## K 300 K

Whence internal gravity waves in the atmosphere have a typical period of 2n N 9min.5 Internal gravity waves are ubiquitous in the atmosphere and are continually excited by, for example, horizontal winds blowing over hills and mountains, and convective plumes buffeting a stable layer above, among other things. On occasion, when the air is nearly saturated, they are made visible by the presence of regular bands of clouds in the crest of each wave (note, not all visible bands of clouds are...

## V

A supercell is a giant cumulonimbus storm with a deep rotating updraft. Supercells can produce large amounts of hail, torrential rainfall, strong winds, and sometimes tornadoes. The close-up view of the supercell thunderstorm in the picture shows a bulging dome of clouds extending above the flat, anvil top. This is caused by a very intense updraft that is strong enough to punch through the tropopause and into the stratosphere. At the time of this photograph, baseball-sized hail was...

## Freshwater Transport By The Ocean

The ocean and atmosphere must move freshwater from regions of excess rainfall to regions with excess evaporation (see Fig. 11.6). Knowledge of water fluxes and transports in the ocean is important for understanding the global hydrological cycle and climate. For example, variability in fresh water fluxes may have played an important role in the ice ages, as will be discussed in Chapter 12. The plot of evaporation minus precipitation in Fig. 11.6 shows that evaporation exceeds precipitation by...

## Physical Characteristics Of The Ocean

The ocean covers about 71 of the Earth's surface and has an average total depth of 3.7 km the distribution of land and sea and the bathymetry of the ocean basins is plotted in Fig. 9.1. A north-south section along the Greenwich meridian through Fig. 9.1 is shown in Fig. 1.2. We see that the ocean basins are highly complex and the bottom topography much more jagged than that of the land surface. See also the bathymetry shown in the hydrographic section of Fig. 9.9. This is because, as we shall...

## Info

Schematic diagram showing the classification of ocean gyres and major ocean current systems and their relation to the prevailing zonal winds. The pattern of Ekman transport and regions of upwelling and downwelling are also marked. FIGURE 10.20. Schematic diagram showing the classification of ocean gyres and major ocean current systems and their relation to the prevailing zonal winds. The pattern of Ekman transport and regions of upwelling and downwelling are also marked. for...

## Preface

The circulation of the atmosphere and oceans is inherently complicated, involving the transfer of radiation through a semi-transparent medium of variable composition, phase changes between liquid water, ice and vapor, interactions between phenomena on scales from centimeters to the globe, and timescales from seconds to millennia. But one only has to look at a picture of the Earth from space, such as that shown in Fig. 1, to appreciate that organizing principles must be at work to bring such...

## The Wind Stress And Ekman Layers

One cannot escape noticing the similarity between the pattern of surface currents in the ocean and that of the low-level winds in the atmosphere. Compare, for example, the pattern of surface elevation of the ocean in Fig. 9.19 with the pattern of surface atmospheric pressure, Fig. 7.27. Winds, through turbulent transfer of momentum across the atmospheric boundary layer, exert a stress on the ocean's surface that drives ocean currents. The surface wind stress can, to a useful degree of accuracy,...

## Problems

Figure 5.5 shows the net incoming solar and outgoing long-wave irra-diance at the top of the atmosphere. Note that there is a net gain of radiation in low latitudes and a net loss in high latitudes. By inspection of the figure, estimate the magnitude of the poleward energy flux that must be carried by the atmosphere-ocean system across the 30 latitude circle, to achieve a steady state. 2. Suppose that the Earth's rotation axis were normal to the Earth-Sun line. The solar flux, measured per unit...

## Q

Where now H is interpreted as the vertical scale of the motion. The horizontal length scale Eq. 7-23 is known as the Rossby radius of deformation. It is the scale at which the effects of rotation become comparable with those of stratification. More detailed analysis shows that on scales smaller than Lp, the pressure adjusts to the velocity field, whereas on scales much greater than Lp, the reverse is true and the velocity adjusts to the pressure. For the values of g and Q appropriate to our...

