Download 3. Vertical Temperature Structure and Stability I. Vertical

3. Vertical Temperature Structure
and Stability
I. Vertical Temperature Profile
1. Absorption of solar radiation (Lambert-Bouguet-Beer Law)
z
dz
I∞
θ
ds
The change in incoming radiation flux (dI) along a path length ds, where
the density of the absorber is ρ, and the absorption coefficient is k, may
be written:
[I and ds are both measured
dI = -k ρ I ds
positive downwards]
dI = -k ρ I ds
z
dz
I∞
θ
ds
I and k both depend on frequency. The units of k are m2kg-1 or area
per unit mass. Therefore, k is sometimes called the absorption cross
section of the gas in question. The pathlength is related to the altitude by:
dz = - cos θ ds
Therefore,
dI
cos θ dz = k ρ I
z
dz
I∞
θ
ds
The optical depth (τ) along the vertical path is defined as:
∫
∞
τ = k ρ dz
[∫ ρ dz=number of particles]
z
which implies that d τ = -k ρ dz and therefore:
This equation has a very simple solution:
I = I∞ e-τ /cosθ
dI
cos θ dτ = -I
I = I∞ e-τ /cosθ
Where I∞ is the downward flux density at the top of the atmosphere.
The incident flux thus decays exponentially along the slant path ds
where the optical depth is given by τ/cosθ. The greater the optical
thickness, the less the transmission. The optical depth is dimensionless,
and has a value of zero outside the earth’s atmosphere, and increases
with increasing depth into the atmosphere. If we consider radiation
normal to the surface, cosθ = 1 then
I(z) = I ∞ e -τ(z)
At the surface,
I(0) = I ∞ e -τ(0)
The transmission T through the atmosphere is simply e-τ. Hence, if T=0.
or the absorption is complete, τ(0) => ∞, I(0) = 0 implying no radiation
gets to the surface. If τ=1 there is 40% transmission; if τ=2, 14%
transmission.
In the atmosphere density decreases exponentially with increasing height:
ρ = ρo e-z/H
where H=RT/g is called the scale height (R, g constants). This is the
height where the density drops to 1/e the surface density. In a well
mixed atmosphere H~8km.
∞
τ(z) =
Therefore,
∫
k ρo e-z/H dz
z
∞
if
τo =
∫ k ρ dz
o
z
then
τ(z) = τo e-z/H
τ(z) = τo e-z/H
Hence as you go up, τ is decreasing with height, and the atmosphere
becomes optically thin. In this case the scale height H refers to the
absorbing gas. In the atmosphere the major gases such as N2 and O2
(CO2) are well mixed up to 120km.
But water vapour, which is a major absorber in the atmosphere is not
well mixed because it is rained out by precipitation. For water vapour
H ~ 2km. Therefore, at z=10 km
τ(z)
τ(0)
= e-10/2 =
1
150
For this reason places at high altitudes are colder than those at sea level.
For example, Denver, Colorado (altitude ~ 2km asl) has only 37% (1/e)
of the water vapour density found at sea level, and therefore can absorb
less heat from the earth’s surface. Similarly, dry regions (Negev) have
colder nights than moist regions (Tel Aviv).
2. Absorption of terrestrial (longwave) radiation
Reradiation from the earth also provides energy at a particular level j:
R(z) = R(0) e-τj(z)
where
∫
z
τj(z) = klw ρ dz
0
klw= long wave absorption coefft.
So at any altitude there is absorption of both long wave and short wave
radiation:
F(z) = I(z) + R(z)
3.
The atmosphere, however, also emits radiation (with an emission
coefficient j):
dL(z) = j ρ dz
for a particular layer
4.
The net heating in a layer is the difference between the incoming and
outgoing radiation (in that layer).
5.
The solution for the atmospheric heating is obtained numerically, for
various gases, integrating over all zenith angles. The spectrum is
divided into a number of regions which are large enough so that the
fine structure of the spatial lines is smoothed out, but small enough
for the Plank function to be regarded as constant.
The ground temperature is approximated as: Tg4 = Te4 (1+3/4 τo
)
The air temperature is approximated as: Ta4 = Te4 (1/2+3/4 τo )
Tg4 = Te4 (1+3/4 τo
)
Ta4 = Te4 (1/2+3/4 τo )
For 50% transmission τ=0.7,
Ta is 26oC colder than Tg. This
enormous temperature gradient in the
lower atmosphere is never observed due
to vertical mixing that occurs in the
atmosphere by transport of heat away
from the surface, cooling the surface and
warming the upper atmosphere.
6. At any height z:
z
Ta4(z) = Te4 (
after mixing
Before
mixing
T
1 3
2 + 4 τ(z))
z
As τ falls off quite fast with increasing
altitude (due to the falloff of water vapor),
the atmosphere becomes isothermal by 10km.
10
T
Ta
Tg
II. Vertical Stability
1. Static Stability
From the conservation of energy, if dh is the applied heat, dU is the
Internal energy, and dW is the work done, then:
dh = dU + dW = cp dT -
dP
ρ
In an adiabatic process, dh=0
cp dT = dP
ρ
In hydrostatic balance dP=- ρgdz (heavier air is below lighter air)
dTd -g
oC/km = γ = dry adiabatic lapse rate
=
=10
cp
d
dz
So if the measured atmospheric lapse rate (change of temp. with height)
exceeds –10oC/km the atmosphere will automatically readjust to preserve
hydrostatic equilibrium – a process called dry convection. There is no
condensation of moisture in this process (no release of latent heat).
