Topographic Steering of the Upper Arctic Ocean Circulation by Deep Flows

2.1 Governing equations

We develop a two-layer model to examine the large-scale circulation in the Nordic Seas and the Central Arctic Ocean. For simplicity we use the f-plane approximation. We focus on the time-mean circulation and variations of the circulation on seasonal and inter-annual timescales. The model describes the flow averaged over a spatial scale that well exceeds the baroclinic Rossby radius and assumes that the leading order flow is in geostrophic balance. Different from the standard quasi-geotropic approximation, however, the model does not require variations in the interface-layer and total depths to be on the order of the Rossby number (see e.g. Allen, 1991). Although formulated on an f-plane the model accordingly shares some features with the planetary geostrophic approximation which includes the advection of the entire buoyancy field (Welander, 1959). The model does not resolve baroclinic eddies but their effect is represented as a diffusion of the upper-layer thickness which reduces its slope and thereby the baroclinicity of the flow. The thickness diffusion, which is equivalent to diffusion of the upper-layer planetary potential vorticity, can alternatively be interpreted as a form drag between the two layers caused by the baroclinic eddies. Potential effects of eddy-induced relative vorticity fluxes are not represented. Further, effects of surface wind forcing and bottom friction are modeled as surface and bottom Ekman transports in the continuity equation. The model includes elements of previous models of Arctic Ocean dynamics (Nøst and Isachsen, 2003; Manucharyan and Spall, 2016; Manucharyan et al., 2016; Meneghello et al., 2020), as will be discussed below.

Figure 2 shows the structure of the two-layer model, with the upper and lower depths denoted by H₁ and H₂, respectively. Note that the layer thicknesses H₁ and H₂ are not allowed to vanish. We assume that the large-scale flow is in geostrophic balance

where f is the Coriolis parameter, k the vertical unit vector and φ₁ and φ₂ are the dynamic pressures. The pressure is hydrostatic and the dynamical pressures in the two layers and the upper-layer thickness are related as

Figure 2

Here g′ is the reduced gravity and g′H₁ is the baroclinic pressure anomaly. By using Eqs. (2–4) the layer velocity difference can be written as

This is the thermal wind velocity relative to the lower layer velocity and it is parallel to contours of H₁.

We consider a flow with a scale that is much smaller than the barotropic Rossby radius which allows the free surface to be regarded as a rigid lid (Gill, 1982). The continuity equations include the vertically-integrated geostrophic transport, Ekman transports associated with surface and bottom stress, and eddy-induced transport represented as a thickness diffusion, or a bolus transport (Rhines and Young, 1982; Manucharyan et al., 2016; Meneghello et al., 2020). The vertically-integrated transports in the layers are

Here, K is the eddy diffusivity, –/+ K∇H₁ the bolus transport in the upper/lower layer, τ_S the surface ocean stress and τ_B the bottom stress, which is represented as a linear drag

where R is a constant friction parameter. The continuity equations are then

Here, w_W represents water-mass transfer between the two layers of different densities due to diabatic processes such as interior mixing and surface buoyancy fluxes (Walin, 1982). For surface forcing w_W equals the surface buoyancy flux divided by g′ and is positive (negative) when the ocean gains (loses) buoyancy; e.g. surface cooling and associated densification of the surface layer may be represented as a volume transfer from the upper to the lower layer. Similarly, vertical mixing in the stratified interior of the ocean may be modelled as a cross interface volume transfer in a layered model (see e.g. Kawase, 1987; McDougall and Dewar, 1998). We will not here explicitly consider effects of diabatically forced w_W but we include this term in the model formulation since surface buoyancy fluxes in parts of the Arctic Ocean may represent a forcing comparable to the wind-driven Ekman surface vertical velocity. For instance, along the Barents Sea Opening and in the West Spitzbergen Current where the surface buoyancy loss is intense (see Figure 3 in Isachsen et al., 2007) w_W may be on the order of 10^–6 m s^–1, which is comparable to typical vertical surface Ekman velocities in the Arctic Ocean (Timmermans and Marshall, 2020).

