ATMOSPHERIC STABILITY

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• ATMOSPHERIC STABILITY
• Hydrodynamic stability
• Parcel method
• Stability criteria
• Dry adiabatic lapse rate
• Pseudo-adiabatic lapse rate

HYDRODYNAMIC STABILITY We consider the matter of
atmospheric hydrostatic stability here because it
is very important in understanding the occurrence
of convection and turbulence in the atmosphere.
An atmosphere in hydrostatic equilibrium may or
may not be stable, just as any equilibrium may or
may not be stable (see the simple dynamical
examples below). The principle we use to
establish equilibrium stability is to consider
the outcome of a small (virtual) perturbation of
the equilibrium state. If the perturbed system
tends to return to its equilibrium, we call it
stable. If it continues to move away from its
equilibrium state, we say it is unstable. If the
perturbed state does not move towards or away
from equilibrium, it is neutral.
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Notes 1) If the system returns to equilibrium by
decaying oscillations, it is said to have focal
stability. If it returns to equilibrium without
oscillations, it is said to have nodal stability
(A.B. Pippard, 1985 Response and stability, p.
11) 2) In the unstable case, the system may move
away from equilibrium continuously, or by ever
increasing oscillations. In linear mathematical
systems, the amplitude of the perturbation may
increase without bound, but in real physical
systems, energy considerations generally prevent
this, with the result that the system may
eventually arrive at a different equilibrium.
PARCEL METHOD When applying these notions to the
atmosphere, we have to decide how to perturb the
hydrostatic equilibrium. Normally, we imagine
doing this by defining a small air parcel and
considering small vertical displacements of it
from its initial location. This is referred to as
the parcel method. If the displaced air parcel
experiences a restoring force, tending to move it
back towards its starting point, we say the
equilibrium is stable. If there is no restoring
force, we call it neutral. Finally if the
displaced parcel experiences a force pushing it
further from its starting point, we say the
equilibrium is unstable (see the sketches below).
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• In the following description of this method, we
  will make a few simplifying assumptions.
• The air parcel is initially in equilibrium with
  its environment, so that they have identical
• pressure, temperature, and density. The
  atmosphere initially is in hydrostatic
  equilibrium.
• 2) The pressure in the air parcel is always
  identical to the pressure in the environment at
  the
• same level as the parcel. If this were not the
  case, there would be horizontal pressure
  gradients,
• leading to horizontal motion, which is of no
  immediate interest when considering hydrostatic
• stability.
• 3) The parcel does not mix with the environment,
  nor is the environment disturbed by the ascent
• of the parcel.
• 4) The air parcel is dry, and the vertical
  displacement is rapid, so that its temperature
  changes
• adiabatically at the dry adiabatic lapse rate,
  ?dg/cp. Note if the air parcel is moist (but
  not
• saturated), the parcel temperature should be
  replaced by the virtual temperature.

STABILITY CRITERIA From the previous sketches,
it is easy to see that hydrostatic stability or
instability depends on the magnitude of the
local environmental lapse rate, relative to the
dry adiabatic lapse rate (parcel lapse rate).
This result may be summarized as follows
(19.1)
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We could also define the stability criteria in
terms of the potential temperature lapse rate of
the environment, relative to the potential
temperature lapse rate of the parcel (which is
zero for adiabatic changes). Hence
(19.2)
If the parcel is saturated, we can obtain the
appropriate stability criteria from Eq. 19.1 by
replacing the dry adiabatic lapse rate with the
pseudo-adiabatic lapse rate, ?s, on the right
hand side. Thus it is important to determine ?d
and ?s.
DRY ADIABATIC LAPSE RATE Although we have
already quoted an expression for ?d above, we
will do the derivation here for the sake of
completeness. Logarithmically differentiating the
definition of potential temperature, we have
(19.3)
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Multiplying by T, and substituting from the
hydrostatic equation and the ideal gas law into
the second term on the right hand side (keeping
in mind that the vertical pressure gradient in
the hydrostatic equation is determined by the
environmental density and hence the
environmental temperature, Te), leads to
(19.4)
PSEUDOADIABATIC LAPSE RATE Recalling that for
cloudy air undergoing a reversible adiabatic
process, the change in specific entropy is given
by Eq. 9.10
(19.5)
Differentiating with respect to height, Eq. 19.5
can be re-arranged to give an expression for
the pseudoadiabatic lapse rate
(19.6)
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Differentiating the relationship between
saturation mixing ratio and saturation vapour
pressure,
, and substituting into the first term of Eq.
