Chapter 8 The Ekman Layer (July 12, 2006) SUMMARY: Frictional forces, neglecte in the previous chapter, are now investigate. Their main effect is to create horizontal bounary layers that support a flow transverse to the main flow of the flui. The numerical treatment of the velocity profiles ominate by friction is illustrate with a spectral approach. 8.1 Shear turbulence Because most geophysical flui systems are much shallower than they are wie, their vertical confinement forces the flow to be primarily horizontal. Unavoiable in such a situation is friction between the main horizontal motion an the bottom bounary. Friction acts to reuce the velocity in the vicinity of the bottom, thus creating a vertical shear. Mathematically, if u is the velocity component in one of the horizontal irections an z the elevation above the bottom, then u is a function of z, at least for small z values. The function u(z) is calle the velocity profile an its erivative u/z, the velocity shear. Geophysical flows are invariably turbulent (high Reynols number) an this greatly complicates the search for the velocity profile. As a consequence, much of what we know is erive from observations of actual flows, either in the laboratory or in nature. The turbulent nature of the shear flow along a flat or rough surface inclues variability at short time an length scales, an the best observational techniques for the etaile measurements of these have been evelope for the laboratory rather than outoor situations. Laboratory measurements of nonrotating turbulent flows along smooth straight surfaces have le to the conclusion that the velocity varies solely with the stress τ b exerte against the bottom, the flui molecular viscosity ν, the flui ensity ρ an, of course, the istance z above the bottom. Thus, u(z) = F(τ b, ν, ρ, z). Dimensional analysis permits the elimination of the mass imension share by τ b an ρ but not present in u, ν an z, an we may write more simply: 217
218 CHAPTER 8. EKMAN LAYER u(z) = F ( ) τb ρ, ν, z. The ratio τ b /ρ has the same imension as the square of a velocity, an for this reason it is customary to efine τb u = ρ, (8.1) which is calle the friction velocity or turbulent velocity. Physically, its value is relate to the orbital velocity of the vortices that create the cross-flow exchange of particles an the momentum transfer. The velocity structure thus obeys a relation of the form u(z) = F(u, ν, z) an further use of imensional analysis reuces it to a function of a single variable: u(z) ( u z ) = F. (8.2) u ν In the presence of rotation, the Coriolis parameter enters the formalism an the preceing function epens on two variables: 8.1.1 Logarithmic profile u(z) u = F ( u z ν, fz ). (8.3) u The observational etermination of the function F in the absence of rotation has been repeate countless times, yieling the same results every time, an it suffices here to provie a single report (Figure 8-1). When the velocity ratio u/u is plotte versus the logarithm of the imensionless istance u z/ν, not only o all the points coalesce onto a single curve, confirming that there is inee no other variable to be invoke, but the curve also behaves as a straight line over a range of two orers of magnitue (from u z/ν between 10 1 an 10 3 ). If the velocity is linearly epenent on the logarithm of the istance, then we can write for this portion of the velocity profile: u(z) u = A ln u z ν + B. Numerous experimental eterminations of the constants A an B provie A = 2.44 an B = 5.2 within a 5% error (Pope, 2000). Traition has it to write the function as: u(z) = u K ln u z ν + 5.2 u, (8.4) where K = 1/A = 0.41 is calle the von Kármán constant 1 The portion of the curve closer to the wall, where the logarithmic law fails, may be approximate by the laminar solution. Constant laminar stress νu/z = τ b /ρ = u 2 implies u(z) = u 2 z/ν there. Ignoring the region of transition in which the velocity profile graually 1 in honor of Theoore von Kármán (1881 1963), Hungarian-born physicist an engineer who mae significant contributions to flui mechanics while working in Germany an who first introuce this notation.
