KUNDU Fluid Mechanics 2 Episode 14 pdf

with height is due to the absorption of solar radiation within the upper layer of the occan. Thc dcnsity distribution in the ocean is also affected by the salinity. However, there is no characteristic variation of salinity with depth, and a decrease with depth is found to be as common as an increase with depth. In most cases, however, the vertical siructurc of density in the ocean is determined mainly by that of temperature, the salinity effects being secondary. The upper 50-200m of ocean is well-mixed, due to thc turbulence generated by the wind, waves, current shear, and the convcctive ovcrturning caused by surface cooling. The temperature gradients decrease with depth, becoming quite small below a depth of 1500 m. There is usually a large temperature gradient in the depth range of 100-500 m. This layer of high stability is called the thermocline. Figure 14.2 also shows the profilc of buoyancy frequency N, defined by where p of course stands for the potential density and po is a constant reference density. The buoyancy frequency reaches a typical maximum value of Nmax - 0.01 s-l (period - lOmin) in the thermocline and decreases both upward and downward. In this section we shall review the relevant equations of motion, which are derived and discussed in Chapter 4. The equations of motion for a stratified medium, observed in a system of coordinates rotating at an angular velocity P with respect to the “ked stars,” are v*u=o, - Du +2Q x u = Vp 1 - -k+F, gP Dt Po Po (14.2) DP - =ol Dt where F is the friction force per unit mass. The diffusive effects in the density equation are omitted in set (14.2) because they will not be considered here. Set (14.2) makcs the so-calledBr~ussinesy approximation, discussed in Chapter 4, Section 18, in which the density variations are neglected everywhere cxcept in the gravity term. Along with other restrictions, it assumes that the vertical scale of the motion is less than the “scale height” of the medium c2/g, where c is the speed of sound. This assumption is very good in the ocean, in which c2/g - 200lan. In the atmosphere it is less applicable, because c2/g - 1Okm. Under the Boussinesq approximation, the principle of mass conservation is expressed by V u = 0. In contrast, the density equation DplDt = 0 follows fromthe nondiffusive heat equation DTIDt = 0 and an incompressible equation of state of the form Splpo = -cwST. (If the density is determined by the concentration S of a constituent, say the water vapor in the atmosphere or the salinity in the ocean, then DplDt = 0 follows from thc nondflusive conservation equation for the constituent in the form DS/ Dt = 0, plus the incompressible equation of state Splpo = BSS.) The equations can be written in tcrms of the pressure and density perturbations from a state of rest. In thc abscnce of any motion, suppose the density and pressure have the vertical distributions P(z) and P(z), where the z-axis is taken vertically upward. As this state is hydrostatic, we must have dP - dz _- - -pg. (14.3) In the presence of a flow field u(x, t), we can write thc density and pressure as (14.4) whcrc p' and pl are the changes from the state of rest. With this substitution, the first two terms on the right-hand side of the momentum equation in (1 4.2) give Subtracting the hydrostatic state (14.3), this bccomes which shows that we can replace p and p in Eq. (14.2) by the perturbation quantities pl and p'. Formulation of the Frictional Term The friction Iorce per unit mass F in Eq. (14.2) needs to be related to the velocity field. From Chapter 4, Section 7, the friction force is givcn by wherc tij is the viscous stress tensor. The stress in a laminar flow is caused by thc molecular exchanges of momcntum. From Eq. (4.41), the viscous stress tensor in an isotropic incompressible medium in laminar flow is given by In large-scale geophysical flows, however, the frictional E0n.c~ are provided by turbu- lent mixing, and Ihe molecular exchanges are negligible. The complexity a€ turbulent behavior makes it impossible to relatc the stress to the velocity field in a simple way. To proceed, then, wc adopt the eddy viscosity hypothesis, assuming that thc turbulent stress is proportional to the velocity gradient field. Geophysical mcdia arc in thc form of shallow stratified layers, in which the vertical velocities are much smaller than horizontal velocities. This means that thc