Atmospheric Stability and Buoyancy
?
We just the covered the large-scale hydrostatic environment…
We now need to understand whether a small-scale moist air parcel
will spontaneously rise or sink through the atmosphere
Thermodynamics
M. D. Eastin
Atmospheric Stability and Buoyancy
Outline:
 Review
 Dry Adiabatic (unsaturated) Processes
 Moist Adiabatic (saturated) Processes
 Concepts Stability and Buoyancy
 Forced vertical motions
 Spontaneous vertical motions
 Atmospheric Stability Analysis
 Criteria for Unsaturated Air
 Criteria for Saturated Air
 Conditional Instability
 Level of Free Convection (LFC)
Thermodynamics
M. D. Eastin
Review of Dry Adiabatic Processes
Basic Idea:
• No heat is added to or taken from the system
which we assume to be an air parcel
dq  cvdT  pdα  0
Parcel
dq  cpdT  dp  0
• Changes in temperature result from either
expansion or contraction
• Many atmospheric processes are dry adiabatic
• We shall see that dry adiabatic process play
a large role in deep convective processes
• Vertical motions
• Thermals
Thermodynamics
M. D. Eastin
Review of Dry Adiabatic Processes
Poisson’s Relation:
• Relates the initial conditions of temperature
and pressure to the final temperature and
pressure during a dry adiabatic process
Tfinal
 p final 

 Tinitial
 pinitial 
Rd
cp
Potential Temperature:
• Special form of Poisson’s relationship
• Compress all air parcels to 1000 mb
• Provides a “standard” pressure level for
comparison of air parcels at different
altitudes
Thermodynamics
 p0 
θ  T 
 p
Rd
cp
M. D. Eastin
Review of Dry Adiabatic Processes
Dry Adiabatic Ascent or Descent:
• Air parcels undergoing dry adiabatic transformations
maintain a constant potential temperature (θ)
• During dry adiabatic ascent (expansion) the parcel’s
temperature must decrease in order to preserve
the parcel’s potential temperature
• During dry adiabatic descent (compression) the
parcel’s temperature must increase in order to
preserve the parcel’s potential temperature
Constant θ
Thermodynamics
M. D. Eastin
Review of Dry Adiabatic Processes
Dry Adiabatic Lapse Rate (Γd):
• Describes how temperature changes
with height for an air parcel moving up
or down during a dry adiabatic process
d 
dT
g
 
  9.8C / km
dz
cp
• Potential temperature is constant
• “Dry Adiabats” on the Skew-T diagram
An air parcel moving
between 1000-700 mb
parallel to a dry adiabat
Δz = 2.7 km
Using Γd we
should expect
ΔT = 26.5ºC
T700 = -12.5ºC
Dry Adiabatic
(Unsaturated)
Thermodynamics
T1000 = 14°C
z700 = 2.8 km
z1000 = 0.1 km
M. D. Eastin
Review of Moist Adiabatic Processes
Saturated Ascent:
• Once saturation is achieved (at the LCL), further ascent produces additional
cooling (adiabatic expansion) and condensation (phase changes) occur
 The parcel now contains liquid water (cloud drops)
 The condensation process releases latent heat that warms the parcel
 This heat partially offsets (cancels out) the expansion cooling
 “Pseudo-adiabats” on Skew-T diagram
Pseudo-adiabat
Moist Adiabatic Ascent
(Saturated)
(a Cloud)
TLCL
Dry Adiabatic Ascent
(Unsaturated)
Thermodynamics
Dry adiabat
Td
T
M. D. Eastin
Review of Moist Adiabatic Processes
Saturated Descent:
• A descending saturated air parcel that contains liquid water (cloud / rain drops)
will experience warming (adiabatic compression)
• The parcel will become temporarily unsaturated → cloud/rain drops evaporate
 The evaporation process absorbs latent heat that cools the parcel
 This cooling partially offsets (cancels out) the compression warming
 “Pseudo-adiabats” on Skew-T diagram
Moist Descent
(Saturated)
(Cloud evaporation)
Moist Descent
(Saturated)
(Rain evaporation)
Thermodynamics
Pseudo-adiabat
Pseudo-adiabat
M. D. Eastin
Concept of Stability
Basic Idea:
Ability of an air parcel to return to is level of origin after a displacement
Thermodynamics
M. D. Eastin
Concept of Stability
Basic Idea:
Ability of an air parcel to return to is level of origin after a displacement
Depends on the temperature structure of the atmosphere
Temperature
Dewpoint
Temperature
Thermodynamics
M. D. Eastin
Concept of Stability
Three Categories of Stability:
Stable:
• Returns to its original position
after displacement
Neutral:
• Remains in new position after
being displaced
Unstable:
• Moves further away from its original
position after being displaced
Thermodynamics
M. D. Eastin
Concept of Stability
Evidence of stability type in the atmosphere:
• The type of cloud depends on atmospheric stability
Stratus – Stable
Thermodynamics
Cumulus – Unstable
M. D. Eastin
Concept of Stability
How is air displaced? Forced Ascent
• Flow over mountains
• Flow over cold and warm fronts
Thermodynamics
M. D. Eastin
Concept of Stability
How is air displaced? Spontaneous Ascent
• Air parcel is warmer than its environment
which means the parcel is “buoyant”
• Air becomes buoyant through “heating”
Warm
Cool
Thermodynamics
Hot
Cool
M. D. Eastin
Concept of Buoyancy
Basic Idea:
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.