8.1. SHEAR TURBULENCE 219 Figure 8-1 Mean velocity profiles in fully evelope turbulent channel flow measure by Wei an Willmarth (1989) at various Reynols numbers: circles Re = 2970, squares Re = 14914, upright triangles Re = 22776, an ownright triangles Re = 39582. The straight line on this log-linear plot correspons to the logarithmic profile of Equation (8.2). (From Pope, 2000) changes from one solution to the other, we can attempt to connect the two. Doing so yiels u z/ν = 11. This sets the thickness of the laminar bounary layer δ as the value of z for which u z/ν = 11, i.e., δ = 11 ν u. (8.5) Most textbooks (e.g., Kunu, 1990) give δ = 5ν/u, for the region in which the velocity profile is strictly laminar, an label the region between 5ν/u an 30ν/u as the buffer layer, the transition zone between laminar an fully turbulent flow. For water in ambient conitions, the molecular viscosity ν is equal to 1.0 10 6 m 2 /s, while the friction velocity in the ocean rarely falls below 1 mm/s. This implies that δ harly excees a centimeter in the ocean an is almost always smaller than the height of the cobbles, ripples an other asperities that typically line the bottom of the ocean basin. Similarly for the atmosphere: the air viscosity at ambient temperature an pressure is about 1.5 10 5 m 2 /s an u rarely falls below 1 cm/s, giving δ < 5 cm, smaller than most irregularities on lan an wave heights at sea. When this is the case, the velocity profile above the bottom asperities no longer epens on the molecular viscosity of the flui but on the so-calle roughness height z 0, such that u(z) = u K ln z z 0, (8.6) as epicte in Figure 8-2. It is important to note that the roughness height is not the average height of bumps on the surface but is a small fraction of it, about one tenth (Garratt, 1992, page 87).
220 CHAPTER 8. EKMAN LAYER Figure 8-2 Velocity profile in the vicinity of a rough wall. The roughness heigh z 0 is smaller than the average height of the surface asperities. So, the velocity u falls to zero somewhere within the asperities, where local flow egenerates into small vortices between the peaks, an the negative values preicte by the logarithmic profile are not physically realize. 8.1.2 Ey viscosity We have alreay mentione in Section 5.2 what an ey iffusivity or viscosity is an how it can be formulate in the case of a homogeneous turbulence fiel, i.e., away from bounaries. Near a bounary, the turbulence ceases to be isotropic an an alternate formulation nees to be evelope. In analogy with Newton s law for viscous fluis, which has the tangential stress τ proportional to the velocity shear u/z with the coefficient of proportionality being the molecular viscosity ν, we write for turbulent flow: τ = ρ 0 ν E u z, (8.7) where the turbulent viscosity ν E supersees the molecular viscosity ν. For the logarithmic profile (8.6) of a flow along a rough surface, the velocity shear is u/z = u /Kz an the stress τ is uniform across the flow (at least in the vicinity of the bounary for lack of other significant forces): τ = τ b = ρu 2, giving an thus ρ 0 u 2 = ρ 0ν E u Kz ν E = Kzu. (8.8) Note that unlike the molecular viscosity, the turbulent viscosity is not constant in space, for it is not a property of the flui but of the flow, incluing its geometry. From its imension ([ν E ] = L 2 T 1 ), we verify that (8.8) is imensionally correct an note that it can be expresse as the prouct of a length by the friction velocity with the mixing length l m efine as ν E = l m u, (8.9) l m = Kz. (8.10) This parameterization is occasionally use for cases other than bounary layers (see Chapter 14).
8.2. FRICTION AND ROTATION 221 The preceing consierations ignore the effect of rotation. When rotation is present, the character of the bounary layer changes ramatically. 8.2 Friction an rotation After the evelopment of the equations governing geophysical motions (Sections 4.1 to 4.4), a scale analysis was performe to evaluate the relative importance of the various terms (Section 4.5). In the horizontal momentum equations [(4.21a) an (4.21b)], each term was compare to the Coriolis term, an a corresponing imensionless ratio was efine. For vertical friction, the imensionless ratio was the Ekman number: Ek = ν E ΩH 2, (8.11) where ν E is the ey viscosity, Ω the ambient rotation rate, an H the height (epth) scale of the motion (the total thickness if the flui is homogeneous). Typical geophysical flows, as well as laboratory experiments, are characterize by very small Ekman numbers. For example, in the ocean at milatitues (Ω 10 4 s 1 ), motions moele with an ey-intensifie viscosity ν E = 10 2 m 2 /s (much larger than the molecular viscosity of water, equal to 1.0 10 6 m 2 /s) an extening over a epth of about 1000 m have an Ekman number of about 10 4. The smallness of the Ekman number inicates that vertical friction plays a very minor role in the balance of forces an may, consequently, be omitte from the equations. This is usually one an with great success. However, something is then lost. The frictional terms happen to be those with the highest orer of erivatives among all terms of the momentum equations. Thus, when friction is neglecte, the orer of the set of ifferential equations is reuce, an not all bounary conitions can be applie simultaneously. Usually, slipping along the bottom must be accepte. Since Luwig Prantl 2 an his general theory of bounary layers, we know that in such a circumstance the flui system exhibits two istinct behaviors: At some istance from the bounaries, in what is calle the interior, friction is usually negligible, whereas, near a bounary (wall) an across a short istance, calle the bounary layer, friction acts to bring the finite interior velocity to zero at the wall. The thickness,, of this thin layer is such that the Ekman number is on the orer of one at that scale, allowing friction to be a ominant force: which leas to ν E Ω 2 1, νe Ω. (8.12) 2 See biography at the en of this chapter.
