James I.N. Introduction to Circulating Atmospheres
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3.3 Thermodynamics
of
fluid motion
69
Net i-r
Flux
Height
Emitting Levels
Net Solar
Flux
Absorbing Level
-s
s
Fig. 3.4. Schematic illustration of
the
vertical variation of
the
downward flux of solar
radiation and the net upward flux of infrared radiation in
a
cloud-free atmosphere.
The difference between incoming and outgoing radiant energy fluxes
in
the lower
atmosphere is balanced by an upward transport of heat by atmospheric motions.
of the heat-transporting motions within
the
atmosphere
and
oceans.
The
difference between the incoming and outgoing radiation
is
balanced
by the
divergence of the heat fluxes. The partitioning
of
the heat fluxes between the
atmosphere
and the
oceans
is
still uncertain, though
it is
generally agreed
that not more than half the heat flux
is
carried by the ocean circulation.
Less generally appreciated
is
the fact that the vertical transport
of
heat
is
also
a
crucial aspect
of
the atmospheric circulation. Since the atmosphere
is
relatively opaque to infrared radiation, radiation escapes to space principally
from upper tropospheric levels. But sunlight
by
and large reaches the lower
troposphere.
At
intermediate levels, there
is an
imbalance between
the net
upward flux
of
infrared radiation and the downward flux
of
solar radiation.
This imbalance
is
compensated
by an
upward dynamical flux
of
heat
by
motions within
the
atmosphere. Fig.
3.4
gives
a
schematic sketch
of the
imbalance
of
vertical heat fluxes.
3.3 Thermodynamics
of
fluid motion
The thermodynamic state
of a
parcel
of
dry
air is
determined
by
fixing
the value
of
any two
of
the thermodynamic state variables. These include
such quantities
as
temperature, pressure, density
or
specific entropy.
The
remaining variables can than
be
calculated from the equation
of
state.
For
70
The
atmospheric heat engine
our purposes, it will be most convenient to describe the state of an air parcel
by specifying its temperature T and potential temperature 6. The specific
entropy s is closely related to 9 by:
s = c
p
\n9, (3.13)
while the pressure is obtained from a form of the equation of state for an
ideal gas, Eq. (1.9):
/
""\
(3.14)
where p
R
is some arbitrary reference pressure, generally taken as
100
kPa for
the Earth's atmosphere.
In order to change the thermodynamic state of the air parcel, heat must
in general be supplied or removed. The only exception is the special class
of 'adiabatic processes'. Many meteorological processes are approximately
adiabatic, because the time scales associated with fluid motion are often fast
compared to the timescales associated with radiative and diffusive transport
of heat. But of course in discussing the global circulation, departures from
adiabatic conditions are crucial in supplying the energy needed to maintain
the circulation against friction. The heat taken up by a unit mass of air
during a thermodynamic process from state A to state B is
Q=
I Tds.
(3.15)
JA
It is helpful to represent such a process on a 'thermodynamic diagram' in
which s is plotted as abscissa and T as ordinate, shown in Fig. 3.5. The heat
supplied to execute the process is simply the area under the curve joining
successive states of the air parcel on the thermodynamic diagram. Note that
if the specific entropy of state B exceeds that of state A, then Q is positive,
that is, heat must be supplied to the air parcel. Conversely, if s(B) < s(A),
heat must be extracted.
Immediately apparent from this diagram is that the heat supplied in
changing from state A to state B cannot be stated uniquely. There are
an infinite number of different paths between the two states, and the area
under each will usually be different. In general, if the entropy is increased
at higher temperature, then more heat will be required. Now suppose that
the air parcel is processed from A to B, and then back to A. As illustrated,
more heat is supplied during the A to B process than is extracted in the
return process. The air parcel has been returned to its initial state, and yet
net heat has been supplied to it. What has happened to this excess heat
energy? The answer is that it has been converted into some other form of
3.3 Thermodynamics of fluid motion
71
T
c
p
ln(0)
Fig. 3.5. Thermodynamic diagram, with s = c
p
\n(6) plotted against T, for a parcel
of air whose state changes from state A to state B.
energy, typically into kinetic energy of motion of the air parcel. Any cyclic
process which takes the air around a clockwise path on the thermodynamic
diagram will generate kinetic energy. A parcel of air executing such a path is
an example of a thermodynamic 'heat engine'. Conversely, an anticlockwise
path will extract more heat than is supplied; the imbalance is made up by
reducing the kinetic energy of the parcel. The cycle is then an example of a
'refrigeration' cycle.
