Atmosphere-Ocean Dynamics

By Adrian E. Gill

A systematic, unifying approach to the dynamics of the ocean and atmosphere is given in this book, with emphasis on the larger-scale motions (from a few kilometers to global scale). The foundations of the subject (the equations of state and dynamical equations) are covered in some detail, so that students with training in mathematics should find it a self-contained text. Knowledge of fluid mechanics is helpful but not essential. Simple mathematical models are used to demonstrate the fundamental dynamical principles with plentiful illustrations from field and laboratory.

• Language: English
• Publisher: Elsevier Science
• Release date: Dec 13, 1982
• ISBN: 9780080570525

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GILL

Physics of the Aurora and Airglow, Vol. 30, Suppl. (C), 1982

ISSN: 0074-6142

doi: 10.1016/S0074-6142(08)60026-1

How the Ocean–Atmosphere System Is Driven

1.1 Introduction

This book is about winds, currents, and the distribution of heat in the atmosphere and ocean. Since these are due to the sun, this first chapter looks at some of the essential processes that determine how the atmosphere and ocean respond to radiation from the sun. Ideally, one would like to be able to deduce this response in all its details from a knowledge of the appropriate properties of the earth and of its ocean and atmosphere, but this is not a simple matter. The nearest approach to a solution of this problem is by means of numerical models, but these still rely to some extent on observations of the real system, e.g., for determining the effects of processes (like those associated with individual clouds) that have a scale small compared with the grid used in the model.

The aim of the numerical models is to include the effects of all processes that play a significant part in determining the response of the ocean–atmosphere system. The aim of this chapter, on the other hand, is to consider only the most basic processes and to show how an equilibrium state can be reached. One such basic process is the absorption of radiation by certain gases (principally water vapor, carbon dioxide, and ozone), and so the greenhouse effect is discussed. The density field that results from radiation processes acting in isolation is not in dynamical equilibrium, because air near the ground is so warm that it is lighter than the air above. Consequently, vertical convection takes place and stirs up the lower atmosphere. Calculations of the equilibrium established when convective and radiative processes are both active is discussed in Section 1.5. These calculations, however, neglect variations in the horizontal, which are, of course, extremely important since they are responsible for the winds and currents that are the main subject of this book. A brief discussion of the effects of horizontal variations is given in Section 1.6. Finally, since radiation is the source of energy for the atmosphere–ocean system, variations in the radiative input are discussed in Section 1.7.

1.2 The Amount of Energy Received by the Earth

Energy from the sun is received in the form of radiation, nearly all the energy being at wavelengths between 0.2 and 4 µm. About 40% is in the visible part of the spectrum (0.4–0.67 µm). The average energy flux from the sun at the mean radius of the earth is called the solar constant S and has the value (Willson R. C. 1984)

     (1.2.1)

(A great variety of units is used for energy flux. The relation between these is given in Appendix 1.) In other words, a 1-m-diameter dish in space could collect enough energy from the sun to run a 1-kW electric heater! Since the earth’s orbit is elliptical rather than circular, the actual energy received varies seasonally by ±3.5% (Kondratyev, 1969, Section 1.1), the maximum amount being received at the beginning of January.

The total energy received from the sun per unit time is

     (1.2.2)

where R is the radius of the earth. Since the area of the earth’s surface is 4πR², the average amount of energy received per unit area of the earth’s surface per unit time is

     (1.2.3)

If the earth’s axis were not tilted, the average flux received would vary from π−1S at the equator to zero at the poles. However, the tilt of the earth (23.5°) results in seasonal variations in the distribution of the flux received. When account is taken of these variations, the average flux received in 1 yr is found to vary with latitude as shown in Fig. 1.1.

Fig. 1.1 The radiation balance of the earth. The upper solid curve shows the average flux of solar energy reaching the outer atmosphere. The lower solid curve shows the average amount of solar energy absorbed; the dashed line shows the average amount of outgoing radiation. The lower curves are average values from satellite measurements between June 1974 and February 1978, and are taken from Volume 2 of Winston et al. (1979). Values are in watts per square meter. The horizontal scale is such that the spacing between latitudes is proportional to the area of the earth’s surface between them, i.e., is linear in the sine of the latitude

is reflected or scattered, so the average flux actually absorbed is

     (1.2.4)

The amount reflected or scattered is about 100 W m−2 at all latitudes, as shown in is called the albedo of the earth and has a value (Stephens et al., 1981) of about

     (1.2.5)

Similarly, the albedo α can be defined for a particular place and particular time as the fraction of the impinging radiation that is reflected or scattered. The reflected light is the light by which the earth may be photographed from space, and such photographs (see Fig. 1.2, which is effectively the result of combining many such photographs to give the mean reflectivity) show that the albedo can vary enormously with such factors as the amount of cloud, and whether the ground is covered by ice or snow. Mars, with no cloud cover, has about half the albedo of the earth, whereas Venus, with total cloud cover, has about twice the albedo of the earth. A quantitative estimate of the degree to which clouds, ice, and snow affect the albedo can be obtained from satellite measurements (Fig. 1.3). The minimum albedo is presumably close to the value in the absence of clouds and of snow-free conditions where these occur. On land, the value is usually about 0.15, with higher values in desert regions (0.2–0.3) and in icy regions, reaching 0.6 in parts of the Antarctic. Comparison of the minimum albedo with the average albedo shows the effect of clouds. For instance, most of the ocean within 40° of the equator has minimum albedo below 0.1, but the average albedo is normally between 0.15 and 0.3. It is clear from these figures that the factors that determine albedo are very important in determining the energy balance of the earth.

