Abstract
 Top of page
 Abstract
 1. Introduction
 2. LargeScale Dynamics
 3. Vorticity Balance
 4. Divergence of the Bottom Geostrophic Flow
 5. Analytical Solutions
 6. Discussion
 Appendix A:: Errors of the Climatological Fields
 Acknowledgments
 References
 Supporting Information
[1] Baroclinic forcing of the barotropic flow in the East Greenland Current in Fram Strait (EGCFS) is studied using theoretical analysis and climatological data. First, the validity of the arrested topographic wave (ATW) vorticity balance is reexamined. Then, a distribution of the net (depthaveraged) relative geostrophic flow and the associated alongisobath variations of the bottom density are estimated from the Polar Science Center Hydrographic Climatology (Seattle, Washington, United States). The effect of the alongisobath bottom density variations is referred to as JEBAR_{b} in order to distinguish it from the famous JEBAR (joint effect of baroclinicity and relief), which involves depthintegrated density variations. It is shown that the JEBAR_{b} is able to maintain a significant bottom geostrophic flow in the climatological EGCFS. Estimates of the JEBAR_{b} at 79°N correspond to an increase of the magnitude of the flow by ∼3 cm s^{−1} over a distance of 200 km along the continental slope. An analytical solution for the flow that is driven by a constant JEBAR_{b} term and satisfies the ATW vorticity balance is then obtained in a domain bounded by two parallel isobaths. Finally, the JEBAR_{b} is related to the JEBAR. It is shown that while the JEBAR_{b} maintains the curl of the Coriolis force associated with the bottom geostrophic flow, the JEBAR maintains the curl of the net Coriolis force associated with the absolute geostrophic flow. In the climatological EGCFS over the continental slope, the JEBAR is nearly equal to the JEBAR_{b}.
1. Introduction
 Top of page
 Abstract
 1. Introduction
 2. LargeScale Dynamics
 3. Vorticity Balance
 4. Divergence of the Bottom Geostrophic Flow
 5. Analytical Solutions
 6. Discussion
 Appendix A:: Errors of the Climatological Fields
 Acknowledgments
 References
 Supporting Information
[2] A forcedplanetary vorticity balance often invoked in the theories of the largescale flow in a flatbottomed ocean is the Sverdrup balance, i.e., a balance between the vortex tube squashing by the surface Ekman pumping and the advection of planetary vorticity by the geostrophic transport [e.g., Pedlosky, 1996]. On the other hand, a geostrophic flow over a sloping bottom in an area with a small variation of the Coriolis parameter is likely to be in a vorticity balance between the bottom Ekman pumping and the vortex tube stretching by the crossisobath bottom geostrophic flow. The motion associated with such a frictionaltopographic vorticity balance is often referred to as the arrested topographic wave (ATW). The original ATW model was constructed by Csanady [1978] from steady equations for a net (depthaveraged), winddriven flow in a coastal zone. Various formulations of the ATW exist. A common feature of the ATW models is that the bottom stress is assumed to be proportional to a velocity. Another assumption is that only the alongisobath component of bottom friction is dynamically relevant, while the bottom topography is simplified to be a plane sloping in one direction (Figure 1). The model equations can then be combined into a parabolic equation for a single variable which, for a constant bottom slope, is an analog of the onedimensional equation for heat conduction. A general, nondimensional form of the equation for a variable �� is
where x and y are the cross and alongisobath coordinates, respectively. Negative of y plays the role of time in the heat conduction analogy, while the term Q is a “source” function depending on the problem at hand. For instance, Csanady's original model resulted in a homogeneous (Q = 0) equation for the surface elevation. The author also considered the problem of freshwater influx at the shore, in which a nonhomogeneous term (Q ≠ 0) appears in the governing equation [Csanady, 1978]. A nonhomogeneous term also appears in the governing equation of other ATW models for flow in a baroclinic ocean. It appears, for instance, in an equation for the alongisobath component of the bottom geostrophic velocity, v_{gb}, derived by Shaw and Csanady [1983]. The equation for v_{gb} was recently used by Schlichtholz [2002] to interpret the bottom geostrophic flow over the continental slope in the East Greenland Current in Fram Strait (EGCFS) using hydrographic data from summer 1984 (MIZEX 84 data) and an inverse model based on these data [Schlichtholz and Houssais, 1999a].
