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 thecondensed water must remain in the parcel. Since the process is adiabatic and reversible, it is alsoisentropic, so that:

€

ds = (c pd+ rtcw )d lnT − Rd d ln p + d

lvrs

T

⎛

⎝ ⎜

⎞

⎠ ⎟= 0 (16.1)

where rt is the total mixing ratio consisting of the sum of the mixing ratio of water vapour andthe mixing ratio of liquid water. However, for a given state of the parcel (defined by T, p), thetotal mixing ratio is not unique because the liquid water mixing ratio depends on the historyof the parcel and not just its current state. Hence it is impossible to draw a reversible, saturatedadiabat uniquely on a tephigram.

Pseudoadiabatic Processes

It is possible to obviate the difficulty alluded to above if we assume all of the condensed liquidwater 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 aninfinite sequence of infinitesimal two-stage processes (see Note below), we may write for the pseudo-adiabatic process:

(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.]€

(c pd+ rscw )d lnT − Rd d ln p + d

lvrs

T

⎛

⎝ ⎜

⎞

⎠ ⎟= 0

Despite the theoretical differences between Eq. 16.1 and 16.2 (i.e., using the saturation mixingratio 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 givenby:

(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:

€

c pdd lnT − Rd d ln p + lvd

rs

T

⎛

⎝ ⎜

⎞

⎠ ⎟= 0

1. Starting at (T,p,r), follow a moist adiabat upwards (actually a dry adiabat) to the saturationpoint (the point at which the adiabat intersects the equisaturated curve with rs=r). This point isalso 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 requiredto evaporate liquid water into it (which had to be derived from its internal energy since theprocess was adiabatic). The final state of the parcel is therefore similar to that which occursduring 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 thanTiw. But in reaching the adiabatic wet-bulb temperature, the liquid water is evaporated into theparcel at temperatures lower than Taw. Since the specific latent heat of vapourization decreaseswith temperature, there is less latent heat required to evaporate the same mass of liquid waterisobarically 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 twoprocesses.

Adiabatic Equivalent Temperature, Tae

The adiabatic equivalent temperature is defined by the following processes on the tephigram:

1. Starting at (T,p,r), ascend along a dry adiabat to the LCL.2. 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 higherthan the initial temperature because of the release of the latent heat of condensation. The finalstate 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 achievethe adiabatic equivalent temperature is possible, at least in principle). The condensation in theprocess leading to the adiabatic equivalent temperature occurs at temperatures below thesaturation temperature, while the condensation in the process leading to the isobaric equivalenttemperature occurs at temperatures above the initial temperature. Because the specific latentheat of condensation decreases with temperature, there is more latent heat released in theadiabatic process than in the isobaric process. Consequently, Tae>Tie, and the difference canbe several degrees.

The adiabatic wet-bulb potential temperature, aw, can be found by simply continuing downthe 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. Theconservative properties are particularly valuable in helping to identify and track air masses.

Property Isobaric warming and cooling

Isobaric evaporation and condensation

Non-saturated 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-equilibriumbetween the vertical pressure gradient force and the force of gravity:

(17.1)

€

1

ρ

∂p

∂z= −g

Note that if we consider only the vertical direction then we would have

For those of you who are not students of dynamical meteorology, this equation can be readilyderived by considering the balance of forces acting on an air parcel of height dz in a column ofunit cross-sectional area, as in the figure below: €

1

ρ

dp

dz= −g

Because gravity is a conservative force (i.e., one for which ), it can be defined as thegradient of a potential function:

€

vg ⋅d

v r ≡ 0∫

€

vg = −

v ∇ψ

You should verify for yourself that if 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 meansea level (our chosen reference height) to a certain height, z. (Recall that work, or energy, equalsforce times distance so that for a unit mass, Eq. 17.2 is just such a statement.)

€

vg = −

v ∇ψ

(17.2)

€

(z) = gdz'

0

z

∫ (17.3)

We can now use to define a new height scale. Let:

€

Z ≡ψ

g0(17.4)

where g0=9.80665 m/s2 is the acceleration due to gravity at mean sea level. Z is the height ingeopotential metres. The chief advantage of using Z rather than the geometric height, z, is thatwe can consider the acceleration due to gravity to be a constant when integrating, since:

€

g0dZ = gdz (17.5)

Having made the distinction between the geopotential height and geometrical height, we willnow 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 ofpotential 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=g0Z).

Consequently, geopotential height is an important concept, and when the term height is used bya 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 (bothmeasured 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 realatmosphere (or a portion of it), under certain circumstances (you should try to figure out whatthese are).

