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Lecture 10: Atmospheric Stability. Outline. Dry adiabatic lapse rates Moist adiabatic lapse rates Conditional instability Buoyancy and CAPE Parcel theory. Introduction. Not all important weather phenomena fit into the assumptions of quasigeostrophic theory Supercell thunderstorms

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outline
Outline
  • Dry adiabatic lapse rates
  • Moist adiabatic lapse rates
  • Conditional instability
  • Buoyancy and CAPE
  • Parcel theory
introduction
Introduction
  • Not all important weather phenomena fit into the assumptions of quasigeostrophic theory
    • Supercell thunderstorms
    • Mesoscale convective systems
    • Tornadoes
  • The dominant energy source for mesoscale disturbances is convection.
  • In this section we examine the types of mesoscale instabilities that lead to convection and examine the role of atmospheric stability in mesoscale weather systems.
rising air parcels
Rising Air Parcels
  • As the parcel rises, it will adiabatically expand and cool
    • Adiabatic: a process where the parcel temperature changes due to an expansion or compression
    • No heat is added or taken away from the parcel
  • The parcel expands since the lower pressure outside allows the air molecules to push out on the parcel walls
  • To compensate, they use up some of their internal energy in the process  parcel cools
sinking air parcels
Sinking Air Parcels
  • As the parcel sinks, it will adiabatically compress and warm
  • The parcel compresses since it is moving into a region of higher pressure.
  • Due to the parcel compression, the air molecules gain internal energy.
  • Therefore, the mean temperature of the parcel increases.
dry adiabatic lapse rate
Dry Adiabatic Lapse Rate
  • Recall that the thermodynamic energy equation can be written in terms of potential temperature
  • For an adiabatic process, we have
  • How does the temperature change with height in a dry adiabatic atmosphere?
dry adiabatic lapse rate1
Dry Adiabatic Lapse Rate
  • Taking the logarithm and differentiating with respect to height gives
  • Using the ideal gas law and the hydrostatic equation, we obtain
  • If is constant with height then
dry adiabatic lapse rate2
Dry Adiabatic Lapse Rate
  • is referred to as the dry adiabatic lapse rate
  • Note that the dry adiabatic lapse rate is a constant, independent of height or temperature
  • gives the rate of change of temperature of a dry air parcel that is being raised (cooling) or lowered (warming) adiabatically in the atmosphere.
dry adiabatic lapse rate3
Dry Adiabatic Lapse Rate
  • When is nonzero, we can write the above expression as
  • is referred to as the environmental adiabatic lapse rate
  • represents the vertical temperature profile observed in the environment at any particular location and time.
  • is measured using radiosonde observations in the atmosphere.
lapse rates and stability
Lapse Rates and Stability
  • There are three types of stability conditions for dry, unsaturated environments.
  • Case 1:
    • Dry air parcel is always cooler than environment
    • This is characteristic of a stable environment
  • Case 2:
    • Dry air parcel has same temperature as environment
    • This is characteristic of a neutrally stable environment
  • Case 3:
    • Dry air parcel is always warmer than environment
    • This is characteristic of an absolutely unstable environment
moist adiabatic lapse rate
Moist Adiabatic Lapse Rate
  • As an air parcel rises, it cools dry adiabatically until it becomes saturated.
  • Further ascent results in condensation.
  • Latent heat is released and the parcel cools at a rate that is less than
  • How does the temperature change with height in a moist adiabatic atmosphere?
moist adiabatic lapse rate1
Moist Adiabatic Lapse Rate
  • It can be shown that the moist adiabatic lapse rate is given as
  • is the change in saturation mixing ratio with temperature.
  • increases as temperature and the amount of water vapor increases.
  • Values of range from 4 K/km near the ground in warm, humid air to 6-7 K/km in the middle troposphere.
equivalent potential temperature
Equivalent Potential Temperature
  • As we showed earlier, potential temperature is conserved for dry adiabatic processes.
  • Using classical thermodynamics, it can be shown that there is a variable that is conserved for moist adiabatic processes called the equivalent potential temperature.
