## 1. Introduction

Improvements in geoid determination enabled Woodworth et al. (2012), Higginson et al. (2015), and Lin et al. (2015) to demonstrate that sea level (SL) along coastal boundaries can differ markedly from the adjacent open ocean (interior). In particular, Higginson et al. (2015) showed that between the Florida Keys and Halifax, Canada, the approximately 1-m northward drop in SL across the Gulf Stream is missing at the coast, replaced by a smaller 20-cm drop some 10° farther south.

While SL (specifically ocean surface dynamic topography) gradients in the deep ocean are approximately in geostrophic balance, the zero normal-flow condition imposed by continents implies this balance does not describe coastal alongshore SL gradients. The threat of rising global SL has motivated the investigation of the drivers of coastal SL globally and is of particular interest along the North American east coast owing to the identification of a SL rise “hot spot” (Sallenger et al. 2012). Advancing our understanding of the basic processes relating coastal to interior SL, particularly where strong western boundary currents and complex bathymetry are present, is fundamental to building confidence in the predictions of numerical models.

For basins modeled with flat bottoms and vertical sidewalls, Stommel (1948) showed that a solution for the circulation could be found by balancing the vorticity added by wind stress with bottom friction. This approach resulted in boundary layers running north–south, which Munk (1950) further developed by replacing bottom friction with lateral friction, a more realistic assumption for flows that do not reach the bottom. Charney (1955) also used horizontal momentum advection to balance vorticity resulting in an additional western inertial boundary layer.

More recently, Minobe et al. (2017) addressed western boundary (coastal) SL for the Munk- or Stommel-type solution with vertical sidewalls and found an equatorward displacement and attenuation in coastal SL relative to the interior SL. Their relationship depends on the meridional integral of mass anomalies in the ocean interior, thus building on the idea that mass input into the boundary layer is transmitted equatorward (Godfrey 1975; Marshall and Johnson 2013). This relationship allows coastal SL at a chosen latitude to be given by contributions of coastal SL at some poleward latitude and the interior SL between the two latitudes. Notably, their relationship also describes coastal SL as being independent of the details of friction. A missing element, however, in this special vertical sidewall case, is the influence of continental shelves and slopes, potentially important given the variable bathymetry along the North American east coast (Pratt 1968).

Csanady (1978) looked at the effect of a linearly sloping bathymetry in a steady *f*-plane barotropic model and showed that alongshore pressure gradients prescribed at the edge of the shelf resulted in the same gradient being present at the coast, beyond some initial insulated region. Wang (1982) and Huthnance (1987) later showed that including a continental slope increased the insulation to thousands of kilometers in scale, and in a more complex model employing stratification Huthnance (2004) found results similar to the barotropic case. For the case of modeling large-scale SL along western boundaries, however, allowing the Coriolis parameter to change and maintaining consistency when applying the boundary condition with the deep ocean are, as will be seen, crucial. This added complexity has contributed to limiting the study of SL in western boundary regions over sloping bathymetry. One notable result comes from Salmon (1998) in his study of linear ocean circulation where sloping bathymetry was described as “advecting” pressure along isobaths and the *β* effect (due to variable Coriolis parameter) “advecting” pressure westward. In referring to “advection” Salmon extended an advection–diffusion analogy that had first been made by Welander (1968), and later Becker and Salmon (1997), regarding the mass transport streamfunction. Although Salmon’s model included both bathymetry and stratification, the assumption of linearity in the equation for density advection resulted in a somewhat artificial role for diapycnal diffusivity to balance any vertical velocity.

The inclusion of bathymetry (in this paper we intend bathymetry to mean sloping bottom topography) in these models resulted in solutions depending explicitly on the bottom friction parameter. As we will show, a consequence of using a western boundary vertical sidewall is that the coastal SL solution is independent of the details of friction because geostrophic flow is always distributed over the same depth range. Indeed, Minobe et al. (2017) list the effects of bathymetry, alterations to the vertical mode structure, and nonlinear advection as areas to explore further. In this paper we study the first two points.

