KUNDU Fluid Mechanics 2E Episode 15 doc

604 thaphpkal lhid I?y~nunk#i time corresponding to ut = 0, n/2, and n. It is clear that the horizontal hodographs are clockwise ellipses, with the major axis h the direction of propagation x, and the axis ratio is f/o. The same conclusion applics for the lower signs in Q. (14.1 10). The particle orbits in the horizontal plane arc therefore identical to those of Poincark waves (Figure 14.16). However, the plane of the motion is no longer horizontal. From the velocity components Eq. (1 4.1 OS), we note that 11 m - = 7- = 7 tanf3, W k (1.4.1 11) wherc 6, = Lan-'(m/k) is thc angle made by the wavenumbcr vcctor K with the horizontal (Figure 14.24). For upward phase propagation, Eq. (1.4.1 1 1) gives u/w = -tanO, so that w is negative if u is positive, as indicated in Figurc 14.24. A three-dimensional sketch of the particle orbit is shown in Figure 14.23b. It is casy to show (Exercise 6) that the phase velocity vector c is in the direction of K, that c and cg arc perpendicular, and that the fluid motion u is parallel to e,; these facts are dernonstratcd in Chapter 7 for internal waves unaffected by Coriolis forces. The velocity vector at any location rotates clockwise with time. Because of thc vertical propagation of phasc, the tips of the instuntuneous vectors also turn with depth. Consider the turning of the velocity vcctors with depth when the phase velocity is upward, so that the deeper currents have a phase lcad over the shallower currents (Figure 14.25). Because the currents at all depths rotate clockwise in rime (whether the vertical component of c is upward or downward), it follows that the tips of the instantaneous velocity vectors should fall on a helical spiral that turns clockwise with depth. Only such a turning in dcpth, coupled with a clockwise rotation of the velocity vectors with time, can result in a phase lead or Lhe deeper currents. In the opposite t, Figure 14.24 Vertical section of an intcmal wavc. Thc hc pudcl hcs uc conrihn~ phisc hcu, with the arrows indicating fluid motion along [he lines. Figure 14.25 Helical spiral traced out by thc lips olinstanltlncous vclocity vectors in an internal wavc with upward phasc speed. IIeavy arrows show the velocity oeclnrs a1 two dcplhs, and light mws indicate .hat thcy arc roltlling clockwisc with he. Note that the instantaneous vectors turn clockwisc with depth. casc of a downwurd phase propagation, the helix turns counterclockwise with dcpth. The direction of turning of the velocity vectors can also be found from Eq. (I 4.1 OS), by considering x = t = 0 and finding u and u at various valucs of z. Discussion of the Dispersion Relation Thc dispcrsion dation (1 4.109) can be written as 2 k2 m2 w - .f2 = -(N2 - w2). (14.112) Tnhoducing tan 8 = m/ k, Eq. (I 4.1 12) becomes w2 = f2 sin20 + N~ cos28: which shows that w is a function of the angle made by the wavenumber with the horizontal and is not a function ofthe magnitude of K. For f = 0 the forementioned expression reduces to w = N cos 8, dcrivcd in Chapter 7, Section 19 without Coriolis forces. A plot of the dispersion relation (14.1 12) is presented in Figure 14.26, showing II) as r? function of k for various values of m. All curvcs pass through the point w = f, which represents inertial oscillations. rnically, N >> f in most of the atmosphere and the ocean. Because of thc widc scparation of the. upper and lower limits of the internal wave rangc f < w < N. various limiting cases are possiblc, as indicatcd in Figure 14.26. They are (1) Highfrequency regime (w - N, hut w < N): In this range f2 is negligible in comparison with w2 in the denominator of the dispcrsion relation (14.:I.W), 1 high frequency (nonrotating) mid frcqucncy (hydrostatic, nonrotaling) low frequency (hydrostalk) Figure 14.26 Dispersion relation for internal wavcs. Thc dillkent regimes are indicakd on thc lefi-hand side of the figure. which reduces to N2k2 , that is, w 2: - m2+k2' k2(N2 - &) w2 mcx Using tan 8 = m/k, this gives w = N cos 8. Thus, the high-frequency inter- nal waves are the samc as the nonrotating internal waves discussed in Chapter 7. hw-jkquency regime (w - f, but o 2 f ): In this range o2 can be neglected in comparison to N2 in the dispersion relation (14.109), which becomes k2N2 that is, w2 21 f2 + k2N2 m cx- "2- f29 m2 Thc low-frequency limit is obtained by making the hydrostatic assumption, that