An Implicit Formula for Boundary Current Separation

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NOTES AND CORRESPONDENCE

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An Implicit Formula for Boundary Current Separation DAVID P. MARSHALL

CLAIRE E. TANSLEY

AND

Department of Meteorology, University of Reading, Reading, United Kingdom 1 May 2000 and 8 January 2001 ABSTRACT Boundary layer separation occurs in classical fluids when the boundary layer is decelerated by an adverse pressure gradient. Here a ‘‘separation formula’’ is derived for downstream variations in the velocity, or pressure, of an ocean boundary current. The formula is implicit in the sense that it requires an a priori knowledge of the path of the streamlines. Three contributing processes are identified: the b effect, vortex stretching, and changes in streamline curvature. The b effect acts always to accelerate western boundary currents but to decelerate eastern boundary currents, the former consistent with continued attachment but the latter consistent with separation. Vortex stretching acts to decelerate anticyclonic slope currents but to accelerate cyclonic slope currents, destabilizing the former but stabilizing the latter. Finally, for coastline curvature to induce separation of a boundary current, it must overcome the stabilizing influences of the b effect and/or vortex stretching. Scaling analysis indicates that the condition for separation for a western boundary current from a vertical sidewall is r,

1 2 U b*

1/2

,

where r is the radius of curvature of the coastline, U is the speed of the boundary current, and b* is the gradient of the Coriolis parameter in the downstream direction.

1. Introduction Boundary layer separation is a ubiquitous feature of high Reynolds number flows in classical, nonrotating fluids. Separation occurs when the nearly inviscid fluid lying just outside the boundary layer encounters an adverse pressure gradient and undergoes an appreciable deceleration (e.g., Batchelor 1967). The onset of separation is associated with a singularity in the attached boundary layer (Goldstein 1948; Stewartson 1970). The removal of this singularity requires a ‘‘triple deck’’ structure, in which the separation of the boundary layer is an integral component of the solution. An excellent review of the triple-deck theory of boundary layer separation is given by Smith (1982). In the ocean there has been much recent interest in understanding the dynamics of boundary current separation, and in particular the separation of the Gulf Stream from the North American coastline at Cape Hatteras [see Dengg et al. (1996) and references therein]. However relatively little attention has been given to Corresponding author address: Dr. David Marshall, Department of Meteorology, University of Reading, P.O. Box 243, Reading RG6 6BB, United Kingdom. E-mail: [email protected]

q 2001 American Meteorological Society

understanding boundary current separation in the language of classical fluid dynamics, that is, in terms of the deceleration of a boundary current through an adverse pressure gradient. An important difference between the ocean and a nonrotating fluid is that pressure contours are aligned with streamlines at leading order in the ocean due to the nearly geostrophic nature of the flow. This difference led Haidvogel et al. (1992) to suggest that adverse pressure gradients in the ocean must differ fundamentally from those in classical flows, where the pressure gradient is often externally imposed. Nevertheless, in their analysis of an idealized quasigeostrophic ocean basin model, Haidvogel et al. found that separation was always coincident with rapid deceleration of the boundary current through a strong adverse pressure gradient, independent of the choice of no-slip or no-stress boundary conditions. More recently Baines and Hughes (1996) studied the separation of a western boundary current in a rotating, barotropic laboratory experiment. They obtained an analytical solution for the attached part of the boundary current to demonstrate the occurrence of an adverse pressure gradient at the point of separation, although they did not solve for the details of the separation itself. Tansley and Marshall (2000) also showed that vortex

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JOURNAL OF PHYSICAL OCEANOGRAPHY

