Environmental Fluid Mechanics - Chapter 9 pot

9 Geophysical Fluid Motions 9.1 INTRODUCTION Geophysical fluid mechanics generally deals with flows of large spatial extent. There are many subjects that fall within this category, including ocean currents, tides, estuaries, coastal flows, and others. In the present chapter we focus on motions for which the rotation of the earth, in particular the Coriolis term, is important in the equations of motion. This condition arises in problems with large spatial scales, which result in Rossby numbers approaching one or less. Recall from Chap. 2 that the Rossby number is defined as the ratio of characteristic velocity to the product of characteristic length and rotation rate. This is derived from the ratio of the relative magnitude of the nonlinear acceleration terms to the Coriolis terms in the equations of motion, as shown in Sec. 2.9. Here we modify this definition slightly to be more consistent with the literature in this field, using Ro D U fL 9.1.1 where Ro D Rossby number, U D characteristic velocity, L D characteristic length, and f D Coriolis parameter,orplanetary vorticity D 2 0 sin , where  0 is the angular rotation rate of the earth and is the latitude. The magnitude of f varies between 1.45 ð 10 4 s 1 at the poles to zero at the equator. Referring back to Table 2.1, Rossby numbers that are sufficiently small that rotation effects become important are associated with large lakes, estuaries, coastal regions, and oceanic currents. Atmospheric motions also are subject to Coriolis effects, but the present discussion focuses on aqueous systems. In addition to the inclusion of the Coriolis terms, an interesting feature of the analysis of fluid motions in very large systems is the relative unim- portance of solid boundaries. This sometimes poses difficulties in specifying boundary conditions, since the location of the boundaries is not well-defined. Copyright 2001 by Marcel Dekker, Inc. All Rights Reserved. The only clear boundary in the deep oceans, for instance, is the air/water inter- face. Boundaries do become important, however, in developing descriptions of general circulation. 9.2 GENERAL CONCEPTS The general equations of motion for geophysical flows consist of the continuity and Navier–Stokes equations for incompressible flow introduced in Chap. 2. For convenience, these are repeated here:  rÐ  V D ∂u ∂x C ∂ v ∂y C ∂w ∂z D 0 9.2.1 ∂  V ∂t C  V Ðr  V C 2   ð  V D  g  1  0  rp C r 2  V9.2.2 where  V D u, v,wis the velocity vector,   is the rotation rate of the earth, g is gravity, p is pressure,  0 is a reference density, and  is kinematic viscosity. Usually the main concern is with horizontal or two-dimensional motions, so that w will usually be assumed to be zero for the present discussion. With w D 0, Eq. (9.2.2) in component form appears as ∂u ∂t C u ∂u ∂x C v ∂u ∂y C w ∂u ∂z  f v D 1  0 ∂p ∂x C r 2 u9.2.3 ∂ v ∂t C u ∂ v ∂x C v ∂v ∂y C w ∂ v ∂z C fu D 1  0 ∂p ∂y C r 2 v 9.2.4 0 Dg  1  0 ∂p ∂z 9.2.5 In the following, the viscous stress term will be replaced with a turbulent stress, with a possibly nonhomogeneous and nonisotropic turbulent eddy diffusivity (see Chap. 5). However, this does not change the basic form of the equation. 9.2.1 Geostrophic Balance Geostrophic flow, as introduced in Eq. (2.9.26), involves a balance between the pressure and Coriolis terms in the equations of motion, that is, f v D 1  0 ∂p ∂x 9.2.6a fu D 1  0 ∂p ∂y 9.2.6b This result is obtained by assuming steady conditions and neglecting the nonlinear acceleration and friction terms in the equations of motion. Models Copyright 2001 by Marcel Dekker, Inc. All Rights Reserved. that neglect the nonlinear accelerations are sometimes referred to as Ekman models. They are mostly appropriate for relatively small values of Ro, which as noted previously expresses the ratio of the magnitudes of the acceleration and Coriolis terms. It should be noted that the geostrophic balance is not valid near the equator, within a latitude of about š3 ° ,wheref becomes very small. An interesting result of Eq. (9.2.6) is that the flow direction is perpen- dicular to the pressure