## 1. Introduction

Poleward undercurrents—poleward subsurface flows over the upper continental slope in coastal-upwelling regimes—are a familiar feature of coastal circulation along eastern boundaries during the upwelling season (Neshyba et al. 1989). These subsurface currents are especially notable in that they flow against the direction of the prevailing winds that drive the seasonal upwelling circulation and its equatorward alongshore surface currents. While these flows can be ephemeral and intermittent in both space and time, compelling observational evidence of coherent poleward undercurrent flow over alongshore scales exceeding 15° of latitude has been presented, for example, by Pierce et al. (2000). Those observations yielded an alongshore mean flow exceeding 0.1 m s^{−1} in the 200–275-m depth range 20 km offshore of the shelf break, with mean alongshore transport 0.8 × 10^{6} m^{3} s^{−1} within the 125–325-m depth range, which may be taken as characteristic of the undercurrent.

There are many hypotheses but no broadly accepted theory for the generation of these flows; consequently, many questions regarding their essential dynamics remain open. A variety of potential mechanisms has been put forth, relying on elements as diverse as turbulent frictional boundary layers (Pedlosky 1974; McCreary 1981), coastal-trapped waves and alongshore variations in wind forcing (Yoshida 1980; Connolly et al. 2014), interaction of alongshore flows with alongshore variations in seafloor topography (Haidvogel and Brink 1986; Samelson and Allen 1987; Brink 2011), and eddy buoyancy fluxes (Cessi and Wolfe 2013). Many of these theories have been posed in highly idealized geometries, with coastal boundaries represented by vertical walls and continental slopes either greatly reduced or entirely absent, and none of them are widely recognized as providing a compelling rationalization of the general phenomenon.

Arguably one of the more successful theoretical constructions, in its ability consistently to generate undercurrent flows in the right locations with the right qualitative and quantitative characteristics, is that proposed by McCreary (1981) and subsequently extended by McCreary et al. (1987, 1992). In this conception, the undercurrent is viewed as a geostrophic current in a stratified interior-ocean domain in which the dominant dynamics are linear planetary wave propagation, subject perhaps also to the influence of turbulent frictional processes. Motion in the interior domain is forced by a coastal boundary condition, which in the inviscid limit is derived from the assumption that a no-normal-flow condition at the coast may be applied directly to the sum of the geostrophic and Ekman flow, with the wind stress effectively balanced everywhere—even at the coastal boundary itself—by offshore Ekman transport. Alternatively, this inviscid condition may be understood as a requirement that the alongshore wind stress at the coastal boundary be balanced by an alongshore pressure gradient, with the cross-shore velocity vanishing at the coast. Unfortunately, these assumptions regarding the coastal boundary condition are not consistent with the standard coastal upwelling momentum balance over the shelf, in which the acceleration of the near-surface alongshore flow takes up the alongshore stress as the coastal boundary is approached, the Ekman transport itself adjusts independently to a no-normal-flow condition at the coast, and no alongshore pressure gradient in response to alongshore-uniform wind stress ensues or is required (e.g., Allen 1973; Samelson 1997). Thus, despite the appealing undercurrent flows produced by this sequence of models, the underlying theory has not proved compelling.

In the present approach, the undercurrent is again viewed as a geostrophic current in a stratified interior-ocean domain, in which the dynamics are dominated by slow-time-scale, linear planetary wave propagation, and the interior motion is again forced by a coastal boundary condition that is related to the alongshore wind stress. However, consideration of the fast-time-scale processes by which the coastal ocean responds to alongshore wind forcing motivates the introduction of a different coastal-boundary condition than had been assumed in these previous works. This boundary condition—which does not involve or require the presence of local alongshore pressure gradients, and which, by construction, is generally consistent with the standard coastal upwelling momentum balance over the shelf—is found to drive an interior response that includes subsurface poleward flow over the upper continental slope under upwelling conditions, with location and amplitude that are generally consistent with observations of poleward undercurrents.

## 2. Model geometry and overview

*x*,

*y*,

*z*):

*x*is zonal and positive onshore,

*y*is meridional and positive poleward, and

*z*is vertical and positive upward. The model geometry (Fig. 1) is semi-infinite in

*x*, of effectively arbitrary extent in

*y*, and bounded above and below in

*z*by the sea surface and the seafloor at

*z*= 0 and

*z*= −

*H*(

*x*), respectively. The eastern coastal boundary is taken to be meridional, so the model geometry is independent of alongshore coordinate

*y*. An exponential profile (e.g., Buchwald and Adams 1968; Allen 1975; Samelson 1997) is used to represent the continental slope and shelf topography, with the seafloor depth

*H*(

*x*) increasing offshore to a maximum, constant value

*H*

_{0}= 4000 m:

*x*= 0 km, but at

*x*=

*x*

_{c}= 16 km, and the depth

*H*

_{b}=

*H*(

*x*= 0) is chosen so that the origin of the zonal coordinate is located at a depth characteristic of the outer continental shelf. In this study’s main computations,

*H*

_{b}=

*H*(

*x*= 0) = 150 m; the depth at the coastal boundary is therefore

*H*(

*x*

_{c}) ≈ 40 m, or roughly the depth of the offshore edge of the inner shelf. One computation uses a flat seafloor offshore of the outer continental shelf instead:

*H*

_{0}= 4000 m is used for the abyssal seafloor depth. Alongshore wind forcing, described in detail in section 3, is applied in the region 0 ≤

*y*≤

*y*

_{0}. A nominal latitude of 40°N is used to set the values of the Coriolis parameter

*f*and its meridional gradient

*β*=

*df*/

*dy*, so that

*f*= 9.4 × 10

^{−5}s

^{−1}and

*β*= 1.75 × 10

^{−11}m

^{−1}s

^{−1}.

Two different time scales of the ocean response to the applied wind forcing are relevant. The fast time scale is associated with the coastal-trapped response to wind stress forcing, while the slow time scale is associated with planetary wave propagation into the interior ocean. Model solutions on the fast and slow time scales are computed separately, and the zonal extents of the domains for these two sets of calculations differ slightly. The domain for the fast time scale (*x*_{0} ≤ *x* ≤ *x*_{c}) is restricted to the continental slope and shelf, with coastal-boundary seafloor depth *H*(*x*_{c}) ≈ 40 m (Figs. 1a,c). The domain for the slow time scale (−∞ ≤ *x* ≤ 0) extends arbitrarily far offshore, but its coastal boundary is placed at the outer continental shelf *x* = 0, where *H* = *H*_{b} = 150 m and a boundary condition derived from the fast-time-scale response is applied (Figs. 1b,d). The fast response is coastal trapped by virtue of the physical dynamics, but the motivation for placing the coastal boundary for the slow-time-scale domain at the outer shelf is not obvious a priori; this is discussed further in sections 3 and 9. For simplicity and analytical accessibility, both sets of motions are treated here in their respective long-wave approximations, for which both sets of waves are nondispersive. The flat-seafloor topography [(2)] is used only for the first slow-time-scale solution (section 5); the other slow-time-scale solutions (section 6) and all of the fast-time-scale solutions (section 3) use (1).

The two models and calculations are linked by the geostrophic sea surface height anomaly along the outer continental shelf poleward of the forcing region, which is computed—in an illustrative manner and under a restrictive assumption—from the fast-time-scale model and then imposed as a boundary condition on the slow-time-scale model. The analysis of the fast, coastal-trapped wave response is limited to a brief review based on classical solutions for the simplified case of barotropic continental shelf waves. The purpose of this exercise is twofold: to illustrate, concretely, some basic aspects of the fast, coastal-trapped response; and to provide an explicit calculation of the effective boundary condition that is applied to the planetary wave model. The fast time response is computed under the approximation that the Coriolis parameter *f* is constant.

