Comment on “Why western boundary currents are diffusive: A link between bottom pressure torque and bolus velocity” by C. Eden and D. Olbers

Ocean Modelling 39 (2011) 416–424

Contents lists available at ScienceDirect

Ocean Modelling journal homepage: www.elsevier.com/locate/ocemod

Comment

Comment on ‘‘Why western boundary currents are diffusive: A link between bottom pressure torque and bolus velocity’’ by C. Eden and D. Olbers Richard J. Greatbatch a,⇑, Chris W. Hughes b a b

Leibniz Institut für Meereswissenschaften an der Universität Kiel, IFM-GEOMAR, Kiel, Germany National Oceanography Centre, Liverpool, UK

a r t i c l e

i n f o

Article history: Received 25 March 2010 Received in revised form 29 November 2010 Accepted 3 December 2010 Available online 10 December 2010 Keywords: Western boundary currents Bottom pressure torque Bolus velocity

a b s t r a c t Eden and Olbers have discussed the relationship between bottom pressure torque and bolus velocity in the western boundary current using the vertically truncated BARBI model approach. Here we revisit this issue using the much simpler residual mean framework. The central role played by a density equation that is linearised about a state of rest is discussed, as well as mechanisms required to maintain the baroclinicity of the western boundary current. We conclude that in the framework being considered by Eden and Olbers, frictional processes must play an important role in the western boundary current dynamics, otherwise the baroclinicity of the current is completely removed by the cross-front mixing effect of the eddies. We also derive the form of the Stommel equation obtained by Eden and Olbers in a manner which clarifies the approximations made by these authors. We argue that for their analysis to be valid, the flow must be concentrated in a shallow layer compared to the ocean depth, there must be no density structure at the sea floor, and any overturning circulation, whether directly wind-driven or as a part of the global thermohaline circulation, must be much smaller than the western boundary current transport. Ó 2010 Elsevier Ltd. All rights reserved.

1. Introduction In a recent paper, Eden and Olbers (2010) have discussed a link between the bolus velocity and bottom pressure torque in the dynamics of western boundary currents. Whereas these authors used the machinery of the BARBI model (Olbers and Eden, 2003), an ocean circulation model based on a truncated series of vertical structure equations, here we use residual mean theory to illustrate, in a simple and transparent framework, the basic underlying ideas and some of their consequences. The traditional view, since the classic papers of Stommel (1948), Munk (1950), has been that vorticity input from the wind forcing in the gyre interior is dissipated in the western boundary current by frictional processes. More recently, Hughes and de Cuevas (2001) have argued that the barotropic dynamics of the gyre circulation can be closed at the western boundary by bottom pressure torque, reducing the dependence on friction. In this scenario, transport across lines of latitude in the western boundary region is achieved primarily by means of vorticity input from the bottom pressure torque. The latter is associated, for a poleward flowing western boundary current such as the Gulf Stream or the Kuroshio, with a downward vertical velocity at the bottom that is a conse⇑ Corresponding author. E-mail addresses: [email protected] (R.J. Greatbatch), [email protected] (C.W. Hughes). 1463-5003/$ - see front matter Ó 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.ocemod.2010.12.001

quence of the vortex stretching of the entire water column resulting from the poleward flow. In a uniform density ocean with variable bottom topography, there is a correspondence to flow that is both poleward and downward when conserving its potential vorticity by following the f/H contours along the continental slope. The idea that friction might not be as important in the dynamics of western boundary currents as previously thought, at least for the barotropic, vertically-averaged flow, was anticipated by Mellor et al. (1982), and later by Greatbatch et al. (1991), when these authors noticed that because the f/H contours at the western boundary run along, rather than into the coast, there appeared to be no need to invoke friction in their diagnostic model calculations (see also Salmon (1998)). By using the BARBI formalism, Eden and Olbers (2010) have reintroduced the idea that what they call diffusion is important in western boundary current dynamics, even when bottom pressure torque dominates the barotropic vorticity balance. The diffusion comes from the thickness diffusivity associated with the Gent and McWilliams (1990) parameterisation for the eddy-induced (or bolus) velocity. Indeed, these authors present a form of Stommel’s (1948) equation in which the thickness diffusivity plays the role of the bottom friction in Stommel’s original equation. At the heart of this approach (although it is not clear in their paper) is the use of a linearised density equation from which all the basic results follow. The structure of this comment is as follows. In Sections 2 and 3 we derive the basic equations and review the linear vorticity

417

R.J. Greatbatch, C.W. Hughes / Ocean Modelling 39 (2011) 416–424

balance in the residual mean framework. Then in Section 4 we discuss the density equation in some detail, highlighting the simplification made by Eden and Olbers (2010) in their theoretical analysis of neglecting the horizontal advection and the diabatic mixing terms. We then go onto discuss mechanisms for maintaining the baroclinicity of the western boundary current and the likely importance of frictional dynamics for maintaining a source of dense water on the western side of the current in the numerical experiments of Eden and Olbers (2010). In Section 5, we present a pared-down derivation of the Stommel equation that appears in Eden and Olbers (2010), starting from the density equation linearised about a state of rest and employing linear vorticity dynamics. A number of assumptions are made along the way which, in turn, are discussed in detail in Section 6 where a detailed derivation of the Stommel equation is presented and critically assessed. Section 7 provides a Summary and Conclusions. 2. The residual mean equations We work with equations for the residual mean velocity whose components are ur (eastward), vr (northward) and wr (upward) in the x, y, z directions, respectively. The horizontal momentum equations then take the form

 f vr ¼  fur ¼ 

1

qo

1

qo

px þ

py þ

1

qo

1

qo

T xz

T yz þ

þ







mug z z ; 

mv g z z ;

ð1Þ ð2Þ

where qo is a representative density for sea water, T = (Tx, Ty) is the horizontal stress associated with small scale turbulence, equal to the wind stress at the surface and the bottom stress at the bottom, and p is the perturbation pressure (hereafter referred to simply as the pressure). The approach of Greatbatch and Lamb (1990) is used to represent the vertical flux of momentum due to the mesoscale eddies (the eddy form drag effect) using a vertical eddy viscosity m given by

m¼K

ð3Þ

; 2

where f is the Coriolis parameter, N is the buoyancy frequency and K is a diffusivity corresponding to the thickness diffusivity of Gent and McWilliams (1990). For the Gent and McWilliams (1990) parameterisation (used throughout this comment), the horizontal velocity (ug, vg) that appears in the vertical mixing term is the geostrophic velocity (Greatbatch and Lamb (1990), Gent et al. (1995)) given by f v g ¼  q1 px ; fug ¼  q1 py . We also have the hydrostatic o o and continuity equations