222 CHAPTER 8. EKMAN LAYER Obviously, is much less than H, an the bounary layer occupies a very small portion of the flow omain. For the oceanic values cite above (ν E = 10 2 m 2 /s an Ω = 10 4 s 1 ), is about 10 m. Because of the Coriolis effect, the frictional bounary layer of geophysical flows, calle the Ekman layer, iffers greatly from the bounary layer in nonrotating fluis. Although, the traitional bounary layer has no particular thickness an grows either ownstream or with time, the existence of the epth scale in rotating fluis suggests that the Ekman layer can be characterize by a fixe thickness. [Note that as the rotational effects isappear (Ω 0), tens to infinity, exemplifying this essential ifference between rotating an nonrotating fluis.] 8.3 The bottom Ekman layer Let us consier a uniform, geostrophic flow in a homogeneous flui over a flat bottom (Figure 8-3). This bottom exerts a frictional stress against the flow, bringing the velocity graually to zero within a thin layer above the bottom. We now solve for the structure of this layer. z Interior u = ū z = 0 Ekman layer u = 0 u(z) Figure 8-3 Frictional influence of a flat bottom on a uniform flow in a rotating framework. In the absence of horizontal graients (the interior flow is sai to be uniform) an of temporal variations, continuity equation (4.21) yiels w/ z = 0 an thus w = 0 in the thin layer near the bottom. The remaining equations are the following reuce forms of
8.3. BOTTOM EKMAN LAYER 223 (4.21a) through (4.21c): fv = 1 ρ 0 p x + ν E + fu = 1 ρ 0 p + ν E 0 = 1 ρ 0 p z, 2 u z 2 2 v z 2 (8.13a) (8.13b) (8.13c) where f is the Coriolis parameter (taken as a constant here), ρ 0 is the flui ensity, an ν E is the ey viscosity (taken as a constant for simplicity). The horizontal graient of the pressure p is retaine because a uniform flow requires a uniformly varying pressure (Section 7.1). For convenience, we align the x axis with the irection of the interior flow, which is of velocity ū. The bounary conitions are then Bottom (z = 0) : u = 0, v = 0, (8.14a) Towar the interior (z ) : u = ū, v = 0, p = p(x, y). (8.14b) By virtue of equation (8.13c), the ynamic pressure p is the same at all epths; thus, p = p(x, y) in the outer flow as well as throughout the bounary layer. In the outer flow (z, mathematically equivalent to z ), equations (8.13a) an (8.13b) relate the velocity to the pressure graient: 0 = 1 ρ 0 p x, fū = 1 ρ 0 p = constant. Substitution of these erivatives in the same equations, which are now taken at any epth, yiels fv f (u ū) = ν E 2 u z 2 = ν E 2 v z 2. (8.15a) (8.15b) Seeking a solution of the type u = ū + A exp(λz) an v = B exp(λz), we fin that λ obeys ν 2 λ 4 + f 2 = 0; that is, where the istance is efine by λ = ± (1 ± i ) 1 = 2νE. (8.16) f
224 CHAPTER 8. EKMAN LAYER Figure 8-4 The velocity spiral in the bottom Ekman layer. The figure is rawn for the Northern Hemisphere (f > 0), an the eflection is to the left of the current above the layer. The reverse hols for the Southern Hemisphere. Here, we have restricte ourselves to cases with positive f (Northern Hemisphere). Note the similarity to (8.12). Bounary conitions (8.14b) rule out the exponentially growing solutions, leaving ( u = ū + e z/ A cos z + B sin z ) (8.17a) ( v = e z/ B cos z A sin z ), (8.17b) an the application of the remaining bounary conitions (8.14a) yiels A = ū, B = 0, or u = ū ( 1 e z/ cos z ) (8.18a) v = ū e z/ sin z. (8.18b) This solution has a number of important properties. First an foremost, we notice that the istance over which it approaches the interior solution is on the orer of. Thus, expression (8.16) gives the thickness of the bounary layer. For this reason, is calle the Ekman epth. A comparison with (8.12) confirms the earlier argument that the bounary-layer thickness is the one corresponing to a local Ekman number near unity. The preceing solution