Transcript Document

ATMOSPHERIC PROCESSES--PSEUDOADIABATIC
• Adiabatic wet-bulb temperature
• Adiabatic equivalent temperature
• Conservative properties
Reversible Saturated Adiabatic Processes
If we consider a closed parcel of rising, cloudy air, the process is reversible since all the
condensed water must remain in the parcel. Since the process is adiabatic and reversible, it is also
isentropic, so that:
ds  (c p d
lv rs 
 rt c w )d lnT  Rd d ln p  d  0
 T 
(16.1)
where rt is the total mixing ratio consisting of the sum of the mixing ratio of water vapour and
the mixing ratio of liquid water. However, for a given state of the parcel (defined by T, p), the
total mixing ratio is not unique because the liquid water mixing ratio depends on the history
 and not just its current state. Hence it is impossible to draw a reversible, saturated
of the parcel
adiabat uniquely on a tephigram.
Pseudoadiabatic Processes
It is possible to obviate the difficulty alluded to above if we assume all of the condensed liquid
water to be continuously removed from the air parcel. The parcel thus becomes an open system
and the process is irreversible. However, if we assume that the process can be divided into an
infinite sequence of infinitesimal two-stage processes (see Note below), we may write for the
pseudo-adiabatic process:
(c p d
lv rs 
 rsc w )d lnT  Rd d ln p  d  0
 T 
(16.2)
[Note: These two-stage processes are an infinitesimal reversible saturated adiabatic expansion
and condensation, followed by the removal of an infinitesimal amount of condensed liquid water,
without changing
 T, p.]
Despite the theoretical differences between Eq. 16.1 and 16.2 (i.e., using the saturation mixing
ratio in 16.2 as opposed to the total mixing ratio in 16.1), the practical difference is small.
Hence an approximation to both the saturated adiabatic and pseudoadiabatic processes is given
by:
rs 
c p d d lnT  Rd d ln p  lv d  0
T 
(16.3)
Integration of Eq. 16.3 leads to the pseudoadiabats on the tephigram (the zebra-stripe curves).
Adiabatic Wet-Bulb Temperature, Taw

The adiabatic wet-bulb temperature is defined by the following process on a tephigram:
1. Starting at (T,p,r), follow a moist adiabat upwards (actually a dry adiabat) to the saturation
point (the point at which the adiabat intersects the equisaturated curve with rs=r). This point is
also known as the lifting condensation level (LCL).
2. From the LCL, descend along a pseudoadiabat to the original pressure, p.
3. The temperature at this point is the adiabatic wet-bulb temperature.
Note that the result of this process is that the original air parcel is now saturated and at the
initial pressure. Its temperature, however, has dropped because of the heat which was required
to evaporate liquid water into it (which had to be derived from its internal energy since the
process was adiabatic). The final state of the parcel is therefore similar to that which occurs
during adiabatic, isobaric evaporation of liquid water, leading to the isobaric wet-bulb
temperature. However, the adiabatic wet-bulb temperature and the isobaric wet-bulb
temperature are not identical. The explanation is a simple one. In reaching the isobaric
wet-bulb temperature, the liquid water is evaporated into the parcel at temperatures higher than
Tiw. But in reaching the adiabatic wet-bulb temperature, the liquid water is evaporated into the
parcel at temperatures lower than Taw. Since the specific latent heat of vapourization decreases
with temperature, there is less latent heat required to evaporate the same mass of liquid water
isobarically than there is adiabatically. As a result, Tiw > Taw, although the difference is
generally only a fraction of a degree. See the sketch below for an illustration of these two
processes.
Adiabatic Equivalent Temperature, Tae
The adiabatic equivalent temperature is defined by the following processes on the tephigram:
1.
2.
Starting at (T,p,r), ascend along a dry adiabat to the LCL.
Continue to ascend along a pseudoadiabat from the LCL, until the air parcel is dry
(practically speaking, you can only go as far as the -50oC isotherm).
3. Descend along a dry adiabat to the original pressure.
4. The temperature at this point is the adiabatic equivalent temperature.
The outcome of this process is a dry air parcel at the initial pressure. Its temperature is higher
than the initial temperature because of the release of the latent heat of condensation. The final
state is thus similar to the final state defined by the isobaric equivalent temperature (although
the process required to achieve the latter is impossible, while the process required to achieve
the adiabatic equivalent temperature is possible, at least in principle). The condensation in the
process leading to the adiabatic equivalent temperature occurs at temperatures below the
saturation temperature, while the condensation in the process leading to the isobaric equivalent
temperature occurs at temperatures above the initial temperature. Because the specific latent
heat of condensation decreases with temperature, there is more latent heat released in the
adiabatic process than in the isobaric process. Consequently, Tae>Tie, and the difference can
be several degrees.
The adiabatic wet-bulb potential temperature, aw, can be found by simply continuing down
the pseudoadiabat to 100 kPa and reading the temperature there. Similarly, the adiabatic
equivalent potential temperature, ae, can be found by continuing down the final dry adiabat
to 100 kPa and reading the temperature there. (See the previous diagram.)
CONSERVATIVE PROPERTIES
The table below indicates which thermodynamic properties are conserved (denoted by “C”)
and which are not conserved (denoted by “N”) under various atmospheric processes. The
conservative properties are particularly valuable in helping to identify and track air masses.
Property
Isobaric
warming and
cooling
Isobaric
evaporation
and
condensation
Nonsaturated
adiabatic
expansion
Saturated adiabatic
expansion
u
N
N
N
C
e, Td
C
N
N
N
q, r
C
N
C
N
Taw, Tae
N
“C”
N
N
aw, ae
N
“C”
C
C

