Bottom Ekman Pumping with Stress-Dependent Eddy Viscosity

Transcription

1 1967 Bottom Ekman Pumping with Stress-Dependent Eddy Viscosity BENOIT CUSHMAN-ROISIN Thayer School o Engineering, Dartmouth College, Hanover, New Hampshire VLADO MALAČIČ Marine Biological Station Piran, Piran, Slovenia (Manuscript received 9 August 1996, in inal orm 4 March 1997) ABSTRACT This paper reconsiders the classic problem o bottom Ekman pumping below a steady geostrophic low by relaxing the assumption o a constant eddy viscosity. It is assumed instead that the eddy viscosity depends on the magnitude o the bottom stress, which itsel is a unction o the geostrophic low. Results show that the vertical Ekman pumping is no longer directly proportional to the relative vorticity o the geostrophic low, but is a ar more complicated unction o the geostrophic low. Speciic examples are discussed, which show that the Ekman pumping rate may be 5% or 1% larger than that predicted by the traditional theory. 1. Introduction It has long been established that the role o vertical riction in the deep ocean is relegated to relatively thin layers, called Ekman layers. One o these is the bottom Ekman layer, which exists to bring the horizontal velocity components rom their inite values in the lower water column to zero at the bottom. The combination o Coriolis and rictional orces in the Ekman layer creates a substantial veering, and there exists in the layer a low component transverse to the current alot. Thus, although that current may be in geostrophic balance and almost nondivergent, the transverse low in the Ekman layer is divergent or convergent. Continuity o luid demands a compensating vertical velocity, which extends through the water column. This is called Ekman pumping. Ekman pumping is important or several reasons. First, it plays a signiicant dynamical role by stretching or squeezing the ocean s interior low; this, in turn, aects the vorticity o the ocean circulation. Second, slow but persistent vertical velocities are essential in transporting chemicals, contaminants (such as radioactive wastes), and nutrients through the water column, and in controlling their residency time. According to the traditional theory, which relies on the assumption o a uniorm eddy viscosity (see, e.g., Pedlosky 1987, chapter 4; Cushman-Roisin 1994, Corresponding author address: Dr. Benoit Cushman-Roisin, Thayer School o Engineering, Dartmouth College, 8 Cummings Hall, Hanover, NH B.Cushman@dartmouth.edu chapter 5), there is a direct proportionality between the transverse velocity component in the Ekman layer and the overlying geostrophic velocity, and it is ound that the Ekman pumping rate is proportional to the relative vorticity o the geostrophic low. Thus, or example, a cyclonic gyre in the ocean s interior is associated with convergent low in the bottom Ekman layer and an upward vertical velocity proportional to the gyre s local relative vorticity. It is well known, however, that the turbulent state o the ocean s bottom boundary layer is such that the eddy viscosity is not vertically uniorm but varies signiicantly with distance above the bottom (Weatherly and Martin 1978). The magnitude o the eddy viscosity also varies horizontally with the local bottom stress. While the vertical proile o the eddy viscosity controls the structure o the current distribution through the Ekman layer, its magnitude controls characteristics such as the thickness o the layer. The magnitude o the transverse transport depends on the overlying geostrophic low speed and on the thickness o the layer, which varies with the eddy viscosity, the bottom stress, and thus the geostrophic low speed itsel. In sum, the transverse transport is more than proportional to the overlying geostrophic current, and the Ekman-pumping velocity cannot be simply proportional to the vorticity o the geostrophic low but must depend on it in some more complicated, nonlinear way. Our objective here is to determine this relationship between Ekman pumping and geostrophic low when a more realistic eddy viscosity parameterization than a constant is adopted American Meteorological Society