If the atmosphere is moist, condensation can occur as the air rises and
cools, releasing latent heat (L) and warming the environment:
-L dws = cp dT + gdz
Therefore,
where w=mixing ratio=
dTm
dz = f(T, w, P) = γm
Temp.
(C)
Pressure
(mb)
-30
0
γm
<
γd with a mean value for γm = 6oC/km
20
ρw
ρdry
1000
500
9.6
6.5
4
8.7
5.1
3.3
These lapse rates indicate how a parcel of air will cool as it rises and
expands in the atmosphere. If the parcel becomes saturated, some
moisture will condense, and the latent heat released will warm the parcel
and thus decrease its cooling rate. If at any level the parcel is warmer than
its environment, it will continue to rise. This condition is called convective
instability – either dry convection, or moist convection.
The likelihood of the atmosphere to convect (mix) depends on the
temperature profile of the observed atmosphere. If the background lapse
rate is γ, then the conditions below indicate whether a parcel of air,
when lifted, will be warmer or cooler than the background at a certain
height.
Air always cools according to the dry or moist adiabatic lapse rate as it
rises.
γ = γd
γ > γd
γ < γd
neutral stability for dry air
absolutely unstable (all rising parcels will be warmer
than the ambient atmosphere)
absolutely stable (all rising parcels will be colder
than the ambient atmosphere)
-15C
-5C
γ = -15oC/km
5C
15C
γ = 5oC/km
-15C
γ = 5oC/km
-5C
γ = 15oC/km
5C
15C
γd > γ > γ m
conditional instability (stable if dry air and unstable if
moist, which is the general condition in the atmosphere)
2. Baroclinic Instability
Baroclinic instability is a result of horizontal temperature gradients that
exist on the earth’s surface due to uneven heating by the sun. The
instability is caused by warmer air entering regions of lower temperature
by advection (horizontal transport). Midlatitude storms form due to
baroclinic instabilities that result from the large temperature gradients
between the warm equatorial latitudes and the cold polar latitudes. If
warm air from a lower latitude is forced to rise at a shallow angle, then
it can be warmer than the background at some location even if the
background is stable to air which rises purely vertically. Baroclinic
instabilities generally depend on there being a latitudinal temperature
gradient, and their role is to mix the air to reduce the horizontal and
vertical temperature gradients.
warm
cold
Low latitudes
High latitudes
3.
The observed lapse rate is a combination of these two effects (vertical
and horizontal mixing). They thus change the appearance of the
temperature profile in the troposphere from that which would exist
in pure radiative equilibrium.
4.
In the stratosphere, ozone absorption causes the temperature to
increase, making the region statically stable.
III. 1-D Models (Radiative-Convective Models)
1.
1-D is altitude – generally globally averaged, yearly averaged
- similar to 0-D models, but radiation explicitly calculated to determine
temperature profile.
One dependent variable (T) but here it is a function of height.
2.
An initial temperature profile is calculated based on the initial
conditions. The model then looks at this temperature profile and
compares it to the dry and moist adiabatic lapse rates. If the calculated
lapse rate exceeds this critical value, the model assumes that the
atmosphere would have reduced it (either by mixing due to convective
or baroclinic instability) and readjusts to the prescribed critical lapse
rate by cooling the lower layer(s) and warming the higher ones. When
the model returns to the critical lapse rate the model is said to be in
radiative-convective equilibrium.
3. In practice the atmosphere is broken up into layers:
dTo(zk)
g ∆qsw(zk)
= cp
dto
∆p
[
+
∆qlw(zk)
∆p
]
∆p is the pressure thickness of the layer, ∆q is the radiative flux at the top
minus that at the bottom of the layer. At the next time step: t1=t0 + ∆t
dTo(zk)
T1(zk) = To(zk) +
dto ∆t
This produces a new temperature profile, which gives new heating rates,
which gives a new temperature profile, etc. The calculations continue
until the net radiative heating rate of each model layer is not subject to
convective adjustment, and the net fluxes at the top of the layers
approaches zero.
4.
Boundary conditions: at the top of the atmosphere, the net short wave
radiation coming into the atmosphere must equal the net long wave
radiation going out. This also applies at the base of the stratosphere,
which is assumed to be in radiative equilibrium (no convective
adjustment being necessary). At the surface, which is in radiativeconvective equilibrium, the net gain by radiation = the net convective
heat loss.
5.
Initial conditions: The variables that determine the fluxes of radiant
energy in the atmosphere include atmospheric gases (primarily H2O,
CO2 and O3), aerosols (sulfate, dust and volcanic), cloud charactistics
(altitude, reflectivity and absorptivity), the surface albedo (snow,
ocean, landcover) and insolation (solar constant, zenith angle).
6.
The temperature in the atmosphere drops off quickly with height
because the concentration of water vapour, the major absorber of
longwave radiation, drops off quickly with height. At 10km the
temperature profile is close to isothermal because of very little water
vapour. At higher altitudes in the stratosphere O3 results in further
heating of the atmosphere.
Effect of Clouds
Homework:
Kasting and Ackerman, 1986: Climatic consequences
of very high CO2 levels in the Earth’s early atmosphere,
Science, 234, 1383-1385.