Lecture 3:Convection

EESC V2100The Climate System

spring 2004

Yochanan Kushnir Lamont Doherty Earth Observatory

of Columbia UniversityPalisades, NY 10964, USA

kushnir@ldeo.columbia.edu

Layers of the

Atmosphere

The atmosphere is divided into layers according to its vertical temperature distribution.

The overall vertical profile of temperature indicates the location of sources and sink of heat in the atmosphere.

surface warming

infrared cooling

Ultraviolet absorption by Ozone

photodissociation of Oxygen & Nitrogen

Thermodynamics of Dry air

Radiative Equilibrium Temperature Profile

PURE RADIATIVE EQUIL.

Convection and the Vertical Profile of Temperature in the Troposphere

• The vertical profile of temperature in the troposphere is determined by a combination of radiative and convective processes.

• The ground warms up by incoming shortwave radiation and by the longwave radiation emitted by atmospheric absorbers. It losses heat through longwave radiation, evaporation, and sensible heat flux. The latter are both linked with convection.

• The warming of the atmosphere by radiative heat trasnfer results in a vertical temperature profile that is convectively unstable, giving rise to vertical mixing that modifies it by transferring heat from the surface to the top of the troposphere.

convection is a from of heat transfer achieved through

fluid vertical motion, turbulence, and mixing.

evaporation is the transformation of water

from the liquid into vapor phase. It requires the expenditure of energy

causing a surface heat loss.

sensible heat flux is the flow of heat from the surface

through dry convection, that is, movement of warm air

up and cold air down

a convectively unstable situation occurs when air near the surface is more buoyant than the air above it rises spontaneously

The process of convection• Convection is the process of vertical heat transfer through vertical exchange of mass,

driven by buoyancy. Convection also distributes moisture vertically.

• Close to the surface in the atmospheric boundary layer, turbulence distributes air parcel up and down creating a well mixed layer in moisture and temperature. This motion can penetrate above the boundary layer (or convect) into the free atmosphere, if conditions favorable to it arise. Convection is a crucial part of the hydrological cycle, and also is, as we shall later see, a driver of the entire atmospheric circulation.

• The process of convection:

- Convection is governed by the vertical movement of fluid (air) parcels.

- If a parcel is buoyant within its environment, upward motion is possible.

- Buoyancy is governed by the density difference between the parcel and the surrounding environment.

- The density difference is governed by temperature of the parcel relative to the environmental temperature, because in the atmosphere pressure is roughly the same horizontally (ideal gas equation in isobaric conditions).

- Thus, if a parcel is warmer than its surroundings it will rise. If it is colder, it will sink.

• In the atmosphere, the process of convection is complicated by the air’s compressibility and the drop in pressure, density, and temperature with heigh. To understand how convection works in the atmosphere we need to consider the compressibility of the atmosphere and the first law of thermodynamics.

• The atmosphere is a compressible fluid - the volume of a unit mass depends on the pressure exerted on it. It implies that the density of gas changes as the pressure on it increases. Two equations sum up the compressible behavior of the atmosphere and dictate how pressure and density change with height: The ideal gas law and the hydrostatic balance equation.

• One of the early discoveries regarding thermodynamics of gases, dating back to the 17th-19th are the laws relating the pressure of a fixed mass of gas and its temperature and volume.

• Pressure is the force exerted per unit area on a surface placed in any arbitrary direction within a volume of gas - it is related to the momentum transfered to the surface by the gas molecules due to their random motion.

• Sir Robert Boyle (1672-1691) found that in constant temperature (isothermal process) the volume of a fixed mass of gas is inversely related to its pressure: reduction of the pressure results in an increase in the volume and vice versa.

• Jacques A. C. Charles (1746-1823) found that in constant pressure (isobaric process) in a fixed mass of gas the volume is directly proportional to the gas absolute temperature. He also showed that holding the mass and volume fixed pressure is directly proportional to the absolute temperature.

Compressibility of the Atmosphere

• Let us summarize these three laws:

Boyle’s Law states that when T (absolute temperature) and m (mass) are constant:

V = constant / p

Here V is the volume and p is the pressure. Charles’s two laws are:

V = constant x T, when p and m are fixed and:

p = constant x T, when V and m are fixed.

• The ideal gas law combine these three laws into one by stating that:

pV = mRT

Here R is the so called gas constant, its value varies from one gas to another.