Fig. 1.2 The geographical distribution of reflectivity for (a) January 1967–1970 and (b) July 1969–1970, as determined from satellite observations. Most of the bright areas in the figure are characterized by persistent cloudiness and relatively heavy precipitation. However, the following exceptions should be noted: areas indicated by X’s denote desert regions where the earth’s surface is highly reflective and areas indicated by Y’s denote regions of persistent low, nonprecipitating cloud decks. Tick marks along the side denote the position of the equator: the Mercator grid lines are spaced at intervals of 5° of latitude and longitude.

[From U.S. Air Force and U.S. Department of Commerce, Global Atlas of Relative Cloud Cover, 1967–1970, Washington, D.C., 1971.]

Fig. 1.3 (a) The average albedo obtained from a composite of 48 months of satellite data obtained between 1964 and 1977.

[From Stephens et al. (1981, Fig. 6).] (b) The minimum albedo of the earth from Nimbus 3 satellite measurements in 1969–1970. [From Raschke et al. (1973, Fig. 23).]

1.3 Radiative Equilibrium Models

Since the ocean–atmosphere system is driven by the sun’s radiation, it is important to know how radiation is affected by the atmosphere and ocean. Detailed discussion may be found in books such as those of Goody (1964), Kondratyev (1969), and Paltridge and Platt (1976). Only the most basic elements will be discussed here.

of the incoming radiation and absorb the remainder. The absorption of energy would cause the surface to warm up until it radiated to space as much energy as it absorbed. When the surface reaches temperature T, the amount of energy E radiated per unit time is given by Stefan’s law

     (1.3.1)

where

     (1.3.2)

For the radiation actually absorbed by the earth (see Fig. 1.1), such an equilibrium would be achieved when the temperature at the equator reached 270 K, the temperature at the South Pole 150 K, and the temperature at the North Pole 170 K. In fact the earth’s surface is much warmer, and the contrast in temperature between the equator and the poles is much less. The difference from the observed surface temperature must be due to the existence of the fluid cover of the earth. This can affect the equilibrium reached in two ways. First, radiation can be absorbed within the atmosphere itself. Second, the atmosphere and ocean can carry heat from one area to another, thereby affecting the balance. In this section, the first effect will be considered in isolation from the second. In subsequent sections, the effect of fluid motion on the equilibrium will be discussed. This fluid motion consists of winds, ocean currents, etc., which will be the main concern of this book.

The radiative equilibrium that would be established in the absence of fluid motion has been calculated by Möller and Manabe (1961), and is discussed by Goody (1964, Chapter 8). The average temperature profile thus obtained is shown by the solid line in Fig. 1.4. In some ways the left-hand version of the figure is more appropriate because it gives equal weight to equal masses of air. In the lower 70% (by mass) of the atmosphere, the main physical factor responsible for the equilibrium reached is the absorption of radiation by the water vapor present in the atmosphere. For their calculations, Möller and Manabe used the observed distribution of water vapor with height. At higher levels, other absorbers such as carbon dioxide and ozone become important. Figure 1.4 shows that the presence of the atmosphere results in much higher ground temperatures than would otherwise be achieved. This is due to the greenhouse effect, which will be discussed in Section 1.4.

Fig. 1.4 The radiative equilibrium solution (solid line) corresponding to the observed distribution of atmospheric absorbers at 35°N in April, the observed annual average insolation for the whole atmosphere, and no clouds. The dashed line shows the effect of convective adjustment to a constant lapse rate of 6.5 K km−1. In (a) the curves are drawn with a scale linear in pressure, i.e., equal intervals correspond to equal masses of atmosphere. In (b) the scale is linear in altitude.

[From Manabe and Strickler (1964, Fig. 4).]

1.4 The Greenhouse Effect

The radiative equilibrium solution shown in Fig. 1.4 has much higher ground temperatures than would exist in the absence of the atmosphere. This is caused by the greenhouse effect, the principle of which can be explained as follows. Consider a greenhouse formed by placing a horizontal sheet of glass above the ground as shown in Fig. 1.5. The glass used is transparent to radiation with wavelengths below 4 µm, but partially absorbs radiation of longer wavelengths. Suppose the glass and ground are initially cold, and then a downward flux I of solar radiation is switched on. This radiation will pass through the glass unattenuated and be absorbed by the ground.

Fig. 1.5 The greenhouse effect. The glass is transparent to short-wave radiation, the net downward flux of which is l. The balancing upward flux of long-wave radiation from the ground is U, a fraction e of this being absorbed by the glass. This warms the glass, causing it to emit a flux B in both directions.

The ground will warm up to a temperature Tg and emit long-wave radiation with an upward flux U given by Stefan’s law:

     (1.4.1)

Practically all the radiation emitted at temperatures typical of the atmosphere has wavelengths above 4 µm (the range is 4–100 µm), so a fraction e of this radiation will be absorbed by the glass. Thus the glass will also warm up and emit radiation. Suppose the flux emitted in each direction is B.

Equilibrium will be reached when the upward fluxes balance the downward fluxes, i.e., when

     (1.4.2)

Solving (1.4.1) and (1.4.2), the result for the ground temperature is

     (1.4.3)

Thus Tg is higher (by up to 19%) than it would be in the absence (e = 0) of the glass. This is the principle on which a greenhouse operates.

The effect can be most easily understood in the extreme case of glass that absorbs all the long-wave radiation (e = 1). Then (Fig. 1.5) I = B, which implies that the glass reaches the same temperature that the ground would have in the absence of glass. Since the underside of the glass is at the same temperature, it radiates a downward flux B of long-wave radiation downward, so the ground receives a total flux of I + B = 2I. Thus by Stefan’s law the ground reaches a temperature that is higher than in the absence of glass by a factor 2¹/⁴ = 1.19. For other nonzero values of e, the ground still receives a back radiation flux B in addition to the short-wave flux I, so it reaches a higher temperature than it would otherwise.

