Salby M.L. Fundamentals of Atmospheric Physics
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160
6
Hydrostatic Equilibrium
cycle and is proportional to the change of the parcel's potential temperature
at the upper and lower boundaries (i.e., to the heat transfer there).
Net heat absorption and work performed by individual air parcels make
the general circulation of the troposphere behave as a heat engine, one driven
thermally by heat transfer at its lower and upper boundaries. Work performed
by individual parcels is associated with a redistribution of mass when air that
is effectively warmer and lighter (namely, when compressibility is taken into
account) at the lower boundary is exchanged with air that is effectively cooler
and heavier air at the upper boundary. The latter represents a conversion of
potential energy into kinetic energy, which maintains the general circulation
against frictional dissipation.
6.5.2 Diabatic Stratification
The idealized behavior just described relies on heat transfer being confined to
the lower and upper boundaries of the layer, where an air parcel resides long
enough for diabatic effects to become important. On longer timescales, as are
typical of vertical motions in the stratosphere, the evolution of an individual
air parcel is not adiabatic. Radiative transfer, which is the primary diabatic
influence outside the boundary layer and clouds, is characterized by cooling
rates of order 1 K day -1 in the troposphere (see Fig. 8.24). Cooling rates as
large as 10 K day -1 occur in the stratosphere and near cloud (Fig. 9.33).
By comparison, the cooling rate associated with adiabatic expansion follows
from (2.33) as
= -F'w, (6.29.1)
where T and z refer to an individual parcel, F' denotes its lapse rate, and
dz
w = (6.29.2)
dt
is the parcel's vertical velocity. A vertical velocity of 0.1
m s -1,
which is rep-
resentative of cumulus and sloping convection, gives (dT/dt)a d = 84 K day -1
under unsaturated conditions and of order 50 K day -1 under saturated con-
ditions. Both are much larger than radiative cooling rates, so heat transfer in
those motions can be ignored to a good approximation. On the other hand, a
velocity of 1 mm s -~, which is representative of vertical motion in the strato-
sphere, is associated with an adiabatic cooling rate of only (dT/dt)a d = 0.84 K
day -1. Heat transfer in such motions cannot be ignored and may even domi-
nate temperature changes associated with expansion and compression of indi-
vidual air parcels.
6.5 Lagrangian Interpretation of Stratification 161
Under those circumstances, an air parcel moving vertically interacts ther-
mally with its environment. Because 0 is no longer conserved, diabatic motion
implies a different thermal structure. Consider a layer characterized by
Fd > F = const, (6.30.1)
the stratification of which is termed subadiabatic: Temperature decreases with
height slower than dry adiabatic. By (6.18), 0 increases with height (compare
Fig. 6.5). Then (6.25) together with (2.37) gives
c>O
6q > 0 dT > 0 (6.30.2)
6q < 0 dT < 0
for F < 0 (temperature increasing with height) and c and 6q with reversed
inequalities for F > 0. Both imply that an air parcel absorbs heat during
ascent and rejects heat during descent, so its potential temperature increases
with height. As illustrated in Fig. 6.8, heat is rejected at high temperature in
upper levels and absorbed at low temperature in lower levels. By the second
law, net heat is rejected over a cycle. Then the first law implies that net work
must be performed on the parcel for it to traverse the circuit. More work
is performed on the parcel during descent than is performed by the parcel
during ascent. 3
Opposite to the troposphere, this behavior is characteristic of the strato-
sphere. Net heat rejection and work performed on individual air parcels make
the general circulation of the stratosphere behave as a refrigerator, one driven
mechanically by waves that propagate upward from the troposphere. By rear-
ranging air, planetary waves exert an influence on the stratosphere analogous
to paddle work. When dissipated, they cause air at middle and high latitudes
to cool radiatively, so individual parcels sink across isentropic surfaces to lower
0 (see Figs. 8.27 and 17.11). Conversely, air at low latitudes warms radiatively,
so parcels then rise to higher 0.
For both, vertical motion is slow enough to make diabatic effects important
and to allow the temperature of individual parcels to increase with height.
This feature of the stratospheric circulation follows from buoyancy. Contrary
to the troposphere, buoyancy in the stratosphere strongly restrains vertical
motion because, on average, warmer (high 0) air overlies cooler (low 0) air.
