## Problems

Consider a zonally symmetric circulation (i.e., one with no longitudinal variations) in the atmosphere. In the inviscid upper troposphere, one expects such a flow to conserve absolute angular momentum, so that DA Dt 0, where A Qa2 cos 2p + ua cos p is the absolute angular momentum per unit mass (see Eq. 8-1) where Q is the Earth rotation rate, u the eastward wind component, a the Earth's radius, and p latitude. symmetric flow that the relation DA Dt 0 is consistent with the zonal component of...

## Info

Convection in a shallow fluid 4.3. Dry convection in a compressible atmosphere 4.3.1. The adiabatic lapse rate (in unsaturated air) 4.4. The atmosphere under stable conditions 4.5.2. Saturated adiabatic lapse rate 4.5.3. Equivalent potential temperature 4.6. Convection in the atmosphere 4.6.2. Where does convection occur 4.7. Radiative-convective equilibrium We learned in Chapters 2 and 3 that terrestrial radiation emanates to space primarily from the upper troposphere, rather than the...

## Cbb

Consider the Atlantic Ocean to be a rectangular basin, centered on 35 N, 10This statement refers to the midlatitude eddies evident in the height variance maps, Fig. 9.19 (bottom). The near-equatorial eddies evident in the surface current variance maps, Fig. 9.22 (bottom), are produced by another mechanism. of longitudinal width Lx 5000 km and latitudinal width Ly 3000 km. The ocean is subjected to a zonal wind stress of the form where ts 0.1Nm-2. Assume a constant value of p df dy appropriate...

## G

Where g is gravity and f the Coriolis parameter. Explain how this equation is consistent with the geostrophic relationship between Coriolis force and pressure gradient. (c) Assuming (for simplicity) that the flow is uniform to a depth D 500 m, and that the flow is zero below this depth, show that the net water transport (volume flux) along the Gulf Stream at this latitude is where 8n is the elevation difference you estimated in part (a). Evaluate this transport. 6. Figure 9.27 shows the...

## K

The energy emitted at different wavelengths for blackbodies at several temperatures. The function Bx (T), Eq. A-1, is plotted. blackbody spectrum. (A brief theoretical background to the Planck spectrum is given in Appendix A.1.1.) It is plotted as a function of temperature in Fig. 2.3. Note that the hotter the radiating body, the more energy it emits at shorter wavelengths. If the observed radiation spectrum of the Sun is fitted to the blackbody curve by using T as a free parameter,...

## The winddriven circulation

The wind stress and Ekman layers 10.1.1. Balance of forces and transport in the Ekman layer 10.1.2. Ekman pumping and suction and GFD Lab XII 10.1.3. Ekman pumping and suction induced by large-scale wind patterns 10.2. Response of the interior ocean to Ekman pumping 10.2.2. Wind-driven gyres and western boundary currents 10.2.3. Taylor-Proudman on the sphere 10.2.4. GFD Lab XIII Wind-driven ocean gyres 10.3. The depth-integrated circulation Sverdrup theory 10.3.1. Rationalization of...

## R

Equation 7-24 expresses the thermal wind relationship in pressure coordinates. By analogy with Eq. 7-8, just as height contours on a pressure surface act as streamlines for the geostrophic flow, then we see from Eq. 7-24 that temperature contours on a pressure surface act as streamlines for the thermal wind shear. We note in passing that one can obtain a relationship similar to Eq. 7-24 in height coordinates (see Problem 9 at end of chapter), but it is less elegant because of the p factors in...

## P

4.18 x 103 3.33 x 105 2.25 x 106 0.999 x 103 10-3 10-6 1.4 x 10-7 J kg-1 K-1 J kg-1 J kg-1 kg m-3 kg m-1 s-1 Temperature (C ) Temperature (C ) FIGURE 9.2. Contours of seawater density anomalies (a p - pref in kgm-3) plotted against salinity (in psu gkg-1 ) and temperature ( C) at the sea surface. Note that seawater in the open ocean has a in the range 20 29 kgm-3, T in the range 0 30 C and S in the range 33 36 psu. The panel on the right zooms in on the region of oceanographic relevance. An...

## The meridional structure of the atmosphere

Radiative forcing and temperature 5.1.3. The energy balance of the atmosphere 5.1.4. Meridional structure of temperature 5.2. Pressure and geopotential height In previous chapters we considered those processes that play a role in setting the vertical distribution of atmospheric properties. Here we discuss how these properties vary horizontally, on the global scale. We shall see that geometrical effects play a major role in setting the observed horizontal distribution. The spherical Earth...

