## 1. Introduction

The continents of the Northern Hemisphere provide boundaries for the oceans that allow the oceanic meridional heat transport to largely be brought about by time-mean geostrophic currents. This can be compared to the Southern Ocean where the lack of boundaries yields a situation where meridional heat transport instead is mainly mediated by eddies (Döös and Webb 1994; Marshall and Speer 2012). Furthermore, the relatively weak stratification of northern high-latitude ocean basins causes the geostrophic currents to follow the underlying bottom topography. Such is the case in the Nordic seas and the Labrador Sea, where the large-scale circulation is dominated by time-mean boundary currents, largely tied to the continental slopes. These basins connect the North Atlantic with the Arctic Ocean; the warm and salty Atlantic water that flows along the eastern boundary of the Nordic seas is the main source water for the Arctic Ocean and thus affects the Arctic climate and sea ice (e.g., Zhang et al. 1998; Orvik and Niiler 2002; Carton et al. 2011). These subpolar seas are also regions of deep-water formation, feeding the lower limb of the Atlantic meridional overturning circulation (AMOC). That the boundary currents play an important role in the production of deep overflow waters from the Nordic seas that form the North Atlantic Deep Water has been suggested by, for example, Mauritzen (1996), Eldevik et al. (2009), and Isachsen et al. (2007). Similarly, Böning et al. (1996) and Spall and Pickart (2001) found in modeling studies that the sinking and formation of the middepth Labrador Sea Water takes place near lateral topographic boundaries.

In the present work we focus on the Nordic seas, situated in the northeastern corner of the North Atlantic between Greenland and Norway (see Fig. 1). The eastern part of the Nordic seas is mainly influenced by warm and saline Atlantic water, while the western part is filled with cold and freshwater of mainly Arctic origin (Mauritzen 1996). There is a clear separation between the two halves along the topographic ridge running south–north through the seas. The Atlantic water enters the Nordic seas over the Greenland–Scotland Ridge and then flows northward in an eastern boundary current, connecting the North Atlantic Ocean with the Arctic Ocean. The current can be seen to follow the contours of the bottom topography (Helland-Hansen and Nansen 1909; Poulain et al. 1996; Jakobsen et al. 2003; Søiland et al. 2008). There are two main branches of the current: one inner, barotropic branch strictly following the Norwegian continental slope [Norwegian Atlantic Slope Current (NwASC)] and one outer frontal branch [Norwegian Atlantic Front Current (NwAFC)]. The bulk of the transport is in the NwASC; Mork and Skagseth (2010) estimate the time-mean volume fluxes in the Svinøy section to be 3.4 and 1.4 Sv (1 Sv ≡ 10^{6} m^{3} s^{−1}), respectively, for the two branches. For a more detailed description of the circulation in the Nordic seas, see, for example, Mork and Skagseth (2010), Orvik and Niiler (2002) and Jakobsen et al. (2003).

In contrast to the stratified, low-latitude ocean, the Sverdrup balance (Sverdrup 1947) does not generally provide a leading-order description of the circulation in the weakly stratified subpolar oceans. Here, the gradients of planetary potential vorticity *f*/*H*, where *f* is the Coriolis parameter, are strongly dominated by the steep topography *H*, causing the flow to trace the isobaths around the basins (e.g., Orvik and Niiler 2002; Jakobsen et al. 2003; Koszalka et al. 2011). The underlying dynamics reflects the conservation of potential vorticity of the low Rossby number flows for which the leading-order bottom velocity is directed along the *f*/*H* contours (Greenspan 1968). Nøst and Isachsen (2003) showed that the leading-order bottom flow along closed *f*/*H* contours are primarily set by a global vorticity balance in which the along-contour, integrated surface wind stress and bottom stress balance each other. The integrated wind stress over the closed *f*/*H* contours in the Nordic seas and the Arctic Ocean tends to be cyclonic (i.e., dominated by an Iceland low pressure), yielding a prevalence of cyclonic, along-isobath bottom currents in the basins (Nøst and Isachsen 2003; Rudels et al. 2000; Aaboe and Nøst 2008). By cyclonic, we refer to flows with shallow water to the right, sometimes also referred to as prograde (see, e.g., Li and McClimans 2000). On the open *f*/*H* contours, which are typically shallower than about 500 m, the bottom velocities on the slope in the Atlantic can be viewed as upstream boundary conditions for the Nordic seas. The downstream changes of the bottom velocities on these *f*/*H* contours are mainly related to bottom buoyancy variations (Walin et al. 2004; Schlichtholz 2007; Spall 2005; Aaboe and Nøst 2008); if the buoyancy decreases downstream, the bottom velocity becomes increasingly cyclonic, making the flow more barotropic (Walin et al. 2004).

There is, however, a profound asymmetry in the tendency to follow *f*/*H* contours between cyclonic (prograde) and anticyclonic (retrograde) flows (e.g., Nøst et al. 2008). One reason is that an anticyclonic current along a continental slope, which flows with shallow water to the left, supports stationary topographic waves. As a result, along-stream bathymetric variations tend to generate stationary waves, which can create significant flow across the depth contours. In fact, most well-known examples of boundary currents that separate, such as the Gulf Stream, are anticyclonic flows (see, e.g., Marshall and Tansley 2001). We note, however, that Björk et al. (2010) report hydrographic data that indicate partial separation of a cyclonic boundary current as it passes over some strong topographic gradients on the Morris Jesup Rise off North Greenland.

As the time-mean flow in the Nordic seas is cyclonic and, to a good approximation, in geostrophic balance, the theory summarized above suggests that the time-mean bottom pressure should essentially be aligned with the topography. In the present work, we analyze the time-mean circulation in the Nordic seas on the basis of hydrographic and altimetric data, with a particular focus on the NwASC [see, e.g., Skagseth et al. (2004) for altimetry in the Svinøy section]. In the data analysis, we partition the dynamic surface pressure into a bottom pressure contribution and a steric height contribution, where the latter is proportional to the depth-integrated buoyancy. Although the main variations of these dynamical fields occur perpendicularly to the isobaths, there are some significant variations along isobaths. For instance, along the isobaths beneath the NwASC the variations in the altimetrically derived surface height suggest cross-isobath volume transport on the order of a few Sverdrups, which flow toward deeper water in some regions and back shoreward in other regions. This could be reflecting errors in the data, but we argue that these along-isobath pressure variations can be explained, at least qualitatively, by the conservation of potential vorticity of a low Rossby number flow that experiences along-stream changes in the steepness and curvature of the topography. This result provides, for example, information on the time-mean exchange of tracers across the depth contours in cyclonic boundary currents, for example, the exchange between the two branches of the Norwegian Atlantic Current (see Raj et al. 2015).