@p ¼ g q0 ; @z ur x þ v r y þ wr z ¼ 0;

ð4Þ ð5Þ

respectively, where q0 is the perturbation density. q0 in turn, satisfies the density equation

q0t þ ur q0x þ v r q0y þ wr ðq þ q0 Þz ¼ Q ;

ð6Þ

where N2 ¼ g qqz is the square of a representative buoyancy freo  which is a function of quency based on the background density q  þ q0 , and Q refers to diabatic mixing proz only, total density q ¼ q cesses (note that we are thinking here in terms of a simple linear equation of state). For later convenience we note that the horizontal components of the Eulerian mean, (u, v), and bolus velocities, (u⁄, v⁄) satisfy the horizontal momentum equations 

1

qo

px þ

1

qo

T xz ;

fu ¼ 

1

qo

py þ

1

qo

T yz

ð7Þ

and 

f v ¼









mug z z ; fu ¼ mv g z z ;

ð8Þ

where (ur, vr, wr) = (u, v, w) + (u , v , w ), and that each of (u, v, w) and (u⁄, v⁄, w⁄) satisfy the continuity equation (5), i.e. ⁄

ux þ v y þ wz ¼ 0;

⁄

⁄

ux þ v y þ wz ¼ 0:

ð9Þ

At both the surface z = 0 and the ocean bottom z = H, the eddy stress, qo(mugz, mvgz), satisfies the boundary condition

ðmug z ; mv g z Þ ¼ 0

ð10Þ

appropriate for the Gent and McWilliams (1990) parameterisation (cf. Greatbatch and Li (2000)). We also have the kinematic boundary condition applied to each of the residual, Eulerian and bolus velocities; at the surface, z = 0:

wr ¼ w ¼ w ¼ 0

ð11Þ

and at the bottom z = H:

wr ¼ ður Hx þ v r Hy Þ;

w ¼ ðuHx þ v Hy Þ;

w ¼ ðu Hx þ v  Hy Þ:

ð12Þ

Use of the kinematic boundary conditions together with (9) leads to the vertically-integrated continuity equation

U x þ V y ¼ 0;

ð13Þ R0

where ðU; VÞ ¼ H ðu; v Þdz. Finally in this section we note that a consequence of the boundary conditions on the eddy stress, (10), is that the vertical integral of the bolus velocity is zero, i.e.

0¼

Z

0

ðu ; v  Þdz:

ð14Þ

H

It follows that

ðU; VÞ ¼

f2 N

f v ¼ 

Z

0

ðu; v Þdz ¼

Z

H

0

ður ; v r Þdz

ð15Þ

H

with the implication that the vertically-integrated transport carried by the residual and Eulerian mean velocities is identical. 3. Derivation of the vorticity balance We begin by vertically integrating the horizontal momentum equations (1) and (2) to give

 qo fV ¼ Hpbx  Ex þ sx ;

ð16Þ

qo fU ¼ Hpby  Ey þ sy ;

ð17Þ

where pb is the bottom pressure, s = (sx, sy) is the difference between surface and bottom stress (we assume this is dominated by R0 wind stress, and neglect bottom stress here), and E ¼ g H zq0 dz is the potential energy per unit area of the water column. (To derive (16) and (17) use is made of the hydrostatic equation (4) to write Rz R0 0 the pressure as p ¼ pb  g H q0 dz . It follows that H px dz ¼ R0 Rz 0 0 @ q dz dz. Integration by parts is then used to write Hpbx  g @x H H R0 R0 Rz 0 0 q dz dz in terms of E, and likewise for H py dz). The vorticity H H @ equation for the vertically-integrated flow is obtained by taking @x of @ (17) and subtracting @y of (16) to obtain

h

i

qo bV ¼ Jðpb ; HÞ þ syx  sxy ;

ð18Þ

df where b ¼ dy comes from the variation of the Coriolis parameter with latitude. Use has also been made of the vertically-integrated continuity equation (13). Variants of equation (18) can be found

418

R.J. Greatbatch, C.W. Hughes / Ocean Modelling 39 (2011) 416–424

in Holland (1973), Greatbatch et al. (1991), Hughes and de Cuevas (2001). In the absence of the bottom pressure torque J(pb, H), e.g. in a flat-bottomed ocean, (18) reduces to the equation for the flatbottomed Sverdrup transport

h

qo bV ¼ syx  sxy

i

ð19Þ

dating back to Sverdrup (1947). In the case of the subtropical gyres that feed the Gulf Stream and Kuroshio, for which the wind stress h i  sy  @ sx < 0 , (19) implies southcurl is predominantly negative @@x @y ward transport. In the classical theories of Stommel (1948), Munk (1950) a frictional torque is added to (19) to allow northward return flow in the western boundary current so that the net northward transport integrated across an ocean basin is zero. Models, on the other hand, including those studied by Eden and Olbers (2010), show the important role played by the bottom pressure torque in closing the gyre circulation at the western boundary (Hughes and de Cuevas, 2001). The bottom pressure torque is associated, for a subtropical gyre, with a downslope Eulerian mean flow at the bottom. One way to see this is to consider the vorticity equation for the Eulerian mean flow which is readily obtained from (7), using (9), to give

bv ¼ fwz þ

1 h

qo

i

T yx  T xy : z

ð20Þ

bV ¼ fwb þ

qo

i

syx  sxy ;

ð21Þ

where the subscript b denotes evaluation at the bottom and (sx, sy) is the surface wind stress (neglecting the bottom stress). Using the fact that the vertical integral of the bolus velocity is zero, V on the left hand side can refer to the northward transport by either the Eulerian mean or the residual mean velocity. Clearly, to have northward flow in the western boundary layer requires (for f > 0) wb < 0, i.e. downslope Eulerian mean flow at the bottom. Furthermore Eqs. (18) and (21) are equivalent with the implication that

qo fwb ¼ Jðpb ; HÞ;