[3] The southward flow in the EGCFS (Figure 2) is an important element of the current system in the Arctic Mediterranean. Not only it evacuates sea ice and fresh surface polar water from the Arctic Ocean, but it is also a means by which intermediate and deep waters from that ocean can get to the Nordic Seas [e.g., Aagaard et al., 1985; Rudels et al., 1999]. An outflow of these waters occurs along the East Greenland Slope. In addition to water masses advected from the Arctic Ocean along the continental slope, the EGCFS receives some water masses locally in Fram Strait, mainly through westward recirculations of Atlantic and deep waters from the West Spitsbergen Current, and also through injection of surface polar water along the East Greenland Polar Front (Figure 2). Local admixtures gradually change the water mass characteristics along the slope [e.g., Schlichtholz and Houssais, 1999b, 2002]. Alongisobath gradients of density are present in the EGCFS near the bottom, as shown by Schlichtholz [2002], and generally in the whole water column, as demonstrated by Schlichtholz and Houssais [1999c] using distributions of potential energy calculated from the MIZEX 84 data. The alongisobath density gradients are dynamically important since they do not only determine the vertical shear of the crossisobath component of the geostrophic flow, but also constitute a driving agent for the barotropic flow. The alongisobath variations of the bottom density appear in a formula for the divergence of the bottom geostrophic flow derived by Shaw and Csanady [1983]. The formula, when combined with the ATW vorticity balance, gives rise to the “source” term in equation (1) for �� = v_{gb}. On the other hand, the alongisobath variations of potential energy appear in the famous JEBAR (joint effect of baroclinicity and relief) term in the vorticity balance for the net flow. The JEBAR effect was introduced to oceanography by Sarkisyan and Ivanov [1971] and interpreted in terms of a crossisobath net baroclinic flow by Mertz and Wright [1992]. Its diagnosis from the MIZEX 84 data gave rise to an analytical model of the EGCFS in which the JEBAR term was balanced by a damping term due to either bottom or internal friction [Schlichtholz and Houssais, 1999c]. That model was the first attempt to explain the barotropic flow in the EGCFS as resulting from local baroclinic influences over a sloping bottom. The subsequent study of Schlichtholz [2002] made it clear that, on the assumption of geostrophy, the bottom flow in the current should be maintained by the effect of alongisobath variations of the bottom density. The effect can be referred to as JEBAR_{b} (joint effect of bottom baroclinicity and relief) in order to stress the fact that it involves baroclinicity at the ocean bottom instead of a total (depthintegrated) baroclinicity.
[4] The present study is intended to give further insight into the dynamics of the EGCFS and tackles, after an introduction of basic equations in section 2, with four problems. The first problem concerns the validity of the ATW balance which is checked in section 3 against observations reported in the literature. The second problem concerns the persistence of the JEBAR_{b}. Till now only the importance of that effect for maintaining the flow in summer 1984 has been evidenced. In section 4, the effect is quantified using climatological data. A short discussion of a relationship between the alongisobath bottom density variations and the distribution of the baroclinic transport is included in that section. Errors of the climatological fields are discussed in Appendix A. The third problem is whether an analytical solution to equation (1) can grasp main features of the bottom flow in the EGCFS. An approximate solution for v_{gb} when Q = const, analogous to the solution for the surface elevation in Csanady's problem of freshwater influx at the shore, was mentioned by Schlichtholz [2002]. That solution is a part of a solution in a domain which is semiinfinite in both x and y [Csanady, 1978]. Here, in section 5, exact solutions are obtained in a domain bounded in x (Figure 1). The fourth problem, discussed in section 6, concerns a relationship between the JEBAR_{b} and JEBAR in general, and in the EGCFS in particular.
2. LargeScale Dynamics
 Top of page
 Abstract
 1. Introduction
 2. LargeScale Dynamics
 3. Vorticity Balance
 4. Divergence of the Bottom Geostrophic Flow
 5. Analytical Solutions
 6. Discussion
 Appendix A:: Errors of the Climatological Fields
 Acknowledgments
 References
 Supporting Information
[5] The “largescale” approximations to the horizontal momentum equation and the mass conservation equation are
where u is the horizontal velocity vector, w is the velocity component in the z (vertical) direction parallel to the local direction of gravity (positive upward), is a vertical unit vector, p is the pressure, ρ_{0}(= 1027 kg m^{−3}) is a reference density, ∇ is the horizontal gradient operator, f is the Coriolis parameter, and τ is the vertical shearing stress. The stress is significant only in thin boundary layers. In the ocean interior, equation (2) is reduced to the balance between the Coriolis acceleration and the horizontal pressure gradient, i.e.,
where the index g stresses the fact that the horizontal flow is geostrophic.
[6] In a homogeneous ocean the geostrophic velocity is depthindependent while in a nonhomogenous ocean a vertical shear of u_{g} appears because the vertical pressure gradient is related to the density, ρ, by
where g(= 9.8 m s^{−2}) is the gravitational acceleration. Eliminating the pressure between the geostrophic equation (4) and the hydrostatic equation (5) yields the famous thermal wind relation. Consequently, the absolute geostrophic velocity at any depth level can be decomposed into a depthindependent bottom contribution, u_{gb}, and a contribution relative to the bottom, u_{gr}, i.e.,
The relative velocity is entirely determined by the density distribution below the given level,
where H is the bottom depth.
[7] The bottom geostrophic flow should be determined, at least partly, by the vertical boundary conditions, and hence the barotropic vorticity equation. Appropriate kinematic conditions for equation (3) are those of nonormal flow,
where the indices s and b denote evaluation at the ocean surface and bottom, respectively. The corresponding dynamical boundary conditions for equation (2) are
where τ_{s} is the wind stress, while r is a constant friction coefficient with dimension of velocity.