2 1968 JOURNAL OF PHYSICAL OCEANOGRAPHY VOLUME 7 A classic parameterization o the eddy viscosity in a sheared boundary layer 1 is (Ellison 1956) u z, (1) where is the eddy viscosity,.4 the von Kármán constant o turbulence, u the so-called riction velocity, and z the distance rom the boundary. The riction velocity is deined rom the boundary stress b according to b u, () where is the luid s density. The virtues o this relatively simple parameterization are its increase with distance rom the boundary and its increase with the amount o shear stress, according to results o laboratory experiments and geophysical observations. It has been used in numerous studies o homogeneous rotating lows: among others, Ellison (1956) in a study o turbulence in the atmospheric boundary layer, Madsen (1977) in a study o the ocean s wind-driven surace Ekman layer, and Nihoul (1977), Lavelle and Mojeld (1983), Ostendor (1984), and Soulsby (199) in studies o tidal currents in shallow well-mixed waters. The solution is in terms o Kelvin Thomson unctions (a variety o Bessel unctions because o the proportionality o to z) and is well known. In all these studies, however, the dependency o the eddy viscosity on the riction velocity is treated parameterically, and no one to our knowledge has yet investigated its eect on Ekman pumping in the deep ocean. To ill this gap, we investigate here how parameterization (1) () or the eddy viscosity modiies the expression or the Ekman pumping generated by bottom riction in the deep ocean. The study o Weatherly and Martin (1978), based on a turbulence-closure model, reveals that the eddy viscosity in the ocean s benthic layer behaves according to (1) near the bottom but reaches a maximum at some height and decreases nearly exponentially with arther distance rom the bottom. An improved parameterization with exponential attenuation has been proposed (Long 1981), but the intractability o the ensuing mathematical problem demands numerical solution. Adopting such reined parameterization in our study would preclude the derivation o an analytical ormula or the Ekman pumping rate, and we decided not to go beyond (1) (). It is also expected that the dierence would be minor since values o the eddy viscosity signiicantly away rom the bottom where the velocity shear is small are nearly irrelevant, and the structure o the boundary layer is essentially governed by the behavior o the eddy viscosity near the bottom where the velocity shear is greatest. This assertion is veriied a posteriori. Our work is structured in the ollowing way. Section briely reviews the solution o the Ekman-layer equa- 1 The turbulent Ekman layer is none other than a sheared boundary layer in the presence o rotation. tions with eddy viscosity parameterized by (1) () and recapitulates its advantages over that with constant eddy viscosity. Section 3 then derives rom this solution a new ormula giving the vertical Ekman-pumping velocity in the ocean s interior in terms o its geostrophic velocity. Several particular cases are considered and discussed. The concluding section 4 highlights the dierences with the predictions o the traditional theory based on constant eddy viscosity. At the end o the paper is a short appendix deining two particular unctions derived rom the Kelvin Thomson unctions that acilitate the analytical developments.. Bottom Ekman layer a. Equations and boundary conditions Traditional Ekman dynamics include a balance between the Coriolis, pressure gradient, and vertical-riction orces. Time derivatives and nonlinear advective terms can also be incorporated (Soulsby 199; Hart 1995), but we shall not to do so in order to restrict the attention to the eect o a varying eddy viscosity on Ekman pumping. Also, our new parameterization o Ekman pumping would be most useul in general circulation studies, when geostrophy is an excellent approximation. Above the Ekman layer, the rictional orce vanishes and the balance o orces reduces to geostrophy; by virtue o the homogeneity o the luid and o the hydrostatic balance, which we also assume, the horizontal pressure gradient is depth independent, and we write u ( ) uz (3a) z z (u ū) uz, (3b) z z where is the Coriolis parameter (assumed constant and positive), u and the horizontal velocity components within the layer ( z), ū and the geostrophic low alot (z ), and u z the variable eddy viscosity presented earlier. We employ partial derivatives with respect to the vertical coordinate z because we consider the geostrophic low (ū, ) to be varying horizontally (in x and y directions) and to induce horizontal variability in u,, and u. Boundary conditions are Top o Ekman layer: u ū,, as z (4) Bottom o Ekman layer: u,, at z z. (5) In the bottom boundary conditions, it is necessary to distinguish the level z z, where z is the roughness height, rom the average bottom level z, because o the singularity o the anticipated logarithmic velocity proile in the near-bottom zone (e.g., Lavelle and Mojeld 1983).