N
N
C
N
THE HYDROSTATIC EQUATION
• Geopotential energy
• Hydrostatic equation
Geopotential Energy
A scale analysis of the vertical equation of motion leads to a very convincing near-equilibrium
between the vertical pressure gradient force and the force of gravity:
1 p
 g
 z
Note that if we consider only the vertical direction then we would have
(17.1)
1 dp
 g
 dz
For those of you who are not students of dynamical meteorology, this equation can be readily
derived by considering
the balance of forces acting on an air parcel of height dz in a column of


unit cross-sectional area, as in the figure below:
Because gravity is a conservative force (i.e., one for which
gradient of a potential function:
g  

 g  dr  0 ), it can be defined as the
(17.2)
You should verify for yourself that if g   then gravity is a conservative force.
Therefore, surfaces of constant potential function, , are perpendicular to gravity and vice versa.


Thus we see that  is the potential
energy per unit mass required to lift an air parcel from mean
sea level (our chosen reference height) to a certain height, z. (Recall that work, or energy, equals
force times distance so that for a unit mass, Eq. 17.2 is just such a statement.)
 (z) 
z
 gdz
'
(17.3)
0
We can now use  to define a new height scale. Let:

Z

g0
(17.4)
where g0=9.80665 m/s2 is the acceleration due to gravity at mean sea level. Z is the height in
geopotential metres. The chief advantage of using Z rather than the geometric height, z, is that
we can consider the acceleration due to gravity to be a constant when integrating, since:
g0 dZ  gdz
(17.5)
Having made the distinction between the geopotential height and geometrical height, we will
now drop the adjective. Although the numerical difference between the two is only about 0.1%
or less in the troposphere, ignoring the difference could mean ignoring a significant amount of
potential energy.

