The shallow water equations describe the flow of a single-layer fluid with constant density, where velocity varies only in the horizontal. The horizontal velocity vector is given by u=ui+vj (i and j are unit vectors in the x and y directions) and the total fluid depth (layer thickness) is D=H+η, where H(x,y) is the fluid depth at rest and η(x,y,t) is the free surface elevation. The governing equations are 2∂tDu+∇⋅Duu+fk×Du=-D∇ϕ+τs-τb and 3∂tD+∇⋅Du=0, where f is the Coriolis parameter (k is the unit vector in the z direction), τs and τb are the surface and bottom kinematic stresses, and ϕ=gη is the dynamic pressure, where g is the gravitational acceleration. A derivative with respect to a variable x is written on the form ∂x, so that ∂t is a time derivative.

Depending on the diagnostic purpose, it can be convenient to rewrite the momentum equation using the identity 4u⋅∇u=ζk×u+∇|u|22, where 5ζ=∂xv-∂yu is the relative vorticity. Using Eqs. () and (), we can thus rewrite Eq. () as 6D∂tu+Dζ+fk×u=-D∇ϕ+|u|22+τs-τb. As we will see, this formulation highlights the explicit role of vorticity fluxes.

In this study we will use a constant Coriolis parameter f=10-4 s⁻¹, appropriate for high-latitude regions where variations in planetary vorticity are small. The bottom frictional stress will be modeled as a linear drag, 7τb=Ru, where R=5×10-4 m s⁻¹ is a constant bottom drag coefficient. This value is chosen based on typical values diagnosed in .

We consider an idealized horizontal domain with extents of Lx=Ly=90 km, discretized using a uniform grid with resolution Δx=Δy=1 km. The domain is periodic in the x-direction. At y=0 and y=Ly, there is a no-flux boundary condition. This configuration permits continuous circulation in the x-direction, analogous to flow around a basin.

The bottom topography consists of a large scale slope in the y-direction, with a smooth transition from a shallow shelf to a deeper basin as y increases. Superimposed are along-slope sinusoidal corrugations in the x-direction that are scaled with the y-derivative of the background profile. The full bathymetry is thus given by H(x,y), and is defined as 8H(x,y)=Hsh+12Hbs-Hsh1+tanh⁡π(y-yc-δ(x,y))W where the corrugation displacement δ(x,y) is specified as 9δ(x,y)=Acorrsin⁡2πxλsech2π(y-yc)W. Here, λ is the wavelength of the corrugations. Since the domain is periodic in x with length Lx the domain must contain an integer number of corrugation wavelengths, i.e. Lx=Nλ for some integer N. The slope has a width W of 30 km, corresponding to 30 grid cells at the model resolution of 1 km, so the continental slope is well resolved. Parameters relevant for the domain and bathymetry are summarized in Table . An illustration of the resulting bathymetry is shown in Fig. . While the Arctic Ocean contains basins that are substantially deeper than the 900 m basin depth used here, the idealized setup is intended to capture the essential dynamics rather than reproduce the detailed Arctic bathymetry.

In this bathymetric setup, flow in the positive x-direction is prograde, while flow in the negative x-direction is retrograde. Prograde flow corresponds here to cyclonic circulation around a basin, with shallower water to the right of the current in this Northern Hemisphere configuration.

The system is forced by an along-channel spatially uniform surface stress that varies sinusoidally in time. It is defined as 10τs=τsxi=τ0sin⁡2πTti, where T is the forcing period, and τ0 is the maximum kinematic stress. In the base case, we set τ0=10-8 m⁴ s⁻². For a reference density of 1000 kg m⁻³, this corresponds to a dynamic stress of 0.1 N m⁻², realistic for typical wind forcing.

To explore the role of forcing time scales, we contrast a short and a long forcing period. For the short-period case, we use T=16 d. This value lies at the longer end of what can be considered synoptic variability, but experiments with shorter periods in our setup exhibit qualitatively similar behavior. We use 16 d as a representative fast-forcing limit, because it produces a clearer signal and facilitates interpretation of the flow response. The long-period case, with T=128 d, represents slowly varying, seasonal-scale forcing. Together, these two limits bracket the essential differences between rapidly and slowly varying forcing. More generally, one relevant parameter is the ratio between the forcing period and the frictional adjustment timescale H/R. Thus, varying T at fixed R captures much of the same dependence that would be obtained by varying R at fixed T. Animations of the flow field resulting from these two forcing periods can be found in the video supplements (ref to 10.5446/74040, and 10.5446/74041, ).