[8] Following one of the procedures outlined by, e.g., Mertz and Wright [1992], the barotropic vorticity equation associated with equations (2), (3), and (8)–(11) can be written as
where the overbar denotes vertical averaging from the surface to the bottom, and w_{Es} is the surface Ekman pumping velocity,
With the last two terms on the lefthand side (LHS) neglected, equation (12) is reduced to the Sverdrup balance, while with the first two terms neglected, it becomes the ATW balance. In the diagnostic context, equation (12) is an equation for u_{gb} whatever is the actual local balance since _{g} = u_{gb} + _{gr}. When the friction term is neglected, the resulting hyperbolic problem can easily be solved for the flow field [e.g., Bogden et al., 1993]. With the friction term included, the problem becomes parabolic. The inverse modeling applied by Schlichtholz and Houssais [1999a] to the MIZEX 84 data is an example of an attempt to solve that problem in a variational manner.
3. Vorticity Balance
 Top of page
 Abstract
 1. Introduction
 2. LargeScale Dynamics
 3. Vorticity Balance
 4. Divergence of the Bottom Geostrophic Flow
 5. Analytical Solutions
 6. Discussion
 Appendix A:: Errors of the Climatological Fields
 Acknowledgments
 References
 Supporting Information
[9] The relative magnitude of the four terms in equation (12) can be expressed as
where W_{s}, W_{β}, W_{t}, and W_{f} are the scales of the surface Ekman suction, “equivalent vertical velocity” due to the meridional geostrophic transport, vertical velocity induced by the crossisobath bottom geostrophic flow, and bottom Ekman pumping, respectively. The last three scales have been defined using the scales of the Coriolis parameter and its gradient, f_{c} and β_{c}, respectively, the scales of the ocean depth and bottom slope, D and s_{0}, respectively, the scales of the cross and alongisobath components of the bottom geostrophic flow, U_{b} and V_{b}, respectively, the associated crossisobath length scale of motion, l_{b}, and the scale of the meridional component of the net geostrophic flow, V_{β}.
[10] All scales on the LHS of equation (14) but W_{s} depend on estimates of the horizontal velocity components. Since the isobaths in the area of the EGCFS nearly coincide with the meridians (Figure 2), we can assume that the alongisobath and meridional flows have a same scale. The magnitude of the nearbottom flow in the EGCFS is on the order of a few centimeters per second, as shown by Foldvik et al. [1988] from yearlong current meter measurements at 79°N starting in the period of the MIZEX 84 experiment. The annual mean speed from the deepest instruments at their moorings FS1 and FS2 located on the East Greenland Slope (Figure 2) was ∼2.5 cm s^{−1}. The deep instruments were placed 25 m and 300 m above the bottom, over the isobaths H_{FS1} ≈ 1100 m and H_{FS2} ≈ 1700 m, respectively. That a typical magnitude of the deep flow in the EGCFS is 2–3 cm s^{−1} seems to be corroborated by recent current measurements. Figure 4 of Fahrbach et al. [2001] shows a vertical transect of the meridional velocity component in Fram Strait at 79°N averaged over the period from September 1997 to August 1999. The transect was based on measurements at 14 mooring sites extending from the western shelf break off Spitsbergen to the East Greenland Shelf break, with the deepest instruments located 10 m above the bottom. According to that figure, the magnitude of the near bottom meridional velocity in the EGCFS exceeds 2 cm s^{−1} over the lower slope and at the shelf break. Therefore a reasonable estimate of V_{b} is 3 cm s^{−1}. Consequently, V_{β} must be ∼6 cm s^{−1}, as inferred from an approximately equal partition of the transport in the EGCFS between bottom and relative contributions [Foldvik et al., 1988; Fahrbach et al., 2001].
[11] Results of calculations of the crossisobath component of the near bottom velocity and its ratio to the alongisobath component in the EGCFS are presented in Table 1. The calculations were based on a smoothed topography and the eastward and northward components of the annual mean near bottom velocity at FS1 and FS2 (see Table 2 of Foldvik et al. [1988]). The bottom depth was calculated by local averaging of the 5min gridded Earth topography data set, ETOPO5 [U.S. National Geophysical Data Center, 1988], over a square about 20 km × 20 km (H_{20}), 50 km × 50 km (H_{50}) or 100 km × 100 km (H_{100}), and then subsampled on a 1° grid. The depth contours in Figure 2 are plotted from the original (unsmoothed) data set (H_{0}). The projection coefficients for the velocity components were calculated from depth differences on the subsampled grid and then interpolated to locations corresponding to the bottom depths H = H_{FS1} and H = H_{FS2}. Since the EGCFS flows in the direction of propagation of the topographic Rossby waves, i.e., with the shallower water to the right when looking downstream, its alongisobath component is negative in the coordinate system in Figure 1. The ratio of the velocity components in Table 1 is negative because the crossisobath component is positive (downslope). The dependence of the estimates in Table 1 on the scale of smoothing of topography is not large, with slightly smaller values of the crossisobath component of the flow for rougher topography. The scale U_{b} would be above 0.4 cm s^{−1} when based on the estimates from FS1, and above 0.1 cm s^{−1} from FS2. A difference between the estimates from FS1 and FS2 may not only reflect a different position of the instruments on the slope, but also their different height above the bottom.