## The ocean and its circulation

Physical characteristics of the ocean 9.1.3. Properties of seawater equation of state 9.1.4. Temperature, salinity, and temperature structure 9.1.5. The mixed layer and thermocline 9.2. The observed mean circulation 9.3. Inferences from geostrophic and hydrostatic balance 9.3.1. Ocean surface structure and geostrophic flow 9.3.2. Geostrophic flow at depth We now begin our discussion of the circulation of the ocean. In this introductory chapter we describe the physical characteristics of...

## Thermodynamic Equation

The equation governing the evolution of temperature can be derived from the first law of thermodynamics applied to a moving parcel of fluid. Dividing Eq. 4-12 by St and letting St > 0 we find DQ Dt is known as the diabatic heating rate per unit mass. In the atmosphere, this is mostly due to latent heating and cooling (from condensation and evaporation of H2O) and radiative heating and cooling (due to absorption and emission of radiation). If the heating rate is zero then DT Dt P- Dp Dt, and,...

## Differentiation Following The Motion

When we apply the laws of motion and thermodynamics to a fluid to derive the equations that govern its motion, we must remember that these laws apply to material elements of fluid that are usually mobile. We must learn, therefore, how to express the rate of change of a property of a fluid element, following that element as it moves along, rather than at a fixed point in space. It is useful to consider the following simple example. Consider again the situation sketched in Fig. 4.13 in which a...

## F

Thus the ageostrophic component is always directed to the right'' of F (in the northern hemisphere). We can readily demonstrate the role of Ekman layers in the laboratory as follows. 7.4.1. GFD Lab X Ekman layers frictionally-induced cross-isobaric flow We bring a cylindrical tank filled with water up to solid-body rotation at a speed of 5 rpm. A few crystals of potassium permanganate are dropped into the tank they leave streaks through the water column as they fall and settle on the base of...

## Moist Convection

We have seen that the atmosphere in most places and at most times is stable to dry convection. Nevertheless, convection is common in most locations over the globe (more so in some locations than in others, as we will discuss in Section 4.6.2). There is FIGURE 4.17. (Top) A satellite image showing dense haze associated with pollution over eastern China. The view looks eastward across the Yellow Sea toward Korea. Provided by the SeaWiFS Project, NASA Goddard Space Flight Center. (Bottom)...

## Ong

Where now YAmax is the magnitude of the mass transport overturning circulation of 6The overturning circulation shown in Fig. 11.30 is derived from a model constrained by obervations, rather than inferred directly from observations, because it is all but impossible to observe Vmoc directly. FIGURE 11.30. The meridional overturning circulation, wmoc, in a model of the global ocean plotted in the (i, z) plane on the left and the (i, 6) plane on the right. Note that on the left the scale over the...

## A4 Laboratory Experiments

The experiments described throughout this text come to life when they are carried out live, either in demonstration mode in front of a class or in a laboratory setting in which the students are actively involved. Indeed a subset of the experiments form the basis of laboratory-based, hands-on teaching at both undergraduate and graduate level at MIT. As mentioned in Section 0.1, the experiments have been chosen not only for their relevance to the concepts under discussion, but also for their...

## Baroclinic Instability In The Ocean

In our discussion of the general circulation of the ocean in Chapter 9 it was emphasized that the mean circulation emerges only after long time-averages. Instantaneously the flow is highly turbulent (see, e.g., Figs. 9.19 and 9.22) and the numerical simulation shown in Fig. 9.24. The sloping 9If h is the thickness of the layer across which the density changes by A a, then multiplying f h by A a prf we arrive at isopycnals evident in Fig. 9.7 suggest that there is available potential energy...

## Paleoclimate

Here we briefly review something of what is known about the evolution of climate over Earth history. Fig. 12.12 lists standard terminology for key periods of geologic time. Study of paleoclimate is an extremely exciting area of research, a fascinating detective story in which scientists study evidence of past climates recorded in ocean and lake sediments, glaciers and icesheets, and continental deposits. Proxies of past climates are myriad, and to the uninitiated at least, can be bizarre...

## Energetics Of The Thermal Wind Equation

The immediate source of kinetic energy for the eddying circulation observed in our baroclinic instability experiment and in the middle-latitude atmosphere, is the potential energy of the fluid. In the spirit of the energetic discussion of convection developed in Section 4.2.3, we now compute the potential energy available for conversion to motion. However, rather than, as there, considering the energy of isolated fluid parcels, here we focus on the potential energy of the whole fluid. It will...

## The Atmospheric Absorption Spectrum

A property of the blackbody radiation curve is that the wavelength of maximum energy emission, Xm, satisfies This is known as Wien's displacement law. Since the solar emission temperature is about 6000 K, the maximum of the solar spectrum is (see Fig. 2.2) at about 0.6 m (in the visible spectrum), and we have determined Te 255 K for the Earth, it follows that the peak of the terrestrial spectrum is at XEmarth 0.6 m x 6000 - 14 Mm. 255 Thus the Earth's radiation to space is primarily in the...