## E

It is this instability that leads to the convective motions discussed above. Using Eq. 4-4, the stability condition can also be expressed in terms of temperature as Note that Eq. 4-6 is appropriate for an incompressible fluid whose density depends only on temperature. Consider now our problem from yet another angle, in terms of energy conversion. We know that if the potential energy of a parcel can be reduced, just like the ball on the top of a hill in Fig....

## Equation Of Motion For A Nonrotating Fluid

The state of the atmosphere or ocean at any time is defined by five key variables (six if we include specific humidity in the atmosphere, or salinity in the ocean). Note that by using the equation of state, Eq. 1-1, we can infer p from p and T. To tie these variables down we need five independent equations. They are 1. the laws of motion applied to a fluid parcel, yielding three independent (x - g-Sx, y --g-Sy, z -g-Sz) (x + g-Sx, y -g-Sy, z --g-Sz) FIGURE 6.2. An elementary fluid parcel,...

## Tma

Where we have inserted numbers setting Te 255K. Thus for every 1 Wm-2 increase in the forcing of energy balance at the surface, Ts will increase by about a quarter of a degree. This is rather small when one notes that a 1 Wm-2 change in surface forcing demands a change in solar forcing of about 6Wm-2, on taking into account geometrical and albedo effects (see Problem 7 at the end of this chapter). A powerful positive climate feedback results from the temperature dependence of saturated water...

## The Taylorproudman Theorem

A remarkable property of geostrophic motion is that if the fluid is homogeneous (p uniform) then, as we shall see, the geostrophic flow is two dimensional and does not vary in the direction of the rotation vector, Q. Known as the Taylor-Proudman theorem, it is responsible for the glorious patterns observed in our dye shirring experiment, GFD Lab 0, shown again in Fig. 7.7. We discuss the theorem here and make much subsequent use of it, particularly in Chapters 10 and 11, to discuss the...

## T

An opaque greenhouse made up of two layers of atmosphere. Each layer completely absorbs the IR radiation impinging on it. FIGURE 2.9. An opaque greenhouse made up of two layers of atmosphere. Each layer completely absorbs the IR radiation impinging on it. layer-by-layer which depends on the vertical distribution of absorbers, particularly H2O, CO2, and O3 (see section 3.1.2) and do the required budgets for each layer and at the surface (we are not going to do this). An incomplete...

## Surface Salinity psu

The annual mean salinity distribution at the surface of the ocean (in psu). Darker green represents salty fluid. Data from Levitus World Ocean Atlas (1994). TABLE 9.4. The dependence of a, aT, and PS on T and S at two levels in the ocean at the surface and at a depth of 1 km. TABLE 9.4. The dependence of a, aT, and PS on T and S at two levels in the ocean at the surface and at a depth of 1 km.

## L

So, if x xa + Sx, where 5x is a small increment in x, then, to first order in Sx, we may write Taylor expansions were often used in Chapter 4 (in the derivation of vertical stability criteria) and in Chapter 6 (in the derivation of the equations that govern fluid motion). Cartesian coordinates are best used for getting our ideas straight, but occasionally we also make use of polar (sometimes called cylindrical) and spherical polar coordinates (see Section A.2.3). The Cartesian coordinate system...

## Geostrophic And Hydrostatic Balance

On the large-scale, water obeys the same fluid dynamics as air so we have already derived the equations we will need Eq. 6-44 applies just as well to the ocean as to the atmosphere. One simplification we can make in application of these equations to the ocean is to recognize that the density varies rather little in the ocean (by only a few see Fig. 9.2), so we can rewrite the horizontal momentum equations Eq. 6-44a,b thus (using our local Cartesian coordinate system see Fig. 6.19) without...

