1. Stability

2. Reading Hess Tsonis Wallace & Hobbs Bohren & Albrecht pp 92 - 106

3. Objectives Be able to provide the definition of stability
Be able to describe the two methods by which air is displaced
Be able to identify the types of clouds that form during either forced ascent or auto-convective ascent

4. Objectives
Be able to describe how saturation mixing ratio affects the pseudo-adiabatic lapse rate
Be able to describe the changes in meteorological parameters during forced ascent or auto-convective ascent

5. Objectives
Be able to determine the stability of an atmospheric layer by comparing the environmental lapse rate with either dry or pseudoadiabatic lapse rates
Be able to describe the concept of static stability
Be able to determine if the atmosphere is statically stable

6. Objectives Be able to describe the concept of potential instability
Be able to determine if the atmosphere is potentially unstable
Be able to identify the buoyancy equation and describe its significance

7. Objectives
Be able to identify the type of motions that the Brunt – Vaisala Frequency describes

8. Meteorological Stability
The ability of the air to return to its origin after displacement

9. Stability
Depends on the thermal structure of the atmosphere

10. Stability
Can be classified into 3 categories
Stable
Neutral
Unstable

11. Stable
Returns to original position after displacement

12. Neutral
Remains in new position after being displaced

13. Unstable
Moves farther away from its original position

14. Stability How is air displaced? Two methods 1.) Forced Ascent
2.) Auto-Convective Ascent

15. Forced Ascent Some mechanism forces air aloft
Usually synoptic scale feature
Cold air
Warm air
Cool Air

16. Forced Ascent Type of clouds Depends on stability
Stable - Stratus Unstable - Cumulus

17. Auto-Convective Ascent
Air becomes buoyant by contact with warm ground
Usually microscale or mesoscale
Cool
Hot
Cool

18. Auto-Convective Ascent
Type of Clouds
Cumulus

19. Parcel Theory Assumptions Thermally insulated from its environment
Temperature changes adiabatically
Always at the same pressure as the environment at that level
Te,P
Tp,P w

20. Parcel Theory Assumptions Hydrostatic equilibrium
Moving slow enough that its kinetic energy is a negligible
Te,P
Tp,P w

21. Stability As parcel rises 1.) Parcel Temperature Changes Unsaturated?
Dry Adiabatic Lapse Rate

22. Dry Adiabatic Lapse Rate

23. Stability Pseudo- Adiabat Mixing Ratio line 4 UnsaturatedParcel
Temperature 3 2
Height (km)
Dry
Adiabat 1
-20
-10 10 20
Temperature (°C)

24. Stability As parcel rises 1.) Parcel Temperature Changes Saturated?
Pseudoadiabatic Lapse Rate

25. Pseudo-Adiabatic Lapse Rate
First Law of Thermodynamics
lv = latent heat of vaporization
dws = change in mixing ratio due to condensation

26. Pseudo-Adiabatic Lapse Rate
Hydrostatic Equation

27. Pseudo-Adiabatic Lapse Rate
Divide by cpdz

28. Pseudo-Adiabatic Lapse Rate
Rearrange

29. Pseudo-Adiabatic Lapse Rate
Varies with dws/dT
Big
Small
4oC km-1 < Gs < 9.8oC km-1

30. Pseudoadiabats Pressure (mb) -60 -50 -40 -30 -20 200 -10 300 400 10 20
Pressure (mb)
400 10 20
500 30
600 40
700
800 2 5 9 14 17 22 25 30
900
1000

31. Stability Pseudo- Adiabat Mixing Ratio line 4 Saturated Parcel
Temperature 3 2
Height (km)
Dry
Adiabat 1
-20
-10 10 20
Temperature (°C)

32. Stability As parcel rises 2.) Environmental Temperature Changes
Environmental Lapse Rate (ge)

33. Stability Pseudo- adiabat (Gs) Mixing Ratio line 4 Parcel Temperature3 2
Height (km)
Environmental
Temperature (ge)
Dry
Adiabat (Gd) 1
-20
-10 10 20
Temperature (°C)

34. Stability Environmental temperature profile depends on many factors
Advection
Sinking Air
Warm Air Advection
Cold Air Advection

35. Stability
Environmental Temperature
Measured by rawinsonde

36. Stability Te Tp Te Tp Te Tp Stability depends on Temperature
Environment
Parcel
Condition of Parcel
Unsaturated Te Tp Te Tp Te Tp

37. Unsaturated
Unstable
Once displaced, continues

38. Unsaturated-Unstable
Dry Adiabat Gd
(Parcel Temperature) 4 3
ge > Gd 2
Height (km)
Environmental
Lapse Rate (ge) 1
-20
-10 10 20
Temperature (°C)

