Internal wave Propagation

what cause internal waves, what causes internal ocean waves, what generates internal waves, internal gravity wave propagation nonlinear internal wave propagation
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An introduction to internal waves T. Gerkema J.T.F. Zimmerman Lecture notes, Royal NIOZ, Texel, 2008Chapter 1 Introduction Wavesareallaroundus,butitisactuallyhardtosaywhatawaveis. Thisisbe- causeitisanimmaterialthing: asignal,acertainamountofenergypropagating through a medium. The medium we consider is water, seawater in particular. Even though water waves as such are immaterial, they are supported by the oscillatory movement of the water parcels, and this indeed forms a way of de- tecting them. But it is always important not to confuse the water motion with the wave itself. The following analogies may help to clarify this point: \A bit of gossip starting in London reaches Edinburgh very quickly, even though not a single individual who takes part in spreading it travels be- tween these two cities. There are two quite di®erent motions involved, that of the rumour, London to Edinburgh, and that of the persons who spread the rumour. The wind, passing over a ¯eld of grain, sets up a wave which spreads out across the whole ¯eld. Here again we must distinguish between the motion of the wave and the motion of the separate plants, which undergo only small oscillations. We have all seen the waves that spread in wider and wider circles when a stone is thrown into a pool of water. The motion of the wave is very di®erent from that of the particles of water. The particles merely go up and down. The observed motion of the wave is that of a state of matter and not of matter itself. A cork °oating on the wave shows this clearly, for it moves up and down in imitation of the actual motion of the water, instead of being carried along by the wave." 16, pp. 104-105 1.1 The ocean's inner unrest Waves at the ocean's surface are a familiar sight. These lecture notes deal with wavesthatpropagatebeneaththesurface; theyaremostlyhiddenfromeyesight. Occasionally, however, they produce a visible response at the ocean's surface. 9An example is shown in Figure 1.1, a photograph taken from the Apollo-Soyuz spacecraft in 1975, when it passed over Andaman Sea, north of Sumatra. Fig. 1.1: A photograph from the Apollo-Soyuz spacecraft, made over Andaman Sea, showing stripes due to internal waves. The stripes stretch over 100 km or more, and have a mutual distance of the order of a few tens of kilometers; they propagate slowly ¡1 (at a speed of about 2 ms ) to the northeast. The stripes can be observed from a ship as well; they appear as long bands of breaking waves, typically about 1 meter high. Spacecraft or satellite pictures showing such stripes have since been obtained from many other locations; an example from the Bay of Biscay is shown in Figure 8.11. Figure 1.2 provides a look into the ocean's interior, and brings us to the origin of the stripes, in this case in Lombok Strait. The echosounder signal shows the elevation and depression of levels of equal density (isopycnals); in the course of just 20 min- 1 utes, they descend more than 100 m, and rise again to their original levels. The isopycnals closer to the surface, however, undergo a much smaller vertical displacement. This is the de¯ning characteristic of internal waves: that their largest vertical amplitudes occur in the interior of the °uid. Internal waves were discovered more than a century ago. One of the ¯rst observations is due to Helland-Hansen & Nansen 41. They found that temper- ature pro¯les may change substantially within the course of just hours (Figure 1.3);theyascribedthistothepresenceof"puzzling waves",anexampleofwhich is shown in Figure 1.4. They stressed the importance of this newly discovered phenomenon: "Theknowledgeoftheexactnatureandcausesofthese"waves"andtheir movements would, in our opinion, be of signal importance to Oceanogra- phy, and as far as we can see, it is one of its greatest problems that most urgently calls for a solution" p. 88 1 The horizontal currents associated with these waves extend to the surface; these sur- face currents modify the roughness of the surface waves, thus rendering the internal waves (indirectly) detectable by satellite remote sensing imagery. 