KUNDU Fluid Mechanics 2 Episode 7 pdf

We shall see in the next section that it has the significance of being the frequency of oscillation if a fluid particle is vertically displaced. Afier substitution of Eq. (7.136), the equations of motion (7.130)-(7.134) become (7.140) (7.14 I) (7.142) (7.143) (7.144) In deriving this set we have also used Eq. (7.135) and replaced the density equation by its linearized form (7.138). Comparing the sets (7.130X7.134) and (7.140b(7.144), we scc that the equatiuns satisfied by the yemi-barion densify and pressure are iden- tical to those satisfied & the totul p and p. In deriving thc equations for a stratified fluid, we have assumcd that p is a function of temperature T and concentration S of a constituent, but not of pressurc. At first this docs not seem to be a good assumption. The compressibility effects in the atmosphere are ccrtaiuly not negligible; even in the ocean the density changes due to the huge changes in the background pressure are as much as 4% which is 40 times the density changes due to the variations of the salinity and temperature. The effects of compressibility, however, can be handled within the Boussinesq approximation if we regard p in the defiilition of N as the background potential density, that is the density distribution from which the adiabatic changes of density due to the changes of prcssure have been subtracted out. The concept oi potential dcnsity is explained in Cliaptcr 1. Oceanographers account for compressibility effects by converting all their density measurements to thc standard atmospheric pressure; thus, when they report variations hi density (what thcy call “sigma lee”) they arc generally reporting variations due only to changes in temperature and salinity. A useful equation for stratified flows is the one involving only UL The u and li can be eliminated by taking the time derivative of the continuity equation (7.144) and using the horizontal momentum equations (7.140) and (7.141). This givcs (7.145) where Vi p’ from Eqs. (7.142) and (7.143) gives a’/ax2 + a2/ay2 is the horizuntal Laplacian operalor. Elimination o€ (7.14) Finally, p’ can be eliminated by laking Vi of Eq. (7.146), and using Eq. (7.145). This gives which can bc written as (7.147) where V’ a2/~x’ + a2/ay2 + iI2/3z2 = Vi + a2/az2 is the three-dimensional Laplacian operator. The w-equation will be used in the following section to dcrive the dispersion relation for internal gravity wavcs. 19. Iritwnal Mimes in a Continmu&- L9rati@d IYuid In this chapter we have considered gravity waves at the surface or at a density dis- continuity; these waves propagate only in the horizontal direction. Because every horizontal direction is alike, such waves are isotmpic, in which only the magnitude of thc wavenumber vector matters. By taking tbe x-axis along the direction of wave propagation, we obtained a dispersion relation w(k) that depends only on the mg- nitude of the wavcnumber. We found that phases and groups propagate in the same direction, although at Merent spccds. I€, on the other hand, the fluid is continuously suatified, then the internal wavcs can propagate in any direction, at any angle to the vertical. Tn such a case the direction of the wavenumber vector becomes important. Consequently, we can no longer treat the wavenumber, phase velocity, and group velocity as scalars. a2 -v2w + N’VAW = 0, at1 Any flow variable q can now be written as i(k.r+/p+nc-cut) - i(K x-cut) SI =90e - 90e where 40 is the amplilude and K = (k, I, m) is the wavenumber vector with com- ponents k, 1, and m in the three Cartesian directions. Wc cxpect that in this case the direction of wavc propagation should matter becausc horizontal directions are basi- cally differcnt from the vertical direction, along which the all-important gravity acts. Internal waves in a continuously stratified fluid therefore nnisotmpic, for which the fresuency is a function of all three components of K. This can be written in the following two ways: w = ~(k. 1. m) = o(K). (7.148) However, the waves are still horizontally isotropic because thc dependence of the wave field on k and I is similar, although the dependence on k and ni is dissimilar. The propagation of internal waves is a bamlinic process, in which the surfaces of constant pressurc do not coincidc with the surfaces of constant density. It was shown in Section 5.4, in connection with the demonstration of Kelvin’s circulation theorem, that baroclinic proccsses generate vorticity. InteinaX waves in a continuously stratiJied jluid are therefam not iirotutional. Waves at a density interface constitute a limiting cme in which all the vorticity is concentrated in the form of a velocity discontinuity rrt the integace. The Lrrplace