In the atmosphere, the absorbing material is distributed continuously in the vertical rather than being confined to a thin sheet. Generalization of the above ideas to this case is straightforward however (Goody, 1964, Section 8.4; Chamberlain, 1978, Section 1.2), and gives temperature profiles for the lower atmosphere that are similar to those of Möller and Manabe. More accurate calculations require the radiative energy to be divided up into many wavebands rather than just two (i.e., long and short waves), and to take account of the absorption in each band separately. Also, reflection and scattering must be allowed for. This depends on the distribution and albedo of clouds and on the albedo of the underlying surface.

An estimate of the radiation balance for the atmosphere is summarized in units. This leaves 70 units of net downward flux of short-wave radiation at the top of the atmosphere, of which 19 units are absorbed in the atmosphere, leaving only 51 units to be absorbed at the surface. There is also a large amount [London and Sasamori (1971) estimate 98 units] of long-wave radiation absorbed at the surface, this representing back radiation from the atmosphere (it is possible for the back radiation to exceed the incident radiation, as a generalization of Fig. 1.5 to several sheets of glass can readily show). The net surface emission (excess of upward over downward radiation) of long-wave radiation is 21 units, the remaining upward flux of 30 units being by convection. The upward flux at the top of the atmosphere is 70 units, as required to balance the short-wave radiation received. The mean surface temperature is that corresponding to the 98 + 51 = 149 units of radiated energy flux at the ground rather than that corresponding to the 70 units emitted at the top of the atmosphere. The latter flux can be more closely identified with a temperature at cloud-top height.

Fig. 1.6 Radiation balance for the atmosphere.

[Adapted from Understanding Climatic Change, U.S. National Academy of Sciences, Washington, D C., 1975, p. 14, and used with permission.]

1.5 Effects of Convection

The radiative equilibrium solution was described in Section 1.3 as the solution that would be obtained in the absence of fluid motion. This statement is not strictly true, however, because the radiative equilibrium solution is based on the observed distribution of water vapor. This distribution is not predetermined, but is the result of a balance that involves fluid motion.

To see how fluid motion can affect the balance, consider an atmosphere that, at some initial time, contained no water vapor, but was in radiative equilibrium. If the atmosphere absorbed no radiation at all, the ground would warm up as in the absence of an atmosphere (see Section 1.3), but the air above would remain cold. Although the system would be in radiative equilibrium, it would not be in dynamic equilibrium because the air warmed by contact with the surface could not remain below the cold air above without convection occurring, as it does in a kettle full of water that is heated from below. The vigorous motion produced carries not only heat up into the atmosphere, but also water vapor produced by evaporation at the surface. The water vapor then affects the radiative balance because of its radiation-absorbing properties, so the final equilibrium depends on a balance between radiative and convective effects and is called radiative–convective equilibrium.

Whether or not convection will occur depends on the lapse rate, i.e., the rate at which the temperature of the atmosphere decreases with height. Convection will only occur when the lapse rate exceeds a certain value. This value can be calculated by considering the temperature changes of a parcel of air that moves up or down adiabatically, i.e., without exchanging heat with the air outside the parcel. As such a parcel rises, the pressure falls, the parcel expands, and thus its temperature falls. The rate at which the temperature falls with height, due to expansion, is called the dry adiabatic lapse rate and has a value of about 10 K/km. If the temperature of the surroundings fell off more quickly with height, a rising parcel would find itself warmer than its surroundings, and therefore would continue to rise under its own buoyancy. In other words, the situation would not be a stable one, and so convection would occur. Convection carries heat upward and thus will reduce the lapse rate until it falls to the equilibrium value, for then convection can no longer occur. Another way of expressing the same ideas is in terms of potential energy. When the lapse rate exceeds the adiabatic value, the potential energy can be reduced by moving parcels adiabatically to different levels. Thus energy is released and is used to drive the convection.

If the atmosphere contained only small amounts of water vapor, convection would only occur if the dry adiabatic lapse rate were exceeded. In practice, the situation is complicated by the fact that air at a given temperature and pressure can only hold a certain amount of water vapor. The amount of water vapor relative to this saturation value is called the relative humidity. When the relative humidity reaches 100%, water droplets condense out of the air, thereby forming clouds. The condensed water ultimately returns to the earth’s surface as precipitation.

This hydrological cycle affects the energy balance of the atmosphere in a number of important ways. First, clouds have an important effect on the total amount of energy absorbed by the atmosphere because they reflect and scatter a significant amount of the incoming radiation (see Section 1.2). Second, the radiation-absorbing properties of water vapor are important in determining the temperature of the lower atmosphere, as discussed in Section 1.3. Third, cooling takes place upon evaporation because of the latent heat required. This heat is released back into the atmosphere when condensation takes place in clouds. The heat transferred by this means is, on average, about 75% of the convective transport (see Fig. 1.6).

The release of latent heat in clouds also affects the conditions under which convection can take place. The amount of water vapor a parcel of air rising adiabatically can hold decreases with height. Thus if the parcel is already saturated with water vapor, latent heat will be released as the parcel rises, so the rate of decrease of temperature with height will be less than for dry air. The rate of decrease with height is called the moist adiabatic lapse rate and has a value that depends on the temperature and pressure. In the lower atmosphere, the value is about 4 deg km−1 at 20°C and 5 deg km−1 at 10°C [for precise values, see List (1951, Table 79)]. The appropriate lapse rate may also be different if ice is formed instead of liquid water (List, 1951, Table 80). A fuller discussion is given in Section 3.8.

The moist adiabatic lapse rate is appropriate for ascending air, but for descending air the story is different. The amount of water vapor a parcel of air can hold increases as the parcel descends, so the parcel is always unsaturated and the dry adiabatic lapse rate is appropriate. Thus in a convecting atmosphere, potential energy may be released where the air is ascending, whereas work is being done against gravity where the air is descending. [For a discussion of convection and models of convection, see Haltiner (1971, Chapter 10) and Holton (1979, Chapter 12).]

