Lecture 3:

The Atmosphere & Its Circulation Part 1

A view of Earth from space over the North Pole. The Arctic ice cap can be seen in the center. The white swirls are clouds associated with atmospheric weather patterns. Courtesy of NASA/JPL.

A mosaic of satellite images showing the water vapor distribution over the globe at a height of 6–10 km above the surface. We see the organization of H2O by the circulation; dry (sinking) areas in the subtropics ( ±30◦) are dark, moist (upwelling) regions of the equatorial band are bright. Jet streams of the middle latitudes appear as elongated dark regions with adjacent clouds and bright regions. From NASA.

Because of the overwhelming importance of gravity, pressure increases downward in the atmosphere and ocean. For gravitational stability, density must also increase downward—as sketched in the diagram—with heavy fluid below and light fluid above. If ρ = ρ(p) only, then the density is independent of T, and the fluid cannot be brought into motion by heating and/or cooling.

In contrast, if ρ depends on both pressure and temperature, ρ = ρ(p, T), then fluid heated by the Sun, for example, can become buoyant and rise in convection. Such a fluid can be energized thermally.

In the non-rotating tank the dye disperses much as we might intuitively expect - have a look at the pictures and movie loop below. Note that the we can also see a side view on the left-hand side of the images below obtained using a mirror sloped at 45 degrees.

Dye-dispersion in non-rotating fluidIn the rotating body of water, by contrast, something glorious happens - have a look at the images and the movie loop below. We see beautiful vertical streaks of dye falling vertically; the vertical streaks become drawn out by horizontal fluid motion in to vertical `curtains' which wrap around one-another. The vertical columns - called `Taylor Columns' after G.I. Taylor who discovered them - are a result of the rigidity imparted to the fluid by the rotation of the tank. The water moves around in columns which are aligned parallel to the rotation vector. Since the rotation vector is directed upward, the columns are vertical. Thus we see that rotating fluids are not really like fluids at all! Note that the movie is recorded in the frame of reference of the tank - i.e. by a camera mounted above the rotating table, rotating at exactly the same speed.

Dye-dispersion in rotating fluid

More Taylor columns

The vertical columns, which are known as Taylor columns after G. I. Taylor who discovered them, are a result of the rigidity imparted to the fluid by the rotation of the tank. The water moves around in vertical columns which are aligned parallel to the rotation vector. It is in this sense that rotating fluids are rigid. As the horizontal spatial scales and timescales lengthen, rotation becomes an increasingly strong constraint on the motion of both the atmosphere and ocean.

On what scales might the atmosphere, ocean, or our laboratory experiment, ‘‘feel’’ the effect of rotation? Suppose that typical horizontal currents (atmospheric or oceanic, measured, as they are, in the rotating frame) are given by U, and the typical distance over which the current varies is L. Then the timescale of the motion is L/U. Let’s compare this with τrot , the period of rotation, by defining a nondimensional number (known as the Rossby number:

Ro = U × τrot/L

If Ro is much greater than one, then the timescale of the motion is short relative to arotation period, and rotation will not significantly influence the motion. If Ro is much lessthan one, then the motion will be aware of rotation.

In our laboratory tank we observe horizontal swirling of perhaps U 1 cms∼ −1 over thescale of the tank, L 30 cm, which is rotating with a period ∼ τtank = 3 s. This yields a Rossbynumber for the tank flow of Rotank = 0.1. Thus rotation will be an important constraint on thefluid motion, as we have witnessed by the presence of Taylor columns in the movies.

Ro for large-scale flow in the atmosphere and ocean.

• ATMOSPHERE (e.g., for a weather system):L 5000 km, ∼ U 10 ms∼ −1, and τrot = 1 day ≈ 105 s, giving Roatmos = 0.2, which suggeststhat rotation will be important.

• OCEAN ( e.g., for the great gyres of the Atlantic or Pacific Oceans): L 1000 km, ∼ U 0.1 ms∼ −1, giving Roocean = 0.01, and rotation will be a controlling factor.