The remainder of the paper is organized as follows: Section 2 describes the decomposition of the geostrophic velocity into a thermal wind and a bottom velocity component and also presents the satellite altimetry and hydrographic data. In section 3, the results of the observational analysis are presented. In section 4, a steady-state, potential vorticity–conserving model of flow over along-stream varying topography is presented, and in the appendix this is complemented with an analysis of a linearized quasigeostrophic model. In section 5, the model is discussed in relation to the observational results. Some broader implications are discussed in the concluding section 6.

## 2. Method and data

### a. Thermal wind and bottom velocity

*ρ*

_{0}is a reference density, and

*ρ*′ is an anomaly. We further define the buoyancy anomaly aswhere

*g*is the gravitational constant. The hydrostatic equation is then given bywhere

*ϕ*is the dynamic pressure. Integrating the hydrostatic equation giveswhere we have used the boundary condition that the pressure at the surface is given by the dynamic surface height

*η*. As defined here,

*η*corresponds to the absolute dynamic topography data, as described below, that among several corrections includes the atmospheric pressure anomaly.

*z*. However, it can be dynamically revealing to decompose the dynamic pressure into one bottom pressure component and one component related to the buoyancy field. Fofonoff (1962) and several subsequent researchers have used such a decomposition of the pressure field. Following Fofonoff, we define a dynamic bottom pressure aswhere

*H*is the depth. Using this, we can write the dynamic pressure in Eq. (4) asthat is, the pressure at level

*z*is given by subtracting the contribution from the buoyancy in the layer between

*z*= −

*H*and

*z*from the bottom pressure. The introduction of the dynamic bottom pressure

*ϕ*

_{B}in Eq. (5) allows the free-surface height to be partitioned as follows:where the steric height contribution

*η*

_{S}is defined asand the contribution related to the dynamic bottom pressure isIn the data analysis, we assume flows that are in geostrophic balance, that is,where

**u**is the horizontal velocity and

**k**is the vertical unit vector.

**u**

_{B}, which is defined aswhere

*b*

_{B}≡

*b*(

*x*,

*y*,

*z*= −

*H*,

*t*) is the buoyancy at the bottom. In Eq. (12), the first term is the thermal wind velocity relative to the bottom, which is referred to as the “baroclinic” flow component by Fofonoff (1962). The bottom velocity is composed of two terms. The second term is the velocity associated with the bottom pressure gradient. This is, however, not a gradient at fixed

*z*; the first term compensates for this, which can be seen by noting that at the bottom ∇

_{z}

*ϕ*= ∇

*ϕ*

_{B}+ (∂

*ϕ*/∂

*z*)∇

*H*, where ∇

_{z}is the gradient at fixed

*z*and that from the hydrostatic relation (∂

*ϕ*/∂

*z*) =

*b*

_{B}. The compensating term is directed along the depth contours and flows anticyclonically (with shallow water to the left) if

*b*

_{B}is positive.

Mathematically, Eq. (13) is well defined, but in data-based calculations care has to be taken as the bottom velocity can be the difference between two potentially large terms [see Aaboe and Nøst (2008) for a computation of bottom velocities]. However, we note that cross-isobath velocities still rely on a gradient at fixed *z*. Here, we do not explicitly consider the along-isobath bottom velocities, and in section 5 we consider only *ϕ*_{B} that governs the cross-isobath flow.

### b. Absolute dynamic topography and bottom topography data

The observational portion of this study is based on a satellite altimeter product produced by Ssalto/Duacs and distributed by AVISO, with support from CNES (http://www.aviso.altimetry.fr/duacs/). The product is called absolute dynamic topography (ADT) and depicts the dynamic sea surface height, that is, the sea surface height due to motion; a geostrophic approach on the ADT gives the surface geostrophic current. We use the global gridded data on a ¼° Cartesian grid provided as daily fields from January 1993 to May 2014. All data presented here are the climatologic time mean over all available data. For some calculations and figures, we use global bathymetry and elevation data (Becker et al. 2009) at a resolution of 0.05° × 0.05°.

The ADT is here used as an observation of *η* in Eq. (7). Accordingly, the ADT could be divided into a steric and a bottom pressure contribution. To this end, a steric height is constructed from hydrographic data.

### c. Hydrography, steric height, and bottom pressure

The hydrographic data are the *World Ocean Atlas 2013* (*WOA13*) climatology of temperature (Locarnini et al. 2013) and salinity (Zweng et al. 2013) on a ¼° Cartesian grid. For comparison, an alternative hydrography is also used and shown in selected figures. This dataset was created using isopycnal mapping by Isachsen and Nøst (2012) and covers the Nordic seas at 50–70-km horizontal resolution.

*H*varies, however, the partitioning of the surface pressure gradient into a steric and a bottom component is no longer unique; lateral gradients of steric height over variable bottom depths gives a contribution proportional to the bottom slope that depends on the choice of the reference buoyancy. For example, by changing the reference buoyancy according to

*b*→

*b*+

*b*

_{ref}, where

*b*

_{ref}is a constant, it follows from Eq. (8) that the steric height and the bottom pressure become

*η*

_{S}→

*η*

_{S}+

*b*

_{ref}

*H*/

*g*and

*η*

_{B}→

*η*

_{B}+

*b*

_{ref}

*H*/

*g*, respectively. By computing the buoyancy relative to a wisely chosen

*z*-dependent reference density

*ρ*

_{0}(

*z*), we can still obtain dynamically relevant steric heights and bottom pressure fields. A sensible requirement is that when the gradient of

*η*is zero, that is, no surface geostrophic flow, the gradients of

*η*

_{S}and

*η*

_{B}should vanish independently, or at least be small, since two large terms of opposite signs would be cumbersome. In a single

*x*–

*z*section, we can achieve this by selecting the reference density profile asthus, the vertical reference density profile equals the bottom density on the slope at each depth. This choice gives

*η*

_{S}= 0 if the isopycnals are horizontal in the section. Furthermore, by definition, the bottom buoyancy

*b*

_{B}will be zero in the section, and there will be no cancelling terms in the bottom velocity equation [Eq. (13)].