## The Thermal Wind Equation

We saw in Section 5.2 that isobaric surfaces slope down from equator to pole. Moreover, these slopes increase with height, as can be seen, for example, in Fig. 5.13 and the schematic diagram, Fig. 5.14. Thus according to the geostrophic relation, Eq. 7-8, the geostrophic flow will increase with height, as indeed is observed in Fig. 5.20. According to T-P, however, dug dz 0. What's going on The Taylor-Proudman theorem pertains to a slow, steady, frictionless, barotropic fluid, in which p p(p)....

## The Greenhouse Effect

The global average mean surface temperature of the Earth is 288 K (Table 2.1). Previously we deduced that the emission temperature of the Earth is 255 K, which is considerably lower. Why We saw from Fig. 2.6 that the atmosphere is rather opaque to IR radiation, so we cannot think of terrestrial radiation as being radiated into space directly from the surface. Much of the radiation emanating from the surface will be absorbed, primarily by H2O, before passing through the atmosphere. On average,...

## The Ocean As A Buffer Of Temperature Change

The oceans have a much greater capacity to store heat than the atmosphere. This can be readily seen as follows. The heat capacity of a slab of ocean of depth h is YO PrefCwh (i.e., density x specific heat x depth, with units of JK-1m-2). Let us compare this with the heat capacity of the atmosphere, which we may approximate by Ya PscpH, where ps is the mean density of air at the surface and H is vertical scale height of the atmosphere (7-8 km). Inserting typical numbers the ocean is one thousand...

## P P

Which is a more general statement of the ''thermal wind'' relation. In the case of constant P, or more precisely in a barotropic fluid where p p(p) and so Vp is parallel to Vp, Eq. 7-19 reduces to 7-14. But now we are dealing with a baroclinic fluid in which density depends on temperature (see Eq. 4-4) and so p surfaces and p surfaces are no longer coincident. Thus the term on the right of Eq. 7-19, known as the baroclinic term, does not vanish. It can be simplified by noting that to a very...

## The Observed Mean Circulation

The global pattern of mean flow at the surface of the ocean is plotted in Fig. 9.13, where the names of the major current systems are also given. The colors separate the circulation patterns into tropical (pink), subtropical (yellow), and subpolar (blue) regimes (inspired by the dynamical discussion to be developed in Chapter 10). Because the detailed patterns are difficult to discern on this global map, regional circulations in the Pacific, Atlantic, and Indian Oceans are also shown in Figs....

## Convection In The Atmosphere

We have seen that the atmosphere is normally stable in the absence of condensation. Hence most convection in the atmosphere is moist convection, accompanied by saturation and hence cloud formation. Downwelling air parcels do not become saturated because descending air FIGURE 4.20. Schematic of convective clouds Cu cumulus Cb cumulonimbus. The condensation level is the level above which q q . Cb clouds have a characteristic ''anvil,'' where the cloud top spreads and is sheared out by strong...

## Physical Properties Of

Some important numbers for Earth's atmosphere are given in Table 1.3. Global mean surface pressure is 1.013 x 105Pa 1013 h Pa. (The hecto Pascal is now the official unit of atmospheric pressure 1 h Pa 102 Pa , although the terminology millibar 1 mbar 1 h Pa is still in common use and will also be used here.) The global mean density of air at the surface is 1.235 kgm-3. At this average density we require a column of air of about 7-8 km high to exert pressure equivalent to 1 atmosphere....

## The thermohaline circulation of the ocean

Air-sea fluxes and surface property distributions 11.1.1. Heat, freshwater, and buoyancy fluxes 11.1.2. Interpretation of surface temperature distributions 11.1.3. Sites of deep convection 11.2. The observed thermohaline circulation 11.2.1. Inferences from interior tracer distributions 11.2.2. Time scales and intensity of thermohaline circulation 11.3. Dynamical models of the thermohaline circulation 11.3.1. Abyssal circulation schematic deduced from Taylor-Proudman on the sphere 11.3.2....

## Observations Of Abyssal Ocean Circulation

It is very hard to test whether the circulation schematic, Fig. 11.16, has parallels in the ocean because the predicted mean currents are so very weak and the variability of the ocean so strong. However, one of the key predictions of Stommel's abyssal theory was that there ought to be deep western FIGURE 11.22. Three photographs charting the evolution of dye from source (white circle) to sink (black circle) using the apparatus shown in Fig. 11.21. The shallow end of the tank is marked with the...

## Understanding The Observed Circulation

The simplest observed global characteristic of the atmosphere is that the tropics are much warmer than the poles. As discussed in Chapter 5, this is a straightforward consequence of the geometry of the Earth the annually averaged incoming solar radiation per unit area of the Earth's surface is much greater at the equator than at the poles, a difference that is compounded by the fact that the polar regions are covered in ice and snow and therefore reflect much of the incoming radiation back to...