*β*of the Coriolis parameter. The associated interior motions are represented in terms of a geostrophic streamfunction

*ψ*. The effective coastal-boundary condition at

*x*= 0 must therefore be expressed in terms of

*ψ*. A basic hypothesis posed here is that the relevant physical quantity that determines this boundary condition is the low-frequency, geostrophically balanced sea surface height disturbance Δ

*ζ*

_{b}over the outer shelf, which is determined by the fast time response to the wind forcing. The sea surface height condition may be converted to a barotropic condition on the streamfunction through the hydrostatic and geostrophic balances,

*g*is the acceleration of gravity. A minor modification will subsequently be made to this condition in order to make it compatible with the seafloor boundary condition for the interior dynamics. In anticipation of the inferred alongshore-uniform structure of the fast response poleward of the forcing region, the boundary condition (3) on

*ψ*is taken to be independent of the meridional coordinate

*y*.

## 3. Fast, coastal-trapped response

The defining assumption regarding the boundary condition (3), which will be seen subsequently to drive the interior circulation, including the subsurface poleward flow, is that the sea surface over the outer continental shelf will be relatively low during sustained equatorward—upwelling-favorable—winds, and relatively high during sustained poleward—downwelling-favorable—winds. This assumption is generally consistent with the well-known tendency of alongshore winds to lower coastal sea level under upwelling conditions and to raise coastal sea level during downwelling conditions. A recent example in the context of the northern California Current region is provided by (Durski et al. 2015), who compare model- and altimeter-based sea surface height anomalies averaged between the coastline and the 200-m isobath over several years; they find mean anomalies reaching −0.15 m in the upwelling season and +0.15 m in the downwelling season (Durski et al. 2015, their Fig. 5).

The linear, fast-time-scale response to alongshore coastal winds is mediated by coastal-trapped waves (e.g., Wang and Mooers 1976; Huthnance 1978; Halliwell and Allen 1984; Battisti and Hickey 1984). A concrete illustration of the fast-time-scale response that leads to the lowered sea surface height under upwelling conditions can be more easily obtained for the simplified case of continental shelf waves, the equivalent of coastal-trapped waves for a homogeneous fluid (Buchwald and Adams 1968; Gill and Schumann 1974; Gill and Clarke 1974; Allen 1975, 1976). In the long-wave approximation, both continental-shelf and coastal-trapped wave dynamics reduce to a set of first-order wave equations for the poleward propagation of the cross-shore eigenmodes, with the gravest modes typically having wave speeds on the order of 2–4 m s^{−1} at midlatitudes.

Consider the response to an alongshore wind stress *τ*^{y} = −*τ*_{0}, independent of cross-shore coordinate *x*, that is applied over the finite alongshore interval 0 < *y* < *y*_{0} = 250 km (Fig. 1a) abruptly at *t* = 0 and then maintained through 7 days (0 < *t* < 7 days). After an initial adjustment, which occurs on a time scale *t* ≈ 1/*f* but is taken to occur instantaneously at *t* = 0, an offshore (for *τ*_{0} > 0) surface Ekman transport balance is established, which absorbs the momentum continuously injected by the wind. In response to the associated coastal Ekman divergence, a long shallow-water gravity (sea surface height) wave propagates rapidly offshore, inducing an onshore flow behind the wave that compensates the Ekman divergence.

The continental shelf wave response is the geostrophic adjustment process associated with this induced, ageostrophic, barotropic onshore flow. The corresponding solution can be computed following the elegant outline given in section 4 of Allen (1976), with *τ*_{0} ≈ 0.1 N m^{−2} chosen so that the associated offshore Ekman transport *U*_{E} = *τ*^{y}/(*ρ*_{0}*f*) = −1 m^{2} s^{−1}, with reference density *ρ*_{0} = 1025 kg m^{−3}. Solutions are obtained explicitly in the slope–shelf region *x*_{0} ≤ *x* ≤ *x*_{c}; they are implicitly completed by an offshore exponential decay in the flat-seafloor region *x* < *x*_{0}, but this offshore decay has only a passive, dynamical role and is otherwise neglected. The full solution is obtained in terms of an infinite sum of cross-shore modes of the continental shelf wave transport streamfunction Ψ(*x*, *y*, *t*), each of which subsequently propagates poleward at its respective modal phase speed, while the onshore flow continues to be forced at the coastal boundary by the Ekman transport associated with the applied stress.

After the onset of forcing and initial adjustment at *t* = 0, the induced barotropic velocity within the latitude band of the forcing 0 < *y* < *y*_{0} is directly onshore. However, none of this cross-shore transport is geostrophic. The first-mode shelf wave disturbance is excited by the vortex stretching associated with this cross-shore transport and propagates poleward at the first-mode phase speed. Its leading edge follows the space–time (*y*–*t*) characteristic that leaves *y* = *y*_{0}, the poleward boundary of the forcing region, at *t* = 0. The region of first-mode cross-shore transport travels poleward with this wave front, and it is confined on the equatorward side by the first-mode characteristic that leaves the equatorward edge of the forcing region *y* = 0 at *t* = 0. When the latter characteristic crosses *y* = *y*_{0}, no first-mode cross-shore transport remains in the forcing region. As it continues poleward, it leaves behind a region of geostrophic transport. This geostrophic transport is purely alongshore, with no cross-shore component and with a cross-shore structure that is obtained by removing the first-mode projection from the initial cross-shore-uniform structure, leaving the sum of all higher-mode projections (Figs. 2a,b).

Continental-shelf wave response to uniform southward wind forcing of amplitude 0.1 N m^{−2}, imposed over the region 0 < *y* < 250 km during 0 < *t* < 7 days, vs cross-shore distance *x* and alongshore distance *y* at time *t* = 8 days. (a) Full solution for transport streamfunction Ψ (10^{6} m^{3} s^{−1}), allowing only first-mode propagation. (b) Geostrophic portion of the solution from (a). (c) Sea surface height anomaly (cm) for the geostrophic portion of the solution from (a), computed from (4). The outer-shelf position *x* = 0 km (solid black line) defined by the 150-m isobath, used as the coastal boundary for the interior domain for the slow-time-scale, planetary wave model, and the poleward boundary *y* = *y*_{0} = 250 km of the forcing region (dashed black) are also shown.

Citation: Journal of Physical Oceanography 47, 12; 10.1175/JPO-D-17-0077.1

Continental-shelf wave response to uniform southward wind forcing of amplitude 0.1 N m^{−2}, imposed over the region 0 < *y* < 250 km during 0 < *t* < 7 days, vs cross-shore distance *x* and alongshore distance *y* at time *t* = 8 days. (a) Full solution for transport streamfunction Ψ (10^{6} m^{3} s^{−1}), allowing only first-mode propagation. (b) Geostrophic portion of the solution from (a). (c) Sea surface height anomaly (cm) for the geostrophic portion of the solution from (a), computed from (4). The outer-shelf position *x* = 0 km (solid black line) defined by the 150-m isobath, used as the coastal boundary for the interior domain for the slow-time-scale, planetary wave model, and the poleward boundary *y* = *y*_{0} = 250 km of the forcing region (dashed black) are also shown.

Citation: Journal of Physical Oceanography 47, 12; 10.1175/JPO-D-17-0077.1

Continental-shelf wave response to uniform southward wind forcing of amplitude 0.1 N m^{−2}, imposed over the region 0 < *y* < 250 km during 0 < *t* < 7 days, vs cross-shore distance *x* and alongshore distance *y* at time *t* = 8 days. (a) Full solution for transport streamfunction Ψ (10^{6} m^{3} s^{−1}), allowing only first-mode propagation. (b) Geostrophic portion of the solution from (a). (c) Sea surface height anomaly (cm) for the geostrophic portion of the solution from (a), computed from (4). The outer-shelf position *x* = 0 km (solid black line) defined by the 150-m isobath, used as the coastal boundary for the interior domain for the slow-time-scale, planetary wave model, and the poleward boundary *y* = *y*_{0} = 250 km of the forcing region (dashed black) are also shown.