stretching, associated with the descent of the deep western boundary current underneath the Gulf Stream, can induce an adverse pressure gradient within the Gulf Stream at Cape Hatteras. The aim of this short contribution is to derive and discuss the implications of an implicit ‘‘separation formula’’ for downstream variations in the velocity, or pressure, of an ocean boundary current. Despite the near coincidence of streamlines and pressure contours, we show that the residual downstream pressure variations are controlled by three large-scale dynamical processes: (i) the b effect, (ii) vortex stretching, and (iii) changes in streamline curvature. In some respects, our analysis is more general than the earlier studies of Baines and Hughes (1996) and Tansley and Marshall (2000), both of which neglected streamline curvature (the Baines and Hughes study was also barotropic). However, in contrast to these studies, our approach is implicit in that it requires an a priori knowledge of the path of the streamlines. Consequently our approach lends itself most naturally to an ‘‘attached flow strategy’’ in which one initially assumes that a boundary current remains attached to a coastline, and then checks for inconsistencies, that is, an adverse pressure gradient, which would imply separation. The note is organized as follows. The separation formula is derived in section 2, the three contributions to the separation formula are discussed in section 3, and, finally, a brief concluding discussion is given in section 4. 2. Derivation of the separation formula We consider a narrow boundary current flowing adjacent to an arbitrary coastline. We assume that the flow is quasi-adiabatic and employ density as a vertical coordinate. For the nearly inviscid fluid lying just outside the no-slip sublayer, the steady-state equations of motion can be written: =M 5 0, r

(1)

= · (hu) 5 0.

(2)

u · =u 1 f k 3 u 1

Here u is the isopycnal fluid velocity, = is the lateral gradient operator evaluated along an isopycnal, f is the Coriolis parameter, M 5 p 1 rgz is the Montgomery potential, p is pressure, r is density, g is the gravitational acceleration, z is height, and h 5 ]z/]r is the isopycnal thickness (or inverse stratification). Mechanical and buoyancy forcing (including the rectified effects of transient eddies) are neglected in the interest of simplifying the mathematical derivation, but can easily be incorporated as additional terms. Boundary layer separation occurs in nonrotating fluids wherever there is an appreciable fall in the velocity of the fluid lying just outside the viscous boundary layer. Here we assume that this result carries over to fluids

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with differential rotation. There is empirical support for this assumption from the numerical experiments of Haidvogel et al. (1992). There is also considerable support from the analyses of attached boundary layers in various flows past cylindrical obstacles (e.g., Merkine 1980; Foster 1985; Page and Johnson 1990). These studies suggest that the dynamics of the viscous boundary layer is essentially unaffected by differential rotation, except through its influence on the larger-scale flow;1 the latter provides the imposed velocity at the outer edge of the viscous boundary layer. Here, therefore, we set ourselves the more modest goal of deriving a formula for downstream velocity variations in the inviscid part of the boundary current. Despite the fact that the pressure contours are nearly aligned with the streamlines in a rotating fluid, taking the inner product of u with (1) gives 2u · =

1 2 25 u·u

u · =M . r

(3)

Hence in a stratified rotating fluid, deceleration of a boundary current is equivalent to fluid parcels encountering an adverse gradient in Montgomery potential; this isopycnal gradient in Montgomery potential is equivalent to an adverse pressure gradient evaluated at constant depth. Consequently, throughout the remainder of this paper we assume that deceleration of a boundary current is synonymous with an adverse pressure gradient, and refer to the two interchangeably. To derive the separation formula we first form an absolute vorticity equation by taking the curl of (1), which we express in the form = · (zu) 1 u · = f 5

f u · =h. h

(4)

Here z 5 k · = 3 u is the vertical component of the relative vorticity. Equation (4) is equivalent to material conservation of the potential vorticity, ( f 1 z )/h. We now assume that the coastline lies to the left of the boundary current and integrate (4) over an area (ABCD) extending across the boundary current as sketched in Fig. 1. The area is bounded by two streamlines, one lying at the outer edge of the viscous boundary layer (AD) and the other at the outer edge of the inertial boundary current (BC), and by two contours normal to streamlines (AB and CD). In the case of a strong boundary current with near-vanishing flow on its flanks, the integration area will be nearly rectangular (Fig. 1a). More generally, the integration area may be distorted by the structure of the streamlines. In the case of a closed 1 This may appear at odds with the asymptotic theory of Ierley and Ruehr (1986) in which rotation introduces an additional term into the Falkner–Skan boundary layer equation. However, at high Reynolds numbers, the rotational term is only dominant within the inertial part of the boundary layer. At scales shorter than the Munk scale, the dominant balance is between viscosity and inertia, as in the other studies cited above.