Another consequence of the nature of moist convection is the distribution of relative humidity in the atmosphere. The mean value must lie between the 100% of the moist air in rising regions and the lower values of the descending regions. A rough approximation to the observed mean distribution (Manabe and Wetherald, 1967) is a relative humidity that decreases linearly with pressure from 77% at the ground to zero at the top of the atmosphere. The relative humidity does not change very much from one season to another, whereas the actual amount of water vapor present varies a great deal.

A problem in modeling the atmosphere is to find a satisfactory way to represent the effects of convection without modeling details of the ascending and descending parcels of air. Radiative–convective models represent the effects of convection in a very simple way. First, they ignore horizontal variations, so that the temperature and other quantities are functions only of altitude (or, equivalently, of pressure). Distributions of the radiation-absorbing gases, carbon dioxide and ozone, of clouds, and of either relative humidity or absolute humidity are fixed, as is the downward flux of short-wave radiation at the top of the atmosphere. An initial temperature distribution is allowed to adjust toward equilibrium, taking account not only of radiative fluxes but also of convective fluxes. Convection is assumed to occur only when the radiative fluxes are tending to increase the lapse rate above a certain critical value. Then an opposing convective flux is introduced that redistributes (but does not add or remove) heat in such a way as to keep the lapse rate at the critical value. The difficulty lies in the choice of the critical value. Usually this is simply chosen to be the observed mean lapse rate of the lower atmosphere, namely, 6.5 deg km−1. The result of such a calculation (Manabe and Strickler, 1964) is shown in Fig. 1.4 and gives quite a good approximation to the observed mean temperature profile. As such, it is an improvement over the pure radiative equilibrium-model, but its limitations should not be forgotten.

1.6 Effects of Horizontal Gradients

In Section 1.5 it was seen that the large vertical temperature gradients that would be produced by radiation acting in isolation result in convection that tends to reduce these gradients. In a similar way, the variations with latitude of the absorbed radiative flux (Fig. 1.1) would lead to large horizontal temperature gradients if radiation acted in isolation. Again fluid motion takes place that tends to reduce these gradients. The nature of these motions depends on dynamical processes, which will be the subject of subsequent chapters.

Intuitively, one might expect the nonuniform heating of the atmosphere to cause rising motion in the tropics and descending motion at higher latitudes. Halley (1686) and Hadley (1735) proposed this type of circulation, which is now known as a Hadley cell (see Section 2.3). A similar circulation might be expected to occur in the ocean, so that the excess heat received in the tropics would be transported poleward in both atmosphere and ocean.

The circulation (in the meridional plane) that actually occurs is known quantitatively (but with limited accuracy) for the atmosphere from observations and is shown in Fig. 1.7. By comparison, the meridional circulation in the ocean is very poorly known, but estimates have been made that at least give an order of magnitude. A brief description of the atmospheric part of the circulation is as follows. The Hadley cell is confined to the tropics. Moist air from the trade wind zone, where evaporation exceeds precipitation, is drawn into the areas of rising motion, which, because they are wet and cloudy, show up as regions of high reflectivity in Fig. 1.2. Important regions of rising motion are over Indonesia and the Amazon and Congo basins. Over the Atlantic and Pacific Oceans, the rising motions tends to be concentrated in a fairly narrow band called the Inter-Tropical Convergence Zone (ITCZ), usually found between 5 and 10° to the north of the equator. It can be seen very clearly as a band of high reflectivity in Fig. 1.2. The regions of descending air are dry, and include in particular the desert regions (marked by X’s in Fig. 1.2), which are found between latitudes 20° and 30°. These show up as regions of high albedo over land in Fig. 1.3. Where the descent is over cold ocean, low nonprecipitating cloud decks (marked Y in Fig. 1.2) are often found.

Fig. 1.7 Streamlines of the mean meridional mass flux in the atmosphere for (a) December-February and (b) June–August. Units are megatons per second (Mt s−1 = 10⁹ kg s−1). The horizontal scale is such that the spacing between latitudes is proportional to the area of the earth’s surface between them, i.e., is linear in the sine of the latitude.

[Adapted from Newell et al. (1972, Vol. 1, p. 45).]

In mid-latitudes, the picture is quite different. Because of the rotation of the earth, the motion produced by the horizontal density gradients is mainly east–west, and there is relatively little meridional circulation. (The observed velocity and temperature distribution is shown in Fig. 7.9.) However, the situation is not a stable one, and large transient disturbances (which appear as cyclones and anticyclones on the weather map) develop. These disturbances are very effective at transporting energy poleward.

The effectiveness of fluid motion in reducing horizontal gradients can be judged from a comparison of the two lower curves in Fig. 1.1. The solid curve shows the variation with latitude of the absorbed flux of radiative energy. In a pure radiative equilibrium (or a radiative–convective equilibrium), the outgoing radiation would be equal to the absorbed radiation at all latitudes. In practice, the outgoing flux of radiative energy, shown by the dashed line in Fig. 1.1, is much more uniform, its departures from the average flux being about one third of those for the absorbed flux. From the difference between the two curves, the amount of energy that must be transported across each circle of latitude by fluid motion can be calculated. The curve so obtained for the northern hemisphere is shown in Fig. 1.8. This curve can be compared with the one for the observed transport of energy by the atmosphere (Oort, 1971; Vonder Haar and Oort, 1973). The difference between the two curves (the shaded region in Fig. 1.8) provides an estimate of the energy transport by the ocean. According to these results, ocean and atmosphere are equally important in transporting energy, the atmosphere being most important at 50°N and the ocean most important at 20°N. There is, however, considerable uncertainty in the measurements, and probable errors are estimated by Vonder Haar and Oort (1973). For instance, the probable error in the transport of energy by the ocean at 20°N is about 70%.