ð22Þ

that is, the vertical component of the Eulerian mean velocity at the bottom is directly related to the bottom pressure torque (this result does not depend on the vertical integral of the bolus velocity being zero and is easily derived from the Eulerian mean equations alone, neglecting only the bottom stress). From the perspective of equation (22), poleward (in the case of the subtropical gyre) transport in the western boundary current is accompanied by stretching of the water column associated with a downslope component of the Eulerian mean vertical velocity along the continental slope at the western boundary. 4. The simplified version of the density equation used by Eden and Olbers (2010) and some of its consequences Rather than work with the full density equation (6), Eden and Olbers (2010) neglect the horizontal advection terms and work, instead, with

q0t þ wr ðq þ q0 Þz ¼ Q :

q0t þ wr q z ¼ 0:

ð23Þ

They justify neglect of the horizontal advection terms ‘‘As a consequence of the small relative magnitudes of the perturbations from  )’’-see the paragraph imme(the) reference buoyancy profile (here q diately following Eq. (3) in their paper. They also put Q = 0 on the grounds that the effect of diapycnal mixing is small compared to the advective effect by mesoscale eddies (see their Appendix B). Noting that if the horizontal advection terms are neglected from

ð24Þ

Eq. (24) is central to the analysis in Eden and Olbers (2010). In particular, in steady state, which is the situation of interest here, we have

z ¼ 0 wr q

ð25Þ

implying that either

wr ¼ 0

ð26Þ

or

q z ¼ 0:

ð27Þ

 z is not zero (by definition) implying that we For a stratified ocean, q must have

wr ¼ 0:

ð28Þ

Eq. (28) says that the vertical component of the residual mean velocity must be zero with the immediate implication that the Eulerian mean vertical velocity w must be exactly balanced by the vertical component of the eddy-induced or bolus velocity w⁄,

w ¼ w

Integrating (20) over the depth of the water column gives

1 h

(6), we can self-consistently drop the perturbation density from the vertical advection term without further loss of generality (as is easy shown using a scale analysis), we arrive at the density equation linearised about a state of rest, i.e.

ð29Þ

a balance that must hold throughout the solution domain, including the western boundary region. The same analysis can be found on page 21 of Eden and Olbers (2010) (the next to last paragraph before the Acknowledgements). Furthermore, in their Fig. 9, Eden and Olbers show a comparison between the vertical components of the Eulerian mean and the bolus velocity at the ocean bottom in their model, indicating a strong tendency for cancellation. To proceed further, we begin by writing (25) in the form

z  z ¼ w q wq

ð30Þ

in order to separate the effects of the Eulerian mean and bolus vertical velocities. To parameterise the bolus velocity, w⁄, Eden and Olbers (2010) use the Gent and McWilliams parameterisation (Gent and McWilliams (1990)) for which

w ¼ rh  ðKLÞ;

ð31Þ

 z and rh is the horizontal gradient where the vector L ¼ rh q0 =q operator (see Eq. (18) in Gent et al. (1995) and note that we have replaced the vertical gradient of the total density by the vertical  , to be consistent with dropping gradient of the reference density, q the vertical gradient of the perturbation density from (6)). For the case of uniform K considered by Eden and Olbers (2010), and using the boundary layer approximation that derivatives across the boundary current are much larger than those along the boundary current, (31) becomes

 z ¼ K q0xx wq

ð32Þ

showing (perhaps surprisingly) that the Gent and McWilliams parameterisation corresponds to a horizontal diffusion of density. Eq. (32) expresses a balance between the vertical advection of the background density by the Eulerian mean flow and the eddy mixing of density across the western boundary current. We now ask how this balance can be maintained in the western boundary region. For the case of a subtropical gyre, a western boundary region is characterised by relatively light water on the eastern side of the western boundary current and relatively dense water in the western side, with an implied horizontal gradient in density across the front, as observed, for example, across the Gulf Stream. Mesoscale eddies try to remove the gradient of density across the

R.J. Greatbatch, C.W. Hughes / Ocean Modelling 39 (2011) 416–424

front, an effect which in Eq. (32) is represented by the horizontal diffusion term on the right hand side. In order to maintain the front in density against the eddy mixing, Eq. (32) implies that there must be a downward Eulerian mean velocity on the east side of the front (where eddy mixing tries to increase the density) and an upward Eulerian mean velocity on the western side of the front (where eddy mixing tries to decrease the density). Interestingly, this is the pattern we see in the vertical component of the Eulerian mean velocity shown for the ocean bottom in Fig. 9 of Eden and Olbers (2010). However, a complication is that for a subtropical gyre western boundary current, it is a consequence of inviscid linear vorticity dynamics that the Eulerian mean vertical velocity must be everywhere downward. To see this we note that in the western boundary current, the vorticity equation for the Eulerian mean flow, that is Eq. (20), is dominated by the beta term and the vortex stretching term, i.e.

bv ¼ fwz :

ð33Þ

For a subtropical gyre, v is poleward from which it follows (for the northern hemisphere, f > 0) that wz > 0. Hence, if v is poleward at all depths, then wz > 0 at all depths and since w = 0 at z = 0, and since w < 0 at the bottom, it follows that w < 0, i.e. downward, at all depths. Fig. 9 in Eden and Olbers (2010) nevertheless clearly shows that the Eulerian mean velocity in their primitive equation model is upward on the western side of the subtropical western boundary current region (indeed, there is an interesting antisymmetry in their plot between the subtropical and subpolar gyre regions). This means that something more than inviscid linear vorticity dynamics is operating in the western boundary region of their model, strongly suggestive that frictional or inertial processes are playing a role in setting the vertical component of the Eulerian mean velocity on the western side of the western boundary current. That frictional processes are indeed important on the western side of the boundary current is clear from Fig. 6 in Eden and Olbers (2010) (frictional processes are also favoured by their standard model configuration in which a vertical sidewall occupies the top 2.5 km depth range). Furthermore, a simple scale analysis shows that w in their Fig. 9 is sufficient in their model to maintain the density gradient across the boundary current in the face of the cross-front mixing implied by the Gent and McWilliams parameterisation. The importance of the upwelling on the western side of the boundary current can be further appreciated by noting that, if it were valid to use a density equation that is linearised about a state of rest, without this upwelling the cross-front mixing by the eddies would completely remove the baroclinicity of the western boundary current. The above analysis has shown that in order to maintain the baroclinic structure of a subtropical gyre, western boundary current, a source of dense water is required to balance the cross-front mixing effect of the eddies. Such a source of dense water can be provided by upwelling of dense water on the western side of the boundary current, as seems to be happening in the models used by Eden and Olbers (2010), or it can come from either or both of two other routes: (i) horizontal advection of dense water (e.g. in the case of the Gulf Stream by the shelf break current that is, in turn, an extension of the cold and fresh Labrador Current, or indeed the Deep Western Boundary Undercurrent) or (ii) diabatic mixing (e.g. convective cooling during the winter months). Both of these sources of dense water have been neglected by Eden and Olbers (2010) in their theoretical analysis but could play an important role in the dynamics of the observed western boundary currents, especially after separation from the coast (e.g. in the Gulf Stream or Kuroshio Extension regions). It is also worth noting that nonlinearity in the vorticity balance (not included in either our analysis or that of Eden and Olbers (2010)) can also potentially contribute