  • The equivalent potential temperature is defined as
  • is the temperature a parcel of air would reach if all the water vapor in the parcel were to condense, releasing its latent heat, and the parcel was brought adiabatically to a standard reference pressure, usually 1000 hPa.
equivalent potential temperature1
Equivalent Potential Temperature
  • can be used to compare both moisture content and temperature of air parcels at different elevations and the trajectory air parcels will take.
  • is used operationally to map out which regions have the most unstable air.
  • Areas of relatively high are often the formation points for thunderstorms and other convective storms.
dew point lapse rate
Dew Point Lapse Rate
  • As an air parcel rises, the moisture content remains the same but the pressure of the parcel varies.
  • This will cause the dew point to gradually decrease with height, producing a dew point lapse rate.
  • Dew point lapse rate ranges from 1.6°C to 2°C/km.
  • Once a parcel is saturated, the dew point lapse rate is equal to the moist adiabatic lapse rate.
  • Variations in dew point lapse rate with height affects the location of the LCL.
absolute stability
Absolute Stability
  • Notice that Tenv will always be greater than the temperature of a given air parcel.
  • Therefore, an air parcel will always be cooler than the environment and will sink back down to the ground.
  • This is an example of absolute stability.
  • The condition for absolute instability is
absolute stability1
Absolute Stability
  • Absolutely stable layers in the atmosphere can form by
    • Radiative cooling
    • Cold air moving at low-levels
    • Warm air moving over cold ground
absolute instability
Absolute Instability
  • Notice that Te will always be less than the temperature of a given air parcel.
  • Therefore, an air parcel will always be warmer than the environment and will sink back down to the ground.
  • This is an example of absolute instability.
  • The condition for absolute instability is
absolute instability1
Absolute Instability
  • 1. Cold air moving is aloft
    • This often occurs when an extra-tropical cyclone passes over head
  • 2. Surface heating
    • This suggests that the atmosphere is most unstable in mid-afternoon
  • 3. Warm air moving in at low levels
    • This often occurs ahead of a cold front
  • 4. Cold air moving over a warm surface
    • An example of this is lake-effect snow
conditional instability
Conditional Instability
  • Note that an unsaturated (saturated) parcel will be cooler (warmer) than the environment
  • This implies that the unsaturated (saturated) parcel will sink (rise).
  • This is an example of conditional instability.
  • The condition for absolute instability is
  • For conditional instability, the parcel is unstable if it's saturated
conditional instability example
Conditional Instability - Example
  • Consider a parcel with a surface temperature and dew point of 30 °C and 14 °C.
  • The parcel is initially forced to rise in an environment where the environmental lapse rate is 8 °C/km up to 8 km.
  • 1 km: The parcel is rising dry adiabatically since it is unsaturated.
  • 2 km: The parcel has just become saturated. This is called the lifting condensation level (LCL).
conditional instability example1
Conditional Instability - Example
  • 3 km: The parcel is now rising moist adiabatically.
  • 4 km: The parcel is still rising moist adiabatically. What happens if the parcel is pushed upward just a little?
  • 5 km and above: The parcel will rise on its own since it is less dense than the surrounding environmental air.
  • This height is called the level of free convection (LFC).
  • The parcel will rise until Tp = Te. This is often referred to as the equilibrium level (EL).
stability of the environment
Stability of the Environment
  • To determine the environmental stability, one must calculate the lapse rate for a sounding.
  • Since the environment is often composed of layers with different stabilities, it is useful to identify these layers and then calculate their lapse rates.
  • Stable dry environment
  • Stable moist environment
  • Unstable dry environment
  • Unstable moist environment
summary of lapse rates
Summary of Lapse Rates
  • Environmental Lapse Rate: The rate at which temperature decreases with height in the environmental air.
  • Dry Adiabatic Lapse Rate:The rate at which temperature decreases with height in an unsaturated, dry air parcel.
  • Moist Adiabatic Lapse Rate:The rate at which temperature decreases with height in a saturated air parcel.
  • Dew Point Lapse Rate: The rate of at which the dew point temperature decreases with height in an unsaturated parcel.
stability and cloud development
Stability and Cloud Development
  • The cloud base is where the parcel reaches saturation, i.e. the LCL
  • The cloud top is where the parcel will no longer be able to rise, i.e. the EL.