Figure 3

Since the geostrophic velocities are non divergent on an f plane, the continuity equations can be written as

Here, we have introduced the surface vertical Ekman velocity

and the depth of the bottom Ekman layer

Note that ${\displaystyle h_B\mathrm{c}\mathrm{u}\mathrm{r}\mathrm{l}({\mathbf{u}}_2)}$ is the bottom vertical Ekman velocity which is upward (downward) for cyclonic (anticyclonic) lower-layer flow.

On the lateral boundaries of the model the normal components of the volume transports M₁ and M₂ (Eqs. 6, 7) need to be specified; we will return to this when specific solutions are considered.

2.2 Summary of governing equations

A convenient way to the describe the present two-layer system is in terms of the upper-layer continuity equation and the vertically-integrated continuity equation, obtained by adding Eqs. (11) and (12). By noting that ${\displaystyle H_1+H_2=H(x,y)}$ is independent of time and that Eq. (5) implies that ${\displaystyle {\mathbf{u}}_1\cdot \mathrm{\nabla }{H}_1={\mathbf{u}}_2\cdot \mathrm{\nabla }{H}_1}$ (i.e., the thermal wind velocity is parallel to H₁), these equations can be written as

Equation (16) is essentially the two-layer and f-plane equivalent of the bottom-velocity equation used by Nøst and Isachsen (2003). Given the wind forcing, this linear equation yields ${\displaystyle {\mathbf{u}}_2}$, which allows us to solve the advective-diffusive equation for H₁ (Eq. 15). Despite that the term ${\displaystyle {\mathbf{u}}_2\cdot \mathrm{\nabla }{H}_1}$ formally is nonlinear, the upper-layer thickness is governed by a linear equation. A similar feature emerges in the beta-plane two-layer model considered by Rhines and Young (1982), for which the depth-integrated flow equation is also linear. We note that the diabatic volume transfer w_W and the Ekman pumping w_E both force the upper-layer thickness. However, w_W does not affect the vertically-integrated continuity equation.

A noteworthy feature is that Eqs. (15,16) only involve the variations in upper-layer thickness and total depth; the average values of these quantities do not enter explicitly. Thus, there is no restriction that the variations of these variables should be small compared to their mean values, which is the case for the standard two-layer quasi-geostrophic model (Pedlosky, 1987; Allen, 1991). However, the Rossby number [U/(fL)] based on the thermal wind velocity (Eq. 5) is required to be small to ensure a leading-order geostrophic flow, which puts an upper limit on the upper layer slope |∇H₁|. For the present Arctic Ocean application (Table 1) this requires that ${\displaystyle |\mathrm{\nabla }{H}_1|\ll 1}$. Furthermore, the mean depth of the layers enter in the layer transports (Eqs. 6,7) and they also set the (two-layer) internal Rossby radius (Gill, 1982), which in turn can affect the eddy diffusivity K (Visbeck et al., 1997; Karsten et al., 2002). We emphasise that the present model does not predict the mean upper-layer thickness which has to be inferred from additional assumptions or observations. For the main aspects of our study, however, the mean upper-layer thickness does not matter.

2.3 Integral relations for the bottom flow over areas bounded by H contours

In the Arctic Ocean the depth contours deeper than about 800 m close themselves within the basin. On such closed depth contours the bottom flow is essentially governed by an integral momentum balance: From Eq. (16) we can derive the Nøst and Isachsen (2003) integral balance for our two-layer and f-plane model. To this end, we integrate Eq. (16) over an area A(H) enclosed by a curve C(H) defined by the depth contour H. By using Gauss’ theorem on the first term in Eq. (16) we obtain

Here n is a vector normal to C and ds the length element along C. The geostrophic balance (Eq. 3) yields ${\displaystyle H{\mathbf{u}}_2\cdot \mathbf{n}ds=-(H/f)d{\mathit{\varphi }}_2}$, implying that the first term on the lefthand side integrates to zero since H/f is constant on C. By using Stokes’ theorem, Eq. (17) can then be written as

where ${\displaystyle d\mathbf{s}=\mathbf{s}ds}$ is the vector length element along C, or alternatively since $w_E=\mathrm{c}\mathrm{u}\mathrm{r}\mathrm{l}({\mathit{\tau }}_S/f)$ as