3 1969 Governing equations (3a,b) orm a ourth-order problem, which together with the our boundary conditions in (4) (5) can be solved uniquely. In that solution, however, the riction velocity u will appear parameterically, and it will be necessary to close the problem by relating this parameter to the magnitude o the bottom stress via (); that is, u b x y, (6) where x and y are the bottom stress components. Since these implicate the velocity components as well as the riction velocity, it is at this stage that a nonlinearity will be introduced in the mathematical ormalism. b. Solution Using the unctions a() and b() derived in the appendix rom the decaying Kelvin Thomson unctions, the solution o (3a,b) that meets the upper boundary conditions (4) is z z u ū Aa Bb (7a) u u z z Ba Ab, (7b) u u where A and B are two constants o integration. These constants are to be determined by applying the remaining two boundary conditions, namely (5). Since the level z z is very close to the average bottom level z (usually less than a millimeter; Soulsby 199), it can be assumed that the dimensionless quantity z /u is much smaller than unity, and the asymptotic behaviors o a and b near the origin can be used [see (A6a) and (A7a)]. It ollows that 1 z ū A ln B (8a) u 4 1 z B ln A, (8b) u 4 where.577 is the Euler constant. The solution is immediate; 4 A ( ū) (9a) (1 ) 4 B (ū ), (9b) (1 ) where the coeicient, deined by 4 1 u ln, (1) z has been introduced or convenience. Note that, because the logarithm is a weak unction o its argument, the coeicient is a weak unction o the riction velocity u. With u remaining as a parameter, the solution is 4 z u ū ( ū)a (1 ) u 4 z (ū )b (11a) (1 ) u 4 z (ū )a (1 ) u 4 z ( ū)b. (11b) (1 ) u Collecting separately the terms proportional to ū and, we can write the same solution in the orm 4(b a) 4(a b) u 1 ū (1a) (1 ) (1 ) 4(b a) 4(a b) 1 ū, (1b) (1 ) (1 ) which reveals the components o the low that are parallel and transverse to the geostrophic current. We note that the transverse component, equal in magnitude to the absolute value o 4(a b)/(1 ) times the speed o the geostrophic current, is not proportional to the latter; indeed, the parameter includes a dependency on the riction velocity, which will shortly be related to the geostrophic current speed. c. Friction velocity We now close the problem and determine the riction velocity u by relating it to the actual bottom stress. First, the bottom-stress components are calculated: u x lim u z z z [ 4u da ( ū) lim (1 ) d y lim u z z z [ ] db (ū ) lim (13a) d 4u da (ū ) lim (1 ) d ] db ( ū) lim, (13b) d

4 197 JOURNAL OF PHYSICAL OCEANOGRAPHY VOLUME 7 FIG.. Ekman spirals or various values o the dimensionless ratio ū/z, where ū is the speed o the geostrophic current above the Ekman layer, the Coriolis parameter, and z is the roughness height. The uppermost and thickest line corresponds to ū/z 1 4, and every successively thinner line to a value 1 times larger, with the lowest and thinnest line corresponding to ū/z 1 8. The point where the spirals curl represents the geostrophic low alot, while the origin is crossed at level z. The branches in the third quadrant, beyond the origin, exist because o the logarithmic nature o the velocity proiles but are physically meaningless. FIG. 1. Variation o the riction velocity u (thin line) and drag coeicient C D (thick line) with the geostrophic current speed ū. where is z/u. The limits (A7b,c) established in the appendix yield u x ( ū) (14a) (1 ) u y (ū ). (14b) (1 ) The riction velocity u can then be derived rom (6) as u u ( ū) (ū ), (1 ) which can be reduced to ū ū u. (15) 1 1 Since the parameter, deined in (1), includes u, ormula (15) is an implicit equation or the riction velocity, in terms o the magnitude o the geostrophic low ū ū, the roughness height z, the Coriolis parameter, and dimensionless constants. The thin curve in Fig. 1 traces the value o u /ū versus ū/z. Credit or relationship (15) between the riction velocity and the geostrophic speed goes to Ellison [1956, his Eq. (15) and his Table ]. d. Bottom stress and drag coeicient Expressions (14a,b) show that the bottom stress consists o a component antiparallel to the