equation can ther-efoiv be used bo describe thejiowfield within each Zayx However; internal waves in a continuous1.y szrutijied jhid cnnnot be described by the Lpluce equution. The first taqk is to derive the dispersion relation. We shall simplify the analysis by assuming that N is depth indcpendent, an assumption that may seem ~mmdistic at fist. hi the ocean, for example, N is large a1 a depth of %200 in and small elscwhere (see Figure 14.2). Figmz 14.2 shows that N .e 0.01 evcrywhere but N is largest between ~200 m and 2km. However, Ihc results obtained by treating N as constant ;ire locdly valid if N varies slowly over the vcrtical wavelength 25r/ni of the motion. The so-called WKB approximation of internal waves, in which such a slow variation of N(z) is not neglected, is discussed in Chaptcr 14. Consider a wave propagating in three dimensions, for which the vertical vcloc- ity is = u,,, ei(k.r+ly+m~-wr) (7.149) where tug is the amplitude of fluctuations. Substituting into the governing equation a2 -V2w + N'VAW = 0, at' gives the dispersion relation (7.147) (7.150) For simplicity of discussion we shall orient the xz-plane so as to contain the wave- number vector K. No generality is lost by doing this because the medium is hoii- zontally isotropic. For this choice of referencc axes wc have 1 = 0; that is, the wave motion is two dimensional and invariant in the y-direction, aid k rcpresents the eiilirc horizontal wavcnumber. We can then write Eq. (7.150) as w= kN kN JW = 7' (7.151) This is the dispersion relation for internal gravity waves and can also be writtcn a,, (7.152) i w = Ncost), where 6, is the anglc between the phase velocity vector c (and therefore K) and the horizontal direction (Figure 7.32). It follows that the Ircquency of an intcmal wave in a stratified fluid depends only on the direction of the wavenumber vector and not on the magnitude of the wavcnumber. This is in sharp contrast with surface and interfacial gravity waves, for which frequency depends only on the magnitude. The frequency lies jn thc range 0 .c w -= N, revealing one important significance of the buoyancy Iequency : N is the mmimirm possible fi-equeiicy if iiiteinul waves in a strutified.fluid. Before discussing the dispersion relation further, let LIS explore particle motion in an incompressible internal wave. Thc fluid motion can be written as 1 (7.153) = ug ei(kx+l.v-mz nrt) k COS0 = - K I/ k /I K and e E’igiu-e 7.32 Basic parameters olinicrnal waves. Note that e and c, are at right angles md havc opposite veaicnl components. plus two sitiiilar expressions .for u and M’. This gives - ikuo ei(tx+ly+nrz-wr) - - - iku. au - ax Thc continuity equation then requires that ku + Iv + niui = 0, that is, (7.154) showing that pcirticle motion isperpeizdicular to the wuvenzmber vector (Figure 7.32). Note that- only two conditions have been used to derive this result, namely the incom- pressible continuity equation and a trigonometric behavior in ull spatial directions. As such, thc rcsult is valid for many other wavc systems that meet these two conditions. These waves are called shear wuves (or transverse waves) because the fluid moves parallcl to the constant phase lines. Surface or interfacial gravity wa17es do not have this property because the field varies exponenriafly in the vertical. We can now intcrpret 8 in thc dispersion relation (7.152) as the angle bctween the phclc motion and the vertical direction (Figure 7.32). The maximum frequency w = N occurs when 8 = 0, that is, when the particles move up and down vertically. This case cornsponds lo m = 0 (sce Eq. (7.15 1)). showing that the motion is independent of the z-coordinate. Thc resulting motion consists of a series of vertical colurmis, all oscillating at the buoyancy frcquency N, the flow field varying in thc horizontal direction only. 4 Figure 7.33 Blocking in shrmgly sb-&licd flow. Thc circular region represents a two-dimensional body with its axis along Ihe y direction. The w = 0 Limit At the opposite cxtreme we have w = 0 when 8 = x/2, that is, when the particle motion is completely horizontal. In this limit our inkrnal wave solution (7.151) would seem to rcquire k = 0, that is, horizontal independcnce of the motion. However, such a conclusion is not valid; purc horizontal motion is not a limiting ca$e of internal waves, and it is necessary to examine the basic equations to draw any conclusion lor this case. An examination of the governing set (7.1.40)-(7.144) shows that a possible steady solution is w = p' = p' = 0, with u aid v any functions of .r and y satisfying (7.155) The z-dependence of u and v is arbitrary. The motion is thercfore two-dimensioixd in the horizontal plane, with the motion in the various horizontal planes