It is clear then that rotation will be of paramount importance in shaping the patternof air and ocean currents on sufficiently large scales. Indeed, much of the structure andorganization seen in Slide#2 (clouds on earth) is shaped by rotation.

TABLE 1.1. Some parameters of Earth.

Earth’s rotation rate Ω 7.27 × 10−5s−1

Surface gravity g 9.81 ms−2

Earth’s mean radius a 6.37 × 106 mSurface area of Earth 4πa2 5.09 × 1014 m2

Area of Earth’s disc πa2 1.27 × 1014 m2

FIGURE 1.1. The thinness (to scale) of a shell of 10km thickness on the Earth of radius 6370 km

A north-south section of topography relative to sea level (in meters) along the Greenwich meridian (0◦ longitude) cutting through Figure below. Antarctica is over 2km high, whereas the Arctic Ocean and the south Atlantic basin are about 5km deep. Note how smooth the relief of the land is compared to that of the ocean floor.

World relief showing elevations over land and ocean depth. White areas over the continents markthe presence of ice at altitudes that exceed 2 km. The mean depth of the ocean is 3.7 km, but depths sometimesexceed 6 km. The thin white line meandering around the ocean basins marks a depth of 4 km. Red line section is shown in figure above.

TABLE 1.2. Some of the most abundant atmospheric constituents.

Chemical Molecular Proportion species weight (g mol−1) by volume

N2 28.01 78%O2 32.00 21%Ar 39.95 0.93%H2O (vapor) 18.02 0.5%∼CO2 44.01 380 ppm

Atmospheric CO2 concentrations observed at Mauna Loa, Hawaii (19.5◦ N, 155.6◦ W). Note the seasonal cycle superimposed on the long-term trend. The trend is due to anthropogenic emissions. The seasonal cycle is thought to be driven by the terrestrial biosphere: net consumption of CO2 by biomass in the summertime (due to abundance of light and heat) and net respiration in wintertime.

We focus on the lowest 50km of the atmosphere, wherein the mean free path of atmospheric molecules is so short and molecular collisions so frequent that the atmosphere can be regarded as a continuum fluid in local thermodynamic equilibrium (LTE).

Air over water in a box at temperature T. At equilibrium the rate of evaporation equals the rate of condensation. The air is saturated with water vapor, and the pressure exerted by the vapor is es, the saturated vapor pressure. On the right we show the mixture comprising dry ‘d’ and vapor ‘v’ components.

e = ρvRvTpd= ρdRdT

p = pd + e

(a) es (in mbar) as a function of T in ◦C (solid curve); (b) T profile for the ‘‘US standard atmosphere’’ at 40◦ N in December.

(a) (b)

• Vapor is in the lowest few km’s

• Tropics is more moist than poles

• Precipitation when moist air cools by convection, causing H2O concentrations back to saturation at the lower T

• Warm climate is more moist

Warm water is poured into a carboy to a depth of 10 cm or so, as shown on the left. We leave it for a few minutes and throw in a lighted match to provide condensation nuclei: small particles on which the vapor can condense. We rapidly reduce the pressure in the bottle by sucking at the top. The adiabatic expansion of the air reduces its temperature and hence the saturated vapor pressure, causing the vapor to condense and form water droplets, as shown on the right.

Condensation nuclei in atmosphere: sulfate aerosols, dust, smoke from fires, ocean salt…

Dawn mist rising from Basin Brook Reservoir, White Mountain National Forest, July 25, 2004.

A photograph of the sound barrier being broken by a US Navy jet as it crosses the Pacific Ocean at the speed of sound just 23 m above the ocean. Condensation of water is caused by the rapid expansion and subsequent cooling of air parcels induced by the shock (expansion/compression) waves caused by the plane outrunning the sound waves in front of it.

If the atmosphere were at rest, or static, then pressure at any level would depend on the weight of the fluid above that level. This balance is called hydrostatic balance.