Bubble in a tank of water
B = Buoyancy Force
Thermodynamics
B
M. D. Eastin
Concept of Buoyancy
Basic Idea:
• Let’s forget the bubble for now…
Tank of water
•Pressure in the tank increases
with depth
L
• Pressure is the force per unit area
exerted by the weight of all the mass
lying above that height
• Identical to our atmosphere
• Water in the tank is in hydrostatic
balance
dp
 w g
dz
Thermodynamics
P
Z
H
M. D. Eastin
Concept of Buoyancy
Basic Idea:
• Water in the tank is in hydrostatic
balance
Tank of water
• At any given point within the
tank the upward directed pressure
gradient force (dp/dz) must balance
the downward directed gravitational
force (-ρwg) imposed by the weight of
the water mass above that point
F  F
dp
 w g
dz
L
-ρwg
P
Z
dp/dz
H
• The water does not move up or down
Thermodynamics
M. D. Eastin
Concept of Buoyancy
Basic Idea:
• Let’s return to our bubble!
Bubble in a tank of water
• If we examine the forces acting along
the black line located at the base of
the bubble:
• On either side of the bubble ( )
the upward and downward
directed forces balance
• At the bubble base ( ), the upward
directed pressure gradient force
is the same, but the downward
directed gravitational force is
different
-ρwg
-ρbg
-ρwg
dp/dz
dp/dz
dp/dz
• The mass of the bubble must be
taken into account (-ρbg)
Thermodynamics
M. D. Eastin
Concept of Buoyancy
Basic Idea:
Option #1:
• If the mass of the bubble is less than
the mass of the water it replaces…
Bubble in a tank of water
b  w
then the pressure gradient force will be
stronger than the gravitational force…
dp
  b g
dz
B
-ρbg
dp/dz
and the bubble will experience an
upward directed buoyancy force (B)
• The bubble will accelerate upward!
Thermodynamics
M. D. Eastin
Concept of Buoyancy
Basic Idea:
Option #2:
• If the mass of the bubble is greater than
the mass of the water it replaces…
Bubble in a tank of water
b   w
then the pressure gradient force will be
weaker than the gravitational force…
dp
  b g
dz
B
-ρbg
dp/dz
and the bubble will experience an
downward directed buoyancy force (B)
• The bubble will accelerate downward!
Thermodynamics
M. D. Eastin
Concept of Buoyancy
Basic Idea: A Different View…
At the moment of Archimedes’ famous discovery
Thermodynamics
M. D. Eastin
Concept of Buoyancy
Basic Idea: Applied to the Atmosphere…
• Large-scale environment is in
hydrostatic balance
dp
  e g
dz
• If the density of a moist air
parcel (ρp) is less than the
density of the environmental
air (ρe) that it displaces, then
the air parcel will experience
an upward directed buoyancy
force (B):
B
ρe
ρp
ρe
p  e
Thermodynamics
M. D. Eastin
Concept of Buoyancy
Basic Idea: Applied to the Atmosphere…
• Large-scale environment is in
hydrostatic balance
dp
  e g
dz
• If the density of a moist air
parcel (ρp) is greater than the
density of the environmental
air (ρe) that it displaces, then
the air parcel will experience
a downward directed buoyancy
force (B):
B
ρe
ρp
ρe
p  e
Thermodynamics
M. D. Eastin
Concept of Buoyancy
Basic Idea: Applied to the Atmosphere…
• Recall from the Ideal Gas Law:
p  ρR d Tv
Warm Air Rises!
virtual temperature of an air parcel
is inversely proportional to density
• If the virtual temperature of a moist
air parcel (Tvp) is greater than that
of the nearby environmental air (Tve),
then the air parcel will experience
an upward directed buoyancy
force (B):
B
Tve
Tvp
Tve
Tp  Te
Thermodynamics
M. D. Eastin
Concept of Buoyancy
Basic Idea: Applied to the Atmosphere…
• Recall from the Ideal Gas Law:
p  ρR d Tv
Cold Air Sinks!