1. HOMOGENEOUS ATMOSPHERE (=constant)

A homogeneous atmosphere is one with constant density. Because the real atmosphere iscompressible, it cannot be homogeneous. By contrast, the ocean is approximately homogeneousbecause its compressibility is about 10-4 times that of air. If we consider an atmospheric layerof 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 finitethickness (unlike the real atmosphere). This thickness, H, can be determined by integratingthe hydrostatic equation 17.1:

€

dpp0

0

∫ = −ρg dz0

H

∫ ⇒ H =p0

ρg=

RT0

g(18.1)

where the subscript “0” denotes mean sea level conditions, and the final equation is obtained bymaking 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

dz= ρR

dT

dz= −ρg (18.2)

from which the lapse rate, -dT/dz, is given by: =g/R~34 K/km. You may recall that this is theautoconvective lapse rate. That is, if the temperature diminished even more rapidly with heightthan this value, the density would begin to increase with height and the result would clearly beconvectively 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, eventhough the atmosphere tries valiantly to vertically redistribute the surface heat by convectiveoverturning, the resulting rate of heat transfer is not great enough to overcome the effects of thesolar 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 inthe diagram below). You can also see this effect above large campfires during the day (as wellas 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 exponentiallywith height, and that the e-folding height (the height at which pressure or density is a factor of1/e times its value at the surface) is equal to the scale height. We begin by substituting the idealgas law into the hydrostatic equation and integrating:

€

dp

dz= −ρg = −

pg

RT⇒

d ln p

dz= −

1

H⇒ p = p0 exp −

z

H

⎛

⎝ ⎜

⎞

⎠ ⎟ (18.3)

The exponential decrease of density with height follows readily from Eq. 18.3 and the ideal gaslaw 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 ascale height of about 7.3 km. Although the real atmosphere is not isothermal, the variation inabsolute 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 forevery 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 − γz (18.4)

with a constant. If the temperature sounding in the real atmosphere can be approximated by apiecewise-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

dz= −

pg

R(T0 − γz)⇒ p = p0

T0 − γz

T0

⎛

⎝ ⎜

⎞

⎠ ⎟

g

γR

(18.5)

Like the homogeneous atmosphere, a constant lapse rate atmosphere has a finite height. At thetop, T=0, p=0, and this implies a thickness of T0/. Since Eq. 18.5 should be valid for all lapserates, we can ask the question: what is the result when 0?

Recalling that we can manipulate Eq. 18.5 as follows:

€

n →∞lim 1+

1

n

⎛

⎝ ⎜

⎞

⎠ ⎟n

= e

€

p = p 1−γz

T0

⎛

⎝ ⎜

⎞

⎠ ⎟

−T0

γz ⎡

⎣

⎢ ⎢ ⎢

⎤

⎦

⎥ ⎥ ⎥

−gz

RT0

⇒ p = p0 exp −gz

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

c p

≅10 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 issupposed 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 thediagram below:

RAWINSONDE HEIGHT COMPUTATIONS

Rawinsonde balloons generally measure T, p, Td at regular intervals (~10s) as they ascend at aspeed 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 surfacedetermines 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 humidityfrom q=r/(1+r), and then the virtual temperature from Tv=T(1+0.87q). Then, beginning with thehydrostatic 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 formp=RdTv, we can integrate the hydrostatic equation to yield:

€

z2 − z1 = −Rd

gTvd ln p

p1

p2

∫ (18.8)

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 rawinsondestation. Thereafter, Eq. 18.8 can be integrated from the surface numerically. Alternatively, onemay use the tephigram to obtain the mean temperature over a finite pressure interval, as in thesketch below. Using this mean temperature, the thickness between significant pressure levelscan then be looked up on the tephigram. Alternatively, on may use the mean value theorem toevaluate Eq. 18.8, giving:

€

z2 − z1 =Rd Tv

gln

p1

p2(18.9)

REDUCTION OF PRESSURE TO MEAN SEA LEVEL

Surface weather charts depict isobars on a zero-height surface (mean sea level). However, notall 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 bepermanent 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 themeasured station pressure by the weight of the air in the hole. Equivalently, inverting Eq. 18.9,we have:

€

p1 = p2 expz2

Rd Tv

⎛

⎝ ⎜

⎞

⎠ ⎟ (18.10)

where p2 is the measured station pressure, z2 is the geopotential height of the rawinsondestation, and is the mean virtual temperature of the air in the hole. The problem is thatthere 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 stationtemperature and the temperature twelve hours ago, and that the lapse rate in the hole is5o C/km.

€

Tv