## K

The Earth radiates energy at the same rate it is received from the Sun. The Earth's emission temperature is 255 K, and that of the Sun is 6000 K. The outgoing terrestrial radiation peaks in the infrared spectrum the incoming solar radiation peaks at shorter wavelengths, in the visible spectrum. infrared (IR). But the atmosphere is strongly absorbing at these wavelengths due to the presence of trace gases principally the triatomic molecules H2O and CO2 which absorb and emit in the...

## The equations of fluid motion

Differentiation following the motion 6.2. Equation of motion for a nonrotating fluid 6.2.1. Forces on a fluid parcel 6.5. Integration, boundary conditions, and restrictions in application 6.6. Equations of motion for a rotating fluid 6.6.1. GFD Lab III Radial inflow 6.6.2. Transformation into rotating coordinates 6.6.3. The rotating equations of motion 6.6.4. GFD Labs IV and V Experiments with Coriolis forces on a parabolic rotating table 6.6.5. Putting things on the sphere 6.6.6. GFD Lab...

## H

Or, dividing through by N2, Eq. 9-6, we find that the slope of isopycnals interior density surfaces must be related to the slope of the free surface by Using our estimate of N2 in Section 9.1.5 and rearranging, we find that g N2H 400 if we assume that H 1 km. Thus we see that for every meter the free surface tilts up, density surfaces must tilt down by around 400 m if deep pressure gradients, and hence deep geostrophic flows, are to be cancelled out. But this is just what we observe in, for...

## Gwb

A schematic of tropospheric temperature profiles showing the dry adiabat, a typical wet adiabat, and a typical observed profile. Note that the dry adiabatic ascent of a parcel is typically cooler than the surroundings at all levels, whereas the wet adiabat is warmer up to about 10 km. The wet and dry lapse rates are close to one another in the upper troposphere, where the atmosphere is rather dry. 6r s is also known as the pseudo-adiabatic lapse rate. destabilized by the presence...

## P

In Section 3.3 we showed that the pressure of an isothermal atmosphere varies exponentially with height. Consider now an atmosphere with uniform potential temperature. Find how pressure varies with height, and show in particular that such an atmosphere has a discrete top (where p 0) at altitude RTo (Kg), where R, k, and g have their usual meanings, and To is the temperature at 1000 mbar pressure. 11. Consider the convective circulation shown in Fig. 4.29. Air rises in the center of the...

## Phs

Warm subtropical columns of fluid expand relative to colder polar columns. Thus the sea surface (measured relative to the geoid) is higher, by about 1 m, in the subtropics than the pole, and is thus greatly exaggerated in this schematic. Pressure gradients associated with the sea-surface tilt are largely compensated by vertical thermocline undulations, of about 400 m, ensuring that abyssal pressure gradients are much weaker than those at the surface. FIGURE 9.20. Warm subtropical...

## M

Is the mass-weighted mean height of the fluid. We can think of (z> as the ''height of the center of mass.'' It is evident from Eq. 8-7 that potential energy can be released only by lowering the center of mass of the fluid. Consider again the stratified incompressible fluid discussed in Section 4.2.2. Suppose, first, that dp dz < 0 (stably stratified) and that there are no horizontal gradients of density, so p is a function of height only, as depicted in Fig. 4.5. In Section 4.2.3 we...

## A3 Use Of Foraminifera Shells In Paleo Climate

A key proxy record of climate is 518O, the ratio of 18O to 16O in the shells of surface (planktonic) and bottom (benthic) dwelling foraminifera which are made of calcium carbonate, CaCO3. The 518O in the shell, measured and reported as where (18O 16O)smpl is the ratio of 18O to 16O in the sample and (18O 16O)sd is the ratio in a standard reference, is controlled by the 518O value of the water and the temperature at which the shell formed. It turns out that 18O is increasingly enriched in CaCO3...