## Planetary Emission Temperature

The Earth receives almost all of its energy from the Sun. At the present time in its evolution the Sun emits energy at a rate of Q 3.87 x 1026 W. The flux of solar energy at the Earth, called the solar constant, depends on the distance of the Earth from the Sun, r, and is given by the inverse square law, So Q 4nr2. Of course, because of variations in the Earth's orbit (see Sections 5.1.1 and 12.3.5) the solar constant is not really constant the terrestrial value TABLE 2.1. Properties of some of...

## The Ocean Heat Budget And Transport

We now turn to the role of the ocean circulation in meridional heat transport. To maintain an approximately steady climate, the ocean and atmosphere must move excess heat from the tropics to the polar regions. We saw back in Fig. 8.13 that the atmosphere SO FIGURE 11.24. Observations of CFCs at a depth of 2 km (contoured). Superimposed in red is a snap-shot for 1983 of the CFC distribution at a depth of 2 km in the North Atlantic, as simulated by a numerical model of ocean circulation and...

## Equations Of Motion For A Rotating Fluid

Equation 6-6 is an accurate representation of Newton's laws applied to a fluid observed from a fixed, inertial, frame of reference. However, we live on a rotating planet and observe winds and currents in its rotating frame. For example the winds shown in Fig. 5.20 are not the winds that would be observed by someone looking back at the Earth, as in Fig. 1. Rather, they are the winds measured by observers on the planet rotating with it. In most applications it is easier and more desirable to work...

## Radiative Forcing And Temperature

The latitudinal distribution of incoming solar radiation at the top of the atmosphere in the annual mean and at solstice is shown in Fig. 5.2. Its distribution is a consequence of the spherical geometry of the Earth and the tilt of the spin axis, depicted in Fig. 5.3. If the Earth's axis did not tilt with respect to the orbital plane, the average incident flux would maximize at a value of Smax So n 435Wm-2 at the equator, and fall monotonically to zero at the poles. Because of the tilt,...

## The Observed Thermohaline Circulation

The time-mean abyssal flow in the ocean is so weak that it cannot be measured directly. However abyssal circulation, and the convective processes forcing it, leaves its signature in the distribution of water properties, from which much can be inferred. 11.2.1. Inferences from interior tracer distributions Water masses modified by deep convection are tagged with T and S values characteristic of their formation region, together with other tracers, such as tritium from the atomic weapon tests of...

## Ro t im

Hence show that it has a spectrum, TmTm, where T*m is the complex conjugate, given by Eq. 12-2. Graph the spectrum using a log-log plot and hence convince yourself that fluctuations with a frequency greater than 1 yO are damped. 3. For the one-layer ''leaky greenhouse'' model considered in Fig. 2.8 of Chapter 2, suppose that, all else being fixed, the atmospheric absorption depends linearly on atmospheric CO2 concentration as where CO2 is CO2 concentration (in ppm), 0 _ 0.734, and ex _ 1.0 x...

## Mechanistic View Of The Circulation

To what extent can we explain the main features of the observed general circulation on the basis of the fluid dynamics of a simple representation of the atmosphere driven by latitudinal gradients in solar forcing The emphasis here is on simple. In reality, the Earth's surface is very inhomogeneous there are large mountain ranges that disturb the flow and large contrasts (e.g., in temperature and in surface roughness) between oceans and continents. In the interests of simplicity, however, we...

## A5 Figures And Access To Data Over The

The vast majority of the data and figures presented in this book were accessed and plotted using the Climate Data Library of the International Research Institute of the Lamont-Doherty Earth Observatory of Columbia University. The library contains numerous datasets from a variety of Earth science disciplines and climate-related topics, which are accessible free over the web see http iridl.ldeo.columbia.edu . Web-based tools permit access to data sets, analysis software for manipulation of the...

## Dynamical Models Of The Thermohaline Circulation

Abyssal circulation schematic deduced from Taylor-Proudman on the Because of the paucity of direct observations of abyssal flow, theory has been an invaluable guide in deducing likely circulation patterns. The starting point for a theoretical deduction are two important inferences from the observations discussed above 1. Dense water is formed at the surface in small, highly localized regions of the ocean in polar seas. Thus the abyssal circulation seems to be induced by local sources,...

## E

FIGURE 4.6. (a) A sketch of the laboratory apparatus used to study convection. A stable stratification is set up in a 50 cm x 50 cm x 50 cm tank by slowly filling it up with water whose temperature is slowly increased with time. This is done using (1) a mixer, which mixes hot and cold water together, and (2) a diffuser, which floats on the top of the rising water and ensures that the warming water floats on the top without generating turbulence. Using the hot and cold water supply we can...