Table 1. CrossIsobath Component u of the NearBottom Velocity in the EGCFS and Its Ratio to the AlongIsobath Component v^{a}Component  H_{0}  H_{20}  H_{50}  H_{100} 


Site FS1 
u, cm/s  0.416  0.425  0.496  0.554 
u/v  −0.164  −0.169  −0.197  −0.221 

Site FS2 
u, cm/s  0.125  0.128  0.152  0.166 
u/v  −0.055  −0.056  −0.066  −0.072 
[12] The scale of the surface Ekman suction in the EGCFS is W_{s} = 2 × 10^{−6} m s^{−1}, as estimated from a longterm mean (1948–2003) distribution of the wind stress in the area (Figure 3). The distribution has been obtained from averaged monthly mean data from the National Centers for Environmental Prediction (NCEP, in Boulder, Colorado, United States) reanalysis [Kalnay et al., 1996]. The scales V_{β} = 6 cm s^{−1}, f_{c} = 1.4 × 10^{−4} s^{−1}, β_{c} = 4.4 × 10^{−12} s^{−1} m^{−1}, and D = 2000 m, yield W_{β} = 4 × 10^{−6} m s^{−1}. Therefore the “equivalent vertical velocity” related to the planetary β effect has a comparable magnitude to the surface Ekman suction. Both Sverdrupian terms seem, however, to be an order of magnitude smaller than the topographic term. With the scale of the bottom depth variation d (Figure 1) equal to 2000 m and l_{b} = 50 km, the scale of the bottom slope (s_{0} = d/l_{b}) is 0.04. This estimate and the smallest estimate of the magnitude of the crossisobath velocity based on Table 1, i.e., U_{b} = 0.1 cm s^{−1}, gives W_{t} = 4 × 10^{−5} m s^{−1}.
[13] The estimates made above indicate that the vorticity balance in the EGCFS should include a bottom friction term to remain in the limit of the largescale dynamics. With the bottom stress parameterized as in equation (11), the current should be in the ATW balance obtained for
The equality of the scales on the LHS of relation (15) implies that the friction coefficient depends on the degree of anisotropy of the flow in the topographic coordinates expressed by the parameter α_{b} = U_{b}/V_{b},
This relationship should hold approximately for any vorticity balance to which the bottom Ekman pumping and topographic vortex tube stretching contribute at the leading order. Figure 4 shows r as a function of α_{b} for external parameters, d and f_{c}, appropriate for the EGCFS. Four particular regimes are marked on the curve. Point R_{1} corresponds to a maximum value of r (≈10 cm s^{−1}) for which the flow remains anisotropic in the sense that
The somewhat arbitrary choice of the limit for α_{b}^{2} is there 0.1. Point R_{2} corresponds to the value of the friction coefficient (r = 1 cm s^{−1}) obtained for the velocity scales used in our scaling, i.e., U_{b} = 0.1 cm s^{−1} and V_{b} = 3 cm s^{−1}, for which α_{b} = 0.03. The ratios of the velocity components in Table 1 indicate that the value of r in the EGCFS should lie in the range between the values at R_{1} and R_{2}. These values are larger than typical values of r found in the literature. For instance, Csanady [1988] cites values of 3–5 × 10^{−4} m s^{−1}. These values fall between the values at points R_{3} and R_{4} in Figure 4. Point R_{3} corresponds to the value of r (≈0.1 cm s^{−1}) for which the terms of the ATW balance would have the same magnitude as the planetary term. In the neighborhood of this point, all terms of equation (12) would significantly contribute to the vorticity balance of the EGCFS. However, such a balance would require a very small magnitude of the crossisobath bottom geostrophic velocity, ∼1 × 10^{−4} m s^{−1}. Such a magnitude would be appropriate also in the case of the topographic Sverdrup balance, i.e., the frictionless limit of equation (12) with a significant contribution from the topographic vortex tube stretching term. A further reduction of the value of α_{b} along the line in Figure 4 by 1 order of magnitude gives r ≈ 1 × 10^{−4} m s^{−1} (point R_{4}). These values, if appropriate, would suggest the Sverdrup balance in the EGCFS. The balance is a priori not possible since w_{Es} is positive while the meridional flow is southward, so that both Sverdrupian terms have the same sign in equation (12). Both are negative. Similarly, the topographic Sverdrup balance can be ruled out on the basis of the sign of the crossisobath flow from Table 1. A downslope motion corresponds to a topographic vortex tube stretching contributing to equation (12) a term of the same sign as the Sverdrupian terms. Therefore a largescale vorticity balance in the EGCFS different than the frictionaltopographic balance would imply that the crossisobath bottom geostrophic velocity in the area of the sites FS1 and FS2 has not only a much smaller magnitude but also a different sign than the estimates in Table 1.