also tells us that there is, in the bounary layer, a flow transverse to the interior flow (v 0). Very near the bottom (z 0), this component is equal to the ownstream velocity (u v ūz/), thus implying that the near-bottom velocity is at 45 egrees to the left of the interior velocity (Figure 8-4). (The bounary flow is to the right of the interior flow for f < 0.) Further up, where u reaches a first maximum (z = 3π/4), the velocity in the irection of the flow is greater than in the interior (u = 1.07ū). (Viscosity can occasionally fool us!) It is instructive to calculate the net transport of flui transverse to the main flow: V = 0 v z = ū 2, (8.19)
8.4. NON-UNIFORM CURRENTS 225 which is proportional to the interior velocity an the Ekman epth. 8.4 Generalization to non-uniform currents Let us now consier a more complex interior flow, namely, a spatially nonuniform flow that is varying on a scale sufficiently large to be in geostrophic equilibrium (low Rossby number, as in Section 7.1). Thus, f v = 1 ρ 0 p x, fū = 1 ρ 0 p, where the pressure p(x, y, t) is arbitrary. For a constant Coriolis parameter, this flow is nonivergent ( ū/ x + v/ = 0). The bounary-layer equations are now f(v v) = f(u ū) = ν E 2 u z 2 ν E 2 v z 2, (8.20a) (8.20b) an the solution that satisfies the bounary conitions aloft (u ū an v v for z ) is ( u = ū + e z/ A cos z + B sin z ) (8.21) ( v = v + e z/ B cos z A sin z ). (8.22) Here, the constants of integration A an B are inepenent of z but, in general, epenent on x an y through ū an v. Imposing u = v = 0 along the bottom (z = 0) sets their values, an the solution is: ( u = ū 1 e z/ cos z ) v e z/ sin z (8.23a) v = ū e z/ sin z ( + v 1 e z/ cos z ). (8.23b) The transport attribute to the bounary-layer structure has components given by U = V = 0 0 (u ū) z = (ū + v) (8.24a) 2 (v v) z = (ū v). (8.24b) 2
226 CHAPTER 8. EKMAN LAYER Figure 8-5 Divergence in the bottom Ekman layer an compensating ownwelling in the interior. Such a situation arises in the presence of an anticyclonic gyre in the interior, as epicte by the large horizontal arrows. Similarly, interior cyclonic motion causes convergence in the Ekman layer an upwelling in the interior. Since this transport is not necessarily parallel to the interior flow, it is likely to have a non-zero ivergence. Inee, U x + V = 0 ( u x + v ) z = 2 ( v x ū ) = 2ρ 0 f 2 p. (8.25) The flow in the bounary layer converges or iverges if the interior flow has a relative vorticity. The situation is epicte in Figure 8-5. The question is: From where oes the flui come, or where oes it go, to meet this convergence or ivergence? Because of the presence of a soli bottom, the only possibility is that it be supplie from the interior by means of a vertical velocity. But, remember (Section 7.1) that geostrophic flows must be characterize by w z = 0, (8.26) that is, the vertical velocity must occur throughout the epth of the flui. Of course, since the ivergence of the flow in the Ekman layer is proportional to the Ekman epth,, which is very small, this vertical velocity is weak. The vertical velocity in the interior, calle Ekman pumping, can be evaluate by a vertical integration of the continuity equation (4.21), using w(z = 0) = 0 an w(z ) = w: w = = 0 ( u x + v 2ρ 0 f 2 p = 1 ρ 0 ) z = ( v 2 x ū νe 2f 3 2 p. (8.27) )