Here, we choose *ρ*_{0}(*z*) to be defined in a section at 66°N, just south of the Vøring Plateau, which has a fairly gentle slope and thus allows for a better resolution of the bottom buoyancy in the gridded data compared with sections having steeper slopes. The bottom value of the density is taken as the lowest available level in the hydrographic data for every location in the section. The bottom density varies slightly upstream and downstream of the 66°N section, implying some bottom buoyancy variations that affect the along-slope bottom velocity [Eq. (13)] [see Aaboe and Nøst (2008) for further discussion].

Figure 2 shows the steric height in the 66°N section computed using both the reference density profile *ρ*_{0}(*z*), described by Eq. (14), and a constant reference density. The steric height based on *ρ*_{0}(*z*) increases monotonically toward the coast and, accordingly, tends to capture the thermal wind relative to the bottom also in shallower waters. The steric height based on a constant reference density, on the other hand, implies thermal wind flow that is southward and strong near the coast and alternates direction over the slope. For depths greater than 1000 m, the two methods yield very similar steric height gradients. Thus, we argue that our specification of *ρ*_{0}(*z*) provides a reasonable definition of the steric height over sloping bottoms that contains relevant dynamical information.

Finally, we emphasize that the variations in steric height and bottom pressure along a depth contour, to be discussed in section 5, are independent of the choice of reference density—it matters only for variations across the depth contours.

## 3. Observational analysis

Here, we present the decomposition of the satellite data of ADT into contributions from steric height and bottom pressure, as discussed theoretically in section 2a. To begin with, we describe briefly some qualitative features of these fields in the Nordic seas and then go on to a more quantitative discussion of the pressure fields along the NwASC.

It is worth noting that Johannessen et al. (2014) made a similar decomposition of the surface height field into a steric and a bottom component. Differences lie in the choice of hydrographic and ADT data (where the ADT was smoothed using an 80-km Gaussian filter) and in how they define the steric height; they use a constant reference depth of 1500 m and present a steric height field where areas shallower than 1500 m are filled in with steric heights with reference depths every 100- to 500-m depth. Our approach instead uses a varying depth but lets the reference buoyancy vary with vertical coordinate. The resulting steric height in Johannessen et al. (2014) is broadly similar to, but smoother than, the one presented here.

Figure 3 shows the time-mean satellite altimetric ADT, the hydrographic steric height, and the resulting bottom pressure in the Nordic seas. The time-mean ADT and its associated geostrophic velocities have been discussed by, for instance, Raj (2013) and Chafik (2014). They, and Chafik et al. (2015), also discuss the variability and the ADT-based eddy kinetic energies, where the highest activity is found in the Lofoten Basin (LB) and on the slope east of it. The mean ADT field roughly outlines a cyclonic surface circulation that tends to be aligned with the topography, and the steepest gradient can be found along the Norwegian continental slope, signifying the geostrophic velocity of the NwASC [cf. Fig. 1 and Mork and Skagseth (2010)].

The steric height, which is proportional to the vertically integrated buoyancy anomaly, has in general stronger along-isobath variations than the ADT. Specifically notable is the maximum in steric height in the Lofoten Basin and the associated contours crossing the isobaths of the continental slope.

While a geostrophic approach on the ADT and the steric height gives the surface and thermal wind velocities, respectively, it should be noted that one needs to be careful in interpreting velocities from the bottom pressure in Fig. 3; as in Eq. (13), a lateral gradient would not directly be a gradient at fixed *z*. However, cross-slope velocities associated with gradients on constant depth (as analyzed in section 5) are still well defined.

From theoretical considerations and data-based simple modeling (Nøst and Isachsen 2003; Aaboe and Nøst 2008), one expects the bottom pressure field in the Nordic seas to be essentially aligned with the topography and associated with cyclonic circulation. This notion is broadly consistent with the bottom pressure shown in Fig. 3. Specifically, the NwASC is traced out by a gradient in the bottom pressure farther northward than in the ADT. To make a more quantitative analysis of the alignment, we estimate the angle between the gradient of the topography and the gradient of the bottom pressure, ADT, and steric height. The alignment is evaluated along depth contours, tracing the NwASC, by constructing distribution functions (not shown) where complete alignment is represented by a zero. The distributions for the bottom pressure and the ADT are found to be centered around zero, while the steric height distribution is shifted and broader. The peak of the distribution for the bottom pressure is generally found to be higher than that of the ADT, indicating that it is somewhat better aligned with the topography.

However, there are some other notable, and more surprising, features in the bottom pressure. To begin with, the bottom pressure indicates a region of anticyclonic (i.e., southward) flow around the 1500-m isobath that extends from around 62°N, 2°W and northward between the two current branches toward the Vøring Plateau (VP). A coincident but weaker southward flow is also present in the ADT. Mork and Skagseth (2010) report weak southward, near-surface velocities in this area from satellite and ADCP data, and the near-bottom current meter data described by Aaboe and Nøst (2008) also include a few measurements of anticyclonic flows (i.e., southwestward flow) in this region around 63°N.

Second, in the deep LB, the bottom pressure suggests an anticyclonic flow at the bottom, collocated with the surface anticyclonic flow seen in the ADT. There resides a semipermanent eddy in the LB that has been discussed before; see, for example, Søiland and Rossby (2013) and Köhl (2007) on the Lofoten Basin eddy and also Orvik (2004) on the deepening of the Atlantic water in the LB. The bottom pressure calculated by Johannessen et al. (2014) suggests, however, a weak cyclonic flow at the bottom in the central LB. Also, Nøst and Isachsen (2003) predicted a cyclonic circulation in the LB with their linear model driven by wind stress. If the bottom circulation is indeed anticyclonic, and hence dissipates anticyclonic vorticity in the bottom boundary layer, then there must be an eddy flux of anticyclonic vorticity into the central LB that is stronger than the time-mean cyclonic vorticity input due to the surface wind stress (Nøst and Isachsen 2003).