gradient. Therefore, on a weather map, isobars are approximately the same as streamlines of the flow, and the streamlines are lines of constant pressure. Also, the quantity p/f 0  can be regarded as a stream function. Figure 9.1 shows a schematic description of flow along an isobar in the northern hemisphere, around centers of high and low pres- sure. The Coriolis force and the pressure gradient are colinear, with opposite directions. The velocity vector is perpendicular to those vectors and creates a counterclockwise angle of 90 ° with the Coriolis force. In the southern hemi- sphere, the velocity acts 90 ° to the left of the Coriolis force, due to the opposite sense of rotation. Figure 9.1 Relationship between isobars and streamlines in atmospheric flows (nor- thern hemisphere). Copyright 2001 by Marcel Dekker, Inc. All Rights Reserved. Figure 9.2 Surface tilt as a result of geostrophic balance (flow is into the page). Another way of looking at the geostrophic balance is to consider a homogeneous fluid with a surface at an angle  to the horizontal direction, as shown in Fig. 9.2. The pressure along the surface is the atmospheric pressure, p a . The acceleration generated as a result of this angle is a x Dg tan Â9.2.7 where the negative sign arises because x is positive to the right. If the fluid is moving at velocity U into the plane of Fig. 9.2, there will be a horizontal component of the Coriolis acceleration with magnitude fU in the positive x direction (northern hemishere). If these two accelerations are in balance, then g tan  D fU ) U D g tan  f 9.2.8 This gives the expected geostrophic velocity, for a given surface slope or, conversely, the expected surface slope for a given flow velocity. Note that the flow is in a direction normal to the pressure gradient, as was shown in Eq. (9.2.6). Recall that in order for the balance that leads to Eq. (9.2.8) to exist, the flow must be uniform and in a straight direction, since no other accelerations are assumed than the pressure gradient and Coriolis terms. It is somewhat surprising to consider the magnitude of the sea surface tilt angle that corresponds to expected velocities in the ocean. For example, if U D 1 m/s, and we assume a latitude of 45 ° ,thentan ¾ D 10 5 , or about 1 cm/km. This is much too small to be measureable. However, measurements in a stratified ocean are much easier. There is almost always some density stratification in the oceans, due to temperature or salinity variations or both. Issues related to stratification are discussed in Chap. 13, but for now consider that the stratification can be idealized as a two-layer system as sketched in Fig. 9.3, which shows a fluid of density  1 flowing over a stagnant layer of density  2 . Since fluid 2 is at rest, its free surface must be horizontal, while the free surface of fluid 1 is tilted due to its motion, making an angle  with the horizontal direction, as previously described. Consider that the interface between the two fluids lies at some angle  i , as shown in the figure. Copyright 2001 by Marcel Dekker, Inc. All Rights Reserved. Figure 9.3 Surface and interface tilt for geostrophic flow in a two-layer ocean. Along any horizontal line drawn in fluid 2, the pressure must be constant. For such a line drawn at depth h, the pressure is given by the hydrostatic relation p h D p a C  2 gh 9.2.9 Now, with x drawn so that x D 0 at the point at which the interface between the two fluids meets the surface, the pressure along the line drawn at depth h can be written as p h D p a C gf 1 tan   tan  i x C 2 h C x tan  i g 9.2.10 Equating Eqs. (9.2.9) and (9.2.10) then results in tan  i Dtan   1  2   1 9.2.11 This shows that the slope of the interface can have a much greater magnitude than the surface slope, depending on the relative values of  1 and  2 .For example, if fresh water (specific gravity D 1) flows over sea water (specific gravity D 1.025), then the interface slope is approximately 40 times as great as the surface slope. In most cases the density difference is less than this, so that the interface slope would likely be even greater. When surfaces of constant