419

to upwelling of dense water on the western side of the boundary current. 5. The link to the Stommel equation An interesting aspect of Eden and Olbers (2010) is the reappearance of the Stommel equation, but with the role of friction in that equation replaced by the eddy diffusion implied by the Gent and McWilliams parameterisation. In this section, we give a pareddown derivation of the Stommel equation that appears in Eden and Olbers (2010), pointing out the main assumptions that are made. A more comprehensive derivation is given in Section 6. For the pared-down version, we assume that the ocean depth H depends only on the coordinate x which, in turn, is measured positive eastwards, and we neglect from the start all viscous stress terms, assuming them to be negligible in the boundary current region. To formulate boundary conditions at the ocean surface, z = 0, and the ocean bottom, z = H, we note that the requirement that the eddy stress be zero, i.e. that mugz = mvgz = 0 at z = 0 and z = H (cf. Eq. (10)) corresponds (since for the Gent and McWilliams parameterisation, (ug, vg) is the geostrophic velocity) to the requirement that either m = 0 or q0x ¼ q0y ¼ 0 at z = 0 and z = H. For the case m – 0, this means that density at the bottom is a function of H only, so we can choose this to be the reference density profile and can therefore put q0 = 0 everywhere at the bottom z = H. The assumption that q0 = 0 at the ocean bottom is clearly a strong assumption which is unlikely to be valid in reality, suggesting that a more appropriate implementation of the zero eddy stress condition would to be to put m = 0. Nevertheless, to derive the Stommel equation it is necessary to put q0 = 0 at z = H, an assumption that occurs throughout Eden and Olbers (2010), either explicitly or implicitly. (We note that the boundary conditions for use with the Gent and McWilliams parameterisation are a source of difficulty – see, for example, Ferrari et al. (2008). For an alternative approach to dealing with the ocean bottom, readers are referred to Greatbatch and Li (2000).) To derive the Stommel equation as it appears in Eden and Olbers (2010), we begin by writing the density equation, once again, in the form (Eq. (32))

 z ¼ K q0xx : wq

ð34Þ

 z and using the identity Multiplying by z=q

 zw ¼

 z2 z2 w  wz 2 2 z

ð35Þ

we obtain

 2  z z2 zgK 0 q : w  wz ¼  2 xx 2 2 q z oN

ð36Þ

We now integrate over the depth of the water column and follow Eden and Olbers (2010) in assuming that K and N2 are independent of depth. We can use integration by parts together with the boundary conditions q0x ¼ 0 and q0 = 0 at z = H to simplify the right hand side. Combining with (21) and (20) (with the stress terms set to zero) we obtain



qo b V 

1 H2

 2 2Kf Exx z2 v dz ¼  2 2 ; H N f H

Z

0

ð37Þ

where E is the potential energy per unit area introduced after Eq. (17). It should be noted that we have already used the boundary condition q0 = 0 at z = H to arrive at (37). If the northward flow is concentrated in a near-surface layer of characteristic depth scale h, then the second term on the left hand side is of order (h/H)2 compared to the first, and can be neglected if

420

R.J. Greatbatch, C.W. Hughes / Ocean Modelling 39 (2011) 416–424

h is shallow compared with H. In such a surface-concentrated flow, the depth-integrated flow must be dominated by the geostrophic flow relative to the bottom. Integrating thermal wind from the bottom up, and using once again the boundary condition q0 = 0 at z = H, results in

We begin by noting that multiplying the linearised density Eq.  z and integrating over depth, gives an equation for the (24) by z=q first moment of the vertical velocity:

qo f ðWy ; Wx Þ ¼ qo f ðU; VÞ ¼ ðEy ; Ex Þ;

g

ð38Þ

2Kf 2

2

H N2

Wxx :

ð39Þ 2

2

Noting that Eden and Olbers (2010) put k2 ¼ H6fN2 , we then have

bWx ¼ 

K 3k2

Wxx

Z

qo

where we have introduced the streamfunction W for the depthintegrated flow. In fact, it is easy to show that the approximation implied by (38), together with the closure assumed by Eden and Olbers (2010) (their Eq. (20)), results in the neglect of the second term in (37) (see also Eq. (67) in the Appendix where this issue is discussed in more detail). Substituting (38) in (37), and neglecting this second term, then gives

bWx ¼ 

6. Detailed derivation of the Stommel equation

ð40Þ

which is the same as Eq. (6) in Eden and Olbers (2010) apart from the fact that we have approximated the equation to the western boundary region and we have not included a friction term (although this could be added). Although this looks like Stommel’s equation, it is worth noting that the introduction of k disguises the fact that it is not a constant-coefficient equation. In fact, because k is proportional to H, the characteristic length scale K/(3bk2) becomes longer as the water becomes shallower. It should be noted that although friction has not been included in the above analysis, friction could be added and, indeed, as we have seen in Section 4, appears to be important for maintaining the baroclinic structure of the density field in the western boundary region in the numerical experiments of Eden and Olbers. A consequence is that the friction term in the original Stommel model cannot simply be replaced by a diffusion term arising from using the Gent and McWilliams parameterisation; more likely both have to be present (and also nonlinearity in a more complete analysis). A key assumption throughout the above analysis is that the northward flow is concentrated in a near-surface layer, an assumption that clearly cannot be made without there being significant baroclinicity in the western boundary current region. Such an assumption is implicit in the use of Eq. (38) to replace E in (37) by qofW. Fig. 4 in Eden and Olbers (2010) shows that, in fact, (38) looks to be reasonable approximation in their primitive equation model. Dropping the second term on the left hand side of (37) is, however, an even stronger requirement for the flow to be surface-confined and, interestingly, is a consequence not only of being able to replace E by qofW (i.e. Eq. (38)) but of, in addition, the closure assumption that is used by the BARBI model (Eq. (20) in Eden and Olbers (2010)). Likewise, the assumption that q0 = 0 at z = H also requires that the flow be confined near the surface. These issues are explored further in the next section where we also show that the Eden and Olbers (2010) approach actually only requires imposing a constraint on a single moment of the vertical velocity, allowing some greater (although somewhat special) flexibility in the density equation than is provided by the linearised form (24). Nevertheless, the strong limitation to surface-trapped flows, and other approximations along the lines of ignoring effects of bottom flows or bottom density, seem somewhat disturbing in a theory which is supposed to describe interactions with topography. For this reason, it seems worthwhile to provide a more systematic derivation of the Stommel equation.