Citation: Journal of Physical Oceanography 47, 12; 10.1175/JPO-D-17-0077.1

As the second mode subsequently propagates poleward from the forcing region, it will cause a contraction of the geostrophic flow toward the coastal boundary. In the long time limit, when all cross-shore modes have propagated past the region of interest, all of the fast-time-scale geostrophic flow ultimately collapses against the coastal boundary, and no sea surface height disturbance remains over the outer shelf. This collapse occurs, essentially, because there is no way to balance vortex compression from onshore flow in the inviscid steady state. Presumably, however, a combination of nonlinear, frictional, and other effects prevent this collapse from occurring in the coastal ocean. The assumption implicit in the boundary condition (3) is that geostrophically balanced, approximately alongshore-uniform sea surface height disturbances Δ*ζ*_{b}, induced by the response to large-scale alongshore wind forcing, remain over the outer continental shelf on time scales that are comparable to the length of the upwelling season. To allow a direct, illustrative calculation of Δ*ζ*_{b} in the accessible context of linear, inviscid, fast-time-scale dynamics, a restrictive case is therefore considered in which only the first continental shelf wave mode is allowed to propagate.

This restriction, while admittedly artificial in some respects, is sufficient to prevent the long-time linear collapse and to sustain a nonzero Δ*ζ*_{b} over the outer shelf. The full solution —for this restricted case in which only the first mode is allowed to propagate and the higher modes are taken to be stationary—thus consists of two separate regions of ageostrophic, cross-shore motion connected by an intermediate region of alongshore geostrophic flow (Figs. 2a,b): the ageostrophic, cross-shore motion occurs just behind the leading edge of the wave front and in the forcing band 0 < *y* < *y*_{0}, while the alongshore extent of the alongshore geostrophic flow regime grows rapidly with time, at a rate equal to the first-mode wave speed.

*ζ*(

*x*) satisfies

*dζ*/

*dx*=

*g*

^{−1}

*f*

*υ*=

*H*

^{−1}

*d*Ψ/

*dx*, and may be conveniently computed for the exponential topography [(1)] through an integral by parts:

*ζ*< 0 over roughly the same region where the geostrophic Ψ < 0, with an outer continental shelf sea surface height anomaly

*ζ*(

*x*= 0) ≈ −0.017 m (Fig. 2c). This lowered, geostrophically balanced, and alongshore-uniform (within the geostrophically adjusted region) sea surface height anomaly

*ζ*(

*x*= 0) over the outer shelf is consistent in form and in sign with the posited sea surface height anomaly Δ

*ζ*

_{b}in the interior boundary condition (3). Its amplitude is approximately one-tenth of the mean disturbances computed by Durski et al. (2015) over the region between the coastline and the 200-m isobath.

The characteristic value Δ*ζ*_{b} = *ζ*(*x* = 0) ≈ −0.017 m computed in this way is used in the alongshore-uniform boundary condition [(3)] for the slow-time-scale solutions described in sections 4–6. It should be clear, however, that this explicit calculation of Δ*ζ*_{b} from the continental shelf wave response is only illustrative; a broader appeal to analyses such as that of Durski et al. (2015) is necessary to justify the boundary condition (3). This point is discussed further in section 9.

## 4. Slow, interior dynamics

*x*= 0; Figs. 1b,d), and are taken to be described by the linear, inviscid, long-wave, quasigeostrophic potential vorticity equation for a continuously stratified fluid on a

*β*plane (e.g., Pedlosky 1987):

*q*is the quasigeostrophic potential vorticity,

*ψ*is the quasigeostrophic streamfunction for the zonal and meridional geostrophic velocities,

*N*(

*z*) is the buoyancy frequency, and

*f*and

*β*are the Coriolis frequency and its meridional gradient. In (5), an exponential

*N*

^{2}profile is used that roughly matches regional mean profiles of the squared buoyancy frequency:

The long-wave equation (5) is a statement of the Sverdrup vorticity relation *βυ* = *f*∂*w*/∂*z*, where the quasi-geostrophic vertical velocity *w* = −(*f*/*N*^{2})∂^{2}*ψ*/*dt dz*. The curl of the imposed wind stress (section 3) vanishes identically at all times, so there is no forcing term in the dynamical equation (5) and no wind-driven Sverdrup transport. Instead, the interior solution is forced through the boundary condition (3), with the characteristic amplitude |Δ*ζ*_{b}| = 0.017 m of the low-frequency, geostrophically balanced, outer-shelf sea surface height disturbance derived from the fast-time-scale, coastal-trapped response as described in section 3. The details of the specification of Δ*ζ*_{b}(*t*) are described along with the respective solutions of (5) for the flat-seafloor and continental-slope geometries. The alongshore-uniform boundary condition (3) applies approximately over the full alongshore extent of the geostrophically adjusted region of the fast solution, poleward of the forcing region (i.e., in *y* > *y*_{0}) and equatorward of the leading-edge fast wave front (Fig. 2c). With the seafloor geometry (1) or (2) also independent of *y*, the interior solution *ψ*(*x*, *z*, *t*) will inherit this alongshore symmetry and should be viewed as an approximate solution of (5) in the corresponding latitudinal range poleward of the fast-solution wind-forcing region (Fig. 1b).

*x*

_{0}= 0 in the flat-seafloor case (2). For the case with continental-slope topography (1), however, topographic variations are comparable to the fluid depth, and bottom flow across topographic contours would induce large vertical velocities over the continental slope if the standard no-normal-flow condition (e.g., Pedlosky 1987) were applied at the sloping bottom. These vertical velocities will be of relative formal order 1/Ro in the Rossby number Ro, and will diverge in the formal quasigeostrophic limit Ro → 0. To maintain the quasigeostrophic ordering, it is necessary instead to require that the flow across topographic contours vanish, which implies that ∂

*ψ*/∂

*y*must vanish at the boundary. Under the further assumption that the forcing and flow vanish sufficiently far equatorward of the region of interest (e.g., for

*y*< 0; Figs. 1a,b), it then follows that

*ψ*remains implied over the entire alongshore extent of the slow-dynamics domain by the condition of vanishing normal velocity. The condition (9) is perhaps unusual, but it has been used or inferred before in the presence of topography (e.g., Samelson 1992; Tailleux and McWilliams 2001). In the initial simpler case of a flat seafloor (2) with no slope topography, the condition (8) is used over the entire domain (i.e., with

*x*

_{0}= 0), and the condition (9) does not apply. Instead, the condition (3) at

*x*= 0 is supplemented by the usual no-normal-flow condition at the vertical wall beneath the outer-shelf depth, which by the previous argument also implies

*x*

_{0}→ 0, in which the continental slope region

*x*

_{0}<

*x*< 0 shrinks into a vertical wall at the domain edge

*x*= 0.

A standard radiation condition is imposed far offshore, requiring that energy propagate zonally only toward −∞ as *x* → −∞, so that no disturbances may enter the domain from offshore. The complete set of boundary conditions for (5), consisting of (3) and (8) with (9) or (10), supplemented by the offshore radiation condition, is thus consistent with solutions for the streamfunction *ψ* that are independent of *y*, which should be viewed as approximate local solutions in the previously identified latitudinal range poleward of the fast-solution wind-forcing region (Fig. 1).

## 5. Results: Flat seafloor

### a. General solution

*x*= 0 consists of an opening to the outer shelf that extends from the sea surface to the outer-shelf seafloor at

*z*= −

*H*

_{b}and a vertical wall that extends from the outer-shelf seafloor at

*z*= −

*H*

_{b}to the flat, abyssal seafloor at

*z*= −

*H*

_{0}. In this geometry,

*x*

_{0}= 0, and the top and bottom boundary conditions are the standard conditions (8) throughout the domain. The baroclinic dynamical equation (5) is then separable and can be solved by decomposition into vertical modes

*P*

_{n}(

*z*),

*n*= {0, 1, 2, …},

*n*= 0) mode is degenerate and independent of

*x*,

*n*≥ 1) modes satisfy

*P*

_{n}are orthogonal in integral over −

*H*

_{0}<

*z*< 0 and may be made orthonormal, with

*P*

_{n}(

*z*= 0) > 0. By (13), the baroclinic field evolves by the offshore propagation of the baroclinic modes, at the respective wave speeds

*c*

_{n}.