We consider SL along the east coast of North America relative to the adjacent interior SL that originates from a wind-driven double gyre corresponding to a SL depression from the subpolar gyre and elevation from the subtropical gyre. Our focus is the effect of bathymetry on coastal SL for a specified ocean interior SL; we are therefore excluding the more local response to near-coastal wind stress. See, for example, Hong et al. (2000), Thompson and Mitchum (2014), Frederikse et al. (2017), and Valle-Levinson et al. (2017) for discussions on the importance of interior ocean wind stress to coastal SL. Although the North Atlantic region provided our motivation, this idealized study would apply equally well to other ocean basins with western boundary currents.

The remainder of this paper is as follows. In section 2 the result of Minobe et al. (2017) is derived from an angular momentum argument to explicitly highlight the importance of bathymetry on coastal SL. In section 3 we formulate a model that includes bathymetry for a single-layer interior and an interior with a decoupled upper layer. In section 4 the effects of the continental shelf and slope on SL are presented, and in section 5 this is extended to a simple stratified case. Section 6 summarizes and highlights implications.

## 2. Vertical sidewall special case

*x*and

*y*are the zonal and meridional coordinates, respectively, subscripts

*x*and

*y*denote partial differentiation,

*f*is the Coriolis parameter, and

*y*, multiplied by

*f*).

This can be interpreted as the interior SL at each latitude contributing to a step up in coastal SL toward the south, at that latitude, which then decays to zero at the equator in a manner proportional to the sine of latitude. The effect of this at the coast is to smooth and reduce the interior signal and shift it toward the equator.

In this special case, the solution can be found without specifying the form of the friction in detail. In fact, all that is needed for the derivation are the assumptions of no normal flow at the western boundary and that friction acts in a western boundary layer. A simpler argument can be made that leads to the same conclusion.

*H*and an applied zonal wind stress

The simpler determination of the western boundary SL, from Eq. (2), illustrates straightforwardly the critical nature of the assumption of vertical sidewalls. The net force on the western boundary is determined by the combination of the SL *H* over which the resulting pressure anomaly acts. With a bathymetric slope at the boundary, this will come to depend crucially on where currents flow. A western boundary current flowing higher up the continental slope will produce a larger SL signal for the same total transport, as the associated boundary pressure signal becomes concentrated in a shallower region, reducing the effective value of *H*. Recirculating currents on the slope can complicate things even further. Note that although we have found a simpler way to derive the Minobe et al. (2017) result, this relies on certain assumptions about interior ocean dynamics, for example, that there is no interaction with bathymetry within the basin to disturb Sverdrup balance and that there is no outflow along the northern boundary, which would imply a nonzero zonal integral of meridional velocity in Eq. (2). By relating coastal SL to nearby interior SL, Minobe et al. (2017) have sidestepped these requirements and produced a valuable result, albeit restricted to the case of a vertical sidewall at the west.

For this reason, it is our aim in this paper to investigate how the presence of a continental shelf and slope alters the relationship between interior ocean dynamics and boundary SL.

## 3. Model formulation

We begin by introducing the conceptual model. Consider the western boundary region and the interior basin as two separate domains where in the interior, between

A Northern Hemisphere coordinate system is oriented with *x* in the zonal and *y* in the meridional, as shown by the schematic in Fig. 1. Note that though *y* increases in the poleward direction, a reference latitude, *y*).

For orientation and as an introduction to the general character of the solutions we will find, an example is shown in Fig. 2. Figure 2a shows SL contours over the combined interior and western domains for a purely zonal wind stress over the interior (producing a double gyre circulation). Figure 2b shows only the SL contours for the western domain where bathymetry is present and where there is no wind stress.

*ρ*, velocity

*g*, inverse barometer corrected SL

*η*, horizontal differential operator

*s*for surface stress and

*b*for bottom friction.

*f*and then taking the projection of the curl in the

*z*coordinate

*r*the friction parameter and

Note that while we assume that the western coastline runs meridionally, the results do generalize to the case where the coastline is at an angle *ϕ* to the meridional. As shown in the appendix of Minobe et al. (2017), a transformation to bathymetry following coordinates [i.e., *f*, so we would expect the main result of such a change to be similar to a latitude-dependent friction coefficient.

Equation (10) requires boundary conditions at the coast *n*. If the vertical sidewall is replaced with sloping bathymetry of cross-shore width *H* the maximum ocean depth.

