Introduction to Atmospheric Dynamics Chapter 2

Home , Lapse rate, Potential temperature

Part 3: Buoyancy and Convection Vertical Structure

This cooling with height is related to the dynamics of the atmosphere. The change of temperature with height is called the lapse rate.

Defnition: The lapse rate is defned as the rate (for instance in K/km) at which temperature decreases with height. @T ⌘@z

Paul Ullrich Introduction to Atmospheric Dynamics March 2014 Lapse Rate

For a dry adiabatic, stable, hydrostatic atmosphere the potential temperature θ does not vary in the vertical direction: @✓ =0 @z

In a dry adiabatic, hydrostatic atmosphere the temperature T must decrease with height. How quickly does the temperature decrease?

R/cp p0 Note: Use ✓ = T p ✓ ◆

Paul Ullrich Introduction to Atmospheric Dynamics March 2014 Lapse Rate The adiabac change in temperature with height is T @✓ @T g = + ✓ @z @z cp

For dry adiabac, hydrostac atmosphere:

@T g = d @z cp ⌘

Defnition: The dry adiabatic lapse rate is defned as the g rate (for instance in K/km) at which the temperature of an d air parcel will decrease with height if raised adiabatically. ⌘ cp

g 1 9.8Kkm cp ⇡

Paul Ullrich Introduction to Atmospheric Dynamics March 2014 Lapse Rate

This profle should be very close to the adiabatic lapse rate in a dry atmosphere.

Paul Ullrich Introduction to Atmospheric Dynamics March 2014 Fundamentals

Even in adiabatic motion, with no external source of heating, if a parcel moves up or down its temperature will change.

• What if a parcel moves about a surface of constant pressure?

• What if a parcel moves about a surface of constant height?

If the atmosphere is in adiabatic balance, the temperature still changes with height.

Adiabatic does not mean isothermal. It means there is no external heating or cooling.

Paul Ullrich Introduction to Atmospheric Dynamics March 2014 Lapse Rate

In fact, the temperature decline here is not the dry adiabatic lapse rate, because the atmosphere (in general) is not dry.

Paul Ullrich Introduction to Atmospheric Dynamics March 2014 Static Stability and Moisture

In general, the atmosphere is not dry.

If air reaches saturation (and the conditions are right for cloud formation), vapor will condense to liquid or solid and release energy (J = 0) 6

Moist (saturated) adiabatic lapse rate: ~ 5K/km

Average lapse rate in the troposphere: ~ 6.5K/km

Paul Ullrich Introduction to Atmospheric Dynamics March 2014 Dry Convection z In accordance with observations, the temperature of the troposphere decreases roughly linearly with height. Cooler z

T Warmer

Paul Ullrich Introduction to Atmospheric Dynamics March 2014 Dry Convection

The Parcel Method

• Consider an arbitrary air parcel sitting in the atmosphere that is displaced up or down.

• Assume that the pressure adjusts instantaneously; that is, the parcel assumes the pressure of the altitude to which it is displaced.

• As the parcel moves its temperature will change according to the adiabatic lapse rate. That is, the motion is without the addition or subtraction of energy (J = 0 in thermodynamic equation)

Paul Ullrich Introduction to Atmospheric Dynamics March 2014 Dry Convection z

Cooler

Question: If the parcel moves up and fnds itself cooler than the environment, what will happen?

Warmer

Paul Ullrich Introduction to Atmospheric Dynamics March 2014 Dry Convection z

Cooler

If the parcel moves up and fnds itself cooler than the environment then it will sink.

Aside: What is its density? Larger or smaller?

Warmer

Paul Ullrich Introduction to Atmospheric Dynamics March 2014 Dry Convection z

Cooler

Question: If the parcel moves up and fnds itself warmer than the environment, what will happen?

Warmer

Paul Ullrich Introduction to Atmospheric Dynamics March 2014 Dry Convection z

Cooler

If the parcel moves up and fnds itself warmer than the environment then it will go up some more.

This is a frst example of “instability” – a perturbation that grows.