*ψ*

_{n}must be computed from the conditions (3) and (10) at the outer-shelf boundary

*x*= 0. The imposed streamfunction

*ψ*

_{b}(

*t*) at the outer-shelf boundary in (3) is independent of depth for −

*H*

_{b}<

*z*< 0, so the modal projections are

*ψ*

_{0}=

*ψ*

_{b}

*H*

_{b}/

*H*

_{0}≪

*ψ*

_{b}< 0, with depth integral

*ψ*

_{0}

*H*

_{0}=

*ψ*

_{b}

*H*

_{b}. The first baroclinic mode will likewise have a negative modal coefficient, representing negative streamfunction values above its zero crossing level and positive values below. In the case of constant

*N*=

*N*

_{0}, this would correspond to negative values in the upper half of the water column (

*z*< −

*H*

_{0}/2), and to positive values in the lower half; with the exponential stratification (7), the zero crossing is displaced upward from middepth. As each successive higher-modal contribution is computed from (15) and added to the sum (11) at

*x*= 0, the effect will be systematically to concentrate the streamfunction disturbance into the upper water column until, in the limit of the infinite sum (11), it is confined to the shallow layer −

*H*

_{b}<

*z*< 0.

The boundary condition may also be interpreted in terms of the quasigeostrophic density perturbation *ρ*′ * = * −(*ρ*_{0}*f*/*g*)∂*ψ*/∂*z*. The imposed discontinuity in the boundary profile *ψ*(*x* = 0, *z*) at *z* = −*H*_{b}, with *ψ* = 0 for *z* = −*H*_{b} and *ψ* = *ψ*_{b} < 0 for *z* > −*H*_{b}, implies a singularity in *ρ*′(0, *z*) at *z* = −*H*_{b} with sign corresponding to a positive density perturbation, and *ρ*′(0, *z*) = 0 for *z* ≠ −*H*_{b}. The successive modal contributions to this singular-density profile at the boundary will begin with a positive gravest-mode density perturbation *ρ*′(0, *z*, *t*) = −(*ρ*_{0}*f*/*g*)*ψ*_{1}(0, *t*)*dP*_{1}(*z*)/*dz* centered at the gravest-mode streamfunction zero-crossing level and continue with higher-mode contributions that concentrate the positive density perturbation at *z* = −*H*_{b}. The infinite sum is necessarily divergent at *z* = −*H*_{b}, but this singularity is localized, and the generalized solution of the initial-boundary-value problem by Fourier decomposition remains valid. For the solution described in section 5b, the first 800 terms of the infinite sum (11) were retained, but all except the first several of these modes were effectively trapped at the coastal boundary.

### b. Response to onset of upwelling winds

For the slow, interior dynamics (5) in the approximately alongshore-uniform region poleward of the wind-forcing region (Fig. 1), the effect of the onset of the steady upwelling winds is felt through (3). As illustrated by the continental shelf wave solution in section 3, this onset of upwelling winds can be represented by an imposed sea surface height disturbance Δ*ζ*_{b} = −0.017 m that is switched on instantaneously at *t* = 0 and then remains constant for *t* > 0; now the implied time of wind-forcing onset precedes the time *t* = 0 of imposed Δ*ζ*_{b} setup by the time (e.g., 8 days, for the illustrative solution in Fig. 2) required for the fast dynamics to establish the outer-shelf geostrophic balance over the extended region poleward of the wind-forcing region.

For this simple case of response to onset of steady forcing, the evolution of the alongshore-uniform, interior, slow-dynamics flow field *ψ*(*x*, *z*, *t*) over a flat seafloor (2) can be entirely understood in terms of the offshore propagation of the sequence of progressively surface-intensified streamfunction structure represented by the accumulating modal contributions (15). Initially, at *t* = 0, the interior streamfunction consists only of a weak barotropic disturbance that has adjusted instantaneously through the rigid-lid approximation and has a slightly lowered sea surface, proportional to *ψ*_{0}. The first baroclinic modal contribution *ψ*_{1}(*x*, *t*)*P*_{1}(*z*) will be zero offshore of the wave front at *x* = *x*_{1} = −*c*_{1}*t*, and equal to the constant boundary value from (15) at the wave front and inshore, in the region −*c*_{1}*t* = *x*_{1} < *x* < 0. Thus, inshore of the wave front, there will be negative streamfunction anomalies in the upper water column and positive anomalies in the lower water column (Figs. 3a,b). The wave fronts associated with the higher modes will continue this pattern; the negative streamfunction values will become successively more surface-intensified behind each higher-mode wave front, with the *n*th front at *x*_{n} = *c*_{n}*t*.

(a),(b) Geostrophic streamfunction *ψ* (km m s^{−1}) and (c),(d) density perturbation *ρ*′ (kg m^{−3}) vs distance offshore and depth, at times (a),(c) *t* = 15 days and (b),(d) *t* = 30 days after the onset of the outer-shelf sea surface height disturbance at time *t* = 0 days, from linear theory for flat seafloor and vertical coastal wall with opening to shelf above *z* = −*H*_{b} = −150 m at *x* = 0. Poleward flow occurs at wave fronts and depths where the streamfunction increases onshore (e.g., as indicated by black circle with enclosed X adjacent to the coastal boundary), while equatorward flow occurs where the streamfunction decreases onshore (e.g., as indicated by black circle with enclosed dot at the first baroclinic-mode wave front). For this solution, the infinite sum in (11) is truncated at 800 modes.

Citation: Journal of Physical Oceanography 47, 12; 10.1175/JPO-D-17-0077.1

(a),(b) Geostrophic streamfunction *ψ* (km m s^{−1}) and (c),(d) density perturbation *ρ*′ (kg m^{−3}) vs distance offshore and depth, at times (a),(c) *t* = 15 days and (b),(d) *t* = 30 days after the onset of the outer-shelf sea surface height disturbance at time *t* = 0 days, from linear theory for flat seafloor and vertical coastal wall with opening to shelf above *z* = −*H*_{b} = −150 m at *x* = 0. Poleward flow occurs at wave fronts and depths where the streamfunction increases onshore (e.g., as indicated by black circle with enclosed X adjacent to the coastal boundary), while equatorward flow occurs where the streamfunction decreases onshore (e.g., as indicated by black circle with enclosed dot at the first baroclinic-mode wave front). For this solution, the infinite sum in (11) is truncated at 800 modes.

Citation: Journal of Physical Oceanography 47, 12; 10.1175/JPO-D-17-0077.1

(a),(b) Geostrophic streamfunction *ψ* (km m s^{−1}) and (c),(d) density perturbation *ρ*′ (kg m^{−3}) vs distance offshore and depth, at times (a),(c) *t* = 15 days and (b),(d) *t* = 30 days after the onset of the outer-shelf sea surface height disturbance at time *t* = 0 days, from linear theory for flat seafloor and vertical coastal wall with opening to shelf above *z* = −*H*_{b} = −150 m at *x* = 0. Poleward flow occurs at wave fronts and depths where the streamfunction increases onshore (e.g., as indicated by black circle with enclosed X adjacent to the coastal boundary), while equatorward flow occurs where the streamfunction decreases onshore (e.g., as indicated by black circle with enclosed dot at the first baroclinic-mode wave front). For this solution, the infinite sum in (11) is truncated at 800 modes.