Such thermodynamic cycles are familiar to the physicist or engineer and are
of value in discussing the performance of practical heat engines. Obviously,
only a fraction of the heat supplied to the air parcel can be converted to
mechanical energy; the ratio of mechanical energy obtained to heat supplied
is called the 'thermodynamic efficiency' of the heat engine:
p
=
Tds
I
(3.16)
Tds
The thermodynamic efficiency of a heat engine operating between states A
and B is a maximum when the entropy is increased at constant temperature
T
B
and reduced at a lower but constant temperature T
A
, the cycle being
closed by adiabatic processes. The path on the thermodynamic diagram is
therefore a rectangle. Such an ideal cycle is called a 'Carnot cycle', and is
illustrated in Fig. 3.6. Real heat engines are always less efficient than an
ideal Carnot engine, and atmospheric motions are very much less efficient.
72
The atmospheric heat engine
T
A
B
c
p
ln(6)
Fig. 3.6. A Carnot cycle, operating between state A and state B.
In the preceding section, it was suggested that tropospheric motions
must transport heat upwards and polewards. The simplest model of such
heat transporting motions would consist of rising motion in the tropics,
poleward motion at upper levels, sinking at high latitudes and return flow
to the equator at lower levels. Cross sections of the observed zonal mean
temperature structure are shown in Section
4.1.
Projecting such an idealized
circulation on to this section, the successive thermodynamic states of an
air parcel can be read off and plotted on the thermodynamic diagram.
The result is a small loop on the thermodynamic diagram, indicating that
such motions would generate kinetic energy; hence they can be maintained
against friction by the continual conversion of heat energy into mechanical
energy. Circulations which are clockwise on the thermodynamic diagram
involve ascent of warmer air and descent of colder air; they are referred to
as 'thermodynamically direct' circulations. Circulations in the reverse sense,
which must be forced mechanically, are termed 'thermodynamically indirect'.
The kinetic energy generated by each circuit of an air parcel of unit mass
is
K=
IT&S.
(3.17)
If
T
C
is the time required for an air parcel to execute a complete circulation
in the meridional plane, then the rate at which kinetic energy per unit mass
is generated is
f.i/ra,
(3.18)
3.4
Observed atmospheric heating
73
The kinetic energy
per
unit mass
is
related
to the
typical wind speed, (7,
by
K
= U
2
/2.
The generation
of
kinetic energy must
be
balanced
by
frictional
dissipation
in the
long term. Using
the
'Rayleigh friction' introduced
in
Section
2.4 as a
schematic
way of
representing
the
complicated friction
processes, the rate
of
destruction
of
kinetic energy
by
friction
is
(3.19)
T
D
being the frictional timescale with
a
typical value
of
around five days
for
the troposphere. Balancing the creation and destruction
of
kinetic energy,
it
follows that
in
the long term mean:
1
'~*= —orU = {^<bTds} .
(3.20)
Purely thermodynamic arguments
can
take
us no
further. Given
x
D
and
T
C
,
a
typical value
of U may be
estimated. Observations
of the
mean
meridional wind suggest that
x
c
is
typically around 140 days, leading
to U
of around
20 m s"
1
. The
global average wind
is
around
14 m s"
1
,
which
is
in
reasonable agreement with this estimate.
But to
predict
U on
purely
theoretical grounds requires consideration
of the
dynamics
of
the flow
as
well
as its
thermodynamics.
3.4 Observed atmospheric heating
The distribution
of
heating and cooling within the atmosphere which drives
the atmospheric circulation will
be
considered
in
this section.
We
must
distinguish between adiabatic changes
of
temperature, which
may
arise
from vertical motions during which
no
heat enters
or
leaves
the air, and
changes which result from heat entering
or
leaving
the air. The
latter
is
sometimes referred
to as
'diabatic warming'
or,
more simply,
as
'heating'.
Adiabatic temperature changes result
in no
transport
of
heat.
On the
other
hand, heating provides
the
sources
and
sinks
of
heat which drive
the
global
circulation. Heating arises from
a
large number
of
processes. Ultimately,
absorption
of
short wave radiation
is the
source
of
heating,
and
emission
of long wave radiation provides cooling.
But
exchanges
of
heat between
different components
of
the climate system, which includes
the
solid Earth,
the oceans and the ice and snow (or 'cryosphere'),
as
well
as
the atmosphere,
are important
for
driving the global circulation.
