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Why western boundary currents are diffusive: A link between bottom pressure torque and bolus velocity
Ocean Modelling 32 (2010) 14–24
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Why western boundary currents are diffusive: A link between bottom pressure torque and bolus velocity Carsten Eden a,*, Dirk Olbers b a b
IFM-GEOMAR, FB I, Ocean Circulation and Climate Dynamics, Düsternbrooker Weg 20, 24105 Kiel, Germany Alfred Wegener Institute for Polar and Marine Research, Postfach 120161, 27515 Bremerhaven, Germany
a r t i c l e
i n f o
Article history: Received 30 January 2009 Received in revised form 1 July 2009 Accepted 10 July 2009 Available online 17 July 2009 Keywords: Western boundary currents Bottom pressure torque Friction Bolus velocity
a b s t r a c t The bottom pressure torque is known to vanish in the interior ocean but to play a dominant role in the western boundary layer in balancing the planetary vorticity on spatial scales larger than the Rossby radius of deformation. In this study, the appearance of the bottom pressure torque and thus any deviation of wind-driven flow from classical Sverdrup balance is locally related in steady state to non-zero bolus velocity and/or friction, under the assumption that horizontal density advection is small compared to the lifting of isopycnals. To first order approximation, the vortex stretching by the vertical bolus velocity is related to the bottom pressure torque. The bolus vortex stretching becomes a significant term in the barotropic vorticity budget of the western boundary layer and is formally equivalent to bottom friction as in the classical models of the wind-driven gyre circulation. Ó 2009 Elsevier Ltd. All rights reserved.
1. Introduction Energetic boundary currents are found on the western side of the individual ocean basins as part of the wind-driven large-scale gyre systems. The location and strength of these western boundary currents can be understood by considering depth-averaged vorticity balances (Stommel, 1948). In the most simplified form of such balances (Stommel, 1948; Munk, 1950), the torques related to the changes in planetary vorticity of a water parcel and the surface wind stress balance each other in the interior of the ocean, i.e. the interior ocean is in Sverdrup balance (Sverdrup, 1947), while frictional torques related to large bottom friction (Stommel, 1948) or lateral friction (Munk, 1950) balance the planetary vorticity change in the western boundary layer. Although these simple vorticity balances rely on rather restrictive assumptions, they have been very successful in explaining the most important aspects of the large-scale wind-driven circulation, such that they became classical concepts in physical oceanography. However, allowing for topographic variations in the depth-averaged (or depth-integrated) vorticity budgets, the so-called bottom pressure torque shows up, resembling a stretching term due to a geostrophically balanced bottom flow across isobaths. When the bottom flow is directed along isobaths or vanishes entirely, the bottom pressure torque also vanishes in the depth-averaged vorticity balance. This appears to be the case for instance in the interior of the North Atlantic (Wunsch and Roemmich, 1985), where * Corresponding author. E-mail address: [email protected] (C. Eden). 1463-5003/$ - see front matter Ó 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.ocemod.2009.07.003
large topographic variations as the mid-oceanic ridge can be found, which only have a small signature in the depth-averaged flow (Willebrand et al., 2001). On the other hand, Saunders et al. (1999) and Hughes and de Cuevas (2001) found in a realistic high-resolution general circulation ocean model that the bottom pressure torque becomes large in the western boundary layer, where it balances to first order the planetary vorticity change, while other torques, in particular the frictional torques play only a minor role. Therefore, the classical, frictional concept of the western boundary layer has to be abandoned. However, although all important topographic effects in the depth-averaged vorticity balance can be described by the bottom pressure torque in an elegant way, it is not possible to obtain a similarly self-consistent simplified model as in the classical concepts by including also the bottom pressure torque in the depth-averaged vorticity budget. This is because bottom pressure is a diagnostic variable containing information about the density stratification, for which no similar simplified depthaveraged balance equation appears to be available. In this study, we use the BARotropic Baroclinic Interaction (BARBI) model (Olbers and Eden, 2003) to better understand the role and physical meaning of the bottom pressure torque in the western boundary layer. BARBI was designed to describe the impact of the (baroclinic) stratification in the depth-averaged, i.e. barotropic vorticity budget, by constructing a budget for the vertically integrated potential energy. Here, we identify the bottom pressure torque as a coupling term between the barotropic vorticity budget and the budget of vertically integrated potential energy and interpret the latter as the budget of a depth-averaged
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baroclinic vorticity involving a baroclinic streamfunction. As already suggested by Olbers et al. (2007) for the Southern Ocean, it turns out that using BARBI it is possible to return to first order to the simple classical concept, by identifying a diffusive torque related to stretching by the vertical bolus velocity as a simple reinterpretation of the frictional torque in the classical barotropic vorticity budget. Here, we discuss the consequences for the basin circulation and western boundary layers. The following section acts as a summary of the main findings of this study. Most of the detailed mathematical derivations which are used in the second section are placed in the appendices. As a further illustration, the results are discussed in Section 3 on the basis of a numerical implementation of the BARBI model and a standard primitive equation model for comparison. We discuss the near cancellation of the bottom pressure torque and its role in the western boundary in the barotropic and baroclinic vorticity budgets. The last section gives a summary and discussion of the results. 2. The role of the bottom pressure torque
in the western boundary layer of the non-eddy-resolving model version. We therefore follow Hughes and de Cuevas (2001) and conclude that for scales larger than the relevant meso-scale eddy scale, the barotropic vorticity balance is given to good approximation by
b@ x w ¼ r s0 þ r P rh þ fric :
2.2. The baroclinic vorticity budget Using the BARBI model concept of Olbers and Eden (2003) (which is detailed in Appendix A), it is demonstrated in C how to construct an approximate baroclinic vorticity budget given by
b@ x / ¼ r s0 þ b=f sðxÞ 2 r P rh K=k2 r2 / :
The curl of the vertically integrated and steady horizontal momentum equation will be called here the barotropic vorticity budget and is given by1
:
:
:
Z
0
u rudz þ fric
ð1Þ
h
where w denotes the streamfunction of the depth-averaged flow R0 with r w ¼ h udz, where P denotes the pressure at the bottom : at z ¼ h, s0 the surface wind stress and where the last term collects frictional torques related to isotropic small-scale turbulence. A more detailed derivation and discussion of the barotropic vorticity budget is given in Appendix A. The balance between the left hand side of Eq. (1), i.e. the planetary vorticity term, and the first term on the right hand side of Eq. (1), i.e. the wind stress curl, represents the classical Sverdrup balance, which is distorted by the bottom pressure torque r P rh and the occurrence of the non-linear tor: que and the torque of the frictional terms in the barotropic vorticity balance. Fig. 1 shows the most important components of the time-mean budget calculated from a realistic high-resolution general circulation model of the North Atlantic, identical the one used in e.g. Eden (2007). On small spatial scales, the barotropic vorticity budget displays a rather noisy pattern with large and fluctuating terms. In particular the non-linear term is important in the budget. However, when averaging Eq. (1) over spatial scales representative for the length scale of meso-scale eddies, which is given by the Rossby radius in mid- to high-latitudes (Eden, 2007), the non-linear term in the budget becomes much less important and the dominant balance is between planetary vorticity, wind stress curl and the bottom pressure torque. As shown by Saunders et al. (1999) and Hughes and de Cuevas (2001), in particular in the western boundary current layer, the bottom pressure torque balances the planetary vorticity for spatial averages larger than 0.5°–1°, while the non-linear term becomes small in the spatially averaged budget. In fact, when we look at the barotropic vorticity budget in a noneddy-resolving version of the numerical model, the non-linear term in the vorticity budget is an order of magnitude less important than the other terms (not shown). This is in particular true 1 We use the following notation: The vector subscript u denotes a clockwise : rotation of the horizontal velocity u ¼ ðu; vÞ by 90°, i.e. u ¼ ðv; uÞ. The same holds for : r ¼ ð@ x ; @ y Þ resp. r ¼ ð@ y ; @ x Þ. Note that r s denotes the curl of the horizontal : : vector s and r P rh ¼ JðP; hÞ the Jacobian of P and h.
:
ð2Þ
The same result is obtained within the planetary geostrophic approximation. It should be noted, however, that scales of frontal structure in the western boundary layer can be of the same order than the meso-scale eddy scale. For those scales, the following discussion will not be applicable, instead we are focussing on the noneddy-resolving, large-scale circulation of the ocean.
2.1. The barotropic vorticity budget
b@ x w ¼ r s0 þ r P rh r
:
:
ð3Þ
where / denotes the streamfunction of the baroclinic transport and where K denotes a lateral (isopycnal thickness) diffusivity and k the Rossby radius. We call / the baroclinic streamfunction, since it describes the geostrophic transport relative to the bottom. In order to derive Eq. (3), a steady state is assumed and a vertical background profile for buoyancy is introduced. As a consequence of small relative magnitudes of the perturbations from that reference buoyancy profile, it is also assumed that the effect of vertical advection of the background and perturbation buoyancy dominates the effect of lateral advection of buoyancy perturbations. The latter effect is therefore neglected in the buoyancy budget, while the advective effect of meso-scale eddy mixing on buoyancy is parameterised following Gent and McWilliams (1990). As discussed in Appendix B, the last term in Eq. (3) is a torque related to the vertical bolus velocity. Note that the lateral diffusivity K corresponds to the isopycnal thickness diffusivity in the Gent and McWilliams (1990) parameterisation and that K=k2 corresponds to the inverse of a time scale of several days. We have also neglected the effect of thermohaline surface forcing and diapycnal mixing, i.e. we focus on wind-driven flow only. Together with the barotropic, Eq. (2), and baroclinic, Eq. (3), vorticity budgets we have the vertically averaged momentum budget, given by Eq. (16) of Appendix A. The system is linear, and is similar to that studied by Salmon (1998). 2.3. The cancellation of the bottom pressure torque Using the baroclinic vorticity budget Eq. (3), it becomes possible to understand the role of the bottom pressure torque within the above formulated limits, which is the aim of the present study. First, it is possible to show that for the case without friction and diffusion the bottom pressure vanishes: Subtracting Eq. (3) from Eq. (2) in order to eliminate the wind stress curl and using the vertically integrated (steady and frictionless) zonal momentum budget given by f @ x w ¼ f @ x / þ h@ x P sðxÞ (compare Eq. (16) of Appendix A) one obtains after some manipulations
r P rf =h3 ¼ 0 :
ð4Þ
This shows that the bottom pressure P becomes constant along 3 contours of f =h . Together with a boundary condition P ¼ 0 somewhere on each such contour, P ¼ 0 everywhere and so the bottom pressure torque is zero everywhere (except for closed contours of 3 f =h which might produce a different, more complicated situation
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Fig. 1. Time mean barotropic vorticity, Eq. (1) in a realistic eddy-permitting model of the North Atlantic Ocean in 109 m=s2 . Shown are planetary vorticity minus wind stress R0 curl (bwx r s) in the left column, the bottom pressure torque (rP rh) in the middle column and the curl of the non-linear term (r h u rudz) in the right column. The individual terms have been box averaged from the 1/12° 1/12° cos / (Mercator) grid (/ denotes here latitude) of the numerical model on a regular 2° 1.3° grid in the upper row, on a 1° 0.6° grid in the middle row and on a 0.5° 0.3° grid in the lower row. After interpolation on the individual grid, all data have been also smoothed over 3 3 grid points.
which we do not consider here). Eq. (4) implies that in the case with weak diffusion and friction, the bottom pressure torque tends to vanish. This appears to be often the case in certain regions of the interior North Atlantic (Fig. 1), leaving behind the classical Sverdrup relation, as discussed by Wunsch and Roemmich (1985) and Willebrand et al. (2001). Note that by implication, we can also conclude that any deviation of wind-driven flow from the classical Sverdrup balance in steady state is introduced by diffusive and frictional torques, which is in turn in accordance to the classical theory by Stommel (1948) and Munk (1950).