The short-period experiment is run for 192 model days, while the long-period experiment is run for 384 model days. These durations are chosen to exceed the system’s frictional adjustment time, and thus let the system lose memory of the initial state. After an initial transient, the model response becomes cycle-to-cycle repeatable, and we therefore analyze only the final full cycle of each simulation. The time step is 4 s, and output is saved every 3 model hours.

To form an equation for the circulation around a closed isobath, we average Eq. () along the contour, everywhere projecting vector components along the contour. This yields 12H∂tu⋅t‾H=Hζu⋅n‾H+τs⋅t‾H-Ru⋅t‾H, where t and n are the tangent and normal unit vectors to the contour, respectively. Vector t points along the contour in the positive x-direction, while vector n is defined to point towards increasing depth. In obtaining Eq. (), we use the identity (k×u)⋅t=u⋅n. The associated planetary vorticity term Hfu⋅n is proportional to the cross-contour volume flux under the f-plane approximation, and therefore vanishes under the rigid-lid assumption when integrated around the re-entrant domain. The pressure and kinetic-energy gradients do not contribute, since H can be factored out and the remaining gradient term integrates to zero around any closed contour.

The first term on the right-hand side of Eq. () represents the depth and contour-integrated flux of relative vorticity across the contour in the offshore direction. The second term is the average of the tangential projection of the surface stress, and the final term represents damping by linear bottom drag.

The linear version of Eq. () is an ordinary differential equation, and can be solved to obtain an analytical model for the circulation around isobaths. Neglecting the vorticity flux term and using the integrating factor exp⁡Rt/H, we get 13u⋅t‾linH=exp⁡-RHtu0⋅t‾H+∫0texp⁡-RHt-t′Flin(t′)Hdt′, where Flin is given by the surface stress, 14Flin=τs⋅t‾H. The first term on the right-hand side of Eq. () is the exponentially decaying contribution from the initial state u0⋅t‾H, while the second term is a convolution between the forcing term Flin and an exponential kernel. This convolution links circulation to the time-integrated forcing, and predicts the circulation to inherently carry a memory of past states over a time-scale H/R. Given our typical value for R, the factor H/R corresponds to frictional spin-down times of 2–21 d across the domain depths 100–900 m.

Equation () corresponds to the simplified linear model of time variable flow along f/H-contours of , but with the additional approximation of a constant f. Equation () was later used by and to estimate wind-driven circulation in realistic Arctic and Nordic Seas simulations and in observations. In this study, Eq. () serves as a linear reference for evaluating the response. In Fig. , we test this linear model against a variation of the baseline experiment in which the forcing amplitude is reduced to 10 % of its standard value. This weak-forcing case is expected to remain close to linear, and indeed shows excellent agreement with the theoretical linear estimate. This test supports the theoretical assumptions and validates our numerical implementation.

We can further utilize Eq. () to estimate the maximum velocity of the linear response to the surface stress given by Eq. (). Because the stress is purely zonal and uniform, its contour-mean projection is reduced by the geometric factor Lx/LC(H). Solving Eq. () for a periodic forcing of frequency ω=2π/T gives the maximum along-contour velocity amplitude 15umax⁡(H)=LxLC(H)τ0R1+ωHR2-1/2, with derivation provided in Appendix .

From Eq. (), we see that for ω≪R/H (long forcing periods), the flow amplitude depends on depth only through the geometric factor, approaching Lx/LC(H)τ0/R. In contrast, for rapidly oscillating forcing with ω≫R/H, the amplitude approaches Lx/LC(H)τ0/(ωH), implying a weaker response with a stronger depth dependence.

extended the linear model, and also included the nonlinear vorticity flux term from Eq. () as a forcing term, so that the circulation could be estimated as 16u⋅t‾H=exp⁡-RHtu0⋅t‾H+∫0texp⁡-RHt-t′F(t′)Hdt′, where 17F=τs⋅t‾H+Hζu⋅n‾H. This formulation requires that cross-contour relative vorticity fluxes Hζu⋅n‾H are diagnosed from data, or otherwise estimated. With that exception, theoretical consideration, including the role of the damping time-scale, remains the same as for the linear model in Eq. ().