8.5. UNEVEN TERRAIN 227 So, the greater the vorticity of the mean flow, the greater the upwelling/ownwelling. Also, the effect increases towar the equator (ecreasing f = 2Ω sin ϕ an increasing ). The irection of the vertical velocity is upwar in a cyclonic flow (counterclockwise in the Northern Hemisphere) an ownwar in an anticyclonic flow (clockwise in the Northern Hemisphere). In the Southern Hemisphere, where f < 0, the Ekman layer thickness must be reefine with the absolute value of f: = 2ν E / f, but the previous rule remains: the vertical velocity is upwar in a cyclonic flow an ownwar in an anticyclonic flow. The ifference is that cyclonic flow is clockwise an anticyclonic flow is counterclockwise. 8.5 The Ekman layer over uneven terrain It is noteworthy to explore how an irregular topography may affect the structure of the Ekman layer an, in particular, the magnitue of the vertical velocity in the interior. For this, consier a horizontal geostrophic interior flow (ū, v), not necessarily spatially uniform, over an uneven terrain of elevation z = b(x, y) above a horizontal reference level. To be faithful to our restriction (Section 4.3) to geophysical flows much wier than they are thick, we shall assume that the bottom slope ( b/ x, b/) is everywhere small ( 1). This is harly a restriction in most atmospheric an oceanic situations. Our governing equations are again (8.20), couple to the continuity equation (4.21), but the bounary conitions are now Bottom (z = b) : u = 0, v = 0, w = 0, (8.28) Towar the interior (z ) : u = ū, v = v. (8.29) The solution is the previous solution (8.23) with z replace by z b: u = ū e (b z)/ ( ū cos z b v = v + e (b z)/ ( ū sin z b + v sin z b ) v cos z b (8.30a) ). (8.30b) We note that the vertical thickness of the bounary layer is still measure by = 2ν E /f. However, the bounary layer is now oblique, an its true thickness, measure perpenicularly to the bottom, is slightly reuce by the cosine of the small bottom slope. The vertical velocity is then etermine from the continuity equation:
228 CHAPTER 8. EKMAN LAYER w z = u x v {( v = e (b z)/ x ū ) [ + 1 + 1 b x b sin z b (ū v) cos z b [ (ū + v) cos z b + (ū + v) sin z b ] (ū v) sin z b ]}, where use has been mae of the fact that the interior geostrophic flow has no ivergence ( ū/ x + v/ = 0). A vertical integration from the bottom (z = b), where the vertical velocity vanishes (w = 0 because u an v are also zero there) into the interior (z + ) where the vertical velocity assumes a vertically uniform value (w = w), yiels w = ( ū b x ) b + v + 2 ( v x ū ). (8.31) The interior vertical velocity thus consists of two parts: a component that ensures no normal flow to the bottom [see (7.10)] an an Ekman-pumping contribution, as if the bottom were horizontally flat [see (8.27)]. The vanishing of the flow component perpenicular to the bottom must be met by the invisci ynamics of the interior, giving rise to the first contribution to w. The role of the bounary layer is to bring the tangential velocity to zero at the bottom. This explains the secon contribution to w. Note that the Ekman pumping is not affecte by the bottom slope. The preceing solution can also be applie to the lower portion of the atmospheric bounary layer. This was first one by Akerblom (1908), an matching between the logarithmic layer close to the groun (Section 8.1.1) with the Ekman layer further aloft was performe by Van Dyke (1975). Oftentimes, however, the lower atmosphere is in a stable (stratifie) or unstable (convecting) state, an the neutral state uring which Ekman ynamics prevail is more the exception than the rule. 8.6 The surface Ekman layer An Ekman layer occurs not only along bottom surfaces but wherever there is a horizontal frictional stress. This is the case, for example, along the ocean surface, where waters are subject to a win stress. In fact, this is precisely the situation first examine by Vagn Walfri Ekman 3. Fritjof Nansen 4 ha notice uring his cruises to northern latitues that icebergs rift not ownwin but systematically at some angle to the right of the win. Ekman, his stuent at the time, reasone that the cause of this bias was the earth s rotation an subsequently evelope the mathematical representation that now bears his name. The solution 3 See biography at the en of this chapter. 4 Fritjof Nansen (1861 1930), Norwegian oceanographer famous for his Arctic expeitions an Nobel Peace Prize laureate (1922).