is, neglecting awlat in the vertical equation of motion. Midfrequency regime (f << w << N): In this range the dispersion relation (14.109) simplifies to k2N2 m 2- 02 ' so that both the hydrostatic and the nonrotating assumptions are applicable. Lee Wave Internal waves arc frequently found in the "lee" (that is, the downstream side) of mountains. In stably stratified conditions, the flow of air over a mountain causes a vertical displacement of fluid particles, which sets up intcmal waves as it moves downslrezun of the mountain. If the amplitude is large and the air is moist, the upward motion causes condensation and cloud formation. Due to the effect of a mean flow, the lee waves are stationary with respect to the ground. This is shown in Figure 14.27, where the westward phase speed is cancelcd 14. lniernal WUMU 607 Pigure 1427 Slrcamlincs in a lee wavc. Thc Lhin line drawn through crests shows that Ihc phase pmpa- gates downward and westward. by the eastward mean flow. We shall detcrmine what wave parameters make this cancellation possible. The frequency of lee waves is much larger than f, so that rotational effects are negligible. The dispersion relation is thercfore N2k2 m2 + k2' w2 = - (14.1 13) Howevcr, we now have to introduce the effects of the mean flow. The dispersion relation (1 4.1 13) is still valid if w is intcrpreted as the intrinsichquency, that is, the frequency measured in a frame of refcrence moving with the mean flow. In a medium moving with a velocity U, the observed frequency of waves at a fixed point is Doppler shifted to where w is the intrinsic .frequency; this is discussed further in Chapter 7, Section 3. For a stationary wavc q) = 0, which requires that the intrinsic frcquency is w = -K U = kU. (Here -K U is positive because K is westward and U is castward.) The dispersion relation (1 4.1 13) tbcn gives If the flow speed U is given, and the mountain introduces a typical horizontal wavenumber k, then the preceding equation determines the vcrtical wavenumber m that gencrates stationary waves. Waves that do not satisfy this condition would radiate away. The energy source of lee waves is at the surface. Thc energy thcrefore must prop- agate upward, and conwquently the phases propagate downward. The intrinsic phase spced is thercfore westward and downward in Figurc 14.27. With his information, we caa detcrmine which way thc constant phase lincs should lilt in a stalionary lee wave. Now that the wave pattern in Figure 14.27 would propagate to the left in the 608 (hph+ul Fluid I~umim absence of a mean velocity, and only with the constant phase lines tilting backwards with height would the flow at larger height lead the flow at a lower hcight. Further discussion of internal waves can be found in Phillips (1 977) and Mu& (1981); lee waves are discussed in Holton (1979). 15. Rmsby Waw To this point we have discussed wave motions that are possible with a constant Coriolis liequency f and found that these waves have fiequcncies larger than f. We shall now consider wave motions that owe heir existence to thc variation of f with latitude. With such a variable f, the equations of motion allow a very important type of wavc motion called the Rossby wavc. Their spatial scales are so large in thc atmosphere that they usually have only a few wavelengths around the entire globe (Figure 14.28). This is why Rossby waves are also called planetary waves. In the ocean, however, their wavelengths are only about 100 km. Rossby-wave .hquencics obey the inequality w << f. Because of this slowness the time derivative terms are an order of mag- nitude smaller than the Coriolis forces and the pressure gradients in the horizontal Figure 14.28 Ohscrved hcight (in decamckm) of tbe 50 kF'a prcrsure surface in thc norzhcrn hemi- sphcrc. The ccnter or the piciurc reprcrjcnts thc north pole. Thc undulutions arc due LO Rossby waves (dm = WIOO). I. T. Houghton, The Physics oj'the Atmosphere, 1986 and reprintcd with Ihc permission ol' Cambridge University Press. cquations of motion. Such nearly geostrophic flows are cdlcd quasi-geusrrophic motions. Quasi-Ciostrophic Vorticity Equation We shall first derivc the governing equation for quasi-geostrophic motions. For sim- plicity, wc shall makc the customary pplane approximation valid for By << .fo, keep- ing in mind that the approximation is not a good one for atmospheric Rossby waves, which havc planetary