## Largescale Atmospheric Energy And Momentum Budget

We have seen two basic facts about the atmospheric energy budget 1. There must be a conversion of energy into kinetic energy, from the available potential energy gained from solar heating, which requires that motions develop that transport heat upward (thereby tending to lower the atmospheric center of mass). 2. Accompanying this upward transport, there must be a poleward transport of heat from low latitudes this transport cools the tropics and heats the polar regions, thus balancing the...

## Winds

We saw in Section 5.2 that because of the pole-equator temperature gradient, isobaric surfaces slope down from equator to pole, inducing a horizontal pressure gradient at upper levels. There is thus a pressure gradient force aloft, directed from high pressure to low pressure, which is from warm latitudes to cold latitudes, as seen in Fig. 5.1. We might expect air to move down this pressure gradient. Hadley5 suggested one giant FIGURE 5.19. The circulation envisaged by Hadley (1735) comprising...

## Radiativeconvective Equilibrium

As we saw in Section 3.1.2, the thermal structure expected on the basis of radiative forcing alone has a temperature discontinuity at the ground, as illustrated in Fig. 3.4 the radiative equilibrium temperature of the ground is considerably warmer than that of the air above. This profile is unstable and convection will occur. Convective motions will transport heat upward from the surface when the air parcels mix with the environment (as they will), they increase the environmental temperature...

## Equator

Schematic meridional cross section of the near-equatorial upwelling induced by a westward wind stress near the equator. Since the upper-layer flow is divergent, mass continuity demands upwelling through the thermocline. we have replaced 2Q by f in Eq. 7-23 of Section 7.3.4 and g g (p1 - p2) p1 is the reduced gravity. Here, near the equator, f is no longer approximately constant for motions of length scale L centered on the equator, f pL and so L2 g 'h p2L2, thus defining the...

## Latitudes

Wnd stress wind stress wind stress wnd stress wind stress wind stress FIGURE 10.25. A schematic of the mechanism by which a large-scale sub-surface horizontal density gradient is maintained in the middle-latitude ocean. Ekman suction draws cold, dense fluid up to the surface in subpolar regions Ekman pumping pushes warm, light fluid down in the subtropics. The resulting horizontal density gradient supports a thermal wind shear. Its baroclinic instability spawns an energetic eddy field which...

## Ape

If z2 - zi is small, where (dp dz)r P1) V 'E (Z2 - Z1) is the mean density gradient of the environmental state. Note that the factor g (z2 - Z1)2 is always positive and so the sign of APE depends on that of (dp dz)E. Hence, if (dp dz)E> 0, rearrangement leads to a decrease in APE and thus to the growth of the kinetic energy of the parcels therefore a disturbance is able to grow, and the system will be unstable. But if (dp dz)E < 0, then APE > 0, and potential energy cannot be released by...

## So 32 34 36 38

FIGURE 9.12. (a) Annual mean T and S profile at 50 W, 30 N in the Atlantic Ocean. The thermocline is clearly evident. The T scale is at the top, the S scale at the bottom. (b) The cycle of T at 50 W, 30 N in the Atlantic Ocean in the Levitus monthly mean climatology. The strong seasonal cycle at the surface has vanished at a depth of 500 m. Data from the Levitus World Ocean Atlas (1994). numbers, 5 x 10-3 s-1. This implies a period for internal gravity waves in the ocean of 2n N 20 min5,...

## Airsea Fluxes And Surface Property Distributions

Heat, freshwater, and buoyancy fluxes As discussed in Chapter 4, atmospheric convection is triggered by warming at the surface. Vertical mass transport is confined to a few regions of strong updrafts driven by deep convection over the warmest oceans and land masses in the tropics, with broader areas of subsidence in between (see the atmospheric mean meridional circulation plotted in Fig. 5.21). In contrast to the atmosphere, the ocean is forced from above by air-sea fluxes. We therefore...