density are parallel to surfaces of constant pressure, the system is said to be in a barotropic state. When these surfaces intersect, the field is baroclinic. It is possible for a barotropic system to be statically stable, but in a baroclinic system there must be motion. The two- layer ocean considered above is an example of a baroclinic field, since motion Copyright 2001 by Marcel Dekker, Inc. All Rights Reserved. was required to generate the Coriolis force to oppose the pressure gradient force. The geostrophic flow that results from a horizontal density gradient is also called a thermal wind. This terminology stems from the usual situation in which the density differences are generated as a result of temperature vari- ations. When there is a horizontal density gradient, the geostrophic flow also develops a vertical shear. This can be seen by considering the system shown in Fig. 9.4, which shows several contours of constant density and contours of contant pressure. Assuming that ∂/∂x < 0, then the density along section 1 is greater than that along section 2. In order to maintain hydrostatic equilib- rium, the weight of columns υz 1 and υz 2 must be equal. Therefore the interval between the two isobars increases with x,orυz 1 <υz 2 . The isobars, as shown in Fig. 9.4, then must be consistent with ∂p/∂x > 0, and their slope increases with increasing z. Following the same arguments as before (coming from the geostrophic balance), the thermal wind is thus seen to be into the plane of Fig. 9.4, and its magnitude increases with z. This phenomenon is clearly demonstrated using Eq. (9.2.6), along with the hydrostatic balance equation in the z-direction. Differentiating equation Figure 9.4 Baroclinic field, showing several contours of constant density and pres- sure. Copyright 2001 by Marcel Dekker, Inc. All Rights Reserved. (9.2.6a) with respect to z and using Eq. (9.2.5) to substitute for the vertical pressure gradient, we obtain ∂ v ∂z D g  0 f ∂ ∂x 9.2.12a Performing a similar operation by differentiating Eq. (9.2.6b) with respect to z gives ∂u ∂z D g  0 f ∂ ∂y 9.2.12b These equations are called the thermal wind equations. They provide the vertical variation of velocities from measurements of the horizontal tempera- ture (density) gradients. The thermal wind, as indicated in Fig. 9.4, is associ- ated with systems in which surfaces of constant pressure and constant density intersect, i.e., the baroclinic case. 9.2.2 Potential Vorticity Another concept useful in the study of large-scale flows is that of conservation of potential vorticity. This is demonstrated by first writing the momentum equations for horizontal motion, neglecting the friction terms. From Eq. (9.2.3), Du Dt D ∂u ∂t C u ∂u ∂x C v ∂u ∂y D f v  1  0 ∂p ∂x 9.2.13 and from Eq. (9.2.4), D v Dt D ∂ v ∂t C u ∂ v ∂x C v ∂v ∂y Dfu  1  0 ∂p ∂y 9.2.14 where (D/Dt) is taken here as the two-dimensional or horizontal material derivative operator. We now differentiate Eq. (9.2.13) with respect to y and subtract the result from the derivative of Eq. (9.2.14) with respect to x, giving D Dt  ∂v ∂x  ∂u ∂y  D  ∂v ∂x  ∂u ∂y  f  ∂u ∂x C ∂ v ∂y   u ∂f ∂x  v ∂f ∂y 9.2.15 where horizontal density variations have been neglected. Now, f is not a function of time or longitude (x), so the last term on the right-hand side of Eq. (9.2.15) may be rewritten as v ∂f ∂y D Df Dt 9.2.16 Copyright 2001 by Marcel Dekker, Inc. All Rights Reserved. Substituting Eq. (9.2.16), and rearranging the terms of Eq. (9.2.15) then leads to D Dt  ∂v ∂x  ∂u ∂y  C f  D  ∂u ∂x C ∂ v ∂y  ∂v ∂x  ∂u ∂y  C f  9.2.17 Recall that the vertical component of vorticity, relative to the chosen coordinate system, is defined by Eq. (2.3.12),  D ω z D ∂v ∂x  ∂u ∂y 9.2.18 Also, if the flow satisfies the two-dimensional continuity equation, then the first term on the right-hand side of Eq. (9.2.17) (in parentheses) is zero, resulting in D Dt  Cf D 0 9.2.19 The sum of the relative vorticity  and the planetary vorticity f is called the absolute vorticity. Equation (9.2.19) states that the absolute vorticity is conserved, following a fluid particle along its path line. If the flow field