• The ideal gas law can also be expressed in terms of the relationship between pressure, density (= ratio between mass and volume, denoted ρ), and temperature:

p = ρRT

The value of R for dry air in the atmosphere is: 287 J kg-1 K-1

Ideal Gas Law

Pressure Changes with Height

Hydrostatic Balance Equation• To find out how pressure changes with height we consider the

balance of forces acting on a volume of air suspended in the atmosphere in hydrostatic equilibrium.

• For this volume (figure) the weight of the air is balanced by the difference between the force of pressure acting on its base and pushing upward and the force of pressure acting on its top, pushing downward.

• In mathematical notation:

−mg = a(p+Δp) − ap = aΔp

Here g is the acceleration of gravity a is the area of the column base, and Δp denotes an incremental increase of pressure.

• Expressing the mass m as the product of density ρ and the column volume aΔz, combining terms on the left hand side and finally dividing by the area a we obtain:

Δp/Δz = −ρg

• Finally substituting for ρ from the ideal gas law we obtain:

Δp/Δz = −gp/RT

p

p+Δp

Δz

ρg

surface area a

The Decrease of Pressure with heightThe last equation is one form of the hydrostatic balance. It can help us understand how pressure changes with height.

Consider the ratio:

H = RT/g

It has dimensions of height (meters) and is called scale height.

Substituting in the hydrostatic balance we get:

Δp/Δz = −p/H

Which means that the rate of the pressure decrease with height is equal to the pressure divided by a scale height, which depends on the average ambient temperature.

In summary, the hydrostatic balance equation states that as we go up in altitude, the pressure decreases proportionally to the ambient pressure, that is increasingly more slowly (exponentially) with height.

percent atmospheric mass below

Pressure as a function of height

Pressure and Atmospheric Mass

Percent atmospheric mass as a function of height

The scale height for the global mean vertically integrated temperature is about 7.6 km

The hydrostatic relationship tells us that the pressure decreases exponentially with height.

It also tell us that the mass of atmospheric mass per unit area, which is the vertical integral of ρdz from the surface to the top of the atmosphere (where p=0), is equal to the surface pressure, ps

divided by the acceleration of gravity, g.

Similarly, the mass lying above any given surface in the atmosphere is determined by the pressure at that level divided by g.

Thus we can tell by the pressure at a given height what percentage of atmospheric mass lies below that surface.

The First Law of Thermodynamics

• The first law of thermodynamics is a statement of conservation of energy in a closed system.

• It states that the heat is added to (or extracted from) a closed system (e.g., an air parcel) is balance by a change of the system’s internal energy plus the work done by the system on its environment.

• In symbolic terms:

ΔQ=ΔE+ΔWwhere Δ denotes an incremental change and Q, E, and W are the heat input, internal energy, and work, respectively.

• Internal energy, ΔE is directly related to temperature:

ΔE = constant x ΔT

Work Done by Expansion

• The work done by an air parcel on the environment as it is being heated (and its temperature rises) is due to the parcel’s expansion (the increase in temperature in constant volume leads to an initial increase in pressure, which causes expansion).

• The work of expansion equals the force times the displacement of the parcel’s “walls”. Thus for each unit area of the surface surrounding an air parcel the work done is pΔx, where p is pressure and Δx is the displacement (see figure).

• Multiply that by the area of the surface of parcel: A and divide by the parcel’s mass m to obtain the total work done by expansion per unit mass:

ΔW = (pΔx)A / m = p ΔV / m

Where ΔV is the change in the parcel’s volume.

• We can also write this as:

ΔW = pΔα• where Δα is the change in specific volume of the gas (α=1/ρ).

• Thus the first law state that heat added to a parcel of gas results in a change in both its temperature and its specific volume (inverse density):

ΔQ = constant x Δ T + p Δα

gas with pressure p

Δx

Specific Heat in constant volume

• In the first law equation impose the conditions of constant volume (for example by putting the parcel of gas in a closed container. Then the gas cannot do any work and the equation becomes:

(ΔQ)constant volume = constant x ΔT

• Rearranging terms we get:

constant = (ΔQ/ΔT)constant volume

• The ratio between the heat input into a unit mass of gas and the change in temperature, when the volume is kept fixed is the specific heat of that gas in constant volume, denoted:

cv

• For dry air cv = 718 J K-1

Kg-1

.

• Thus we write the first law of thermodynamics for the atmosphere as:

ΔQ = cv ΔT + pΔα

Specific Heat in constant pressure

• The the ideal gas law:

RT = pαimplies that an increase in temperature can lead to a change in pressure and volume.