0

H

zq0t N

2

dz ¼

Z

0

zwr dz:

ð41Þ

H

For the case in which N2 is constant, the left hand side is equal to Et/ (qoN2) and the first moment of wr is related simply to changes in E, which is the first moment of q0 , allowing for a neat closure in the time-dependent BARBI formulation. However, since we are most interested in the steady state, it is more straightforward simply to say that we are imposing a constraint only on the first moment of wr (which, from (41), must be zero in the steady state), irrespective of whether N2 is constant. In fact, in what follows, only the first moment of wr is required to be zero, allowing for the possibility that there might be other terms in the density equation (e.g. arising from horizontal advection or diabatic mixing) that also have zero first moment. Although this is a rather special requirement, it does represent a significant relaxation of the constraint on wr imposed by the steady state linearised density equation (25). In particular, if there are no eddies or diffusion, the constraint on wr imposed by (25) becomes a constraint on the Eulerian mean velocity, w, implying for a stratified ocean that w = 0 at all depths in steady state (cf. (26) with the bolus velocity, w⁄ = 0). The wind-driven circulation is then confined to a thin layer at the surface (as can be seen by putting w = 0 in (20); also referred to as the ‘‘delta-function jet’’ by Rhines and Young (1982), see also Salmon (1998)). Imposing a constraint only on the first moment of w allows for finite thickness flows that can extend beneath the surface layer and are restricted by only a single integral constraint, rather than a constraint at each depth. In the Appendix we show (see Eq. (69)), starting from the equations for the Eulerian mean velocity given by (7), and making no further approximations (in particular, no use is made of the density equation or of the assumptions that q0 and its horizontal derivatives are zero at the ocean bottom) that

qo

Z

0

H

zw dz ¼

1 bE3x Jðp ; H3 =f Þ  2 þ r  6 b 6f

Z

0

H

z2 T z dz; 2f

ð42Þ

R0 where E3 ¼ g H z3 q dz, and the stress term is negligible if all viscous stresses are confined to a narrow surface layer. As the derivation in the Appendix shows, a consequence of the closure used in Eden and Olbers (2010) is the neglect of the E3 term in (42), which, as can be seen from Eq. (66), is equivalent to neglecting the gradient of E3 in comparison to H2 times the gradient of E. This is equivalent to assuming that horizontal density gradients (and hence @/@z of the geostrophic currents) occur predominantly at a shallow depth compared to the total ocean depth. Such an approximation is clearly invalid for any situation in which the flow is dominated or strongly influenced by near-bottom currents and associated density gradients – indeed the whole foundation of the BARBI model is unsuited to situations in which higher moments of density are at least as important as the first moment, E. It does, however, allow for a western boundary current (for example) in which the vertical shear occurs predominantly near the surface, but a shear-free component of the current penetrates to the bottom, allowing bottom pressure torques to arise. Returning to (42), we note that if the first moment of w is constrained to be zero (in particular in steady state and in the absence of eddies so that the residual and Eulerian mean vertical velocities are identical), and if the stress term and the E3 term are neglected (the latter as in the closure used by Eden and Olbers (2010)), then J(pb, H3/f)  0. This is the case ‘‘without friction and diffusion’’

421

R.J. Greatbatch, C.W. Hughes / Ocean Modelling 39 (2011) 416–424

discussed by Eden and Olbers (2010) in which bottom pressure is constant along contours of H3/f, and with the implication that the constant is zero if J(pb, H3/f)  0 remains valid right up to the eastern boundary point (at the equator) from which these contours originate. A bottom pressure signal can arise if there are closed H3/f contours (or if the neglected quantities become important somewhere), but it is a little surprising to see H3/f rather than the more usual H/f contours arising here. In fact, this is an artifact of the particular truncation that is chosen in BARBI. Eq. (69) (see the appendix) is just one of an infinite series of equations for the nth moment of w (see equation (70)), each of which involves terms like J(pb, Hn+2/f) and E(n+2)x, with the implication that the relevant characteristic contours depend on the moment of w that is being constrained. The chosen truncation (neglecting E3) used in BARBI is therefore complementary to the choice of constraint on w (i.e. the first moment). Furthermore, this truncation is the one which prevents the occurrence of bottom pressure signals in the inviscid, no eddies limit, with the exception of the special case of closed H3/f contours. The constraint on just one moment of w thus allows greater flexibility than constraining w to be zero at all depths. Derivation of the Stommel equation, however, requires further approximations to be made, running the risk of throwing away this greater flexibility. Neglecting the viscous stress terms, (42) can be written (see (66)) as

qo

Z

2

0

zw dz ¼ H

"

#

H bEx bE3x 2qo bWx þ  2 : 6f f fH

ð43Þ

Reintroducing the eddies, we note from (41) that it is not the first moment of w that is constrained to be zero in steady state, but rather the first moment of wr = w + w⁄. This means that the Eulerian component on the left hand side of (43) must be balanced by a bolus velocity w⁄, which can be written in terms of density using (31). Balancing the first moments of w and w⁄ then gives

" # Z 0 H2 bEx bE3x K zr2 q0 dz; 2qo bWx þ  2 ¼ g 2 6f f H N fH

ð44Þ

where the r operator is two-dimensional in the horizontal. Three more approximations must be made in order to derive the Stommel equation. The first is, once again, to neglect E3 (the closure assumption used by the BARBI model). The second is to set K/N2 to a constant, allowing it to be taken outside the integral. Associated with this approximation is the effect, discussed in Section 5, of the boundary condition setting horizontal density gradients to zero at the bottom, thus allowing r2 also to be taken outside the integral. If, instead, K is allowed to go to zero at the boundary, more complicated terms involving bottom density also arise. Neglecting this possibility, (44) reduces to

