Lecture 10: Atmospheric Stability

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 • 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 • 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 • 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 • 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 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 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 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 • 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 • 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 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 • 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 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 • 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 • 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 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 • 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 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 • 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 • 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 - 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 • 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 • 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 • 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 • 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 • 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 • 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 • 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 • 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

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.

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 • 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 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 • 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 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 • 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 • 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 • Mixing of environmental air into a rising air parcel slows the parcel by reducing its buoyancy and upward momentum.

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

Lecture 10: Atmospheric Stability