Consider now taking *H*, which makes contact with the bathymetry at a distance

We now have two different modeling scenarios. In the single-layer case (Fig. 3a), the boundary condition is *υ* is zero and in geostrophic balance),

^{1}Equation (9) can then be written as an analog advection–diffusion equation:with the “diffusion coefficient” defined as

*η*to be “advected” tangentially to the streamlines of

*f*(decreasing

*f*allows SL contours to cross isobaths) indicates why constant

*f*-plane models would suggest greater bathymetric insulation between coast and interior; that is, constant

*f*does not allow the effectiveness of bathymetry to steer SL to change with latitude. Note that, although we are using a beta plane in order to simplify the geometry as far as possible, Eq. (9) and the advection–diffusion analogy hold exactly on a sphere, so there will be no qualitative difference in the more general case, although the insulating effect of topography will increase at higher latitudes as

*β*reduces.

In the context of thermal fluids, a nondimensional Péclet number (Pe) is often defined as a measure of the relative importance of advection and diffusion with respect to unidirectional thermal energy transport; Pe greater than unity implies advection is dominant and Pe less than unity that diffusion is dominant. In our analogy we have defined an analogous “Péclet number”

In terms of coastal SL, the purely “advective” part of SL transport is invariant to scale (following *r* and/or decreasing the scales of the bathymetry (*H* and

It is important to note that the parameter *y*.

As will be demonstrated in the next section, the theory suggests two independent controls on the contribution of interior SL to coastal SL: first through the parameter

## 4. Coastal SL parameter study

In this section we present solutions of the advection–diffusion Eqs. (13)–(16). Section 4a looks at the effect of the “Péclet number”

*Y*is the latitudinal extent of the domain.

A piecewise linear function is used for

Bathymetry is defined by a piecewise linear function in *x* on the basis that it gives the simplest yet most illustrative means of studying the effects of including a continental shelf and slope. In Figs. 3a and 3b we define two extra parameters: depth at the shelf break *S*. We take these parameters as nondimensional (*H* and

Before looking at the dependence of SL on ^{−1}, and some value of the friction parameter *r*, which can be considered as a linear approximation of quadratic friction (Gill 1982). Two values for *r* used in the literature, ^{−1} (Chapman and Brink 1987; Xu and Oey 2011) and ^{−1} (Csanady 1978; Huthnance 2004), give an illustrative parameter value for

Equations (13) to (16) will now be solved using a Crank–Nicholson finite-difference scheme with nondimensional resolution

### a. Sea level dependence on —Single layer

In this subsection we use the Stommel-type model (Fig. 3a), where *H* is the depth scale of the ocean. We take

Figure 5 gives SL in the western domain for three values of

From our advection–diffusion analogy, Fig. 5d (

Focusing on coastal SL *P*_{a} = 0.1, 1, 10, and 100. The coastal SL in each case can be described as a smoothed version of the interior SL with an equatorward displacement and an attenuation that in general increases with displacement; both increase as ^{−1}). Increasing the friction parameter, and/or decreasing the scale of the combined shelf and slope, increases the penetration of SL to the coast.

The displaced and attenuated SL depression shown in Fig. 6a supports the result presented by Higginson et al. (2015) where the interior ocean SL tilt (the transition from SL depression to elevation where the Gulf Stream heads offshore) is observed at the coast displaced equatorward by 10° of latitude and attenuated from 1 m to 20 cm. The result here suggests that equatorward displacement of the tilt would be reduced in the following circumstances: 1) the combined width of the shelf and slope are reduced, 2) the depth to the foot of the slope is reduced, and 3) bottom drag is increased. The same is implied for the magnitude of the tilt. Note that while Higginson et al. (2015) do not comment on overall bathymetric scale, they do speculate that the width of the continental shelf [i.e., the definition of

An important result can be demonstrated by looking at the limit

### b. Sea level dependence on —Upper layer ( )

In this subsection we model an upper layer of the ocean (Fig. 3b) where *H* is the scale for the thickness of the upper layer.