10Fig. 1.2: Isopycnal movements associated with the passage of an internal wave, ob- served with an echosounder in Lombok Strait, covering the upper 250 m of the water column. Colours indicate levels of backscatter and can be used to distinguish levels of density. Horizontal is time; the spacing between two vertical lines corresponds to 6 minutes. From 82. AlthoughtheinternalwavesshowninFigure1.2areperhapsunusuallylarge, their presence as such is not at all unusual; they are a ubiquitous phenomenon in the ocean (and in the atmosphere as well). Internal waves provide the `inner unrest' in the oceans, at time scales ranging from tens of minutes to a day. These oscillations are a nuisance when one attempts to establish the ocean's `background state' (i.e. patterns of large-scale circulation, tracer distribution etc.). This was already recognized by Helland-Hansen & Nansen. The puzzling waves, they noted, "makeitmuchmoredi±cultthanhashithertogenerallybeenbelieved,to obtaintrustworthyrepresentationsofthevolumesofthedi®erentkindsof water; they certainly cannot be attained by observations at a small num- berofisolatedstations,chosenmoreorlessatrandom. Suchirregularities, great or small, are seen in most vertical sections where the stations are su±ciently numerous and not too far apart. The equilines (isotherms, isohalines, as well as isopyknals) of the sections hardly ever have quite regular courses, but form bends or undulations, like waves, sometimes great, sometimes small." p. 87 Frommeasurementsmadeatanyonemomentitisthusimpossibletodeduce whatthebackgroundisopycnallevelsandcurrentvelocitiesare; thebackground state is continually being perturbed by internal-wave activity, which may pro- duce isopycnal variations of the order of 100 m, and current velocities of tens of ¡1 cms . An example of such a variation is shown in Figure 1.5. Only prolonged measurements allow for a meaningful estimate of the 'mean background state'. Before we continue our discussion on internal waves, we ¯rst take a closer look at the medium in which they propagate. Two properties are of primary 11Fig. 1.3: Temporal changes in temperature pro¯les, at two di®erent locations. The curves were constructed on the basis of the measurements shown in dots. Pro¯les a' and a": northeast of Iceland, on August 5, 1900; variations at 20 m depth were measured during about 2 1/2 hours, but sometimes rapid changesoccurred in just ¯ve minutes. Pro¯les b and b': north of the Faeroes, on 25-26 July, 1900. From 41. Fig. 1.4: Isopycnal variations with time. The dominant period is about half a semi- diurnaltidalperiod. Forcomparison,openandblackcircleshavebeenadded,denoting the spacing between high and low waters. From 41. importance: 1) the vertical strati¯cation in density, and 2) the diurnal rotation. 1.2 Restoring forces Waves in °uids owe their existence to restoring forces; these forces push parcels that are brought out of their equilbrium position, back towards that position, thus bringing them into oscillation. Sound waves, for example, exist due to compression; here pressure (gradients) act as the restoring force. In internal waves, tworestoring forces are at work: 1) buoyancy (i.e. reduced gravityin the 12Fig. 1.5: Variation of the depth of the Chlorophyll maximum (shaded region) due to the presence of internal waves, in the Bay of Biscay. Horizontal is time (in hours), but the variations are both temporal and spatial, since the measurements were made from a moving ship. From 43. ocean's interior), due to the ocean's strati¯cation, and 2) the Coriolis force, due to the Earth's diurnal rotation. 1.2.1 The ocean's strati¯cation ± The bulk of the ocean is very cold; the ocean's mean temperature is only 3.5 C. The variation with depth, however, is large: below 1000 m depth, temperature ± islessthan5 C,butintheupper200mitrisesstrongly,especiallyinthetropics (seeFigure2.2a),andduringsummeratmid-latitudes. Togetherwithvariations in salinity (Figure 2.2b), this determines in-situ density ½, being a function of pressure, temperature, and salinity (Figure 2.2e). The steady increase of in- situ density with depth does not in itself guarantee that the water column is gravitationally stable. As explained in Chapter 3, the stability of the water column is determined by ³ ´ ½ 1 2 2 N =g ¡ ; (1.1) 2 p c s where p is pressure, c the speed of sound (Figure 2.2f), and g the acceleration s 2 duetogravity. Thewatercolumnisstablystrati¯edif N 0. ThequantityN is called the Brunt-VÄaisÄalÄa or buoyancy frequency; its unit is radians per second (but also common are cycles per hour, or cycles per day). A typical distribution of N in the ocean is shown in Figure 1.6. It shows ¡4 ¡2 ¡1 thatN varies greatly: from O(10 ) in the deepest layers to O(10 ) rads in theupper200m. Thelatterregionincludesthe thermocline, correspondingtoa peakinN duetotherapiddecreaseoftemperaturewithdepth; thethermocline 13¡1 Fig. 1.6: The strati¯cation N (in rads ), derived from temperature and salinity ± pro¯les in the Paci¯c Ocean, for a south-north section near 179 E (WOCE section P14, fromtheFijiIslandsintotheBeringSea, July/August1993). Adaptedfrom30. has a permanent character in the tropics and is seasonal at mid-latitudes. We see from Figure 1.6 that N decreases again in the upper 50 m or so, the upper mixed layer, which is due to the mixing by the wind. ¡1 Intheatmosphere, valuesrangefrom0.01 inthetroposphere to0.02rads in the stratosphere. Both in the ocean and atmosphere, N becomes locally very small in turbulently mixed, convective layers. 1.2.2 The Earth's diurnal rotation The Earth undergoes a diurnal rotation on its axis. After one full rotational period it regains the same orientation with respect to the `¯xed stars'; this period of 23 h 56 min 4 s (=86164 s) is called a sidereal day, d . It is distinct sid fromthesolarday(i.e.24hours)because,astheEarthtraversesitspatharound 2 the sun (in what as such is a translational motion), it takes slightly more than the sidereal day to regain the same orientation with respect to the sun which is what de¯nes the solar day. The Earth angular velocity thus is 2¼ ¡5 ¡1 ­= =7:292£10 rads : d sid 2 The Earth's diurnal rotation is prograde; if it were retrograde, the solar day would be shorter than the sidereal day. 14(Note that the last two decimals would be di®erent if one mistakenly uses the solar day.) We can now express the vectorial character of the diurnal rotation as ­, aligned to the axis of rotation (pointing northward), and with magnitudej­j= ­. To ¯nd the e®ects of rotation at a certain latitude Á, we can decompose the vector as indicated in Figure 1.7. Thus we ¯nd the Coriolis frequencies f =2­sinÁ; f =2­cosÁ: (1.2) These components determine the Coriolis force, which is formed by the outer product of 2­ with velocity (see Chapter 2). The Coriolis force acts as a purely de°ecting force: it never initiates a motion (the force does no work since it is perpendicular to velocity), it only de°ects an existing motion. Fig. 1.7: Decomposition of the rotation vector ­ at latitude Á, giving rise to the Coriolis components f =2­ and f =2­ . ? k Theperpendicularitywithvelocityhasstillanotherconsequence: thecompo- nentf,beingitselfhorizontal(seeFigure1.7),de°ectsdownwardmovingparcels eastward(e.g.ifyoudropastonefromatower,itwillundergoaslightde°ection to the east), and produces an upward force on eastward moving parcels (i.e. the weight of an eastward moving object is reduced, the so-called EÄotvÄos e®ect). In either case, there is a vertical direction involved. Now, the currents in the ocean are predominantly horizontal, due to the fact that the ocean constitutes a thin layer compared to the Earth's radius. This diminishes the importance of f; the e®ect of f usually far exceeds that of f, despite the fact that f and f as such are of similar magnitude at mid-latitudes. We continue this discussion in later chapters, but for the moment we assume that f is negligible, so that the e®ects of the Earth's diurnal rotation are represented solely by f. Notice that f is negative in the Southern Hemisphere. We have thus established two fundamental frequencies, N and f, each of them associated with a restoring force. These two restoring forces, gravity and the Coriolis force, lie at the heart of the phenomenon of internal waves; this is 15re°ected by the fact that N and f are key parameters in internal-wave theory. We note that in most parts of the ocean, N exceedsjfj. Finally, a few words on nomenclature. Internal waves for which only gravity actsastherestoringforce,arecalledinternalgravitywaves;thissituationoccurs for example in laboratory experiments on a non-rotating platform, or in the ocean for waves at frequencies much higher thanjfj, in which case the Coriolis forcecan beneglected. Conversely, ifonlytheCoriolis forceisat work, theyare called gyroscopic (or inertial) waves; this situation occurs in neutrally strati¯ed layers (N = 0). Finally, if both forces are at work as is commonly the case they are called internal inertio-gravity waves. 