cxchange of momentum across a horizontal surfacc is much weaker than that across a vcrtical surface. We expect then that the vedcal eddy viscosity u, is much smallcr than the horizontal eddy viscosity UH, and we assume that the turbulent stress components have the form (1 4.5) The difficulty with set (14.5) is that the exprcssions for txz and tJZ depend on the fluid rotution in the vertical plane and not just the deformation. In Chaptcr 4, Section 10, we saw hat a requirement for a constitutive equation is that the stresses should be inde- pendent of fluid rotation and should dcpend only on thc deformation. Therefom, rxz should depend only on the combination (air/& + aw/ax), whcreas thc expression in Eq. (14.5) depends on both deformation and rotation. A tensorially correct gco- physical treatment of the frictional terms is discussed, for example, in Kamenkovich (1967). However, the assumed form (14.5) lcads to a simple fornulation for viscous effects, as we shall see shortly. As the eddy viscosity assumption is of questionable validity (which Pedlosky (197 1 ) describes as a "rather disreputable and dcsperak atlcmpt"), there does not secm to be any purposc in formulating the stress-strain relation in more complicated ways merely to obey he requirement of invariance with respcct to rotation. With the assumed form for the turbulent strcss, the components of the frictional force fi = atij/i1xj become Estimates of the eddy cocfficients vary greatly. Typical suggcsted values are v, - 10m2/s and vH - lo5 m2/s for thc lower atmosphere, and u, - 0.01 m2/s and VH - 100 m2/s for the uppcr ocean. In comparison, thc molecular values are u = 1.5 x m2/s for air and u = 1W6 m2/s for water. 4. Appmrimatc! LiipationsJor a Thin Layer on The atmosphere and the Ocean are very thin layers in which the depth scale of flow is a few kilometers, whereas the horizontal scale is of the order of hundreds, or even thousands, of kilometers. The trajectories of fluid elements are very shallow and the vertical velocities are much smaller than the horizontal vclocities. In fact, the continuity equation suggests that the scale of the vertical velocity W is related to that or the horizontal velocity U by WH u L' where H is the depth scale and L is the horizontal length scale. Stratification and Coriolis effects usually constrain the vertical velocity to be even smaller than U H/L. Large-scale geophysical flow problems should be solved using spherical polar coordinates. If, however, the horizontal length scales are much smaller than the radius of the earth (= 6371 km), then the curvature of the earth can be ignored, and the motion can be studied by adopting a bcul Cartesian system on a tangent plane (Figure 14.3). On this plane we take an xyz coordinate system, with x increasing eastward, y northward, and z upward. The corresponding velocity components are u (eastward), v (northward), and w (upward). a Rotaling Sphem The earth rotates a1 a rate !J = 2~ rad/day = 0.73 x s-', around the polar axis, in an counterclockwise sense looking from above the north pole. From Figure 14.3, the components of angular velocity of thc carh in the local figure 143 Local Cartesian coordinates. Thc x-axis is inm he plane of the pnpcr. 28 x U= whcre we have defined ij k 0 2Rcos8 2C2sin8 uu W (14.8) to be twice the vertical component of 8. As vorticity is twice the angular velocity, f is called the pluncrary vorticity. More commonly, f is referred to as the Coriolis purumeier, or thc Curiolisfkequency. It is posilivc in the northern hemisphere and negative in the southern hcmispherc, varying from f1.45 x lo4 s-' at the poles to zero at the equator. This makes sense, since a person standing at the north pole spins around himself in an counterclockwise sense at a rate S2, whereas a person standing at the equator does not spin around himsclf but simply translates. The quantity Ti = 27c/f, is called the incrtilrlperiud, for reasons that will bc clear in Section I 1. The vertical componcnt of the Coriolis force, namely -2Ru cos 8, is generally negligiblc compared to the dominant terms in the vertical equation of motion, namely gp'/fi and p;'(ap'/az). Using Eqs. (14.6) and (14.7), the equations of motion (14.2) reducc to Du 1 ilp + uH ( a2u + a%) a2u - - fu = Dt pu ax ax2 ay a22 + vv-, (14.9) These are the equations of motion for a thin shell on a rotating earth. Note that only the vertical component of the earth's angular velocity appears as a consequence of thc flatness of the fluid trajectories. f-Plane Model The Coriolis parameter f = 2S2 sin 0 varics with latitude 0. However, we shall see later that this variation is important only for phenomena having very long timc scales (several weeks) or very long length scales (thousands of kilometers). For many pur- poses we can assume f to be a constant, say fo = 2S2 sin&, where & is the central latitude of the region under study. A model using a conqtant Coriolis parameter is called an.f-pZane model. /?