8.6. SURFACE EKMAN LAYER 229 was originally publishe in his 1902 octoral thesis an again, in a more complete article, three years later (Ekman, 1905). In a subsequent article (Ekman, 1906), he mentione the relevance of his theory to the lower atmosphere, where the win approaches a geostrophic value with increasing height. z z = 0 Ekman layer Win stress Sea surface (u, v) Interior (u, v) = (ū, v) Figure 8-6 The surface Ekman layer generate by a win stress on the ocean. Let us consier the situation epicte in Figure 8-6, where an ocean region with interior flow fiel (ū, v) is subjecte to a win stress (τ x, τ y ) along its surface. Again, assuming steay conitions, a homogeneous flui, an a geostrophic interior, we obtain the following equations an bounary conitions for the flow fiel (u, v) in the surface Ekman layer: f (v v) = ν E 2 u z 2 + f (u ū) = ν E 2 v z 2 (8.32a) (8.32b) Surface (z = 0) : u ρ 0 ν E z = v τx, ρ 0 ν E z = τy (8.32c) Towar interior (z ) : u = ū, v = v. (8.32)
230 CHAPTER 8. EKMAN LAYER Figure 8-7 Structure of the surface Ekman layer. The figure is rawn for the Northern Hemisphere (f > 0), an the eflection is to the right of the surface stress. The reverse hols for the Southern Hemisphere. The solution to this problem is 2 [ z u = ū + ρ 0 f ez/ τ cos( x π ) 4 2 [ ( z v = v + ρ 0 f ez/ τ x sin π ) 4 ( z τ y sin π )] 4 (8.33a) ( z + τ y cos π )], (8.33b) 4 in which we note that the eparture from the interior flow (ū, v) is exclusively ue to the win stress. In other wors, it oes not epen on the interior flow. Moreover, this win-riven flow component is inversely proportional to the Ekman-layer epth,, an may be very large. Physically, if the flui is almost invisci (small ν, hence short ), a moerate surface stress can generate large rift velocities. The win-riven horizontal transport in the surface Ekman layer has components given by U = V = 0 0 (u ū) z = 1 ρ 0 f τy (v v) z = 1 ρ 0 f τx. (8.34a) (8.34b) Surprisingly, it is oriente perpenicular to the win stress (Figure 8-7), to the right in the Northern Hemisphere an to the left in the Southern Hemisphere. This fact explains why icebergs, which float mostly unerwater, systematically rift to the right of the win in the North Atlantic, as observe by Fritjof Nansen.
8.7. REAL GEOPHYSICAL FLOWS 231 Figure 8-8 Ekman pumping in an ocean subject to sheare wins (case of Northern Hemisphere). As for the bottom Ekman layer, let us etermine the ivergence of the flow, integrate over the bounary layer: 0 ( u x + v ) z = 1 [ ρ 0 x ( τ y f ) ( τ x f )]. (8.35) At constant f, the contribution is entirely ue to the win stress since the interior geostrophic flow is nonivergent. It is proportional to the win-stress curl an, most importantly, it is inepenent of the value of the viscosity. It can be shown furthermore that this property continues to hol even when the turbulent ey viscosity varies spatially (see Analytical Problem 8-7). If the win stress has a non-zero curl, the ivergence of the Ekman transport must be provie by a vertical velocity throughout the interior. A vertical integration of the continuity equation, (4.21), across the Ekman layer with w(z = 0) an w(z ) = w yiels 0 ( u w = + x + v ) z = 1 [ ( ) τ y ( )] τ x ρ 0 x f f = w Ek. (8.36) This vertical velocity is calle Ekman pumping. In the Northern Hemisphere (f > 0), a clockwise win pattern (negative curl) generates a ownwelling (Figure 8-8a), whereas a counterclockwise win pattern causes upwelling (Figure 8-8b). The irections are opposite in the Southern Hemisphere. Ekman pumping is a very effective mechanism by which wins rive subsurface ocean currents (Pelosky, 1996; see also Chapter 20). 8.7 The Ekman layer in real geophysical flows The preceing moels of bottom an surface Ekman layers are highly iealize, an we o not expect their solutions to match actual atmospheric an oceanic observations closely
232 CHAPTER 8. EKMAN LAYER Figure 8-9 Comparison between observe currents below a rifting ice floe at 84.3 N an theoretical preictions base on an ey viscosity ν E = 2.4 10 3 m 2 /s. (Reprinte from Deep-Sea Research, 13, Kenneth Hunkins, Ekman rift currents in the Arctic Ocean, p. 614, 1966, with kin permission from Pergamon Press Lt, Heaington Hill Hall, Oxfor 0X3 0BW, UK) (except in some cases; see Figure 8-9). Two factors, among others, account for substantial ifferences: turbulence an stratification. It was note at the en of Chapter 4 that geophysical flows have large Reynols numbers an are therefore in a state of turbulence. Replacing the molecular viscosity of the flui by a much greater ey viscosity, as performe in Section 4.2, is a first attempt