This shows that integrated around a closed depth contour, the Ekman transports in the surface and bottom layers are equal but of opposite signs. As f is constant this means that a bottom flow is established, associated with a bottom stress that balances the wind stress when integrated around a closed contour. Note that by Stokes’ theorem the contour integrated wind stress (divided by f) is proportional to the area-integrated surface Ekman velocity.

3.1 Scaling analysis

We will now make a scaling analysis of the two-layer model (Eqs. 15, 16), focusing on wind-forced flows on closed depth contours. For this purpose we will use some parameters characterising the Arctic Ocean which are listed in Tab. 1. For simplicity we take w_W = 0 and the thickness diffusivity K to be constant. However, since the diabatic velocity w_W is generally smaller than the wind-forced Ekman velocity w_E, the present scaling yields results that apply also for non zero w_W which forces the upper-layer in a similar way as w_E (see Eq. 15). Following Nøst and Isachsen (2003) we assume that the lower-layer flow is essentially aligned with the depth contours and that the integrated along depth-contour bottom stress and the wind stress are of the same order: see the integral relation Eq. (19). This gives the following along-isobath velocity scale for the lower-layer (say ${\displaystyle {\widehat{v}}_2}$)

where ${\displaystyle {\widehat{\tau }}_S}$ is the averaged value of the wind stress integrated along the isobaths. It is useful to introduce a scale for the area-mean surface Ekman velocity within a closed depth contour, say ${\displaystyle {\widehat{w}}_E}$. The mean Ekman velocity is related to the mean along-isobath wind stress as (see Eqs. 18, 19)

where L_G, the basin or gyre length scale, is defined as gyre perimeter length divided by twice the area: for a circular basin L_G is the gyre radius. We estimate the scale of the cross-slope flow in the lower layer (say ${\displaystyle {\widehat{v}}_{C2}}$) from Eq. (15): by assuming that the vertical velocity induced by the cross-slope flow balances the vertical surface Ekman velocity we obtain

Thus, the cross-slope velocity decreases with increasing bottom slope; ${\displaystyle |\mathrm{\nabla }H|\sim {10}^{-2}}$ is a typical bottom slope. The ratio between the scales of the cross- and along-slope flows, ${\displaystyle {\widehat{v}}_{C2}/{\widehat{v}}_2}$, defines a non-dimensional parameter:

where δH is the depth variation (∼3 km) and L_S is the continental slope width (∼300 km). Thus, ε is of the order of h_B/δH or smaller, and with h_B on the order of a few meters gives ε values on the order of 10^–3 (see Tab. 1).

The eddy (thickness) fluxes can balance Ekman pumping in the upper-layer continuity equation but not close the depth-integrated continuity equation, which when integrated around a closed depth yields a balance between surface and bottom Ekman layers (Eq. 18). Note that we neglect Reynolds stresses but retain eddy thickness fluxes, represented as diffusion of the upper-layer thickness. Reynolds stresses would enter the vertically-averaged continuity equation and provide a mechanism that can alter the bottom flow by redistributing the wind stress input horizontally.

In the upper-layer (Eq. 15) we accordingly assume that the Ekman velocity, proportional to the curl of wind stress, and the thickness diffusion are of the same order (Manucharyan et al., 2016). This yields the following scale for the upper-layer thickness variations

Using this scale for H₁ and Eq. (21) in the thermal wind relation (Eq. 5), we get the following scale for the velocity difference between the layers

The ratio between scales of thermal wind velocity (Eq. 25) and the lower layer velocity (Eq. 20) defines a non-dimensional parameter:

There are two limiting regimes:

1. σ ≪ 1: Barotropic flow without thermal wind shear. Here ${\displaystyle {\mathit{\varphi }}_1\approx {\mathit{\varphi }}_2}$. This limit is approached for instance when the thickness diffusivity K becomes large or when g′ or R is small.