19.6, then substituting the hydrostatic equation
and ideal gas law into the second term of Eq.
19.6, and finally using the chain rule
and the Clausius-Clapeyron
equation, leads to
(19.7)
Since ?lv/cp?1500 K, the denominator in Eq. 19.7
always exceeds the numerator (for meteorological
conditions), with the result (which we know
already from the tephigram), that ?s??d . The
equality arises if the air is dry (rs0), in
which case the ratio in Eq. 19.7 becomes unity.
At high pressures and low temperatures, we also
have ?s ?d (see the tephigram to confirm this).
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• ATMOSPHERIC STABILITY II
• Vertical oscillations in a stable atmosphere
• Brunt-Väisälä frequency

VERTICAL OSCILLATIONS IN A STABLE ATMOSPHERE A
stable atmosphere exhibits focal stability. That
is, air parcels displaced from equilibrium
will return to it by means of decaying
oscillations. When friction is small, as in the
free atmosphere, the oscillations may continue
for a long time. This process is of some
intrinsic interest, inasmuch as some internal
gravity waves in the atmosphere arise from the
oscillation of air parcels displaced from
equilibrium by flow over topography. We will
calculate the frequency of these oscillations by
solving the equation of motion of the displaced
air parcel. Since the environment is in
hydrostatic equilibrium, by assumption, its
equation of motion is simply the hydrostatic
equation
(20.1)
Where the subscript e refers to the
environment. Unsubscripted variables will refer
to the air parcel. The equation of motion of the
air parcel is given by Newtons second law for a
unit mass of air
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(20.2)
If we assume that the parcel and environmental
pressures at the same level are identical,
then Eq. 20.2 may be written
(20.3)
Substituting from Eq. 20.1 leads to
(20.4)
The term on the right hand side is called the
buoyancy, and it is clearly positive (leading
to upwards acceleration) if the specific volume
of the parcel exceeds the specific volume of
the environment. It is clearly negative if the
parcels specific volume is less than that of the
environment, and the resulting acceleration will
be downwards. Buoyancy is often
written (approximately) as g?v/v, g??/?, or
g?T/T, where the meaning of the deltas can be
ascertained by substituting either the ideal gas
law or v1/? into Eq. 20.4. Buoyancy is also
sometimes referred to as reduced gravity, for
obvious reasons. Since we have Assumed that ppe,
an application of the ideal gas law leads to
v/veT/Te. This result can be
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substituted into Eq. 20.4 to give
(20.5)
Assuming the parcel displacement is small, we may
write
(20.6)
where the subscript 0 refers to the initial
point of the air parcel. Substituting from Eq.
20.6 into Eq. 20.5, we have, finally
(20.7)
BRUNT-VÄISÄLÄ FREQUENCY Eq. 20.7 leads
independently to the stability criteria which we
fomulated in the previous lecture (see Eqs.
19.1, 19.2). Here, however, we will use it to
solve for the frequency of oscillations in the
stable case. Note that in form the equation is
identical to that describing a spring (with a
particular spring constant) undergoing periodic
oscillations. Let us assume a solution of the
form . Then upon
substitution into Eq. 20.7
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and solving for ?
(20.8)
This quantity is the Brunt-Väisälä period. It is
the period of small amplitude, vertical,
gravity- controlled oscillations in a stable
atmosphere. The Brunt-Väisälä frequency is
1/?. By logarithmically differentiating the
defining equation for potential temperature, and
making use of the hydrostatic equation and the
ideal gas law, it may be shown that
(20.9)
Hence, the Brunt-Väisälä frequency may also be
written as
(20.10)
As an example, if ?e9oC/km then ? 1000s. In a
wind of 10 m/s, the vertical oscillations
will give rise to waves of wavelength about 10
km. If the conditions are right, these may be
visible as a regular pattern of lenticular
clouds, which form in the updraft sections of the
waves.