Although we mention these interesting features, the limitations of the data and the present approach require further work to assess their significance. Thus, we stress that these regional features should be interpreted cautiously and do not pursue them further here.

### Along-slope steric height variations

As mentioned above, the ADT, and in particular the steric height, indicate cross-isobath transport along sections of the NwASC. Also, Chafik et al. (2015) found along-stream variations in their ADT-based volume transport estimate of the NwASC in the 500- to 900-m depth interval. To examine this further, we construct an along-isobath measure of the pressure variations, where the data are interpolated to depth contours of 500, 700, and 900 m along the NwASC. The results are shown in Fig. 4, revealing two local maxima along the slope. These are encountered near the Svinøy section, at about 64°N, and off the Lofoten Islands at about 70°N. The along-slope variations are found in both the steric height and the bottom pressure, although less accentuated in the latter. This suggests that the somewhat weaker along-isobath variations of the bottom pressure are amplified toward the surface by the baroclinic pressure field, resulting in a larger response in the surface pressure. This corresponds well to the notion of stronger topographic control of the bottom velocity and that the steric height, or the thermal wind, is accountable for most of the cross-isobath transport.

Figure 4 also shows the steric height and the bottom pressure calculated from the alternative isopycnal hydrography, revealing broadly similar along-slope variations.

From the perspective of an essentially isobath-following current, the downstream local maxima in depth-integrated buoyancy along the NwASC, suggested in Fig. 4, are somewhat unexpected. Stationary topographic waves are not present in cyclonic boundary currents and thus cannot account for the variations, which must have a different cause. Possible reasons for the variations in depth-integrated buoyancy are

- shortcomings in the analysis and erroneous data, for example, errors from uncertainties in the geoid (Johannessen et al. 2014; Rio et al. 2011); and
- the time-mean flow essentially follows the isobaths, but eddy-induced buoyancy fluxes and/or surface buoyancy fluxes cause the along-stream buoyancy to vary.

We will return to these possibilities in section 6. In the next section, however, we will show that the variations in bottom pressure and steric height along the isobaths can be explained qualitatively by a model of an essentially adiabatic and isobath-following current that conserves its potential vorticity.

## 4. A model along isobaths

We examine if an inviscid adiabatic model can explain some aspects of the along-stream surface height variations that are suggested by the data. We consider a cyclonic boundary current where the flow is characterized by a small Rossby number and essentially traces the topography. However, conservation of potential vorticity generally requires some cross-isobath velocities where the slope and curvature of the topography changes in the along-stream direction. For a cyclonic boundary current, where there are no stationary topographic waves, the resulting dynamics can be investigated using a local perturbation analysis as will be illustrated below. In contrast, for an anticyclonic boundary current that has the coast to the left, there exist nonlocal, free, stationary topographic waves, which makes a local perturbation analysis unfeasible (see appendix).

*f*-plane dynamics of a steady inviscid flow in hydrostatic balance, governed by the momentum equationwhere

*ζ*≡ ∂

*υ*/∂

*x*− ∂

*u*/∂

*y*is the vertical component of the relative vorticity, and

*w*is the vertical velocity. The pressure obeys the hydrostatic relation [Eq. (3)], and the buoyancy equation iswhere

*b*here includes the full vertical variation of the buoyancy field. The continuity equation is given bysubject to the boundary conditionswhere we assume a rigid lid.

*Q*along streamlines (Allen 1991; Vallis 2006):where

**v**≡ (

*u*,

*υ*,

*w*) andWe will here use a version of the equation for the vertical component of the vorticity, which is given by

**k**⋅ ∇×[Eq. (16)]:where we have omitted the terms related to the curl of

*w*∂

**u**/∂

*z*as they are negligible in the low Rossby number regime studied below.

*H*/

*L*is the aspect ratio. Assuming that Ro ≪ 1, we make a series expansion of Eq. (16) in this quantity (see, e.g., Allen 1991; Vallis 2006). To the zeroth order, we obtain in dimensional formAs the zeroth-order horizontal flow is divergence free, that is, ∇ ⋅

**u**

_{0}= 0, the continuity equation [Eq. (18)] reduces toThus, the rigid-lid boundary condition [Eq. (19)] implies that

*w*

_{0}= 0. The bottom boundary condition does not enter until the first order.

*H*is the depth variation and

*δ*is of the same order as the Rossby number, implying a weak slope, then the bottom boundary condition [Eq. (19)] becomes to the first orderThis is the quasigeostrophic limit where the lowest-order bottom geostrophic velocity is related to

*w*

_{1}. However, the continental slopes generally have order one depth variations, implying that

*δ*~ 1. In this case, Eq. (29) givesimplying that the leading-order bottom velocities are aligned with the topography, andThus, the ageostrophic velocity enters in the boundary condition at

*w*

_{1}. We will here examine this regime. However, for stratified flows, we will show that the quasigeostrophic equations yield similar results if the slope parameter

*δ*is larger than the Rossby number but still smaller than unity, an issue that is further discussed in the appendix.

### a. The homogeneous limit

*f*/

*H*:with

**u**

_{0}according to Eq. (23). This gives (taking

*f*constant)that is, to the lowest order the flow follows the topography, and the vertical velocity is zero.

*f*is constant and

**u**

_{0}⋅ ∇

*H*= 0, equivalentlyPhysically, Eq. (37) states that the first-order velocity will be directed toward deeper (shallower) depths—corresponding to lower (higher) values of the zeroth-order potential vorticity

*f*/

*H*—when the zeroth-order relative vorticity increases (decreases) downstream along the isobaths. This iterative approximation of potential vorticity conservation is illustrated in Fig. 5; to the leading order, the flow follows strictly the topography as required by Eq. (34). However, where the slope steepens or the isobaths curve, anomalies in relative vorticity arise along the isobaths (Fig. 5a). Equation (37) relates these relative vorticity anomalies to first-order, cross-isobath velocities, which induce compensating changes in the planetary vorticity

*f*/

*H*. This yields conservation of potential vorticity to the first order [see Eq. (36)]. The sum of the zeroth- and first-order flow yield the transport streamfunction depicted by the gray lines in Fig. 5b. As discussed by Nøst et al. (2008), the conservation of potential vorticity implies that in low Rossby number cyclonic boundary currents, the curvature of the streamlines tend to be smaller than that of the topography. Thus, the streamlines tend to follow a somewhat low-passed filtered topography [see LaCasce et al. (2008) and the appendix].