maximum or minimum, i.e. we expect differences between w and / at those latitudes, which, however, should not exceed relative order oðL=aÞ. On the other hand, we have seen in Fig. 1 that in the western boundary the bottom pressure torque is large, pointing towards the importance of diffusion and friction in the western boundary layer. It is, however, still necessary that / and w are similar: Using again the vertically integrated (steady and frictionless) momentum balance f rw ¼ rðf /Þ þ hrP s0 (see Eq. (16) of Appendix A) the bottom pressure torque becomes
hrP r h ¼ f rðw /Þ r h /b@ x h þ s0 r h 2.4. The similarity of barotropic and baroclinic streamfunction Second, we demonstrate that a vanishing bottom torque implies that the barotropic and baroclinic streamfunction are getting similar, i.e. / w: Scaling shows that the magnitude of the wind stress curl, r s0 , in the baroclinic vorticity budget Eq. (3) is a factor : ðxÞ a=L 1 larger than the term b=f s0 , where a denotes the earth radius and L the scale of the spatial variations of the wind stress. Therefore, baroclinic and barotropic vorticity budgets become identical for vanishing diffusivity and viscosity to order oðL=aÞ, with the consequence of similar streamfunctions w and /. HowðxÞ ever, it should be noted that the neglect of b=f s0 in comparison with r s0 is not really permitted near a latitude of wind stress :
:
:
:
ð5Þ
Scaling shows that the first term on the right hand side of Eq. (5) is a factor a=L 1 larger than the other two, where L now denotes (the maximum of) the scale of the spatial variations of wind stress, streamfunction and topography. In fact, the leading order term in the bottom pressure torque in Eq. (5) is also a factor a=L 1 larger when compared in the barotropic and baroclinic vorticity budgets, Eqs. (2) and (3), with the planetary vorticity term and the wind stress curl. Since the bottom pressure torque shows up with opposite sign in both budgets, Eqs. (2) and (3), it is therefore hard to see how the pressure torque can be balanced without implying viscous and diffusive torques much larger (of factor a=L 1) than the planetary vorticity in both budgets. On the other hand, a small
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C. Eden, D. Olbers / Ocean Modelling 32 (2010) 14–24
0 50°
(a)
(b) −1000
40°
−2000
−3000 30° −4000
−5000
20° −0.2
0
0.2
0°
10°
20°
30°
ðxÞ
Fig. 2. (a) Zonal component of surface wind stress, s0 , in N=m2 as a function of latitude which is used in the idealised experiments with BARBI and the primitive equation model. (b) Ocean bottom depth in m as a function of longitude for a ¼ 0:2, a ¼ 0:5 and a ¼ 0:8 in Eq. (10).
difference between w and / (of order L=a 1 relative to the order of w and /) can be balanced in both budgets by planetary vorticity and diffusive and viscous torques of similar (or smaller) magnitude. We consider the latter case as the relevant one for the ocean.
2
@ t E þ hU rEh
2
¼ N 20 =6U rh þ N20 =2r u0 þ K r2 E 2
@ t U þ f U ¼ hrP rE þ s0 þ Ah r U :
2
@ t u0 þ f u0 ¼ h =3ðrE s0 Ah r2 UÞ þ Ah r2 u0 :
2.5. The baroclinic Stommel equation To zero order approximation, we can therefore assume that w / both in the interior and the western boundary layer, and by combining Eqs. (2) and (3) to eliminate the bottom pressure torque we arrive to first order (in L=a 1) approximation at the simplified barotropic (and baroclinic) vorticity budget
b@ x w ¼ r s0 Kk2 =3r2 w þ fric :
ð6Þ
Note that in Eq. (6) we have neglected for consistency the term ðxÞ bf 1 s0 . The factor Kk2 =3 represents the inverse of a time scale on the order of days and the related term in Eq. (6) acts similar to bottom friction. Eq. (6) is thus formally similar to the barotropic vorticity budget by Stommel (1948) for a flat bottom, who introduced large bottom friction to balance the return flow in the western boundary layer. In Eq. (2), however, the return flow is balanced by the bottom pressure torque (and also by frictional torques). By comparing with Eq. (6), we see that the bottom pressure torque was replaced to first order approximation by the diffusive torque in the baroclinic vorticity budget. Somehow surprisingly, we have rediscovered Stommel’s original vorticity budget with a physically rather different motivation. Eq. (6) was called the ‘‘baroclinic Stommel equation” by Olbers et al. (2007).2 Note that we have interpreted the diffusive torque in the baroclinic Stommel equation Eq. (6) as the torque related to the vertical bolus velocity. 3. Results and interpretation 3.1. Idealised numerical model The theoretical results found in the previous section are illustrated using idealised numerical models, i.e. a numerical implementation of the BARBI model (Olbers and Eden, 2003) and a standard primitive equations model in identical configuration. The BARBI model which is solved numerically is given by
2 In Olbers et al. (2007) a factor 1/2 is missing in the diffusive torque of the baroclinic Stommel equation, i.e. the third term on the left hand side of Eq. (17) of Olbers et al. (2007).
ð7Þ ð8Þ ð9Þ
while the other model is based on the standard primitive equations (PE). Note that the meaning of all variables in the BARBI model are discussed in Appendix A; here, we just note that it resembles a two mode system, i.e. the model contains a barotropic mode, given by the depth-averaged velocity U, and a single baroclinic vertical mode, given by the vertical density moment E and baroclinic velocity moment u0 . Note also that the budget for the vertical density moment E in the numerical BARBI model, given by Eq. (7), includes the effect of horizontal advection by the barotropic flow which was neglected in the foregoing theoretical consideration in Section 2, while advection by the horizontal part of the baroclinic flow remains neglected. We solve both sets of equations (BARBI and PE) in a sector domain ranging 32° in longitude and from 20°N to 52°N resembling a mid-latitude ocean basin. This ocean is driven by a classical, cosine-shaped, eastward wind stress s0 ¼ 0:2 N=m2 ðcosð2pð/ 20 Þ=32 Þ; 0Þ. The zonal component of s0 is shown in Fig. 2a. There is a strong background stratification, represented by a constant stability frequency N 0 ¼ 2 103 s1 ; meso-scale density mixing is parameterised following Gent and McWilliams (1990) with thickness diffusivity of K ¼ 5000 m2 =s and numerical dissipation is given by harmonic lateral friction with viscosity Ah ¼ 5000 m2 =s and no-slip boundary conditions. The ocean depth
h i hðxÞ ¼ 5000 m 1 a expððx 16o Þ2 =ð5o Þ2 Þ h i 1 a expððx þ 1o Þ2 =ð5o Þ2 Þ
ð10Þ
features an idealised north/south mid-ocean ridge in the middle of the domain and an idealised continental shelf at the western boundary, i.e. a reasonable situation for an ocean basin and resembling the North Atlantic Ocean. The size of the topographic variations, shown in Fig. 2b, are varied from a ¼ 0:2 to a ¼ 0:8 but is set to a ¼ 0:5 if not otherwise noted. Horizontal resolution is 0.5° 0.5° and the time step is 0:5 h. The numerical code of the BARBI model can be accessed at http://www.ifm-geomar.de/~barbi and the primitive equations code at http://www.ifm-geomar.de/ ~spflame. The latter was configured as close as possible to the BARBI model. The primitive equation model contains 50 vertical levels with 100 m thickness and the topographic variations are discretised on this vertical grid. Horizontal resolution, boundary and initial
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C. Eden, D. Olbers / Ocean Modelling 32 (2010) 14–24
(a)
(b)
(c)
Fig. 3. Streamfunction w in Sv in steady state in a simulation of the BARBI model with N ¼ 0 s1 (left), a simulation with a flat bottom (middle), and the standard model simulation with topographic variations and stratification. Also shown are isobaths (lines).
conditions, viscous and diffusive parameters are set as in the BARBI model. The numerical BARBI model features the familiar near cancellation of the impact of the topographic variations by the stratification on the barotropic flow as discussed first by Holland (1973). Fig. 3 shows three different model solutions in steady state, i.e. one without stratification, i.e. setting N 0 ¼ 0 s1 (Fig. 3a), a solution without topographic variations, i.e. setting h ¼ 2000 m (Fig. 3b) and a solution including stratification and topographic variations. In the simulation without stratification only a weak barotropic flow develops which apparently shows a strong impact of the topographic variations on the flow. Switching on the stratification, the barotropic circulation gets stronger and the flow gets close to the circulation in the simulation with flat bottom. This cancellation of the topographic variations will be discussed in detail in the next section. When comparing to the primitive equation model, Fig. 4, it becomes obvious that the BARBI model is in this configuration indeed a very good approximation to the primitive equation model in terms of streamfunction w and vertical density moment E (or baroclinic streamfunction /). The same holds for the bottom pressure torque, and also for the other terms in the barotropic and the baroclinic vorticity budgets, which we have diagnosed in the primitive equations model for comparison with the BARBI model (not shown). In particular the form of the diffusive torque related to the vertical bolus velocity in the baroclinic vorticity budget (where
we have neglected topographic variations and horizontal advection as discussed in Appendix B) matches well the effect of the diagnosed (complete) bolus velocity in the primitive equation model. There are, however, slight differences over the mid-ocean ridge and near the western boundary visible in Fig. 4. This is due to the truncation in BARBI, which contains only the first vertical density moment. Using two vertical moments the solution is getting closer to the primitive equation model (not shown) as discussed by Olbers and Eden (2003). However, note that it is impractical to use more than the first couple of vertical modes in BARBI, because of the slow convergence of the basis functions zn for large n. 3.2. Numerical model results It was argued above that scaling of the components of the bottom pressure torque in the barotropic and baroclinic vorticity budget implies a strong similarity between / and w, which is in fact featured by the numerical example. Fig. 5 shows the components of the bottom pressure torque as given by Eq. (5). The figure shows the simulation with BARBI but note that the primitive equations 1 model is very similar (not shown). The terms fh rw r h (shown : 1 in Fig. 5a) and fh r/ r h (not shown) are indeed very similar : and their difference (shown in Fig. 5b) one order of magnitude 1 smaller than the individual terms. The remaining term, bh /@ x h (shown in Fig. 5c), is of similar magnitude to the difference of the former two terms (note that s0 r h ¼ 0 in the idealised :
(a)
(b)
(c)
(d)
Fig. 4. Streamfunction w in Sv (upper row) and first vertical density moment E in m3 =s2 (lower row) in BARBI (left column) and a primitive equations model (right column) in a setup as close as possible to the BARBI model.
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C. Eden, D. Olbers / Ocean Modelling 32 (2010) 14–24
(a)
(b)
(c)
1
1
1
Fig. 5. Components of the bottom pressure torque in the BARBI model, i.e. fh rw rh (a in 109 m2 =s2 ) and fh rðw /Þ rh (b in 109 m2 =s2 ) and bh /@ x h (c in 109 m2 =s2 ). Note the different color scale in (a).
(a)
(b)
(c)
(d)
(e)
(f)
(g)
(h)
Fig. 6. Cumulative zonal integrals of the components of the barotropic vorticity budget Eq. (2) (upper row) and baroclinic vorticity budget Eq. (3) (lower row). All zonal R integrals are divided by b and shown in Sv. Upper row: barotropic streamfunction w (a), transport related to wind stress curl b1 dxr s0 (b), bottom pressure torque R R R ðxÞ b1 dxrP rh (c) and frictional torque b1 dxfric (d). Lower row: baroclinic streamfunction / (e), baroclinic transport related to wind stress curl b1 dxðr s0 þ bf 1 s0 Þ R R (f), bottom pressure torque 2b1 dxrP rh (g) and meso-scale eddy mixing b1 dxKk2 r2 / (h). Also shown are contours of the inverse Rossby radius 1=k in (a) and (e).
configuration). As expected from the theoretical discussion in the previous section, we find that the barotropic and baroclinic streamfunctions become similar such that the bottom pressure torque becomes of similar magnitude as the other terms in the baroclinic and barotropic vorticity budgets. However, although reduced in magnitude by the similarity of w and /, the bottom pressure torque still plays a dominant role in the barotropic and baroclinic vorticity budgets. Fig. 6(upper row) shows the individual terms of Eq. (2) in the numerical BARBI model (the primitive equation model is very similar, not shown) as cumulative zonal integrals which are also divided by b, such that all terms add up to form the barotropic streamfunction (left hand side of Eq. (2)). In the interior, the most important contribution to the transport is given by the wind stress curl, i.e. the flow is more or less in Sverdrup balance. The contribution of the bottom pressure term becomes larger over the mid-ocean ridge and in particular important near the western boundary. Here, it largely contributes to the compensating transports of the interior flow, while the frictional torques are only important in a thin boundary layer. Note that more than half of the return flow in the western boundary layer is made by the bottom pressure torque. Fig. 6(lower row) shows the corresponding cumulative zonal integrals of the baroclinic vorticity budget Eq. (3) in the numerical BARBI model (the primitive equation model is very similar, not shown). As expected, the planetary vorticity, b/x , and the wind ðxÞ stress forcing, r s0 þ b=f s0 are very similar to the corresponding : terms in the barotropic vorticity budget Eq. (2), while the diffusive torques, K=k2 r2 /, and the bottom pressure torque, 2 r P rh, are : balancing each other to zero order.