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After making use of the divergence theorem, the integrated vorticity budget can be written

[ ] [ ] [E ] y2 2

C

2

B

5

y2 2

EE

D

1

A

y2 dn R

CD

1 BA

EE

b*y dA

ABCD

f u · =h dA. h

(6)

ABCD

Here b* 5 ] f /]s is the gradient of the Coriolis parameter in the downstream direction, and [g ]DA [ g D 2 g A. Now consider a typical ocean boundary current such as the Florida Current. Taking a typical boundary current width of d ; 10 5 m, a conservative boundary current velocity of U ; 10 21 m s 21 , b* ; 10 211 m 21 s 21 , and evaluating the integral over a downstream distance of Ds ; 10 5 , we obtain an estimate for the magnitude of the b term:

EE

b*y dA ; b*UdDs ; 1022 m 2 s22 .

ABCD

By comparison, the points B and C are defined to lie outside the boundary current where we take the velocity to be at least an order of magnitude smaller, U ; 10 22 m s 21 . This gives an upper bound for the magnitude of the first term on the left-hand side of (6), FIG. 1. The absolute vorticity equation (4) is integrated over the shaded area ABCD extending across a boundary current. Here the light solid lines represent streamlines and the heavy solid line represents the coast. The shaded area is bounded by two segments parallel to streamlines (AD and BC) and two segments perpendicular to streamlines (AB and CD). In the case of a strong boundary current with near-vanishing flow on its flanks, the integration area will be nearly rectangular (a). However, in the case of a closed gyre it is sensible to take points B and C as coincident with the stagnation point at the center of the gyre (b). The result is expressed in natural coordinates (s, n), where s is a unit vector tangent to streamlines and n is a unit vector normal to streamlines. In these natural coordinates the horizontal velocity is u 5 y s.

gyre, it would make sense for points B and C to both coincide with the stagnation point at the center of the gyre such that the segment BC vanishes (Fig. 1b). In evaluating the integral, we adopt natural coordinates (s, n) where s is the unit vector tangent to the streamlines and n is the unit vector normal to streamlines [by convention, n points to the left of s; see Holton (1992) or Dutton (1995) for further details]. In this natural coordinate system the velocity is y s, and the absolute vorticity is given as

z5

y ]y 2 , R ]n

[] y2 2

C

& U 2 ; 1024 m 2 s22 , B

which can therefore be neglected.2 A similar scaling reveals that the second term in (6) is potentially of the same order as the b term and must be retained. Thus, we obtain the approximate balance:

[] y2 2

D

ø A

EE [E

b*y dA 2

ABCD

1

EE

f u · =h dA h

ABCD

]

y2 dn R

CD

.

(7)

BA

Equation (7) indicates that the downstream speed of a boundary current, or equivalently its pressure, is dependent on three distinct processes: the b effect, vortex stretching, and changes in streamline curvature. Finally separation will occur whenever the fluid lying just outside the viscous boundary layer undergoes an appreciable deceleration; that is, y D2 , y A2 . The condition for separation of a boundary current with a coastline to its left is therefore

(5)

where R is the radius of curvature of the streamlines (with positive values denoting deflection to the left).

2 Note that this term vanishes identically if B and C are taken to coincide with the stagnation point at the center of the gyre, as sketched in Fig. 1b.

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TABLE 1. The character of the boundary current can be varied by changing the signs of b* and f. The table corresponds to the case in which the coastline is to the left of the boundary current; WBC refers to a western boundary current and EBC to an eastern boundary current.

b* . 0 f .0 f ,0

EE

b* , 0

Poleward WBC Equatorward WBC

b*y dA 2

ABCD

EE

ABCD

Equatorward EBC Poleward EBC

f u · =h dA 1 h

[E ] y2 dn R

, 0.

CD

BA

(8)

In the case that the coastline is to the right of the boundary current, the sign of the inequality is reversed. 3. Implications for the separation of ocean boundary currents We now discuss the implications of each of the three terms in (8) for the separation of ocean boundary currents. Here we consider only the case in which the coastline lies to the left of the boundary current, as assumed in deriving (8). However the conclusions reached below apply equally in the case that the coastline lies to the right of the boundary current. The use of natural coordinates makes the application of the separation formula to both western and eastern boundary currents straightforward simply by varying the sign of b* and to poleward and equatorward flowing boundary currents simply by varying the sign of f ; details are given in Table 1. a. b effect The sign of the planetary vorticity term in (8) is dependent on the sign of b* since y is positive-definite in natural coordinates. For a western boundary current with a coastline to its left, b* . 0 (see Table 1), and hence the planetary vorticity gradient acts to accelerate the current. Conversely for an eastern boundary current with a coastline to its left, b* , 0, and hence the planetary vorticity gradient acts to decelerate the current. The same conclusions hold when the coastline is to the right of the boundary current. This leads to the somewhat surprising, but general, conclusion that the b effect always inhibits the separation of western boundary currents, but encourages the separation of eastern boundary currents. This result is consistent with a wide range of observations and model results. Boundary currents are known to form freely on the western margins of basins, whereas they tend to be unstable on the eastern margins of basins. The traditional explanations are in terms of steady-state vorticity budgets (Stommel 1948; Munk 1950) and the westward propagation of Rossby waves (Pedlosky