1.3 Origins of internal waves Where does the ubiquitous `inner unrest' originate from? As it turns out, there are two principal sources of internal waves. One is the atmospheric disturbance of the ocean's upper mixed layer by the wind; this was already recognized by Helland-Hansen & Nansen 41: "It is a striking fact, and apparently not merely an accidental one, that by far the greatest "waves" of this kind in our sections, occurred in 1901, when the atmosphere was unusually stormy; and it appears probable that the "waves" in that year might have been due to stirring of the water masses, caused by disturbances in the atmosphere." p. 88 As the wind resides, variations of the base of the mixed layer slowly evolve towards equilibrium, in a process called geostrophic adjustment 33. During this process, internal waves are emitted, predominantly at frequencies close to jfj, the inertial frequency. These waves are called near-inertial waves; they are usually clearly present in internal-wave spectra, as a peak centered around jfj. They form indeed the most energetic part of the internal-wave spectrum. Notwithstanding their importance, it would seem that a comprehensive under- standing of their generation and propagation is still lacking. This is very di®erent for the other source of internal waves, also at low frequencies: the internal tides. They are formed by the °ow of barotropic (i.e surface) tides over sloping bottom. Theoriginofbarotropictidesthemselvesliesintheastronomicaltide-generating forces: thegravitationalpullbythemoonand,toalesserextent,thesun. These forces, together with the diurnal rotation of the Earth, produce the barotropic tides,whichtraversetheoceansassurfacewaves(anexampleisshowninFigure 7.1). This movement acts as drag to the moon, and thus slows down its angular velocity. Conservationofangularmomentumimpliesthatthemoonmustrecede fromtheEarth. Thishasbeencon¯rmedbyobservations: thedistancebetween 3 themoonandEarthincreasesby3.8cmperyear. Fromthis, onecancalculate 3 Measured using laser beams re°ecting from mirrors that were placed on the moon during theApollo 11 mission, in July 1969, and later missions. The drag not only retards the moon's 16how much energy goes into the barotropic tides in the ocean (the amount going into tides in the atmosphere and the Earth's mantle is small by comparison): 12 about3.5TWforalltidalcomponentstogether(1TeraWatt=10 Watt). The barotropic tide, in turn, loses its energy mostly by bottom friction in shallow seas, but also for a signi¯cant part, about 30% (1 TW), over ridges in the `open ocean' (and for another, yet unknown part, over the continental slopes); here the energy is transferred to internal tides. This is illustrated in Figure 1.8. Fig. 1.8: Regions where dissipation of the semi-diurnal lunar barotropic tide (M ) 2 occurs,determinedusingdatafromsatellitealtimetry. Thereisaclearcorrespondence with bottom topography; noticeable dissipation occurs over, for example, the Mid- AtlanticRidgeandtheHawaiianRidge. Theresultsarelessreliableinshallowregions (because of uncertainties in the estimates of tidal currents), where errors may lead to spots of negative values (in blue). From 15. The idea behind this process is as follows. Barotropic tidal currents are predominantlyhorizontal(U),butoverbottomslopesaverticalcomponentmust arise (Urh, with bottom topography h), which, like the horizontal component, oscillates at the tidal frequency. This vertical tidal current brings isopycnal surfaces into oscillation; they are periodically lifted up and pulled down. These verticaloscillationsactasawavemaker,emittingwavesattheforcingfrequency: the internal tides. We may compare this process with that of wave generation in a stretched string or rope: if it is forced into vertical oscillation at one point, waves are generated which propagate away from that point. But how do internal tides propagate away from the region of forcing? This brings us to what is perhaps the most remarkable (and in any case the most counter-intuitive) property of internal waves: their energy propagates at once horizontally and vertically, quite unlike surface waves, whose energy propagates only horizontally. The di®erence is due to the di®erent nature of the strati¯- cation supporting the waves. Surface waves owe their existence to the sharp change in density between air and water, which is restricted to the surface and so is their energy propagation. Internal waves, on the other hand, owe movement around the Earth, but also lengthens the terrestrial day. The combined e®ect of lunar and solar tides amounts to an increase of 2.4 milliseconds per century (see 7, p. 249). 