39. Unsaturated
Neutral
Once displaced, stays

40. Unsaturated-Neutral Dry Adiabat Gd (Parcel Temperature) 4 3 ge = Gd 2
Height (km)
Environmental
Lapse Rate (ge) 1
-20
-10 10 20
Temperature (°C)

41. Unsaturated
Stable
Once displaced, returns

42. Unsaturated-Stable 4 ge < Gd 3 Dry Adiabat Gd (Parcel Temperature)2
Height (km)
Environmental
Lapse Rate (ge) 1
-20
-10 10 20
Temperature (°C)

43. Stability A real environmental sounding sometimes combines all threege Gd
A real environmental sounding sometimes combines all three
Evaluate each layer

44. Stability Te Tp Te Tp Te Tp Stability depends on Condition of Parcel
Saturated Te Tp Te Tp Te Tp

45. Saturated-Unstable Pseudoadiabat (Gs) 4 3 ge > Gs 2 Height (km)
Environmental
Lapse Rate (ge) 1
-20
-10 10 20
Temperature (°C)

46. Saturated-Neutral Pseudoadiabat (Gs) 4 3 ge = Gs 2 Height (km)
Environmental
Lapse Rate (ge) 1
-20
-10 10 20
Temperature (°C)

47. Saturated-Stable Environmental Lapse Rate (ge) 4 3 2 Height (km)
ge < Gs 1
Pseudoadiabat
(Gs)
-20
-10 10 20
Temperature (°C)

48. Stability Dry (or Unsaturated) ge > Gd ge = Gd ge < Gd
Temperature
Height
Height
Height
Temperature
Temperature
ge > Gd
ge = Gd
ge < Gd
Dry Unstable
Dry Neutral
Dry Stable

49. Stability Saturated ge > Gs ge = Gs ge < Gs Saturated Unstable
Temperature
Height
Height
Height
Temperature
Temperature
ge > Gs
ge = Gs
ge < Gs
Saturated Unstable
Saturated Neutral
Saturated Stable

50. Stability Combine to simplify Absolutely Unstable Dry Neutral
Conditionally Unstable
Saturated Neutral
Absolutely Stable

51. Conditionally Unstable
Temperature
Height
Height
Height
Temperature
Temperature
ge > Gd
ge = Gd
ge < Gd
ge > Gs
Absolutely Unstable
Dry Neutral
Conditionally Unstable
Height
Height
Temperature
Temperature
ge = Gs
ge < Gs
Saturated Neutral
Absolutely Stable

52. Stability Conditional Instability
Saturated
Adiabatic
Lapse
Rate (Gs)
Conditional Instability
Depends upon whether the parcel is dry or saturated
Environmental
Lapse Rate (ge) 4 3
Height (km) 2
Dry
Adiabatic
Lapse Rate (Gd ) 1
-20
-10 10 20
Temperature (°C)

53. Stability Conditional Instability Unsaturated Parcel Stable 4 3
Adiabatic
Lapse
Rate (Gs)
Conditional Instability
Unsaturated Parcel
Stable
Environmental
Lapse Rate (ge) 4 3
Height (km) 2
Dry
Adiabatic
Lapse Rate (Gd ) 1
-20
-10 10 20
Temperature (°C)

54. Stability Conditional Instability Saturated Parcel Unstable 4 3
Adiabatic
Lapse
Rate (Gs)
Conditional Instability
Saturated Parcel
Unstable
Environmental
Lapse Rate (ge) 4 3
Height (km) 2
Dry
Adiabatic
Lapse Rate (Gd ) 1
-20
-10 10 20
Temperature (°C)

55. Other Types of Stability
Static Stability
Potential (or Convective) Instability

56. Stability Static Stability
The change of potential temperature with height

57. Static Stability
Atmosphere is said to be statically stable if potential temperature increases with height
Typical of atmosphere

58. Static Stability Bigger q Pressure (mb) Smaller q Temperature (oC) -40
1000
900
800
700
600
500
300
200
400
Temperature (oC) 30 40 20 10
-10
-20
-30
-40
-50
-60
Smaller q
Bigger q

59. Static Stability
Large
Temperature Inversions
Tropopause
Stratosphere

60. Strong Static Stability Strong Static Stability
-60
-50
-40
-30
-20
200
Bigger q
Strong Static Stability
-10
300
Pressure (mb)
400 10 20
500 30
600 40
700
800
Strong Static Stability
900
Smaller q
1000
Temperature (oC)

61. Potential Instability
The state of an unsaturated layer (or column) of air in the atmosphere
Either
Wet bulb potential temperature (qw) or
Equivalent potential temperature (qe)
Decreases with elevation

62. Potential Instability
Also Called Convective Instability

63. Potential Instability
Common when dry layer tops a warm, humid layer
Low level southerly flow
Upper level southwesterly flow
Warmer & Dry
Warm & Moist

