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IACETH
Inst
itute
for
Atm
osph
eric
and
Clim
ate
Sci
enc
e
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
IACETH
Inst
itute
for
Atm
osph
eric
and
Clim
ate
Sci
enc
e
Stability Conv. inst. Horizontal forces Slantwise displacement
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
Ulrike Lohmann (IACETH) Parcel Buoyancy and Atmospheric Stability ETH, Nov 9, 2005
IACETH
Inst
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Atm
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eric
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Clim
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Sci
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e
Stability Conv. inst. Horizontal forces Slantwise displacement
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
Ulrike Lohmann (IACETH) Parcel Buoyancy and Atmospheric Stability ETH, Nov 9, 2005
IACETH
Inst
itute
for
Atm
osph
eric
and
Clim
ate
Sci
enc
e
Stability Conv. inst. Horizontal forces Slantwise displacement
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)
Ulrike Lohmann (IACETH) Parcel Buoyancy and Atmospheric Stability ETH, Nov 9, 2005
IACETH
Inst
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Stability Conv. inst. Horizontal forces Slantwise displacement
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
Ulrike Lohmann (IACETH) Parcel Buoyancy and Atmospheric Stability ETH, Nov 9, 2005
IACETH
Inst
itute
for
Atm
osph
eric
and
Clim
ate
Sci
enc
e
Stability Conv. inst. Horizontal forces Slantwise displacement
Stability criteria for moist air
Ulrike Lohmann (IACETH) Parcel Buoyancy and Atmospheric Stability ETH, Nov 9, 2005
IACETH
Inst
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Atm
osph
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Clim
ate
Sci
enc
e
Stability Conv. inst. Horizontal forces Slantwise displacement
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
Ulrike Lohmann (IACETH) Parcel Buoyancy and Atmospheric Stability ETH, Nov 9, 2005
IACETH
Inst
itute
for
Atm
osph
eric
and
Clim
ate
Sci
enc
e
Stability Conv. inst. Horizontal forces Slantwise displacement
Ulrike Lohmann (IACETH) Parcel Buoyancy and Atmospheric Stability ETH, Nov 9, 2005
IACETH
Inst
itute
for
Atm
osph
eric
and
Clim
ate
Sci
enc
e
Stability Conv. inst. Horizontal forces Slantwise displacement
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.
Ulrike Lohmann (IACETH) Parcel Buoyancy and Atmospheric Stability ETH, Nov 9, 2005
IACETH
Inst
itute
for
Atm
osph
eric
and
Clim
ate
Sci
enc
e
Stability Conv. inst. Horizontal forces Slantwise displacement
Effect of lifting on stability in dry air
Ulrike Lohmann (IACETH) Parcel Buoyancy and Atmospheric Stability ETH, Nov 9, 2005
IACETH
Inst
itute
for
Atm
osph
eric
and
Clim
ate
Sci
enc
e
Stability Conv. inst. Horizontal forces Slantwise displacement
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.
Ulrike Lohmann (IACETH) Parcel Buoyancy and Atmospheric Stability ETH, Nov 9, 2005
IACETH
Inst
itute
for
Atm
osph
eric
and
Clim
ate
Sci
enc
e
Stability Conv. inst. Horizontal forces Slantwise displacement
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.
Ulrike Lohmann (IACETH) Parcel Buoyancy and Atmospheric Stability ETH, Nov 9, 2005
IACETH
Inst
itute
for
Atm
osph
eric
and
Clim
ate
Sci
enc
e
Stability Conv. inst. Horizontal forces Slantwise displacement
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:
Ulrike Lohmann (IACETH) Parcel Buoyancy and Atmospheric Stability ETH, Nov 9, 2005
IACETH
Inst
itute
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Atm
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Sci
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e
Stability Conv. inst. Horizontal forces Slantwise displacement
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.