17Fig. 1.9: The path of an internal tidal beam generated over the continental shelf break in the Bay of Biscay. The depth of maximum vertical isopycnal excursion was determinedatvarioushorizontalpositionsbyCTDyoyoing;thesedepthsareindicated by circles. They follow a path that coincides with the theoretical path of internal-tide propagation (dashed line). From 69. their existence to the strati¯cation of the ocean's interior, which is smoothly distributed over the vertical (see Figure 1.6), and so energy is carried from one depth level to the other. A vivid illustration of the path of energy propagation is shown in Figure 1.9: the internal tide generated over the continental slope propagates into the deep ocean, following a diagonal path. The arrows indicate the vertical extent of the energy, showing that the internal tide propagates in a beam-like manner. To summarize, there are two main generation mechanisms: atmospheric forcing,andbarotropictidal°owovertopography. Bothgeneratelow-frequency waves. However,interactionsamongthesewavesleadtointernalwavesathigher frequencies. As a result, internal waves are found at all frequencies between jfj and N, although those at low frequencies dominate the spectrum. 181.4 Dissipation and mixing IntheprevioussectionwehaveseenthattheEarth-moonsystemlosesenergyto barotropic tides, which, in turn, lose part of their energy to internal tides. The questionthenariseswheretheirenergygoes. Wehavealsoseenthatmuchofthe internal-wave energy originates from the upper layer of the ocean (near-inertial waves generated by the wind, internal tides generated over the continental shelf break). However, the vertical component in their energy propagation opens up the possibility that their energy, though originating from the upper layer, may ¯nally be dissipated in the abyssal ocean. This, indeed, seems to be what is happening. The precise pathways to dissipation are yet to be established quantitatively, but the general picture has become clearer in recent years, see Figure 1.10. Internal waves can become unstable due to the presence of a background shear ¯eld, leading to internal- wave breaking and mixing. Fig. 1.10: Sketch of the pathway of internal-wave energy: from its origin, by the wind and by tidal °ow over topography, to dissipation as small-scale mixing. From 21. Figure 1.10 thus illustrates how energy is transferred to smaller scales. Sur- prisingly, this has important implications for the large-scale ocean circulation. For a large part, this circulation is wind-driven, but part of it consists in a sinking of cold water at high latitudes (deep convection), speci¯cally in the Labrador, Greenland and Weddell Seas; this water spreads horizontally over the ocean basins, hence the low temperatures in the deep layers at all latitudes. If this were the only factor determining the ocean's vertical temperature distri- bution, one would ¯nd low temperatures extending upwards until the ocean's most upper layer, where direct warming by the sun takes place. In reality, the temperature gradient is much more gradual (see Figure 2.2a). This shows that 19there must be downward mixing of heat (Figure 1.11). The combined e®ects of downwardmixinganddeepconvectionkeeptheoceaninastationarystate. The mixing is thought to be largely due to internal waves; at any rate, the numbers are consistent. The estimate of the required energy input into mixing is 2 TW; near-inertial waves and internal tides each contribute about 1 TW. Fig. 1.11: Near-inertial waves (not depicted), along with internal tides generated over bottom topography by the barotropic tidal current, feature in the deep ocean. These internal waves can lead to turbulence and mixing. This mixing plays a role in main- tainingagradualtransitionbetweenthesun-warmedsurfacelayeroftheoceanandthe upwelling cold, dense water formed at high latitudes. T(z) denotes the temperature pro¯le as a function of depth z. From 20. 