scales. Although Rossby waves are frequently supcrposed on a mean flow, we shall derive he equations without a mean flow, and superpose a uniform mean flow at thc end, assuming thal thc perturbations are small and that a lincar superposition is valid. The first step is to simplify the vorticity equation for quasi-geos3ophic motions, assuming that the vebcit): is geoutmphic tu the lowest order. The small departures from gcostrophy, however, arc important because they determine the evolution of the flow with time. We start with tbc shallow-water potential vorticity equation which can bc written as We now expand the matcrial derivativc and substitute h = H + 17, where H is the uniform undisturbed depth of the layer, and q is the surface displaccment. This gives (14.1 14) Here, wc have used Dj/Dt = v(d.f/dy) = Bv. We have also replaccd f by .fn in thc second term bccause the /I-planc approximation neglects the variation of f except when it involvcs df/dy. For small perturbations we can neglect the quadratic nonlinear terms in Eq. (14.114)$ obtaining (14.1 15) This is the linearizcd form of the potential vorticity equation. Its quasi-geostrophic ver- sion is obtained if wc substitute the approximatc geostrophic cxpressions lor vclodty: (14.1 16) From this the vorticity is found as f=- 6 (a% -+- ;;) : .fo axz so that the vorticity equation (14.1 15) becomes Denoting c = a, this becomes (14.117) This is the quasi-geostrophic form of the linearized vorticity equation, which governs the flow of large-scale motions. The ratio c/fo is recognized as the Rossby radius. Note that wehavenot set av/at = O,inEq.(14.115)duringthederivationofEq. (14.117), although a strict validity of the geostrophic relations (14.1 16) would require that the borizontal divergence, and hence aq/at, be zero. This is because the departure from strict geostrophy determines the evolution af the flow described by Eq. (14.117). We can therefore use the geostrophic relations for velocity everywhere except in the horizontal divergence term in the vorticity equation. Dispersion Relation Assume solutions of the form We shall regard w as positive; the signs of k and I then determine the direction of phase propagation. A substitution into the vorticity equation (14.1 17) gives I I k2 + Iz + ft/c2' i3k I ( 14.1 1 8) This is the dispersion relation for Rossby waves. The asymmetry of the dispersion relation with rcspect to k and I signifies that the wavc motion is not isotropic in the horizontal, which is expected because of the j?-effect. Although we have dcrived it for a single homogeneous layer, it is equally applicable to stratified flows if c is replaced by the corresponding intenzul value, which is c = for the reduced gravity model (see Chapter 7, Section 17) and c = NH/nn for the nth mode of a continuously stratified model. For the barompic mode c is vq large, and f-/c2 is usually negligible in the denominator of Eq. (14.1 18). The dispersion relation w(k, I) in Eq. (14.118) can be displayed as a surface, cslking k and X along the horizontal axes and w along the vertical axis. The section of this surface along I = 0 is indicated in the upper panel of Figure 14.29, and sections of the surface for three values of w are indicated in the bottom pancl. The contours of constant w are circles because the dispersion relation (14.118) can bc written as 1=0 cgx>o I cgxco I I -3 -2 -1 kc& of0 = 0.2 -3 -2 -1 /I I i- WO Pc 0.5 nondispersive region 0 IC - fo 2 1 kc fo k'igure 142Y Dispersion rclation f~~(k. I) lor a Rorsby wave. The upper panel shows fr) vs k lor 1 = 0. Rcgions ol' posilivc and ncgntivc pup velocity cRx are indicated. Thc lowcr pancl shows n plan vicw of the surface m(k. I), showing conlours olconsiant w on a kl-plane. The values of ofo/,%: for the thrcc circlcx are 0.2, G.3, and 0.4. Amws perpendicular to contours indicatc directions or group vclocity vcctor E*. A. E. Gill, Armfh.;phcn~-Or.cun Dynamics, 1982 und rcprintcd wilh the permission of Academic 1'1~s and Mn. Helen Saunders-Gill. The definition or group velocity Bw .Bw c, = i- + J- ak 31' shows that the group velocity vector is the gradient or win the wavenumber space. Thc dircction of cg is thcrefore perpendicular to the w contours, as indicated in the lower panel of Figurc 14.29. For I = 0, the maximum .frequency and zero group speed are attained at kc/Jo = - 1 , comsponding to % fo/Bc = 0.5. Thc maximum frequency is much