When integrated along straight transects, the same system is more conveniently described by a different momentum balance. Integrating Eq. () along the channel at fixed y, we obtain a budget for the depth-integrated x-momentum, 18∂tHu‾y=-∂y(Hvu)‾y-H∂xϕ‾y+τsx‾y-Ru‾y. The first term on the right-hand side represents the convergence of depth-integrated flux of x-momentum. The second term measures the correlation between water depth and pressure gradients. By using the periodicity of the integration path, we can rewrite this as -H∂xϕ‾y=ϕ∂xH‾y. The term thus also measures the correlation between the pressure and the slope of the corrugations. This term is the topographic form stress, that is the net pressure force on the corrugated bottom. Importantly, nonzero form stress requires a phase offset between pressure and the underlying bottom corrugations. Without such an offset, the integral vanishes. The third term represents the integrated wind forcing in the x-direction, and the final term accounts for linear bottom drag opposing the flow. As for the analysis along isobaths, the term associated with planetary vorticity (third term on left hand side of Eq. ) becomes proportional to the cross-contour volume flux, and thus integrates to zero under the rigid-lid approximation.

While not inherently apparent from the expression ϕ∂xH‾y, topographic form stress term is typically stronger for retrograde flow than for prograde flow. Although the expression for topographic form stress appears linear, this asymmetric response still relies on nonlinear flow dynamics. In particular, relative vorticity advection modifies the amplitude and phase of the pressure field relative to the corrugations. During retrograde flow, this nonlinear modification becomes stronger, increasing both the pressure amplitude and the phase offset required for nonzero form stress. The form stress is therefore enhanced during retrograde phases, even though the budget term itself appears as a linear pressure–topography correlation. This flow-dependent asymmetry can be connected to standing Rossby waves arrested by retrograde flow, as shown for example by , who developed a quasi-geostrophic standing-wave theory for predicting the strength of topographic form stress. We refer to for the full theoretical derivation, but repeat some key results here. For a mean along-slope flow U, mean depth H0, and corrugations given by h‾sin⁡(kx), the topographic form stress is given by 19ϕ∂xH‾y=12h‾H02RUf2/k2U+ck2+RkH2, where ck is the phase speed of a Rossby wave with wavenumber k. Since Rossby waves travel with shallower water to the right in the Northern Hemisphere, ck is strictly positive. While Eq. () is for a steady, quasi-geostrophic system, it illustrates some key points; when retrograde flow arrests Rossby waves with a wavelength comparable to the bottom corrugation scale, corresponding to U=-ck, a standing wave that resonates with topography can form and the associated topographic form stress increases considerably in strength. Only far from resonance do we expect topographic form stress to be approximately symmetric for prograde and retrograde flow velocities. Further, Eq. () indicates that the topographic form stress depends on the square of the topographic corrugation amplitude, and therefore increases with the height of the topography. We will return to these points in Sect. , where we use arrested wave theory to understand some of the simulation results.

Finally, as shown explicitly in Appendix , if one additionally takes the mean of Eq. () over time and y, the acceleration term and periodic wind stress term both vanish. Additionally, under the assumption of no momentum fluxes through the walls, the integrated flux divergence terms also vanishes. What remains is a domain-wide balance between topographic form stress and the frictional stresses. This global balance, previously noted by , thus reveals that for oscillatory forcing with zero time mean the only way to achieve a net-zero total mass transport through the channel – or around an isolated ocean basin – is that it contains topographic variations that can give rise to a topographic form stress.

The two momentum budgets above emphasize different dynamics in addition to surface stress and bottom frictional stress: vorticity fluxes when the balance is evaluated along depth contours and momentum flux convergence together with topographic form stress when evaluated along straight transects. Such arbitrariness related to choice of integration path is unsatisfactory. However, as it turns out, both frameworks can be shown to reflect the same dynamics under QG assumptions, namely a down-gradient flux of PV across the slope.