  • An estimation of the base of a convective cloud is given by:
lifting mechanisms
Lifting Mechanisms
  • Q: How can an air parcel be lifted up to the LFC in a given environment?
  • There are three basic lifting mechanisms
    • Frontal lifting
    • Surface convergence
    • Topographic lifting
frontal lifting
Frontal Lifting
  • Another lifting mechanism is by fronts.
  • If air is lifted into a stable layer:
    • Stratus or Nimbostratus clouds often results (common along warm fronts).
  • If air is lifted into a conditionally unstable layer:
    • Cumulus or Cumulonimbus often result (common along cold fronts)
surface convergence
Surface Convergence
  • Another lifting mechanism is due to convergence of air near the surface.
  • If air converges to a given location near the surface, it must go up.
  • This is common at the center of an extra-tropical cyclone.
topographic lifting
Topographic Lifting
  • If air being forced over a topographically barrier is stable, then wave clouds often form.
  • Lenticular clouds are an example.
  • Wave clouds are often aligned in “waves” and are often visible in satellite imagery.
buoyancy
Buoyancy
  • One way to assess static instability is to begin with the vertical momentum equation
  • Defining a horizontally homogeneous base state pressure and density field that is in hydrostatic balance, we can rewrite the above equation as
  • Rearranging terms gives
buoyancy1
Buoyancy
  • The first term on the RHS is the vertical perturbation PGF.
  • The second term on the RHS is the buoyancy force
  • The buoyancy force is what causes the static instability of air parcels and drives the vertical circulation.
  • The buoyancy force is attributable to density variations within an atmospheric column.
buoyancy2
Buoyancy
  • The buoyancy force can be approximated by
  • Thus, when an air parcel is warmer (cooler) than its environment, a positive (negative) buoyancy force exists, resulting in upward (downward) acceleration.
cape and cin
CAPE and CIN
  • Another way to assess the static stability in the atmosphere is through the convective available potential energy (CAPE).
  • CAPE is the amount of energy an air parcel would have if lifted a certain distance vertically through the atmosphere.
  • We define CAPE as
cape and cin1
CAPE and CIN
  • The amount of energy needed to lift a parcel to its LFC is called convective inhibition.
  • CIN is the amount of energy required to overcome the negative buoyant energy that the environment exerts on an air parcel.
cape and vertical velocity
CAPE and Vertical Velocity
  • To relate CAPE to the updraft velocity of a parcel, let’s assume that
    • The air parcel is thermally insulated from its environment
    • The air parcel remains at the same pressure as the environmental air
  • These assumptions reduce the vertical momentum equation to
cape and vertical velocity1
CAPE and Vertical Velocity
  • Multiplying both sides by gives
  • Integrating over the time required to travel from the LFC to the EL gives
  • As CAPE increases, buoyancy does work on a given air parcel, causing it to increase its kinetic energy.
limitations of parcel theory
Limitations of Parcel Theory
  • The above expression is an overestimation of the actual speed of air parcels because parcel theory neglects three basic processes
    • Vertical perturbation pressure gradient force
    • Entrainment and mixing
    • The effects of condensation
the effects of the vertical perturbation pgf
The Effects of the Vertical Perturbation PGF
  • Generally speaking, the vertical perturbation pressure gradient is not negligible in the vertical momentum equation and tends to offset the acceleration induced by the buoyancy force.
  • An upward-directed (downward-directed) buoyancy force associated with (cold air) warm air tends to be associated with a downward-directed (upward-directed) perturbation pressure gradient.
  • Thus, air parcels tend not to rise as fast as one would expect based on the consideration of the buoyancy force alone.
the effects of entrainment
The Effects of Entrainment
  • Mixing of environmental air into a rising air parcel slows the parcel by reducing its buoyancy and upward momentum.
the effects of condensation
The Effects of Condensation
  • Parcel theory also neglects
    • The presence of hydrometers or condensate (parcel theory assuming pseudoadiabatic ascent)
    • The effects of freezing
    • The effects of compensating subsidence
  • For these reasons, the approximate vertical velocity of the updraft is