smaller than the Coriolis frcquency. For examplc, in the ocean the ratio ~,,,~~/.fo = 0.5#?c/fi is of order 0.1 for the barotropic mode, and of order 0.001 for a baroclinic mode, taking a typical rnidlalitudc value of fo - 1 0-4 s ' , a barotropic gravity wave speed of c - 200 m/s, and a baroclinic gravity wave spccd of c - 2 m/s. The shortest period of midlatitude baroclinic Rossby waves in the ocean can therefon be more than a ycar. The eastward phase speed is (14.1 19) The negative sign shows that the phase propagation is always westward. Thc phase spcedrcaches amaxhum when kZ+Z2 + 0, comsponding to very large wavelengths represented by the region near the origin of Figure 14.29. In this rcgion the waves are nearly nondispersive and have an easlward phase speed With = 2 x lo-" m-I s-l, a typical baroclinic value of c - 2m/s, and a mid- latitude value of fo - lo4 s-l, this gives c, - m/s. At these slow speeds thc Rossby waves would takc ycars to cross the width of the ocean at midlatitudes. The Rossby waves in the Ocean are therefore more important at lower latitudes, where hey propagatc faster. (The dispersion relation (14.1 18), howevcr, is not valid within a latitude band of 3" from the equator, for then the assumption of a near geoslrophic balance breaks down. A Merent analysis is needed in the tropics. A discussion of the wave dynamics of thc tropics is given in Gill (1982) and in the review paper by McCreary (1 985).) In the atmosphere c is much larger, and consequently the Rossby waves propagate hkr. A typical large atmospheric disturbance can propagate aq a Rossby wave at a speed of several meters pcr second. Frcqucntly, the Rossby waves are superposcd on a strong eastward mean current, such as the atmospheric jet stream. If U is thc speed of this eastward current, then thc observed eslslward phase speed is B k2 + l2 + :ji/c2 ' c,=u- (14.120) Stationary Rossby waves can therefore form when the eastward cmnt cancels the westward phase spccd, giving c, = 0. This is how stationary waves are formed down- stream of the topographic step in Figure 14.21. A simple expression for thc wavelength results if we assume 1 = 0 and the flow is barotmpic, so that f,'/c' is negligible in m. (14.1.20). hi^ gives u = p/kZ lor stationary solutions, so-thzlt the wavelength is 2nm. Finally, notc that we have been rather cavalier in deriving the quasi-geostrophic vorticity equation in this section, in thc sense that we have substituted the approximate geostrophic cxpressions for velocity without a formal ordering of the scales. Gill (I 982) has given a more precise derivation, cxpandjng in terms of a small paramem. Another way to justify the dispersion rclation (14.1 18) is to obtain it fiom the general dispersion rclation (14.76) derived in Section 10: w3 - c20(k’ + 12) - .fi;w - c2Bk = 0. (14.1 2 1) For w << f, the first term is negligible compared to the third, reducing Eq. (14.121) to Eq. (14.1 18). 16. Bumhpic lnxtabilily In Chaptcr 12, Scction 9 we discussed the inviscid stability of a shear flow U(y) in a nonrotating system, and demonstrated that a necessary condition for its instability is that d2U/dy2 must change sign somewhere in the flow. This was called Rayleigh’s pin1 of injlecrion criterion. In terms of vorticity 4 = -dU/dy, the criterion states that di/dy must change sign somewhere in the flow. We shall now show that, on a rotating earth, the criterion requires that d(i + f)/dy must change sign somewhere within the flow. Consider a horizontal current U (4’) in a medium of uniform density. In the absence of horizontal density gradients only the barotropic mode is allowed, and U(y) does not vary with depth. The vorticity equation is (1 4.1 22) This is identical to the potential vorticity equation D/Dr[(C + f )/h] = 0, with the added simplification that the layer depth is constant because 111 = 0. Lct thc total flow be decomposed into background flow plus a disturbance: u = U(y) + u’, I v=li. The total vorticity is then wherc wc havc dcfined the perturbation streamfunction , u= I a+ w u’ = -_ ily ax Substituting into Eq. (14.122) and linearizing, we obtain the perturbation vorticity cquaticin (14.123) [...]... Khines, P B (1975) “Waves and turbulence on B p-planc.”Juurnal oJFluidMechanics 6Y 417443 Taylor, G I (1 915) .“Hrldy motion in Ihc almosphcrc.” Philusophicul Tmnsacrifmqfrhe Riiyul Society o f London A 215 1-26 Williams, G P (1979).