Work performed on the stratosphere against the opposition of buoyancy leads
to a redistribution of mass, when heavier air at lower levels is exchanged with
lighter air at upper levels. The latter represents a conversion of kinetic energy
3 In practice, this paradigm can be pushed only so far because stratospheric air must eventually
enter the troposphere to complete the thermodynamic cycle (refer to Fig. 17.9), which makes the
stratosphere an open system.
162
6 Hydrostatic Equilibrium
(a)
t
q<O
..... -~;,
,~' ....
/ #
I ......i~ I
/ ../"
/ .... ."/ / q > 0
/ ...:
# ! I O Im'
I ...........'" I
q <0/'~
...I //
/ /
/~.~. _ ~/_ .~_ /
t q>O
(b)
I
§ /=
/ .' T
j
....
1 /
IF<0 ..:
' /
"~,.~ /
q<o~ /
I
...
§ :
/ "
t
q<O
I
!
~q>O
I
I
I
I
I
/
/
/
~ -t~ "--~ -~ J
t q>O
it,
T
Figure
6.8 As in Fig. 6.7,
but for an air
parcel whose vertical motion is diabatic and
whose temperature increases with height.
into potential energy, which is dissipated thermally through LW emission to
space.
Stratification with
r > r~,
(6.31)
which is termed
superadiabatic,
can also be interpreted in terms of diabatic
motion of individual air parcels. However, such stratification will be seen to be
mechanically unstable. Buoyancy then reinforces vertical motion, so it occurs
on timescales too short for diabatic effects to be important. Through convec-
tive overturning, a superadiabatic layer quickly adjusts to adiabatic stratifica-
tion, for reasons developed in Chapter 7.
Problems
163
Suggested Reading
Atmospheric Thermodynamics (1981)
by Iribarne and Godson contains a thor-
ough discussion of atmospheric statics.
Problems
6.1.
6.2.
6.3.
6.4.
6.5.
6.6.
6.7.
Calculate the discrepancy between geometric and geopotential height,
exclusive of anisotropic contributions to gravity, up to 100 km.
Show that the polytropic potential temperature in Problem 2.9 is con-
served under the diabatic conditions described in Sec. 6.5.2.
Use (2.27) and replace z by -In p to relate the area circumscribed in
Fig. 6.7 to the work performed by the air parcel during one thermody-
namic cycle.
Approximate the thermodynamic cycle in Fig. 6.7 by one comprised
of two isobaric legs and two adiabatic legs to show that (a) net work
is performed during a complete cycle and (b) the work performed is
proportional to the change of temperature associated with isobaric heat
transfer.
Show that an atmosphere of uniform negative lapse rate need not have
an upper bound.
For reasons developed in Chapter 12, large-scale horizontal motion is
nearly tangential to contours of geopotential height and given approxi-
mately by the geostrophic velocity
1( o )
8~
,
where v = (u, v) denotes the horizontal component of motion in the
eastward (x) and northward (y) directions and 9 is the geopotential
along an isobaric surface. Suppose that satellite measurements provide
the three-dimensional distributions of temperature and of the mixing ra-
tio of a long-lived chemical species. From those measurements, construct
an algorithm to determine 3-dimensional air motion on large scales at
elevations great enough to ignore variations in surface pressure.
An upper level depression, typical of extratropical cyclones during early
stages of development, affects isobaric surfaces aloft but leaves the sur-
face pressure distribution undisturbed. Consider the depression of iso-
baric height associated with the temperature distribution
= -- [ ~2 -'~ (~ -- ~0)2]
T(A, 4~, ~) -- T + T cos(4~) + T' exp - L2 9 f(~),
where T, T, and T' reflect global-mean, zonal-mean, and disturbance
contributions to temperature, respectively, ~ - -ln(p/po), with P0 =
164
6 Hydrostatic Equilibrium
6.8.
6.9.
6.10.
6.11.
1000 mb, is a measure of elevation,
{-cos
0 ~>~T,
with
~r = --In(100/1000)
reflecting the tropopause elevation. The sur-
face pressure
Ps = Po
remains constant, ~b 0 = 7r/4, L = 0.25 rads, and
T = 250 K, T = 30 K, and T' -- 5 K. (a) Derive an expression for
the 3-dimensional distribution of geopotential height. (b) Plot the verti-
cal profiles of disturbance temperature and height in the center of the
anomaly. (c) Plot the horizontal distribution of geopotential height for
the midtropospheric level ~ =
~r/2,
as a function of A and q~ > 0.