Citation: Journal of Physical Oceanography 47, 12; 10.1175/JPO-D-17-0077.1

For typical midlatitude eastern-boundary *N*^{2} profiles, *c*_{1} ≈ 1–2 cm s^{−1} ≈ 1–2 km day^{−1}, so the first mode will propagate roughly 50–100 km offshore in 50 days. The second mode will propagate about one-fourth as far offshore, and the third mode only one-ninth as far offshore, roughly 5–10 km. Higher modes will remain effectively trapped at the boundary over time scales on the order of 50–100 days. The progressive surface intensification (toward *z* ≥ −*H*_{b}) of negative streamfunction values will thus remain substantially incomplete in the ocean interior over time scales comparable to the length of the upwelling season, and near the coastal boundary will extend to depths comparable to the uppermost zero crossing of a relatively low mode, which will be much deeper than the outer- shelf depth *z* = −*H*_{b}.

*t*= 0, the streamfunction is discontinuous and the alongshore velocity is singular at each wave front, but the associated alongshore transport associated with each mode has a finite value that is equal to the streamfunction difference across the front:

*n*th-mode wave front at

*x*

_{n}=

*c*

_{n}

*t*, and

*ψ*

_{n}< 0 for all

*n*with first (shallowest) zero crossing below

*z*= −

*H*

_{b}, and the sign of the alongshore transport contribution from each such mode at the given depth

*z*is thus opposite to the sign of

*P*

_{n}(

*z*). Therefore, above the first zero crossing of any such mode, the streamfunction contribution for that mode will be negative, corresponding to equatorward transport

*V*

_{n}< 0. By the same token, just below the first zero crossing of any given mode, the transport contribution will be positive, corresponding to poleward transport

*V*

_{n}> 0.

Geostrophic streamfunction *ψ* (km m s^{−1}) at depth *z* = −200 m vs distance offshore at times *t* = 15 days (dashed line) and *t* = 30 days (solid line) after the onset of the outer-shelf sea surface height disturbance at time *t* = 0 days, from linear theory with flat seafloor and vertical coastal wall with opening to shelf above *z* = −*H*_{b} = −150 m (Fig. 3). The model poleward undercurrent flow is represented by the region of positive onshore streamfunction gradient adjacent to the boundary *x* = 0 km, where *ψ* = 0.

Citation: Journal of Physical Oceanography 47, 12; 10.1175/JPO-D-17-0077.1

Geostrophic streamfunction *ψ* (km m s^{−1}) at depth *z* = −200 m vs distance offshore at times *t* = 15 days (dashed line) and *t* = 30 days (solid line) after the onset of the outer-shelf sea surface height disturbance at time *t* = 0 days, from linear theory with flat seafloor and vertical coastal wall with opening to shelf above *z* = −*H*_{b} = −150 m (Fig. 3). The model poleward undercurrent flow is represented by the region of positive onshore streamfunction gradient adjacent to the boundary *x* = 0 km, where *ψ* = 0.

Citation: Journal of Physical Oceanography 47, 12; 10.1175/JPO-D-17-0077.1

Geostrophic streamfunction *ψ* (km m s^{−1}) at depth *z* = −200 m vs distance offshore at times *t* = 15 days (dashed line) and *t* = 30 days (solid line) after the onset of the outer-shelf sea surface height disturbance at time *t* = 0 days, from linear theory with flat seafloor and vertical coastal wall with opening to shelf above *z* = −*H*_{b} = −150 m (Fig. 3). The model poleward undercurrent flow is represented by the region of positive onshore streamfunction gradient adjacent to the boundary *x* = 0 km, where *ψ* = 0.

Citation: Journal of Physical Oceanography 47, 12; 10.1175/JPO-D-17-0077.1

In addition, if the accumulated transport value of the total streamfunction *ψ*, constructed from the sum of modes across each offshore wave front, remains equatorward below the outer-shelf depth *z* = −*H*_{b} as the boundary at *x* = 0 is approached, this value must adjust rapidly to zero as the boundary is approached. This zone, just below the outer-shelf depth *z* = −*H*_{b}, of rapid variation from negative toward zero values of the streamfunction will itself consist of a coastal boundary layer of subsurface poleward flow. The poleward flow thus consists of the sum of poleward-flow contributions from just below the first zero crossings of all of the higher, slow-moving modes that are effectively trapped close to the boundary by their successively slower offshore propagation speeds (Fig. 4).

The interior response may also be interpreted in terms of the corresponding density perturbations *z* = −*H*_{b}, so that the interior density distribution slowly approaches the singular-boundary profile (Figs. 3c,d). Because the quasigeostrophic vertical velocity is proportional to the rate of change of the quasigeostrophic density perturbation, each successive density increase at the modal wave fronts is accompanied by an instantaneous effective upward displacement that is in Sverdrup vorticity balance with the corresponding alongshore transport (16) given by the streamfunction difference across the wave front. From this point of view, the interior response that drives the poleward subsurface flow may be understood as an upwelling planetary wave that brings the interior toward equilibrium with the boundary condition (3), which, despite its barotropic origin as lowered sea surface height over the outer shelf, appears to the interior flow to include the singular density perturbation at *z* = −*H*_{b}. An infinite number of modes is required to represent the singularity, but *c*_{n} → 0 as *n* → ∞, so the singularity remains trapped at the boundary for any finite time interval.

This simplified model with flat-seafloor geometry (2) is amenable to analytical solution—apart from the computation of the vertical mode structures *P*_{n}(*z*) for general *N*(*z*)—and yields an appealing dynamical explanation for poleward undercurrent flows on upwelling eastern boundaries. In this model, the poleward flow adjacent to the boundary arises from inviscid processes, which can be understood as upwelling planetary wave disturbances that are emitted, or forced, by the low-frequency, geostrophically balanced, coastal-trapped flow over the outer shelf. The existence of the poleward flow is dependent upon the planetary wave adjustment being incomplete, as the alongshore flow would be entirely concentrated above *z* = −*H*_{b} in the fully adjusted state—equivalent to the infinite modal sum in the representation of the boundary condition—while the associated disturbance streamfunction and cross-shore pressure gradients at depths *z* = −*H*_{b} would vanish. The adjustment is incomplete because the offshore propagation of the higher baroclinic modes is slow relative to the time scales of upwelling winds. While the interior circulation is ultimately driven by geostrophically balanced sea surface height disturbances along the interior-domain boundary, there are no alongshore pressure gradients in this alongshore-uniform limit. The model undercurrent is itself geostrophic, with flow balanced not by alongshore, but by cross-shore pressure gradients.

The simplified geometry in this formulation, however, removes one of the dominant physical features of the coastal ocean: the continental-slope topography, which is known to have a fundamental effect on the character of coastal circulation. The essential roles played in this flat-seafloor geometry by the vertical coastal boundary and the offshore propagation of flat-seafloor planetary wave modes make it unclear how, and even if, this model will translate to a geometry that includes a realistic representation of continental-slope topography.

## 6. Results: Variable depth seafloor with continental slope

### a. Numerical formulation and solution

*x*and

*z*, with the surface and seafloor boundary conditions (8) and (9). In the region of sloping topography

*x*

_{0}<

*x*< 0, the inversion for

*ψ*from

*q*can be accomplished with two vertical integrals, the first taken downward from the surface and the second taken upward from the bottom:

*x*=

*x*

_{0}, the depth-averaged part of

*ψ*(

*x*=

*x*

_{0},

*z*,

*t*) scatters into the barotropic mode and propagates offshore instantaneously under the rigid-lid approximation:

*x*<

*x*

_{0}, this leaves the baroclinic part,

*ψ*/∂

*x*in (5) must be computed differently at

*z*and

*t*have been dropped to simplify notation. In both regions, ∂

*ψ*/∂

*x*in (5) is otherwise computed numerically by a first-order upstream (relative to the offshore wave propagation tendency) scheme of the form (21), evaluated for a finite gridsize

*h*=

*δx*. The problem is solved on a regular

*x*–

*z*grid, with uniform grid spacing

*δx*= 1 km,

*δz*= 5 m, using a second-order, predictor-corrector time-stepping scheme with time step

*δt*= 0.0625 days = 1.5 h.