  H2 bEx K ¼  2 r2 E: 2qo bWx þ 6f f N

ð45Þ

The final approximation is to set qobWx  b Ex/f (cf. Eq. (38)), which leads to a Stommel equation in E or, approximately, in W. This last approximation is perhaps the most interesting, and is worth discussing in detail. From the depth-integrated zonal momentum equation (16), we can see that the approximation of replacing qobWx by bEx/f is equivalent to neglecting both the wind stress and the bottom pressure term Hpbx. To examine this issue in more detail, we integrate (16) from x = x0 at the eastern edge of the western boundary region to x = e on the eastern boundary, to give

W0 ¼

E0 1 þ qo f f qo

Z

e

w

sx dx ¼

E0

qo f

þ Q E;

ð46Þ

where QE is the net northward Ekman transport, and subscript 0 means the value at x = x0 (note that the contribution of the frictional stress from the western boundary region itself has been assumed to be negligible, enabling the integral involving the stress in (46) to go all the way across the basin). We have neglected the bottom pressure term consistent with the idea that east of the western boundary region, the flow is everywhere above the topography, which follows from neglect of the eddies in the ocean interior. In order for qoWx and Ex/f to be approximately equal in the boundary region, and given the boundary condition W = 0 and E = 0 at the western boundary (the latter resulting from either depth or boundary density tending to zero), it is clear that the Ekman transport QE must be much smaller than the western boundary current transport W0. We can calculate the western boundary current transport from Sverdrup balance in the interior (Eq. (19)), to give

W0 ¼ 

1 qo b

Z

e w

½syx  sxy dx ¼

1 @y qo b @y

Z

e

w

sx dx ¼ 

ðfQ E Þy ; b

ð47Þ

where terms involving the alongcoast wind stress have been neglected for simplicity. Then, for Ekman transport to be negligible compared with W0, we clearly need

  f  jfQ E j   ðfQ E Þy : b

ð48Þ

In fact, the left hand side is zero where Ekman transport is zero (typically near the gyre centre), where it can obviously be neglected, but the opposite is true where (fQE)y = 0, near the gyre boundary. Simple scale analysis clearly cannot tell us that the Ekman transport contribution can be neglected everywhere, the question is over what fraction of the gyre it can be neglected. As a concrete example, we can calculate the Ekman and boundary current transports for a case in which zonally-integrated stress is a sine function of latitude. For a gyre of width 20°, centred at 30°N, we find that the boundary current is larger than Ekman flux over 87% of the gyre, but is ten times larger than Ekman flux over only 31% of the gyre. The approximation is therefore strongly convincing only in the central third of the gyre, although this fraction increases as the gyre scale decreases, or the gyre moves further from the equator. There is, however, more to this approximation, as we can see from (16). The bottom pressure term may be negligible in the interior, but not in the boundary region. In the boundary region we have, neglecting wind stress, qofWx = Ex + Hpbx, so equating qofWx and Ex requires the bottom pressure term to remain negligible here too. In fact, in the zonal integral, we know that the bottom pressure term must balance the stress term (Hughes and de Cuevas (2001)), so the criterion for neglecting the bottom pressure term in the western boundary region becomes the same as that discussed above for neglecting the Ekman transport in the gyre interior. However, it is possible for Hpbx to be positive in part of the boundary current and negative in another part, so that it could be an important term even when its integral is small. To see what this means in physical terms, recall that the zonal integral balance between wind stress and the bottom pressure term (form stress) represents, in z-coordinates, a meridional overturning circulation: a southward Ekman flux is balanced by a geostrophic northward return flow, which requires a negative pressure anomaly on the western slope (compared with the eastern slope). The scaling given above effectively assumes that the geostrophic return flow is single-signed: to the north at all depths for a subtropical gyre. In a situation with a meridional overturning circulation (in z-coordinates, but driven by thermohaline processes) in addition to that driven by Ekman flow, as in the North Atlantic, the relevant comparison to the western boundary transport is no longer the Ekman transport, but the overturning transport (multiplied by two, since the

422

R.J. Greatbatch, C.W. Hughes / Ocean Modelling 39 (2011) 416–424

horizontal length scale is now at most half of the width of the boundary region with the flow going one way in one part of the slope and the other in another part of the slope). In this case, neglecting the bottom pressure term becomes a poor approximation if the overturning circulation is comparable to half the total western boundary current transport, as is probably the case in the North Atlantic (e.g. Schmitz et al. (1992)). The effect of the approximations made by Eden and Olbers (2010) in deriving the Stommel equation is therefore, effectively, to remove any three-dimensional character from the flow. The flow must be concentrated in a shallow layer compared to the ocean depth, there must be no density structure at the sea floor, the effect of diffusion must not excite any three-dimensional structure, and any overturning circulation, whether directly wind-driven or as a part of the global thermohaline circulation, must be much smaller than the western boundary current transport. In this sense, the Eden and Olbers (2010) Stommel equation is a weak-topography extension of flat-bottomed ocean dynamics. In particular, for a flat-bottomed ocean it is possible to use separation into vertical normal modes. In fact, as long as the eddy diffusivity, K, is spatially uniform, then, as shown by McCreary (1981), separation into vertical normal modes proceeds exactly as described in Gill (1982). The equations for the horizontal structure associated with each mode are then the linear shallow water equations, but including a Rayleigh friction term that depends on K. For the steady-state residual mean equations used here (see Section 2), the corresponding shallow water equations are

 f v r ¼ g gx þ X 

f 2K ug ; c2

fur ¼ g gy þ Y 

ur x þ v r y ¼ 0;

f 2K vg; c2 ð49Þ

where (X, Y) is the projection of the wind forcing term onto the vertical normal mode, and c is the shallow water gravity wave speed for the mode in question (ur, vr, ug, vg are here projections of the corresponding horizontal velocity components onto the vertical normal mode). These equations are, for quasi-geostrophic dynamics, the same as those studied by Stommel (1948). The width of the 2 boundary layer at the western boundary is Kf and corresponds dic2 b rectly to the boundary layer width implied by (40) above with N2H2/2 = c2.