2. σ ≫ 1: Surface intensified flow with weaker bottom flow. Here ${\displaystyle {\mathit{\varphi }}_1\approx g^{\prime }{H}_1}$. This limit is approached, for instance, when the bottom friction coefficient R or g′ becomes large.

Typical values in the Arctic Ocean of σ are on the order of one (see Tab. 1), indicating that the surface flow should generally be about twice as strong as the bottom flow. Note that in principle the present model can yield local regions where the bottom flow is stronger or directed opposite to the surface flow. In reality this may occur along the continental slopes in the Canada Basin where the bottom flow theoretically is expected to be determined by the global integrated along-isobath cyclonic surface wind stress, but local anticyclonic surface wind stress and buoyancy forcing may drive a near-surface flow in the opposite direction (Nøst and Isachsen, 2003; Timmermans and Marshall, 2020).

The wind forcing can be either steady or vary with a frequency ω and we put the time on non-dimensional form using the timescale ω^–1. Using the length scale L_G and the flow scales Eqs. (20,24) to put the continuity equations (Eqs. 15, 16) on non-dimensional form, we obtain

Here, ${\displaystyle \mathrm{c}\mathrm{u}\mathrm{r}\mathrm{l}({\mathit{\tau }}_S{)}_{\ast }}$ is the non-dimensional wind forcing and we have introduced the non-dimensional parameters

The parameter δ measures the ratio between the diffusive timescale (${\displaystyle L_G^2/K\sim 10}$ years) and the time scale of the wind forcing. The Peclet number Pe measures the relative importance of advection and diffusion of the upper-layer thickness: when Pe is large advection dominates over diffusion. Note that Pe is large if the wind forcing is large or the diffusivity is small.

Table 1 gives estimates of σ, Pe and ε in the Arctic Ocean. These non-dimensional numbers depend on the values of the model parameters R and K which account for unresolved nonlinear processes. Thus, the R and K values listed here involve uncertainties and somewhat different values are reported in the literature (see e.g. Nøst and Isachsen, 2003; Isachsen and Nøst, 2012; Manucharyan et al., 2016; Meneghello et al., 2018). Nevertheless, Table 1 suggests that in the Arctic Ocean σ is O(1), Pe is O(10) and ε^–1 is O(1000).

3.2 Lower layer dynamics

In the limit where ε ≪ 1 the bottom flow follows the depth contours to the lowest order (Nøst and Isachsen, 2003). In this case, Eq. (28) (on dimensional form) becomes to the lowest order

This implies that the lower-layer pressure is a function of H alone: φ₂ = φ₂(H) and thus ${\displaystyle {\mathbf{u}}_2=f^{-1}\frac{d{\mathit{\varphi }}_2}{dH}\mathbf{k}\times \mathrm{\nabla }H}$, showing that the bottom flow along H contours varies in proportion with |∇H|. To determine the φ₂(H) in this limit one can use the integral relation Eq. (18) that applies on closed H-contours: see also Eq. (A4). As a result, the lower-layer flow depends only on the value of the integrated wind stress along the contour, or equivalently the area-integrated surface Ekman pumping. Thus, to lowest order in ε the lower-layer flow is insensitive to local variations of the surface Ekman pumping. Nøst and Isachsen (2003) reported numerical solutions of Eq. (28), which confirm the notion that the lower-layer flow is well aligned with the bathymetry when ε is small.

Using observed surface wind stress and hydrography, Nøst and Isachsen (2003) and Aaboe and Nøst (2008) calculated the bottom flow in the Arctic Ocean. They took the variations of f into account which gives an integral relation [analogous to Eq. (18)] for the bottom flow on closed f/H contours. However, since the variations of f/H in the Arctic Ocean are dominated by depth variations the present f-plane model (where H contours replaces f/H contours) has qualitatively the same dynamics.