Fig. 1.8 The northward transport of energy (in units of petawatt = 10¹⁵ W) as a function of latitude. The outer curve is the net transport deduced from radiation measurements. The white area is the part transported by the atmosphere and the shaded area the part transported by the ocean. The lower curve denotes the part of the atmospheric transport due to transient eddies and is the mean of the monthly values from Oort (1971, Table 3). The horizontal scale is such that the spacing between latitudes is proportional to the area of the earth’s surface between them, i.e., is linear in the sine of the latitude.

[From Vonder Haar and Oort (1973).]

In calculating the energy transport by the atmosphere from observations, a distinction can be made between the energy transported by the mean (time-averaged) circulation and the energy transported by transient motions. If this calculation is done for each month in turn and the results are averaged, the curve in the unshaded part of Fig. 1.8 is obtained. In latitudes where the transport by the atmosphere is important, the transient motions account for most of the transport. This observation is the basis for simple equilibrium models [e.g., Stone (1972); see also Held and Suarez (1978)] in which the radiative heat flux is balanced not by small-scale convection, as in radiative–convective equilibrium models, but by energy fluxes due to large-scale transient motions (such as cyclones and anticyclones). These motions transport heat vertically as well as horizontally [see Palmén and Newton (1969, Chapter 2)], so calculations of both vertical and horizontal gradients can be made.

The method of estimating the transports due to the large-scale transient motions is beyond the scope of the present chapter, but the concept is important. The structure of the atmosphere and ocean depends on the motions driven by radiation and their effectiveness in redistributing heat. If the effect of the dominant energy-transporting mechanism can be estimated in some simple way, one hopes that reasonable estimates of basic features such as the mean horizontal and vertical temperature gradients of the atmosphere can be obtained.

1.7 Variability in Radiative Driving of the Earth

Since the present state of the ocean and atmosphere is a result of their response to the radiation received from the sun, one would like to know what variability there is in this driving. The total amount of radiation incident on the earth in 1 year depends only on the output of radiation from the sun, which is measured by the solar constant S, whose present value is given by (1.2.1). Measurements since the 1920s (Drummond, 1970) show no variations larger than the probable measurement errors, so S cannot have varied more than 1 or 2% in that time. Thus the hypothesis that S is constant, as suggested by the name solar constant, is consistent with observations to date, although other possibilities are not ruled out.

The amount of radiation incident at a particular point on the earth does, however, vary enormously between day and night and from season to season, and these variations are of obvious importance to life as we know it. Since the emphasis in this book is on periods larger than a day, daily variations will not be discussed explicitly. However, it is important to realize that the existence of daily variations can affect the state of the atmosphere over longer periods, the magnitude of the effect depending on the amplitude of the daily variations. An example of such an effect is the mixing of the lower atmosphere. In summer especially, the ground can become very hot during the day, causing strong convection that stirs up a considerable depth of air. The air is not unmixed at night, so the net effect is substantially different from that which would be achieved with uniform radiation.

Seasonal variations are due to (i) the tilt of the earth’s axis relative to the plane of its orbit (at present 23.5°) and (ii) the ellipticity of the earth’s orbit. The ellipticity is such that the total amount of radiation incident on the earth varies by ±3.5%, with the maximum in early January. The consequent changes with latitude and time of the incident radiation are given by List (1951, Tables 132 and 134), whereas Stephens et al. (1981) give the observed changes in outgoing radiation. These are smaller than the changes in incident radiation, so there is a net gain of energy between October and March when the earth is nearer the sun, and a net loss in the remainder of the year. The variations show a marked asymmetry between the two hemispheres because of the different proportions of land and sea, changes over the latter being relatively small.

The existence of seasonal variations has important effects on the mean state of the atmosphere and ocean, the magnitude of the effect depending on the amplitude of the variations. This fact has been demonstrated by numerical experiments of Wetherald and Manabe (1972). They began with an ocean–atmosphere model driven by the annual mean radiation and then changed to seasonal forcing. The mean state was changed thereby, e.g., surface temperatures in high latitudes were greater and the mean north-south temperature gradient in the atmosphere was reduced. [The sensitivity, e.g., to changes in CO2 content, is also affected—see Wetherald and Manabe (1981).] The most important contributing factor was found to be the melting of snow in high latitudes in summer, thus reducing the net albedo. Another factor was found to be the development of a warm surface layer in the ocean in summer, giving a higher mean sea-surface temperature.

The fact that seasonal variations affect the mean state of the ocean-atmosphere system is the basis of an astronomical theory of climate change due to Milankovich (1930, 1941). Because of perturbations caused by other planets, the tilt of the earth’s axis varies between 22 and 24.5°, and the eccentricity of the earth’s orbit changes, the time scales of these changes being 10⁴–10⁵ years. The net radiation incident over a year is altered very little, but the distribution in time and space is changed. The eccentricity varies sufficiently for the amplitude of seasonal variations in the incident radiation to change between 0 and 15% and the time of the maximum also changes. The effects of these changes on the incident radiation are given by Berger (1979), and the theory is discussed by Imbrie and Imbrie (1979) and Monin (1972, Chapter 4). Periods during which the amount of radiation received in summer over the high-latitude continental areas of the northern hemisphere was small appear to coincide with ice ages. Geological evidence in support of the theory is discussed by Hays et al. (1976) and Imbrie and Imbrie (1980).