(d) Plot a vertical section at latitude ~b 0, showing the elevations ~: of
isobaric surfaces as functions of A.
Under the conditions of Problem 6.7, how large must the anomalous
temperature T' be for the disturbance to form a cut-off low at the mid-
tropospheric level ~: =
~r/2?
Unlike midlatitude depressions, hurricanes are warm-core cyclones that
are invariably accompanied by a trough in surface pressure. Consider a
hurricane associated with the temperature distribution
T(A, ~b, ~:) = T + T'(A, q~)e -'~,
where T and T' reflect zonal-mean and disturbance contributions to
temperature, ~: =
-ln(p/po),
with P0 = 1000 mb, is a measure of
elevation,
m
T'(A, ~b)-
Tar
~b)exp[asrs(A, ~b)],
in which
~:,(A, 4~) =0.1 exp - L2 ,
with a -1 = ~:r/2, and ~:r = In(100/1000) reflects the surface and
tropopause pressures, respectively, and L = 0.1 rads. (a) Derive an
expression for the 3-dimensional distribution of geopotential height.
(b) Plot the vertical profiles of disturbance temperature and height in
the center of the anomaly. (c) Plot the horizontal distribution of geopo-
tential height for the midtropospheric level ~: = ~:r/2. (d) Plot a vertical
section at the equator, showing the elevations ~: of isobaric surfaces as
functions of A.
Show that subadiabatic stratification requires an individual parcel moving
vertically to, in general, absorb (reject) heat as it ascends (descends).
Surface analyses produced by the weather service plot pressure adjusted
to the common reference level, mean sea level (MSL), so that different
locations can be compared meaningfully. (a) Derive an approximate
Problems 165
6.12.
expression for the pressure at MSL in terms of the local surface height
Zs,
surface pressure
Ps,
and the global-mean surface temperature T -
288 K. (b) A surface analysis shows a ridge with a maximum of 1040 mb
over Denver, which lies at an elevation of 5300 ft above MSL. What
surface pressure was actually recorded there?
Deep convection inside a region of the tropics releases latent heat to
achieve a column heating rate of 250 W m -2. (a) If, uncompensated,
that heating would produce a temperature perturbation that is invari-
ant with height up to the tropopause, calculate the implied change of
thickness for the 1000- to 100-mb layer after 1 day (compare Fig. 1.30).
(b) What process, in reality, compensates for the tendency of tropo-
spheric thickness implied in part (a)?
Chapter 7
Hydrostatic Stability
The property responsible for the distinctly different character of the tro-
posphere and stratosphere is vertical stability. Although forces are never far
from hydrostatic equilibrium, vertical motions are introduced by forced lift-
ing over elevated terrain and through buoyancy. Buoyantly driven motion is
related closely to the stability of the atmospheric mass distribution, which is
shaped by transfers of energy between the earth's surface, the atmosphere,
and deep space. By promoting convection in some regions and suppressing it
in others, vertical stability controls a wide range of properties.
7.1 Reaction to Vertical Displacement
Hydrostatic equilibrium and compressibility lead to density decreasing with
height regardless of temperature structure (Fig. 6.5). Thus, mean stratification
invariably has lighter air configured over heavier air, which suggests stability
with respect to vertical displacements. Were air incompressible, this would
indeed be the case.
Consider an air parcel inside the layer shown in Fig. 7.1. In a linear stability
analysis, we will use this parcel to establish how the layer reacts to small
disturbances from equilibrium. Although the analysis focuses on an individual
parcel, bear in mind that stability refers to the layer as a whole. Suppose
the parcel is subjected to a virtual displacement 6z' = z', where primes will
distinguish properties of the parcel from those of its environment. Through
expansion or compression, the parcel automatically adjusts to the environmen-
tal pressure to preserve mechanical equilibrium. Thus,
p'= p
during the displacement. Further, so long as the displacement occurs on a
timescale short compared to that of heat transfer with the environment, the
parcel's evolution may be regarded as adiabatic.