We start with the fixed-y momentum budget and consider the first two terms on the right-hand side of Eq. (), namely the convergence of depth-integrated momentum flux and the topographic form stress. We define their sum as Fq‾y, and use the identity in Eq. () to re-write it as 20Fq‾y=Hζv‾y-H∂xϕ+12|u|2‾y. For small Rossby numbers, geostrophic scaling implies that the pressure term dominates the kinetic energy contribution, |ϕ|≫12|u|2, allowing the latter to be neglected.

We now also use a second requirement for QG, namely that the total depth can be written as H(x,y)=H0-h(x,y). Here, H0 denotes some mean depth and h represents the height of variable bottom topography, assumed to be small compared to H0. Substituting this decomposition into Eq. (), and using the assumption that |h|≪H0, then yields 21Fq‾y=H0ζv‾y-h∂xϕ‾y. And, again, assuming that velocities are geostrophic to lowest order, fv=∂xϕ, we obtain 22Fq‾y≈H0vζ‾y+vfhH0‾y, or 23Fq‾y≈H0vq‾y, where 24q=ζ+fH0h is quasi-geostrophic PV QGPV; seeSect. 5.3.1. Thus, under QG assumptions, the combined effect of nonlinear advection and topographic form stress in the fixed-y momentum balance can be interpreted as a depth-integrated flux of potential vorticity across constant-y contours seefor a treatment starting from QG.

In the depth-following framework the association with PV fluxes is trivial; since h is zero the sole contribution to the QG PV flux is the relative vorticity flux. So under QG scaling, the momentum equation integrated along both the fixed-y and fixed-H paths take the form 25H0∂tu⋅t‾=H0uq⋅n‾+τs⋅t‾-Ru⋅t‾, where the overbar here denotes an average either along fixed-y or constant depth contours. Equation () corresponds to the Transformed Eulerian Mean (TEM) form of the depth-integrated momentum equation seeSect. 10, for the formulation in the multi-layer shallow-water system with, as we emphasize here, the topographic form stress generating the thickness contribution to the PV flux. Further, according to Eq. (), a flux of total PV down (up) the topographic slope, that is down (up) the PV gradient, either accelerates a prograde (retrograde) flow or maintains a steady prograde (retrograde) flow against bottom friction.

Do the nonlinear deviations identified in realistic Arctic Ocean simulations also emerge in a stripped-down, idealized setting? To address this question, we apply the same diagnostic used for the realistic simulation in Fig.  to the idealized model experiments. Specifically, we examine scatterplots of the simulated mid-slope circulation against linear estimates given by Eq. (), for both long and short forcing periods (Fig. ). The circulation is diagnosed along the mid-slope isobath (highlighted in Fig. ), allowing direct comparison with linear theory. Perfect agreement with the linear estimates would place points along the 1:1 line, similar to what we see in the analysis of nearly linear simulations in Fig. . As a reminder, we discard an initial spin-up period after which the response is cycle-to-cycle repeatable and independent of the initial state.

From Fig. , we see that the deviations from linear theory can be reproduced, though their expression depends on forcing period. In particular, the long forcing period (Fig. a) yields a clear prograde–retrograde asymmetry, with a clear bend in the scatter, while the short forcing period (Fig. b) shows mainly a shift of the data upwards relative to the 1:1 line, i.e., a prograde shift. Note, however, that maximum circulation strength also differs considerably between the two experiments, a point we will return to in Sect. . In brief, only the long-period forcing generates a response whose amplitude is strong enough to arrest topographic Rossby waves matching the corrugation wavelength, greatly enhancing the nonlinear response. Thus, while both forcing periods exhibit the same underlying nonlinear mechanism, it becomes much more pronounced when arrested waves resonate with the bathymetry. Further, both forcing periods show signs of hysteresis, but it is most pronounced in the long forcing case. In the long forcing case, data separate into two distinct branches, one near linear theory and one displaced in the prograde direction. In contrast, the short forcing case exhibits only a weak loop around an uniformly shifted line. These differences point to damping timescales relative to forcing timescales playing an important role in shaping the nonlinear response.

