* Reading Assignments:

All Sections

8. Hydrostatic Stability

Describe the states of vertical stratification of atmosphere:

* Stable equilibrium

* Unstable equilibrium

* Neutral equilibrium

8.1 Parcel method

)(2

2

g

dt

zd

zT

g

dt

zd

)(2

2

The vertical acceleration of a unit mass air parcel is proportionalto its displacement and the difference between the parcel andenvironmental lapse rates.

)(2

2

TTT

g

dt

zd

8.2 Stability Categories

8.2.1 Stability in terms of temperature lapse rate

* For unsaturated air

stable

neutral

unstable

d

* For saturated air

stable

neutral

unstable

s

* For both unsaturated and saturated air

1) d Absolute unstable

2) s Absolute stable

3) ds Conditionally unstable

Stable for unsaturated air

Unstable for saturated air

8.2.2 Moisture Dependence

8.2.3 Stability in terms of potential temperature

stable

neutraldz

d

unstable

0

)( dTdz

d * For unsaturated air

* For saturated air

stable

neutraldz

d

unstable

e

0

8.3 Implications for vertical motion

022

2

zNdt

zd

The vertical momentum equation for an unsaturated air parcel

)( dT

g

dz

dgN

N is called the Brunt-Väisäillä frequency.

If the atmospheric stratification is stable, N > 0.

)sin(NtAz

)cos(NtANdt

zdw

N

2

Meteorology 341 Homework (8)

)( ds 3. An air parcel, which has LCL (Lifting Condensation Level) at height of hc ,

ascends in a conditionally unstable environment , show the height of LFC (Level of Free Convection) hf

s

sdcf hh

ds and ,, are constants.

1. Problem 8 on page 201

2. Problem 15 on page 201

8.4 Conditional instability

)(

)ln)(()(2

1

pRA

pdTTRpPp

p

c

LFC

p

pLFCc pdTTRwwCAPE )ln)((2

1

2

1 22

Potential Energy:

Convective Available Potential Energy:the total potential energy available for conversion to kinetic energy

Kinetic Energy:

PK

)(

)ln)(()(2

1

pRA

pdTTRpPp

p

Potential Energy::

Conditional instability is classified into three types:

1) Stable type: no positive area

3) Pseudo latent type: negative area is larger than positive area

2) Real latent type: positive area is larger than negative area

1)

2)

3)

dzT

TTgwwK

z

z

2

1

)()(

2

1 21

22

When the height is used as the vertical coordinate, the kinetic energy for the air parcel within a layer is defined as

T wwhere and are the air parcel’s temperature and vertical velocity, respectively. If the temperature lapse rate of the environment is 12oC/km from surface to 500m and the surface temperature is 27oC, find the kinetic energy for an unsaturated air parcel in this layer.

8.5 Effect of entrainment on ascending parcels

dz

dm

m

w

T

TTgw

dz

d 22

2

1

eHdz

dm

m

11

is the ratio of entraining mass per unit height over the air mass.

is defined as the mass entrainment height.eH

8.6 Modification of stability due to the displacement of layer

* Potential Instability

)1)(( 112 pA

Apd

* Vertical compression and expansion

Stability decreases

Instability decreases

No change

ppAA ,For

121

121

121

,)3

,)2

,)1

dd

d

d

thenIf

thenIf

thenIf

8.7 Layer (slice) method

One assumption in the parcel method is that the environmentremains undisturbed.

The layer method removes this assumption by allowing thecompensatory subsidence for the ascent of an air mass.

The stability criterion:

t

T

t

TT du

* Both ascending and descending air is unsaturated

))(1())(( dd

uuddu m

mwwwT

* Ascending air is saturated and descending air is unsaturated

)()()( d

d

usuddsu m

mwwwT

8.8 Stabilizing and destabilizing effects

* External heat transfer