virtual temperature of an air parcel
is inversely proportional to density
• If the virtual temperature of a moist
air parcel (Tvp) is less than that of
the nearby environmental air (Tve),
then the air parcel will experience
a downward directed buoyancy
force (B):
B
Tve
Tve
Tve
Tp  Te
Thermodynamics
M. D. Eastin
Concept of Buoyancy
Mathematical Definition of Buoyancy:
• See your text for the full derivation
 Tvp  Tve 

B  g 
 Tve 
Buoyancy Force
(Virtual Temperature Form)
• Other commonly used forms that are roughly equivalent…
Temperature Form
 Tp  Te 

B  g 
 Te 
Thermodynamics
Potential
Temperature Form
 θp  θe 

B  g 
 θe 
Virtual Potential
Temperature Form
 θ vp  θ ve 

B  g 
 θ ve 
M. D. Eastin
Atmospheric Stability Analysis
Basic Idea: Unsaturated Air
Use the observed atmospheric temperature profile to determine
the stability of an unsaturated air parcel after vertical displacement
Assume: Upward displacement
• Compare Γd to the observed
lapse rate (Γ)
• Will the new parcel temperature
be colder than, warmer than,
or equivalent to the nearby
environment?
Thermodynamics
Height
• The air parcel will always cool at
the dry adiabatic lapse rate (Γd)
Γd
Γ
(parcel)
(environment)
Temperature
M. D. Eastin
Atmospheric Stability Analysis
Stable:
  d
Parcel becomes
colder
than nearby
environment
Downward
Buoyancy
Force
Parcel will
return to
original
location
Height
Criteria for Unsaturated Air Parcel:
Γ
Γd
Temperature
  d
Parcel becomes
equivalent to
the nearby
environment
No
Buoyancy
Force
Parcel will
remain
at new
location
Height
Neutral:
Γ
Γd
Temperature
  d
Parcel becomes
warmer
than nearby
environment
Upward
Buoyancy
Force
Parcel will
move
further away
from original
location
Height
Unstable:
Γd
Γ
Temperature
Thermodynamics
M. D. Eastin
Atmospheric Stability Analysis
Application: Unsaturated Air
Compare the observed lapse rate (Γ)
(temperature change with height)
to the local dry adiabatic lapse rate (Γd)
Temperature
Neutral
  d
Unstable
  d
Stable
  d
Thermodynamics
M. D. Eastin
Atmospheric Stability Analysis
Application: Unsaturated Air
Compare the observed lapse rate (Γ)
(temperature change with height)
to the local dry adiabatic lapse rate (Γd)
G
Temperature
F
E
D
C
B
Thermodynamics
A
M. D. Eastin
Atmospheric Stability Analysis
Basic Idea: Saturated Air
Use the observed atmospheric temperature profile to determine
the stability of a saturated air parcel after vertical displacement
Assume: Upward displacement
• The air parcel will always cool at the
pseudo-adiabatic lapse rate (Γs)
Γs
• Compare Γs to the observed
lapse rate (Γ)
• Will the new parcel temperature
be colder than, warmer than,
or equivalent to the nearby
environment?
Thermodynamics
Height
(parcel)
Γ
(environment)
Temperature
M. D. Eastin
Atmospheric Stability Analysis
Stable:
  s
Parcel becomes
colder
than nearby
environment
Downward
Buoyancy
Force
Parcel will
return to
original
location
Height
Criteria for Saturated Air Parcel:
Γ
Γs
Temperature
  s
Parcel becomes
equivalent to
the nearby
environment
No
Buoyancy
Force
Parcel will
remain
at new
location
Height
Neutral:
Γ
Γs
Temperature
  s
Parcel becomes
warmer
than nearby
environment
Upward
Buoyancy
Force
Parcel will
move
further away
from original
location
Height
Unstable:
Γs
Γ
Temperature
Thermodynamics
M. D. Eastin
Atmospheric Stability Analysis
Application: Saturated Air
Compare the observed lapse rate (Γ)
(temperature change with height)
to the local pseudo-adiabatic lapse rate (Γs)
Temperature
Neutral
  s
Unstable
  s
Stable
  s
Thermodynamics
M. D. Eastin
Atmospheric Stability Analysis
Application: Saturated Air
Compare the observed lapse rate (Γ)
(temperature change with height)
to the local pseudo-adiabatic lapse rate (Γs)
G
Temperature
F
E
D
C
B
A
Thermodynamics
M. D. Eastin
Atmospheric Stability Analysis
Combined Criteria for Moist Air (either saturated or unsaturated):
Absolutely Unstable:
Saturated
parcel becomes
warmer
than nearby
environment
Γd
Height
Γ  Γd  Γs
Unsaturated
parcel becomes
warmer
than nearby