*B*

_{1}[and

*ϕ*

_{1}through Eq. (26)] vary along the depth contours. We start by inserting Eq. (35) into Eq. (23), giving thatThe properties of the scalar triple product, stating that

**a**⋅ (

**b**×

**c**) =

**b**⋅ (

**c**×

**a**) =

**c**⋅ (

**a**×

**b**), can be utilized so that

**u**

_{1}⋅ ∇

*H*can be written with all terms proportional to

**k**× ∇

*H*. Inserting Eqs. (25) and (38) into Eq. (37) then yieldswhere we have introduced the vectorwhich is parallel to the isobaths. The quantity within the square brackets is invariant along the zeroth-order streamlines, that is, along constant

*H*. For a cyclonic flow, having shallower water to the right (

*ϕ*

_{0}decreasing offshore), implying that

*dϕ*

_{0}/

*dH*< 0,

*B*

_{1}will be higher where

*ζ*

_{0}is anomalously positive.

_{1}, describing the vertically integrated, first-order flow, can be obtained from Eq. (25) by integrating it vertically and multiplying with ∇

*H*. Some manipulations then yieldThus, the streamfunction is proportional to the vertically integrated Bernoulli function; this is a result that also applies for baroclinic flows if

**u**

_{0}is parallel to the isobaths at all depths.

*ζ*

_{0}and kinetic energy along the depth contours are related to variations in the curvature ∇

^{2}

*H*and slope |∇

*H*| of the bottom topography.

### b. The stratified case

As in the homogeneous case, we consider flows that are strongly steered by the topography. However, even in the limit of a small Rossby number, the presence of stratification allows for surface-intensified flows that are decoupled from the topography. We will therefore focus on a class of flows that to the lowest order are aligned with the bottom topography throughout the water column, that is, the magnitude of the velocity can vary vertically, but the velocity vector is parallel to the isobaths at all depths; this is sometimes referred to as equivalently barotropic flow (e.g., Killworth 1992). We will first make a Rossby number expansion of the stratified equations to describe the flow variations along curving isobaths. We then briefly illustrate that similar results can be obtained also in the quasigeostrophic framework in a limit of weak stratification and moderate bottom slopes.

*w*

_{0}is zero, which implies that the buoyancy [Eq. (17)] to this order becomesTo satisfy Eq. (43), the horizontal gradient of

*b*

_{0}has to be parallel to that of

*ϕ*

_{0}. Here, we consider zeroth-order velocity fields that at all depths are aligned with the local isobaths, that is,

*ϕ*

_{0}=

*ϕ*

_{0}(

*H*,

*z*) and

*b*

_{0}=

*b*

_{0}(

*H*,

*z*). This choice automatically satisfies the bottom boundary condition [Eq. (31)]. Accordingly, there can be a vertical velocity shear tied to the horizontal variation of

*b*

_{0}, but the shear is by our specific choice constrained to be locally parallel to the isobaths. As a consequence, the zeroth-order flow conserves the total volume transport between the isobaths. From Eqs. (20) and (21), it follows that potential vorticity conservation to the zeroth order is given bystating that the vertical isopycnal spacing is constant along the isobaths.

*w*

_{1}is given by Eq. (32). The vertical velocity

*w*

_{1}can be calculated from the vorticity equation [Eq. (27)] asSince the vertical velocity

*w*

_{1}is determined by the zeroth-order fields, Eq. (45) together with the hydrostatic relation [Eq. (3)] and the first-order momentum equation [Eq. (25)] allow us to calculate the first-order pressure

*ϕ*

_{1}. Below, we present a method for solving this set of equations. An interpretation of the results in terms of potential vorticity conservation is given in section 4c.

*z*= −

*H*) can be written asNote that the quantities in the square brackets in Eqs. (48) and (50) are invariant along the depth contours. Equation (48) is a linear, first-order differential equation in

*z*for

*ϕ*

_{1}, where the inhomogeneous term

*F*represents the buoyancy advection due to the zeroth-order fields. An explicit solution will be presented after we have briefly commented on how quasigeostrophic versions of the results presented above can be obtained.

#### Quasigeostrophic limit

*N*(

*z*), and

*b*now represents a small perturbation on the background stratification (e.g., Allen 1991; Vallis 2006). A key parameter in this regime is the Burger numberwhere

*L*is the horizontal length scale of the flow. The Burger number measures the ratio between vertical and horizontal buoyancy advection. As a result, the QG buoyancy equation becomes to the zeroth order in the Burger number identical to the zeroth-order Rossby number buoyancy equation [Eq. (43)] (see, e.g., Vallis 2006).

*δ*~ Ro, which gives the boundary condition in Eq. (30). The value of

*δ*given by Eq. (52) is larger than Ro, though still taken to be small compared to one and yields in a Burger number expansion the zeroth- and first-order bottom boundary conditions in Eqs. (31) and (32). In the appendix, a simple, linear QG model is presented where

*δ*can be smoothly increased to yield the regime with strong topographic steering specified by Eq. (52).

*b*

_{0}are small and make the substitutionneglect all the |

**u**

_{0}|

^{2}terms, for example,

*B*

_{1}→

*ϕ*

_{1}, and take

**u**

_{1}=

*f*

^{−1}

**k**× ∇

*ϕ*

_{1}. We also must replace the potential vorticity equation [Eq. (44)] with the QG version

### c. A simple illustrative case

*ϕ*

_{1}= 0 and

*B*

_{1}= 0. For notational convenience, we introduce the along-isobath change in kinetic energy relative to the upstream locationwhere

*s*is an along-isobath coordinate and

*B*

_{1}=

*ϕ*

_{1}+ Δ

*K*

_{1}. We also introduce the analogously defined along-isobath variation in relative vorticity Δ

*ζ*

_{0}.

