Figure 1
Alignment of the time-mean surface flow with the bathymetry; the surface flow is based on satellite altimetry (Mulet et al., 2021). Here, the alignment is defined as the cosine of the angle between the surface flow and the depth contours; see Eq. (1). Positive (negative) values show flow with shallow water to the right (left). The figure shows the alignment along selected depth contours in the range ∼800–4000 m; the Canada Basin (CB), the Eurasian Basin (EB) and the Norwegian Sea (NS) are indicated. The ∼ 800 m depth contour separates between shallow contours, extending into the Atlantic, and deeper locally closed contours. The bathymetry has been smoothed with a filter that suppresses variation on scales below around 100 km. For a flow field that is randomly oriented relative to the bathymetry the angle α will be uniformly distributed between 0 and 2π; in this case the mean of |cos(α)| is π/2 ≈ 0.6. Similar results are obtained on f/H contours (not shown). Note that the Transpolar Drift, directed from CB towards EB, crosses topography in the Central Arctic Basin (Rudels et al., 2012; Haine et al., 2015; Timmermans and Marshall, 2020).
Figure 2
The two layer model, with subscript 1 (2) denoting the upper (lower) layer. The rigid lid approximation (Gill, 1982) is made and the total depth is time independent. The dynamic pressure in the layers are related as (Eq. 4) where φ1 is proportional to the sea surface height anomaly which is much smaller than the variations in H1. The red horizontal arrows show Ekman transports in the surface and bottom layer, respectively, and the blue wavy arrows show eddy-induced volume transports that are proportional to |∇H1|; see Eqs. (6,7). The red dashed vertical arrows represent wind-forced vertical Ekman velocity (wE) and transfer between the layers (wW) due to diabatic processes. The surface Ekman suction/pumping velocity wE and wW force the upper layer in similar ways (Eq. 15).
Figure 3
Pattern of asymmetric Ekman pumping [Eq. (42)] (a) and associated asymmetric upper-layer thickness H1a(r,θ) for varying Peclet numbers (b–d): Pe = 0 (b), Pe = 10 (c), and Pe = 100 (d). Here H1a is approximated by the first 50 terms in the infinite series defined by Eq. (A20). The amplitude of H1a(r,θ) decreases with Pe but is here normalised to range between –1 and +1. The white lines indicate circles () that coincide with the depth contours.
Table 1
Order of magnitude estimates of some model parameters in the Canada Basin (CB) and the Norwegian Sea (NS) (data taken from Nøst and Isachsen, 2003; Isachsen and Nøst, 2012; Meneghello et al., 2018; Timmermans and Marshall, 2020). The most uncertain quantities are the frictional parameter R and the eddy diffusivity K which are representations of unresolved non-linear processes. These uncertainties propagate into the derived model scales and parameters , , σ, Pe and ε. In the Norwegian Sea the area integrated wind stress is cyclonic whereas in the Canada Basin it is anticyclonic. The main differences between the two regions are a weaker Ekman pumping – due to partly land fast sea ice – and a stronger stratification in the Canada Basin. See Figure 7 in Timmermans and Marshall (2020) for the spatial distribution of the Ekman pumping in the Arctic Ocean.
| BASIN | CB | NS |
|---|---|---|
| R (m s–1) | 2·10–4 | 2·10–4 |
| K (m2 s–1) | 103 | 103 |
| LG (km) | 500 | 300 |
| Rossby radius (km) | 15 | 10 |
| g′ (m s–2) | 6 · 10–2 | 3 · 10–2 |
| (m2 s–2) | 5 · 10–6 | 15 · 10–6 |
| (10–6m s–1) | 0.1 | 0.5 |
| (m) | 25 | 45 |
| (cm s–1) | 2 | 6 |
| 1 | 0.5 | |
| 10 | 20 | |
| 10–3 | 10–3 |
Figure 4
Upper-layer pressure φ1 (Eq. 46), which is proportional to the upper-layer geostrophic stream function, for two combinations of the parameter σ (Eq. 26) and the Peclet number. Here wa/wm = 4, implying that there is Ekman downwelling (upwelling) where x > –1/4 (). The amplitude of φ1 is normalised to be one at its maximum. The white lines show circles () that coincide with the depth contours, and dashed white lines show the zero contour of φ1.
Figure 5
Alignment of the upper layer flow with the bottom topography as a function of the Peclet number for the flow field given by Eq. (46): see Figure 4. The topographic alignment is defined as the area mean of |cos(α)| (Eq. 1); perfect alignment corresponds to 1. Black, red, and blue lines correspond to σ = 0.5, σ = 1, and σ = 10, respectively. Solid (dashed) lines correspond to a wind forcing asymmetry – measured by wa/wm – of 4 (8). For this flow field the alignment for Pe = 0 and large values of σ approaches ≈ 0.8 when wa/wm = 4 and ≈ 0.7 when wa/wm = 8. If α – the angle between the isobaths and upper-layer flow – is randomly distributed in a domain then the area average of |cos(α)| is 2/π ≈ 0.6.
Figure 6
Topographic alignment of the upper layer flow as a function of the Peclet number and the parameter σ. The flow is given by Eq. (46) with wa/wm = 4. In this case the alignment for Pe = 0 becomes ≈ 0.8 for large values of σ, corresponding to a strongly baroclinic flow. When σ approaches zero the flow becomes barotropic and fully aligned with the topography.
Figure 7
Solutions of upper-layer thickness H1A(x,y) forced by an imposed gradient at the gyre boundary (r = 1) where H1A = cos(θ) = x: see Eq. (A22). The solutions depict how a northward ‘Atlantic Water’ flow in the upper layer is affected by a cyclonic isobath-following flow that is forced by wind-stress over the gyre. For large vales of the Peclet number the flow becomes confined to boundary layers with a thickness on the order of Pe–1/2 (see Eq. A24). Note that since the upper-layer thickness is only advected by the lower layer velocity the shape of H1A(x,y) depends solely on the Peclet number based on the lower layer: Pe = vGLG/K (Eq. 39).
Figure 8
Solutions of upper-layer dynamic pressure φ1 forced by an imposed boundary northward flow of Atlantic Water and a uniform Ekman surface velocity (wE > 0): see Eq. (41). For all solutions, Pe = 20 and σ = 0.5. The panels shows different ratios of the imposed Atlantic Water velocity vA and the lower-layer gyre velocity vG: see Eq. (38). The solutions are normalised to range between –1 and +1.
Figure 9
Time-mean dynamic surface height, based on satellite altimetry (Mulet et al., 2021), in the Norwegian Sea (NS) and the Lofoten Basin (LB). The dynamic height is proportional to the upper-layer dynamic pressure φ1 in the model. The black lines show depth contours. Figure 8 shows model solutions representing an idealised Norwegian Sea case.