Per unit volume, Newton's second law for the parcel is
d2z ' 3p'
p' = - ~, (7.l.l)
dt 2 -P' g ~z'
166
7.1 Reaction to Vertical Displacement
167
dz'
pdS
z~ fb = (P-
p,P')g
p, T, z, p
Figure 7.1 Schematic of an air parcel, the properties of which are distinguished by primes,
inside a designated layer. The specific buoyancy force
fb
results from an imbalance between the
parcel's weight and the net pressure force acting over its surface.
which follows from a development analogous to the one leading to (1.16). The
momentum balance for the environment is simply hydrostatic equilibrium:
Op
(7.1.2)
0=
pg Oz"
Subtracting (7.1) and incorporating mechanical equilibrium yields for the par-
cel
p~ d 2 z'
- (p - p')g,
(7.2.1)
which may also be argued from Archimedes' principle (Problem 7.3). Per unit
mass, the parcel's momentum balance is then described by
dt 2 = p, g
= fb,
(7.2.2)
where
fb
is the (specific) net buoyancy force acting on the parcel. With (7.1.2),
the buoyancy force is recognized as an imbalance between the parcel's weight
and the vertical component of pressure force acting over its surface.
168
7 Hydrostatic Stability
Consider how the displaced parcel's temperature varies in relation to that
of the environment. With the gas law, (7.2.2) may be expressed
d2z '
( T' - T)
(7.3)
dt 2 - g T "
For small displacements from its undisturbed elevation, the parcel's temper-
ature decreases with height at the parcel lapse rate: F' = Fa (Fs) under
unsaturated (saturated) conditions. Thus,
T' = T o - F'z' (7.4.1)
describes the first-order variation of the parcel's temperature from its undis-
turbed value T o , which equals the environmental temperature at the parcel's
undisturbed elevation. On the other hand, the temperature of the surround-
ings decreases with height at the local environmental lapse rate F, so the
corresponding variation of environmental temperature is
T = T o - Fz'. (7.4.2)
The parcel's lapse rate is determined by its thermodynamics. Under adia-
batic conditions, F' is independent of the environmental lapse rate F because
the parcel is then thermally isolated. The environmental lapse rate, on the
other hand, is determined by the history of air residing in the layer (e.g., by
where that air has been and what thermodynamic influences have acted on it).
Incorporating (7.4) transforms the vertical momentum balance (7.3) into
dZz'
= g(v- r')z'. (7.5)
dt 2 T
Hence, the specific buoyancy force experienced by the parcel is proportional to
its displacement z' and to the difference between the parcel and environmental
lapse rates.
7.2 Stability Categories
If the parcel is unsaturated,
F'= F d. (7.6.1)
Since F' is then constant, (7.4.1) is also valid for finite displacements. If the
parcel is saturated,
F'= F s, (7.6.2)
which implies the local saturated lapse rate at the undisturbed elevation.
7.2
Stability Categories
169
7.2.1 Stability in Terms of Temperature
For each of the preceding conditions, three possibilities exist (Fig. 7.2)"
1. F < F': Environmental temperature decreases with height slower
than the displaced parcel's temperature. Then (7.5) implies
1 d2z '
<0.
z' dt 2
The parcel experiences a buoyancy force
fb
that opposes the displace-
ment z'. If it is displaced upward, the parcel becomes heavier than its
surroundings and thus negatively buoyant, whereas it becomes positively
buoyant and lighter than its surroundings if the parcel is displaced down-
ward. This buoyancy reaction constitutes a
positive restoring force,
one
that acts to restore the system to its undisturbed elevationmirrespective
of the sense of the displacement. The atmospheric layer is then said to
be
hydrostatically stable
and the environmental temperature profile is said
to be of
positive stability.
2. F- F" Environmental temperature decreases with height at the
same rate as the parcel's temperature. Under these circumstances,
1 d2z '
O,
z' dt 2
so the buoyancy force fb vanishes and the displaced parcel experiences
no restoring force. The atmospheric layer is then said to be
hydrostatically
neutral
and the environmental temperature profile is said to be of
zero
stability.
(a) Stable (F < F')
~m' 5 m
fb < 0 ~ "~ ~~
~'\lfb > 0
T'>T
(b) Neutral (F = F')
F ,~,
fb=0
(c) Unstable (F > F')
T'>TI fb >0
k
k
fb<O~
T' <T
Ira, Ira,
T T T
Figure
7.2 Buoyancy reaction experienced by a displaced air parcel, in terms of the envi-
ronmental lapse rate F and the parcel lapse rate F', which equals F d (Fs) under unsaturated
(saturated) conditions. Stability categories: (a) F < F', (b) F = F', and (c) F > F'.