## Moisture

As discussed in Sections 1.3.2 and 4.5, the moisture distribution in the atmosphere is strongly controlled by the temperature distribution the atmosphere is moist near the surface in the tropics where it is very warm and drier aloft and in polar latitudes where it is cold. As shown in Fig. 5.15, the specific humidity, defined in Eq. 4-23, reaches a maximum (of around 18gkg-1) at the surface near the equator and decreases The control by temperature of the specific humidity distribution can be...

## Further Reading

The reader is referred to Holton (2004) and especially to Wallace and Hobbs (2006) for more detailed discussions of atmospheric thermodynamics, where many of the more exotic thermodynamic variables are defined and discussed. Emanuel (1994) gives a very thorough and advanced treatment of atmospheric convection. FIGURE 4.28. Schematic of temperature profiles before and after convection in the laboratory experiment GFD Lab II. The initial T profile, increasing linearly with height, is returned to...

## Rt

Where H is the vertical scale height, Eq. 3-9. For an isothermal atmosphere (with constant scale height), z varies as ln p, which of course is just another way of saying that p varies exponentially with z (see Eqs. 3-7 and 3-8). By integrating Eq. 5-1 vertically, we see that the height of a given pressure surface is dependent on the average temperature below that surface and the surface pressure, ps, thus where we have set z(ps) 0. The geopotential height of the surface is defined by Eq. 5-2,...

## Convection In Water

Objects that are lighter than water bounce back to the surface when immersed, as has been understood since the time of Archimedes (287-212 BC). But what if the object is a parcel3 of the fluid itself, as sketched in Fig. 4.4 Consider the stability of such a parcel in an incompressible liquid. FIGURE 4.4. A parcel of light, buoyant fluid surrounded by resting, homogeneous, heavier fluid in hydrostatic balance, Eq. 3.3. The fluid above points Ai, A, and A2 has the same density, and hence, as can...

## Dt

Where cw is the heat capacity of water (see Table 9.3), and Q and E are, respectively, the turbulent vertical flux of heat and freshwater driven by air-sea exchange, convection, ice formation, and vertical mixing in the ocean (as sketched in Fig. 9.11). At the surface Q Qnet, the net heat flux across the sea surface (see Eq. 11-5) and E Esurface E - P (evaporation minus precipitation, including that due to river runoff and ice formation processes) is the net fresh water flux across the sea...

## S

Apparatus used to illustrate the driving of deep ocean circulation by localized sinking of fluid. A sloping base is used to represent the influence of sphericity on Taylor columns as in GFD XIII. The 50 cm square tank is filled with water and set rotating anticlockwise at a rate of Q 5 rpm. (The sense of rotation is thus representative of the northern hemisphere). Dyed water, supplied via a funnel from an overhead bucket, flows slowly into the tank through a diffuser located in...

## A1 Derivations

A blackbody is a theoretical construct that absorbs 100 of the radiation that hits it. Therefore it reflects no radiation and appears perfectly black. It is also a perfect emitter of radiation. Planck showed that the power per unit area, per unit solid angle, per unit wavelength, emitted by a black body is where h is Planck's constant, c is the speed of light, k is Boltzmann's constant and A is the wavelength of the radiation. Figures 2.2 and 2.3 are plots of Ba(T) against wavelength for...

## Pressure And Geopotential Height

We have seen that it is warmer in the tropics than at higher latitudes. We will now describe how, through hydrostatic balance, this warmth leads to expansion of tropical air columns relative to polar air columns and hence meridional pressure gradients. It is these pressure gradients that induce fluid accelerations and hence winds. It is customary in meteorology to use pressure rather than height as the primary vertical coordinate. Some conceptual reasons will become clear in Chapter 6. Since,...

## Balanced flow

The geostrophic wind in pressure coordinates 7.1.2. Highs and lows synoptic charts 7.1.3. Balanced flow in the radial-inflow experiment 7.2. The Taylor-Proudman theorem 7.2.1. GFD Lab VII Taylor columns 7.3.1. GFD Lab VIII The thermal wind relation 7.3.2. The thermal wind equation and the Taylor-Proudman theorem 7.3.3. GFD Lab IX Cylinder ''collapse'' under gravity and rotation 7.3.4. Mutual adjustment of velocity and pressure 7.3.5. Thermal wind in pressure coordinates 7.4....