[14] That the friction coefficient in the EGCFS might be O(1 cm s^{−1}) is also supported by the results of the analytical and inverse modeling applied to the MIZEX 84 data. Only for such a large value of r, the magnitude of the transport in the current diagnosed from the analytical model of Schlichtholz and Houssais [1999c] agreed with observations. The value r = 1 cm s^{−1} was then used in the inverse model by Schlichtholz and Houssais [1999a]. Noteworthy is the fact that the vorticity balance could not be closed in the inverse model without friction or with r much smaller than 1 cm s^{−1}. Of course, prescribing a uniform value to the friction coefficient is an imperfect parameterization. The coefficient can be interpreted on the basis of the quadratic friction law for an instantaneous flow as
where c_{D} is a drag coefficient which depends on the roughness of the ocean bottom, and V_{m} is the speed of a highfrequency perturbation superimposed on the largescale flow [e.g., Csanady, 1976, 1988]. A typical value of c_{D} is in the range 10^{−3}–10^{−2} according to Csanady [1976]. In this range of c_{D}, r equal to O(1 cm s^{−1}) would correspond to V_{m} = O(1–10 m s^{−1}), which is a too high value. According to Table 5 of Foldvik et al. [1988], the kinetic energy of the fluctuating part of the near bottom flow at FS1 and FS2 is only 2–3 times larger than the kinetic energy of the corresponding mean flow. With V_{m} = 5 cm s^{−1}, c_{D} should be 0.2 for r to be 1 cm s^{−1}. This implies that the “roughness” of the East Greenland Slope might be larger than suggested by the magnitude of the drag coefficient cited in studies of shallow seas or shelf circulation. Perhaps this is not surprising in view of equation (16), which shows that the friction coefficient for a slope current in the ATW balance should experience a much larger friction that a shelf current at the same latitude and with the same degree of horizontal anisotropy because of the much larger scale of the bottom depth variation.
[15] A conclusion that can be drawn from this section is that either (a) our estimates of the velocity components do not represent, even roughly, the true components of u_{gb} in the EGCFS, (b) the current cannot be described by the largescale dynamics alone, or (c) the current is in the frictionaltopographic vorticity balance with a relatively large drag or friction coefficient. We will assume the last alternative and further consider implications of the ATW balance. Using an approximation f = f_{c}, the balance reads as
In view of relation (17), equation (19) can be reduced to a relation between the crossisobath component of the bottom geostrophic flow, u_{gb}, and the crossisobath variation of the alongisobath component of that flow. On a fplane and for the bottom sloping only in one direction, i.e., H = H(x), we have
where s = dH/dx. A downslope motion at the top of the bottom Ekman layer generates a topographic vortex tube stretching which should be canceled out by the bottom Ekman pumping. The latter, according to equation (20), corresponds to an upslope increase of the magnitude of the (negative) alongisobath velocity, as shown in Figure 5 depicting a core in the current flowing with the shallower water to the right. According to the estimates in Table 1, FS1 and FS2 either were both located in the sector u_{gb} > 0 of the schematic in Figure 5 or FS1 was located in a shallower core while FS2 in a deeper one. In any case, at an extreme of v_{gb}, u_{gb} should vanish for equation (20) to hold. Therefore, even if the ATW balance is dominant, there should be locations in the EGCFS where other terms come into play.
4. Divergence of the Bottom Geostrophic Flow
 Top of page
 Abstract
 1. Introduction
 2. LargeScale Dynamics
 3. Vorticity Balance
 4. Divergence of the Bottom Geostrophic Flow
 5. Analytical Solutions
 6. Discussion
 Appendix A:: Errors of the Climatological Fields
 Acknowledgments
 References
 Supporting Information
[16] Although the frictionaltopographic vorticity balance involves the bottom geostrophic flow, it does not explicitly indicate any forcing mechanism for that flow. The latter should depend on the density distribution as it follows from consideration of the Coriolis acceleration acting on the bottom flow over a sloping bottom,
The righthand side (RHS) of equation (21) can be expressed in terms of the bottom pressure, p_{b}, and bottom density, ρ_{b}, using the identity
and evaluating the hydrostatic equation (5) at the bottom, so that we have
Acting with the curl operator ( · ×) on equation (23) yields
where J is the Jacobian operator in the horizontal plane. By equation (24), the curl of the Coriolis force acting on the bottom geostrophic flow is maintained by alongisobath bottom density variations, an effect called JEBAR_{b} in the introduction. On a fplane and for the bottom sloping only in one direction (Figure 1), equation (24) is reduced to the formula for the divergence of u_{gb} considered by Shaw and Csanady [1983],
The alongisobath bottom density variations are related to the distribution of the net relative geostrophic velocity. The latter can be expressed in terms of the potential energy per unit area, ρ_{0}χ, and the bottom density [e.g., Schlichtholz and Houssais, 1999c],
The static density profile, ρ_{r}, related hydrostatically to the corresponding pressure profile, p_{r}, is irrelevant for the calculation. For convenience, we will treat ρ_{b} and χ as the anomalies with respect to their static values, i.e., ρ_{b} = (ρ − ρ_{r})∣_{z=−H} and
Equation (26) can be rewritten as a formula for the total Coriolis acceleration acting on the relative flow,
Now acting with the curl operator on equation (28) gives
Therefore the JEBAR_{b} can be interpreted in terms of the curl of the total Coriolis force associated with the relative geostrophic flow which, by definition, is equal to the divergence of the relative transport multiplied by the Coriolis parameter. The two quantities, JEBAR_{b} and ∇ · (fH_{gr}), differ by a factor equal to the ocean depth, and have an opposite sign.