decoupled from each othcr. This is why clouds in the upper atmosphere seem to move in flat horizontal sheets, as often observed in airplane flights (Gill, 1982). For a similar I-eason a cloud pattern pierced by a mountain peak soinetimes shows Kurman vurrex streets, a two-dimensional feature; see ihe striking photograph in Figure 10.18. A i-eslriction of strong stratification is necessary for such almost horizontal flows, for Eq. (7.143) suggests that the vertical motion is small if N is large. The forcgoing discussion leads to the interesting phenomcnon of blocking in a strongly stratified fluid. Coiisidcr a two-dimensional body placed in such a fluid, with its axis horizontal (Figure 7.33). The two dimensionality of the body requires av/8y = 0, so that Ihc continuity Eq. (7.155) rcduces to au/ax = 0. A horizontal layer of fluid ahcad af thc body, bounded by tangcnts abovc and below it, is therefore blocked. (For photographic evidencc see Figure 3.18 in the book by ?inner (1973).) This happens bccause lhc strong stratification suppresses the M: ficld and pi-events the fluid horn going around and over thc body. In the casc of isotropic gravity wavcs at a free surface and at a density discontinuity, we found hat c and c, are in the same dircction, although their rnagnitudcs can bc diflerent. This couclusioii is no longer valid for thc anisoiropic intcrnal wavcs in a continuously stratified fluid. In fact, as we shall see shortly, lhcy are peipendiculur to each olhcr, violating all our intuitions acqilired by obscrving surface gravity waves! In three dimensions, the dcfinition cg = dw/dk has to be generalized to (7.156) where i,. i,, i, are the unit vectors in the three Cartesian dimtions. As in the preceding section, we orient thc sz-plane so that the wavcnumber vecmr K lies in this plane and 1 = 0. Substituting Eq. (7.151). this givcs Nm c - -(i,m -iik). g- K3 The phase vclocitv is (7.1 57) (7.158) whmc K/ K represents the unit vcctor in the direction of K. (Note that c # i, (m/k) + iL(w/ntj, as explained in Section 3.) It lbllows fromEqs. (7.157) and (7.158) that (7.159) showing thal phase and group velocity veclors are peipeidiculai: Equations (7.157) and (7.1 58) show that the horizontal components of c and cg are in the same direction, while thcir vcrtical components are equal and opposite. In fact, c and cg form two sides of a right-angled triangle whose hypotenuse is horizontal (Figuie 7.34). Consequently. thc phase velocity has an upward component when thc pup velocity has a downward component, and vice versa. Equations (7.154) and (7.159) are consistent because c and K are parallel and cg and u are parallel. The fact that c and cg arc pcrpendicular, and havc opposite vertical components, is illustrated in Figure 7.35. It shows that the phasc lines are propagating toward the left and upward, whereas the wave groups are propagating to the left and downward. Wave cmsts are constantly appearing at one cdge 01 the group, propagating through the g~vup, and vanishing at the other cdge. The group velocity here has the usual significance of being the velocity ofprop- agation of energy of a certain sinusoidal Component. Supposc a source is oscillating at 1requcncy w. Thcn its energy will only be found radially outward along four beams C Figure 7 W Oricnhtion ofphc md gmup velocity in inkriirl wavcs. Figurr! 7.35 Illustration of phase and group propagation in internal waves. Positions of a wave group at two timcs are shown. Thc phase line PF’ at time tl propagates to PP at tz. oriented at an angle 0 with the vertical, where cos 8 = o/N. This has been verified in a laboratory experiment (Figure 7.36). The source in this casc was a vertically oscillatiiig cylinder with its axis perpendicular to the planc of paper. The €Teguency was w < N. The light and dark lines in the photograph are lines of constant density, inade visible by an optical technique. The experiment showcd that the cnergy radiated along four beams that became morc vertical as the frequency was increased, which agrees with cos0 = o/N. 23. Knergy C‘omsideradiorix of lirlcinal Maues in a Stra@kd Fluid In this section we shall derive the various cormnonly used expressions for potential energy of a continuously stratified fluid, and show that they are equivalent. We then show that the encrgy flux p‘u is cg times the wave energy. A mechanical energy equation for internal waves can be derived from Eqs. (7.140)-(7.142) by multiplying the first equation by pou, the sccond by pov, the third by fiw, and summing the results. This gives $& + v2 + w2) + gp‘w + v (p’u) = 0. (7.160) 1 Hem the continuit yequation has beenused to write u ap’/ax+v ap’/iIy+w = V (p’u), which reprcsents thc net work done by pressure €orces. Another