1.5 Overview As a guide through later chapters, we may use Figure 1.9 and the questions it raises: \the facts which call for explanation". To answer most of these ques- tions, it su±ces to consider linear theory, i.e. the theory of small-amplitude internal waves (Chapters 5 and 6). This theory explains the remarkable diago- nalpropagationaswellasthere°ectionfromthebottom. ExaminingFigure1.9 more closely, we see that the beam becomes slightly steeper in deeper waters, i.e. refraction occurs; this, too, is explained by linear theory. At the origin of the beam lies the barotropic tidal °ow over a slope; this generation mechanism is studied in Chapter 7. Not visible in Figure 1.9 is what happens after the beam has re°ected from the bottom. Other observations, to be discussed later, show that the beam, with upward energy propagation, ¯nally impinges on the seasonal thermocline (in the upper 100 m of the water column); this gener- ates high-frequency high-amplitude internal waves, called internal solitons. To describe these waves, which are beyond the assumption of small amplitudes, nonlinear theory is required (Chapter 8). Firstofall,however,weneedtoestablishthebasicequationsofinternal-wave 20theory (Chapter 2), put the notion of strati¯cation in an exact form (Chapter 3), and discuss the approximations underlying internal-wave theory (Chapter 4). To do this properly, we also need to examine carefully the thermodynamic principles that form part of the governing equations. Further reading Althoughtheselecturenotesaremeanttobeself-contained,itisofcourseuseful to consult other literature as well; here we give some suggestions for further reading. More references follow in later chapters as appropriate. Most textbooks on ocean physics or dynamical meteorology pay some at- tention to internal waves. Three older textbooks deal exclusively with internal waves in the ocean: Krauss 47, Roberts 72 and Miropol'sky 58. Of these, thethird is the most advancedtext. The second is probably the most accessible and also provides an admirably complete reference list of the literature up to 1975. Alotofusefulmaterialoninternalwavescanbefoundinthetextbookby Leblond&Mysak48. Chaptersoninternalwavescanbefoundinthebooksby Turner 84, Phillips 67, and Lighthill 51; on gyroscopic (i.e. inertial) waves, see Greenspan 35. See Vlasenko et al. 88 for a recent monograph on the modelling of internal tides. The review papers by Garrett & St. Laurent on deep-ocean mixing 24 and by Garrett & Kunze on internal tides 23 provide a valuable account of the current understanding of these subjects. On short internal waves and internal-wave spectra, see the review paper by Munk 61. We focus on the ocean, and will only in passing discuss internal waves in the atmosphere. More on this subject can be found in the textbook by Gossard & Hooke 34, and the review paper by Fritts & Alexander 19. 2122Chapter 2 The equations of motion 2.1 Introduction In early 1913, Vilhelm Bjerknes gave his inaugural lecture at the University of Leipzig, which was titled \Die Meteorologie als exakte Wissenschaft" (Meteo- rologyasanexactscience). Init, hedrewattentiontothefactthatthephysical laws governing the motions of the atmosphere, together form a closed set; i.e. there are as many equations as unknowns. Meteorology, Bjerknes argues, has thus become an exact science. This o®ers the prospect, at least in principle, that a solution to the equations may be obtained, which would provide a math- ematical description, and even prediction, of the motions in the atmosphere 5. Fig. 2.1: Vilhelm Bjerknes (1862-1951), and the front page of his inaugural lecture. The variables in question are the three velocity components, pressure, den- sity, temperature and humidity (or, for the ocean: salinity). They feature in 23the following laws, seven in total: 1-3: the three momentum equations; 4: conservation of mass; 5: the equation of state; 6-7: the two laws of thermodynamics. If we consider the ocean instead of the atmosphere, the laws remain the same except the equation of state, which is speci¯c for the medium in question. We only brie°y discuss the laws 1-4, for they belong to the standard ma- terial in textbooks on (geophysical) °uid dynamics; they form the subject of Section 2.2. Much more attention needs to be paid to the equations relating to thermodynamics, 5-7, for two reasons. First, thermodynamic aspects are usu- ally dealt with cursorily in the oceanographic literature; as a result, neither the meaning nor the importance of thermodynamics is properly conveyed. Second, a recent development calls for a new approach. This development is the usage of the so-called Gibbs potential in ocean physics. At ¯rst, it may seem to make things more complicated, but once grasped, it brings out the structure of ther- modynamics,andthewaythermodynamicsenterstheequationsofmotion,more clearly than would otherwise be attainable. We discuss this in Sections 2.3 and 2.4. Finally, in Section 2.5, we arrive at the equations governing internal-wave dynamics. 