Physics of the Aurora and Airglow, Vol. 30, Suppl. (C), 1982

ISSN: 0074-6142

doi: 10.1016/S0074-6142(08)60027-3

Transfer of Properties between Atmosphere and Ocean

2.1 Introduction

As stated in the introduction to Chapter 1, one would like to determine the response of the atmosphere–ocean system to the known radiative input from the sun given only the physical properties of air and water, the distribution of land and sea, and other such basic information. Some of that information is given in this chapter, as Section 2.2 discusses the differences between the physical properties of air and water that make their mutual boundary of such importance. The density difference is obviously significant, but contrasts in optical properties are also important since they result in the thermal driving of the ocean being effectively at the surface.

Processes that are responsible for transfer of heat and moisture across the air–sea boundary are briefly discussed in Section 2.4, along with formulas used for calculating the rates of transfer. These can be used to calculate global budgets of heat, moisture, and momentum, which are examined in three different sections. First, the angular momentum budget of the atmosphere is discussed in Section 2.3, this having some historical interest in connection with the Hadley circulation. The moisture budget (hydrological cycle) is discussed in Section 2.5, and the heat budget of the ocean is discussed in Section 2.6. Finally, the thermohaline or buoyancy-driven circulation of the ocean is considered in Section 2.7.

2.2 Contrasts in Properties of Ocean and Atmosphere

Water is very much denser than air. The density of air varies with temperature, pressure, and humidity (see Chapter 3), typical surface values being 1.2–1.3 kg m−3(0.0013 tonne m−3), whereas the sea is some 800 times more dense (1025 kg m−3 = 1.025 tonne m−3 at the surface). Thus the interface between air and water is very stable because of the strength of the gravitational restoring force when it is displaced from its equilibrium position. Typical displacements observed in surface waves are of order 1 m. Because of the stability of the interface, the two media do not mix in any significant way (whitecaps and spray are only found close to the interface), so transfers of properties between the two media must take place through a well-defined interface. This contrasts with the atmosphere, for instance, where heat transfer can take place through a plume of hot air rising hundreds of meters and then mixing with the surrounding air. Obviously, such a plume cannot cross the ocean surface, and this is one reason why the air–sea transfer processes need special consideration.

The existence of the interface affects the radiation balance because it reflects radiation. The fraction α of solar radiation reflected is a function of the angle of incidence and of the surface roughness (Kraus, 1972, Section 3.2). Typical values of α are indicated by the satellite measurements of minimum albedo shown in Fig. 1.3b. It is assumed that the minimum albedo approximates the value that would be obtained in the absence of clouds, and so is close to the surface value. At latitudes below 30°, values less than 0.1 are found. At higher latitudes, the values increase with latitude because of the progressive reduction in the angle between the sun’s rays and the surface.

Fig. 2.3 Mean sea-level pressure (mb) for January [(a) and (b)] and July [(c) and (d)]. The northern hemisphere data are from Crutcher and Meserve (1970), and the southern hemisphere data are from Taljaard et al. (1969).

There is not only a discontinuity in density at the ocean surface, but also a discontinuity in optical properties that has important consequences for the radiation balance. Consider first the solar radiation impinging on the atmosphere. According to Fig. 1.6, only 19% of this is absorbed within the atmosphere. At the surface, a fraction α is reflected. What happens to the remainder that enters the ocean? This represents about 51% (see Fig. 1.6) of the radiation entering the outer atmosphere. Unlike the atmosphere, the ocean absorbs solar radiation very rapidly. The rate of absorption varies with wavelength and with the amount of suspended material (Kraus, 1972, Section 3.2). The total energy (in the range of wavelengths appropriate to solar radiation) falls off exponentially with depth. Typical decay rates are such that about 80% (Jerlov 1968, Table 21 and Fig. 50) is absorbed in the top 10 m. In coastal areas where a lot of suspended material is present, the absorption rate can be much greater. A more detailed discussion can be found in Jerlov’s (1968) book.

Fig. 2.6 A map showing the annual rainfall over the ocean.

[Courtesy of C. Dorman.]

In the atmosphere, long-wave radiation is absorbed much more rapidly than solar radiation, the principal absorber being water vapor. It is hardly surprising, therefore, that long-wave radiation in the ocean is absorbed very rapidly indeed. The result is that the emission (and absorption) of long-wave radiation takes place from a very thin layer, less than 1-mm thick (McAlister and McLeish, 1969).

The density contrast between air and water means that the mass of the ocean is very much greater than (270 times) that of the atmosphere. The mass per unit area of the atmosphere is approximately 10⁴ kg m−2 (10 tonne m−2), and since the acceleration due to gravity is about 10 ms−2, the weight per unit area, or surface pressure, is about

A mere 10-m depth of ocean has the same weight per unit area, so the pressure increases by 1 bar every 10 m. For this reason, oceanographers often express pressures in decibars (dbar) since 1 dbar ≈ 1 m in depth (see Section 3.5).

The large difference in mass between air and water also implies a large difference in heat capacity. In fact, the specific heat (heat capacity per unit mass) of water is four times that of air, so a mere 2.5-m depth of water has the same heat capacity per unit area (10⁷ J m−2 K−1) as the whole depth of the atmosphere. In other words, the heat required to raise the temperature of the atmosphere by 1 K can be obtained by changing the temperature of 2.5 m of water by the same amount (or of 25 m by 0.1 K or of 250 m by 0.01 K). Heat can also be stored in latent form, and this same amount of heat can be used to evaporate 4 mm of water or to melt 30 mm of ice. (Values of latent heats and specific heats from which these figures are derived are given in Appendixes 3 and 4.) The importance of latent heat can be seen when it is considered that evaporation rates in the tropics are of order 4 mm per day, corresponding to changing the temperature of the atmosphere by 1 K per day. This is consistent with cooling rates by radiation, which are of order 1 K. per day (Riehl, 1979).