*b*

_{0}is a function of

*z*only, and

*db*

_{0}/

*dz*=

*N*

^{2}is a constant, that is, the zeroth-order flow is purely barotropic. In this special case, the first-order conservation of potential vorticity [cf. Eq. (20)] is simply given bywhich can be recognized as steady QG potential vorticity conservation. The first-order solutions obtained from Eqs. (48)–(50) areSince the zeroth-order flow is barotropic in this case, these first-order fields are similar to the ones discussed in relation to Fig. 5; the quasigeostrophic versions result when Δ

*K*

_{0}is set to zero. The baroclinic component of the solution is related to the terms proportional to

*N*

^{2}; if

*N*

^{2}is set to zero the purely barotropic case is recovered. Thus, as discussed in section 3, the free-surface height can be partitioned into a barotropic bottom pressure component and a steric height component. Here, the first-order variations in buoyancy and steric height result from vertical advection of the background stratification; downward vertical velocities, occurring where Δ

*ζ*

_{0}is positive, result in positive buoyancy anomalies. Equation (56) shows that this is a consequence of potential vorticity conservation.

Figure 6 illustrates Eq. (60), for which *dϕ*_{0}/*dH* is a constant and the slope becomes narrower halfway along the depth contours in the domain. Note that Eq. (60) is defined along depth contours, so that the anomalies in the solution are compared to upstream values at the same depth at *x* = 0. The flow is essentially a jet concentrated to the steepest part of the slope, with cyclonic (anticyclonic) vorticity on the outer (inner) part of the slope; this yields the vorticity anomaly pattern shown in Fig. 5a, having a somewhat stronger anticyclonic anomaly. The surface height anomaly in Fig. 6, on the other hand, has a larger positive than negative amplitude, which is a result of the *H*-dependent terms in Eq. (60). To estimate possible magnitudes of the along-isobath surface height changes, we have chosen parameters broadly representative of the Norwegian Atlantic Slope Current (see the caption of Fig. 6). The slope |∇*H*| ranges from about 2% to 4% along the coast, a variation that is even larger along the NwASC. In the resulting surface height anomaly, the changes in kinetic energy play a minor role except for near the coast. Thus, where a continental slope narrows, one typically expects to see the surface height increase along the isobaths on the outer slope. On the inner slope, the surface height can be expected to decrease but with a magnitude that is smaller than the surface height changes on the outer slope.

Along the Norwegian coast, the buoyancy decreases offshore, that is, ∂*b*_{0}/∂*H* < 0. As discussed in the appendix, this amplifies the surface height anomaly *η*_{1}. From Fig. 5, it is clear that where Δ*ζ*_{0} is positive, the corrected streamlines are displaced toward deeper water. Since ∂*b*_{0}/∂*H* < 0, this is associated with an offshore transport of light water, creating an additional positive buoyancy anomaly that is in phase with the one arising from the vertical advection.

## 5. Intercomparison of model and observations

According to the model presented in the sections above, along-slope variations in relative vorticity and kinetic energy give rise to variations in sea surface height. To examine this in the observational data, the relative vorticity is computed from the ADT according to *ζ* = *g*∇^{2}*η*_{ADT}/*f*. Figure 7 shows the ratio of this and the Coriolis parameter *f*, representing the Rossby number. It can be seen to trace out the slope current as well as the frontal outer branch, with a negative (positive) value on the right (left) flank. It can also be noted that the relative vorticity in Fig. 7 displays an anticyclone in the LB (see previous mentioning of the Lofoten Basin eddy).

*ζ*/

*f*≃ 0.1. To make an estimate of the associated surface height anomaly expected from Eq. (60), we assume numbers typical for the NwASC:The resulting surface height anomaly is 3 cm, which is the same order of magnitude as the along-slope variations in the ADT (Fig. 4).

To further compare theory and observations, we simulate the along-slope first-order correction to the free-surface height, *η*_{1} in Eq. (60), by interpolating the ADT to the 700-m depth contour, which lies on the continental slope all along the NwASC. The result is shown in Fig. 8, together with the gradient (slope) and the Laplacian (curvature) of the topography, interpolated to the same isobath. We find a correlation of the ADT with both the slope and the Laplacian, with correlation coefficients of 0.6 and 0.7, respectively; thus, the observed surface height anomalies along the slope are located where the topography steepens and/or curves, as predicted by the model. The maxima in the gradient and the Laplacian are close to, but not exactly, collocated and thus reinforce each other’s effect on the vorticity of the flow to some extent. A geostrophic approach of the ADT in Fig. 8 suggests a cross-contour flow: out from the coast where the ADT increases along the slope and toward the coast where it decreases. In other words, to compensate for the variations in relative vorticity due to the topographic features, the geostrophic current moves, in this case, toward deeper water (cf. Fig. 5b).

For a more quantitative measure, the magnitude of the along-slope variations in the Laplacian of the topography is estimated to be 3 × 10^{−7} m^{−1}. Together with the numbers in Eq. (61), Eq. (60) with Eq. (42) implies a sea surface height variation of about 2 cm in response to the observed along-slope variations in the curvature. The steepness of the topography similarly adds a few centimeters to this estimate. This gives a surface elevation of the same order of magnitude as the observed variations in ADT.

The choice of depth contour for the along-slope representation of the data also affects the structure and correlation to some degree. For example, the correlation of the ADT with the Laplacian does not seem very sensitive to the choice of depth contour, keeping about 0.7 for the 500- and 900-m contours as well, while the correlation with the steepness of the slope is more sensitive and exhibits a slightly higher correlation on the 900-m contour and a weak (<0.4) correlation coefficient on the 500-m contour.

Another discrepancy between the model and observations is the predicted negative anomaly in surface height on the inshore side of the current in Fig. 6, which is not found in observations. Some simplifications are made in Figs. 5 and 6, for example, *dϕ*_{0}/*dH* is considered as a constant and only the curvature term is taken into account, so we expect some discrepancies. We can also note that the response of the surface height to the vorticity anomaly in Fig. 5a is amplified on the deeper contours because of the *H*-dependent terms in Eq. (60). This is further amplified if horizontal buoyancy gradients are accounted for (see appendix). We can thus expect a positive surface anomaly on the deep flank to be the dominant response.