Even a 10 metre height discrepancy corresponds to an error in potential energy that could
convert into a wind error of almost 15 m/s (use 0.5v2=g0Z).
Consequently, geopotential height is an important concept, and when the term height is used by
a meteorologist, it is safe to assume that geopotential height is what is meant.
The table below shows a comparison between geometric height and geopotential height (both
measured from mean sea level) at 40oN.
Geometric height (km)
Geopotential height (km)
Acceleration due to gravity
(m/s2)
0
0.
9.807
1.0
1.000
9.798
10.0
9.986
9.771
90.0
88.758
9.531
200.0
193.928
9.214
500.0
463.597
8.186
MODEL ATMOSPHERES
• Model atmospheres
• Rawinsonde height computations
• Reduction of pressure to sea level
Model Atmospheres
We will consider here four idealized atmospheres and examine the lapse rates of pressure,
density, and temperature within them. Each of these atmospheres is a suitable model of the real
atmosphere (or a portion of it), under certain circumstances (you should try to figure out what
these are).
1.
HOMOGENEOUS ATMOSPHERE (=constant)
A homogeneous atmosphere is one with constant density. Because the real atmosphere is
compressible, it cannot be homogeneous. By contrast, the ocean is approximately homogeneous
because its compressibility is about 10-4 times that of air. If we consider an atmospheric layer
of thickness 100 m, the variation of density across it is only about 1%. Hence, such a layer
(or a thinner one) could be considered to be approximately homogeneous. Ignoring this caveat,
if we were to consider the entire atmosphere to be homogeneous, it would have to have a finite
thickness (unlike the real atmosphere). This thickness, H, can be determined by integrating
the hydrostatic equation 17.1:
0
H
 dp  g  dz  H 
p0
0
p0 RT0

g
g
(18.1)
where the subscript “0” denotes mean sea level conditions, and the final equation is obtained by
making use of the ideal gas law. Using appropriate values leads to a value of H~8 km.
H is known as the scale height.

The temperature lapse rate in a homogeneous atmosphere can be deduced by equating the
vertical pressure gradient expressions from the ideal gas law and the hydrostatic equation (17.1):
dp
dT
 R
  g
dz
dz
(18.2)
from which the lapse rate, -dT/dz, is given by: =g/R~34 K/km. You may recall that this is the
autoconvective lapse rate. That is, if the temperature diminished even more rapidly with height

than this value, the density
would begin to increase with height and the result would clearly be
convectively unstable. It is possible to get such high lapse rates (or even higher ones) very close
(i.e., within a few centimetres) to a sun-warmed surface. Even over the distance of our bodies,
lapse rates can be generally quite large over a hot surface such as a road. In such cases, even
though the atmosphere tries valiantly to vertically redistribute the surface heat by convective
overturning, the resulting rate of heat transfer is not great enough to overcome the effects of the
solar heating. Hence it is possible, very near a hot surface, to have a density profile which
increases with height. This can lead to refraction of light in the atmosphere in such a way as to
give rise to the so-called superior mirage (e.g., apparent pools of water on a hot surface, as in
the diagram below). You can also see this effect above large campfires during the day (as well
as the effect of temperature on sound speed if you listen carefully to the people across the fire
from you!).
2. ISOTHERMAL ATMOSPHERE (T=constant)
We shall show that, in an isothermal atmosphere, the pressure and density decrease exponentially
with height, and that the e-folding height (the height at which pressure or density is a factor of
1/e times its value at the surface) is equal to the scale height. We begin by substituting the ideal
gas law into the hydrostatic equation and integrating:
 z 
dp
pg
d ln p
1
 g  

   p  p0 exp 
 H 
dz
RT
dz
H
(18.3)
The exponential decrease of density with height follows readily from Eq. 18.3 and the ideal gas
law for constant temperature.
By definition, the temperature lapse rate is zero in an isothermal atmosphere.
In the real atmosphere, the pressure decreases approximately exponentially with height, with a
scale height of about 7.3 km. Although the real atmosphere is not isothermal, the variation in
absolute temperature is only about 20% over the region of meteorological interest (the
troposphere and the stratosphere), with the result that an isothermal approximation is “not bad”
(at least for some purposes). A more useful number, however, is the half-height of the
atmosphere, which is about 5 km. This means that the pressure falls by a factor of about 2 for
every 5 km increase in height.
3. CONSTANT LAPSE RATE ATMOSPHERE
A constant lapse rate atmosphere is one in which the temperature varies linearly with height:
T  T0  z
(18.4)
with  a constant. If the temperature sounding in the real atmosphere can be approximated by a
piecewise-linear function (i.e., a sequence of straight lines), then a constant lapse rate model
can be applied to each of these linear layers. Integrating the hydrostatic equation gives:
 dp
g
R
T0  z 
pg

 p  p0 

dz
R(T0  z)
T
 0 
(18.5)
Like the homogeneous atmosphere, a constant lapse rate atmosphere has a finite height. At the
top, T=0, p=0, and this implies a thickness of T0/. Since Eq. 18.5 should be valid for all lapse
rates, we can ask the question: what is the result when 0?