3.3 Upper layer dynamics

The scaling analysis suggets that Pe is on the order of 10 (Table 1), implying that advection as well as eddy diffusion can play roles for the upper-layer dynamics. It is instructive to considering the limiting cases of the advective regime (Pe ≫ 1) and the diffusive regime (Pe ≪ 1). Here we assume that δ is on the order of unity or smaller, implying that we are considering variations of the wind forcing on time scales on the order of decades or longer (see Manucharyan and Spall, 2016, for a discussion on eddy-induced diffusive adjustment time scales). With this restriction on the forcing time scales the two limiting regimes of the upper-layer flow are (on dimensional form):

3.4 Extension to a multi-layer model

It is possible to extend the present two-layer model to a model including several layers (Pedlosky, 1987). A key feature is that vertically-integrated continuity relation is still given by the equivalent to Eq. (16), with u₂ replaced by the velocity of the lowest layer velocity. This is because it is only the top- and bottom-vertical Ekman velocities and the bottom vertical velocity that enter the geostrophic vertically-integrated [or summed for a layer system] continuity equation on an f-plane (Nøst and Isachsen, 2003).

If the parameter ε (Eq. 23) – which is independent of the vertical model structure – is small then the velocity in the lowest layer will be aligned with the H(x, y)-contours. Further if the Peclet number (Eq. 29) is large then the flow in the layers above the lowest one will be closely aligned with the H(x, y)-contours provided that a generalised version of the parameter σ (Eq. 26) is on the order of unity. Thus, in the limit where Pe and ε^–1 are large the qualitative features of a multi-layer model are similar to those of the present two-layer model, but a multi-layer model has a vertical flow structure that depends on how g′ and K vary between the layers.

4.1 Isobath-following flow components

To begin with, we determine the isobath-following flow components which are functions of r alone. It is straightforward to obtain v₂ from the integral relation (see Eq. 18) and integrating v₂ radially gives φ₂. The result is

where φ₂ is specified up to a constant of integration. Note that v₂ describes solid body rotation, a feature that arises because w_m is constant but is independent of the form of H(r). We also define a gyre velocity scale as

which yields the following Peclet number Pe = v_GL_G/K.

The isobath-aligned part of H₁, forced by ${\displaystyle {\overline{w}}_E}$, can be determined from Eq. (37); by integrating the equation twice in the radial direction, we obtain

Here H₀ is the upper-layer thickness in the gyre center (r = 0). Note that for positive (negative) w_m, H₁ increases (decreases) away from the gyre centre.

By using the lower-layer flow specified by Eq. (38) and the hydrostatic relation (Eq. 4) in combination with Eq. (40), we obtain the circular symmetric (isobath-aligned) parts of v₁ and φ₁

where φ_1s is specified up to a constant of integration, and σ is defined by Eq. (26).

Qualitatively, the Eqs. (38, 41) characterise the integral balance of the along-isobath flow also for realistic non-circular bottom topography [see Eqs. (A4, A11)]: the integrated surface wind stress balances the integrated bottom stress (Figure 2). Further, the surface wind stress is transferred from the upper to the lower layer by an eddy-induced form stress, which is represented as a diffusion of upper-layer thickness. This acts to reduce the thermal-wind shear between the layers. However, a thermal-wind shear (proportional to the upper-layer slope) is needed to sustain the vertical stress transfer, and the parameter σ measures how large the shear and hence interface slope become in a steady state. This balance can also be interpreted as a situation with a vanishing residual circulation in the cross-isobath direction (Marshall and Speer, 2012; Manucharyan et al., 2016); the surface and bottom Ekman transports have equal magnitude and opposite direction, and within each layer they are canceled by the eddy-induced transports (represented a thickness diffusion). This implies that the radial component of the layer volume transports M₁ and M₂ (Eqs. 6,7) vanish. Thus, solutions given by Eqs. (38, 41) admit a solid vertical gyre boundary to be placed at a constant radius.