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Images taken from the website of Bruce Sutherland
(http//taylor.math.ualberta.ca/bruce )
Internal waves over Edmonton (A. Mehta)
Internal waves viewed from space.
Internal waves launched by flow through Strait
of Gibraltar.
More internal waves from space.
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• ATMOSPHERIC STABILITY III
• Conditional instability
• Stability changes caused by lifting air masses
• Convective instability

CONDITIONAL INSTABILITY Up to this point, we
have examined atmospheric hydrostatic stability
by considering the virtual, vertical displacement
of dry air parcels. But what if they are
saturated (I.e., cloudy)? Certainly, the vertical
displacement of cloudy air occurs in the
atmosphere. A little thought about this problem
will quickly lead to the conclusion that the
stability criteria for this case are analogous to
those for the dry case, except that the parcels
dry adiabatic lapse rate is replaced by
the pseudo-adiabatic lapse rate. Thus, for a
saturated atmosphere, the stability criteria are
(21.1)
Since ?s lt ?d it is thus easier to obtain
instability with saturated air than with dry
air. Supposing now that we dont know whether
the virtual parcel of air we displace will be
saturated or unsaturated. But we would
nevertheless like to say something about the
stability of
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the atmosphere. NOTE Keep in mind that
stability is a property of the atmosphere, not of
the virtual parcels of air we choose to displace
from their equilibrium. What can we say? If ? lt
?s then the atmosphere will be stable for both
saturated and unsaturated Parcel displacements.
We call this absolute stability. Similarly, if ?
gt ?d the atmosphere will be Unstable for both
saturated and unsaturated parcel displacements.
This is absolute instability. If the
environmental lapse rate lies between the
pseudoadiabatic and the dry adiabatic
value, Stability will depend on whether saturated
or unsaturated parcel displacement occurs. This
is Known as conditional instability. We summarize
these results below. NOTE Conditional
instability here, which is a form of static
instability, should not be confused with
conditional instability of the second kind
(CISK), which is a form of dynamic instability
occurring in the tropics. CISK arises out of a
cooperative interaction (positive feedback)
between cumulus convection and a large-scale
disturbance. In a hurricane, for example,
cumulus convection provides the latent heat
energy that is converted into kinetic energy and
drives the hurricane winds. In turn, the
hurricane provides moisture convergence near the
surface, which feeds the cumulus convection.
(21.2)
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The sketch below illustrates what will happen
when a parcel of air is lifted through a
large vertical displacement in an atmosphere with
a conditionally unstable lapse rate. NOTE Up to
this point we have been considering only small
vertical displacements of air parcels, and this
is all that is needed in order to determine the
local stability of the atmosphere. But we know
that real atmospheric processes (e.g.,
convection) lead to large vertical displacement
of air parcels. So it is quite realistic to
consider the consequences of such displacements.
Some physical mechanism will be required to lift
the parcel through the stable region. This could
be orographic ascent, or ascent arising from a
combination of divergence above and convergence
below. In the sketch above, the parcel is
initially at point A. As it is lifted, it
is colder than its environment, so work must be
done in order to lift it to the level of free
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convection (LFC). This work is the negative area
on the tephigram and is hatched. After reaching
the LFC, the parcel becomes warmer than its
environment and continues to rise pseudoadiabatica
lly. The environment now performs work on the
parcel, and this is the positive area on the
tephigram (also hatched). Part of this work will
be required to overcome friction as the parcel
rises through its environment, and part will be
converted into kinetic energy in the updraft of
the resulting cloud. NOTE This is not molecular
friction. Much of it is the results of
entrainment of ambient air (with low vertical
momentum) into the cloud. If you observe
closely, you can see how this happens at the edge
of a cloud as cloudy elements engulf clear air
and drag it into the cloud. Some of it is due to
turbulent drag, and finally some of it will be
due to the drag of the precipitation particles.
If we were to convert all of the positive area
into kinetic energy, this would give an upper
limit to the maximum possible updraft in the
cloud.
PROOF THAT THE WORK REQUIRED TO LIFT AN AIR
PARCEL FROM POINT A TO POINT B EQUALS THE
NEGATIVE AREA ON THE TEPHIGRAM BETWEEN
THE ENVIRONMENT AND PARCEL SOUNDINGS. The work
done in lifting a unit mass through the vertical
distance from A to B is, from integrating
WorkForce ? Distance,
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Substituting from the equation of motion (Eq.