geostrophic cur- This implicit relation can, however, be inverted, and ū/z can be expressed as an explicit unction o u /ū. In the process, a square root must be taken; physics dictate that the root corresponding to the lower velocity values be selected. rent (acting as a retarding orce) and a transverse component. The ratio o these components is 1, which yields the angle o delection between the geostrophic current and its associated bottom stress. From the magnitude o the bottom stress, we can derive a drag coeicient by writing b x y CDū, (16) with 4 C D. (17) (1 ) Since contains a weak dependency upon u according to (1), with u itsel depending upon the speed o the geostrophic low ū according to (15), this drag coeicient is not constant but a unction o the dimensionless, Rossby-like number ū/z. [In the atmospheric boundary layer literature, this number is occasionally called the surace Rossby number (e.g., Garratt 199, p. 44)]. The variation is shown by the thick line in Fig. 1. Expression (17) is not new but was irst derived by Ellison (1956) and again by Lavelle and Mojeld (1983). These last authors also compared its predictions with a variety o estimated values derived rom measurements taken in the ocean s benthic boundary layer and ound good agreement (see their Fig. 9). This agreement spurs us on to derive later an expression or the Ekman pumping under the present choice o eddy-viscosity parameterization. e. Veering angle and thickness o Ekman layer Figure shows velocity hodographs according to solution (11a,b), in which the parameter and the riction velocity u are related to the geostrophic speed ū, the roughness height z, and the Coriolis parameter by (1) and (15). Since this solution contains the ree dimensionless number ū/z, there is a amily o hodographs, all Ekman spirals but o various widths. Regardless o the value o ū/z, the transverse low (to the let in the Northern Hemisphere, to the right in the Southern Hemisphere) hardly exceeds 1% o the geostrophic

5 1971 low, unlike in the traditional solution according to which it reaches 3%. 3 I the veering angle is deined as the angle between the current as z approaches zero and the geostrophic current alot, Eqs. (1a b) yield 4(a b) 1 tan lim. (18) (1 ) 4(b a) z Table 1 lists selected values or a wide range o Rossbylike numbers. As we can readily note, all values are signiicantly lower than the traditional value o 45. In numerical simulations o the oceanic bottom boundary layer with a turbulence-closure model, Ezer and Weatherly (199) ound values o 1 with s 1 (latitude 4.5 N), z m, and a geostrophic velocity above the Ekman layer o 1 cm s 1. For the same parameter values, the present model yields an angle o 8.6. A question that arises naturally is: How thick is the Ekman layer? Taking the top o the layer as the level above which the vector velocity does not dier rom the geostrophic current by more than 5%, we write (u ū) ( ) ū at z d, where.5 and d is the nominal layer thickness. Use o solution (11a,b) yields an implicit equation or d: d d (1 ) a b. (19) u u 16 Table 1 lists solutions or ive dierent values o the Rossby-like number ū/z. Beore closing this section, we return to our choice o parameterization o eddy viscosity and show that the ever-increasing value o with z (while a more realistic parameterization would have a saturating or even decaying value ar above the bottom) has no perceptible impact on the solution. There are several ways to prove this assertion. The irst one is to remark that all properties derived rom the present solution (riction velocity, drag coeicient, veering angle, layer thickness, and, later, Ekman pumping) depend solely on the properties (A7a,b,c) and (A8a,b) o the building-block unctions a() and b(). While the behavior near the origin (A7a,b,c) is directly controlled by the logarithmic singularity, which would remain unchanged as long as (z) behaves like z near the origin, the integral properties (A8a,b) too do not depend on the detailed structure o z but only on the requirement that the unctions vanish at ininity and again on the behavior near the origin. 