By varying the value of K relative to the value of Ah we can change the relative importance of the frictional torque and the bottom pressure torque in the barotropic vorticity budget for the western boundary return flow. Fig. 7 shows the barotropic streamfunction w, the cumulative integrals of bottom pressure torque and the frictional torque in a simulation of the BARBI model using K ¼ 1000 m2 =s and K ¼ 15000 m2 =s. In the simulation with the small (large) K, the barotropic streamfunction gets larger (smaller), the bottom pressure torque smaller (larger) and the bottom pressure torque contributes relatively less (more) to the return flow in the western boundary layer than the frictional torque. In the individual cases, however, barotropic and baroclinic streamfunctions continue to be very similar, which holds in particular also for the zero order balance between the torques related to vertical bolus velocity and bottom pressure in the baroclinic vorticity budget. Note also that the western boundary layer width of w and / becomes larger for increasing values of K. This is related to the similarity of the bolus torque with the effect of bottom friction: Increasing the value of K is equivalent to increasing the bottom friction coefficient, which is in turn proportional to the boundary layer width in the classical model of Stommel (1948). However, we do not see a linear relationship between boundary layer width and K as suggested by the classical model, which means that lateral friction in the barotropic vorticity budget is also important in the experiments with the BARBI model. A similar impact on the magnitude of the bottom pressure torque and its relative importance compared to friction is given
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C. Eden, D. Olbers / Ocean Modelling 32 (2010) 14–24
(a)
(b)
(c)
(d)
(e)
(f)
Fig. 7. Cumulative zonal integrals of the components of the barotropic vorticity budget Eq. (2) for a simulation of the BARBI model with K ¼ 1000 m2 =s (upper row) and a simulation with K ¼ 15000 m2 =s (lower row). All zonal integrals are divided by b and shown in Sv: Barotropic streamfunction w (a and d), transport related to the bottom R R pressure torque b1 dxrP rh (b and e) and frictional torque b1 dxfric (c and f). Also shown are contours of the inverse Rossby radius 1=k in (a) and (d).
(a)
(b)
(c)
(d)
(e)
(f)
Fig. 8. Cumulative zonal integrals of the components of the barotropic vorticity budget Eq. (2) for a simulation of the BARBI model with different topographic variations, i.e. with a ¼ 0:2 (upper row) and a simulation with a ¼ 0:8 (lower row). All zonal integrals are divided by b and shown in Sv: Barotropic streamfunction w (a and d), transport R R related to the bottom pressure torque b1 dxrP rh (b and e) and frictional torque b1 dxfric (c and f). Also shown are contours of the inverse Rossby radius 1=k in (a) and (d).
by the height of the topographic variations. Fig. 8 shows the components of the barotropic vorticity budget in a simulation with smaller and larger topographic obstacles, by setting a ¼ 0:2 and a ¼ 0:8 in Eq. (10) respectively instead of a ¼ 0:5 as before. As expected, when increasing (reducing) the variations in h the magnitude of the bottom pressure torque becomes larger (smaller) and the frictional torques less (more) important. In the numerical experiments, we found that finite lateral friction is necessary to obtain a stable integration at horizontal resolution of the experiments shown here. Only by increasing the horizontal resolution it was possible to reduce the lateral viscosity further (not shown), such that the diffusive torques are able to balance more and more of the western boundary flow. Extrapolating the numerical experiments, it appears therefore possible to balance the western boundary flow entirely by the effect of density mixing, without the need of invoking frictional torques. However, mechanical dissipation is still needed in the energy budget to balance the wind work as discussed in Appendix B.
4. Summary and discussion In the barotropic vorticity balance relevant for spatial scales larger than meso-scale eddies, Eq. (2), the bottom pressure torque tends to vanish in the interior ocean Wunsch and Roemmich (1985), Willebrand et al. (2001). In consequence, the classical Sverdrup balance between wind stress curl and planetary vorticity change holds in those regions. On the other hand, the bottom pressure torque plays a dominant role in the western boundary layer in determining the barotropic volume transport (Saunders et al., 1999; Hughes and de Cuevas, 2001). It was the aim of this study to better understand the role of the bottom pressure torque and to relate it to simple physical mechanisms. Using the BARBI model of Olbers and Eden (2003), we have formulated a baroclinic vorticity budget, Eq. (3), which determines the baroclinic volume transport (relative to the bottom), and which looks very similar to its barotropic counterpart, but in which the bottom pressure torque shows up with opposite sign. This baroclinic vorticity budget is valid in steady state and for situations
C. Eden, D. Olbers / Ocean Modelling 32 (2010) 14–24
in which the horizontal advection of perturbation density is small compared to vertical lifting of perturbation and background density. Combining the baroclinic with the barotropic vorticity budget it is possible to show that for vanishing friction and diffusion, the bottom pressure torque cancels such that the classical Sverdrup balance holds. In consequence, we are able to conclude that any deviation from the Sverdrup balance is related to effects of density diffusion and friction, in accordance to the classical theory by Stommel (1948) and Munk (1950). For the case without diffusion and friction and vanishing bottom pressure torque, we found that barotropic and baroclinic streamfunction become identical to first order approximation. This similarity also extents to the case with finite bottom pressure torque, since for a large difference between the streamfunctions, the bottom pressure torque would be an order of magnitude larger than the planetary vorticity change and could only be balanced by unrealistically large frictional or diffusive torques. Using the similarity between both streamfunctions, it is possible to eliminate the bottom pressure torque from both vorticity budgets, which then contains only torques related to wind stress forcing, friction and density diffusion. The latter acts similar to bottom friction in the classical Stommel model (Stommel, 1948) of the wind-driven circulation. We have therefore rediscovered the classical Stommel model with, however, a different physical meaning of the frictional torque. Using the BARBI model, we associate to first order approximation the vortex stretching by the vertical bolus velocity of the Gent and McWilliams (1990) parameterisation with the bottom pressure torque. This vortex stretching becomes a significant term in the barotropic vorticity budget of the western boundary layer and formally equivalent to bottom friction as in the simple model by Stommel (1948). It is able to balance a large part of the return flow in the western boundary layer, but it is not related to friction as in Stommel’s original model but rather is related to advective stirring of density. The relation between vertical bolus velocity and bottom pressure torque can also be considered using the Residual Mean Theory of Andrews et al. (1987). The density budget becomes in this formulation
@ t q þ u rq þ w @ z q ¼ 0
ð11Þ
with the horizontal, u ¼ u þ ub , and vertical w ¼ w þ wb residual velocity and the horizontal ub ¼ @ z B and vertical wb ¼ r B bolus : : velocity. If one neglects the effect of the horizontal advection, as we have done in the derivation of the baroclinic vorticity budget, it becomes clear in w ¼ w þ wb ¼ 0 for steady state. At the bottom, the (geostrophically balanced) Eulerian vertical velocity is related to the bottom pressure torque, fwjz¼h ¼ rP r h, and so the vertical :
(a)
21
bolus velocity, fwb jz¼h ¼ rP r h. It becomes obvious, how a : non-vanishing vertical bolus velocity at the bottom, is related to a non-vanishing bottom pressure torque. Fig. 9 shows the bottom pressure torque related to the Eulerian mean vertical bottom velocity, fwjz¼h and to the vertical bolus velocity at the bottom fwb jz¼h in the primitive equation model. Both cancel indeed to a large extent. We have shown above how the vertical bolus velocity at the bottom translates to a torque in the baroclinic and barotropic vorticity budget, which is then formally identical to bottom friction as in the original model by Stommel (1948). Acknowledgements The authors thank Richard Greatbatch for his helpful discussions. We also thank Chris Hughes for the helpful review and for pointing out the cancellation of the bottom pressure torque in Eq. (4). Appendix A. The BARBI model The BARBI model is based on the relevant equations of motion for the oceanic circulation on scales much larger than the Rossby radius, given by the planetary geostrophic approximation. They are repeated here for completeness
@ t u þ f u ¼ rp þ @ z s þ F :
ð12Þ
@ z p ¼ g q
ð13Þ
r u þ @zw ¼ 0 @ t q þ u rq þ w@ z q ¼ Q
ð14Þ ð15Þ
The notation is standard, i.e. u denotes horizontal velocity, w vertical velocity, p pressure, q density and the vector subscript u : a rotation of the horizontal velocity u ¼ ðu; vÞ by 90°, i.e. u ¼ ðv; uÞ. For simplification, the mean balances of heat and salt : are combined into a thermohaline balance for density, with a density source term Q related to turbulent meso-scale mixing and stirring, diapycnal turbulent micro-scale processes and effects from compressibility and the non-linear equation of state. The horizontal momentum equation is also supplemented with vertical turbulent friction given by the vertical stress vector s and the lateral friction F. The vertical stress tensor represents effects of microscale turbulence while F the effect of meso-scale eddies on the horizontal momentum. Deviating from the planetary geostrophic approximation, we also retained the local time derivative of the horizontal momentum, i.e. we allow for gravity waves and short Rossby waves in addition to long Rossby waves. Note that the rigid-lid boundary condition is employed here.
(b)
Fig. 9. fwjz¼h (a) and fw jz¼h (b) in 109 m=s2 in the primitive equation model, where w denotes the Eulerian mean vertical velocity and w the vertical eddy advection (bolus) velocity.
22
C. Eden, D. Olbers / Ocean Modelling 32 (2010) 14–24
Vertical integration of the horizontal momentum equation Eq. (12) leads to
@ t U þ f U ¼ hrP rE þ s0 sb þ
Z
:
0
Fdz
ð16Þ
h
R0 with the horizontal transport vector U ¼ h udz, the wind stress s0 and the stress sb at the bottom of the ocean at z ¼ h. Note that the bottom depth h is allowed to vary. By partial integration and use of the hydrostatic relation Eq. (13), the vertically integrated pressure gradient splits into two terms, related to the vertically integrated R0 potential energy, E ¼ g h zqdz, and the bottom pressure, P ¼ pjz¼h . The latter can be eliminated from the momentum equation Eq. (17) by dividing by h and taking the curl
R0 with F 0 ¼ @ z s þ F 1=h h Fdz. By the occurrence of the term Enþ2 in Eq. (19) it becomes obvious that the BARBI model is given by a coupled infinite hierarchy of density and baroclinic velocity moments. A closure of this hierarchy was proposed in Olbers and Eden (2003) based on the consideration of wave properties. For the simplest closure, Eq. (19) becomes 0
2
@ t u02 þ f u2 ¼ h =3 ðrE s0 þ sb Þ þ
Z
:
0
z2 F 0 dz
ð20Þ
h
This model consists of the barotropic momentum U, the vertically integrated potential energy E1 and the second baroclinic velocity moment u02 . For simplicity, we will drop the subscripts at E1 and u02 in the following.
r h1 r@ t w þ r w rfh1 ¼ rh1 r E þ r h1 ðs0 sb Þ :
:
1
þ r h :
Z
:
Appendix B. Diffusive and viscous closures in BARBI
0
Fdz
ð17Þ
h
with the barotropic streamfunction w, defined by the relation r: w ¼ U. Note that the vector subscript at the gradient operator r ¼ ð@ x ; @ y Þ denotes again rotation by 90°, i.e. r: ¼ ð@ y ; @ x Þ. The first term on the right hand side of Eq. (17) is related to the combined effect of topographic variations and stratification and is often called the ‘‘JEBAR” (Joint Effect of Baroclinicity and Relief) term as introduced by Sarkisyan and Ivanov (1971). Knowledge of E would allow the computation of the streamfunction w from Eq. (17), i.e. would allow to close the model (with additional closures for sb and F). This is the basic idea of BARBI and the starting point for its formulation. Note that an alternative formulation of the barotropic vorticity budget is obtained by taking the curl of Eq. (17) directly (i.e. without devision by h), which yields Eq. (1). Note also that in the latter form the bottom pressure P shows up while in the formulation Eq. (17), the potential energy E shows up. As discussed in detail by Olbers and Eden (2003), using the density budget Eq. (15) it is possible to derive a tendency equation for R0 the vertical density moments3 En ¼ g h zn q0 dz with n ¼ 1; 2; . . . n given by n
n
@ t En þh U rEn h
ðnþ1Þ
þð1Þn
N20 h N2 U rh 0 r u0nþ1 ¼ nþ2 nþ1
Z
0
zn Qdz
h
ð18Þ To derive Eq. (18) the density q in Eq. (15) was decomposed into a background state and perturbation q0 , similar as in quasi-geostrophic theory. The background state was chosen for simplicity as linear with depth represented by the constant stability frequency N 0 (although arbitrary functions of depth are possible in a refined BARBI model version). In Eq. (18), the effect of the advection of the density perturbation by the baroclinic flow, u0 ¼ u U=h, has been neglected. This effect was shown to be small as long as perturbations in density remain small compared to the background density. All other advective effects in Eq. (18) are exact, in particular the lifting of the background density by the baroclinic flow is represented in Eq. (18) by the divergence of R0 the baroclinic velocity moments u0nþ1 ¼ h znþ1 u0 dz, for which the following budget can be derived exactly
@ t u0nþ1
0
þf u
:
nþ1
i 1 h nþ1 ¼ ð1Þn h ðrE1 s0 þ sb Þ þ rEnþ2 nþ2 Z 0 þ znþ1 F 0 dz ð19Þ h
3 Note that the definition of E using the perturbation density q0 instead of full density q as above does not affect the momentum equation Eq. (17) or the barotropic vorticity budget Eq. (17).