is required to satisfy the full continuity constraint, then Eq. (9.2.1) gives ∂u ∂x C ∂ v ∂y D ∂w ∂z 9.2.20 The right-hand side of this equation can be related to the rate of stretching of a column of fluid of thickness H,by ∂w ∂z D 1 H DH Dt 9.2.21 Introducing this result into Eq. (9.2.17) then gives D Dt  Cf D 1 H DH Dt  Cf 9.2.22 Dividing both sides by H, we obtain 1 H D Dt  C f  1 H 2 DH Dt  Cf D D Dt   Cf H  D 0 9.2.23 where the quantity  Cf/H is called the potential vorticity. This last result shows that, for frictionless incompressible flow, potential vorticity is conserved following a fluid particle. For steady flows the particle paths are the same as the streamlines, so potential vorticity is thus conserved along streamlines. Use of this concept, along with geostrophic flow assumptions and other results Copyright 2001 by Marcel Dekker, Inc. All Rights Reserved. such as the Bernoulli equation (Chap. 2) has formed the basis for a number of theoretical models of ocean currents. 9.3 THE TAYLOR–PROUDMAN THEOREM A number of laboratory experiments have been performed, usually using rotating tables, to simulate different aspects of low-Rossby-number flows. One of the more interesting experiments of this type involves simulation of geostrophic flow of a homogeneous fluid and produces direct observations of Taylor columns, as explained below. This experiment involves a tank of fluid that is rotated at a steady angular speed . The rotation speed is sufficiently high that the Coriolis force is much larger than the acceleration terms, and conditions of geostrophic equilibrium may be assumed. In regions that are not affected by the friction induced by the bound- aries, the equations for geostrophic equilibrium in the horizontal directions, and hydrostatic conditions in the vertical direction, are given by Eqs. (9.2.6a), (9.2.6b), and (9.2.5), respectively. Note, however, that f D 2 for the condi- tions of the experiment. By differentiating Eq. (9.2.6a) with respect to y and Eq. (9.2.6b) with respect to x and subtracting, we find 2  ∂u ∂x C ∂ v ∂y  D 0 9.3.1 Using Eq. (9.2.20), and since  6D 0, Eq. (9.3.1) implies ∂w ∂z D 0 9.3.2 Also, by differentiating each of Eqs. (9.2.6a) and (9.2.6b) with respect to z and substituting Eq. (9.2.5) for the vertical pressure gradient results in ∂u ∂z D ∂ v ∂z D 0 9.3.3 Since there is no vertical motion, the angular velocity vector is oriented in the z-direction. Equations (9.3.2) and (9.3.3) indicate that the velocity vector does not vary with z, so we may conclude that steady, slow motions in a rotating, homogeneous, inviscid fluid are two-dimensional. This result is called the Taylor–Proudman theorem. It was obtained theoretically by Proudman in 1916. Soon afterwards, Taylor proved this theorem using an experimental setup as sketched in Fig. 9.5. A tank full of fluid was rotating as a solid body. A small cylinder was slowly dragged along the bottom of the tank and dye was released at point A, above the cylinder and slightly ahead of it. The thread Copyright 2001 by Marcel Dekker, Inc. All Rights Reserved. Figure 9.5 Schematic diagram of Taylor’s experiment. of dye divided at the point S, while appearing to belong to a column of fluid extending over the depth of the cylinder. This column of fluid is called a Taylor column. Taylor’s experiment indicated that bodies moving slowly in a strongly rotating system of homogeneous fluid carry along their motion in a two-dimensional column of fluid. Copyright 2001 by Marcel Dekker, Inc. All Rights Reserved. [...]... York McLellan, Hugh J., 196 5 Elements of Physical Oceanography Pergamon Press, Oxford Munk, W H., 195 0 On the wind driven ocean circulation J Meteorol 7: 79 93 Neuman, Gerhard and Piersol Willard, J., 196 6 Principles of Physical Oceanography Prentice Hall, Englewood Cliffs, NJ Pedlosky, Joseph, 198 7 Geophysical Fluid Dynamics, 2 ND Ed Springer-Verlag, New York Rossby, C G., 193 2 A generalization of the... two horizontal velocity components are fv D A ∂2 u ∂z2 9. 4.1 and ∂2 v 9. 4.2 ∂z2 where A is a horizontal eddy diffusivity, which is assumed to be constant By combining the constants into one term, we define fu D A 2 D f A 9. 