Parcel Buoyancy and Atmospheric Stability

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Stability Conv. inst. Horizontal forces Slantwise displacement

Parcel Buoyancy and Atmospheric Stability

Ulrike Lohmann

ETH Zurich

Institut fur Atmosphare und Klima

ETH, Nov 9, 2005

Ulrike Lohmann (IACETH) Parcel Buoyancy and Atmospheric Stability ETH, Nov 9, 2005

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Stability criteria for dry air

I Denote ambient lapse with γ and that of the parcel with Γ.

I Connection between ∂Θ∂z and static stability:

1

Θ

∂Θ

∂z=

1

T(Γ− γ) (1)

I γ < Γ ↔ ∂Θ∂z > 0

I γ = Γ ↔ ∂Θ∂z = 0

I γ > Γ ↔ ∂Θ∂z < 0


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I Relate buoyancy (FB) to stability:

d2z

dt2=

dw

dt= FB = g

(T − T

′

T ′

)(2)

where T = temperature of the air parcel, T′= ambient temperature,

FB = buoyant force per unit mass.

I This can be related to Θ:

d2z

dt= − g

Θ

(∂Θ

∂z

)z ≡ −N2z (3)

where N =√

(g/Θ)(∂Θ/∂z) = Brunt-Vaisala frequency (s−1):

I N2 = 0

I N2 > 0

I N2 < 0


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Saturated adiabatic lapse rate

I Rate of change in temperature with height of a parcel of air

undergoing a pseudoadiabatic or saturated (=reversible)

adiabatic process.

I Starting from the 1. law of thermodynamics:

dq = cpdT − αdp (4)


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I .

I Thus, the saturated adiabatic lapse rate Γs is defined as:

Γs ≡ −dT

dz=

Γd

1 + Lcp

dwsdT

(5)

I Γs is not constant, but depends on p and T.

I Since dwsdT > 0 → Γs < Γd


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Stability criteria for moist air


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WAVE CLOUDS

Mountain WaveLee Wave

Wind

StableAir

LCL

(STABLE)

CLOUDS - "Lenticular"

LCL

Wind

"Rotor Cloud"Föhn Wall Cloud

"Jump"Turb

ulent

"Hydraulic Jump"

Figure: Houze’s cloud atlas


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Convective instabilityI Let weight of air column be p1 - p2. Consider vertical displacement with

mass remaining constant → ∆p = p1 − p2 = constant

∆p = gρ∆z (6)

I since ρ decreases with height, lifting must result in stretching → changes

stability

I Consider change in Θ over small height:

δΘ =∂Θ

∂zδz (7)

I δΘ is const for adiabatic lifting. If δz increases, ∂Θ∂z

must decrease → air

becomes unstable.

I Exception: air with neutral stability for which ∂Θ∂z

= 0 before and after

displacement.

I Initially unstable layer becomes less unstable and initially stable layer

becomes less stable.

I → lifting makes the lapse rate tend toward the dry adiabatic.


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Effect of lifting on stability in dry air


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Convective instability

I Note: lifting a column of moist air, initially stable may be

made unstable or conditionally unstable by lifting.

I ∂Θw∂z > 0

I ∂Θw∂z = 0

I ∂Θw∂z < 0

I where Θw is the wet-bulb potential temperature, which is

defined as the intersection of the pseudoadiabat through p with

the isobar p = 1000 hPa.


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Convective instability

Note: convective

instability has to do

with lifting of layers and

should not be confused

with conditional

instability, which

applies to an

undisplaced layer.


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Horizontal restoring forces

I So far, stability conditions were limited to vertical displacement

of an air parcel or a layer of air.

I In the atmosphere, instability can also occur when air is

displaced in a slantwise direction.

I Major horizontal forces: Coriolis force and horizontal pressure

gradient force.

I Coriolis force:


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Geostrophic wind and geostrophic wind shear

I because the geostrophic wind is related to the horizontal

pressure gradient, any change of the pressure gradient with

height implies a vertical variation of the geostrophic wind.

I This situation often prevails in the atmosphere because of a

nonuniform temperature distribution in the horizontal.


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Thermal wind

The variation of the geostrophic wind with height, geostrophic

wind shear or thermal wind, can be obtained by differentiating

the above equations with respect to z:

I neglected here: density variations with height

I geostrophic wind shear is related to the horizontal gradient of

Θ. I.e., if Θ is not uniform in the horizontal, the geostrophic

wind will change with height.


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Slantwise convection characteristics

I Banded clouds and precipitation

I Sometimes associated with extratropical fronts

I Single or multiple bands; isolated or embedded

I Length: 100 km to > 500 km; width 5-40 km

I Bands observed in regions where the atmosphere is

gravitationally stable


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Slantwise displacement

I Occurs in the atmosphere when a horizontal temperature

gradient exists.

I If displacement occurs over large enough area then Coriolis

force comes into play.

I Assume that isentropic surfaces (constant Θ) are tilted.

I Suppose parcel of air at point A is in equilibrium with the

environment (same T, Θ, p, u, v).

I Next suppose parcel is slantwise displaced to B.