It remains to specify the vertical structure of the outer-shelf boundary condition (3) at *x* = 0. A possible choice would be to require this transport to be depth independent, as in the flat-seafloor case. However, this would induce a discontinuity at the bottom that would be difficult to handle numerically. A convenient alternative is to project the condition on the first local vertical mode solution of (14), computed for a constant *H* = *H*_{b} with the modified bottom boundary condition (9). With (9), there is no depth-independent mode; instead, all of the modes have nonzero vertical integral and can contribute to a depth-integrated transport. For constant *N*, for example, the first modified mode has quarter-sine vertical structure, with the node at the seafloor: *ψ* at *z* = −*H*_{b} (section 5), is made for the calculations described here. Without this effective smoothing, the boundary profile projections on higher modes would give additional, slowly propagating contributions that complicate the structure of the solution near the outer shelf, but otherwise do not affect the qualitative response; this is illustrated by the flat-seafloor solution (section 5), for which 800 modes were retained and the effective discontinuity was relatively well represented.

### b. Response to onset of upwelling winds

Numerical solutions of the slow-time-scale equations (5)–(9) with (17)–(21) were computed with boundary condition (3) imposed with the vertical structure *x* = 0, for *H*_{b} = 150 m. The solutions show an offshore streamfunction structure that is generally consistent with that implied by the outer-shelf boundary condition (Figs. 5a,b). However, they show also the development of a subsurface poleward flow over the upper slope that is generally consistent with the flat-seafloor solution (Figs. 5c,d and 6). The density surfaces over the slope show the characteristic deformations that are associated with these geostrophic velocities: elevation of the surfaces over the upper slope, and weak depression of surfaces over the midslope (Fig. 7). These results confirm that the most basic dynamics of the subsurface poleward flow in this model are qualitatively unchanged when a representative continental-slope topography is included.

(a),(b) Geostrophic streamfunction *ψ* (km m s^{−1}) and (c),(d) alongshore velocity *υ* (m s^{−1}) vs distance offshore and depth, at times (a),(c) *t* = 45 days and (b),(d) *t* = 90 days after the onset of upwelling winds at time *t* = 0 days, from linear theory for exponential slope and shelf topography, with opening to shelf at *x* = 0 km, where the seafloor has depth *z* = −*H*_{b} = −150 m. The model poleward undercurrent flow is represented by the subsurface regions with *υ* > 0 in (c),(d).

Citation: Journal of Physical Oceanography 47, 12; 10.1175/JPO-D-17-0077.1

(a),(b) Geostrophic streamfunction *ψ* (km m s^{−1}) and (c),(d) alongshore velocity *υ* (m s^{−1}) vs distance offshore and depth, at times (a),(c) *t* = 45 days and (b),(d) *t* = 90 days after the onset of upwelling winds at time *t* = 0 days, from linear theory for exponential slope and shelf topography, with opening to shelf at *x* = 0 km, where the seafloor has depth *z* = −*H*_{b} = −150 m. The model poleward undercurrent flow is represented by the subsurface regions with *υ* > 0 in (c),(d).

Citation: Journal of Physical Oceanography 47, 12; 10.1175/JPO-D-17-0077.1

(a),(b) Geostrophic streamfunction *ψ* (km m s^{−1}) and (c),(d) alongshore velocity *υ* (m s^{−1}) vs distance offshore and depth, at times (a),(c) *t* = 45 days and (b),(d) *t* = 90 days after the onset of upwelling winds at time *t* = 0 days, from linear theory for exponential slope and shelf topography, with opening to shelf at *x* = 0 km, where the seafloor has depth *z* = −*H*_{b} = −150 m. The model poleward undercurrent flow is represented by the subsurface regions with *υ* > 0 in (c),(d).

Citation: Journal of Physical Oceanography 47, 12; 10.1175/JPO-D-17-0077.1

Geostrophic streamfunction *ψ*(km m s^{−1}, thick line) at depth *z* = −200 m and depth-integrated geostrophic transport streamfunction Ψ [10^{6} m^{3} s^{−1}( = 1Sv), thin line] vs distance offshore, at times (a) *t* = 45 days and (b) *t* = 90 days after the onset of the outer-shelf sea surface height disturbance at time *t* = 0 days, from linear theory for exponential slope and shelf topography, with opening to shelf at *x* = 0 km, where the seafloor has depth *z* = −*H*_{b} = −150 m. The model poleward undercurrent flow is represented by the 200-m streamfunction (thick line) gradient immediately adjacent to the intersection of the 200-m depth surface with the seafloor at *x* ≈ −6 km. The depth-integrated transport propagates westward at effectively infinite speed after it reaches the flat-seafloor region at *x* ≈ −40 km.

Citation: Journal of Physical Oceanography 47, 12; 10.1175/JPO-D-17-0077.1

Geostrophic streamfunction *ψ*(km m s^{−1}, thick line) at depth *z* = −200 m and depth-integrated geostrophic transport streamfunction Ψ [10^{6} m^{3} s^{−1}( = 1Sv), thin line] vs distance offshore, at times (a) *t* = 45 days and (b) *t* = 90 days after the onset of the outer-shelf sea surface height disturbance at time *t* = 0 days, from linear theory for exponential slope and shelf topography, with opening to shelf at *x* = 0 km, where the seafloor has depth *z* = −*H*_{b} = −150 m. The model poleward undercurrent flow is represented by the 200-m streamfunction (thick line) gradient immediately adjacent to the intersection of the 200-m depth surface with the seafloor at *x* ≈ −6 km. The depth-integrated transport propagates westward at effectively infinite speed after it reaches the flat-seafloor region at *x* ≈ −40 km.

Citation: Journal of Physical Oceanography 47, 12; 10.1175/JPO-D-17-0077.1

Geostrophic streamfunction *ψ*(km m s^{−1}, thick line) at depth *z* = −200 m and depth-integrated geostrophic transport streamfunction Ψ [10^{6} m^{3} s^{−1}( = 1Sv), thin line] vs distance offshore, at times (a) *t* = 45 days and (b) *t* = 90 days after the onset of the outer-shelf sea surface height disturbance at time *t* = 0 days, from linear theory for exponential slope and shelf topography, with opening to shelf at *x* = 0 km, where the seafloor has depth *z* = −*H*_{b} = −150 m. The model poleward undercurrent flow is represented by the 200-m streamfunction (thick line) gradient immediately adjacent to the intersection of the 200-m depth surface with the seafloor at *x* ≈ −6 km. The depth-integrated transport propagates westward at effectively infinite speed after it reaches the flat-seafloor region at *x* ≈ −40 km.

Citation: Journal of Physical Oceanography 47, 12; 10.1175/JPO-D-17-0077.1

Total linear density ^{−3}) vs distance offshore and depth, at times (a) *t* = 45 and (b) *t* = 90 days after the onset of the outer-shelf sea surface height disturbance at time *t* = 0 days, from linear theory for exponential slope and shelf topography, with opening to shelf at *x* = 0 km, where the seafloor has depth *z* = −*H*_{b} = −150 m. The reference density *ρ*_{0} = 1025 kg m^{−3}, and the undisturbed density profile *ρ*_{0}/*g* times the squared buoyancy frequency (7) with respect to *z*.

Citation: Journal of Physical Oceanography 47, 12; 10.1175/JPO-D-17-0077.1

Total linear density ^{−3}) vs distance offshore and depth, at times (a) *t* = 45 and (b) *t* = 90 days after the onset of the outer-shelf sea surface height disturbance at time *t* = 0 days, from linear theory for exponential slope and shelf topography, with opening to shelf at *x* = 0 km, where the seafloor has depth *z* = −*H*_{b} = −150 m. The reference density *ρ*_{0} = 1025 kg m^{−3}, and the undisturbed density profile *ρ*_{0}/*g* times the squared buoyancy frequency (7) with respect to *z*.