4.2 Beaufort Gyre example: asymmetric w_E forcing

Here we consider a ‘Beaufort Gyre’ like case (Timmermans and Toole, 2023), with anti-cyclonic surface winds and area-integrated downward Ekman pumping over a basin with closed depth contours, yielding a deeper upper layer in the central gyre. Since the model is linear the analysis also applies to a gyre with cyclonic wind forcing, for instance the Greenland Sea Gyre. Although the area-integrated Ekman velocity is downward there are regions in the Beaufort Gyre where the Ekman velocity locally is upward (Timmermans and Toole, 2023). Our aim is to examine how asymmetries in Ekman pumping, associated with along-isobath variations in w_E, force upper-layer flow across the isobaths.

We non-dimensionalise the radial coordinate with the gyre radius L_G and consider a w_E anomaly (see Figure 3a) given by

Note that ${\displaystyle x=r\cos (\theta )}$, so ${\displaystyle w_E^{\prime }}$ decreases linearly in the x-direction. Further, we use the scale ${\displaystyle w_m{L}_G^2/(4K)}$ to non-dimensionalise H₁: the isobath-aligned solution (Eq. 40) then becomes

where we have taken H_1s to be zero on the gyre boundary (r = 1). The upper-layer thickness is expressed as

where H_1a is the asymmetric part related to ${\displaystyle w_E^{\prime }}$. To conserve mass the normal component of the upper-layer geostrophic flow must be zero on the gyre boundary. This is ensured if we require that the asymmetric part H_1a(r, θ) is zero at r = 1.

The upper-layer solution to Eq. (35) related to ${\displaystyle w_E^{\prime }}$ (Eqs. 42,44) can be written as an infinite sum: see Eq. (A20). It turns out that the solution is qualitatively approximated by the first term in the series, which yields

Here J₁ is the Bessel function, k₁ ≈ 3.83 the first zero of J₁, a₁ ≈ 1.3 and ϕ is a phase factor (describing rotation of the field due to the anticyclonic v₂ velocity in the present example). Note that Pe = v_GL_G/K where v_G is defined by Eq. (39).

As is illustrated in Figure 3, the spatial form of H_1a depends on the Peclet number: its over-all radial structure remains roughly similar but the pattern is rotated by the angle ϕ in the clockwise direction. In the diffusive limit, where Pe = 0, ϕ = 0, and in the advective limit, where Pe approaches infinity, ϕ = π/2. The diffusion as well as the advection act to damp spatial variations forced by ${\displaystyle w_E^{\prime }}$ in the H_1a field. For a fixed diffusivity the amplitude of H_1a decreases with increasing Pe (or v₂): for large Pe, Eq. (45) shows that H_1a ∼ Pe^–1. This can also be inferred from Eq. (35) which in this limit reduces to a balance between advection of upper-layer thickness and the Ekman forcing.

We now go on to examine the factors controlling the alignment of the upper-layer flow with the topography. For this purpose we put the pressure on non-dimensional form using the scale ${\displaystyle w_m{f}^2{L}_G^2/(4R)}$ [see Eq. (38)] which yields the non-dimensional form ${\displaystyle {\mathit{\varphi }}_2=H_{1s}(r)}$ where ${\displaystyle H_{1s}(r)}$ is given by Eq. (43). The non-dimensional upper-layer pressure equals ${\displaystyle {\mathit{\varphi }}_1={\mathit{\varphi }}_2+\sigma H_1}$ [see Eqs. (4,41)], and is given by

To begin with, we note that when σ is small the flow becomes barotropic and hence the upper-layer flow aligns with the topography. This limit is approached when the stratification (g′) is weak or when the ratio between R and K is small. For fixed values of g′ and R, a large value of the eddy diffusivity K yields a small σ value. The resulting limit represents one of efficient vertical momentum transfer via the parameterised baroclinic eddy interfacial stress. This results in weak horizontal gradients in H₁ and hence weak thermal wind difference between the layers. In the opposite limit, when ${\displaystyle \sigma \gg 1}$ and ${\displaystyle {\mathit{\varphi }}_1\approx \sigma (H_{1s}+H_{1a})}$, the cross-isobath flow is set by the (normalised) asymmetry of the wind forcing (w_a/w_m) and the Peclet number (see Eq. 45). This limit is approached, for instance, when the bottom friction parameter R is large, which causes a weak lower-layer flow.