3 This percentage is the maximum value o e z sinz, the ratio o the transverse velocity to the geostrophic speed in the traditional Ekmanlayer theory (obtained or z /4). Thereore, values o away rom the origin are inconsequential. Another way o arriving at the same conclusion is to consider numerical values. Since the decay o velocity deicit (velocity minus geostrophic low) at the top o the boundary layer is exponential, the 5% departure criterion truly captures most o the extent o the boundary layer. The values listed in Table 1 can then be used to estimate the inal values o (z) that matter, and with d ( )u /, we ind successively top u d (.94.65) u / ( ) ū /. With a large ū value o about 1 cm s 1 and 1 4 s 1, the range is cm s 1, all in the realm o inerred values or the ocean bottom. [For example, in the Celtic Sea with u cms 1, Soulsby (199) estimated eddy viscosity values as large as 11 cm s 1 where the parameterization (z) u z still holds.] 3. Ekman pumping The volumetric transport carried by the velocity deicit in the boundary layer has components given by U (u ū) dz 4u ( ū) a() d (1 ) V [ ( ) dz ] (ū ) b() d (a) 4u ( ū) b() d (1 ) [ ] (ū ) a() d. (b) Integral relations (A8a,b) yield u (ū ) U, (1 ) u ( ū) V. (1a,b) (1 ) This Ekman layer transport is perpendicular to the bottom stress vector given by (14a,b) and is at an angle tan 1 to the let o the geostrophic current (in the Northern Hemisphere, since we chose to be positive). More importantly, it is not a linear but a nearly quadratic unction o the geostrophic current. The Ekman-pumping vertical velocity in the geo-

6 197 JOURNAL OF PHYSICAL OCEANOGRAPHY VOLUME 7 TABLE 1. Veering angle and thickness o the Ekman layer or various values o the geostrophic current speed and roughness height. Surace Rossby number ū z Veering angle Ekman-layer thickness d.663 u ū z.489 u ū z.374 u ū z.94 u ū z u ū z strophic interior is obtained rom a vertical integration o the three-dimensional continuity equation: [ ] U V w x y u (ū ) u ( ū), () x 1 y 1 where the intermediate variables u and are nonlinearly related to the geostrophic low (ū, ) via Eqs. (1) and (15). Formula () or the Ekman-pumping rate is the main result o this paper. It diers rom the traditional expression or the Ekman pumping (e.g., Pedlosky 1987, sections 4 5; Cushman-Roisin 1994, sections 5 3): ū w, (3) x y where is the eddy viscosity, then assumed to be a constant. Because the logarithmic unction in (1) prevents us rom solving (1) (15) explicitly or u and, itis diicult to quantiy precisely the dierence between the predictions o () and (3). However, because the logarithm is a weak unction o its argument, the parameter given by (1) may be approximated to a constant. With this approximation the riction velocity u becomes simply proportional to the geostrophic speed ū, according to (15). (Note that the variation o the ratio u /ū shown in Fig. 1 is displayed on a logarithmic linear scale; thus, or a ixed value o the roughness height z, variations o the geostrophic speed within an order o magnitude imply minor variations o the ratio u /ū.) The corresponding expression or the Ekmanpumping rate is w ū(ū ) (1 ) x ] ū( ū). (4) y Then, taking the derivatives o the individual actors and recalling that a geostrophic low is nondivergent on the plane (i.e., ū /x /y), we obtain [ 4 3/ 4 ū ū w 3/ (1 ) x y [ ] 4 ū ū (ū ) ū ( ū) ū. (5) 3/ (1 ) ū x x y y The irst term o (5), that on the upper line, is proportional to the vorticity o the geostrophic low and can be readily identiied with the single term o the traditional expression (3). Identiication o the respective coeicients yields the resulting bulk eddy viscosity: 8 u bulk, (6) (1 ) which can be interpreted as the product o the von Kármán constant, the riction velocity u, and an eective height proportional to u /. This comes as no surprise since the ratio u / is known to provide the depth scale o the turbulent Ekman layer (Csanady and Shaw 198) and that scaling was already apparent in the dimensionless argument o the unctions a and b in (11a, b) above. O much greater interest is the second term o (5), that on the lower line. This term has the same order o magnitude as the irst one and is by no account negligible. Yet, it has no counterpart in the classic theory.