-Plane Model The variation of f with latitude can bc approximatcly represented by expanding .f in a Taylor series about the central latitude 00: f = fo + 8% (14.10) where we defined 2~2 COS eo R' Here, we have used f = 2S2 sin 8 and dO/dy = 1/R, where the radius of the carth is nearly R = 6371 lan. A model that takes into account the variation of the Coriolis parameter in thc simplified form f = fo + By, with p as constant, is called a B-plane model. Consider quasi-steady large-scale motions in the atmosphere or the ocean, away from boundaries. For these flows an excellent approximation for thc horizontal equilibrium is a balance between thc Coriolis force and the pressure gradient: (14.11) Here we have neglccted the nonlinear acceleration terms, which are of order U2/L, in comparison to the Coriolis force -f U (U is the horizontal velocity scale, and L is the horizonla] length scale.) The ratio of the nonlincar term to thc Coriolis term is callcd the Rossby number: Nonlinear acceleralion U2/L U - Ro. Rossby number = Coriolis force fU .fL For a typical atmospheric value of U - 10m/s, f - s-’, and L - loOokm, the Rossby number turns out to bc 0.1. Thc Rossby numbcr is even smaller for many flows in the occan, so that the neglect of nonlinear terms is justified for many flows. The balance of forces represented by Eq. (14.1 I), in which the horizontal pressure gradients arc balanccd by Coriolis forces, is called a geostrophic balance. In such a system thc velocity distribution can be determined from a measured distribution of thc pressure field. The geostrophic equilibrium brcaks down near the equator (within a latitude belt of f3’), whcre f becomes small. It also brcaks down if the frictional cffects or unsteadiness bccome important. Vclocities in a geostrophic flow arc perpcndicular to the horizontal pressure gradient. This is becausc Eq. (14.11) implies that (iu +jv) Vp = - Po 1 .f ( -i- E + j- ic) (i$+.i$)=u. Thus, the horizontal velocity is along, and not across, the lines of constant pressure. If f is rcgarded as constant, then thc geostrophic balance (14.1 1) shows that p/fpo can bc regarded as a smamfunction. The isobars on a weather map are therefore nearly the slrcamlines of the flow. Figure 14.4 shows the geostrophic flow around low and high prcssure centers in thc northern hemisphcre. Herc the Coriolis force acts to thc right of the velocity vcctor. This requircs the flow to be counterclockwise (viewed from above) around a low prcssure region and clockwise around a high pressure region. The scnse of circulation is opposite in the southern hemispherc, where the Coriolis force acts to the left of the velocity vector. (Frictional forces bccome important at lower levels in the atmosphere and rcsult in a flow partially acmss the isobars. This will be discussed in Section 7, where we will see that the Bow around a low pressure center spirals inwurd due to frictional effects.) The flow along isobars at first surprises a reader unfamiliar with the cffects of Ihc Coriolis force. A question commonly asked is: How is such a motion seL up? A typical manner of establishmcnt of such aflow is as follows. Consider a horizontally converging flow in thc surface laycr of the occan. The convergent flow sets up the sea surface in the form of a gentle “hill:’ with the sea surfacc dropping away from the ccnter of the hill. A fluid particle starting to move down the “hill” is deflected to the right in the northern hemisphere, and a steady statc is reachcd when thc particle finally movcs along thc isobars. Thermal Wind In thc presence of a horizontal gradient of density, thc geostrophic velocily devclops a vertical shear. Consider a situation in which the density contours slope downward Figure 14.4 Gcustrophic flow murid lour and high prcssure centers. Thc pressure force (-Vp) is indi- cated by a thin wow, and hc Coriolis fmc is indicated by a thick mw. 1 2 X Figure 