20.4) where the last equality arises from
the use of the hydrostatic equation and the ideal
gas law (keeping in mind that we are assuming the
pressure in the environment equals the pressure
in the air parcel at the same level). Making use
of the definition of potential temperature,
and keeping in mind that the parcel and
environmental temperatures are the same at points
A and B, we have finally This is just the
negative area on the tephigram, illustrated in
the previous sketch.
If an air parcel rising from the surface gives
rise to a positive area on the tephigram
sounding (see sketch below), it is likely that
air parcels rising from some finite layer above
the surface will also give rise to a positive
area. This layer of air near the surface is said
to have latent instability. The top of the layer
with latent instability occurs at the point where
rising parcels cease to produce a positive area
on the sounding. Even though there may be a
layer with latent instability near the surface,
there will be no cumulus cloud formation unless
there is some means to produce the work necessary
to overcome the lower negative area on the
sounding. This may be accomplished by surface
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heating, which will tend to modify the lowest
portion of the sounding, making it dry
adiabatic. The convective condensation level
(CCL) is the point of intersection of this dry
adiabat with an equisaturated curve whose value
represents the average mixing ratio over the
lowest 50 mb. The convective temperature, Tc, is
the surface temperature at the bottom of the dry
adiabat, extending downwards from the CCL. It is
an estimate of the surface temperature that must
be achieved in order to cause the onset of
cumulus convection on a particular day.
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The positive area on a tephigram is often used to
predict the intensity (severity) of
convection. Since areas are not so easy to
measure, various indices have been developed that
give a measure of this area. One of the earliest,
and still used, is the Showalter Index. It is
defined as the difference between the parcel and
environment temperatures at 500 mb, for parcels
rising from 850mb. STABILITY CHANGES CAUSED BY
LIFTING AIR MASSES LIFTING A DRY AIR MASS As an
air mass rises or subsides (descends) in the
atmosphere, its stability can change as
a consequence of changes in its lapse rate
(since, as we have seen above, the local lapse
rate determines the local stability). We will
begin by considering the case of a dry air mass,
and examine a layer within it defined by two
isobaric surfaces (so that the mass of the layer
is fixed). If we use the vertical lapse rate of
potential temperature to examine stability (Eq.
19.2), then we need to examine how ??/?z varies
during lifting. Since the process is dry, by
assumption, ?? will not change. However, ?z will
increase as a result of expansion of the air mass
in the vertical. The result is that the magnitude
of the potential temperature gradient will be
reduced. Thus, whatever the initial stability, or
instability, it will be reduced as lifting moves
the lapse rate closer to neutral. Conversely,
subsidence in the atmosphere increases stability
or instability, by moving the lapse rate further
from neutral (?z decreases as the air mass is
compressed during descent). See the following
sketch for an illustration of how this occurs.
You may wish to consider an example on the
tephigram for yourself.
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LIFTING AN AIR MASS WHICH BECOMES SATURATED In
the three following sketches, we show that when
an initially isothermal layer is lifted to
a Level where it becomes entirely saturated, its
final stability depends on the lapse rate of
the Adiabatic equivalent potential temperature
(or the adiabatic wet-bulb potential
temperature). NOTE Remember that ?ae is
conserved during adiabatic lifting, so that it
doesnt matter whether we speak of the initial
or the final lapse rate of the adiabatic
equivalent potential temperature. They are the
same. Moreover, you can convince yourself, by
considering examples with other initial lapse
rates, that the result is independent of the
initial temperature lapse rate, and hence
independent of the initial stability of the layer.
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In summary, a layer that is lifted until it is
completely saturated has a stability, after
lifting, that is independent of its initial
stability, but depends on the initial vertical
gradient of adiabatic equivalent potential
temperature, as follows
(21.3)
Just to confuse matters, some people refer to
this as potential stability or instability, so be
prepared to encounter both names. As an example
of the use of this concept, consider
orographic flow leading to cloud formation. If
the air mass is potentially stable, we would
expect stratiform clouds to ensue, and possible
steady rainfall. If the air mass is potentially
unstable, cumuliform clouds and showers are
likely.