To illustrate how the Peclet number and the parameter σ affects the alignment of the upper-layer flow with the topography we consider a case where w_a/w_m = 4. This is a somewhat more extreme case than the observed Ekman pumping asymmetry in the Beaufort Gyre, where ${\displaystyle w_a/w_m\approx 2}$ (see Figure 5 in Timmermans and Toole, 2023): the case w_a/w_m = 4 corresponds to surface Ekman downwelling where x > – 1/4 and upwelling further to the east where ${\displaystyle x<-1/4}$. Figure 4 shows φ₁ for combinations of two different values of σ and Pe. Qualitatively, the figure shows that the alignment of the upper-layer flow increases with Pe and decreases with σ.

Figure 4

To qualitatively measure the alignment between the upper-layer flow and the bottom topography we use the cosine of the angle between the flow and the depth contours defined by Eq. 1. Figure 5 shows the area-mean of |cos(α)| as a function of the Peclet number for the flow defined by Eq. (46) for three different values of σ and for two cases of wind-forcing asymmetries: w_a/w_m = 4 and w_a/w_m = 8. As expected, the alignment between the flow and the topography increases with the Peclet number but decreases with increasing values of σ. This is because the contribution of the asymmetric component H_1a to upper-layer flow increases with σ as the flow becomes more surface intensified (see Eq. 46). Stronger wind-forcing asymmetry, measured by w_a/w_m, decreases the topographic alignment. For the present asymmetric wind-forcing (Eq. 42), the alignment approaches a minimum of ${\displaystyle \approx 0.66}$ when w_a/w_m becomes large for non zero values of σ. In this limit the upper-layer flow (φ₁) is proportional to H_1a. Notably for large values of Pe, the upper-layer flow becomes highly aligned with the depth contours, implying that ${\displaystyle {\mathbf{u}}_2\cdot \mathrm{\nabla }{H}_1\approx 0}$.

Figure 5

Figure 6 shows the topographic alignment of the upper-layer flow as function of the Peclet number and the parameter σ, for the flow defined by Eq. (46) with w_a/w_m = 4. When σ becomes small the flow becomes nearly barotropic and aligned with the topography. In this limit the alignment is weakly dependent on the Peclet number. The dependence of the alignment on σ is strongest when the Peclet number is small. The qualitative features of Figure 6 are similar to those for other values of w_a/w_m (not shown). The detailed features of Figure 6, however, can be expected to change slightly with the specific spatial form of the asymmetric forcing ${\displaystyle w_E^{\prime }}$ (Eq. 42).

Figure 6

We emphasise that the bathymetry in the real Beaufort Gyre, embedded in the Canada Basin, is complex (see e.g., Figure 1 and Timmermans and Marshall, 2020). Thus, the circular basin geometry used here is an extreme idealisation. Further, only depth contours deeper than around 2700 m close themselves locally in the Canada Basin, implying that shallower depth contours extend in the Arctic Ocean beyond the conventional geographical definition of the Canada Basin (Jakobsson, 2002). These features should be kept in mind when comparing our model results with the real Beaufort Gyre. Nevertheless, the present idealised model solutions should give some qualitative indications of how the real near-surface flow in Beaufort Gyre is influenced by the bathymetry via the Peclet number and the parameter σ (see Figures 5, 6). Further, Figure 4 suggests that the surface height maximum (proportional to φ₁) should, for increasing Pe and decreasing σ, be shifted away from the maximum in downward Ekman pumping (located at x = 1 in Figure 4) and towards the ‘gyre centre’ which in the present model context would correspond to the location of the maximum basin depth in a non-circular basin. Observational estimates of the surface Ekman pumping and the surface height field in the Beaufort Gyre (see e.g., Figures 4 and 5 in Timmermans and Toole, 2023) suggest some degree of topographic influence of surface Beaufort Gyre flow on large scales, but also indicate local regions of significant cross isobath flow.