## Conservation Of Mass

In addition to Newton's laws there is a further constraint on the fluid motion conservation of mass. Consider a fixed fluid volume as illustrated in Fig. 6.4. The volume has dimensions (Sx, Sy, Sz). The mass of the fluid occupying this volume, p Sx Sy Sz, may change with time if p does so. However, mass continuity tells us that this can only (x - j Sx, y - j Sy, z - -g- Szj (x + -g- Sx, y - -g- Sy, z - g- Szj FIGURE 6.4. The mass of fluid contained in the fixed volume, pSx Sy Sz, can be changed...

## Dry Convection In A Compressible Atmosphere

Before we can apply the foregoing ideas to atmospheric convection, we must take into account the fact that the atmosphere is a compressible fluid in which p p(p, T) specifically, since the atmosphere closely obeys the perfect gas law, p p RT. For now we will assume a dry atmosphere, deferring consideration of the effects of moisture until Section 4.5. The parcel and environmental pressure, temperature, and density at z z1 in Fig. 4.5 are p1 p(z1), T1 T(z1), and p1 p1 RT1. The real difference...

## Effects Of Stratification And Topography

Our analysis of the wind-driven circulation in Section 10.2 assumed the ocean to have constant density, whereas (see, e.g., Fig. 9.7) the density of the ocean varies horizontally and with depth. In fact, the variation of density with depth helps us out of a conceptual difficulty with our physical interpretation in terms of the Taylor-Proudman theorem on the sphere presented in Section 10.2.3. We described how the Taylor columns of fluid must lengthen in the subtropical gyres to accommodate...

## Pref dX f dy f

The Ekman pumping velocity defined in Eq. 10-8 depends on the curl of (Twind f). Note, however, that typically Twind varies much more than f, and so the pattern of wEk is largely set by variations in Twind. We can estimate the magnitude of wEk as follows. Figure 10.2 shows that Twind changes from +0.1Nm-2 to -0.1Nm-2 over 20 of latitude, or 2000 km. Thus Eq. 10-8 suggests i in i inn im * v'vk v.-1 mivj h 'w ni li ii mi -hi h< ihi'i i < 11 FIGURE 10.11. The global...

## Enso

Atmospheric variability The Southern Oscillation As Walker noted, the Southern Oscillation shows up very clearly as a see-saw in FIGURE 12.8. Schematic of the feedback inherent in the Pacific Ocean-atmosphere interaction. This has become known as the Bjerknes feedback. FIGURE 12.8. Schematic of the feedback inherent in the Pacific Ocean-atmosphere interaction. This has become known as the Bjerknes feedback. sea level pressure (SLP) across the tropical Pacific basin. When SLP is higher than...

## Nll

Where, as before, (z) is defined by Eq. 8-8 and M byEq. 8-6. We can apply the ideas discussed in the previous section to a compressible fluid if, as discussed in Chapter 4, we think in terms of the distribution of potential temperature, rather than density, since the former is conserved under adiabatic displacement. Consider Fig. 8.11, in which the distribution of 0 in the atmosphere, 0 increasing upward and equatorward as in Fig. 5.8, is schematically shown. Again, we suppose FIGURE 8.11. Air...

## Geometry

The Earth is an almost perfect sphere with mean radius a 6370 km, a surface gravity field g 9.81 ms-2, and a rotation period of TEarth 24 h, equivalent to an angular velocity Q 2n rEarth 7.27 x 10-5 s-1 (see Table 1.1). The atmosphere which envelops the Earth is very thin it fades rapidly away TABLE l.l. Some parameters of Earth. TABLE l.l. Some parameters of Earth.