environment
Γs
Γ
Temperature
Dry Neutral:
Unsaturated
parcel becomes
equivalent to
the nearby
environment
Saturated
parcel becomes
warmer
than nearby
environment
Γd
Height
Γ  Γd  Γs
Γs
Γ
Temperature
Thermodynamics
M. D. Eastin
Atmospheric Stability Analysis
Combined Criteria for Moist Air (either saturated or unsaturated):
Conditionally Unstable:
Saturated
parcel becomes
warmer
than nearby
environment
Γ
Height
Γd  Γ  Γs
Unsaturated
parcel becomes
colder
than nearby
environment
Γd
Γs
Temperature
 The vertical temperature profile at most locations in our atmosphere
is conditionally unstable
• This is an important special case that we will return to in a little bit…
Thermodynamics
M. D. Eastin
Atmospheric Stability Analysis
Combined Criteria for Moist Air (either saturated or unsaturated):
Moist Neutral:
Saturated
parcel becomes
equivalent to
the nearby
environment
Γ
Height
Γd  Γ  Γs
Unsaturated
parcel becomes
colder
than nearby
environment
Γs
Γd
Temperature
Absolutely Stable:
Saturated
parcel becomes
colder
than nearby
environment
Γs
Height
Γ d  Γs  Γ
Unsaturated
parcel becomes
colder
than nearby
environment
Γ
Γd
Temperature
Thermodynamics
M. D. Eastin
Atmospheric Stability Analysis
Application: Moist Air
Compare the observed lapse rate (Γ)
(temperature change with height)
to the local dry adiabatic lapse rate (Γd)
and the pseudo-adiabatic lapse rate (Γs)
E
D
C
B
A
Thermodynamics
M. D. Eastin
Atmospheric Stability Analysis
Conditional Instability:
• Unsaturated air parcels experiencing a small vertical displacement will be
stable and experience a downward buoyancy force
 However, if the unsaturated parcel can experience enough forced ascent
with a large vertical displacement, the parcel may become saturated and
reach an altitude at which it becomes warmer than its local environment
Td
T
Where will a parcel
starting at the surface
become buoyant
due to forced ascent?
Lift the surface parcel
Thermodynamics
M. D. Eastin
Atmospheric Stability Analysis
Conditional Instability:
• Unsaturated air parcels experiencing a small vertical displacement will
stable and experience a downward buoyancy force
 However, if the unsaturated parcel can experience enough forced ascent
with a large vertical displacement, the parcel may become saturated and
reach an altitude at which it becomes warmer than its local environment
Td
T
Altitude at which
parcel first becomes
warmer than the
environment
LCL
Thermodynamics
M. D. Eastin
Atmospheric Stability Analysis
Level of Free Convection (LFC):
Definition: Altitude at which a lifted air parcel first becomes warmer than
the nearby environment (acquires an upward buoyancy force)
and begin to accelerate upward without additional forced ascent
Td
T
Level of Free
Convection
(LFC)
LCL
Thermodynamics
M. D. Eastin
Atmospheric Stability Analysis
Application: Find the Level of Free Convection (LFC)
Find the LFC
for the
surface air parcel
Thermodynamics
M. D. Eastin
Atmospheric Stability Analysis
Application: Find the Level of Free Convection (LFC)
Find the LFC
for the
surface air parcel
Thermodynamics
M. D. Eastin
Atmospheric Stability and Buoyancy
Summary:
• Review
• Dry Adiabatic (unsaturated) Processes
• Moist Adiabatic (saturated) Processes
• Concepts Stability and Buoyancy
• Forced vertical motions
• Spontaneous vertical motions
• Atmospheric Stability Analysis
• Criteria for Unsaturated Air
• Criteria for Saturated Air
• Conditional Instability
• Level of Free Convection (LFC)
Thermodynamics
M. D. Eastin
References
Houze, R. A. Jr., 1993: Cloud Dynamics, Academic Press, New York, 573 pp.
Markowski, P. M., and Y. Richardson, 2010: Mesoscale Meteorology in Midlatitudes, Wiley Publishing, 397 pp.
Petty, G. W., 2008: A First Course in Atmospheric Thermodynamics, Sundog Publishing, 336 pp.
Tsonis, A. A., 2007: An Introduction to Atmospheric Thermodynamics, Cambridge Press, 197 pp.
Wallace, J. M., and P. V. Hobbs, 1977: Atmospheric Science: An Introductory Survey, Academic Press, New York, 467 pp.
Thermodynamics
M. D. Eastin