The large heat capacity of the ocean is of importance for seasonal changes. Although in the long term each hemisphere loses by radiation about as much heat as it receives, this is not true of an individual season. The excess heat gained in summer is not transported to the winter hemisphere, but is stored in the surface layers (100 m or so) of the ocean and returned to the atmosphere in the winter (Palmén and Newton, 1969, Chapter 2). Because of this ability to store heat, the ocean surface temperature changes by much smaller amounts than the land surface, which cannot store much heat. This contrast between land and sea shows up vividly in Fig. 2.1, which shows the seasonal range in temperature at the earth’s surface. Although the outlines of the continents are not drawn in, their position is quite clear. Thermal storage in the ocean is also important at longer time scales, and therefore is of significance for climatic variations.

Fig. 2.1 Annual range of monthly mean temperatures at the earth’s surface.

[Adapted from Monin (1975, p. 203).]

2.3 Momentum Transfer between Air and Sea, and the Atmosphere’s Angular Momentum Balance

How are the winds produced, and what determines their distribution? In offering an explanation for the trade winds found in the tropics, Halley (1686, p. 165) pointed out that the driving force is the Action of the Suns Beams upon the Air and Water. This produces a dynamic effect, namely, "that according to the Laws of Staticks, the Air which is less rarified or expanded by heat, and consequently more ponderous, must have a Motion towards those parts thereof, which are more rarified, and less ponderous, to bring it to an Æquilibrium." Thus Halley had in mind a steady situation in which there is a balance between the forcing effect of radiation, which tends to produce horizontal density differences, and the dynamic effects, which tend to reduce the differences.

But as the cool and dense Air, by reason of its greater Gravity, presses upon the hot and rarified, ’tis demonstrative that this latter must ascend in a continued stream as fast as it Rarifies, and that being ascended, it must disperse it self to preserve the Æquilibrium; that is, by a contrary Current, the upper Air must move from those parts where the greatest Heat is: So by a kind of Circulation, the North-East Trade Wind below, will be attended with a South Westerly above, and the South Easterly with a North West Wind above (Halley, 1686, p. 167).

Such a circulation in the meridional plane is now known to exist in the tropics (see Fig. 1.7) and Halley’s explanation of the circulation is essentially correct. However, this meridional circulation is now called the Hadley circulation. This appears to be because Halley’s explanation of the easterly component of the trade winds was incorrect, whereas Hadley (1735) gave an explanation that is much closer to the truth. He pointed out that because of the rotation of the earth, the speed of the equator about the earth’s axis is greater than that of the tropics (23.5° latitude) by some 2083 miles per day. Thus air at rest relative to the earth at 23.5° would, in the absence of friction, acquire a westward velocity of 2083 miles per day at the equator. Since velocities this large are not observed, it is to be considered, that before the Air from the Tropicks can arrive at the Equator, it must have gained Some Motion Eastward from the Surface of the Earth or Sea, whereby its relative Motion will be diminished, and in several successive Circulations, may be supposed to be reduced to the Strength it is found to be of. Thus I think the N.E. Winds on this Side of the Equator, and the S.E. on the other Side, are fully accounted for (Hadley, 1735, p. 61). Halley (1686, facing p. 151) produced the first comprehensive map of these winds over the tropical Atlantic and Indian Oceans based on his own observations and information obtained from a multitude of Observers.

The principle Hadley appealed to was that of conservation of angular momentum, which applies in the absence of friction. Hadley (1735, p. 62) also stated, The N.E. and S.E. Winds within the Tropicks must be compensated by as much N.W. and S.W. in other Parts, and generally all Winds from any one Quarter must be compensated by a contrary Wind some where or other; otherwise some Change must be produced in the Motion of the Earth round its Axis. This statement is not correct with respect to the northward component of the wind, but the principle Hadley was referring to is clear enough. The net rate of exchange of angular momentum between the atmosphere and the underlying surface must be zero, otherwise the angular momentum of the atmosphere would be continually increasing or decreasing. The quantitative expression of this principle may be derived as follows. Suppose the average eastward force (or rate of transfer of eastward momentum) per unit area acting on the earth’s surface at latitude ϕ is

Then the average torque (or rate of transfer of angular momentum) per unit area about the earth’s axis is

where a is the radius of the earth. The area of a zonal strip between latitudes ϕ and ϕ + dϕ is 2πa² cos ϕ dϕ, so the torque on this strip is

The net torque on the earth’s surface (or the net rate of exchange of angular momentum between the atmosphere and the underlying surface) must vanish, so

     (2.3.1)

This is the quantitative expression of the principle that Hadley appealed to.

The force of the atmosphere on the underlying surface may be exerted in two different ways. One is the force exerted on irregularities in the surface associated with pressure differences across the irregularities. The second is by viscous stresses. The irregularities on which forces are exerted may vary in size from mountain ranges like the Andes down to trees, blades of grass, and ocean surface waves. When the irregularities are small enough (as is the case over the ocean), the associated force per unit area added to the viscous stress is called the surface stress, or wind stress. Since the earth’s surface is mainly ocean, it is not surprising that (2.3.1) is approximately true with τx(ϕ) being the average eastward wind stress over the ocean at latitude ϕ [other contributions to the angular momentum exchange are discussed by Newton (1971)]. Figure 2.2 shows estimated values of the average eastward wind stress τx(ϕ) as a function of latitude. The latitudinal axis is drawn linear in

Fig. 2.2 The average eastward stress on the ocean surface as a function of latitude [values are from Eyre (1973)]. The spacing of latitudes is such that the distance between two nearby latitudes is proportional to the square of the cosine of the latitude. With this scale the area under the curve would be zero if the average rate of transfer of momentum from the atmosphere were the same over the land as over the sea at each latitude.

so that the area under the curve would be zero if (2.3.1) were exactly correct. Note that there is a westward stress in the trade wind zone (latitudes below 30°) and therefore an eastward stress is required at higher latitudes to give an overall balance. The eastward stress is associated with the prevailing westerly (i.e., eastward) winds at those latitudes. The reason that westerly winds should be found in these latitudes is not particularly straightforward and is discussed in relation to the angular momentum balance in Chapter 13 [see also Lorenz (1967, 1969)].