The closures for the sub-grid effects represented by Q, F, s and sb in the BARBI model have to be specified with care since they become important for the western boundary layers. The turbulent density mixing Q in Eq. (18) can be split into diapycnal mixing effect and an advective effect by meso-scale eddies. We assume that the former is small compared to the latter such that we neglect it. We also neglect any effects from the non-linear equation of state and compressibility effects of the perturbation density. For the advective effect of meso-scale eddy mixing on density, we employ the Gent and McWilliams (1990) parameterization in which the eddy advection (bolus) velocity is given by a streamfunction B, parameterized as
B ¼ K s; vb ¼ @ z ; B wb ¼ r B :
:
ð21Þ
:
where K is equivalent to the thickness diffusivity of the Gent and McWilliams (1990) parameterization, where s ¼ rq=@ z q denotes the vector of the isopycnal slopes, and where vb and wb denote the horizontal and vertical components of the bolus velocity respectively. The barotropic part of the bolus velocity vanishes because of the boundary conditions B ¼ 0 at top and bottom. In agreement to the treatment in Eq. (18), we care about the lifting of the background stratification by the baroclinic vertical bolus velocity only, i.e. we assume that
Z
0
zQ dz
h
Z
0
h
zN 2 r B dz g :
Z
0
r K rzq dz K r2 E
ð22Þ
h
For simplicity, a flat bottom,4 constant K and qz N2 =g was assumed in Eq. (22). It turns out that the dominant effect of advective meso-scale eddy density mixing on vertically integrated potential energy E is given by diffusion of E with a diffusivity identical to the thickness diffusivity of the Gent and McWilliams (1990) parameterization. Note that in an integral over a closed domain no sink term due to the meso-scale eddy density mixing shows up in Eq. (18) by specifying vanishing gradients of E normal to the boundaries. However, on the other hand, we know that the net effect of the parameterisation is given by a conversion of mean (available) potential energy to eddy energy (Gent et al., 1995), i.e. it is possible to show that
Z V
z r ðvb qÞþðwb qÞz dV
Z A
ðrK rE
Z
0
Ks2 N2 ÞdA¼
h
Z
Ks2 N2 dV
V
ð23Þ where V ðAÞ denotes a closed volume (area) and where again a flat bottom was assumed. The term on the right hand side of Eq. (23) is sign definite and a sink for E, resembling the conversion of mean potential energy to eddy energy, mimicking the effect of baroclinic 4
Or, equivalently, vanishing diffusivity K at top and bottom.
23
C. Eden, D. Olbers / Ocean Modelling 32 (2010) 14–24
instability. In Olbers et al. (2007) it was proposed to parameterise this exchange with eddy energy as linear damping of E. However, it turns out that this exchange term is locally much less important than the diffusive term related to the lifting of the background stratification by the vertical bolus velocity, such that we will simply neglect the exchange term here. Note that this neglection is consistent with the neglection of the effect of advection of perturbation density by the baroclinic flow in BARBI, resembling a second order term when density perturbations remain small compared to the background. For the frictional closures it is less clear how to proceed. We assume that vertical friction in the interior can be neglected, except for the surface where the vertical turbulent stress vector s connects to the surface wind stress s0 . We also neglect the bottom stress sb , because we will see below that in many cases the deep flow over the bottom topography will vanish, such that the bottom stress would also vanish in that case. On the other hand, the model needs some kind of dissipation of kinetic energy which becomes clear considering the energy cycle in BARBI. The barotropic kinetic energy is given by K b ¼ U 2 =ð2hÞ and by using Eqs. (18) and (16) the global domain integrals of K b and E can be derived as
Z @t
Z
Z 2 1 U rh þ dx h U s0 Z Z 0 2 1 þ dx h U F dz@ t
2
dx K b ¼
2
dx E ¼
Z
Z
2
2
dx Eh
h 2
dx Eh
2
U rh
ð24Þ
Exchange between E and K b is given by barotropic density advection over topography, dissipation is due to friction and energy is supplied by the wind work. Note that baroclinic kinetic energy is not considered here, since it can be shown that it does not exchange energy with E as long as the effect of advection of perturbation density by the baroclinic flow is set to zero in the budget for E. This effect was assumed and shown to be locally neglectable in BARBI as long as the perturbation density remains small compared to the background density. Note also that there is no energy exchange with eddy energy, since we have neglected the dissipation effect by the Gent and McWilliams (1990) parameterisation of E in Eq. (23). It becomes clear that in the balance of total mechanical energy in BARBI, K b þ E, a steady state can only be established when the wind work is balanced by some kind of mechanical dissipation. We implement in BARBI harmonic lateral friction in combination with a no-slip boundary condition, i.e. F ¼ Ah r2 u and approximate
Z
0
h
F dz Ah r2 U
and
Z
0
2
z2 F 0 dz Ah ðr2 u0 h =3 r2 UÞ
ð25Þ
h
where we have again neglected the spatial dependency of h for simplicity. However, it should be noted that harmonic lateral friction is problematic, since it is well known that mean momentum is not mixing down mean momentum gradients in turbulent geophysical flow (Starr, 1968). In contrast, mixing of potential vorticity should be implemented (Marshall, 1984), but it remains unclear at the moment how this could be done in a consistent way. On the other hand, the formulation using harmonic friction is convenient for numerical reasons since it provides a sink for grid noise such that we follow common practise and use it here as well. Appendix C. Baroclinic vorticity budget To discuss the cancellation of the effect of topographic variations and the corresponding similarity between w and E we need to further simplify the BARBI model by filtering gravity and short
Rossby waves. By neglecting the time derivative and friction for u0 it is possible to combine the budgets for E and u0 , i.e. 2
2
@ t E þ N20 =6 ðr E rh f 1 r w rh Þ :
:
2
¼ N20 =6 r h f 1 s0 þ K r2 E
ð26Þ
:
By this truncation, the BARBI model contains only long baroclinic Rossby waves and barotropic Rossby waves, while baroclinic gravity and short baroclinic Rossby waves are filtered, consistent with the planetary geostrophic approximation. Note that this approximation becomes invalid near the equator but for the present purpose it will turn out to be convenient. Note that we have also neglected the barotropic advection of E since it was shown to be small in Olbers and Eden (2003), which we can confirm in our experiments here as well (not shown). We also rewrite now the barotropic vorticity budget Eq. (17) as 1
b@ x w ¼ r s0 þ fric h ðf rw rE þ s0 Þ r h :
:
ð27Þ
Note that we have neglected the time derivative of w which filters also short barotropic Rossby waves and that fric denotes the torque related to effects of the harmonic lateral friction. The last term in the barotropic vorticity budget collects all torques given by the topographic variations and is often called the bottom pressure torque, since from the barotropic momentum budget we find
hrP ¼ f rw rE þ s0
ð28Þ
(note that P denotes here in fact the bottom pressure in the steady and frictionless system). By rewriting Eq. (26) we can identify the bottom pressure torque also in the budget of the vertical density moment ðxÞ
k2 @ t / þ b@ x / ¼ r s0 þ bf 1 s0 þ 2rP r h Kk2 r2 / :
:
ð29Þ
with the (squared) internal Rossby radius of the BARBI model 2 2 k2 ¼ N6fh2 and the definition E=f ¼ /. Since / is related to the geostrophic transport relative to the bottom,5 we will call / the baroclinic transport streamfunction and Eq. (29) the baroclinic vorticity budget. Note that in the last term of Eq. (29), we have ignored the latitudinal dependency of the Coriolis parameter. It becomes obvious that the bottom pressure torque is the coupling between the barotropic and baroclinic vorticity budgets. This coupling becomes only active for sloping topography. Note that the bottom pressure torque in the baroclinic vorticity equation can be traced back to the effect of the lifting of the background density by the barotropic and baroclinic flow in the budget of the perturbation density. Correspondingly, this lifting plays the role of a vorticity change due to stretching in the barotropic and baroclinic vorticity budgets. Note that the bottom pressure term shows up with opposite sign in Eqs. (27) and (29) relative to the wind stress curl and the planetary vorticity. References Andrews, D.G., Holton, J.R., Leovy, C.B., 1987. Middle Atmosphere Dynamics. Academic Press. Eden, C., 2007. Eddy length scales in the North Atlantic. J. Geophys. Res. 112 (C06004). doi:10.1029/2006JC003901. 5 From the depth-integrated momentum balance, Eq. (16), we see that the total geostrophic transport U ðgÞ is given by f U ðgÞ ¼ hrP þ rE. Since the geostrophic : bottom velocity uðgÞ jh is given by f u ðgÞ jh ¼ rpjh ¼ rP þ g q0 jh rh, the geo: strophic transport relative to the bottom, which we name the baroclinic transport, becomes f ðU ðgÞ h u ðgÞ jh Þ ¼ rE g q0 jh rh. Besides a small contribution by the : : perturbation density at the bottom, q0 jh , in case of topographic variations, the baroclinic transport is therefore given by the streamfunction / ¼ E=f . Note that the bottom pressure torque rP r h is apparently given by the geostrophic bottom flow : directed across isobaths.
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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. Parameterizing eddy-induced tracer transports in ocean circulation models. J. Phys. Oceanogr. 25, 463–474. Holland, W.R., 1973. Baroclinic and topographic influences on the transport in western boundary currents. Geophys. Fluid Dyn. 4, 187–210. Hughes, C.W., de Cuevas, B.A., 2001. Why western boundary currents in realistic oceans are inviscid: a link between form stress and bottom pressure torques. J. Phys. Oceanogr. 31 (10), 2871–2885. Marshall, D., 1984. Eddy-mean-flow interaction in a barotropic ocean model. Quart. J. R. Meterolog. Soc. 110 (465), 573–590. Munk, W.H., 1950. On the wind-driven ocean circulation. J. Atmos. Sci. 7, 80–93. Olbers, D., Eden, C., 2003. A model with simplified circulation dynamics for a baroclinic ocean with topography. Part I: waves and wind-driven circulations. J. Phys. Oceanogr. 33, 2719–2737. Olbers, D., Lettmann, K., Timmermann, R., 2007. Six circumpolar currents – on the forcing of the Antarctic Circumpolar Current by wind and mixing. Ocean Dyn. 57, 12–31. Salmon, R., 1998. Linear ocean circulation theory with realistic bathymetry. J. Mar. Res. 56, 833–884.