4.3 Norwegian Sea example: boundary forced asymmetries

We go on to consider a different situation in which advection of the upper-layer thickness by the lower-layer velocity can affect the bathymetric alignment of the upper-layer flow. Specifically, we consider a highly simplified model of the flow in a region with locally closed depth contours in the Norwegian Sea (see Figure 9) which again is idealised as a circular basin. In this example we take the upper layer to represent the Atlantic Water in the eastern Nordic Seas (Hansen and Østerhus, 2000). We assume that the flow is driven by local surface Ekman pumping, due to cyclonic wind forcing, as well as large-scale atmospheric-ocean interactions that drive a general northward flow of Atlantic Water through the domain in the upper layer (Walin et al., 2004; Spall, 2010). For the sake of simplicity, however, the Atlantic flow is here imposed as a boundary condition on an outer depth contour in the ‘gyre’. Thus, the ‘gyre boundary’ is taken to be a closed depth contour at the edge of a local basin in this case. We specify the inflow as well as the outflow in the upper layer on the ‘gyre boundary’ which is a fairly drastic simplification since the distribution of the outflow is likely to be affected by the locally wind-forced flow. However, this simplification allows analytical solutions that should illuminate aspects of the dynamics. As wind forcing we use a constant upward Ekman velocity in the gyre, associated with the isobath-following flow specified by Eqs. (38, 40, 41).

We consider a constant northward Atlantic Water flow, with the velocity v_A, in the upper layer through the domain, which is imposed as a boundary condition at the gyre boundary (see Figure 7a). Note that Atlantic Water is now assumed to flow in the upper layer across the depth contours at the ‘gyre boundary’ causing φ₁ to vary along the boundary, whereas in the Beaufort Gyre example we assume zero upper-layer flow at the gyre boundary (i.e. φ₁ is taken to be constant). The associated H₁ field is obtained as a solution to the homogeneous advection-diffusion equation (Eq. 35) and it is denoted H_1A: see Eq. (A29). The amplitude of H_1A is set by v_A and its shape depends on the Peclet number: Pe = v_GL_G/K, where v_G is the wind-forced lower layer flow (Eq. 39). The field φ₁ is given as the sum of the isobath-following contribution (Eq. 41) and the contribution due to H_1A; the non-dimensional φ₁ then takes the form

Figure 7

Figure 7 shows the asymmetric upper-layer thickness solution H_1A (Eq. A29). In the absence of local cyclonic wind forcing v₂ and Pe are zero. In this case the homogeneous solution H_1A with Pe = 0 gives the whole field, which is a constant northward flow associated with an east–west gradient in H_1A. When the wind forcing is increased H_1A is distorted by the cyclonic flow. For very high Pe the variations of H_1A become confined to thin layers at the gyre boundary and H_1A becomes nearly constant in the interior gyre.

Figure 8 shows φ₁ given by Eq. (47) for increasing values of v_A/v_G. Here we take Pe = 20 and σ = 0.5, values that roughly characterise the Norwegian Sea. Further, taking observed northward Atlantic Water velocities, which are in the range between 0.05 and 0.15 m s^–1 (Hansen and Østerhus, 2000), we estimate that v_A/v_G is on the order of unity. When v_A/v_G is increased the upper-layer flow crosses the depth contours to an increasing degree but the lower-layer flow remains, by design, aligned with the depth contours. Further, the lowest value of φ₁ in the gyre, which is at the gyre centre when v_A/v_G = 0, becomes displaced towards the south–west when v_A/v_G increases.

Figure 8

Figure 9 shows the observed time-mean dynamic surface height in the eastern Nordic Seas. The dynamic height reveals a general northward flow of surface waters in the eastern Norwegian Sea and in the Lofoten Basin. In the Norwegian Sea there is also a tendency of a cyclonic circulation of the surface flow which has some qualitative similarities with the model flow fields shown in Figure 8b–c. For instance, the minimum surface pressure is displaced south–west of the deepest part (‘centre’) of the gyre.

Figure 9