Warmer

Paul Ullrich Introduction to Atmospheric Dynamics March 2014 Dry Convection Assume that temperature is a linear function of height:

@T T = T z = s @z

So if we go from z to z + Δz, then the change in T of the environment is

T =[T (z + z)] [T z]= z s s

Paul Ullrich Introduction to Atmospheric Dynamics March 2014 Dry Convection

If we go from z to z + Δz, then the change in T of the air parcel will follow the dry adiabatic lapse rate

T = T (z + z) T (z) p p p =(T (z) z) T (z) p d p = z d

If we go from z to z + Δz, then the change in T of the environment will follow the environmental lapse rate

T = T (z + z) T (z) env env env =(T (z) z) T (z) env env = z

Paul Ullrich Introduction to Atmospheric Dynamics March 2014 Dry Convection Stable: If the temperature of parcel is cooler than environment.

But since at the initial position of the parcel it has the same temperature as the environment,

Tp < Tenv

Then using T = z and T = z p d env z< z d

In a stable atmospheric environment, the lapse rate satisfes < d

Paul Ullrich Introduction to Atmospheric Dynamics March 2014 Dry Convection Unstable: If the temperature of parcel is warmer than environment.

Tp >Tenv

But since at the initial position of the parcel it has the same temperature as the environment,

Tp > Tenv

Then using T = z and T = z p d env z> z d

In an unstable atmospheric environment, the lapse rate satisfes > d

Paul Ullrich Introduction to Atmospheric Dynamics March 2014 Dry Convection

Stability criteria from physical argument

d > Stable

d = Neutral

d < Unstable

Adiabatic lapse rate Environmental lapse rate

Important: A compressible atmosphere is unstable if temperature decreases with height faster than the adiabatic lapse rate.

Paul Ullrich Introduction to Atmospheric Dynamics March 2014 Dry Convection z

Cooler z

T Warmer

Paul Ullrich Introduction to Atmospheric Dynamics March 2014 Dry Static Stability

Environment is in hydrostatic balance, no acceleration:

1 @p 0= env g ⇢env @z

But our parcel experiences an acceleration:

2 Dw D z 1 @pp = 2 = g Dt Dt ⇢p @z

Assume instantaneous adjustment of parcel pressure

Paul Ullrich Introduction to Atmospheric Dynamics March 2014 Dry Static Stability

But our parcel experiences an acceleration:

Dw D2z g⇢ ⇢ ⇢ = = env g = g env p Dt Dt2 ⇢ ⇢ p ✓ p ◆ penv pp ⇢env = Ideal gas law ⇢p = RdTenv RdTp

D2z T T = g p env Dt2 T ✓ env ◆

Paul Ullrich Introduction to Atmospheric Dynamics March 2014 Dry Static Stability

D2z T T = g p env Dt2 T ✓ env ◆ T = T (z ) (z z ) p 0 d 0 T = T (z ) (z z ) env 0 0

D2z g = ( )(z z ) Dt2 T (z ) (z z ) d 0 0 0 1 1 1 Binomial expansion = T (z0) (z z0) T (z0) 1 (z z0) T (z0) 1 (z z ) 1+ 0 ⇡ T (z ) T (z ) 0 ✓ 0 ◆

Paul Ullrich Introduction to Atmospheric Dynamics March 2014 Dry Static Stability

2 D z g g 2 2 = ( d)(z z0)+ 2 (z z0) Dt T (z0) T (z0)

For small displacements ignore quadratic terms:

D2z g 2 ( d)(z z0) Dt ⇡ T (z0)

Defne z = z z 0

D2z g 2 + (d )z =0 Dt T (z0)

Paul Ullrich Introduction to Atmospheric Dynamics March 2014 Dry Static Stability

D2z g 2 + (d )z =0 Dt T (z0)

This is an ordinary diﬀerential equation in the variable z . Do you recognize this equation and its solutions?

g( ) d > 0 T (z0) These cases g(d ) correspond to Three cases: =0 stable, neutral T (z0) and unstable g( ) atmospheres. d < 0 T (z0)