Among the various mechanisms which are important are
the
following:
(i) Exchanges of heat with the underlying surface, which may either be the ground
74
The
atmospheric heat engine
or
the
ocean surface. Much incoming sunlight reaches
the
surface directly,
and
heats
it up.
This heat
may
find
its way
back into
the
atmosphere
as a
result
of turbulent transports through
the
atmospheric boundary layer. Solid ground
has
a
rather small heat capacity,
and so the
sunlight reaching
a
solid surface
finds its
way
into
the
atmosphere rather quickly.
The
oceans,
on the
other
hand, have
a
huge heat capacity,
and
they therefore represent
an
important
reservoir
of
heat
in the
climate system.
It is
generally assumed that much
of
the memory
of
the climate system
on
timescales longer than
a
month
or so is
due
to
heat stored
in the
oceans.
(ii) Most
of
the solar energy reaching
the
Earth's surface goes
to
evaporate water,
rather than
to
raise temperatures. Because
of the
very large latent heat
of
evaporation
of
water,
and the
fact that most
of the
Earth's surface
is
moist,
very large amounts
of
heat
can be
taken
up in
this way. Measurements reveal
that
as
much
as
90%
of
the incident sunlight
at the
Earth's surface evaporates
water. Even
in
arid regions, evaporation takes
up
some
10% of the
absorbed
incident radiation.
(iii) Water vapour
in the
atmosphere acts
as a
means
of
storing heat which
can be
released later. As the
air
circulates,
it
may rise
and
cool;
if
it becomes saturated,
water vapour condenses
and
may rain
out of
the air. This condensation releases
large amounts
of
latent heat. Indeed,
in the
tropical atmosphere,
the
heating
is
dominated
by the
release
of
latent heat
in
rain clouds.
(iv) Most
of
the cooling
of
the atmosphere
is due to
long wave radiation, though
some heat
can be
extracted locally
by
contact with
a
colder underlying surface.
The transmission
of
long wave radiation through
the
atmosphere
is
highly
variable, being affected
by the
humidity
of
the
air and its
cloudiness.
Measuring all these different processes is a formidable task, and certainly
cannot be done routinely over the entire volume of the atmosphere. The
parametrizations included in a sophisticated numerical forecast model repres-
ent an attempt to estimate the heating directly in terms of the observed scale
fields of temperature, wind, moisture, and so on. But, as we have seen in
the last chapter, such parametrizations are less reliable than one would wish,
and are sometimes subject to considerable errors. It must be concluded
that the thermal forcing of the global circulation cannot be observed or
calculated directly with any great accuracy. What can be done is to infer
the net heating from the large scale temperature and wind fields, and their
changes over long periods. As we have seen, these fields can be monitored
reasonably accurately by the operational observing network.
Take the time average of the thermodynamic equation, Eq (1.12); the
result is:
A0
T
+
v
•
V0 +
a>
— +
V
•
VW + —oTW = I. (3.21)
dp dp
3A Observed atmospheric heating 75
Here, A0
T
represents the change of 9 over the averaging period
T.
Provided
T
is reasonably large, this term will be small, especially for the DJF and JJA
periods. The second and third terms represent, respectively, the horizontal
and vertical advection of potential temperature by the time average motions.
The fourth and fifth terms represent the divergence of the transient eddy
fluxes of
9.
All these terms must balance the heating. Accordingly, Eq. (3.21)
can be rearranged as a diagnostic expression for J2. Calculating J2 in this
way frequently involves much cancellation between the various transport
terms,
which tend to be of comparable magnitude but varying sign. Such a
calculation of J is called a 'residual method'. Note that it cannot distinguish
between the various physical processes which produce heating, but only
determines the net heating. Its reliability depends crucially upon accurate
estimates of the vertical velocity field. These may be suspect, especially in
the tropics and in the stratosphere.
Figure 3.7 shows the zonal mean heating,
[Q]
— (p/p
R
)
K
[£]
9
calculated
using this residual method. It is based upon just six years of ECMWF
analyses (since there is evidence that difficulties with initialization led to
underestimates of the tropical vertical velocity, and hence of the tropical
heating, during the first few years of operational analysis and forecasting
at the centre). The general pattern throughout the troposphere generally
conforms to the qualitative account given earlier in this chapter. Away from
the deep tropics, there is heating near the ground. This heating is
dominated by the turbulent transport of heat out of the ground, which is
heated by direct insolation. Regions of cooling, dominated by long wave
radiation to space, occupy much of the middle and upper troposphere.