64. Potential Instability
Layered Lifting (lowest 100 mb)

65. Potential Instability
Bottom of Layer
New Temp.
LCL

66. Potential Instability
Top of Layer
New Temp.
LCL

67. Potential Instability
Layer’s Old Lapse Rate

68. Potential Instability
Layer’s New Lapse Rate

69. Potential Instability
Compare Change in Layer
More Unstable

70. Potential Instability
Lifting Destabilizes Layer
Cold air
Warm air
Cool Air

71. Stability
Now the math! But don’t cry ..

72. Archimedes’ Principle
The buoyant force exerted by a fluid on an object in the fluid is equal in magnitude to the weight of fluid displaced by the object.
Archimedes
287 – 211 BC

73. Archimedes’ Principle
‘Square’ bubble in a tank of water B
B = buoyancy force

74. Archimedes’ Principle
Water pressure in tank increases with depth B p z

75. Archimedes’ Principle
Water is in hydrostatic equlibrium B

76. Archimedes’ Principle
Force on bottom of ‘bubble’
Fbottom

77. Archimedes’ Principle
Force on top of ‘bubble’
Ftop
Fbottom

78. Archimedes’ Principle
Buoyancy Force B
Ftop
Fbottom

79. Archimedes’ Principle
Horizontal Pressure Differences Balance

80. Archimedes’ Principle
Pressure Difference Between Top & Bottom
ptop
pbottom

81. Archimedes’ Principle
Combine Equations B
ptop
pbottom

82. Buoyancy Similar to parcel of air in atmosphere At Equilibrium
Density of Parcel Same as Density of Environment

83. Buoyancy
Density Difference Results in Net Buoyancy Force B

84. Buoyancy
Density Difference Results in Net Buoyancy Force B

85. Buoyancy
Net Buoyancy Force B

86. Buoyancy
Divide by mass a

87. Buoyancy a
Ideal gas law

88. Buoyancy a Parcel Theory Assumption
Pressure inside parcel same as environmental pressure
Not valid for large accelerations

89. Buoyancy a
Rearrange

90. Buoyancy a
Temperature difference results in parcel acceleration

91. Stability a Parcel Temperature T0
Cools at the dry adiabatic lapse rate a z
Tp= parcel temp
T0= sfc temp.
Gd= dry adiabatic lapse rate
z= height above sfc
-Gdz T0

92. Stability a Environmental Temperature T0 T0 Measured by radiosonde z
-gez
Te= environmental temp
T0= sfc temp.
ge= environmental lapse rate
z= height above sfc
-Gdz T0 T0

93. Stability
Substitute

94. Stability Dry (or Unsaturated) ge > Gd ge = Gd ge < Gd
Temperature
Height
Height
Height
Temperature
Temperature
ge > Gd
ge = Gd
ge < Gd
Dry Unstable
Dry Neutral
Dry Stable

95. Stability Saturated ge > Gs ge = Gs ge < Gs Saturated Unstable
Temperature
Height
Height
Height
Temperature
Temperature
ge > Gs
ge = Gs
ge < Gs
Saturated Unstable
Saturated Neutral
Saturated Stable

96. Stability
Initially stable air is displaced

97. Stability
How will it react once displacing force is removed?

98. Brunt – Vaisala Frequency
Define coefficient of z as N

99. Brunt – Vaisala Frequency
Substitute
Hey ... this looks like a differential equation!

100. Brunt – Vaisala Frequency
For stable conditions
where
where A & B are constants of integration

101. Brunt – Vaisala Frequency
Buoyancy Oscillation
Brunt – Vaisala Frequency
Brunt – Vaisala Period

102. Brunt – Vaisala Frequency
Theory behind buoyancy oscillations in atmosphere
Orographic

103. Brunt – Vaisala Frequency
Theory behind buoyancy oscillations in atmosphere
Temperature (°C)
Height (km) 2 4 6 8
CCL 10 12
Convective

104. Static Stability The change of potential temperature with height
Proved empirically
Pressure (mb)
1000
900
800
700
600
500
300
200
400
Temperature (oC) 30 40 20 10
-10
-20
-30
-40
-50
-60
Smaller q
Bigger q

105. Static Stability
Mathematically ...
Take the logarithm

106. Static Stability
Differentiate

107. Static Stability
Multiply by T

108. Static Stability
Substitute Ideal Gas Law

109. Static Stability
Hydrostatic Equation

110. Static Stability
Divide by CpTdz

111. Static Stability
Remember

112. Potential Temperature Increases in a Statically Stable Atmosphere
Static Stability
For stable atmosphere (ge < Gd )
Potential Temperature Increases in a Statically Stable Atmosphere