To calculate the ocean currents that are produced by the wind, the detailed distribution of stress with position on the earth’s surface is required. The pattern of surface winds away from the equator can be obtained from surface pressure maps (Fig. 2.3), whereas the tropical wind distributions are shown in Figs. 11.24, 11.28, and 11.29. Features like the trade winds, intertropical convergence zone, and westerly wind belts can be clearly seen in these figures. Sources of more detailed information are listed in Appendix 5.

2.4 Dependence of Exchange Rates on Air–Sea Velocity, Temperature, and Humidity Differences

Winds are produced in the atmosphere in response to radiative forcing. These winds transfer momentum to the ocean, producing ocean currents. By what processes is the momentum transferred and on what do the transfer rates depend? This is an important question, about which much has been written (Kraus, 1972, 1977; Garratt, 1977; Liu et al, 1979; Charnock, 1981; Lumley and Panofsky, 1964), and this section is intended to be but a brief introduction.

The radiative forcing of the atmosphere produces pressure gradients that result in wind speeds of order 10 m s−1. If there were no momentum transfer to the lower boundary (i.e., no frictional contact between the atmosphere and surface), such velocities would be expected right down to the surface. However, there is frictional contact at the surface. This means that at solid boundaries, the air in immediate contact with the boundary is constrained to have zero velocity. Thus a velocity gradient or shear exists near the ground. (An example of the way wind speed varies with height is shown in Fig. 2.4.) The shear flow, however, is not stable because small disturbances can grow to make the flow turbulent. The turbulent eddies (which are responsible for the gusty nature of the wind) modify the shear, but over a sufficiently long time, a well-defined mean velocity can be determined for each value of z, the distance above the ground. (Typical averaging times required are of the order of minutes for points a few meters above the ground.) In the region of substantial shear, momentum is transferred downward by bodily movement of parcels of air, i.e., by fast-moving parcels moving downward and slow moving parcels moving upward. If u is the horizontal component of velocity, w the vertical component, and ρ the density, then the vertical flux of horizontal momentum per unit area is ρuw, so the mean value of this quantity over a sufficiently large area or sufficiently large time is equal to the mean stress τ.

Fig. 2.4 –1 hr averages obtained from anemometers on a mast. Values at 50 m and above were obtained by tracking a pilot baloon released in the same general area.

[From Clarke et al. (1971, data p. 307, 0900 hr).]

As the ground is approached, the shear increases in inverse proportion with the distance from the ground. (This law can be deduced on dimensional grounds from the assumption that the shear depends only on τ, ρ, and distance z from the ground. It implies a logarithmic mean velocity profile.) The inverse law holds only sufficiently close to the ground where the shear is strong because other effects become important when the shear gets weak. For instance, if the lapse rate (see Section 1.5) is large enough to produce convection, turbulence due to convection will become more important than turbulence due to shear at some level.

In order to relate the stress τ to the wind speed u, it is necessary to specify the height at which the wind is measured. Once this is done, it follows on dimensional grounds that the relationship between τ and u can be put in the form

     (2.4.1)

where cD is a dimensionless coefficient called the drag coefficient. Its value over solid surfaces depends on the roughness of the surface and can also depend on the lapse rate. Values for different types of surface are known from measurement.

The ocean is not a solid surface, but surface velocities are still very much less than those in the atmosphere (typically about 3% of the velocity at 10 m). This is basically due to the density difference, for the same momentum can be carried in water with much smaller velocities. Hence the shear over the ocean is just as large as over the land, and turbulence is produced in the same way. However, measurements over the ocean are more difficult than over land, and less is known about how the drag coefficient varies, particularly at high wind speeds. As mentioned in Section 2.3, transfer across the surface can be due to pressure differences across irregularities (in this case waves) or to viscous stresses. The pressure differences across the waves can increase the amplitude of the waves, and waves can carry momentum away without any mean motion of the fluid. However, it seems likely that most of the momentum transferred during a storm is used to drive currents (Manton, 1972).

The drag coefficient cD for the ocean surface is found to increase with wind speed. Values for low speeds are around 1.1 × 10−3. For speeds over 6 ms−1 a linear relation between cD and u is often used, e.g., S. D. Smith (1980) suggests

     (2.4.2)

Alternatively, the data can be fitted by a relation obtained on dimensional grounds by Charnock [see Charnock (1981)]. This makes a quantity called the roughness length (see Section 9.5) proportional to the length scale, which can be obtained from τ, ρ, and g. The drag coefficient is then given by

     (2.4.3)

where κ and a are constants (called the von Kármán and Charnock constants, respectively), and z is the anemometer height (normally 10 m). Wu (1980) suggests the values

     (2.4.4)

An alternative formula for cD is suggested by Liu et al. (1979).

Because the eddying motion in the 10 m or so of air near the surface is caused by shear rather than by buoyancy differences (cf. Section 1.5), the rates of transfer of sensible heat and of moisture depend on the wind speed. The heat and moisture are transferred by bodily movement of parcels of air. The direction of transfer usually involves hot and moist fluid being carried upward and relatively cold and dry air being transferred downward. Like the shear, the temperature and humidity gradients increase as the surface is approached, also in inverse proportion with the distance from the surface. Assuming that the upward heat flux Qs depends on (i) the wind speed u, (ii) the difference between the sea temperature Ts and the air temperature Ta at the standard level, and (iii) the heat capacity ρacp per unit volume of the air, the