To clarify how the hysteresis patterns in Fig.  arise, we turn to time-series of circulation, shown in Fig. . For convenience of presentation, we shift t=0 to the beginning of a retrograde forcing period. Throughout, we refer to intervals during which the linear estimate predicts negative circulation as the retrograde phase, and to intervals with predicted positive values as the prograde phase. For the long forcing period (Fig. a), the circulation initially follows the linear prediction as the system enters the retrograde phase. Deviation from linear theory then sets in, first gradually, then it abruptly becomes more pronounced (approximately 25 d into retrograde forcing), leading to a strong departure from linear estimates. At this point, the system enters a regime where retrograde circulation no longer strengthens, despite increasing forcing. We will refer to this behavior as “saturation”. Animations of the flow and relative vorticity fields show that saturation coincides with the emergence of transient eddies (see video supplement [ref to 10.5446/74040, ], or snapshot of the relative vorticity field in Fig. ). The animated relative vorticity field exhibits a pattern that is periodic in the along-slope direction at the corrugation wavelength, with eddies forming on scales smaller than the topographic wavelength. This pattern is also apparent in the snapshot (Fig. ). As the system moves into the prograde phase in Fig. a, the circulation gradually converges back toward the linear estimate. This behavior is consistent with the forcing period being long compared with the frictional damping time (about 12 d at mid-slope), meaning that the system has time to approach distinct prograde and retrograde states as the forcing is slowly evolving.

For the short forcing period, by contrast, the forcing changes on a timescale comparable to the damping time. The circulation therefore cannot fully adjust to any given forcing level before the forcing reverses. The flow behaves more in a low-pass sense, which explains the more compact hysteresis loop in Fig. b. In this case, the prograde offset varies only weakly over the cycle, because the short forcing period smooths the response and suppresses the development of stronger phase-dependent asymmetries.

The nonlinear contribution relevant for depth-following circulation is a flux of relative vorticity across the isobath (Eq. ). In Fig. , the dashed lines show estimates where diagnosed relative vorticity fluxes have been included as a forcing term (Eq. ). For both forcing periods, including relative vorticity fluxes in the estimates almost entirely removes the discrepancy with the simulations. This agreement indicates that the observed nonlinear deviations from linear theory can indeed be described as the consequence of a single nonlinear mechanism, namely a cross-contour flux of relative vorticity. This supports that Eq. () describes the full response of the system.

As shown in Sect. , if QG scaling applies, both the depth-following and constant-y frameworks may be interpreted in terms of fluxes of quasi-geostrophic potential vorticity (QGPV) across the slope. Even though depth variations in our model are large (on the order of 300 m mid-slope for the base case), we test this interpretation by diagnosing the cross-slope QGPV flux in both frameworks and comparing its spatial structure with that of the integrated bottom frictional stress. According to theory Eq. (), the time-averaged QGPV flux and bottom frictional stress should balance. Figure  shows these two terms for the long (a) and short (b) forcing periods, with results from the depth-following and constant-y frameworks shown together for direct comparison. For both constant-y and constant-H contours (isobaths), we calculate and plot the flux of actual QGPV, as defined in Eq. (). As described in Sect. , in the depth-following framework the QGPV flux is entirely accomplished by the relative vorticity flux, that is, the flux plotted earlier in (a) and (b) in Fig. . For the constant-y framework we also plot the sum of the momentum flux convergence and the topographic form stress (MFC+TFS). Under QG scaling, their sum should constitute a net flux of QGPV across the slope. There are discrepancies between the actual QGPV flux and MFC+TFS, primarily confined to the steepest part of the slope, but overall the agreement suggests that the PV-flux interpretation is a useful one.

In both forcing regimes and for both integration paths, the time-mean QGPV flux is directed offshore, down the large-scale PV gradient. It also balances the bottom frictional stress associated with the time-mean residual flow along the slope. The cross-slope structure of the diagnosed QGPV flux shows strong agreement between the depth-following and constant-y frameworks, with comparable magnitudes and spatial organization across the slope. This agreement suggests that water columns approximately maintain their PV as they are transported from a given constant-y contour to the equivalent isobath, which undulates around that same y position. This, in turn, implies that the integrated thickness flux through the constant-y contour is converted, via stretching and squeezing, into a relative vorticity flux, adding to the total relative vorticity flux through the isobath. In other words, some fraction of the relative vorticity flux through an isobath carries the imprint of the thickness flux that passes through the associated constant-y contour. This thickness flux through the corrugations in the slope – which is a linear process – is thus essential to obtaining a net down-gradient PV flux across either integration path. However, the resulting prograde bias is not purely a linear effect, since the asymmetry between prograde and retrograde phases still depends on nonlinear flow dynamics, as discussed in Sect. .