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1965). While the present result does not constitute a theory for the western intensification of the ocean circulation, it does provide an alternative explanation for the relative stability of western boundary currents compared with eastern boundary currents. This result is also consistent with laboratory and numerical studies of flow past a cylinder on a b plane. The planetary vorticity gradient delays separation downstream of a cylinder in eastward flow, whereas it enhances separation downstream of a cylinder in westward flow (Merkine 1980; Boyer and Davies 1982; Foster 1985; Merkine and Brevdo 1986; Page and Johnson 1990; Tansley and Marshall 2001, manuscript submitted to J. Phys. Oceanogr.). Finally, this result is consistent with an tendency for western boundary currents to overshoot their ‘‘natural’’ separation latitude at the line of zero wind stress curl. A nice illustration is provided by the numerical study of Cessi et al. (1987) in which anticyclonic wind stress forcing is applied over the southern portion of a rectangular basin with no forcing to the north. Nevertheless the western boundary current extends to the northernmost margin of the basin without separating at the line of vanishing wind stress curl. b. Vortex stretching In practice ocean boundary currents do not separate from vertical sidewalls, but from sloping shelves. Stern (1998) has presented a theory for the separation of the Gulf Stream involving the bunching together of bathymetric contours at Cape Hatteras, which forces the Gulf Stream off the continental shelf and into deeper water. ¨ zgo¨kmen et al. (1997) The numerical experiments of O also emphasize the importance of a western boundary current crossing bathymetric contours in order to separate from a continental shelf. For a slope current to separate it must move offshore, involving vortex stretching. In the case that the coastline lies to the left of the current, the effect of vortex stretching in (8) is to decelerate the current if f . 0, corresponding to an anticyclonic slope current, but to accelerate the current if f , 0, corresponding to a cyclonic slope current. The same conclusions hold when the coastline is to the right of the current. Consequently vortex stretching enhances the separation of anticyclonic slope currents (such as the Florida Current), but inhibits the separation of cyclonic slope currents (such as the deep western boundary current, which remains attached throughout its passage around the North Atlantic). Finally, we note that vortex stretching may provide a localized forcing in the separation balance. The specific example of the Deep Western Boundary Current descending beneath the Gulf Stream at Cape Hatteras is discussed in some detail by Tansley and Marshall (2000); the interested reader is referred to that manuscript for further details.

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FIG. 2. Schematic diagram corresponding to the scaling analysis for separation of a western boundary current due to coastline curvature. The boundary current, speed U and width d, initially flows along a straight coastline, but suddenly encounters a region of coastline curvature, radius r. The distance over which the curvature changes is Ds. By geometrical considerations, Ds ; r.

c. Coastline curvature In order for coastline curvature to decelerate a boundary current and induce its separation, it is necessary to overcome the stabilizing influences of the b effect and/ or vortex stretching. This contrasts with the situation in classical fluid flows where boundary layers tend to separate wherever there is favorable boundary curvature. As an illustration, here we consider the idealized problem of a western boundary current, speed U and width d, flowing along a vertical sidewall at the western margin of an flat-bottomed ocean basin. The coastline is initially straight, but the boundary current suddenly encounters a cape of positive curvature, radius r, as sketched in Fig. 2. We apply the separation formula, (8), across the downstream distance over which the curvature changes, Ds. Through geometrical considerations, this downstream distance should scale as Ds ; r. We initially assume that the boundary current remains attached and estimate the magnitude of the terms involving b and streamline curvature on the right-hand side of (8). The b term scales as

EE

b*y dA ; b*UdDs,

(9)

and the curvature term scales as

[E

]

12

y2 1 dn ; U 2dD . R R

(10)

For a net deceleration, and hence separation of the boundary current, we require the curvature term (10) to be larger than the b term (9). Setting D(1/R) 5 1/r (since the radius of curvature is infinite downstream of the cape), the condition for separation is therefore r,

1 2 U b*

1/2

.