145 indicated by solid lincs; and contours of constant dcnsiiy tlre indicated bjf dashed lincs. with x, the contours at lower levcls represenling higher density (Figure 14.5). This implies that ijp/ax is negativc, so lhal the density along Section 1 is larger than that along Section 2. Hydrostatic equilibrium requires that thc weights of columns Szr and Sz2 are equal, so that he separation across two isobars increases with x, that is ,%ed wind, indicated by heavy mws pointing into the plane of papcr. Isohm arc 8z2 > dz,. Consequently, the isobaric surfaces must slope upward with x, with the slopc increa$ing with height, rcsulting in a positive ap/ax whose magnitude increases with height. Since the geostrophic wind is to thc right of the horizontal pressure force (in the northern hemisphere), it follows that the geostrophic velocity is into the planc of the paper, and its magnitude increases with height. This is casy to demonstrate from an analysis of the geostrophic and hydrostatic balance aP 0 = - gp. az (14.12) (14.13) (14.14) Eliminating p between Eqs. (14.12) and (14.14), and also between Eqs. (14.13) and (1 4.14), we obtain, respectively, (14.15) Metcomlogisls call these the thermal wind equations because they give the vertical variation cd wind from measurements of horizontal tcrnperature gradients. The ther- mal wind is a baroclinic phcnomenon, because the surfaces of constant p and p do not coincide (Figure 14.5). Taylor-Proudman Theorem A striking phenomenon occurs in the geosmphic 00w of a homogeneous Ruid. It can only be observed in a laboratory experiment because stratification effects cannot be avoided in natural flows. Consider then a laboratory experiment in which a tank of fluid is steadily rotated at a high angular speed S2 and a solid body is movcd slowly along the bottom of the tank. The purpose of making large and the movcment of the solid body slow is to make the Coriolis force much largcr than the acceleration terms, which must be made negligible for geostrophic equilibrium. Away from the frictional effects of boundaries, the balancc is therefore geostrophic in the horizonta1 and hydrostatic in the vertical: 1 ap pax' -2nv = 1 aP 2nu = , P BY 1 ap 0 = -gR. p az (14.16) (1 4.17) (14.1X) It is useful to define an Elanan number as the ratio of viscous to Coriolis forces (per unit volume): viscous force pvU/L2 v Coriolisforce pfU fL2 Ekman numbcr = - - -E. Under thc circumstances already described here, both Ro and E are small. equations gives Elimination of p by cross differentiation between the horizontal momentum 2Q (; + E) = 0. Using the continuity equation, this gives aW - =o. az (1 4.19) Also, differentiating Eqs. (14.16) and(14.17) withrespccttoz, andusing Eq. (14.18), we obtain Equations (14.19) and (14.20) show that ( 14.20) ! au I az - =o, (14.21) showing that the velocity vector cannot vary in the direction of P. In othcr words, steady slow motions in a rotating, homogeneous, inviscid fluid are two dimensional. This is the Taylor-Proudinan theorem, ht derived by Proudman in 19 16 and demon- strated experimentally by Taylor soon afterwards. In Taylor’s expcriment, a tank was ma& to rotate as a solid body, and a small cyhdcr was slowly draggcd along the bottom of the tank (Figure 14.6). Dye was introduced from point A above the cylinder and directly ahead of it. In a nomotat- ing fluid the water would pass over the top of the moving cylinder. In the rotating experimcnt, however, the dyc divides at a point S, as if it had bccn blocked by an upward extension of the cylinder, and flows around this imaginary cylinder, called the Taylor column. Dye releascd from a point B within the Taylor column remained there and moved with the cylinder. The conclusion was that the flow outside he upward cxtension of the cylinder is the same as if the cylinder extended across the entire water depth and that a column of water directly above the cylinder moves with it. The motion is two dimensional, although the solid body does no1 extend across the enhe water depth. Taylor did a second experiment, in which he dragged a solid body puraZleZ to the axis of rotation. In accordance with awl& = 0, he observed that a column of fluid is pushed ahead. The lateral velocity components u and v were zero. In both of these experiments, there are shear layers at the edge of the Taylor column. [...]... boundary layer is d2u -f v = v,*-, (14. 