Eq. (38), which Eden and Olbers (2010) must make in order to derive their Stommel equation, is equivalent to the more stringent requirement that the flow itself be concentrated near the surface. These requirements can only be met in flows for which the overturning circulation, whether directly wind-driven or as a part of the global thermohaline circulation, must be much smaller than the western boundary current transport. In effect, the Stommel equation requires that the circulation be surface concentrated and layerwise two-dimensional, restrictions which are unlikely to apply to realistic western boundary currents, casting doubt over the usefulness of the analysis presented in Eden and Olbers (2010). We have also shown that a consequence of inviscid, linear vorticity dynamics is that the Eulerian mean vertical velocity must be downward in a poleward flowing western boundary current, a conclusion that is not consistent with the vertical velocity shown in Fig. 9 of Eden and Olbers (2010), and also does not allow for the maintenance of the baroclinic structure of the western boundary current in the face of the cross-front mixing by the eddies. This suggests that frictional dynamics (or possibly nonlinearity) must be important in the numerical experiments of Eden and Olbers (2010). We noted that in the case of a subtropical gyre, a source of dense water is required to maintain the baroclinicity of the boundary current against the cross-front mixing by the eddies. Such a source of dense water can result from upwelling that is induced either by friction (as seems to be the case in the numerical experiments of Eden and Olbers (2010)) or nonlinear vorticity advection. In the case of the observed western boundary currents, either or both of two other routes are also possible: (i) horizontal advection of dense water (e.g. in the case of the Gulf Stream by the shelf break current that is, in turn, an extension of the cold and fresh Labrador Current, or the Deep Western Boundary Current) or (ii) diabatic mixing (e.g. convective cooling during the winter months). Both of these sources of dense water have been neglected by Eden and Olbers (2010) in their theoretical analysis. Acknowledgments This work has been funded by IFM-GEOMAR and the UK Natural Environment Research Council as part of the Oceans 2025 science programme. We are grateful to two anonymous reviewers for insightful comments that led to many improvements in this comment.

7. Summary and conclusions Appendix A In this note, we have used the residual mean equations to reexamine the link between bottom pressure torque and bolus velocity that has been introduced by Eden and Olbers (2010). The residual mean approach makes it clear that the key constraint, resulting from the steady state, linearised density equation (25), is on the vertical component of the residual mean velocity, thereby linking the vertical components of the Eulerian mean and bolus velocities. Using the Gent and McWilliams (Gent and McWilliams (1990)) parameterisation, Eden and Olbers (2010) derive a form of Stommel’s equation (Stommel (1948)) in which the role of friction is replaced by the lateral diffusion associated with the crossfront mixing by eddies. We have rederived the Stommel equation using the residual mean approach and pointing out the many approximations that have been made by Eden and Olbers (2010). We note that, in fact, the constraint on the vertical velocity has only been applied to the first moment of the vertical residual mean velocity. We have shown how this weaker constraint, together with the Olbers and Eden (2003) closure assumption, retains the requirement that the vertical velocity be zero at the bottom in the absence of eddies (except along closed contours of H3/f), but requires the assumption that any vertical shear in the flow be concentrated near the surface. We also show how the additional assumption given by

Starting with the Eulerian mean, planetary geostrophic momentum equation (i.e. (7)):

qo f k  u ¼ rp þ T z

ð50Þ

and taking the curl gives

bv  fwz ¼

1

qo

r  Tz;

ð51Þ

where, in this appendix, we will interpret r as the two-dimensional horizontal gradient, and r as the vertical component of the curl, such that r  T ¼ ½T yx  T xy . Alternatively, integrating (50) over depth, this becomes (cf. (16) and (17)):

qo f rW ¼ rE þ Hrpb  s;

ð52Þ

where s is the difference between surface and bottom stress, W is a streamfunction for the depth-integrated flow, pb is pressure at the R0 R0 Rz 0 bottom, and E ¼ H q0 gz dz ¼ g H H q0 dz dz. The zonal component of this is

423

R.J. Greatbatch, C.W. Hughes / Ocean Modelling 39 (2011) 416–424

qo f Wx ¼ Ex þ Hpbx  sx :

ð53Þ

We can take the curl of (52) after dividing by either 1, f or H, to obtain three vorticity (or potential vorticity) equations:

qo bWx ¼ Jðpb ; HÞ þ r  s;

bEx s ¼ Jðpb ; H=f Þ þ r 

f2

f

ð54Þ

s : H

Z

2

H wb zw dz ¼   2 H

0

2

H

z wz dz ¼ 2

Z

0

H

2

2

ðH  z Þwz dz 2

ð57Þ

and from (51), we can rewrite this as

Z

0

zw dz ¼

H

1 2f

Z

0



1 ðH2  z2 Þ bv  r  T z dz

qo

H

ð58Þ

which gives

Z



Z 0 H2 1 b z2 v dz bWx  r  s  2f H 2f qo Z 0 1 r z2 T z dz: þ 2qo f H

0

zw dz ¼

H

0

zw dz ¼

H

ð59Þ



Z 0 H2 2 1 b bWx  r  s  z2 v 0 dz 2f H 2f 3 qo Z 0 1 þ r z2 T z dz: 2qo f H

@ @x

Z

z

H

qo f v 1 ¼ 

g H

0

qdz0  T xz ;

H

@ @x

Z

z

ð61Þ

q dz0 dz 

H

sx H

¼

E x  sx ; H

qo f v

@ ¼ g @x

Z

z

E x  sx  H

0

T xz

0

@ 2 z @x

qdz 

H

ð63Þ

H

2

Z

0

z2 v dz ¼

H

2 1 bEx bWx þ 3 3 qo f

ð67Þ

qo

" # H2 bE3x zw dz ¼ 2Jðpb ; HÞ þ f Jðpb ; H=f Þ  2 6f H fH Z 0 2 z Tz þr dz H 2f

Z

0

ð68Þ

Z

0

zw dz ¼

H

1 bE3x Jðp ; H3 =f Þ  2 þ r  6 b 6f

Z

0

H

z2 T z dz: 2f

ð69Þ

Note that we have made no approximations in deriving (69) beyond the initial assumption of planetary geostrophy implied by (50). Finally, we note that in fact the whole procedure follows through in an analogous manner for any moment of w, giving the general equation

q0

Z

0

zn w dz ¼

H

! " bEðnþ2Þ x 1 ðHÞnþ2  J pb ; ðn þ 1Þðn þ 2Þ f f2 # Z 0 nþ1 z Tz þ ðn þ 2Þr  dz ; f H