• Expressed mathematically an incremental change in ideal gas temperature is given by:

R ΔT = p Δα + α Δp

• Using this relationship to substitute for pΔα in the first law we get:

ΔQ = (cv + R)ΔT -- α Δp

• In constant pressure the last term in the expression above becomes zero leading to the relationship:

(ΔQ/ΔT)constant pressure = cv + R

•• We term this quantity specific heat in constant pressure cp:

cp=R+cv

• For dry air cp=1004 J K-1 Kg-1.

Adiabatic Lapse Rate• We are now close to a position where we can make a statement on the effect of

convection on the atmosphere.

• The First Law of Thermodynamics is now expressed as:

ΔQ = cpΔT -- αΔp

• We add one additional change by using the hydrostatic equation:

--αΔp = --(1/ρ)Δp = gΔz

• This yields the relationship:

ΔQ = cp ΔT + g Δz

• Consider now a parcel being displaced vertically in the atmosphere without any addition or extraction of heat, that is: ΔQ=0, also referred to as adiabatic motion. According to the first law it must satisfy: cpΔT = -- g Δz, which implies that: ΔT/Δz = -- g/cp.

• Thus the rate of change of temperature in a parcel, displaced vertically in an adiabatic process is a constant, equal to −9.8x10-3 K m-1 or −9.8°C per kilometer.

• An air parcel rising/sinking adiabtically in the atmosphere will cool/warm at a constant rate of 9.8°C/Km. This rate is referred to as the dry adiabatic lapse rate and denoted Γd (note

that Γd is defined as -- ΔT/Δz = g/cp).

Atmospheric Stability• In a dry atmosphere (one that has not reached saturation - see below) a comparison

between the environmental vertical temperature profile and the dry adiabatic one determines atmospheric satiability with respect to convection.

• If the temperature in the environment decreases more rapidly than the dry adiabatic lapse rate, that is: Γe > Γd, then the atmosphere is said to be UNSTABLE to convection.

• If, on the other hand, Γe < Γd, then the atmosphere is said to be STABLE to convection.

• If Γe = Γd, then the atmosphere is said to be NEUTRAL.

• Another way to determine the convective stability of the atmosphere is to plot the change of potential temperature with height. If dΘ/dz < 0, the atmosphere is unstable. If dΘ/dz > 0, the atmosphere is stable.

The hypothetical profile plotted in blue to the right has lines of constant slope of -10 K/km to help determine the stability of the atmosphere.

Environmental Temperature Profile

0

2

4

6

8

10

12

-70 -60 -50 -40 -30 -20 -10 0 10 20 30

Temperature (°C)

Heig

ht (

Km)

Thermodynamics of Moist air

Moisture in the Atmosphere• Most of the water in the atmosphere is carried around in the form of vapor.

• The source of atmospheric water is the evaporation from Earth’s surface, mainly the ocean. When it happens, evaporation draws latent heat from the surface.

• As water vapor move through the atmosphere they carry that latent heat carried with them, ready to be released once condensation occurs.

• The atmosphere’s moisture content is measured in several ways, falling into two categories, absolute and relative.

- An absolute measure of atmospheric water content is the partial pressure of the gaseous state of water or vapor pressure. It is usually denoted e and measured in milibars. Atmospheric vapor pressure is limited by the saturation vapor pressure, denoted es. The saturation vapor pressure depends mainly on the temperature of the

air (see graph next slide).

- Specific humidity q is another absolute measure, which entails the ratio between the mass of water vapor (mv in grams) per kg of dry air (md), i.e., q= mv/md. The saturation specific humidity is denoted q*.

- A relative measure is the relative humidity, which is the ratio of actual vapor pressure to saturation vapor pressure, expressed in %: RH=e/es x 100 and also: RH=q/qs x 100.

- Finally, the dew point temperature, Td is the temperature at which condensation occurs when the air is cooled at constant pressure (see next slide). When a parcel of air warms without adding moisture, the dew point temperature will remain unchanged.

This graph is based on measurements of humidity made in Black Rock Forrest, New York

Measurements of Vapor Pressure

Dew point temperature

The air parcel has a ~50% relative humidity

The humidity of a parcel can be increased to saturation

by evaporating water into it or by cooling the parcel to its dew point

Saturation vapor

pressure

Actual vapor

pressure

Lapse Rate in Moist Air• When a parcel of moist air rises, it cools adiabatically until the air reaches its dew point,

condensation occurs and latent heat is released into the rising parcel.

• The process is no longer adiabatic, but because the latent heat is released into the rising parcel, modifying its lapse rate in a quantifiable manner, we term to this process pseudo- or moist-adiabatic.