2.2 Fluid mechanics The forces that feature in the momentum equations governing °uid motions are pressure gradients, gravity, frictional forces, and external forces. Together they determine the acceleration that °uid parcels undergo in coordinate sys- temswhichareatrestorinuniformrectilinearmotionwithrespectto`absolute space', the `¯xed stars'. The Earth, however, spins on its axis, and thus rotates with respect to the `¯xed stars'. If we use a coordinate system that co-rotates with the Earth, we have to add two apparent forces: the Coriolis and the cen- trifugal force. The momentum equations then become Du 1 = ¡ rp¡r© +F ¡2­£u¡­£(­£r): (2.1) g Dt ½ Here we use a right-handed orthogonal Cartesian coordinate system which has its origin at the centre of the Earth, the x and y axes span the equatorial plane, and the z axis pointstowardsthe northpole. TheEarth spinsatangular ¡5 ¡1 velocity ­=7:29210 rad s on the z axis, and ­=(0;0;­); the position of the °uid parcel is denoted by r =(x;y;z). The velocity ¯eld is denoted by the vector u; ½ is density; p pressure; © the gravitational potential; and F denotes g 24any frictional or external forces, which need not be speci¯ed here. D=Dt andr denote the material derivative and gradient: ³ ´ D = +u¢r; r= ; ; : Dt t x y z The centrifugal force, the last term on the right-hand side of (2.1), can be 1 2 2 2 writtenas(minus)thegradientofthepotential© =¡ ­ (x +y ), andhence c 2 we can write (bringing the Coriolis force, the penultimate term in (2.1), to the left-hand side, as is customary) Du 1 +2­£u = ¡ rp¡r(© +© )+F : (2.2) g c Dt ½ Thecoordinatesystemadoptedhereisinconvenientinthatwewouldratherhave the origin at the surface of the Earth, and the axes oriented to it in a natural way. Before this can be done, we have to ¯nd an appropriate representation of the shape of the Earth. On long geological time scales the Earth is not quite a solid object; rather like a °uid, it has adjusted itself to the state of rotation. It thus has taken an oblate shape such that the gradient of the geopotential © (i.e. the gravitational plus centrifugal potential: © = © +© ) has no components tangential to the g c surface. In other words, the surface of the Earth coincides with a level of constant geopotential. This is important for the dynamics of the atmosphere and ocean, since otherwise a °uid parcel would experience a tangential force duetothegeopotential. Thislevelofconstantgeopotentialcloselyresemblesan ellipsoid of revolution. The ellipticity is small (about 0.08), which suggests that we may pretend the surface of the earth to be spherical (with radius R¼ 6371 km). In this new representation, the Earth's surface (now a sphere) should act as a level of constant geopotential; otherwise the dynamics would become distorted. The equations of motion can now be cast in terms of spherical coordinates. A simpler form can however be obtained if the phenomena of interest are so smallthatthecurvatureoftheEarth'ssurfacebecomesinsigni¯cant; thisyields the so-called f-plane approximation. The corresponding equations can be de- rived either by employing a local approximation to the equations in spherical coordinates (see 48), or directly from (2.2) by moving the origin of the Carte- sian coordinate system to the position of interest (r = R, Á = Á , say), and 0 then tilting it such that the x,y plane becomes tangential to the Earth's sur- face (with x pointing eastward, y northward, and z positive in the outward radial direction). Since this new coordinate system is at rest with respect to theoriginalsystem, theequationsremainthesame, exceptthat ­nowbecomes ­ = ­(0;cosÁ;sinÁ), and the gradient of the geopotential r© = (0;0;g). The momentum equations thus become Du 1 +2­£u=¡ rp¡gz+F ; (2.3) Dt ½ 25where z is the unit vector in the z direction (positive, outward). The f-plane derivesitsnamefromthecommonnotation2­=(0;f;f),inwhichf =2­cosÁ and f =2­sinÁ are regarded constant. A following order of approximation would lead to the so-called ¯-plane, in which variations of the Coriolis parameter with latitude are taken into account while metric terms are still ignored. This amounts to replacing f by f +¯y, 0 with f constant and ¯ = (2­cosÁ)=R; to ensure the conservation of angular 0 momentum, f should be taken constant, as on the f-plane, see 38. To complete the mechanical part, we need an equation that expresses the conservation of mass; in local Cartesian coordinates it reads D½ +½r¢u=0: (2.4) Dt We now have four equations in total, but ¯ve unknowns: three velocity com- ponents, pressure, and density. Hence the set is not closed, unless density were simplyassumedtobeconstant(incompressible°uid). However,variationsinthe density ¯eld are essential to the existence of internal waves, so we have to com- plete the set in a di®erent way: by including thermodynamic principles, which provide relationships between thermodynamic state variables such as pressure and density. This will be elaborated on in the remainder of this chapter. 