33) dz2 f u = vvwhere we have replaced -p-'(dp/dy) boundary conditions are d2v dz2 + fU, ( 14. 34) by f U in accordance with Eq (14. 32) The u=U, v=O u=O, v=O asz+x., ati:=O, (14. 35) (14. 36) where 1: i s taken vertically upward from the solid surface MultiplyingEq. (14. 34) by I and adding Eq (14. 33, the equations of motion become : d2V if - - - -(V - U), dz2 (14. 37) v, +... i, whcre 2, i,and fi are the complex amplitudes, and the real part of the right-hand side is meant Thcn Eq (1 4.45) gives -ioP - f i r = -ikgfi, + f i = -ilgij: + i H(kP + l e ) = 0 -iwi -iw$ (14. 77) ( 14. 78) (14. 79) Solving for P and ir between Eqs (14. 77) and (14. 78), we obtain (14. 80) t: = - ( -gfik + o Z ) if w z - f2 Substituting these in Eq. (14. 79), we obtain w2 - f 2 = gH(k2 + P) (14. 81) This... equations in (14. 45) and using the continuity equation to eliminate allla?.This gives -a2u at2 (-+E), f av = g H - a , 3t au ax (14. 72) ax (14. 73) Now take a / a t of Eq (14. 73) and use &. (14. 72) , to oblain +g at3 H a ax (e+ g)] = g H - a2 ayat ax (e+ E) (14. 74) ax To eliminate u, we first obtain a vorticity equation by cross differentiating and subtracting the momentum equations in & (14. 45): (- -... of motion (14 .22 ) and (14 .23 ) gives -f-dv dz -, v d2w, dz2 ' du -f-==- d2m, dz (14. 31) dz2 The right-hand si& of these equations represent diffusion of vorticity Without Coriolis forces this diffusion would cause a thickening of thc viscous layer The presence of planetary rotation, however, means that vertical fluid lines coincide with thc planctary vortcx lincs Thc tilting of vertical fluid lines,... (14 .24 ) and (14 .25 )can be combined a pv,(dV/dz) = t at z = 0, f o s rm whicb Eq (14 .28 )givcs A= tJ(1 - i ) 2PVv - Substitution of this into EQ (14 .28 ) givcs the vclocity components Thc Swedish oceanogaphcr Ekman worked out this solution in 1905 The solution is shown in Figure 14. 7 for the case of Ihc northern hemisphere, in which f is positive The vclocities at various depths are plotted in Figure 14. 74... 1 (14. 56) which is rhc differentialequalion governingthe vertical structureof the normal modes Equation (14. 56) has the so-called Sturm-Liouville form, for which the various solutions are orthogonal Equation (14. 55) also gives Substitutionof Eqs (14. 52) -( 14. 54) into Eqs (1 4.47)- (14. 5 1) finally gives the normal mode equations 1 ap,, au, av, (14. 57) -+-+ =o, ay c; at ax (14. 58) (1 4.59) (14. 60) (14. 61)... first worked out by Kelvin (Gill, 19 82, p 197) A plot of Eq (14. 82) is shown in Figure 14. 15 It is seen that the waves are dispersive except for w > f when Eq (14. 82) gives d 2: g H K 2 , > so that the propagation speed is w / K = The high-frequency limit agrees with our previous discussion of surface gravity waves unaffectcd by Coriolis forces a 't f K Figure 141 5 Dispersion relations for Poincar6... conditions (1.4.35) and (14. 36) in terms of the complex velocity are V=U V=O asz+m, atz=O (14. 38) (1 4.39) The particular solution of Eq (14. 37) is V = U The total solution is, thcrefore, v = ~ ~ - I l - i ) z / f i+ B ,(l+i)z/a + u, (14. 40) ,/m where 6 To satisfy Eq (14. 38), we must have B = 0 Condition (14. 39) gives A = -U The velocity componentsthen become (14. 41) According to Eq (14. 41), the tip of... gives the normal mode equations 1 ap,, au, av, (14. 57) -+-+ =o, ay c; at ax (14. 58) (1 4.59) (14. 60) (14. 61) Once Eqs (14. 57)- (14. 59) have been solved for Unr t;n and p l l , the arnpltudes pn and wn can be obtained from Eqs (14. 60) and (14. 61) The set (14. 57X14.59) is identical to the set (14. 45) governing the motion of a homogeneous layer, provided pn is identified wt g q and c,’ is identified with gH... k is the eastward wavenumber and I is the northward wavenumber Then Eq.(1 4.75) givcs w.' - c2wK2- f t w - c2Bk = 0, (1.4.76) m + whcre K 2 = k2 1 and c = ' It can bc shown that all roots of Eq (14. 76) are rcal, two of the roots bcing superincnial (w > f) and thc third being subinertial (w < f < ) Equation (14. 76)is thecompletedispersionrelation for linear shallow-water equations In various parametric . = - gp. az (14. 12) (14. 13) (14. 14) Eliminating p between Eqs. (14. 12) and (14. 14), and also between Eqs. (14. 13) and (1 4 .14) , we obtain, respectively, (14. 15) Metcomlogisls. these, the z-derivative of the equations of motion (14 .22 ) and (14 .23 ) gives -f dv d2w, - v,- dz dz2 ' du d2m, -f-== dz dz2 (14. 3 1) The right-hand si& of these equations. balance