The heating is strong and fills the entire depth of the troposphere in the deep
tropics. This is largely a signature of deep cumulus convection, releasing
latent heat throughout the tropical troposphere. Separate bands of relatively
deep heating are found in the midlatitudes. These are where active midlati-
tude depression systems lead to enhanced precipitation and release of latent
heat.
Comparison of
the
DJF and JJA cross sections reveals the seasonal cycle of
heating. The maximum tropical heating rates are in the summer hemisphere.
The midlatitude heating is in the winter hemisphere. The large maximum
at
70 °S
in JJA is probably spurious, and is a result of extrapolation of
the wind and temperature fields beneath the high Antarctic ice sheet.
There are substantial differences between the midlatitude heating rates in
the two hemispheres. We will return to a discussion of some of the
differences between the storm depression latitudes in each hemisphere in
Chapter 5.
76
The atmospheric heat engine
-60S -30S 0 30N 60N
LATITUDE
90N
0
LATITUDE
Fig. 3.7. Latitude-pressure cross sections of the time and zonal mean heating, [Q],
based on six years of ECMWF data. Contour interval
0.2
K day"
1
, positive values
shaded, (a) DJF. (b) JJA.
Regional variations of the heating are illustrated if Q is integrated with
respect to height. Figure 3.8 shows the quantity
g Jo
(3.22)
3.5 Problems 11
Since p
s
/g is the mass per unit area of a column of the atmosphere reaching
from the surface to infinity, {Q} has the dimensions ofWm"
2
and is therefore
directly comparable with the insolation and the long wave radiative fluxes
discussed earlier in the chapter. In the DJF season, the largest heating is
concentrated in the Indonesian region. There are strong maxima over the
equatorial continents of Africa and South America. Two midlatitude 'storm
track' regions show up as heating maxima over the Pacific and Atlantic
Oceans. As might be expected, the largest cooling is close to the winter
pole. The maximum heating or cooling rates are generally no more than
100
Wm~
2
, although the Indonesian maximum reaches
225
Wm~
2
over a
small area. This is quite small compared with the global average absorbed
heat flux of around
240
W
m~~
2
,
showing that a considerable fraction of the
incoming solar flux is simply re-radiated by the surface, without heating
the atmosphere. In JJA, the largest heating rates are found north of the
equator, and the maxima associated with the midlatitude storm tracks are
considerably weaker. The most striking feature is the large maximum close
to the Tibetan plateau. This represents huge latent heat releases as the moist
winds of the Asian summer monsoon impinge on the Himalayan mountain
ranges.
3.5 Problems
3.1 Use the following data to estimate the radiative equilibrium timescale for
Venus, Earth and Mars. In each case, compare this timescale to the planet's
solar day and its year.
Ps
s
a
c
p
Wm~
2
ms"
2
JK^kg"
1
Venus
9MPa
2550
0.76
8.9
567
Earth
100 kPa
1370
0.29
9.8
1004
Mars
700 Pa
583
0.15
3.7
567
3.2 The radiative timescale given in Eq. (3.11) was derived for a surface
with negligible thermal capacity, whose temperature adjusts instantly to
changes of the radiation incident upon it. If the surface has a finite thermal
capacity, C
g
say, write down equations for the evolution of perturbations AT
g
and AT
a
to the temperature of the ground and the atmosphere, respectively,
and hence deduce how such perturbations will change in time. If C
g
is very
78
The atmospheric heat engine
(b)
Fig.
3.8.
Vertically integrated
net
heating based
on six
years
of
ECMWF data.
Contour interval 50Wm~
2
, with positive values shaded,
(a) DJF. (b) JJA.
much larger than c
p
p
s
/g = C
a
, show that two timescales, corresponding to
the equilibrium timescales for the surface and the atmosphere, characterize
the evolution of the temperature.
3.3 Suppose that the radiative equilibrium temperature of an atmosphere
T
E
varies sinusoidally with period T
S
and amplitude AT
E
. Determine the
time between the maximum T
E
and the maximum atmospheric temperature,
and the amplitude of the atmospheric temperature fluctuation.
3.4 A typical air parcel circulates from low levels in the tropics to high
levels in the polar regions and back on a timescale of
120
days.
Assume a
friction timescale of five days, and a typical windspeed of
15
m s"
1
. Assuming,