4.3 Note that has units of length 1 Using Eq (9. 4.3), Eqs (9. 4.1) and (9. 4.2) are rewritten as d2 u 9. 4.4 i2 2 v D 2 dz and d2 v 9. 4.5 i 2u D i 2 dz p where i D 1 and ordinary derivatives... directions are, respectively, 1 Mx D u dz D 0 V0 L p 2 9. 4. 19 and 1 My D dz D 0 0 Copyright 2001 by Marcel Dekker, Inc All Rights Reserved 9. 4.20 The mass flux in the x-direction also can be written using Eq (9. 4.17) for V0 , giving Mx D y0 f 9. 4.21 Thus, curiously, there is zero net mass transport in the direction of the wind, and all of the transport is at 90 ° to the right of the wind direction This Ekman... ∂v ∂y C ∂ ∂z Az ∂v ∂z 9. 5.2 From Chap 2, these can be rewritten in terms of shear stresses as f vD ∂p ∂ xx ∂ xy ∂ xz C C C ∂x ∂x ∂y ∂z 9. 5.3 and f uD ∂ yy ∂ yz ∂p ∂ yx C C C ∂y ∂x ∂y ∂z 9. 5.4 Integrating each of Eqs (9. 5.3) and (9. 5.4) vertically from z D 0 to z D L (Ekman layer depth) gives fMy D ∂Txy ∂P ∂Txx C C C ∂x ∂x ∂y 9. 5.6 xz zD0 and fMx D ∂Tyy ∂P ∂Tyx C C C ∂y ∂x ∂y 9. 5.7 yz zD0 where Mx and... 2001 by Marcel Dekker, Inc All Rights Reserved D y A ) C2 D y A 9. 4.13 To simplify the notation, define a length scale L, so that p 2 2A 1/2 D LD f 9. 4.14 Using this definition, the final results for the velocity components are written by substituting for C2 and C4 into Eqs (9. 4 .9) and (9. 4.10), respectively, giving u D V0 exp L z cos z 9. 4.15 z 9. 4.16 4 L 4 L and v D V0 exp L z sin where V0 is the magnitude... values versus Â, at intervals of 10° Solution As defined in Eq (9. 1.1), f D 2 sin  where is the angular velocity of the earth, D 2 D 7.27 ð 10 24 ð 3600 5 s 1 By applying this value, we obtain the following table:  10° 20° 30° 40° 50° 60° 70° 80° 90 ° f ð 104 0.252 0. 497 0.727 0 .93 5 1.114 1.2 59 1.366 1.432 1.454 Unsolved Problems Problem 9. 3 The usual equation used to estimate geostrophic velocities... in the y-direction at the surface is given by y0 D y jzD0 D A dv dz 9. 4.11 zD0 Note that x D 0 implies that du/dx D 0 at z D 0, since A is constant and A 6D 0 Since both velocities vanish for large z, we conclude that C1 D 0 Then, by differentiating Eqs (9. 4 .9) and (9. 4.10) with respect to z, using the shear boundary conditions, we obtain du dz D C2 p sin C4 C cos C4 D 0 ) C4 D 2 D C2 p zD0 9. 4.12 4... torque imposed upon the water column as it moves to regions of different f The rate of change of f with latitude is usually written as ∂f Dˇ ∂y 9. 5 .9 and models that consider constant values of ˇ are called beta-plane models The first term on the right-hand side of Eq (9. 5.8) is the vertical component of the curl of the wind stress vector and gives a measure of the torque exerted about a vertical axis by... considered to be functions of z only Now, define D u C iv, so that by adding Eqs (9. 4.4) and (9. 4.5), we obtain d2 Di dz2 2 Copyright 2001 by Marcel Dekker, Inc All Rights Reserved 9. 4.6 This equation has a solution, p iz C K2 exp D K1 exp p 9. 4.7 iz where K1 and K2 are complex constants given by K1 D C1 exp iC1 , K2 D C2 exp 9. 4.8 iC2 where C1 and C2 are constants that must be determined from known boundary... to zero in Eq (9. 5.8) Also, why do the shear stresses vanish at z D L? Copyright 2001 by Marcel Dekker, Inc All Rights Reserved Problem 9. 13 Calculate different values of ˇ that would be applicable over different ranges of latitudes, from the equator to the poles Problem 9. 14 Formulate a finite difference expression that could be used to solve Eq (9. 5.8) Assume the first term on the left-hand side is . written as ∂f ∂y D ˇ 9. 5 .9 and models that consider constant values of ˇ are called beta-plane models. The first term on the right-hand side of Eq. (9. 5.8) is the vertical compo- nent of the curl. flow satisfies the two-dimensional continuity equation, then the first term on the right-hand side of Eq. (9. 2.17) (in parentheses) is zero, resulting in D Dt  Cf D 0 9. 2. 19 The sum of the relative. condi- tions of the experiment. By differentiating Eq. (9. 2.6a) with respect to y and Eq. (9. 2.6b) with respect to x and subtracting, we find 2  ∂u ∂x C ∂ v ∂y  D 0 9. 3.1 Using Eq. (9. 2.20),