• In the diagram to the right, air with a temperature of 35°C and a dew point temperature of 26°C is pushed upward from the surface (for example, by being pushed against a mountain range).

• After rising for about 1 km it will cool adiabatically and will reach a temperature equal to its dew point and therefore will become saturated.

• From here on the rising air will cool more slowly as the condensational heating compensates for part of the adiabatic cooling.

• As long as the air rises, it will continue to be saturated with the excess water vapor converting to liquid in the form of cloud droplets which could grow large enough to fall down as precipitation.

Condensational Heating in Rising Air • From the first law of thermodynamics, expressed as the change of temperature with height

we have:

−(cpΔT/Δz + ΔQ/Δz) = g,

• where the release of latent heat is represented by ΔQ.

• The heating due to condensation is equal to the mass of water released into a unit mass of air, that is the change in saturation specific humidity Δq*, times the latent heat of evaporation/condensation (L = 2.26 x 106 J kg-1), namely:

ΔQ = L Δq*

• Since we are interested in the lapse rate, we want to know how ΔQ change with height, or ΔQ/Δz.

• From the above, this is related to how q* changes with height, or to L(Δq*/Δz).

• We can break down the last expression into the change of q* due to the change in temperature times the change in temperature due to the change in height:

ΔQ/Δz = L(Δq*/Δz) = L(Δq*/ΔT)(ΔT/Δz)

z

TT+ΔT

ΔZ

q*+Δq*q*

T

Calculating the Pseudo-Adiabatic Lapse Rate

• Combining the last equation with the first law equation above, we get:

−[cp+ L(Δq*/ΔT)](ΔT/Δz) = g

• Thus the pseudo-adiabatic lapse rate is given by:

Γs = − ΔT/Δz = g [cp + L(Δq*/ΔT)]-1

• We can express the moist, pseudo-adiabatic lapse rate in terms of the dry adiabatic lapse rate by substituting Γd for g/cp in the last equation:

Γs = Γd [1 + (L/cp)(Δq*/ΔT)]-1

• The value of (L/cp)(Δq*/ΔT) is always larger or equal zero therefore the pseudo adiabatic

lapse rate Γs is always smaller or equal than the dry adiabatic counterpart Γd and is not

constant.

• As the atmosphere gets colder it also is able to carry less water and Γs approaches the

value of Γd.

• An approximate value for Γd is -6.5 °C km-1

Formation of Clouds and Lifting Condensation Level

Effects of Mountains on Local Climate

Moist convection can explain the local climate effect of mountains, namely the tendency for large mountain ranges to have excess precipitation on the upwind side and a desert or rain shadow on the downwind side. The former is due to the lifting of the incoming air by the mountain. The latter is due to the warming of the rising air due to latent heat release.

Thermodynamic (Stuve) Diagram

The Stuve diagram on the left has temperature on the abscissa (x) and

pressure on the ordinate (y). The diagonal (yellow) lines are dry

adibats, slanting at a rate of -9.8K per km. The solid green lines are moist-adiabates, slanting at the

pseudo-adiabatic rate, which is a fundtion of the local temperature and pressure. The dashed green

lines have equal saturation specific humidity.

The plot to the right shows data collected in a meteorological station

by launching a weather balloon carrying an instrument package (thermometer, barometer and

hygrometer). The red line is the measured temperature profile, the

black line is the measured dew-point temperature profile.

Plot of a station soundingIn this sounding the first ~950 meters are saturated and display a pseudo-adiabatic lapse rate.

Above the saturated layer lies an inversion: a layer in which the temperature of the environment increases with height, making it very stable to convection.

The sounding suggests that fog and low visibility can develop near the surface.

Above the inversion the air is dry and above a level of about 5 km displays a dry adiabatic lapse rate. This is indicative of sinking motion (subsidence) in which air is dynamically pressed downward warming at a rate of 9.8 K per km.

Smoke Height (km): base=0.1, top=0.15, smoke composition, no precip, process: stable ascent, spreading out under a stable layer.

Rain, Thunderstorms, Severe Weather

In this sounding the entire troposphere is saturated,

displaying a pseudo-adiabatic lapse rate. This is indicative of the presence of deep clouds in

the sky above the meteorological station.

With such deep clouds it is plausible that the station experience heavy rainfall

(showers) and thunderstorms.

Cumulonimbus Height (km): base=0.5, top=5, cloud droplets composition, heavy precip, possibly hail and

thunderstorms, process: unstable moist ascent.