FT = − (p + δp)δA

FB = pδA

Fg = −gM

Mass of the cylinder is M = ρδAδzGravitational force Fg = −gM = −gρδAδzPressure force at top face FT = −(p + δp)δAPressure force at bottom face FB = + pδA

Net force Fg + FT + FB = 0 δp + gρδz = 0:∂p/∂z + gρ = 0 Hydrostatic Equation

An Application of the Hydrostatic Equation:∂p/∂z = −gp/RT (using hydrostatic equation ∂p/∂z = -gρ and the Gas Law: p = ρRT)

Assume temperature is constant = T0, then we have an “isothermal” atmosphere∂p/∂z = −gp/RT0 = −p/Hs; Hs = RT0/g = the Scale HeightSo, p(z) = ps exp(−z/Hs).or, z = Hs ln(ps/p).If we choose a representative value T0 = 250 K, then Hs = 7.31 km. Therefore, for example, in such an atmosphere p is 100 hPa, or one tenth of surface pressure, at a height of z = Hs × (ln 10) = 16.83 km.

The isothermal assumption is actually not too bad, if we choose an appropriate Hs, as shown in the figure at right, which shows observed profile of pressure (solid) plotted against a theoretical profile (dashed) based on the above equation for p(z) with Hs = 6.8 km.

For isothermal atmosphere (from previous slide),p(z) = ps exp(−z/Hs).

The corresponding density is then, from the Gas Law:ρ(z) = [ps/(RTo)].exp(−z/Hs).

Mass of the atmosphere (per unit m2) can be estimated: = (ps/g).[1 - exp(−z/Hs)].

Set z = ∞: Total mass/m2 = (ps/g).

Because of the exponential decay with height, most of the mass of the atmosphere is very near the surface. Thus 80% of it is below z80% where:

1 - exp(−z80%/Hs) = 0.8Therefore, z80% = -ln(0.2)Hs 1.6 × 7.3km 12 km;

i.e. about 80% of the mass of the atmosphere is below an altitude of about 12 km, i.e. within the troposphere.

Review for Quiz (Mar/20/09:00-10:00AM)

1. What is the age of the earth and how did Kelvin err in estimating it, and why was that so important to the scientific debate of his time?2. What is the approximate rate of warming of the earth’s surface in oC/decade since ~1900?3. Since ~1900, the warming rate may be categorized into 4 phases or periods – describe each period.4. Since ~2000, surface temperatures appear to have stopped rising – warming hiatus – has human-induced warming stopped?5. Describe (a) the “normal conditions” of SST over the equatorial Pacific; (b) El Nino conditions and (c) La Nina conditions.6. Describe as fully as you can how El Nino might be triggered?7. How do scientists measure the strengths of EL Nino or La Nina?8. What is OLR and explain how it may be used to detect the height of clouds.9. Without greenhouse effects, the earth surface would be colder, what is the temperature, explain using Stefan’s black-body radiation law?10. Construct a simple model which describes the greenhouse effects, and estimate the resulting surface temperature.11. What is albedo and explain why it is one of the reasons that Arctic ice-melt accelerates in the past 10~20 years?12. Describe the possible consequences of Arctic ice-melt to the jet stream and northern hemisphere climate.13. Why can Arctic ice-melt lead to a slowing-down of the jet stream?14. Derive or write down the formula for the Rossby number Ro .15. Explain how the magnitude of Ro can describe if a given atmospheric or oceanic flow can “feel” the earth’s rotation.16. Explain how the behaviors of atmospheric or oceanic flow can be very different on equator and over the poles?17. Do the different behaviors (in “16”) explain why typhoons or hurricanes do not develop near the equator? Explain as best as you can.18. Derive or write down the formula for the hydrostatic equation?19. Explain why for large-scale atmospheric or oceanic flow the hydrostatic equation is valid?20. Then also explain why over the front-gate hill of National Central University, the hydrostatic equation is not likely to apply to describe the

wind that blows over the hill?21. Find out what the densities of air and water are.22. On earth’s surface, we experience the weight of air some 10 km high. Estimate how deep into the ocean do we need to dive in order to

experience the same pressure.