Sarkisyan, A.S., Ivanov, V.F., 1971. Joint effect of baroclinicity and bottom relief as an important factor in the dynamics of sea currents. Akad. Nauk. Atmopsh. Oceanic Phys. 7 (2), 173–188. Saunders, P.M., Coward, A.C., de Cuevas, B.A., 1999. Circulation of the Pacific Ocean seen in a global ocean model: ocean circulation and climate advanced modelling project (OCCAM). J. Geophys. Res. 104 (C8), 18281–18299. Starr, V.P., 1968. Physics of Negative Viscosity Phenomena. McGraw-Hill, New York, USA. Stommel, H., 1948. The westward intensification of wind-driven ocean currents. Trans. Am. Geophys. Union 29 (2), 202–206. Sverdrup, H.U., 1947. Wind-driven currents in a baroclinic ocean; with application to the equatorial currents of the Eastern Pacific. Proc. Natl. Acad. Sci. USA 33 (11), 318–326. Willebrand, J., Barnier, B., Böning, C., Dieterich, C., Killworth, P., LeProvost, C., Jia, Y., Molines, J.M., New, A.L., 2001. Circulation characteristics in three eddypermitting models of the North Atlantic. Prog. Oceanogr. 48 (2–3), 123– 161. Wunsch, C., Roemmich, D., 1985. Is the North Atlantic in Sverdrup balance? J. Phys. Oceanogr. 15 (12), 1876–1880.
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Ocean Modelling journal homepage: www.elsevier.com/locate/ocemod
Why western boundary currents are diffusive: A link between bottom pressure torque and bolus velocity Carsten Eden a,*, Dirk Olbers b a b
IFM-GEOMAR, FB I, Ocean Circulation and Climate Dynamics, Düsternbrooker Weg 20, 24105 Kiel, Germany Alfred Wegener Institute for Polar and Marine Research, Postfach 120161, 27515 Bremerhaven, Germany
a r t i c l e
i n f o
Article history: Received 30 January 2009 Received in revised form 1 July 2009 Accepted 10 July 2009 Available online 17 July 2009 Keywords: Western boundary currents Bottom pressure torque Friction Bolus velocity
a b s t r a c t The bottom pressure torque is known to vanish in the interior ocean but to play a dominant role in the western boundary layer in balancing the planetary vorticity on spatial scales larger than the Rossby radius of deformation. In this study, the appearance of the bottom pressure torque and thus any deviation of wind-driven flow from classical Sverdrup balance is locally related in steady state to non-zero bolus velocity and/or friction, under the assumption that horizontal density advection is small compared to the lifting of isopycnals. To first order approximation, the vortex stretching by the vertical bolus velocity is related to the bottom pressure torque. The bolus vortex stretching becomes a significant term in the barotropic vorticity budget of the western boundary layer and is formally equivalent to bottom friction as in the classical models of the wind-driven gyre circulation. Ó 2009 Elsevier Ltd. All rights reserved.
1. Introduction Energetic boundary currents are found on the western side of the individual ocean basins as part of the wind-driven large-scale gyre systems. The location and strength of these western boundary currents can be understood by considering depth-averaged vorticity balances (Stommel, 1948). In the most simplified form of such balances (Stommel, 1948; Munk, 1950), the torques related to the changes in planetary vorticity of a water parcel and the surface wind stress balance each other in the interior of the ocean, i.e. the interior ocean is in Sverdrup balance (Sverdrup, 1947), while frictional torques related to large bottom friction (Stommel, 1948) or lateral friction (Munk, 1950) balance the planetary vorticity change in the western boundary layer. Although these simple vorticity balances rely on rather restrictive assumptions, they have been very successful in explaining the most important aspects of the large-scale wind-driven circulation, such that they became classical concepts in physical oceanography. However, allowing for topographic variations in the depth-averaged (or depth-integrated) vorticity budgets, the so-called bottom pressure torque shows up, resembling a stretching term due to a geostrophically balanced bottom flow across isobaths. When the bottom flow is directed along isobaths or vanishes entirely, the bottom pressure torque also vanishes in the depth-averaged vorticity balance. This appears to be the case for instance in the interior of the North Atlantic (Wunsch and Roemmich, 1985), where * Corresponding author. E-mail address: [email protected] (C. Eden). 1463-5003/$ - see front matter Ó 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.ocemod.2009.07.003
large topographic variations as the mid-oceanic ridge can be found, which only have a small signature in the depth-averaged flow (Willebrand et al., 2001). On the other hand, Saunders et al. (1999) and Hughes and de Cuevas (2001) found in a realistic high-resolution general circulation ocean model that the bottom pressure torque becomes large in the western boundary layer, where it balances to first order the planetary vorticity change, while other torques, in particular the frictional torques play only a minor role. Therefore, the classical, frictional concept of the western boundary layer has to be abandoned. However, although all important topographic effects in the depth-averaged vorticity balance can be described by the bottom pressure torque in an elegant way, it is not possible to obtain a similarly self-consistent simplified model as in the classical concepts by including also the bottom pressure torque in the depth-averaged vorticity budget. This is because bottom pressure is a diagnostic variable containing information about the density stratification, for which no similar simplified depthaveraged balance equation appears to be available. In this study, we use the BARotropic Baroclinic Interaction (BARBI) model (Olbers and Eden, 2003) to better understand the role and physical meaning of the bottom pressure torque in the western boundary layer. BARBI was designed to describe the impact of the (baroclinic) stratification in the depth-averaged, i.e. barotropic vorticity budget, by constructing a budget for the vertically integrated potential energy. Here, we identify the bottom pressure torque as a coupling term between the barotropic vorticity budget and the budget of vertically integrated potential energy and interpret the latter as the budget of a depth-averaged
15
C. Eden, D. Olbers / Ocean Modelling 32 (2010) 14–24
baroclinic vorticity involving a baroclinic streamfunction. As already suggested by Olbers et al. (2007) for the Southern Ocean, it turns out that using BARBI it is possible to return to first order to the simple classical concept, by identifying a diffusive torque related to stretching by the vertical bolus velocity as a simple reinterpretation of the frictional torque in the classical barotropic vorticity budget. Here, we discuss the consequences for the basin circulation and western boundary layers. The following section acts as a summary of the main findings of this study. Most of the detailed mathematical derivations which are used in the second section are placed in the appendices. As a further illustration, the results are discussed in Section 3 on the basis of a numerical implementation of the BARBI model and a standard primitive equation model for comparison. We discuss the near cancellation of the bottom pressure torque and its role in the western boundary in the barotropic and baroclinic vorticity budgets. The last section gives a summary and discussion of the results. 2. The role of the bottom pressure torque
in the western boundary layer of the non-eddy-resolving model version. We therefore follow Hughes and de Cuevas (2001) and conclude that for scales larger than the relevant meso-scale eddy scale, the barotropic vorticity balance is given to good approximation by
b@ x w ¼ r s0 þ r P rh þ fric :
2.2. The baroclinic vorticity budget Using the BARBI model concept of Olbers and Eden (2003) (which is detailed in Appendix A), it is demonstrated in C how to construct an approximate baroclinic vorticity budget given by
b@ x / ¼ r s0 þ b=f sðxÞ 2 r P rh K=k2 r2 / :
The curl of the vertically integrated and steady horizontal momentum equation will be called here the barotropic vorticity budget and is given by1
:
:
:
Z
0
u rudz þ fric
ð1Þ
h
where w denotes the streamfunction of the depth-averaged flow R0 with r w ¼ h udz, where P denotes the pressure at the bottom : at z ¼ h, s0 the surface wind stress and where the last term collects frictional torques related to isotropic small-scale turbulence. A more detailed derivation and discussion of the barotropic vorticity budget is given in Appendix A. The balance between the left hand side of Eq. (1), i.e. the planetary vorticity term, and the first term on the right hand side of Eq. (1), i.e. the wind stress curl, represents the classical Sverdrup balance, which is distorted by the bottom pressure torque r P rh and the occurrence of the non-linear tor: que and the torque of the frictional terms in the barotropic vorticity balance. Fig. 1 shows the most important components of the time-mean budget calculated from a realistic high-resolution general circulation model of the North Atlantic, identical the one used in e.g. Eden (2007). On small spatial scales, the barotropic vorticity budget displays a rather noisy pattern with large and fluctuating terms. In particular the non-linear term is important in the budget. However, when averaging Eq. (1) over spatial scales representative for the length scale of meso-scale eddies, which is given by the Rossby radius in mid- to high-latitudes (Eden, 2007), the non-linear term in the budget becomes much less important and the dominant balance is between planetary vorticity, wind stress curl and the bottom pressure torque. As shown by Saunders et al. (1999) and Hughes and de Cuevas (2001), in particular in the western boundary current layer, the bottom pressure torque balances the planetary vorticity for spatial averages larger than 0.5°–1°, while the non-linear term becomes small in the spatially averaged budget. In fact, when we look at the barotropic vorticity budget in a noneddy-resolving version of the numerical model, the non-linear term in the vorticity budget is an order of magnitude less important than the other terms (not shown). This is in particular true 1 We use the following notation: The vector subscript u denotes a clockwise : rotation of the horizontal velocity u ¼ ðu; vÞ by 90°, i.e. u ¼ ðv; uÞ. The same holds for : r ¼ ð@ x ; @ y Þ resp. r ¼ ð@ y ; @ x Þ. Note that r s denotes the curl of the horizontal : : vector s and r P rh ¼ JðP; hÞ the Jacobian of P and h.
:
ð2Þ
The same result is obtained within the planetary geostrophic approximation. It should be noted, however, that scales of frontal structure in the western boundary layer can be of the same order than the meso-scale eddy scale. For those scales, the following discussion will not be applicable, instead we are focussing on the noneddy-resolving, large-scale circulation of the ocean.
2.1. The barotropic vorticity budget
b@ x w ¼ r s0 þ r P rh r
:
:
ð3Þ
where / denotes the streamfunction of the baroclinic transport and where K denotes a lateral (isopycnal thickness) diffusivity and k the Rossby radius. We call / the baroclinic streamfunction, since it describes the geostrophic transport relative to the bottom. In order to derive Eq. (3), a steady state is assumed and a vertical background profile for buoyancy is introduced. As a consequence of small relative magnitudes of the perturbations from that reference buoyancy profile, it is also assumed that the effect of vertical advection of the background and perturbation buoyancy dominates the effect of lateral advection of buoyancy perturbations. The latter effect is therefore neglected in the buoyancy budget, while the advective effect of meso-scale eddy mixing on buoyancy is parameterised following Gent and McWilliams (1990). As discussed in Appendix B, the last term in Eq. (3) is a torque related to the vertical bolus velocity. Note that the lateral diffusivity K corresponds to the isopycnal thickness diffusivity in the Gent and McWilliams (1990) parameterisation and that K=k2 corresponds to the inverse of a time scale of several days. We have also neglected the effect of thermohaline surface forcing and diapycnal mixing, i.e. we focus on wind-driven flow only. Together with the barotropic, Eq. (2), and baroclinic, Eq. (3), vorticity budgets we have the vertically averaged momentum budget, given by Eq. (16) of Appendix A. The system is linear, and is similar to that studied by Salmon (1998). 2.3. The cancellation of the bottom pressure torque Using the baroclinic vorticity budget Eq. (3), it becomes possible to understand the role of the bottom pressure torque within the above formulated limits, which is the aim of the present study. First, it is possible to show that for the case without friction and diffusion the bottom pressure vanishes: Subtracting Eq. (3) from Eq. (2) in order to eliminate the wind stress curl and using the vertically integrated (steady and frictionless) zonal momentum budget given by f @ x w ¼ f @ x / þ h@ x P sðxÞ (compare Eq. (16) of Appendix A) one obtains after some manipulations
r P rf =h3 ¼ 0 :
ð4Þ
This shows that the bottom pressure P becomes constant along 3 contours of f =h . Together with a boundary condition P ¼ 0 somewhere on each such contour, P ¼ 0 everywhere and so the bottom pressure torque is zero everywhere (except for closed contours of 3 f =h which might produce a different, more complicated situation
16
C. Eden, D. Olbers / Ocean Modelling 32 (2010) 14–24
Fig. 1. Time mean barotropic vorticity, Eq. (1) in a realistic eddy-permitting model of the North Atlantic Ocean in 109 m=s2 . Shown are planetary vorticity minus wind stress R0 curl (bwx r s) in the left column, the bottom pressure torque (rP rh) in the middle column and the curl of the non-linear term (r h u rudz) in the right column. The individual terms have been box averaged from the 1/12° 1/12° cos / (Mercator) grid (/ denotes here latitude) of the numerical model on a regular 2° 1.3° grid in the upper row, on a 1° 0.6° grid in the middle row and on a 0.5° 0.3° grid in the lower row. After interpolation on the individual grid, all data have been also smoothed over 3 3 grid points.