## 6. Summary and discussion

In this study, we examine the dynamics of the NwASC, a cyclonic boundary current in the Nordic seas, with satellite altimetry as well as hydrographic data. First, we make a decomposition of the time-mean dynamic sea surface height as observed by satellites into a steric height and a dynamic bottom pressure, where the latter ideally should capture the topographic steering of the NwASC along the Norwegian continental slope. The presented bottom component essentially exhibits the anticipated topographic alignment along the slope, but regional exceptions are also present.

Second, we further examine the topographic steering of the NwASC. Analysis of the time-mean altimetric and hydrographic data indicates dynamically nonnegligible variations and even local maxima in surface pressure and depth-integrated buoyancy along the isobaths in the core of the NwASC. This is somewhat remarkable since stationary topographic waves are not supported in cyclonic flows. To seek an explanation of these variations, we develop a simple adiabatic steady-state model of a cyclonic stratified boundary current over a topographic slope in the limit of small Rossby numbers. To the lowest order, the flow follows the isobaths, implying that changes in the along-stream topography cause changes in velocity and relative vorticity. Conservation of potential vorticity requires cross-isobath excursions, associated with vertical stretching or compression of fluid elements, to compensate for the changes in relative vorticity. The result is that along-stream topographic features that cause a local increase (decrease) in relative vorticity are associated with positive (negative) anomalies in bottom pressure and steric height and thus also in sea surface height. The model predicts along-isobath variations of a few centimeters along the NwASC and seems to qualitatively fit the observational data.

The observational analysis has been limited to the Nordic seas and the NwASC, but the theoretical ideas are directly transferable to cyclonic boundary currents in other subpolar basins such as the Labrador Sea.

*Q*is the surface buoyancy flux, the overbar denotes a time mean, and the prime denotes time-dependent eddies. Assuming that the horizontal velocity

Baroclinic eddies play an important role in the heat balance of the NwASC by fluxing heat from its core into a pool of water with lower mean advection between the two main branches of the Norwegian Atlantic Current (e.g., Isachsen et al. 2012). The intensity of the eddy field varies along stream in the NwASC, possibly reflecting varying degrees of baroclinic instability as the mean current speed and the topographic slope change (Spall 2010; Isachsen 2011; Poulin and Flierl 2005). Isachsen et al. (2012) estimate heat flux divergences in the NwASC due to the horizontal mean flow and eddies using surface drifter data, satellite data, and an eddy-permitting ocean circulation model. The heat flux divergence is examined along a path of the NwASC defined by its surface temperature maximum, which differs slightly between the data and the model simulation and does not strictly follow an isobath. A main result emerging both from the observational data and the model simulation is that the horizontal mean flow advection warms the current core at a rate that is comparable to the cooling due to the eddy flux divergence: the air–sea heat flux plays a secondary role in the heat budget. Thus, it is plausible that along-isobath variations in the eddy fluxes can cause local steric height maxima. Indeed, the maxima in the present data analysis are encountered in regions with more intense eddy activity (see, e.g., Isachsen et al. 2012). However, the estimates of the eddy heat flux divergence along the NwASC presented by Isachsen et al. (2012) are monotonic, acting to reduce the buoyancy. Thus, their study suggests that the eddy fluxes are not the primary mechanism for creating the along-isobath steric height maxima in the NwASC.

As mentioned in section 3a, the gridded altimetric and hydrographic data have limited spatial resolution and could give partly erroneous impressions of along-stream variations. However, comparable variations are found in satellite-based ADT as well as in the two hydrographic datasets (*WOA13* and the isopycnal climatology), which gives some confidence in the findings. In summary, the presented theoretical model in combination with the qualitative observational analysis of the time-mean circulation in the eastern Nordic seas indicate that the stationary anomalies in sea surface height and steric height along the NwASC can result from cross-isobath flow, governed by conservation of potential vorticity.

As a concluding note, the Lofoten Basin is a subarctic hot spot in many ways, and numerous studies have focused on its rather spectacular heat content (cf. the steric height in Fig. 3) and the semipermanent anticyclone residing in the deep basin and its possible connection to the NwASC (e.g., Volkov et al. 2013; Rossby et al. 2009; Orvik 2004). Several studies have suggested that anticyclonic eddies, which shed off from the slope current off the Lofoten Islands and drift westward, supply the heat to the central basin and maintain the semipermanent anticyclonic vortex (e.g., Köhl 2007; Raj et al. 2015). However, it is interesting to note that the present theory predicts that the hydrographic structure along a slope current should become deeper where the topographic slope increases. The simple example illustrated in Fig. 6 has some broad similarities with the hydrographic signature in the Lofoten Basin. It is an open question whether the hydrographic deepening occurring in the present simple model can have some seeding role in the complex chain of events that appear to control the dynamics of the Lofoten Basin.

This work was supported by a grant from the Swedish National Space Board. The authors also thank Jonas Nycander for interesting discussions as well as two anonymous reviewers for valuable comments and suggestions.

# APPENDIX

## A Quasigeostrophic Linear Stratified Model

Here, we analyze the linear dynamics of a stratified steady flow over a gently sloping bottom using the quasigeostrophic equations. The purpose is to illustrate a few dynamical issues mentioned briefly in the main text, including some differences between steady flows that have shallow water to the right and left, dynamical low-pass filtering of the bottom topography, and how the flow becomes topographically steered when the Burger number Bu [Eq. (51)] is small and the bottom slope moderate. At the end, we discuss how vertical shear acts to magnify the surface pressure response to downstream changes in steepness or curvature of the isobaths.