(11)

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The form of this relation corresponds exactly to the ‘‘b parameter’’ that controls downstream separation of eastward flow past a circular cylinder (Tansley and Marshall 2001, manuscript submitted to J. Phys. Oceanogr.). Given typical values for the Gulf Stream of b* ; 10 211 m 21 s 21 and U ; 0.4 m s 21 , we require r , 200 km. The first point at which the 200-m isobath (an excellent proxy for the path of the Gulf Stream prior to its separation) undergoes such curvature of the correct sign is indeed at Cape Hatteras, the Gulf Stream separation point. We anticipate that a similar argument can be constructed for the separation of a cyclonic slope current at a cape (for example, the shedding of meddies by the Mediterranean Undercurrent at Cape St. Vincent). 4. Concluding remarks In this short contribution we have derived a vorticity equation relating changes in the downstream speed, or pressure, of a boundary current to the b effect, changes in streamline curvature, and vortex stretching. This is motivated by the observation that boundary layer separation occurs in classical high Reynolds number fluid flows whenever the fluid just outside the viscous boundary layer undergoes an appreciable deceleration. The numerical experiments of Haidvogel et al. (1992) suggest that this result carries over to rotating fluids, irrespective of the choice of no-slip or no-stress boundary conditions. Our main findings are R The b effect accelerates western boundary currents, consistent with continued attachment, whereas it decelerates eastern boundary currents, consistent with separation. R Vortex stretching acts to decelerate anticyclonic slope currents but to accelerate cyclonic slope currents, destabilizing the former but stabilizing the latter. R For coastline curvature to induce separation of a boundary current, it must overcome the stabilizing influences of the b effect and/or vortex stretching. The condition for separation for a western boundary current from a vertical sidewall is r , (U/b*)1/2 where r is the radius of curvature of the coastline, U is the speed of the boundary current, and b* is the gradient of the Coriolis parameter in the downstream direction. In contrast to Baines and Hughes (1986), our approach is inherently implicit in that it requires a priori knowledge of the path of the streamlines. Nevertheless, there are many instances in which a useful first approximation to the path of the streamlines can be obtained by assuming that a boundary current remains attached to a coastline. One can then check for consistency through the absence of adverse pressure gradients. Conversely if an adverse pressure is found, then separation can be expected to occur. One issue we have not addressed is the boundary layer dynamics linking

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the deceleration of a boundary current to its separation. As mentioned earlier, the analyses of Merkine (1980), Foster (1985), and Page and Johnson (1990) suggest that the dynamics of the viscous boundary layer is essentially unaffected by differential rotation. The effect of b is then purely to modify the larger-scale circulation, which in turn affects the advection of vorticity within the boundary layer and the boundary stress. In the above studies, separation is identified with the singularity in the attached boundary layer at the point that the boundary stress vanishes (Goldstein 1948; Stewartson 1970). However, a complete theory for boundary layer separation in a rotating fluid will most likely require a generalization of the triple-deck theory developed for classical fluids (Smith 1982). A further issue we have neglected is the role of lateral boundary conditions. The experiments of Haidvogel et al. (1992) suggest that the coincidence of boundary current separation with a strong adverse pressure gradient is a robust feature of both no-slip and no-stress solutions. Nevertheless numerical gyre solutions do show sensitivity of separation location to the choice of lateral boundary condition, for example no-slip conditions can lead to premature separation (e.g., Haidvogel et al. 1992). However, virtually all such results have been obtained at relatively low-to-moderate Reynolds numbers, whereas our approach is only valid in the high Reynolds number limit. Numerical experiments carried out by Ierley, Pedlosky, and Young (reported in Pedlosky 1996, pp. 87–88) suggest that no-slip solutions increasingly resemble the no-stress limit as the Reynolds number is increased. Convergence to a unique solution at high Reynolds numbers has yet to be achieved, but it would appear that premature separation is not a characteristic property of such flows. It is also worth noting that there may be implicit dependence of separation on the lateral boundary condition through its impact on the large-scale streamlines, which would in turn modify the various terms in the separation condition (8). In conclusion, we hope this paper will stimulate a renewed interest in the separation of ocean boundary currents from the perspective provided by classical fluid dynamics. In particular we hope that the separation formula will encourage a quantitative assessment of the many theories that have been proposed to explain the separation of boundary currents, both in the ocean and in numerical ocean models. Acknowledgments. We wish to thank Steve Belcher for directing us towards the articles of Frank Smith on the triple-deck theory of boundary layer separation. We are also grateful to Maarten Ambaum, Roger Samelson,