“Planclary circulalions: 2 Thc Jovian quaui-geostrophic regime.” Journal ofAlmospheric Sciences 3 6 932-968 Chapter 15 Aerodynamics 629 630 Control slaraCcs 632... (Figure 15. 3) direction of flight wing tip Figure 15. 2 Wing planform geometry L pitch axis Figure 15. 3 Aircraft axcs 632 Aervd-niw Control Surfaces The aircraft is controlled by the pilot by moving certain control surfaces described in the following paragraphs Aileron:Thesc are poaions of each wing near the wing tip (Figure 15. 1).joined to the main wing by a hinged connection, as shown in Figure 15. 4... counterclockwisein Figurc 15. 10, which means that it musl leave behind a clockwise circulation around thc airfoil To see this, imagine that the fluid is stationary and the airfoil is moving to the left Consider a material circuit ABCD,made up of the same fluid particles and large enough to enclose both thc initial and final locations of the airfoil (Figure 15. 1 1) Initially Folledup shear layer Figure 15. 10 Formation... of undisturbcd Bight and a drag.forceD in the direction 01 flight (Figure 15. 7) T stcady lcvcl flight the drag is balanced by the thrust of n the engine, and the lift equals the weight of the aircraft These forces are exprcsscd , , mailing edge Figure 15. 6 Airli)il gc:)mctry l 6 i 15. 7 Forces on an airfoil -2 -1 0 Figure 15. 8 Distribution of the pressurc coefficient ovcl an airfoil Thc upper plrncl... that it can be rotated downward to incrcase h e lift (Figure 15. 5) A further €unction of the flap is to increase both lift and drag during landing Modemjet transports also have “spoiled’ on the top surface of each wing Whcn raised slightly, they separate the boundary layer early on part of the top of the wing h Figure 15. 4 The ailcmn Fiyre 15. 5 The flap and this decreaTes its lift They can be dcploycd... relevant here It is assumed that the readcr is familiar with that chapter 2 TlieAhvraJl and lix Conlmlx Alhough a book on fluid mechanics is not the proper place for describing an aimaft and its controls, we shall do this here in the hope that the reader will find it interesting Figure 15. 1 showsthree views of an aircraft The body oCthe aircraft, which houses the passengers and other payload, is called... if y < 180’ and infinite if y > 180’ (see Figure 6.4) In thc upper two panels of Figure 15. 9 the fluid goes from the lower to the upper side by turning around the trailing edge, so that y is slightly less than 360“ The resulting vclocity at the trailing edge is therefore infinitc in the uppcr two pancls of Figurc 15. 9 In thc bottom panel, on the othcr hand, the trailing edge is a stagnationpoint hccausc... Consider an airfoil starting f o rest in a real fluid The flow immediately after starting is irrolalional everywhere, bccause the vorticity adjacent to the surface has not yet diffuscd outward The velocity at this stage has a near discontinuity adjaccnt to the surface The flow has no circulation, and resembles the pattern in the upper panel of Figure 15. 9 The fluid goes around the trailing edge with a very... w’p’ A negative w’p’ means that on the average h e lightcr fluid rises and the heavier fluid sinks By such an inkrchangc thc center of gravity of the system, and therefore its potential energy, is lowered The interesting point is that this cannot happen in a stably stratified system with horizontd density surfaces; in that case an exchange of fluid particles raises the potential energy Moreover, a basic... exchange of fluid particles takes lighter particles upward (and northward) and denser particles downward (and southward) Such an interchange would tend to make the density rm surfaces more horizontal, releasing potential energy f o h e mean density field with a consequent growth of the perturbation energy This type of convection is called sloping convection According to Figure 14.33 the exchange of fluid . 1940s by Bjerknes et af. and is considered one of the major triumphs of geophysical fluid mechanics. Our presentation is essentially based on the review article by Pedlosky (1971) average of w’p’ by w’p’. A negative w’p’ means that on the average he lightcr fluid rises and the heavier fluid sinks. By such an inkrchangc thc center of gravity of the system, and. stratified conditions, the flow of air over a mountain causes a vertical displacement of fluid particles, which sets up intcmal waves as it moves downslrezun of the mountain. If