[17] The divergence of fH_{gr} can be split into planetary and nonplanetary contributions, so that
By equation (30), the alongisobath bottom density variations should be significant if the divergence of the relative geostrophic transport is significant and is not compensated by the advection of planetary vorticity by that transport. On a fplane, it is the divergence of the relative transport alone which drives the bottom geostrophic flow,
as schematically depicted in Figure 1 of Shaw and Csanady [1983] for the case of a localized dense water formation over a sloping bottom. If the JEBAR_{b} is not entirely compensated by the crossisobath variation of the crossisobath component of u_{gb}, the scale of the flow is determined by the scale of the alongisobath bottom density variations, Θ. From equation (25), we obtain
To obtain estimates of baroclinic variables, the density has been computed form the annual mean temperature and salinity fields of the Polar Science Center Hydrographic Climatology (PHC, Seattle, Washington, United States). The PHC, available at the standard depth levels on a 1° horizontal grid, is a composite of two heavily smoothed climatological data sets [Steele et al., 2001]. In the EGCFS area (see the box in Figure 2), the “effective” smoothing scale is 150–450 km and the accuracy of the density field in the deep layer is ∼0.01 kg m^{−3} (see Appendix A for details). The JEBAR_{b} directly calculated from the gradients of the PHC density at the maximum unmasked standard level depth (H_{PHC}) and the gradients of H = H_{PHC} is quite noisy in general. This should be at least partly attributed to a low vertical resolution of the data. Alternatively, J(ρ_{b}, H) can be computed using the bottom depth obtained from the ETOPO5 and the PHC density interpolated or extrapolated on that depth. However, the raw version of the ETOPO5 retains smallscale variations which may be incompatible with the smooth PHC fields for a description of the largescale dynamics. On the other hand, a large smoothing of topography requires interpolation or extrapolation of the density to depths much different from H_{PHC} which may result in an excessive alongisobath gradient of ρ_{b} at some locations. To test this, topographies with a different degree of smoothness (H_{0}, H_{20}, H_{50}, and H_{100} from section 3) have been used. A minimum rootmeansquare (RMS) of J(ρ_{b}, H) in the EGCFS area is obtained for H_{50}. It is smaller by 30% (3 times) than the corresponding RMS for H_{100} (H_{0}). The RMS of the JEBAR_{b} for H_{PHC} slightly exceeds that for H_{100}.
[18] The distributions of ρ_{b} and χ for the bottom topography which most restricts the magnitude of the JEBAR_{b} (H_{50}) are presented in Figures 6a and 6b, respectively. The distributions show a contrast between the lighter waters of the Arctic Ocean in the north and the denser waters of the Greenland Sea in the south. Both, the gradient of χ and variations of ρ_{b} contribute significantly to the relative flow, as shown by the distribution of the two contributions to _{gr} from equation (26) in Figure 7. The sum of these contributions results in a generally southward flow, with a magnitude of ∼2 cm s^{−1} (Figure 8a). Three cores of a comparable magnitude can be identified in the flow at 79°N, one over the continental slope, one at the shelf break, and another on the shelf. Consequently, the largest values of the relative transport are found over the slope (Figure 8b). The westward flow in the southeastern part of the area represents a northern recirculating branch of the cyclonic circulation in the Greenland Sea.
[19] The divergence of the relative geostrophic transport is the main contributor to the JEBAR_{b} in the EGCFS. The planetary contribution is 2 orders of magnitude smaller, as shown in Figure 9, where the relative magnitude of the two terms on the RHS of equation (30) is plotted as a function of longitude at 79°N. Actually, the relative transport is generally convergent. Over the slope, the convergence is ∼1 × 10^{−4} m s^{−1} at 79°N and 2–3 times larger at 78°N (Figure 10a). By equation (29) with f = f_{c}, a convergent relative transport implies that the JEBAR_{b} is positive (Figure 10b).
[20] Since H appears as a factor in the relationship between the JEBAR_{b} and the divergence of the relative transport, J(ρ_{b}, H) and ∇ · (H_{gr}) have a different crossisobath distribution. At 79°N, for instance, the magnitude of ∇ · (H_{gr}) has a maximum over the lower slope, while the JEBAR_{b} has a maximum at the shelf break. The location of the maximum convergence of H_{gr} on the crossisobath section moves upslope when looking southward (Figure 10a). This shift results, for instance, in large values of J(ρ_{b}, H) over the entire slope at 78°N (Figure 10b). Details of the distributions depend also on the degree of smoothing of topography. For instance, in the versions with no or small smoothing (H_{0} and H_{20}), two maxima of the JEBAR_{b} appear in the vicinity of the shelf break at 79°N, a stronger one on the shelf side and a weaker one on the slope side (Figure 11). A similar distribution is found for a large smoothing (H_{100}). The maximum on the upper slope disappears in the version with a moderately smoothed topography (H_{50}).