which we do not consider here). Eq. (4) implies that in the case with weak diffusion and friction, the bottom pressure torque tends to vanish. This appears to be often the case in certain regions of the interior North Atlantic (Fig. 1), leaving behind the classical Sverdrup relation, as discussed by Wunsch and Roemmich (1985) and Willebrand et al. (2001). Note that by implication, we can also conclude that any deviation of wind-driven flow from the classical Sverdrup balance in steady state is introduced by diffusive and frictional torques, which is in turn in accordance to the classical theory by Stommel (1948) and Munk (1950).
maximum or minimum, i.e. we expect differences between w and / at those latitudes, which, however, should not exceed relative order oðL=aÞ. On the other hand, we have seen in Fig. 1 that in the western boundary the bottom pressure torque is large, pointing towards the importance of diffusion and friction in the western boundary layer. It is, however, still necessary that / and w are similar: Using again the vertically integrated (steady and frictionless) momentum balance f rw ¼ rðf /Þ þ hrP s0 (see Eq. (16) of Appendix A) the bottom pressure torque becomes
hrP r h ¼ f rðw /Þ r h /b@ x h þ s0 r h 2.4. The similarity of barotropic and baroclinic streamfunction Second, we demonstrate that a vanishing bottom torque implies that the barotropic and baroclinic streamfunction are getting similar, i.e. / w: Scaling shows that the magnitude of the wind stress curl, r s0 , in the baroclinic vorticity budget Eq. (3) is a factor : ðxÞ a=L 1 larger than the term b=f s0 , where a denotes the earth radius and L the scale of the spatial variations of the wind stress. Therefore, baroclinic and barotropic vorticity budgets become identical for vanishing diffusivity and viscosity to order oðL=aÞ, with the consequence of similar streamfunctions w and /. HowðxÞ ever, it should be noted that the neglect of b=f s0 in comparison with r s0 is not really permitted near a latitude of wind stress :
:
:
:
ð5Þ
Scaling shows that the first term on the right hand side of Eq. (5) is a factor a=L 1 larger than the other two, where L now denotes (the maximum of) the scale of the spatial variations of wind stress, streamfunction and topography. In fact, the leading order term in the bottom pressure torque in Eq. (5) is also a factor a=L 1 larger when compared in the barotropic and baroclinic vorticity budgets, Eqs. (2) and (3), with the planetary vorticity term and the wind stress curl. Since the bottom pressure torque shows up with opposite sign in both budgets, Eqs. (2) and (3), it is therefore hard to see how the pressure torque can be balanced without implying viscous and diffusive torques much larger (of factor a=L 1) than the planetary vorticity in both budgets. On the other hand, a small
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C. Eden, D. Olbers / Ocean Modelling 32 (2010) 14–24
0 50°
(a)
(b) −1000
40°
−2000
−3000 30° −4000
−5000
20° −0.2
0
0.2
0°
10°
20°
30°
ðxÞ
Fig. 2. (a) Zonal component of surface wind stress, s0 , in N=m2 as a function of latitude which is used in the idealised experiments with BARBI and the primitive equation model. (b) Ocean bottom depth in m as a function of longitude for a ¼ 0:2, a ¼ 0:5 and a ¼ 0:8 in Eq. (10).
difference between w and / (of order L=a 1 relative to the order of w and /) can be balanced in both budgets by planetary vorticity and diffusive and viscous torques of similar (or smaller) magnitude. We consider the latter case as the relevant one for the ocean.
2
@ t E þ hU rEh
2
¼ N 20 =6U rh þ N20 =2r u0 þ K r2 E 2
@ t U þ f U ¼ hrP rE þ s0 þ Ah r U :
2
@ t u0 þ f u0 ¼ h =3ðrE s0 Ah r2 UÞ þ Ah r2 u0 :
2.5. The baroclinic Stommel equation To zero order approximation, we can therefore assume that w / both in the interior and the western boundary layer, and by combining Eqs. (2) and (3) to eliminate the bottom pressure torque we arrive to first order (in L=a 1) approximation at the simplified barotropic (and baroclinic) vorticity budget
b@ x w ¼ r s0 Kk2 =3r2 w þ fric :
ð6Þ
Note that in Eq. (6) we have neglected for consistency the term ðxÞ bf 1 s0 . The factor Kk2 =3 represents the inverse of a time scale on the order of days and the related term in Eq. (6) acts similar to bottom friction. Eq. (6) is thus formally similar to the barotropic vorticity budget by Stommel (1948) for a flat bottom, who introduced large bottom friction to balance the return flow in the western boundary layer. In Eq. (2), however, the return flow is balanced by the bottom pressure torque (and also by frictional torques). By comparing with Eq. (6), we see that the bottom pressure torque was replaced to first order approximation by the diffusive torque in the baroclinic vorticity budget. Somehow surprisingly, we have rediscovered Stommel’s original vorticity budget with a physically rather different motivation. Eq. (6) was called the ‘‘baroclinic Stommel equation” by Olbers et al. (2007).2 Note that we have interpreted the diffusive torque in the baroclinic Stommel equation Eq. (6) as the torque related to the vertical bolus velocity. 3. Results and interpretation 3.1. Idealised numerical model The theoretical results found in the previous section are illustrated using idealised numerical models, i.e. a numerical implementation of the BARBI model (Olbers and Eden, 2003) and a standard primitive equations model in identical configuration. The BARBI model which is solved numerically is given by
2 In Olbers et al. (2007) a factor 1/2 is missing in the diffusive torque of the baroclinic Stommel equation, i.e. the third term on the left hand side of Eq. (17) of Olbers et al. (2007).
ð7Þ ð8Þ ð9Þ
while the other model is based on the standard primitive equations (PE). Note that the meaning of all variables in the BARBI model are discussed in Appendix A; here, we just note that it resembles a two mode system, i.e. the model contains a barotropic mode, given by the depth-averaged velocity U, and a single baroclinic vertical mode, given by the vertical density moment E and baroclinic velocity moment u0 . Note also that the budget for the vertical density moment E in the numerical BARBI model, given by Eq. (7), includes the effect of horizontal advection by the barotropic flow which was neglected in the foregoing theoretical consideration in Section 2, while advection by the horizontal part of the baroclinic flow remains neglected. We solve both sets of equations (BARBI and PE) in a sector domain ranging 32° in longitude and from 20°N to 52°N resembling a mid-latitude ocean basin. This ocean is driven by a classical, cosine-shaped, eastward wind stress s0 ¼ 0:2 N=m2 ðcosð2pð/ 20 Þ=32 Þ; 0Þ. The zonal component of s0 is shown in Fig. 2a. There is a strong background stratification, represented by a constant stability frequency N 0 ¼ 2 103 s1 ; meso-scale density mixing is parameterised following Gent and McWilliams (1990) with thickness diffusivity of K ¼ 5000 m2 =s and numerical dissipation is given by harmonic lateral friction with viscosity Ah ¼ 5000 m2 =s and no-slip boundary conditions. The ocean depth
h i hðxÞ ¼ 5000 m 1 a expððx 16o Þ2 =ð5o Þ2 Þ h i 1 a expððx þ 1o Þ2 =ð5o Þ2 Þ
ð10Þ
features an idealised north/south mid-ocean ridge in the middle of the domain and an idealised continental shelf at the western boundary, i.e. a reasonable situation for an ocean basin and resembling the North Atlantic Ocean. The size of the topographic variations, shown in Fig. 2b, are varied from a ¼ 0:2 to a ¼ 0:8 but is set to a ¼ 0:5 if not otherwise noted. Horizontal resolution is 0.5° 0.5° and the time step is 0:5 h. The numerical code of the BARBI model can be accessed at http://www.ifm-geomar.de/~barbi and the primitive equations code at http://www.ifm-geomar.de/ ~spflame. The latter was configured as close as possible to the BARBI model. The primitive equation model contains 50 vertical levels with 100 m thickness and the topographic variations are discretised on this vertical grid. Horizontal resolution, boundary and initial
18
C. Eden, D. Olbers / Ocean Modelling 32 (2010) 14–24
(a)
(b)
(c)
Fig. 3. Streamfunction w in Sv in steady state in a simulation of the BARBI model with N ¼ 0 s1 (left), a simulation with a flat bottom (middle), and the standard model simulation with topographic variations and stratification. Also shown are isobaths (lines).
conditions, viscous and diffusive parameters are set as in the BARBI model. The numerical BARBI model features the familiar near cancellation of the impact of the topographic variations by the stratification on the barotropic flow as discussed first by Holland (1973). Fig. 3 shows three different model solutions in steady state, i.e. one without stratification, i.e. setting N 0 ¼ 0 s1 (Fig. 3a), a solution without topographic variations, i.e. setting h ¼ 2000 m (Fig. 3b) and a solution including stratification and topographic variations. In the simulation without stratification only a weak barotropic flow develops which apparently shows a strong impact of the topographic variations on the flow. Switching on the stratification, the barotropic circulation gets stronger and the flow gets close to the circulation in the simulation with flat bottom. This cancellation of the topographic variations will be discussed in detail in the next section. When comparing to the primitive equation model, Fig. 4, it becomes obvious that the BARBI model is in this configuration indeed a very good approximation to the primitive equation model in terms of streamfunction w and vertical density moment E (or baroclinic streamfunction /). The same holds for the bottom pressure torque, and also for the other terms in the barotropic and the baroclinic vorticity budgets, which we have diagnosed in the primitive equations model for comparison with the BARBI model (not shown). In particular the form of the diffusive torque related to the vertical bolus velocity in the baroclinic vorticity budget (where
we have neglected topographic variations and horizontal advection as discussed in Appendix B) matches well the effect of the diagnosed (complete) bolus velocity in the primitive equation model. There are, however, slight differences over the mid-ocean ridge and near the western boundary visible in Fig. 4. This is due to the truncation in BARBI, which contains only the first vertical density moment. Using two vertical moments the solution is getting closer to the primitive equation model (not shown) as discussed by Olbers and Eden (2003). However, note that it is impractical to use more than the first couple of vertical modes in BARBI, because of the slow convergence of the basis functions zn for large n. 3.2. Numerical model results It was argued above that scaling of the components of the bottom pressure torque in the barotropic and baroclinic vorticity budget implies a strong similarity between / and w, which is in fact featured by the numerical example. Fig. 5 shows the components of the bottom pressure torque as given by Eq. (5). The figure shows the simulation with BARBI but note that the primitive equations 1 model is very similar (not shown). The terms fh rw r h (shown : 1 in Fig. 5a) and fh r/ r h (not shown) are indeed very similar : and their difference (shown in Fig. 5b) one order of magnitude 1 smaller than the individual terms. The remaining term, bh /@ x h (shown in Fig. 5c), is of similar magnitude to the difference of the former two terms (note that s0 r h ¼ 0 in the idealised :
(a)
(b)
(c)
(d)
Fig. 4. Streamfunction w in Sv (upper row) and first vertical density moment E in m3 =s2 (lower row) in BARBI (left column) and a primitive equations model (right column) in a setup as close as possible to the BARBI model.
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C. Eden, D. Olbers / Ocean Modelling 32 (2010) 14–24
(a)
(b)
(c)
1
1
1
Fig. 5. Components of the bottom pressure torque in the BARBI model, i.e. fh rw rh (a in 109 m2 =s2 ) and fh rðw /Þ rh (b in 109 m2 =s2 ) and bh /@ x h (c in 109 m2 =s2 ). Note the different color scale in (a).