interpreta- tion is that V - (p’u) is the divergence of the enerKyJIux p’u, which inust change the 21. Energy coneidcrniionv of Internal W- in a Stra@fkd Fkid 25 1 r Figure 73 Waves generated in a stratified fluid of uniform buoyancy frequency N = 1 rad/s. The forcing agency is a horizontal cylinder, with its axis perpendicular to the plane of the paper, oscillathg vertically at frequency w = 0.71 rads. With w/N = 0.71 = cos8, this agrees with the observed angle of 8 = 45" made by the beams with the horizontal. The vertical dark line in the upper half of the photograph is the cylinder support and should be ignored. The light and dark radial lines represent contours of constant p' and are therefore constant phase lines. The schematic diagram below the photograph shows the directions of E and e, for the four beams. Reprinted with the permission of Dr. T. Neil Stevenson, University of Manchester. wave energy at a point. As the first term in Eq. (160) is the rate of change of kinetic energy, we can anticipate that the second term gp’w must be the rate of changc of potential energy. This is consistent with the energy principle derived in Chapter 4 (see Eq. (4.62)), except that pJ and p’ replace p and p because we have subtracted the mean state of rest here. Using the density equation (7.143), the rate of change of potential energy can be written as (7.161) which shows that the potential energy per unit volume must be the positive quan- tity E,, = gZpR/2poN2. The potential energy can also be expressed in terms of the displacement f of a fluid particle, given by w = a(/at. Using the density equation (7.143), we can write _- apt N2Po af at g at’ which requires that N%f pJ = g The potential energy per unit volume is therefore (7.162) (7.163) This expression is consistent with our previous result from Eq. (7.106) for two infinitely deep fluids, for whichthe average potential energy of the entire wakrcolumn per unit horizontal area was shown to be &?2 1 - P1)6a2, (7.164) wherc the interface displacement is of the form f = a cos(kx - ob) and (pz - p1) is the density discontinuity. To see the consistency, we shall symbolically represent the buoyancy frequency of a density discontinuity at z = 0 as (7.165) where S(z) is the Dirac delta function. (As with other relations involving the delta function, Eq. (7.165) is valid in the integral sense, that is, the integral (across the origin) of the last LWO terms is cqual because S(z) dz = 1.) Using Eq. (7.165), a vertical integral of Eq. (7.163), coupled with horizontal averaging over a wavelength, gives Eq. (7.164). Nok that for surface or interfacial waves Ek and E, represent kinetic and potential energies o€ the entire water column, per unit horizontal area. In a continuously stratified fluid, lhcy represent energies pr unit volume. We shall now demonstrate that the average kinetic and potential energies are equal for internal wave motion. Substitute periodic solutions [u, UI, pJ, p’] = [i, 6, jj, 81 ei(kx+mz-cur). Thcn all vari:tblcs can be expmsed in terms of M’: p’ = wn’m ,j, ci(kx-tn,ni-wr) k3 I (7.166) u = ’’ ,jj ei(kr+m;-or) k where y’ is derived from Eq. (7.145), p‘ from Eq. (7.143), and 26 from Eiq. (7.140). The averagc kinetic energy pcr unit volume is therefore (7.167) whcre we have uscd the fact that the average of cos2 x over a wavclcngth is 1/2. Thc avcrage potential ciiergy per unit volume is (7.168) whcrc we have usccl p” = 6’ N4p,2/2w2g2, found from Eq. (7.166) after taking its real part. Use of thc dispersion rclation w2 = k2N2/(k2 + m’) shows that Ek = Ep. (7.169) which is a general result for small oscillations of a conservative system without Coriolis forces. The total wave cncrgy is (7.170) Last. we shall show that e, times thc wave encrgy equals thc energy flux. The average eitci-gy flux across a unit arca can be found hn Eq. (7.166): (7.1 71 j Using Eqs. (7.157) and (7.170). group velocity times wave energy is Nni K3 c,E = -[[i,m - i;k] which reduces to Eq. (7.171) on using the dispcrsion relation (7.1Sl). Tt follows that (7.172) I I F=c,E. . I This result also holds for surface or intcrfacial gravity waves. However, in that case F reprcsents the flux per unit width pcrpendicular to the propagation direction (inte- grated over thc cnlire depth), and E represents the energy per unit horizontal area. In Eq. (7.1 72), on die othcr hand, F is die flux per unit ma, and E is the encrgy per unit volume. 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This givcs (7. 145) where Vi p’ from Eqs. (7. 1 42) and (7. 143) gives a’/ax2 + a2/ay2 is the. o€ (7. 14) Finally, p’ can be eliminated by laking Vi of Eq. (7. 146), and using Eq. (7. 145). This gives which can bc written as (7. 1 47) where V’ a2/~x’ + a2/ay2 + iI2/3z2 =