The general behavior of SL in this case is qualitatively similar to the single-layer case, and the “advection–diffusion” analysis of the previous subsection holds. There is, however, a distinct quantitative difference in coastal SL. In Fig. 6b, the solid lines show coastal SL for *P*_{a} = 0.1, 1, 10, and 100 (this is the upper-layer counterpart to the single-layer version; Fig. 6a), and it is clear that displacement and attenuation of the interior SL is reduced. This is particularly noticeable for

This result suggests that, consistent with the results of Csanady (1978), it is possible to have greater penetration of interior ocean SL than the vertical sidewall limit permits. However, there is a subtlety that is being missed in this case: the “interior” SL should be imposed on the ocean side of the boundary where bottom friction is zero, but in using Eq. (10) we are effectively imposing a value on the slope side of that boundary.

*R*to be continuous, constant over the shelf and slope (between

The extent to which the frictional boundary layer extends offshore now depends on how

*h*is constant), and recalling that

The jump in SL is required to conserve depth-integrated mass flux. A discontinuity in bottom stress implies a discontinuity in offshore Ekman flux, which therefore implies a discontinuity in the onshore geostrophic flow, and hence a jump in SL. This is also a problem with section 10 of Csanady (1978). In that paper the coastal influence of a linear meridional SL slope is considered with the conclusion that the entire amplitude of the slope penetrates to the coast. There is, however, no way to connect this solution to a frictionless ocean interior, without invoking a step in sea level.

The upper-layer model appears to allow greater penetration of the interior SL signal because it is effectively using a larger-amplitude interior SL signal. In fact the upper- and single-layer models are the same, except that the upper-layer model implicitly uses a larger-amplitude interior SL. To demonstrate this point, the dashed lines in Fig. 6b show the coastal SL for the case with a vertical sidewall when the equivalent interior SL, calculated from Eq. (25) or (26), is used. The dashed curves show that the vertical sidewall solutions remain the limit of penetration as in Fig. 6a.

### c. Coastal SL and bathymetric configuration

In reality continental shelves and slopes have varied proportions (configurations), and so we look now at the dependence of SL on

Changing the relative proportions of the shelf and slope requires the location of the shelf break to change without changing the combined depth and width of the shelf and slope. This simply means keeping *S* to vary between zero and one. For example, by increasing

So far we have looked at the penetration of interior SL at the coast for specific values of *S*, and

In the following we focus on a single reference point of the coastal SL signal to investigate attenuation and displacement. For this we choose the coastal SL minimum and define it as

In Figs. 7c and 7d we plot attenuation and displacement of *H*) and *S* held constant. We use *S* (the shelf width relative to the combined width of shelf and slope

Figures 7c and 7d show that displacement and attenuation are maximized in the approximate region

Figures 7e and 7f show that for *S* increases, that is, as the shelf becomes wider. For *S* also decrease attenuation and displacement.

As a whole, the results of Fig. 7 show that penetration of interior SL to the coast increases rapidly (nonlinearly) as

The strong dependence of the solution on geometry and scale raises the question of the effect of model resolution on coastal SL; for example, a 1° ocean model has perhaps only one or two grid points on the combined shelf and slope. Assuming, for example,

It is clear that the solutions do depend on the geometry of the shelf and slope, as well as the overall scales and the friction parameter; in the next section we extend the model by considering a 1.5-layer interior. The following analysis will use dimensional quantities.

## 5. Dimensional model with 1.5 layers

It is more realistic to assume background stratification will alter the vertical mode structure and change how the flow interacts with bathymetry. In this section we create a simple stratified model by allowing the upper-layer depth along the interior boundary to be nonuniform; that is,

^{−3}and apply a zonal wind stress that varies meridionally:where

^{−2}) is the amplitude (see Fig. 8 for the wind stress profile). The interior domain is of width 4500 km with constant top-layer depth at the eastern boundary of

We then take SL along the westernmost edge of the reduced-gravity interior model and use it as the interior boundary condition for the western domain

For the western domain, we represent an upper-layer thickness that changes with latitude by allowing the depth *h* in the model developed in section 3 to vary alongshore; that is, *h* is defined by projecting the upper-layer thickness at the interior boundary, which changes in *y*, up to the slope. The effect of this change on the theory developed in section 3 is that the path along which SL is “advected” changes to reflect the modified

We consider two different cases. In the first case we allow only a slight latitudinal variability in the thermocline thickness. This relates to weak interior gyres (solid lines in Fig. 8). In the second case we allow a larger latitudinal variation in the upper-layer thickness. This relates to stronger interior gyres (dashed lines in Fig. 8). In the latter case, we note that because of the larger latitudinal variation of *h*, there is a reversal in the direction of *U*, the zonal “advecting” velocity, in the northern part of the subpolar gyre. This results in a somewhat artificial frictional boundary layer extending to the northeastern corner (not shown).