7 1973 In order to highlight some o the dierences, we explore our particular cases, namely, those o a unidirectional sheared current, a solid-body rotation, a zero-vorticity circular low, and a more complex vortex low. For a unidirectional shear low ū, (x), we readily obtain 4 d d w, (7) 3/ (1 ) dx dx where the irst and second terms correspond respectively to the irst and second terms o (5). Thereore, in the case o a unidirectional shear low, the new expression or the Ekman-pumping rate predicts a value twice as large as that predicted by the traditional theory. Physically, the aster the geostrophic low, the greater the transverse velocity and also the greater the bottom stress, the more energetic the turbulence, and the thicker the Ekman layer (thickness proportional to the local geostrophic velocity, and not constant as the traditional theory would have it). Thus, the transverse volumetric transport is not linearly but quadratically depending on the geostrophic current. This quadraticity accounts or the actor. In their numerical simulations o an oceanic bottom boundary layer, Ezer and Weatherly (199) discussed Ekman pumping under a lateral velocity gradient o 3.3 cm s 1 over a distance o 1 km (i.e., d /dx s 1 ) when the turbulent eddy viscosity reached values o about 7 cm s 1 (at 4.5 N, where s 1 ). While the traditional theory would have predicted, according to (3) a vertical Ekman-pumping velocity o 1.73 m/day, Ezer and Weatherly ound a value o 3.5 m/day, that is, almost exactly double, in perect agreement with our result. For a solid-body rotation, we take ū y, x, ū r, where is a constant (positive or a cyclone, negative or an anticyclone in the Northern Hemisphere), and r is the radical distance x y. Straightorward calculations yield 4 r w ( ), (8) 3/ (1 ) where again the irst and second terms map those o (5). In this case, we note that the present theory predicts an Ekman-pumping rate 5% greater than that predicted by the traditional theory. In the case o a zero-vorticity circular low, we take ū y/r, x/r,ū/r, and ind 4 w. (9) 3/ (1 ) r r Thus, while the traditional theory would have predicted no Ekman pumping (except or an ininite value at the center r, where a vorticity singularity exists), the new theory predicts a nonzero Ekman pumping at all distances (and an even stronger singularity at the center). The sign is such that a cyclonic low in the Northern Hemisphere ( ) is accompanied by a downward pumping at inite distances (and an ininite upward pumping at the center). Physically, the absence o vorticity leads to a transverse velocity in the Ekman layer that is not divergent (hence zero pumping according to classic theory), but as the geostrophic velocity decays with radial distance, so does the Ekman-layer thickness. So, unlike the velocity, the volumetric transport is divergent. This divergence is responsible or the Ekman pumping according to the new theory. A combination o the previous two cases yields a more realistic vortical low with a nearly solid-body rotation in the center and a zero-vorticity, potential low at large radial distances: y x ū,, R r R r r ū, (3) R r where sets the polarity and strength o the vortex, and R the distance over which the transition rom solid-body rotation turns into potential low. Again, r is the radial variable, equal to x y. Calculations yield 4 r w 3/ (1 ) (R r ) R (R r ), (31) (R r ) (R r ) where again the irst and second terms correspond to those o (5). Figure 3 compares the prediction o the present theory (sum o both terms) with that o the traditional theory (irst term only). Note the signiicant change in amplitude in the inner part o the vortex and the reversal along its rim. In the general case o a circular low with azimuthal velocity V(r) o arbitrary radial proile, the Ekmanpumping velocity is [ ] 4 V(r) dv V dv w, (3) 3/ (1 ) dr r dr where the term proportional to the relative vorticity dv/ dr V/r would be the sole contribution o the traditional theory. We note that the new theory predicts a larger value (or smaller negative value) where V increases radially, a lower value (or greater negative value) where V decreases radially, and an equal value where V reaches an extremum. For a cyclonic vortex with azimuthal velocity reaching a single maximum at inite distance R rom the center, the new theory predicts a stronger upward pumping within the circle o radius R, an equal pumping at R, and a weaker upward pumping or possibly a downward pumping beyond R. Similarly, or an anticyclonic vortex with azimuthal velocity reaching a single maximum at distance R rom the center, the new