to recognize the enhance transfer of momentum in a turbulent flow. However, in a shear flow such as in an Ekman layer, the turbulence is not homogeneous, being more vigorous where the shear is greater an also partially suppresse in the proximity of the bounary where the size of turbulent eies is restricte. In the absence of an exact theory of turbulence, several schemes have been propose. At a minimum, the ey viscosity shoul be mae to vary in the vertical (Masen, 1977) an shoul be a function of the bottom stress value (Cushman-Roisin an Malačič, 1997). A number of schemes have been propose (see Section 4.2), with varying egrees of success. Despite numerous isagreements among moels an with fiel observations, two results nonetheless stan out as quite general. The first is that the angle between the near-bounary velocity an that in the interior or that of the surface stress (epening on the type of Ekman layer) is always substantially less than the theoretical value of 45 an is foun to range between 5 an 20 (Figure 8-10). See also Stacey et al. (1986). The secon result is a formula for the vertical scale of the Ekman-layer thickness: 0.4 u f, (8.37)
8.7. REAL GEOPHYSICAL FLOWS 233 Figure 8-10 Win vectors minus geostrophic win as a function of height (in meters) in the maritime friction layer near the Scilly Isles. Top iagram: Case of warm air over col water. Bottom iagram: Case of col air over warm water. (Aapte from Roll, 1965) where u is the turbulent friction velocity efine in (8.1). The numerical factor is erive from observations (Garratt, 1992, Appenix 3). Whereas 0.4 is the most commonly accepte value, there is evience that certain oceanic conitions call for a somewhat smaller value (Mofjel an Lavelle, 1984; Stigebrant, 1985). Taking u as the turbulent velocity an the (unknown) Ekman-layer epth scale,, as the size of the largest turbulent eies, we write ν E u. (8.38) Then, using rule 8.12 to etermine the bounary-layer thickness, we obtain 1 ν E f 2 u f, which immeiately leas to (8.37). The other major element missing from the Ekman-layer formulations of the previous sections is the presence of vertical ensity stratification. Although the effects of stratification are not iscusse in etail until Chapter 11, it can be anticipate here that the graual change of ensity with height (lighter flui above heavier flui) hiners vertical movements, thereby reucing vertical mixing of momentum by turbulence; it also allows the motions at separate levels to act less coherently an to generate internal gravity waves. As a consequence, stratification reuces the thickness of the Ekman layer an increases the veering of the velocity vector with height (Garratt, 1992, Section 6.2). For a stuy of the oceanic win-riven Ekman
234 CHAPTER 8. EKMAN LAYER layer in the presence of ensity stratification, the reaer is referre to Price an Sunermeyer (1999). The surface atmospheric layer uring aytime over lan an above warm currents at sea is frequently in a state of convection because of heating from below. In such situations, the Ekman ynamics give way to convective motions, an a controlling factor, besies the geostrophic win aloft, is the intensity of the surface heat flux. An elementary moel is presente later (Section 14.7). Because Ekman ynamics then play a seconary role, the layer is simply calle the atmospheric bounary layer. The intereste reaer is referre to books on the subject by Stull (1988), Sorbjan (1989), Zilitinkevich (1991) or Garratt (1992). 8.8 Numerical simulation of shallow flows The theory presente up to now largely relies on the assumption of a constant turbulent viscosity. For real flows, however, turbulence is rarely uniform, an ey-iffusion profiles must be consiere. Such complexity reners the analytical treatment teious or even impossible, an numerical methos nee to be employe. f 2....................... k τ u h ν E (z) Figure 8-11 A vertically confine flui flow, with bottom an top Ekman layers bracketing a non-uniform velocity profile. The vertical structure can be calculate by a one-imensional moel spanning the entire flui column eventhough the turbulent viscosity ν E(z) may vary in the vertical. To illustrate the approach, we reinstate non-stationary terms an assume a vertically varying ey-viscosity (Figure 8-11) but retain the hyrostatic approximation (8.13c) an continue to consier a flui of homogeneous ensity. The governing equations for u an v are u t fv v t + fu = 1 p ρ 0 x + ( z = 1 ρ 0 p + z 0 = 1 ρ 0 p z. ν E (z) u ) z ( ν E (z) v ) z (8.39a) (8.39b) (8.39c) From the last equation it is clear that the horizontal pressure graient is inepenent of z.