Hodograph • Wind turns with height from approximately 45o at the surface to purely geostrophic at the top of the Ekman Layer Theory Measurements

  39. DONE DONE Atmospheric Boundary Layer Topics for today: • Flow in the boundary layer (Ekman layer) • (How to represent the effects of turbulence?) • (How much does the wind turn? How far above the surface will it be roughly geostrophic?) • Vertical motion near the surface • (Convergence  rising motion… how fast?) • Spin-down time for midlatitude cyclones • (How quickly will a storm decay and why?)

  40. DONE DONE Atmospheric Boundary Layer Topics for today: • Flow in the boundary layer (Ekman layer) • (How to represent the effects of turbulence?) • (How much does the wind turn? How far above the surface will it be roughly geostrophic?) • Vertical motion near the surface • (Convergence  rising motion… how fast?) • Spin-down time for midlatitude cyclones • (How quickly will a storm decay and why?)

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

  42. Vertical Motion in the Boundary Layer • Turbulent drag causes wind to deviate from geostrophic near the surface • Causes convergence into the center of low pressure systems • Mass conservation requires rising motion • How fast will the air be rising?

  43. Vertical Motion in the Boundary Layer • Use Ekman equations (vg = 0) • Flux toward center is ρ0v • To get the mass transport out of the boundary layer (and by extension w) • Integrate ρ0v from surface to top of Ekman layer • Use the continuity equation to get w • (See Holton pp. 131-132 for details…)

  44. Vertical Motion in the Boundary Layer Vertical motion proportional to geostrophic vorticity! Greater ζg leads to:  greater wind velocity  greater friction • greater inward mass flux  greater upward mass flux

  45. Vertical Motion in the Boundary Layer • Think about a strong extratropical cyclone • ∂u/∂y ≈ ∂v/∂x ≈ (20 m/s)/(100 km) • f = 10-4 s-1 • Km = 5 m2/s • w(De) ≈ 3 cm/s

  46. DONE DONE DONE Atmospheric Boundary Layer Topics for today: • Flow in the boundary layer (Ekman layer) • (How to represent the effects of turbulence?) • (How much does the wind turn? How far above the surface will it be roughly geostrophic?) • Vertical motion near the surface • (Convergence  rising motion… how fast?) • Spin-down time for midlatitude cyclones • (How quickly will a storm decay and why?)

  47. DONE DONE DONE Atmospheric Boundary Layer Topics for today: • Flow in the boundary layer (Ekman layer) • (How to represent the effects of turbulence?) • (How much does the wind turn? How far above the surface will it be roughly geostrophic?) • Vertical motion near the surface • (Convergence  rising motion… how fast?) • Spin-down time for midlatitude cyclones • (How quickly will a storm decay and why?)

  48. z, vertical x, east Effects of Friction and Turbulence on Midlatitude Cyclones • Think about the circulation in the Ekman layer (from the side)

  49. L H y, north x, east Effects of Friction and Turbulence on Midlatitude Cyclones • Think about the circulation in the Ekman layer (from the top) Opposes cyclonic (counterclockwise) circulation in a low pressure system

  50. Spindown Timescale • “Secondary circulation” forced by effects of friction causes relatively rapid spin-down of cyclones • What is the time-scale? • Remember from last time—occlusion = change from baroclinic to barotropic disturbance

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