8 1974 JOURNAL OF PHYSICAL OCEANOGRAPHY VOLUME 7 FIG. 3. Radial distributions o the Ekman-pumping vertical velocity or the vortex low (31), according to the present theory (thick line) and traditional theory (thin line). The present theory predicts an upward velocity that is 41% greater and a weak downward velocity at some distance away rom the vortex center. (The vertical axis scale is arbitrary, and only relative variations are signiicant.) theory predicts a stronger downward pumping within the circle o radius R, an equal pumping at R, and a weaker downward pumping or possibly an upward pumping beyond R. 4. Conclusions Our study has revisited the theory o the bottom Ekman layer when the eddy viscosity is made dependent on both the vertical distance rom the bottom and the low magnitude (via the riction velocity); it then recapitulated earlier results and evaluated the resulting Ekman-pumping vertical velocity. A major result is that, because o the dependency o the eddy viscosity on the low itsel, the Ekman-layer thickness increases nearly linearly with the overlying geostrophic current speed, and the volumetric low carried by the velocity deicit in the Ekman layer is more than proportional to the geostrophic current. From the divergence/convergence o this transverse low, we derived a new expression or the Ekman-pumping vertical velocity, which is no longer simply proportional to the relative vorticity o the geostrophic current but is a more complicated unction o its components and their horizontal derivatives. The ull expression is given in () with u and to be derived rom (1) and (15). In practice, however, an approximation can be made, and our new ormulation can be implemented as ollows: Select a bottom roughness height z (usually taken as 5% 1% o the actual height o roughness elements), a value or the Coriolis parameter, and a representative value (scale) o the riction velocity; use (1) to obtain a reerence value o ; then use (4) to obtain the Ekman-pumping rate. In applying the approximate version o our new expression (i.e., by neglecting the weak variation o with the riction velocity), we noted that the Ekman-pumping rate in a unidirectional shear low is twice that predicted by the traditional theory and is 5% greater in the case o solid-body rotation. For a potential low (zero relative vorticity), our new expression yields a inite Ekmanpumping rate instead o zero. We explored how a variation in the ormulation o the rictional orce impacts the magnitude o the Ekman-pumping vertical velocity. This is by no means the sole modiication that can be brought to the traditional bottom Ekman-layer theory. Recently, Hart (1995) considered how the ageostrophic terms in the horizontal-momentum equations modiy the Ekmanpumping rate, and he, too, obtained a new expression. It remains now to combine the two modiications in a uniied theory, which would rely on neither the assumption o geostrophy or the assumption o constant eddy viscosity, and would be ar superior to the traditional theory. Other theoretical generalizations should consider sloping bottoms. In a inal remark, we ask to what extent the ideas and results presented here apply in the presence o vertical stratiication and in the presence o convection. In particular, one could explore their application to the atmospheric boundary layer. Such a task alls beyond the scope o the present article and is matter or uture work. Acknowledgments. This research was initiated when the authors participated in a three-week course on environmental physics organized by the International Centre or Theoretical Physics in Trieste, Italy. The authors wish to express their gratitude to the Centre or its hospitality and to the ollowing participants or their helpul suggestions: A. Alonso-Maleta, N. Baybag, and L. Hapolu-Balas. The work was inished at Dartmouth College under Grant N o the U.S. Oice o Naval Research. APPENDIX Kelvin-Thomson Functions The purpose o this appendix is to derive rom the Kelvin Thomson unctions o zeroth order two particular unctions that orm the natural building blocks or the solution o the problem at hand. Some useul properties o these unctions are also briely established. The Kelvin unctions o order zero are the real and imaginary parts o the solutions w() to the ollowing complex dierential equation: d d dw iw, (A1) d