2.3 A brief introduction to thermodynamics This section provides a r¶esum¶e of thermodynamic principles, with a view to oceanphysicsandmeteorology. ItservesasapreparationforSection2.4, where the set of governing equations is completed. In line with common usage in thermodynamics, we use here speci¯c volume º =1=½, instead of density ½. 2.3.1 Fundamentals Thermodynamics deals with transitions from one thermodynamic state to an- other. In the simplest case, both are equilibrium states; states, that is, which would remain unchanged if the system were isolated. Equilibrium states are de- ¯ned by a certain number of thermodynamic state variables (such as pressure, temperature, density, internal energy or entropy). For example, the equilibrium state of an ideal gas is de¯ned by two such variables; pressure p and temperature T, say. All other state variables are then a function of those two; such a functional relationship is called an equation of state. Speci¯c volume º, for instance, is given by the following expression: R T d º(p;T) = ; (2.5) p 26inwhichR =R =m,wheremisthemass(inkg)of1mol,andR theuniversal d ¤ ¤ ¡1 ¡1 1 gas constant: R = 8:31 J K mol . The constant R thus depends on the ¤ d ¡1 ¡1 2 type of gas. For dry air one ¯nds R =287 J K kg . d Despitethefactthattheactualstateoftheocean,oratmosphere,considered in its entirety, is far removed from thermodynamic equilibrium, the concept of thermodynamic equilibrium still proves very useful in the geophysical context. This is because we can adopt the `local equilibrium assumption' 46, x15.1. Its meaningismosteasilygraspedbysupposingtheopposite: thatitwerenotvalid. This would be the case if speci¯c volume º were not only dependent on tem- perature and pressure, like in (2.5), but also on spatial gradients of temperature and pressure. This would call for an extended non-equilibrium thermodynam- ics. The `local equilibrium assumption' amounts to assuming that such spatial gradients are negligible; this allows us to apply, locally, the thermodynamic equilibrium relations, such as the equation of state. In thermodynamics two types of processes are distinguished: according to whether they are reversible or irreversible. In a reversible process all interme- diate states are equilibrium states, whereas in an irreversible process they are not. Strictlyspeaking, theformerisnotaprocess(sincetheonlywaytochange an equilibrium state is by bringing the system out of equilibrium), but rather a chain of disconnected equilibrium states. Nevertheless, it is often useful to consider quasi-static processes, as practical approximations to reversible pro- cesses, which take place slowly enough for the state to be always very close to thermodynamic equilibrium; one can then assume that the equation of state is valid throughout the process. In the applications, discussed here and in the following chapter, we assume this to be the case. Besides state variables, which characterize the state of a system indepen- dently of how it came into that state, there are also quantities heat and work which are the exact reverse in that they do not refer to a state, but to the way in which one state transforms into another. The First Law of thermodynamics connects the two; in it, the state variable (speci¯c) internal energy ² is postu- lated, which can change either by heat (dQ) or by work done on the system (dW): d²=dQ+dW : ¡1 All quantities are here taken per unit of mass, hence the dimension J kg . If, after some process, a system returns to its initial state (a cyclic process), then the First Law guarantees that ² takes again its original value; but in the course of the process heat may have been partly converted into work the principle of a heat engine. For quasi-static processes, the work done on the system can be expressed as dW = ¡pdº: IntheSecondLawanotherstatevariable, (speci¯c)entropy ´, ispostulated, 1 23 1 mol contains 6:02£10 molecules. 2 78% N , 21% O , 1% Ar, with respective molmasses (in grams) of 28, 32 and 40. 2 2 27

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