Figure 9a shows the interior boundary SL, the new coastal SL, and the vertical wall solution for the weak interior gyre case. We show in addition the corresponding solution for the single-layer model with

Vertical mode interaction allows the thickness of the upper layer to be redistributed such that it decreases over a poleward portion of the interior. This decrease enables the interior SL over this poleward portion to penetrate farther toward the coast before making contact with the bathymetry; this can increase penetration of the subpolar SL depression. On the other hand, the upper layer thickens toward the equator suggesting a decrease in penetration of the subtropical SL elevation. In effect our “Péclet number” is changing with latitude, smallest where the upper-layer thickness is thinnest. The reversal of the characteristic direction in the strong gyre case means the validity of this solution is questionable. This raises questions about SL penetration when a linear approximation may not be appropriate for modeling thermocline depth. We leave this investigation for future studies.

## 6. Discussion and conclusions

We have shown that the assumption of a vertical sidewall at the coast within a western boundary allows coastal SL to be independent of layer thickness and the friction parameter and that the vertical sidewall solution is a special limit case for the more general problem that includes sloping bathymetry.

A *β*-plane theory has been developed for a general bathymetry that is uniform alongshore showing that interior SL transmits to the coast analogously to the steady “advection–diffusion” of a thermal fluid. For an interior SL originating from a wind-driven double gyre, corresponding to a coastal SL depression from the subpolar gyre and elevation from the subtropical gyre, the theory demonstrates that ocean interior sea level can penetrate to the coast having been attenuated and displaced equatorward. The analogy describes SL as being “advected” along *β* effect steering (“advecting”) contours westward. For bathymetry that tends to zero at the coast and Coriolis parameter that vanishes at the equator, the interior SL does not register at the coast in the limit of no friction (though technically a friction stress is required at the singularity at the coastal equator point). The addition of alongshore friction, however, introduces cross-shore “diffusion” and allows SL contours to cross

A nondimensional “Péclet number” (

A distinction is drawn between a single-layer interior and an interior with a decoupled upper layer of uniform thickness that makes contact with the continental slope at a distance

Independently of the overall scales accounted for in

The results and analysis presented here have implications for our understanding of the drivers of coastal SL. Higginson et al. (2015) showed that the 1-m difference in interior SL across the location where the Gulf Stream moves into deep water is represented at the coast by an attenuated and equatorward displaced version. They noted that this was not explained by *f*-plane theoretical models, which suggest that oceanic SL features should not penetrate to the coast over the observed alongshore distance. The *β*-plane model developed here explains why a displaced and attenuated tilt in coastal SL should be expected and that, for example, an increased interior SL due to a weakened subpolar gyre (decreased tilt) would affect the coast.

Higginson et al. (2015) also suggested that the position of the coastal tilt might be explained by the narrow shelf at the Florida Straits. This study has shown that topography that is well approximated by a vertical wall (

The results and analysis presented here suggest that how bathymetry is configured and how finely it is resolved, in addition to the representation of bottom friction, are potentially quite important to ocean models focusing on SL in western boundaries. While the linear model used here has been intentionally simple, many additions can be made, notably the impact of including momentum advection and including time dependence to explore shorter-time-scale SL adjustments in a more sophisticated numerical model.

We thank the reviewers for helping to improve this manuscript with their suggestions. This work was supported by the Natural Environment Research Council (Anthony Wise: NE/L002469/1), (Chris W. Hughes: NE/K012789/1), and (Jeff A. Polton: NE/L003325/1).

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^{1}

Note added in proof: this “advection” velocity was described earlier by Tyler and Käse (2000).