9 1975 where is real and nonnegative, and i is the imaginary unit (Abramowitz and Stegun 197, 9.9) [Kelvin unctions are occasionally called Thomson unctions (Nosova and Basu 1961); recall that William Thomson was Lord Kelvin s earlier name.] Since (A1) is a second-order equation, it admits two independent complex solutions; one solution grows exponentially and the other vanishes or large values o. The exponentially growing solution inds no place in our boundary-layer analysis and is discarded. Separating the decaying solution into its real and imaginary parts, we write in traditional notation: w() ker () i kei (), (A) and note the ollowing dierential relations obtained rom (A1): d dker kei () (A3a) d d d dkei ker (). (A3b) d d Functions better suited to our present problem are obtained with the change o variable /4; a() ker (), b() kei (), (A4) which obey the dierential equations d d d d da b (A5a) d db a, (A5b) d and vanish or large values o. From the properties o the Kelvin unctions ker (x) and kei (x) (Abramowitz and Stegun 197, and 9.9.1; Nosova 1961, p. 6), we derive the ollowing expansions near the origin: 1 a() ln (, ln) (A6a) b() ln (1 ) (, ln), (A6b) 4 where is the Euler constant (.577). Thus, the unction a has a logarithmic singularity at the origin, while its companion unction b does not. The preceding expressions yield the ollowing values at the origin: da 1 b( ), lim, 4 d db lim. (A7a,b,c) d Integration o the dierential equations (A5a, b) rom zero to ininity with the use o the limits (A7b, c) and o the property that a and b vanish at ininity provides integral values: 1 a() d, b() d. (A8a,b) Finally, we note that the unctions a and b are linearly independent. Indeed, i they were not, we could write Aa Bb C, where A, B, and C are nonzero constants. Application o the dierential operator d(d/d)/d once and twice yields Ab Ba and Aa Bb, respectively. Since a and b are not zero everywhere, it ollows that this system or A and B must have a zero determinant. Since this determinant is A B, it ollows that A B, rom which also ollows C. Thus, there is no linear relationship between the two unctions. REFERENCES Abramowitz, M., and I. A. Stegun, 197: Handbook o Mathematical Functions. 1th ed. Dover, 146 pp. Csanady, G. T., and P. T. Shaw, 198: The evolution o a turbulent Ekman layer. J. Geophys. Res., 85, Cushman-Roisin, B., 1994: Introduction to Geophysical Fluid Dynamics. Prentice Hall, 3 pp. Ellison, T. H., 1956: Atmospheric turbulence. Surveys in Mechanics, G. I. Taylor Anniversary Volume, G. K. Batchelor and R. M. Davies, Eds., Cambridge University Press, Ezer, T., and G. L. Weatherly, 199: A numerical study o the interaction between a deep cold jet and the bottom boundary layer o the ocean. J. Phys. Oceanogr.,, Garratt, J. R., 199: The Atmospheric Boundary Layer. Cambridge University Press, 316 pp. Hart, J. E., 1995: Nonlinear Ekman suction and ageostrophic eects in rapidly rotating luids. Geophys. Astrophys. Fluid Dyn., 79, 1. Lavelle, J. W., and H. O. Mojeld, 1983: Eects o time-varying viscosity on oscillatory turbulent channel low. J. Geophys. Res., 88, Long, C. E., 1981: A simple model or time-dependent stably stratiied turbulent boundary layers. Dept. o Oceanography, University o Washington, Seattle, Special Rep. 95, 3 pp. [Available rom University o Washington, WB-1, Seattle, WA ] Masden, O. S., 1977: A realistic model o the wind-induced Ekman boundary layer. J. Phys. Oceanogr., 7, Nihoul, J. C. J., 1977: Three-dimensional model o tides and storm surges in a shallow well-mixed continental sea. Dyn. Atmos. Oceans,, Nosova, L. N., and P. Basu, 1961: Tables o Thomson Functions: Their First Derivatives. Pergamon Press, 4 pp. Ostendor, D. W., 1984: The rotary bottom boundary layer. J. Geophys. Res., 89, Pedlosky, J., 1987: Geophysical Fluid Dynamics. d ed. Springer- Verlag, 71 pp. Soulsby, R. L., 199: Tidal-current boundary layers. The Sea, B. LeMehaute and D. M. Hanes, Eds., Wiley-Interscience, Weatherly, G. L., and P. J. Martin, 1978: On the structure and dynamics o the oceanic bottom boundary layer. J. Phys. Oceanogr., 8,