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and two anonymous reviewers for comments that lead to a significantly improved manuscript. The financial support of the Natural Environment Research Council (GR3/10157) and, previously, the U.K. Meteorological Office (met1b/2170) is gratefully acknowledged. REFERENCES Baines, P. B., and R. L. Hughes, 1996: Western boundary current separation: Inferences from a laboratory experiment. J. Phys. Oceanogr., 26, 2576–2588. Batchelor, G. K., 1967: An Introduction to Fluid Dynamics. Cambridge University Press, 615 pp. Boyer, D. L., and P. A. Davies, 1982: Flow past a circular cylinder on a b-plane. Philos. Trans. Roy. Soc. London, 306A, 533–556. Cessi, P., G. Ierley, and W. Young, 1987: A model of the inertial recirculation driven by potential vorticity anomalies. J. Phys. Oceanogr., 17, 1640–1652. Dengg, J., A. Beckmann, and R. Gerdes, 1996: The Gulf Stream separation problem. The Warmwatersphere of the North Atlantic Ocean, W. Kraus, Ed., Gebruder Borntrager, 253–289. Dutton, J. A., 1995: Dynamics of Atmospheric Motion. Dover, 617 pp. Foster, M. R., 1985: Delayed separation in eastward, rotating flow on a b-plane. J. Fluid Mech., 155, 59–75. Goldstein, S., 1948: On laminar boundary layer flow near a point of separation. Quart. J. Mech. Appl. Math., 1, 43–69. Haidvogel, D. B., J. C. McWilliams, and P. R. Gent, 1992: Boundary current separation in a quasigeostrophic, eddy-resolving ocean circulation model. J. Phys. Oceanogr., 22, 882–902. Holton, J., 1992. An Introduction to Dynamic Meteorology. Academic Press, 511 pp. Ierley, G. R., and O. G. Ruehr, 1986: Analytic and numerical solutions of a nonlinear boundary-layer problem. Stud. Appl. Math., 75, 1–36. Merkine, L. O., 1980: Flow separation on a b-plane. J. Fluid Mech., 99, 399–409. Munk, W. H., 1950: On the wind-driven ocean circulation. J. Meteor., 7, 79–93. ¨ zgo¨kmen, T. M., E. P. Chassignet, and A. M. Paiva, 1997: Impact O of wind forcing, bottom topography, and inertia on midlatitude jet separation in a quasigeostrophic model. J. Phys. Oceanogr., 27, 2460–2476. Page, M. A., and E. R. Johnson, 1990: Flow past cylindrical obstacles on a beta-plane. J. Fluid Mech., 221, 349–382. Pedlosky, J., 1965: A note on the western intensification of the ocean circulation. J. Mar. Res., 23, 207–209. ——, 1996: Ocean Circulation Theory. Springer-Verlag, 453 pp. Smith, F. T., 1982: On the high Reynolds number theory of laminar flows. IMA J. Appl. Math., 28, 207–281. Stern, M. E., 1998: Separation of a density current from the bottom of a continental shelf. J. Phys. Oceanogr., 28, 2040–2049. Stommel, H., 1948: The westward intensification of wind-driven ocean currents. Trans. Amer. Geophys. Union, 29, 202–206. Stewartson, K., 1970: Is the singularity at separation removable? J. Fluid Mech., 44, 347–364. Tansley, C. E., and D. P. Marshall, 2000: On the influence of bottom topography and the Deep Western Boundary Current on Gulf Stream separation. J. Mar. Res., 58, 297–325. ——, and ——, 2001: Flow past a cylinder on a b plane, with application to Gulf Stream separation and the Antarctic Circumpolar Current. J. Phys. Oceanogr., in press.