[21] The magnitude of J(ρ_{b}, H) in the EGCFS over the slope, estimated from Figure 10b to be ∼2 × 10^{−9} kg m^{−4} at 79°N, corresponds to a density variation of 0.01 kg m^{−3} over a distance of ∼200 km. This estimate is comparable to an estimate of the variation of ρ_{b} along the isobath H = 2000 m obtained from quasisynoptic data by Schlichtholz [2002]. According to equation (32), Θ = 0.01 kg m^{−3} yields V_{b} = 3 cm s^{−1}, which is in agreement with the observations of a nearbottom flow at 79°N.
5. Analytical Solutions
 Top of page
 Abstract
 1. Introduction
 2. LargeScale Dynamics
 3. Vorticity Balance
 4. Divergence of the Bottom Geostrophic Flow
 5. Analytical Solutions
 6. Discussion
 Appendix A:: Errors of the Climatological Fields
 Acknowledgments
 References
 Supporting Information
[22] Observations show that the alongisobath component of a nearbottom flow in the EGCFS is larger than the crossisobath component. We have seen in section 3 that, on the assumption of an ATW balance, the degree of anisotropy of the flow is determined by the magnitude of the friction coefficient. On the other hand, the analysis in section 4 shows that the scale of the flow itself is determined by the magnitude of the alongisobath bottom density variations. Since JEBAR_{b} is significant while the ATW balance is a likely vorticity balance in the EGCFS, it is judicious to combine both pieces of information to learn more about the current. Eliminating the crossisobath component of the bottom geostrophic flow between equations (20) and (25) gives a closed equation for the alongisobath component. In nondimensional form, we have
Once equation (33) is solved for v_{gb}, u_{gb} can be obtained from equation (20). In the case of a constant bottom slope (s = 1), equation (33) is reduced to equation (1) with �� = v_{gb} and Q = ∂ρ_{b}/∂y, i.e.,
[23] Consider equation (34) in a domain limited in the crossisobath direction by 0 < x < 1 (0 < x < l_{b} in dimensional units, Figure 1). The problem is an analog of the problem of heat conduction in a slab, i.e., a solid bounded by two parallel planes [Carslaw and Jaeger, 1959]. The baroclinic “source” term acts here on v_{gb} as the internal heat source acts on the temperature of the slab. If alongisobath density variations are not zero only in the “backward” portion of the region (y > 0), their effect on the “forward” portion (y < 0) should translate into a prescribed distribution of v_{gb} at an “initial” section (y = 0), v_{0}(x). In general, v_{0} is a result of all physical phenomena in the “backward” region as the temperature distribution in a conducting body at an initial time is a result of past events. A mathematically analogous problem, but in a semiinfinite domain in x, was analyzed at length by Csanady [1978]. If we here assume that v_{gb} vanishes at the edges x = 0 and x = 1 of the “forward” region, the distribution of v_{gb} at y < 0 resulting from an inflow (negative v_{0}) or outflow (positive v_{0}) alone is obtained by solving equation (34) with
The solution of the analogous heat conduction problem, i.e., with ends of a slab kept at zero temperature, is given by Carslaw and Jaeger [1959, p. 93]. Adapting that solution to the present problem yields
[24] According to equation (36), the details of the “initial” distribution of v_{gb} should gradually be lost following the “forward” region in −y. At large distances compared to π^{−2}(π^{−2}α_{b}^{−1}l_{b} in dimensional units), v_{gb} should tend to zero. An implication for the EGCFS is that the flow leaving the strait to the south should be more influenced by the local conditions in Fram Strait than by the inflow along the slope from the Arctic Ocean. An indirect evidence for that is a gradual change of water mass characteristics along the continental slope, already mentioned in the introduction.