1
Recalling that lim 

n 
1 
  e we can manipulate Eq. 18.5 as follows:
n 
n
gz
T0  RT
0

z


 z 
p  p1 
 T0 







 gz 
 p  p0 exp

 RT0 
as for an isothermal atmosphere.

4. DRY ADIABATIC ATMOSPHERE
A dry adiabatic atmosphere is a special case of a constant lapse rate atmosphere, and has:
d 
g
 10
cp
K /km
(18.6)
5. U.S. STANDARD ATMOSPHERE

The U.S. Standard Atmosphere
is an atmospheric model created by a committee in 1962. It is
supposed to represent the annual average characteristics of the real atmosphere in mid-latitudes
(40N) up to an altitude of 32 km. It consists of a sequence of constant lapse rate layers, as in the
diagram below:
RAWINSONDE HEIGHT COMPUTATIONS
Rawinsonde balloons generally measure T, p, Td at regular intervals (~10s) as they ascend at a
speed of about 5 m/s. This gives roughly 50 m resolution in the vertical, or 5 mb near the surface.
In order to determine the local height of isobaric surfaces (the height field on an isobaric surface
determines the geostrophic wind field, and hence is a very essential meteorological variable)
the sounding data can be integrated using the hydrostatic equation. This is done as follows.
First, Td is used to determine the mixing ratio r=rs(Td). Then we determine the specific humidity
from q=r/(1+r), and then the virtual temperature from Tv=T(1+0.87q). Then, beginning with the
hydrostatic equation (17.1) but now taking z to be the geopotential height, and hence g=g0, the
constant mean sea-level value. Substituting from the ideal gas law for moist air in the form
p=RdTv, we can integrate the hydrostatic equation to yield:
R
z2  z1   d
g
p2
 T d ln p
v
(18.8)
p1
where z2-z1 is known as the thickness between the pressure levels p1 and p2.
The geopotential height at the surface can be determined once and for all for each rawinsonde
station. Thereafter,Eq. 18.8 can be integrated from the surface numerically. Alternatively, one
may use the tephigram to obtain the mean temperature over a finite pressure interval, as in the
sketch below. Using this mean temperature, the thickness between significant pressure levels
can then be looked up on the tephigram. Alternatively, on may use the mean value theorem to
evaluate Eq. 18.8, giving:
Rd Tv p1
z2  z1 
ln
g
p2

(18.9)
REDUCTION OF PRESSURE TO MEAN SEA LEVEL
Surface weather charts depict isobars on a zero-height surface (mean sea level). However, not
all observing stations are at sea level, so the observed station pressure must be adjusted
(normally, this means increased) to mean sea level. If one did not do this, there would be
permanent low pressure areas over mountainous and hilly terrain, which would make it
difficult to identify the transient cyclonic systems which are associated with weather.
In order to perform the necessary adjustment, we imagine that a hole is drilled in the ground
(with a cross-section of 1 square metre) down to mean sea level. Then we increment the
measured station pressure by the weight of the air in the hole. Equivalently, inverting Eq. 18.9,
we have:
 z2 
p1  p2 exp

Rd Tv 
(18.10)
where p2 is the measured station pressure, z2 is the geopotential height of the rawinsonde
station, and Tv is the mean virtual temperature of the air in the hole. The problem is that
there is no hole, and hence no way to measure this mean virtual temperature. So we suppose,

for this purpose, that the temperature at the top of the hole is the average of the current station
temperature and the temperature twelve hours ago, and that the lapse rate in the hole is

5o C/km.