1. Lecture 10 Static Stability

2. General Concept An equilibrium state can be stable or unstable Stable equilibrium: A displacement induces a restoring force i.e., system tends to move back to its original state i.e., system tends to move back to its original state Unstable equilibrium: A displacement induces a force that tends to drive the system even further away from its original state

4. A More Realistic Scenario Equilibrium

5. Small Displacement

15. Stable.

16. Large Displacement

21. Unstable.

22. Idea of Previous Slides There may be a critical displacement magnitude displacement < critical  stable displacement > critical  unstable (More about this shortly)

23. Atmospheric Stability Unsaturated Air

24. Consider a vertical parcel displacement,  z Assume displacement is (dry) adiabatic Assume displacement is (dry) adiabatic Change in parcel temperature = -  d  z Denote lapse rate of environment by 

25. zz Parcel T = T 0 T=T 0 -  d  z Environment T = T 0 T = T 0 -  z Temp of displaced parcel  temp of environment

26. Two Cases Parcel temp. > environment temp.  parcel less dense than environment  parcel less dense than environment  parcel is buoyant  parcel is buoyant Parcel temp. < environment temp.  parcel denser than environment  parcel denser than environment  parcel is negatively buoyant  parcel is negatively buoyant

27. Lapse Rates T parcel = T 0 -  d  z T env = T 0 -  z T parcel > T env if T 0 -  d  z > T 0 -  z   >  d   >  d T parcel < T env if T 0 -  d  z < T 0 -  z   <  d   <  d

28. Stability  >  d  resultant force is positive  parcel acceleration is upward (away from original position)  equilibrium is unstable  <  d  resultant force is negative  parcel acceleration is downward (toward original position)  equilibrium is stable

29. Graphical Depiction Temperature z Temp of rising parcel Stable lapse rate Unstable lapse rate

30. Saturated Air Recall: Vertically displaced parcel cools/warms at smaller rate Call this the moist-adiabatic rate,  m Call this the moist-adiabatic rate,  m Previous analysis same with  d replaced by  m Equilibrium stable if  <  m Equilibrium stable if  <  m Equilibrium unstable if  >  m Equilibrium unstable if  >  m

31. General Result Suppose we don’t know whether a layer of the atmosphere is saturated or not  >  d   >  m  equilibrium is unstable, regardless Equilibrium is absolutely unstable Equilibrium is absolutely unstable  <  m   <  d  equilibrium is stable, regardless Equilibrium is absolutely stable Equilibrium is absolutely stable

32. Continued Suppose  m <  <  d Layer is stable if unsaturated, but unstable if saturated Equilibrium is conditionally unstable

33. mm dd  Absolutely unstable Absolutely stable Conditionally unstable

34. Application If a layer is unstable and clouds form, they will likely be cumuliform If a layer is stable and clouds form, they will likely be stratiform

35. Example: Mid-Level Clouds Suppose that clouds form in the middle troposphere Unstable  altocumulus Stable  altostratus

36. Altocumulus

37. Altostratus

38. Deep Convection Previous discussion not sufficient to explain thunderstorm development Thunderstorms start in lower atmosphere, but extend high into the troposphere

39. Physics Review: Energy h Object at height h

40. Physics Review: Energy h Remove support: Object falls

41. Physics Review: Energy Let z(t) = height a time t z(t)

42. It Can Be Shown … kinetic energy potential energy (v = speed) As object falls, potential energy is converted to kinetic energy.

43. Available Potential Energy Object may have potential energy, but it may not be dynamically possible to release it

44. h Technically, PE = mgh, but lower energy state is inaccessible. The energy is unavailable.

45. Energy Barriers h a b To get from a to b, energy must be supplied to surmount the barrier. hbhb Energy needed: mgh b

46. Energy Barriers h a b Now, ball can roll down hill.

47. Energy Barriers h a b Amount of PE converted to KE: mg(h + h b ) Net release of energy: mg(h + h b ) – mgh b = mgh hbhb

48. CAPE, CIN CAPE: Convective Available Potential Energy (Positive area) (Positive area) CIN: Convective Inhibition (Negative area at bottom of sounding) (Negative area at bottom of sounding)

49. Negative area Positive area Dry adiabat Saturated adiabat LCL Sounding

50. CAPE, CIN CIN is the energy barrier CAPE is the energy that is potentially available if the energy barrier can be surmounted

51. Isolated Severe Thunderstorms Suppose CIN and CAPE are large Consider a population of incipient thunderstorms Few of these storms will surmount the energy barrier, however … Those that do will have a lot of energy available.

52. Sudden Outbreaks of Severe Weather Start with a high energy barrier (large CIN)

53. Sudden Outbreaks of Severe Weather Now, suppose energy barrier decreases.

54. Sudden Outbreaks of Severe Weather Disturbances that previously couldn’t overcome the barrier now can. If CAPE is large, storms could be severe.

55. Level of Free Convection (LFC) Level of Free Convection (LFC): When a parcel ceases to be colder and denser than surrounding air (environment), and instead becomes positively buoyant. On a thermo diagram, this occurs when the moist adiabat being followed by the parcel crosses from the cold side of the environmental profile to the warm side. The level at which this crossover occurs is the LFC.

56. Equilibrium Level (EL) Thermals will continue to rise until their temperature matches that of the environment. The level at which this occurs is called the equilibrium level (EL). Also called level of neutral buoyancy. At that point, parcels may overshoot a little, becoming colder than environment and ultimately falling back to their EL.

57. Convective Inhibition (CIN) Buoyant force is negative by definition, minus sign in front of integral. T(z) temp of rising parcel, T’(z) temp of environment at same level. f B = Buoyancy force

58. Convective Inhibition (CIN) Invoking hydrostatic approximation, and ideal gas law:

59. CAPE Once the parcel has overcome the energy barrier, CIN, and reached its LFC, CAPE is the energy that may be released by resulting buoyant ascent. CIN represents the energy barrier to initiation of free convection. CAPE is the maximum possible energy that can be released after CIN has been overcome. [J /kg]