Small differences remain, which are likely related to geometric factors: a depth-following contour is generally curved and longer than a straight y-transect, leading to differences in bottom frictional stresses – and associated balancing QGPV fluxes. Nevertheless, the interpretation remains that both frameworks, integrated along constant-H and constant-y, represent a systematic down-gradient QGPV flux that maintains the residual prograde slope circulation.

So far, all simulations considered in this study have included topographic corrugations. The above results suggest that the thickness flux through these corrugations, or equivalently, the topographic form stress, is an essential ingredient in the integrated momentum budgets and in the establishment of time-mean currents. A direct theoretical implication of the absence of such along-slope depth variations, and thus of topographic form stress, is that oscillatory wind forcing with zero time mean cannot produce a domain-integrated residual flow, that is, a non-zero net mass transport through our channel (Eq. ). To verify this theoretical constraint, we now examine a set of simulations in which the slope is smooth. These simulations share the same large-scale geometry and mean slope as the corrugated cases but lack depth variations along constant-y contours. In this configuration, analyses integrated along constant-y and along depth-following contours coincide.

We first examine the response to the standard along-slope wind forcing (Eq. ), shown in Fig. a. In this case, the surface stress is spatially uniform and purely along-slope. We find that the response is well described by linear theory: the flow oscillates in time according to linear analytical estimates (Eq. ) and produces no residual mean flow at any point along the slope. This is illustrated by the zero time-mean of all momentum terms in panel (iii).

The purely linear response for every isobath may reflect the fact that the applied stress is everywhere parallel to the slope and does not induce any cross-slope motion. In the presence of topographic corrugations, however, the same along-slope forcing is observed to generate cross-slope flow. To probe whether cross-slope motion alone alters the system’s response, we therefore perform an additional experiment in which a cross-slope flow component is forced explicitly by the wind, while keeping the topography smooth.

This modified surface stress, denoted τcs, is constructed to be tangential to the bathymetric contours of the corrugated slope configuration used in the other experiments, while retaining the standard along-slope stress amplitude. It takes the form 26τcs=τsxi+τsx∂xH∂yHj, with H and τsx as defined in Eqs. () and (). The curl of this wind stress varies on horizontal scales comparable to those of the bathymetric corrugations in the corrugated-slope experiments, representing an Ekman-driven cross-slope motion analogous to that induced by flow–topography interactions. Importantly, as in the above, the time mean is zero.

Figure b shows the resulting momentum balances for this experiment. Here, the circulation response takes the form of a persistent time-mean alternating jet structure over the slope (panel iii), with a prograde jet onshore and a retrograde jet offshore. This pattern is maintained by local convergence and divergence of momentum fluxes, which drive departures from the linear solution at individual depths. The sign of these fluxes alternates across the slope, such that their contributions cancel in the domain integral. In this sense, the system permits local momentum redistribution and the formation of along-slope jets, but without corrugations there is no mechanism by which the oscillatory wind forcing can set up a time-mean net mass transport through the channel.

This flow structure can also be interpreted through the lens of potential vorticity rearrangement. Cross-slope motions driven by the time-variable wind induce local transport across the background PV gradient, leading to a redistribution of PV and the emergence of alternating along-slope jets. This pattern resembles jets associated with PV staircase formation , albeit without an explicit eddy field. Here, it is the wind forcing itself, rather than turbulent eddies, that mediates the cross-PV gradient transport.

In earlier sections, we found that saturation strongly influences the retrograde response to long forcing periods in the presence of corrugations, whereas saturation is absent for short-period forcing. Earlier studies have shown that strong topographic form stress occurs when topographic Rossby waves with wavelengths comparable to the bathymetric corrugations are arrested by the mean flow . Arrest corresponds to U=-ck in Eq. (). We thus hypothesize that saturation, which is associated with strong topographic form stress (Fig. c), is connected to such arrest. In this section, we first focus on the response to the long forcing period and investigate the mechanism responsible for the observed saturation, before we return to why saturation is absent in the short period runs.