(a)
(b)
(c)
(d)
(e)
(f)
(g)
(h)
Fig. 6. Cumulative zonal integrals of the components of the barotropic vorticity budget Eq. (2) (upper row) and baroclinic vorticity budget Eq. (3) (lower row). All zonal R integrals are divided by b and shown in Sv. Upper row: barotropic streamfunction w (a), transport related to wind stress curl b1 dxr s0 (b), bottom pressure torque R R R ðxÞ b1 dxrP rh (c) and frictional torque b1 dxfric (d). Lower row: baroclinic streamfunction / (e), baroclinic transport related to wind stress curl b1 dxðr s0 þ bf 1 s0 Þ R R (f), bottom pressure torque 2b1 dxrP rh (g) and meso-scale eddy mixing b1 dxKk2 r2 / (h). Also shown are contours of the inverse Rossby radius 1=k in (a) and (e).
configuration). As expected from the theoretical discussion in the previous section, we find that the barotropic and baroclinic streamfunctions become similar such that the bottom pressure torque becomes of similar magnitude as the other terms in the baroclinic and barotropic vorticity budgets. However, although reduced in magnitude by the similarity of w and /, the bottom pressure torque still plays a dominant role in the barotropic and baroclinic vorticity budgets. Fig. 6(upper row) shows the individual terms of Eq. (2) in the numerical BARBI model (the primitive equation model is very similar, not shown) as cumulative zonal integrals which are also divided by b, such that all terms add up to form the barotropic streamfunction (left hand side of Eq. (2)). In the interior, the most important contribution to the transport is given by the wind stress curl, i.e. the flow is more or less in Sverdrup balance. The contribution of the bottom pressure term becomes larger over the mid-ocean ridge and in particular important near the western boundary. Here, it largely contributes to the compensating transports of the interior flow, while the frictional torques are only important in a thin boundary layer. Note that more than half of the return flow in the western boundary layer is made by the bottom pressure torque. Fig. 6(lower row) shows the corresponding cumulative zonal integrals of the baroclinic vorticity budget Eq. (3) in the numerical BARBI model (the primitive equation model is very similar, not shown). As expected, the planetary vorticity, b/x , and the wind ðxÞ stress forcing, r s0 þ b=f s0 are very similar to the corresponding : terms in the barotropic vorticity budget Eq. (2), while the diffusive torques, K=k2 r2 /, and the bottom pressure torque, 2 r P rh, are : balancing each other to zero order.
By varying the value of K relative to the value of Ah we can change the relative importance of the frictional torque and the bottom pressure torque in the barotropic vorticity budget for the western boundary return flow. Fig. 7 shows the barotropic streamfunction w, the cumulative integrals of bottom pressure torque and the frictional torque in a simulation of the BARBI model using K ¼ 1000 m2 =s and K ¼ 15000 m2 =s. In the simulation with the small (large) K, the barotropic streamfunction gets larger (smaller), the bottom pressure torque smaller (larger) and the bottom pressure torque contributes relatively less (more) to the return flow in the western boundary layer than the frictional torque. In the individual cases, however, barotropic and baroclinic streamfunctions continue to be very similar, which holds in particular also for the zero order balance between the torques related to vertical bolus velocity and bottom pressure in the baroclinic vorticity budget. Note also that the western boundary layer width of w and / becomes larger for increasing values of K. This is related to the similarity of the bolus torque with the effect of bottom friction: Increasing the value of K is equivalent to increasing the bottom friction coefficient, which is in turn proportional to the boundary layer width in the classical model of Stommel (1948). However, we do not see a linear relationship between boundary layer width and K as suggested by the classical model, which means that lateral friction in the barotropic vorticity budget is also important in the experiments with the BARBI model. A similar impact on the magnitude of the bottom pressure torque and its relative importance compared to friction is given
20
C. Eden, D. Olbers / Ocean Modelling 32 (2010) 14–24
(a)
(b)
(c)
(d)
(e)
(f)
Fig. 7. Cumulative zonal integrals of the components of the barotropic vorticity budget Eq. (2) for a simulation of the BARBI model with K ¼ 1000 m2 =s (upper row) and a simulation with K ¼ 15000 m2 =s (lower row). All zonal integrals are divided by b and shown in Sv: Barotropic streamfunction w (a and d), transport related to the bottom R R pressure torque b1 dxrP rh (b and e) and frictional torque b1 dxfric (c and f). Also shown are contours of the inverse Rossby radius 1=k in (a) and (d).
(a)
(b)
(c)
(d)
(e)
(f)
Fig. 8. Cumulative zonal integrals of the components of the barotropic vorticity budget Eq. (2) for a simulation of the BARBI model with different topographic variations, i.e. with a ¼ 0:2 (upper row) and a simulation with a ¼ 0:8 (lower row). All zonal integrals are divided by b and shown in Sv: Barotropic streamfunction w (a and d), transport R R related to the bottom pressure torque b1 dxrP rh (b and e) and frictional torque b1 dxfric (c and f). Also shown are contours of the inverse Rossby radius 1=k in (a) and (d).
by the height of the topographic variations. Fig. 8 shows the components of the barotropic vorticity budget in a simulation with smaller and larger topographic obstacles, by setting a ¼ 0:2 and a ¼ 0:8 in Eq. (10) respectively instead of a ¼ 0:5 as before. As expected, when increasing (reducing) the variations in h the magnitude of the bottom pressure torque becomes larger (smaller) and the frictional torques less (more) important. In the numerical experiments, we found that finite lateral friction is necessary to obtain a stable integration at horizontal resolution of the experiments shown here. Only by increasing the horizontal resolution it was possible to reduce the lateral viscosity further (not shown), such that the diffusive torques are able to balance more and more of the western boundary flow. Extrapolating the numerical experiments, it appears therefore possible to balance the western boundary flow entirely by the effect of density mixing, without the need of invoking frictional torques. However, mechanical dissipation is still needed in the energy budget to balance the wind work as discussed in Appendix B.
4. Summary and discussion In the barotropic vorticity balance relevant for spatial scales larger than meso-scale eddies, Eq. (2), the bottom pressure torque tends to vanish in the interior ocean Wunsch and Roemmich (1985), Willebrand et al. (2001). In consequence, the classical Sverdrup balance between wind stress curl and planetary vorticity change holds in those regions. On the other hand, the bottom pressure torque plays a dominant role in the western boundary layer in determining the barotropic volume transport (Saunders et al., 1999; Hughes and de Cuevas, 2001). It was the aim of this study to better understand the role of the bottom pressure torque and to relate it to simple physical mechanisms. Using the BARBI model of Olbers and Eden (2003), we have formulated a baroclinic vorticity budget, Eq. (3), which determines the baroclinic volume transport (relative to the bottom), and which looks very similar to its barotropic counterpart, but in which the bottom pressure torque shows up with opposite sign. This baroclinic vorticity budget is valid in steady state and for situations
C. Eden, D. Olbers / Ocean Modelling 32 (2010) 14–24
in which the horizontal advection of perturbation density is small compared to vertical lifting of perturbation and background density. Combining the baroclinic with the barotropic vorticity budget it is possible to show that for vanishing friction and diffusion, the bottom pressure torque cancels such that the classical Sverdrup balance holds. In consequence, we are able to conclude that any deviation from the Sverdrup balance is related to effects of density diffusion and friction, in accordance to the classical theory by Stommel (1948) and Munk (1950). For the case without diffusion and friction and vanishing bottom pressure torque, we found that barotropic and baroclinic streamfunction become identical to first order approximation. This similarity also extents to the case with finite bottom pressure torque, since for a large difference between the streamfunctions, the bottom pressure torque would be an order of magnitude larger than the planetary vorticity change and could only be balanced by unrealistically large frictional or diffusive torques. Using the similarity between both streamfunctions, it is possible to eliminate the bottom pressure torque from both vorticity budgets, which then contains only torques related to wind stress forcing, friction and density diffusion. The latter acts similar to bottom friction in the classical Stommel model (Stommel, 1948) of the wind-driven circulation. We have therefore rediscovered the classical Stommel model with, however, a different physical meaning of the frictional torque. Using the BARBI model, we associate to first order approximation the vortex stretching by the vertical bolus velocity of the Gent and McWilliams (1990) parameterisation with the bottom pressure torque. This vortex stretching becomes a significant term in the barotropic vorticity budget of the western boundary layer and formally equivalent to bottom friction as in the simple model by Stommel (1948). It is able to balance a large part of the return flow in the western boundary layer, but it is not related to friction as in Stommel’s original model but rather is related to advective stirring of density. The relation between vertical bolus velocity and bottom pressure torque can also be considered using the Residual Mean Theory of Andrews et al. (1987). The density budget becomes in this formulation
@ t q þ u rq þ w @ z q ¼ 0
ð11Þ
with the horizontal, u ¼ u þ ub , and vertical w ¼ w þ wb residual velocity and the horizontal ub ¼ @ z B and vertical wb ¼ r B bolus : : velocity. If one neglects the effect of the horizontal advection, as we have done in the derivation of the baroclinic vorticity budget, it becomes clear in w ¼ w þ wb ¼ 0 for steady state. At the bottom, the (geostrophically balanced) Eulerian vertical velocity is related to the bottom pressure torque, fwjz¼h ¼ rP r h, and so the vertical :
(a)
21
bolus velocity, fwb jz¼h ¼ rP r h. It becomes obvious, how a : non-vanishing vertical bolus velocity at the bottom, is related to a non-vanishing bottom pressure torque. Fig. 9 shows the bottom pressure torque related to the Eulerian mean vertical bottom velocity, fwjz¼h and to the vertical bolus velocity at the bottom fwb jz¼h in the primitive equation model. Both cancel indeed to a large extent. We have shown above how the vertical bolus velocity at the bottom translates to a torque in the baroclinic and barotropic vorticity budget, which is then formally identical to bottom friction as in the original model by Stommel (1948). Acknowledgements The authors thank Richard Greatbatch for his helpful discussions. We also thank Chris Hughes for the helpful review and for pointing out the cancellation of the bottom pressure torque in Eq. (4). Appendix A. The BARBI model The BARBI model is based on the relevant equations of motion for the oceanic circulation on scales much larger than the Rossby radius, given by the planetary geostrophic approximation. They are repeated here for completeness
@ t u þ f u ¼ rp þ @ z s þ F :
ð12Þ
@ z p ¼ g q
ð13Þ
r u þ @zw ¼ 0 @ t q þ u rq þ w@ z q ¼ Q
ð14Þ ð15Þ
The notation is standard, i.e. u denotes horizontal velocity, w vertical velocity, p pressure, q density and the vector subscript u : a rotation of the horizontal velocity u ¼ ðu; vÞ by 90°, i.e. u ¼ ðv; uÞ. For simplification, the mean balances of heat and salt : are combined into a thermohaline balance for density, with a density source term Q related to turbulent meso-scale mixing and stirring, diapycnal turbulent micro-scale processes and effects from compressibility and the non-linear equation of state. The horizontal momentum equation is also supplemented with vertical turbulent friction given by the vertical stress vector s and the lateral friction F. The vertical stress tensor represents effects of microscale turbulence while F the effect of meso-scale eddies on the horizontal momentum. Deviating from the planetary geostrophic approximation, we also retained the local time derivative of the horizontal momentum, i.e. we allow for gravity waves and short Rossby waves in addition to long Rossby waves. Note that the rigid-lid boundary condition is employed here.
(b)
Fig. 9. fwjz¼h (a) and fw jz¼h (b) in 109 m=s2 in the primitive equation model, where w denotes the Eulerian mean vertical velocity and w the vertical eddy advection (bolus) velocity.