ð70Þ

of which (69) can be considered to be a special case for n = 1 and (55) a special case for n = 1 (in the latter case the corresponding moment of w reduces to w at the surface which is zero). References

Z

0

z2 v 0 dz ¼ g

H

Z

H

Z

Z

H2 q dz0 dz  z2 T xz dz  3 H H z

0

 ðEx  sx Þ:

ð64Þ

Integrating by parts, this becomes

qo f

b

ð62Þ

and

qo f

" # H2 bEx bE3x s 2qo bWx þ  2  2r  s  f r  6f f f fH Z 0 2 z Tz dz: ð66Þ þr H 2f

which demonstrates how neglecting E3 and making the assumption that qoWx = Ex/f is equivalent to ignoring the second term on the left hand side of (67) (see also Eq. (37) and the discussion thereon). That neglect requires that v be concentrated in a layer which is within a small distance of the surface compared to the total depth H. Continuing with the main derivation, we now note that, in (66), we can use (54) to substitute for qobWx  r  s, and (55) to substitute for bEx/f2  r  (s/f) to give

ð60Þ

so that 0

H

bWx 

qo

such that @v1/@z = @v/@z, where v is given by the zonal component of (50). We then have v 0 ¼ v 1  v 1 , where v 1 is the depth-averaged value of v1, and

Z

zw dz ¼

which can be rewritten as

We can calculate the term involving v0 by defining

qo f v 1 ¼ g

0

As an aside at this point we note that ignoring all the stress terms (appropriate in an inviscid boundary current), and neglecting the E3 term (this is the closure used by Eden and Olbers (2010) in their Eq. (20)), (66) and (59) lead to

Separating v into its depth average v ¼ Wx =H and deviation from the average v0 , this can be rewritten as

Z

Z

ð56Þ

(We do not use (56) in what follows but include it for completeness.) Now, using the surface boundary condition w = 0, we can write the depth integral of zw as 0

qo

ð55Þ

;

qo JðW; f =HÞ ¼ JðE; 1=HÞ þ r 

Z

T xz associated with bottom stress, so we will retain this term for completeness. Substituting (65) into (60), and regathering terms involving stress (i.e. s and T), we obtain

Z

0

H

z2 v 0 dz ¼

  Z 0 H2 E3x x   E þ s z2 T xz dz; x 3 H2 H

ð65Þ

R0 where E3 ¼ g H z3 q dz. In Eq. (20) of Eden and Olbers (2010), the closure assumption is shown to consist of the neglect of E3 in this equation. It is worth noting at this point that, if Tz is significant only in a near surface Ekman layer of thickness hek, then the term in T xz is a factor of 3(hek/H)2 smaller than the term in sx, and therefore usually negligible. However, this is not the case for any contribution to

Eden, C., Olbers, D., 2010. Why western boundary currents are diffusive: a link between bottom pressure torque and bolus velocity. Ocean Modell. 32, 14–24. Ferrari, R., McWilliams, J.C., Canuto, V., Dubovikov, D., 2008. Parameterisation of eddy fluxes at ocean boundaries. J. Climate 31, 2770–2789. Gent, P.R., McWilliams, J.C., 1990. Isopycnal mixing in ocean circulation models. J. Phys. Oceanogr. 20, 150–155. Gent, P.R., Willebrand, J., McDougall, T.J., McWilliams, J.C., 1995. Parameterising eddy-induced tracer transports in ocean circulation models. J. Phys. Oceanogr. 25, 463–474. Gill, A.E., 1982. Atmosphere-Ocean Dynamics. Academic Press. pp. 662. Greatbatch, R.J., Lamb, K.G., 1990. On parameterising vertical mixing of momentum in non-eddy resolving ocean models. J. Phys. Oceanogr. 20, 1634–1637. Greatbatch, R.J., Fanning, A.F., Goulding, A.D., Levitus, S., 1991. A diagnosis of interpentadal circulation changes in the North Atlantic. J. Geophys. Res. 96, 22009–22023. Greatbatch, R.J., Li, G., 2000. Alongslope mean flow and an associated upslope bolus flux of tracer in a parameterisation of mesoscale turbulence. Deep Sea Res. 47, 709–735.

424

R.J. Greatbatch, C.W. Hughes / Ocean Modelling 39 (2011) 416–424

Holland, W.R., 1973. Baroclinic and topographic influences on the transport in western boundary currents. Geophys. Fliud. Dyn. 4, 187–210. Hughes, C.W., de Cuevas, B.A., 2001. Why western boundary currents in realistic ocean models are inviscid: a link between form stress and bottom pressure torques. J. Phys. Oceanogr. 31, 2871–2885. J. Geophys. Res., 106, 2713–2722. McCreary, J., 1981. A linear stratified model of the equatorial undercurrent. Roy. Soc. London Phil. Tran. 298, 603–635. 16721–16738. Mellor, G.L., Mechoso, C.R., Keto, E., 1982. A diagnostic model of the general circulation of the Atlantic Ocean. Deep Sea Res. 29, 1171–1192. Munk, W., 1950. On the wind-driven ocean circulation. J. Meteorol. 7, 79–93. Olbers, D., Eden, C., 2003. A model with simplified circulation dynamics for a baroclinic ocean with topography. Part 1: Waves and wind-driven circulations. J. Phys. Oceanogr. 33, 2719–2737.

Rhines, P.B., Young, W.R., 1982. A theory of wind-driven ocean circulation. 1. Midocean gyres. J. Mar. Res. 40 (suppl.), 559–596. Salmon, R., 1998. Linear ocean circulation theory with realistic bathymetry. J. Mar. Res. 56, 833–884. Schmitz, W.J., Thompson, J.D., Luyten, J.R., 1992. The Sverdrup circulation for the Atlantic along 24oN. J. Geophys. Res. 97, C5. doi:10.1029/92JC00417. Stommel, W.H., 1948. The westward intensification of wind-driven ocean currents. Trans. Amer. Geophys. Union 29, 202–206. Sverdrup, H.U., 1947. Wind-driven currents in a baroclinic ocean; with application to the equatorial currents of the eastern Pacific. Proc. Nat. Acad. Sci. 33, 31 8–326.