## A2 Mathematical Definitions And Notation

We assume that the function r(x) can be expressed as an expansion at the point xA in terms of an infinite series thus r(x) Co + ci (x - xa) + C2 (x - xa)2 + C3 (x - xa )3 + . . ., where the cs are constants. To find c0 we set x xA to yield c0 r(xA). To find c1 we differentiate once with respect to x and then set x xA to yield c1 (dr dx)A, where the subscript A indicates that dr dx is evaluated at x xA. Carrying on we see that we can write (dmr dxm)A m cm and hence

## Chemical Composition Of The Atmosphere

Air is a mixture of permanent gases (N2, O2) in constant ratio together with minor constituents (see Table 1.2). The molecular weight of the mixture that makes up air is 28.97, so that 22.4 liters of air at standard temperature and pressure (STP T 273 K and p 1013 h Pa) weighs 28.97 g. The composition of air is a direct consequence of the supply of elements from the Earth's interior and the presence of life on the surface. Photosynthesis by plants makes O2 nitrogenous compounds from living...

## The general circulation of the atmosphere

Understanding the observed circulation 8.2. A mechanistic view of the circulation 8.2.1. The tropical Hadley circulation 8.2.2. The extra tropical circulation and GFD Lab XI Baroclinic instability 8.3. Energetics of the thermal wind equation 8.3.1. Potential energy for a fluid system 8.3.2. Available potential energy 8.3.3. Release of available potential energy in baroclinic instability 8.3.4. Energetics in a compressible atmosphere 8.4. Large-scale atmospheric energy and momentum budget...

## The Nature Of Convection

Convection in a shallow fluid When a fluid such as water is heated from below (or cooled from above), it develops overturning motions. It may seem obvious that this must occur, because the tendency of the heating (or cooling) is to make the fluid top-heavy.1 Consider the shallow, horizontally infinite fluid shown in Fig. 4.2. Let the heating be applied uniformly at the base then we may expect the fluid to have a horizontally uniform temperature, so T T(z) only. This will be top-heavy...

## Da

Which yields our transformation rule for the operator D acting on a vector. Setting A r, the position vector of the particle in the rotating frame, we arrive at Eq. 6-24. To write down the rate of change of velocity following a parcel of fluid in a rotating frame, ( D1) , we set A > uin in

## Surface Wind Stress Nm2

Annual mean wind stress on the ocean. The green shading and contours represent the magnitude of the stress. Stresses reach values of 0.1 to 0.2 Nm-2 under the middle-latitude westerlies, and are particularly strong in the southern hemisphere. The arrow is a vector of length 0.1 Nm-2. Note that the stress vectors circulate around the high and low pressure centers shown in Fig. 7.27, as one would expect if the surface wind, on which the stress depends, has a strong geostrophic...

## F

Where f is the Coriolis parameter, Eq. 6-42, and h is the thickness of the layer in the direction of gravity measured in the vertical, as sketched in Fig. 10.23. FIGURE 10.22. Upper-layer Taylor column in a two layer idealization of the ocean moving over topography, such as the mid-Atlantic ridge, confined to the lower layer. FIGURE 10.22. Upper-layer Taylor column in a two layer idealization of the ocean moving over topography, such as the mid-Atlantic ridge, confined to the lower layer....

## 1

In middle latitudes (say near 45 see Table 6.1), f 2Q V2 1.03 x 10-4s-1. So given our typical numbers, Ro 0.1 and we see that the Rossby number in the atmosphere is small. We will find in Section 9.3 that Ro 10-3 for large-scale ocean circulation. The smallness of Ro for large-scale motion in the free atmosphere and ocean4 implies that the acceleration term in Eq. 6-43 dominates the Coriolis term, leaving

## Vertical Structure Of Pressure And Density

Using the equation of state of air, Eq. 1-1, we may rewrite Eq. 3-3 as In general, this has not helped, since we have replaced the two unknowns, p and p, by p and T. However, unlike p and p, which vary by many orders of magnitude from the surface to, say, 100 km altitude, the variation of T is much less. In the profile in Fig. 3.1, for example, T lies in the range 200-280 K, thus varying by no more than 15 from a value of 240 K. So for the present purpose, we may replace T by a typical mean...

