Stability

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

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

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

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

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

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

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

Stability Depends on the thermal structure of the atmosphere

Stability Can be classified into 3 categories Stable Neutral Unstable

Stable Returns to original position after displacement

Neutral Remains in new position after being displaced

Unstable Moves farther away from its original position

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

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

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

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

Auto-Convective Ascent Type of Clouds Cumulus

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

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

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

Dry Adiabatic Lapse Rate

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

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

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

Pseudo-Adiabatic Lapse Rate Hydrostatic Equation

Pseudo-Adiabatic Lapse Rate Divide by cpdz

Pseudo-Adiabatic Lapse Rate Rearrange

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

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

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

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

Stability Pseudo- adiabat (Gs) Mixing Ratio line 4 Parcel Temperature 3 2 Height (km) Environmental Temperature (ge) Dry Adiabat (Gd) 1 -20 -10 10 20 Temperature (°C)

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

Stability Environmental Temperature Measured by rawinsonde

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

Unsaturated Unstable Once displaced, continues

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)

Unsaturated Neutral Once displaced, stays

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)

Unsaturated Stable Once displaced, returns

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)

Stability A real environmental sounding sometimes combines all three ge Gd A real environmental sounding sometimes combines all three Evaluate each layer

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

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

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

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

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

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

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

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

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)

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)

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)

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

Stability Static Stability The change of potential temperature with height

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

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

Static Stability Large Temperature Inversions Tropopause Stratosphere

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)

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

Potential Instability Also Called Convective Instability

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

Potential Instability Layered Lifting (lowest 100 mb)

Potential Instability Bottom of Layer New Temp. LCL

Potential Instability Top of Layer New Temp. LCL

Potential Instability Layer’s Old Lapse Rate

Potential Instability Layer’s New Lapse Rate

Potential Instability Compare Change in Layer More Unstable

Potential Instability Lifting Destabilizes Layer Cold air Warm air Cool Air

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

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

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

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

Archimedes’ Principle Water is in hydrostatic equlibrium B

Archimedes’ Principle Force on bottom of ‘bubble’ Fbottom

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

Archimedes’ Principle Buoyancy Force B Ftop Fbottom

Archimedes’ Principle Horizontal Pressure Differences Balance

Archimedes’ Principle Pressure Difference Between Top & Bottom ptop pbottom

Archimedes’ Principle Combine Equations B ptop pbottom

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

Buoyancy Density Difference Results in Net Buoyancy Force B

Buoyancy Density Difference Results in Net Buoyancy Force B

Buoyancy Net Buoyancy Force B

Buoyancy Divide by mass a

Buoyancy a Ideal gas law

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

Buoyancy a Rearrange

Buoyancy a Temperature difference results in parcel acceleration

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

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

Stability Substitute

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

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

Stability Initially stable air is displaced

Stability How will it react once displacing force is removed?

Brunt – Vaisala Frequency Define coefficient of z as N

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

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

Brunt – Vaisala Frequency Buoyancy Oscillation Brunt – Vaisala Frequency Brunt – Vaisala Period

Brunt – Vaisala Frequency Theory behind buoyancy oscillations in atmosphere Orographic

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

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

Static Stability Mathematically ... Take the logarithm

Static Stability Differentiate

Static Stability Multiply by T

Static Stability Substitute Ideal Gas Law

Static Stability Hydrostatic Equation

Static Stability Divide by CpTdz

Static Stability Remember

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