Citation: Journal of Physical Oceanography 47, 12; 10.1175/JPO-D-17-0077.1

Total linear density ^{−3}) vs distance offshore and depth, at times (a) *t* = 45 and (b) *t* = 90 days after the onset of the outer-shelf sea surface height disturbance at time *t* = 0 days, from linear theory for exponential slope and shelf topography, with opening to shelf at *x* = 0 km, where the seafloor has depth *z* = −*H*_{b} = −150 m. The reference density *ρ*_{0} = 1025 kg m^{−3}, and the undisturbed density profile *ρ*_{0}/*g* times the squared buoyancy frequency (7) with respect to *z*.

Citation: Journal of Physical Oceanography 47, 12; 10.1175/JPO-D-17-0077.1

As the disturbances propagate into deeper water, the flow weakens as it spreads over a greater depth range; the disturbance appears trapped at the boundary, but the depth-integrated flow is seen to propagate offshore steadily (Figs. 8 and 6). The poleward flow develops initially at depth over the uppermost slope and slowly extends offshore and down the slope as time proceeds (Figs. 5c,d and 8c,d). As in the flat-seafloor case, the offshore and downward planetary wave propagation induces geostrophic flow beneath the outer-shelf depth *z* = −*H*_{b}, and the associated cross-shore pressure gradients adjacent to the boundary support the subsurface poleward flow.

As in Fig. 5, but for −100 < *x* < 0 km, and with contour ranges limited to reveal the weak offshore structure.

Citation: Journal of Physical Oceanography 47, 12; 10.1175/JPO-D-17-0077.1

As in Fig. 5, but for −100 < *x* < 0 km, and with contour ranges limited to reveal the weak offshore structure.

Citation: Journal of Physical Oceanography 47, 12; 10.1175/JPO-D-17-0077.1

As in Fig. 5, but for −100 < *x* < 0 km, and with contour ranges limited to reveal the weak offshore structure.

Citation: Journal of Physical Oceanography 47, 12; 10.1175/JPO-D-17-0077.1

The dynamics of the depth-integrated and near-surface flow nonetheless differ significantly from the flat-seafloor case because of the absence over the slope of the barotropic mode. Although the solution is not separable, the offshore propagation of the depth-integrated flow can be heuristically understood to be accomplished by a gravest baroclinic-mode structure: this will tend to propagate at a phase speed close to, but faster than, that of the standard first baroclinic mode, rather than by the barotropic mode, which has effectively infinite phase speed in the rigid-lid limit. This gravest baroclinic-mode structure will propagate faster than the higher vertical-mode structures that describe the subsurface poleward flow. The region of depth-integrated, equatorward transport thus propagates offshore relatively rapidly, while the slowly propagating, subsurface poleward flow remains close to the boundary (Fig. 6). When the depth-integrated transport reaches the base of the continental slope at *x* = *x*_{0}, it is scattered following (18) into the rigid-lid barotropic mode and propagates westward across the flat seafloor at an effectively infinite speed.

## 7. Response to time-dependent winds

### a. The time-dependent outer-shelf condition

The response to upwelling winds was computed above for (5)–(8) with (17)–(21) under the assumption that the interior geostrophic flow must match a steady, time-mean, sea surface height disturbance at a point on the outer continental shelf, with this point fixed at the 150-m isobath in the solutions described here. It is interesting to explore the behavior of the model under conditions in which the upwelling wind-forcing—or, more precisely, the imposed low-frequency, outer-shelf sea surface height disturbance—relaxes, or reverses. It is assumed that the same balance (3) holds when the imposed disturbance vanishes, in which case the inshore-domain boundary *x* = 0 acts like a vertical wall, at which a no-normal-flow condition is effectively enforced. When the upwelling winds reverse to downwelling, the sea surface height disturbance at the outer shelf is taken to reverse also, so that the outer-shelf boundary condition induces a positive geostrophic streamfunction disturbance with the vertical structure of the gravest local baroclinic mode.

### b. Relaxation of upwelling winds

To illustrate the response to relaxation of upwelling winds, the imposed sea surface height disturbance Δ*ζ*_{b} is taken to vanish after *t* = 90 days, and the solution computed above through *t* = 90 days is extended for an additional 90 days to *t* = 180 days, with Δ*ζ*_{b} = 0 during 90 < *t* < 180 days. In the absence of the boundary forcing, the interior disturbance propagates slowly offshore, and the poleward subsurface flow lifts off of the bottom, where it is progressively replaced by weak equatorward flow (Fig. 9). By *t* = 180 days—90 days after the cessation of the upwelling winds (i.e., after the vanishing of the sea surface height disturbance Δ*ζ*_{b})—the alongshore flow has approximately reversed relative to the *t* = 90 days pattern at the end of the upwelling phase, but with poleward flow in the upper water column over the upper slope that is much weaker than the corresponding equatorward flow at the end of the upwelling phase (Figs. 8b,d and 9b,d).

As in Fig. 8, but for (a),(c) *t* = 135 days and (b),(d) *t* = 180 days, respectively (a),(c) 45 days and (b),(d) 90 days after the removal of the outer-shelf sea surface height disturbance at time *t* = 90 days.

Citation: Journal of Physical Oceanography 47, 12; 10.1175/JPO-D-17-0077.1

As in Fig. 8, but for (a),(c) *t* = 135 days and (b),(d) *t* = 180 days, respectively (a),(c) 45 days and (b),(d) 90 days after the removal of the outer-shelf sea surface height disturbance at time *t* = 90 days.

Citation: Journal of Physical Oceanography 47, 12; 10.1175/JPO-D-17-0077.1

As in Fig. 8, but for (a),(c) *t* = 135 days and (b),(d) *t* = 180 days, respectively (a),(c) 45 days and (b),(d) 90 days after the removal of the outer-shelf sea surface height disturbance at time *t* = 90 days.

Citation: Journal of Physical Oceanography 47, 12; 10.1175/JPO-D-17-0077.1

### c. Oscillatory, annual-period wind forcing

*ζ*

_{b}(

*t*) is taken to have sinusoidal form with annual period,

*ζ*

_{b}(

*t*), the time point

*t*= 0 days is set at the beginning of the first upwelling phase, which ends with the first transition to downwelling conditions at

*t*= 182.5 days, with subsequent years repeating this pattern. For the solutions described here, (5)–(8) with (17)–(21) are again integrated from rest at

*t*= 0 days, forced as before at the outer-shelf boundary point by the now time-dependent sea surface height (22).

Although the model is inviscid, it allows the radiation of disturbances westward out of the domain; consequently, the response to the time-dependent boundary forcing (22) approaches a periodic, limit-cycle solution, with the same annual period as the forcing. The response is effectively asymptotic and periodic within 2 years after initiation of the sinusoidal forcing (*t* = 730 days). The model is linear, and the upwelling and downwelling phases of the forcing are equal and opposite, so the phases of the response are also equal and opposite. The flow halfway through the upwelling phase (Figs. 10a,c, 11a, and 12a,c), at *t* = 3 × 365 + 91.25 days, is similar over the upper slope to that from the constant-forcing solution after 90 days of sustained upwelling (Figs. 5b,d, 6b, and 8b,d). In response to the oscillatory forcing, the flow reverses after a half year, so that at times *t* = 3 × 365 + 273.75 days, the near-surface flow is poleward and the subsurface flow is equatorward (Figs. 10b,d, 11b, and 12b,d). This latter pattern resembles the structure of the flow at *t* = 180 days in the wind-relaxation case (Figs. 9b,d), but with stronger near-surface poleward flow and exact rather than approximate flow reversal relative to the upwelling phase. The deformation of the density surfaces again shows the characteristic patterns of upward deflection under upwelling, downward deflection under downwelling over the upper slope, and weak opposite deflection over the midslope (Fig. 13).