## D

Is the depth-integrated meridional transport. Equation 10-17 is the result we seek. It is known as the Sverdrup relation6 and relates the vertically-integrated meridional flow to the curl of the wind stress. The key assumption that must be satisfied for its validity in addition to Ro < < 1 is that mean flow in the deep ocean must be sufficiently weak (well supported by observation), so that both frictional stress on the ocean bottom and vertical motion are negligibly small. Note the close...

## Zonal Average Zonal Wind ms

Meridional cross-section of zonal-average zonal wind (ms 1) under annual mean conditions (top), DJF (December, January, February ) (middle) and JJA (June, July, August) (bottom) conditions. w BON FIGURE 5.20. Meridional cross-section of zonal-average zonal wind (ms 1) under annual mean conditions (top), DJF (December, January, February ) (middle) and JJA (June, July, August) (bottom) conditions.

## Lmlm

Wffx sao aey< M cap- wk smk joor c* o'K M sook FIGURE 5.8. The zonally averaged potential temperature in (top) the annual mean, averaged over (middle) December, January, and February (DJF), and (bottom) June, July, and August (JJA). Zonal-Average Moist Potential Temperature (K) 90 s 60 s 30 s 0 30 n 60 n 00 n 90 s 60 s 30 s 0 30 n 60 n 00 n 240 K 260 K 260K 300 K 320 K 3 K 3fi0'K 300 K 400 K 420 K 440K 480 K 480 K S00 K FIGURE 5.9. The zonal average, annual mean equivalent potential...

## It

If the origin of our inertial coordinate system lies at the center of our dish, then the above can be written out in component form thus + Q2x 0 + Q2y 0 dt dt where the subscript n means inertial. This should be compared to the equation of motion in the rotating frame see Eq. 6-34. The solution of Eq. 6-37 satisfying our boundary conditions is

## Ocean Eddies

Figures 9.14 to 9.16 show currents averaged over many (about 20) years of observations. However, just as in the atmosphere (in fact, even more so) the picture we have described of the general circulation of the ocean, while appropriate to the time-averaged flow, is inadequate for describing the instantaneous flow. There are large variations of currents and of surface height that, instantaneously, can mask the time-averaged picture. The altimetric and drifter data can be analyzed to yield...

## Cxn

From its definition, Eq. 4-17, we can see that 9 is the temperature a parcel of air would have if it were expanded or compressed adiabatically from its existing p and T to the standard pressure p0. It allows one, for example, to directly determine how the temperature of an air parcel will change as it is moved around adiabatically if we know its 9, all we need to know at any instant is its pressure, and then Eq. 4-17 allows us to determine its temperature at that instant. For example, from the...

## Latitudinal Variations Of Climate

Putting everything together, we can depict the atmospheric wind systems in the upper and lower troposphere schematically as in Fig. 8.15. As we have remarked at various points in this chapter, the structure of the circulation dictates more than just the pattern of winds. In the near-equatorial regions, the convergence of the trade winds is associated with frequent and intense rainfall, as is characteristic of the deep tropics. It is here, for example, that the world's great tropical rainforests...

## Figurl

Sectional area of a Taylor column, A, projected on to the surface of the sphere, where p is the latitude. area measured perpendicular to Q (which, as discussed above, does not change in time), and d is its length parallel to Q, then since A' _ A sin p is the area of the Taylor column projected onto the surface of the sphere over which fluid is being pumped down from the Ekman layer at rate wEk. Note that the minus sign ensures that if wEk < 0 (pumping down into the ocean), then Dd Dt > 0 so...

## 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...

## 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...

## 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...

## 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,...