Ulrike Lohmann (IACETH) Parcel Buoyancy and Atmospheric Stability ETH, Nov 9, 2005
IACETH
Inst
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Atm
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Stability Conv. inst. Horizontal forces Slantwise displacement
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.
Ulrike Lohmann (IACETH) Parcel Buoyancy and Atmospheric Stability ETH, Nov 9, 2005
IACETH
Inst
itute
for
Atm
osph
eric
and
Clim
ate
Sci
enc
e
Stability Conv. inst. Horizontal forces Slantwise displacement
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
Ulrike Lohmann (IACETH) Parcel Buoyancy and Atmospheric Stability ETH, Nov 9, 2005
IACETH
Inst
itute
for
Atm
osph
eric
and
Clim
ate
Sci
enc
e
Stability Conv. inst. Horizontal forces Slantwise displacement
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.
Ulrike Lohmann (IACETH) Parcel Buoyancy and Atmospheric Stability ETH, Nov 9, 2005
IACETH
Inst
itute
for
Atm
osph
eric
and
Clim
ate
Sci
enc
e
Stability Conv. inst. Horizontal forces Slantwise displacement
Ulrike Lohmann (IACETH) Parcel Buoyancy and Atmospheric Stability ETH, Nov 9, 2005
IACETH
Inst
itute
for
Atm
osph
eric
and
Clim
ate
Sci
enc
e
Stability Conv. inst. Horizontal forces Slantwise displacement
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:
Ulrike Lohmann (IACETH) Parcel Buoyancy and Atmospheric Stability ETH, Nov 9, 2005
IACETH
Inst
itute
for
Atm
osph
eric
and
Clim
ate
Sci
enc
e
Stability Conv. inst. Horizontal forces Slantwise displacement
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)
Ulrike Lohmann (IACETH) Parcel Buoyancy and Atmospheric Stability ETH, Nov 9, 2005
IACETH
Inst
itute
for
Atm
osph
eric
and
Clim
ate
Sci
enc
e
Stability Conv. inst. Horizontal forces Slantwise displacement
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)
Ulrike Lohmann (IACETH) Parcel Buoyancy and Atmospheric Stability ETH, Nov 9, 2005
IACETH
Inst
itute
for
Atm
osph
eric
and
Clim
ate
Sci
enc
e
Stability Conv. inst. Horizontal forces Slantwise displacement
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:
Ulrike Lohmann (IACETH) Parcel Buoyancy and Atmospheric Stability ETH, Nov 9, 2005
IACETH
Inst
itute
for
Atm
osph
eric
and
Clim
ate
Sci
enc
e
Stability Conv. inst. Horizontal forces Slantwise displacement
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
Ulrike Lohmann (IACETH) Parcel Buoyancy and Atmospheric Stability ETH, Nov 9, 2005
IACETH
Inst
itute
for
Atm
osph
eric
and
Clim
ate
Sci
enc
e
Stability Conv. inst. Horizontal forces Slantwise displacement
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 →
Ulrike Lohmann (IACETH) Parcel Buoyancy and Atmospheric Stability ETH, Nov 9, 2005
IACETH
Inst
itute
for
Atm
osph
eric
and
Clim
ate
Sci
enc
e
Stability Conv. inst. Horizontal forces Slantwise displacement
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.
Ulrike Lohmann (IACETH) Parcel Buoyancy and Atmospheric Stability ETH, Nov 9, 2005
IACETH
Inst
itute
for
Atm
osph
eric
and
Clim
ate
Sci
enc
e
Stability Conv. inst. Horizontal forces Slantwise displacement
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.
Ulrike Lohmann (IACETH) Parcel Buoyancy and Atmospheric Stability ETH, Nov 9, 2005
IACETH
Inst
itute
for
Atm
osph
eric
and
Clim
ate
Sci
enc
e
Stability Conv. inst. Horizontal forces Slantwise displacement
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
Ulrike Lohmann (IACETH) Parcel Buoyancy and Atmospheric Stability ETH, Nov 9, 2005