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C. Eden, D. Olbers / Ocean Modelling 32 (2010) 14–24
Vertical integration of the horizontal momentum equation Eq. (12) leads to
@ t U þ f U ¼ hrP rE þ s0 sb þ
Z
:
0
Fdz
ð16Þ
h
R0 with the horizontal transport vector U ¼ h udz, the wind stress s0 and the stress sb at the bottom of the ocean at z ¼ h. Note that the bottom depth h is allowed to vary. By partial integration and use of the hydrostatic relation Eq. (13), the vertically integrated pressure gradient splits into two terms, related to the vertically integrated R0 potential energy, E ¼ g h zqdz, and the bottom pressure, P ¼ pjz¼h . The latter can be eliminated from the momentum equation Eq. (17) by dividing by h and taking the curl
R0 with F 0 ¼ @ z s þ F 1=h h Fdz. By the occurrence of the term Enþ2 in Eq. (19) it becomes obvious that the BARBI model is given by a coupled infinite hierarchy of density and baroclinic velocity moments. A closure of this hierarchy was proposed in Olbers and Eden (2003) based on the consideration of wave properties. For the simplest closure, Eq. (19) becomes 0
2
@ t u02 þ f u2 ¼ h =3 ðrE s0 þ sb Þ þ
Z
:
0
z2 F 0 dz
ð20Þ
h
This model consists of the barotropic momentum U, the vertically integrated potential energy E1 and the second baroclinic velocity moment u02 . For simplicity, we will drop the subscripts at E1 and u02 in the following.
r h1 r@ t w þ r w rfh1 ¼ rh1 r E þ r h1 ðs0 sb Þ :
:
1
þ r h :
Z
:
Appendix B. Diffusive and viscous closures in BARBI
0
Fdz
ð17Þ
h
with the barotropic streamfunction w, defined by the relation r: w ¼ U. Note that the vector subscript at the gradient operator r ¼ ð@ x ; @ y Þ denotes again rotation by 90°, i.e. r: ¼ ð@ y ; @ x Þ. The first term on the right hand side of Eq. (17) is related to the combined effect of topographic variations and stratification and is often called the ‘‘JEBAR” (Joint Effect of Baroclinicity and Relief) term as introduced by Sarkisyan and Ivanov (1971). Knowledge of E would allow the computation of the streamfunction w from Eq. (17), i.e. would allow to close the model (with additional closures for sb and F). This is the basic idea of BARBI and the starting point for its formulation. Note that an alternative formulation of the barotropic vorticity budget is obtained by taking the curl of Eq. (17) directly (i.e. without devision by h), which yields Eq. (1). Note also that in the latter form the bottom pressure P shows up while in the formulation Eq. (17), the potential energy E shows up. As discussed in detail by Olbers and Eden (2003), using the density budget Eq. (15) it is possible to derive a tendency equation for R0 the vertical density moments3 En ¼ g h zn q0 dz with n ¼ 1; 2; . . . n given by n
n
@ t En þh U rEn h
ðnþ1Þ
þð1Þn
N20 h N2 U rh 0 r u0nþ1 ¼ nþ2 nþ1
Z
0
zn Qdz
h
ð18Þ To derive Eq. (18) the density q in Eq. (15) was decomposed into a background state and perturbation q0 , similar as in quasi-geostrophic theory. The background state was chosen for simplicity as linear with depth represented by the constant stability frequency N 0 (although arbitrary functions of depth are possible in a refined BARBI model version). In Eq. (18), the effect of the advection of the density perturbation by the baroclinic flow, u0 ¼ u U=h, has been neglected. This effect was shown to be small as long as perturbations in density remain small compared to the background density. All other advective effects in Eq. (18) are exact, in particular the lifting of the background density by the baroclinic flow is represented in Eq. (18) by the divergence of R0 the baroclinic velocity moments u0nþ1 ¼ h znþ1 u0 dz, for which the following budget can be derived exactly
@ t u0nþ1
0
þf u
:
nþ1
i 1 h nþ1 ¼ ð1Þn h ðrE1 s0 þ sb Þ þ rEnþ2 nþ2 Z 0 þ znþ1 F 0 dz ð19Þ h
3 Note that the definition of E using the perturbation density q0 instead of full density q as above does not affect the momentum equation Eq. (17) or the barotropic vorticity budget Eq. (17).
The closures for the sub-grid effects represented by Q, F, s and sb in the BARBI model have to be specified with care since they become important for the western boundary layers. The turbulent density mixing Q in Eq. (18) can be split into diapycnal mixing effect and an advective effect by meso-scale eddies. We assume that the former is small compared to the latter such that we neglect it. We also neglect any effects from the non-linear equation of state and compressibility effects of the perturbation density. For the advective effect of meso-scale eddy mixing on density, we employ the Gent and McWilliams (1990) parameterization in which the eddy advection (bolus) velocity is given by a streamfunction B, parameterized as
B ¼ K s; vb ¼ @ z ; B wb ¼ r B :
:
ð21Þ
:
where K is equivalent to the thickness diffusivity of the Gent and McWilliams (1990) parameterization, where s ¼ rq=@ z q denotes the vector of the isopycnal slopes, and where vb and wb denote the horizontal and vertical components of the bolus velocity respectively. The barotropic part of the bolus velocity vanishes because of the boundary conditions B ¼ 0 at top and bottom. In agreement to the treatment in Eq. (18), we care about the lifting of the background stratification by the baroclinic vertical bolus velocity only, i.e. we assume that
Z
0
zQ dz
h
Z
0
h
zN 2 r B dz g :
Z
0
r K rzq dz K r2 E
ð22Þ
h
For simplicity, a flat bottom,4 constant K and qz N2 =g was assumed in Eq. (22). It turns out that the dominant effect of advective meso-scale eddy density mixing on vertically integrated potential energy E is given by diffusion of E with a diffusivity identical to the thickness diffusivity of the Gent and McWilliams (1990) parameterization. Note that in an integral over a closed domain no sink term due to the meso-scale eddy density mixing shows up in Eq. (18) by specifying vanishing gradients of E normal to the boundaries. However, on the other hand, we know that the net effect of the parameterisation is given by a conversion of mean (available) potential energy to eddy energy (Gent et al., 1995), i.e. it is possible to show that
Z V
z r ðvb qÞþðwb qÞz dV
Z A
ðrK rE
Z
0
Ks2 N2 ÞdA¼
h
Z
Ks2 N2 dV
V
ð23Þ where V ðAÞ denotes a closed volume (area) and where again a flat bottom was assumed. The term on the right hand side of Eq. (23) is sign definite and a sink for E, resembling the conversion of mean potential energy to eddy energy, mimicking the effect of baroclinic 4
Or, equivalently, vanishing diffusivity K at top and bottom.
23
C. Eden, D. Olbers / Ocean Modelling 32 (2010) 14–24
instability. In Olbers et al. (2007) it was proposed to parameterise this exchange with eddy energy as linear damping of E. However, it turns out that this exchange term is locally much less important than the diffusive term related to the lifting of the background stratification by the vertical bolus velocity, such that we will simply neglect the exchange term here. Note that this neglection is consistent with the neglection of the effect of advection of perturbation density by the baroclinic flow in BARBI, resembling a second order term when density perturbations remain small compared to the background. For the frictional closures it is less clear how to proceed. We assume that vertical friction in the interior can be neglected, except for the surface where the vertical turbulent stress vector s connects to the surface wind stress s0 . We also neglect the bottom stress sb , because we will see below that in many cases the deep flow over the bottom topography will vanish, such that the bottom stress would also vanish in that case. On the other hand, the model needs some kind of dissipation of kinetic energy which becomes clear considering the energy cycle in BARBI. The barotropic kinetic energy is given by K b ¼ U 2 =ð2hÞ and by using Eqs. (18) and (16) the global domain integrals of K b and E can be derived as
Z @t
Z
Z 2 1 U rh þ dx h U s0 Z Z 0 2 1 þ dx h U F dz@ t
2
dx K b ¼
2
dx E ¼
Z
Z
2
2
dx Eh
h 2
dx Eh
2
U rh
ð24Þ
Exchange between E and K b is given by barotropic density advection over topography, dissipation is due to friction and energy is supplied by the wind work. Note that baroclinic kinetic energy is not considered here, since it can be shown that it does not exchange energy with E as long as the effect of advection of perturbation density by the baroclinic flow is set to zero in the budget for E. This effect was assumed and shown to be locally neglectable in BARBI as long as the perturbation density remains small compared to the background density. Note also that there is no energy exchange with eddy energy, since we have neglected the dissipation effect by the Gent and McWilliams (1990) parameterisation of E in Eq. (23). It becomes clear that in the balance of total mechanical energy in BARBI, K b þ E, a steady state can only be established when the wind work is balanced by some kind of mechanical dissipation. We implement in BARBI harmonic lateral friction in combination with a no-slip boundary condition, i.e. F ¼ Ah r2 u and approximate
Z
0
h
F dz Ah r2 U
and
Z
0
2
z2 F 0 dz Ah ðr2 u0 h =3 r2 UÞ
ð25Þ
h
where we have again neglected the spatial dependency of h for simplicity. However, it should be noted that harmonic lateral friction is problematic, since it is well known that mean momentum is not mixing down mean momentum gradients in turbulent geophysical flow (Starr, 1968). In contrast, mixing of potential vorticity should be implemented (Marshall, 1984), but it remains unclear at the moment how this could be done in a consistent way. On the other hand, the formulation using harmonic friction is convenient for numerical reasons since it provides a sink for grid noise such that we follow common practise and use it here as well. Appendix C. Baroclinic vorticity budget To discuss the cancellation of the effect of topographic variations and the corresponding similarity between w and E we need to further simplify the BARBI model by filtering gravity and short
Rossby waves. By neglecting the time derivative and friction for u0 it is possible to combine the budgets for E and u0 , i.e. 2
2
@ t E þ N20 =6 ðr E rh f 1 r w rh Þ :
:
2
¼ N20 =6 r h f 1 s0 þ K r2 E
ð26Þ
:
By this truncation, the BARBI model contains only long baroclinic Rossby waves and barotropic Rossby waves, while baroclinic gravity and short baroclinic Rossby waves are filtered, consistent with the planetary geostrophic approximation. Note that this approximation becomes invalid near the equator but for the present purpose it will turn out to be convenient. Note that we have also neglected the barotropic advection of E since it was shown to be small in Olbers and Eden (2003), which we can confirm in our experiments here as well (not shown). We also rewrite now the barotropic vorticity budget Eq. (17) as 1
b@ x w ¼ r s0 þ fric h ðf rw rE þ s0 Þ r h :
:
ð27Þ
Note that we have neglected the time derivative of w which filters also short barotropic Rossby waves and that fric denotes the torque related to effects of the harmonic lateral friction. The last term in the barotropic vorticity budget collects all torques given by the topographic variations and is often called the bottom pressure torque, since from the barotropic momentum budget we find
hrP ¼ f rw rE þ s0
ð28Þ
(note that P denotes here in fact the bottom pressure in the steady and frictionless system). By rewriting Eq. (26) we can identify the bottom pressure torque also in the budget of the vertical density moment ðxÞ
k2 @ t / þ b@ x / ¼ r s0 þ bf 1 s0 þ 2rP r h Kk2 r2 / :
:
ð29Þ
with the (squared) internal Rossby radius of the BARBI model 2 2 k2 ¼ N6fh2 and the definition E=f ¼ /. Since / is related to the geostrophic transport relative to the bottom,5 we will call / the baroclinic transport streamfunction and Eq. (29) the baroclinic vorticity budget. Note that in the last term of Eq. (29), we have ignored the latitudinal dependency of the Coriolis parameter. It becomes obvious that the bottom pressure torque is the coupling between the barotropic and baroclinic vorticity budgets. This coupling becomes only active for sloping topography. Note that the bottom pressure torque in the baroclinic vorticity equation can be traced back to the effect of the lifting of the background density by the barotropic and baroclinic flow in the budget of the perturbation density. Correspondingly, this lifting plays the role of a vorticity change due to stretching in the barotropic and baroclinic vorticity budgets. Note that the bottom pressure term shows up with opposite sign in Eqs. (27) and (29) relative to the wind stress curl and the planetary vorticity. References Andrews, D.G., Holton, J.R., Leovy, C.B., 1987. Middle Atmosphere Dynamics. Academic Press. Eden, C., 2007. Eddy length scales in the North Atlantic. J. Geophys. Res. 112 (C06004). doi:10.1029/2006JC003901. 5 From the depth-integrated momentum balance, Eq. (16), we see that the total geostrophic transport U ðgÞ is given by f U ðgÞ ¼ hrP þ rE. Since the geostrophic : bottom velocity uðgÞ jh is given by f u ðgÞ jh ¼ rpjh ¼ rP þ g q0 jh rh, the geo: strophic transport relative to the bottom, which we name the baroclinic transport, becomes f ðU ðgÞ h u ðgÞ jh Þ ¼ rE g q0 jh rh. Besides a small contribution by the : : perturbation density at the bottom, q0 jh , in case of topographic variations, the baroclinic transport is therefore given by the streamfunction / ¼ E=f . Note that the bottom pressure torque rP r h is apparently given by the geostrophic bottom flow : directed across isobaths.
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C. Eden, D. Olbers / Ocean Modelling 32 (2010) 14–24
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