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I If no condensation occurs, Θ is conserved and the temperature of the

parcel T (y , z) at B is:

T +

(dT

dp

)dp = T +

κT

p

(∂p

∂yδy +

∂p

∂zδz

)(8)

I The ambient temperature (Θ 6= const.) at B is given by:

I The excess temperature of the displaced parcel over the ambient air

is:

I The buoyancy force on the displaced parcel is:


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I In addition to the buoyancy force, the parcel is subject to a

horizontal restoring force because the Coriolis and horizontal

pressure forces acting on the parcel are out of balance at the

new position.I If the parcel moves from A to B in time δt , the Coriolis force

changes the x-component of its velocity by an amount

∆v = f v δt = f δy . At its new position, the Coriolis force

CFx on the parcel is therefore increased by the amount

∆CFx = f ∆v = f 2δy .I The tilt of the isentropic surfaces indicates that the horizontal

pressure gradient force must vary with height. From the

thermal wind equation, the change of the pressure force from A

to B is given by:I

∂

∂y

(−1

ρ

∂p

∂y

)δy +

∂

∂z

(−1

ρ

∂p

∂y

)δz = f

∂ug

∂yδy + f

∂ug

∂zδz (9)


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Generalized equation for parcel displacement

I Because the parcel is in equilibrium at A, the net horizontal restoring

force FH at B is given by the difference between the incremental

changes in the Coriolis force + the horizontal pressure gradient force:

I

FH = f

[∂ug

∂zδz −

(f − ∂ug

∂y

)δy

](10)

I Thus, the equation of motion of the parcel along its direction of

displacement, with distance denoted by ∆, is therefore:

I

d2∆

dt2= FBsinβ + FHcosβ (11)

= −g

[1

Θ

∂Θ

∂zδz +

1

Θ

∂Θ

∂yδy

]sinβ (12)

+f

[∂ug

∂zδz −

(f − ∂ug

∂y

)δy

]cosβ (13)


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Generalized equation for parcel displacement

d2∆

dt2= −g

[1

Θ

∂Θ

∂zδz +

1

Θ

∂Θ

∂yδy

]sinβ (14)

+f

[∂ug

∂zδz −

(f − ∂ug

∂y

)δy

]cosβ (15)

I LHS: acceleration of air parcel

I first term on RHS: buoyancy force

I second term on RHS: pressure gradient force, ug geostrophic wind.

I for δ y = 0 and β = 90 , the buoyancy force, as discussed before, is

the only force left:


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Symmetric instability

I a type of slantwise instability, that occurs if the parcel is

displaced along an isentropic surface, so that buoyancy force

vanishes and the only restoring force is FH .

I Symmetric instability can be responsible for the mesoscale

banded structure of precipitation associated with midlatitude

frontal systems.

Source: http : //meted .ucar .edu/mesoprim/bandedprecip/print.htm#4.4


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Symmetric stability

I

d2∆

dt2= f δy cosβ

∂ug

∂z

[δz

δy−

(f − ∂ug

∂y )

∂ug

∂z

](16)

I first term in brackets: slope of the isentropic surface.

I second term: slope of absolute vorticity (ratio of horizontal to

vertical component of absolute vorticity) and f = 2 Ω sinΦ.

I slope of isentropic surface < slope of absolute vorticity →I slope of isentropic surface = slope of absolute vorticity →I slope of isentropic surface > slope of absolute vorticity →


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Baroclinic instability

I another type of slantwise instability, which occurs if only the

generalized buoyancy force is included (no Coriolis effect).

I

d2∆

dt2= −g

(1

Θ

∂Θ

∂z

)δy sinβ

[δz

δy−

(−

∂Θ∂y

∂Θ∂z

)](17)

= g

(1

Θ

∂Θ

∂z

)δy sinβ

[(−

∂Θ∂y

∂Θ∂z

)− δz

δy

](18)

I first term in brackets: slope of the air parcel displacement

I second term: slope of the isentropic surface.


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Define stability in statically stable atmosphere (∂Θ∂z > 0):

I slope of isentropic surface < slope of parcel displacement →I slope of isentropic surface = slope of parcel displacement →I slope of isentropic surface > slope of parcel displacement →I This instability mechanism, first investigated by Charney

(1947) and Eady (1949) is often met in the atmosphere at

midlatitudes

I It is firmly established that this kind of instability is responsible

for the formation of midlatitude cyclones and the associated

widespread cloud and precipitation.


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Summary of instabilities

gravitational symmetric

dry absolute instability symmetric instabilitydΘdz < 0 dΘ

dz

∣∣Mg < 0 ;

dMg

dx |Θ < 0

−dTdz > Γd −dT

dz

∣∣Mg > Γd

∣∣Mg

cond. conditional instability cond. symmetric inst. (CSI)dΘesdz < 0 dΘes

dz

∣∣Mg < 0 ;

dMg

dx |Θes < 0

Γm < −dTdz < Γd Γm

∣∣Mg < −dT

dz

∣∣Mg

< Γd

∣∣Mg

conv. convective instability potential symmetric instability

(= potential inst.)dΘedz < 0 dΘe

dz

∣∣Mg < 0 ;

dMg

dx |Θe < 0where Θes = saturation equivalent potential temperature


Parcel Buoyancy and Atmospheric Stability

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