[25] To find a solution in the presence of baroclinic forcing relevant for the EGCFS, take the case of a “sink” (increase of the bottom density with −y). Assume for simplicity that the alongisobath density variation is uniform in the “forward” region (∂ρ_{b}/∂y = −1), and that there is no inflow/outflow at y = 0. Assume also, as before, that the alongisobath flow is zero at the shallow and deep edges of the region. The problem to be solved is equation (34) with
The problem can be rewritten as
with
where v*_{gb} = v_{gb} + γ. The equation for the auxiliary variable v*_{gb} is homogeneous, while the “sink” term of the original problem appears in the “initial” condition for v*_{gb}. The solution for v*_{gb} is obtained from equation (36) with v_{0} = γ. The final solution, v_{gb} = v*_{gb} − γ, is
[26] The distributions of v_{gb} and u_{gb} = ∂v_{gb}/∂x are shown in Figure 12. The symmetric boundary conditions result in a symmetric distribution of both variables. The magnitude of v_{gb} has a maximum at the “midslope” position (x = 0.5) and increases with −y at any x = const except for the edges of the region, where it is zero (Figure 12a). As a consequence, u_{gb} is zero at x = 0.5 and, for any y = const, its magnitude is the largest at the edges x = 0 and x = 1 (Figure 12b). Positive (negative) values of u_{gb} on the deeper (shallower) half of the region correspond to a downslope (upslope) geostrophic motion and, therefore, to the vortex tube stretching (squashing) compensated by an Ekman pumping (suction). Since u_{gb} is an analog of the heat flux, a positive u_{gb} at the deeper edge and a negative u_{gb} at the shallower edge correspond to a heat gain through both ends of the conducting slab. v_{gb} is negative everywhere in the “forward” region as should be the temperature of a conducting slab with a uniformly distributed internal heat sink and with zero temperature initially everywhere and at ends all the time. A uniform internal heat sink would reduce continuously the temperature of the slab to the moment when it would be compensated by a heat gain through the ends of the slab. Similarly, the increase of the magnitude of the alongisobath flow weakens with −y, so that the solution approaches a limit which is quadratic in x (v_{gb} = −γ) and corresponds to a linear change of the crossisobath flow, i.e., u_{gb} = x − 0.5 at large −y.
[27] The above solution incorporates most essential dynamical features of the EGCFS, i.e., a southward flow and the presence of a core in the crossisobath section. Of course, there may be more cores in the current, but there may also be local extremes in the crossisobath distribution of the JEBAR_{b} (Figure 11). In addition, the forcing varies along the slope (Figure 10b), so that the real flow in the EGCFS is certainly more complex than in the simple solution. A remarkable feature of the solution is the increase of the magnitude of the flow along isobaths in the downstream direction. A southward increase of the magnitude of the bottom geostrophic flow in the EGCFS is a feature clearly appearing in the inverse model applied to the MIZEX 84 hydrographic data [Schlichtholz, 2002]. Of course, that feature cannot be verified by velocity observations at a single crossisobath section. Unfortunately, the historical and presentgoing longterm moorings in Fram Strait are limited to a single section.
6. Discussion
 Top of page
 Abstract
 1. Introduction
 2. LargeScale Dynamics
 3. Vorticity Balance
 4. Divergence of the Bottom Geostrophic Flow
 5. Analytical Solutions
 6. Discussion
 Appendix A:: Errors of the Climatological Fields
 Acknowledgments
 References
 Supporting Information
[28] The analytical solutions from the previous section clearly demonstrate the importance of the JEBAR_{b} for flow over a frictional sloping ocean bottom. While, in the absence of that forcing, the bottom flow is gradually retarded by friction in the direction of propagation of the topographic Rossby waves, the presence of the JEBAR_{b} can accelerate the flow in that direction. This is likely to happen in the EGCFS under the restrictions on the validity of the ATW vorticity balance mentioned in section 3. It should also be stressed that the inference of the significance of the JEBAR_{b} for maintaining a divergent bottom geostrophic flow in the EGCFS, made in section 4, does not rely on particular assumptions on the vorticity balance or friction details.
[29] A question remains about the relationship between the JEBAR_{b} and the JEBAR. It has been shown, in section 4, that the JEBAR_{b} can be decomposed into a planetary term (negligible in the EGCFS) and a term in the divergence of the relative transport. The JEBAR_{b} can also be decomposed into a term involving the divergence of f_{gr} and a term involving the crossisobath component of _{gr},
The second term on the RHS of equation (41) is the JEBAR term written in the form derived by Mertz and Wright [1992],
Therefore the JEBAR and the JEBAR_{b} differ from each other if the curl of the net Coriolis acceleration associated with the relative geostrophic flow is significant. We have
which, using equation (24), yields
So the JEBAR results from the curl of the net Coriolis force acting on the absolute geostrophic flow which, on a f plane, is proportional to the divergence of _{g}. This interpretation of the JEBAR was not emphasized in the literature.
[30] In the EGCFS, the JEBAR and JEBAR_{b} are equal (or nearly so) over the continental slope, as demonstrated in Figure 13, where both terms and their difference is plotted as a function of longitude at 79°N using the climatological density distribution in the version with a smoothed topography (H_{50}). Therefore the divergence of the net geostrophic flow in the EGCFS over the slope should be achieved through the divergence of the bottom flow. The figure also shows that the divergence of the net relative flow becomes important at the slope base and, especially, at the shelf break. Answering the question whether this is a real feature or an artifact resulting from calculation of the density gradients in areas of a drastic change in the bottom slope from climatological data is beyond the scope of the study. If we assume that it is a real feature, we should conclude that the divergence of _{gr} competes with the divergence of u_{gb} at the edges of the continental slope since the JEBAR_{b} is there larger than the JEBAR. However, even at the edges of the slope, the JEBAR remains positive (_{gr} is shoreward). The overall crossisobath distribution of the term is parabolic, with a maximum at the upper slope, where it attains a value of ∼2 × 10^{−11} s^{−2}. The value corresponds to an increase of the magnitude of _{g} of ∼6 cm s^{−1} over a distance comparable to the latitudinal extent of the EGCFS, which is ∼400 km.