*U*

_{0}and Λ are constants, and

*B*=

*B*(

*y*) is the basic-state horizontal buoyancy. The vorticity and the buoyancy equations, governing the deviation from the background state, are given byThe bottom depth is defined bywhere

*h*is a small anomaly on a linearly sloping bottom. A key assumption is that the deviations from the mean depth

*H*

_{0}are small enough that the bottom boundary condition can be evaluated at

*z*= −

*H*

_{0}. The boundary conditions on the vertical velocity at the surface and the bottom arewhere we make the rigid-lid approximation and neglect the small term

**u**⋅ ∇

*h*. By using the hydrostatic and geostrophic relations, the equations and boundary conditions can be formulated in terms of the pressure

*ϕ*alone. Combining Eqs. (A2) and (A3) and taking

*N*as constant yields the potential vorticity equationBy using Eq. (A3), the bottom boundary condition in Eq. (A5) can be written asand the surface boundary condition can be written as

### a. Vertical structure and the difference between cyclonic and anticyclonic mean flow

*h*(

*x*,

*y*) ∝ exp(

*i*

**k**⋅

**x**), where

**k**is the horizontal wave vector [or formally taking the Fourier transform of

*h*(

*x*,

*y*) and the equations], the solution to Eq. (A6) with its boundary conditions can be written aswhere

*h*(

**k**) is the wavenumber representation of the topographic perturbation and

*χ*≡ |

**k**|

*N*/

*f*.

*χH*

_{0})

^{2}is a Burger number based on the horizontal scale

*L*~ |

**k**|

^{−1}of the topography

*h*. When this parameter is large, the response is bottom trapped with a negligible surface pressure signature; the ratio between surface and bottom pressure is proportional to exp(−

*χH*

_{0}). We are here interested in the opposite limit where

*χH*

_{0}≪ 1, that is, when the horizontal scale of the topographic perturbation is larger than the baroclinic Rossby radius. By expanding the denominator in Eq. (A10), retaining terms up to the second order in

*χH*

_{0}, we obtain, after some simplifications, an expression for the surface pressure in this limitwhich, using an inverse Fourier transform, corresponds to the potential vorticity–like equationin

*x*–

*y*coordinates. Here, we have definedwhere

*ε*is the ratio between the basic-state vertical velocity shear and the bottom velocity. In the definition of Γ

^{2}, we recognize two length scales: the wavelength of stationary barotropic topographic waves in an anticyclonic mean flow [(

*H*

_{0}|

*U*

_{0}|)/(

*Sf*)]

^{1/2}and the baroclinic Rossby radius

*NH*

_{0}/

*f*. The sign of Γ

^{2}determines the character of the solutions to Eqs. (A11) and (A12). When Γ

^{2}< 0, the solutions are wavelike and of hyperbolic character; in this case, Eq. (A12) describes forced steady topographic Rossby waves [see Holton (2004, section 7.7) for a discussion of the barotropic case]. On the other hand, when Γ

^{2}> 0, the solutions are trapped to the topographic perturbations and have elliptic character. In the barotropic limit, that is, when

*N*= 0 and

*ε*= 0, the sign of

*U*

_{0}alone determines whether the solutions are of hyperbolic (

*U*

_{0}< 0, anticyclonic mean flow having the shallow water to the left) or elliptic character (

*U*

_{0}> 0, cyclonic mean flow having the shallow water to the right).

### b. Topographic steering and low-pass filtering at small Bu

*L*)

^{2}≫ 1, the dominant balance in Eq. (A12) is

*ζ*∝ −

*h*, that is, anomalies in relative vorticity balance water depth changes. Here,

*ϕ*(

*x*,

*y*, 0) becomes small, implying that the net flow approximately is the mean flow

*h*rather than around them. In the opposite regime, where (Γ/

*L*)

^{2}≪ 1, the dominant balance in Eq. (A12) is

*ϕ*(

*x*,

*y*, 0) ∝

*h*, that is, the surface pressure and water depth anomalies are aligned and the flow follows the depth contours. If

*ε*~ 1 or smaller, we haveAs we have already assumed that Bu ≪ 1, (Γ/

*L*)

^{2}will be small if Ro/

*δ*≪ 1; in the quasigeostrophic analysis in section 4, we required that Ro/

*δ*~ Bu. Thus, to the zeroth order in Bu, Eq. (A12) gives

*ϕ*(

*x*,

*y*, 0) =

*hfU*

_{0}(1 +

*ε*)/

*S*. Combining this with the pressure associated with the mean flow

*yU*

_{0}(1 +

*ε*)/

*f*, we get an isobath-aligned “zeroth-order” surface pressure field:To the first order in Bu, Eq. (A12) gives

*ϕ*

_{1}(

*x*,

*y*, 0) = Γ

^{2}∇

^{2}

*hfU*

_{0}(1 +

*ε*)/

*S*, or equivalently if we use Eq. (A15),

*γ*and

*N*

^{2}are positive constants, and we have definedHere, Φ

_{H}and

*σ*are positive constants, and

*σH*is the ratio between the bottom-to-top baroclinic shear in velocity and the bottom velocity. Some calculations show that the first-order surface pressure field iswhere we for simplicity have omitted terms involving the zeroth-order kinetic energy. By noting that (

*H*

_{0}

*U*

_{0}

*f*)/

*S*corresponds to

*H*Φ

_{H}and that

*ε*corresponds to

*σH*, we see that this first-order surface pressure field is essentially identical to the one given in Eq. (A16). The main difference in the underlying assumptions behind Eqs. (A16) and (A19) is that the latter is formally valid for order one variations in the depth

*H*. Yet, it is important to note that they yield the same parametric dependence on the basic-state parameters, suggesting that quasigeostrophic theory gives qualitatively correct results, even when the bottom slope becomes steep.

Finally, we note that a positive vertical shear in the zeroth-order flow, which is measured here by *ε* or *σH*, amplifies the first-order surface pressure response [see Eqs. (A16) and (A19)]. This can be interpreted in terms of potential vorticity conservation, as the horizontal buoyancy gradients modify the potential vorticity gradients on the boundaries (Bretherton 1966). Alternatively, the amplification can be seen as a result of buoyancy advection, which brings lighter water offshore where the relative vorticity is anomalously positive. Using the near-surface horizontal buoyancy gradients along the Norwegian Atlantic Slope Current, we find *σ*^{−1} is on the order of 1000 m; that is, along the 1000-m isobath *σH* should be on the order of unity. Thus, the effects of the horizontal buoyancy gradient should be of leading-order importance for the amplitude in the along-isobath pressure variations.

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