As in Fig. 5, but for (a),(c) *t* = 3 × 365 + 91.25 days and (b),(d) *t* = 3 × 365 + 273.75 days, respectively halfway through the (a),(c) upwelling and (b),(d) downwelling phases of the annual-period forcing (22).

Citation: Journal of Physical Oceanography 47, 12; 10.1175/JPO-D-17-0077.1

As in Fig. 5, but for (a),(c) *t* = 3 × 365 + 91.25 days and (b),(d) *t* = 3 × 365 + 273.75 days, respectively halfway through the (a),(c) upwelling and (b),(d) downwelling phases of the annual-period forcing (22).

Citation: Journal of Physical Oceanography 47, 12; 10.1175/JPO-D-17-0077.1

As in Fig. 5, but for (a),(c) *t* = 3 × 365 + 91.25 days and (b),(d) *t* = 3 × 365 + 273.75 days, respectively halfway through the (a),(c) upwelling and (b),(d) downwelling phases of the annual-period forcing (22).

Citation: Journal of Physical Oceanography 47, 12; 10.1175/JPO-D-17-0077.1

As in Fig. 6, but for (a),(c) *t* = 3 × 365 + 91.25 days and (b),(d) *t* = 3 × 365 + 273.75 days, respectively, halfway through the (a),(c) upwelling and (b),(d) downwelling phases of the annual-period forcing (22).

Citation: Journal of Physical Oceanography 47, 12; 10.1175/JPO-D-17-0077.1

As in Fig. 6, but for (a),(c) *t* = 3 × 365 + 91.25 days and (b),(d) *t* = 3 × 365 + 273.75 days, respectively, halfway through the (a),(c) upwelling and (b),(d) downwelling phases of the annual-period forcing (22).

Citation: Journal of Physical Oceanography 47, 12; 10.1175/JPO-D-17-0077.1

As in Fig. 6, but for (a),(c) *t* = 3 × 365 + 91.25 days and (b),(d) *t* = 3 × 365 + 273.75 days, respectively, halfway through the (a),(c) upwelling and (b),(d) downwelling phases of the annual-period forcing (22).

Citation: Journal of Physical Oceanography 47, 12; 10.1175/JPO-D-17-0077.1

As in Fig. 10, but for −100 < *x* < 0 km, and with contour ranges limited to reveal the weak offshore structure.

Citation: Journal of Physical Oceanography 47, 12; 10.1175/JPO-D-17-0077.1

As in Fig. 10, but for −100 < *x* < 0 km, and with contour ranges limited to reveal the weak offshore structure.

Citation: Journal of Physical Oceanography 47, 12; 10.1175/JPO-D-17-0077.1

As in Fig. 10, but for −100 < *x* < 0 km, and with contour ranges limited to reveal the weak offshore structure.

Citation: Journal of Physical Oceanography 47, 12; 10.1175/JPO-D-17-0077.1

As in Fig. 7, but for (a) *t* = 3 × 365 + 91.25 days and (b) *t* = 3 × 365 + 273.75 days, respectively, halfway through the (a) upwelling and (b) downwelling phases of the annual-period forcing (22).

Citation: Journal of Physical Oceanography 47, 12; 10.1175/JPO-D-17-0077.1

As in Fig. 7, but for (a) *t* = 3 × 365 + 91.25 days and (b) *t* = 3 × 365 + 273.75 days, respectively, halfway through the (a) upwelling and (b) downwelling phases of the annual-period forcing (22).

Citation: Journal of Physical Oceanography 47, 12; 10.1175/JPO-D-17-0077.1

As in Fig. 7, but for (a) *t* = 3 × 365 + 91.25 days and (b) *t* = 3 × 365 + 273.75 days, respectively, halfway through the (a) upwelling and (b) downwelling phases of the annual-period forcing (22).

Citation: Journal of Physical Oceanography 47, 12; 10.1175/JPO-D-17-0077.1

The offshore structure of the periodically forced solution contains a complex pattern of vertical and horizontal planetary wave radiation and interference (Fig. 12). A small amount of potential vorticity damping, in the form of the small term −*rq* on the right-hand side of the first equation of (5), is sufficient to eliminate this interference without disrupting the response over the upper continental slope (Figs. 14 and 15).

(a),(b) Geostrophic streamfunction *ψ* (km m s^{−1}) and (c),(d) alongshore velocity *υ* (m s^{−1}) vs distance offshore and depth, as in Fig. 12, but with potential vorticity damping −*rq* added to the right-hand side of the first equation of (5), with *r* = 1/100 days^{−1}.

Citation: Journal of Physical Oceanography 47, 12; 10.1175/JPO-D-17-0077.1

(a),(b) Geostrophic streamfunction *ψ* (km m s^{−1}) and (c),(d) alongshore velocity *υ* (m s^{−1}) vs distance offshore and depth, as in Fig. 12, but with potential vorticity damping −*rq* added to the right-hand side of the first equation of (5), with *r* = 1/100 days^{−1}.

Citation: Journal of Physical Oceanography 47, 12; 10.1175/JPO-D-17-0077.1

(a),(b) Geostrophic streamfunction *ψ* (km m s^{−1}) and (c),(d) alongshore velocity *υ* (m s^{−1}) vs distance offshore and depth, as in Fig. 12, but with potential vorticity damping −*rq* added to the right-hand side of the first equation of (5), with *r* = 1/100 days^{−1}.

Citation: Journal of Physical Oceanography 47, 12; 10.1175/JPO-D-17-0077.1

As in Fig. 10, but with potential vorticity damping −*rq* as in Fig. 14.

Citation: Journal of Physical Oceanography 47, 12; 10.1175/JPO-D-17-0077.1

As in Fig. 10, but with potential vorticity damping −*rq* as in Fig. 14.

Citation: Journal of Physical Oceanography 47, 12; 10.1175/JPO-D-17-0077.1

As in Fig. 10, but with potential vorticity damping −*rq* as in Fig. 14.

Citation: Journal of Physical Oceanography 47, 12; 10.1175/JPO-D-17-0077.1

As in Fig. 5, but for reversed (downwelling) wind stress and weak stratification representative of winter conditions, with ^{−2} in (7).

Citation: Journal of Physical Oceanography 47, 12; 10.1175/JPO-D-17-0077.1

As in Fig. 5, but for reversed (downwelling) wind stress and weak stratification representative of winter conditions, with ^{−2} in (7).

Citation: Journal of Physical Oceanography 47, 12; 10.1175/JPO-D-17-0077.1

As in Fig. 5, but for reversed (downwelling) wind stress and weak stratification representative of winter conditions, with ^{−2} in (7).

Citation: Journal of Physical Oceanography 47, 12; 10.1175/JPO-D-17-0077.1

The wintertime reversal to equatorward subsurface flow (Figs. 5b,d, 6b, and 15b,d) is perhaps unexpected, as such a reversal is not a generally accepted characteristic of observed eastern-boundary undercurrents. One possible rationalization of this discrepancy is the dependence of the response on the upper-ocean stratification, which is generally much weaker in the winter than in the summer. When the model stratification is reduced by setting ^{−5} s^{−1} in (7), for example, the amplitude of the subsurface response is substantially reduced (Fig. 16). An alternative possibility is that the baroclinic mode does reverse in winter, but other mechanisms contribute to a strong, barotropic, poleward flow that overwhelms the baroclinic reversal and maintains poleward subsurface flow.

## 8. The role of wind stress curl

The solutions thus far have been obtained under the assumption that the alongshore wind stress is uniform in the cross-shore direction. It is well known that upwelling winds along some sections of the Northern California Current System increase systematically offshore, giving a positive wind stress curl over the shelf and slope (e.g., Perlin et al. 2004), and it is sometimes suggested that this positive wind stress curl may drive a poleward alongshore flow through Sverdrup dynamics. This problem may be addressed in the present framework by considering the fast-time, coastal-trapped response to the wind stress curl forcing.

*y*<

*y*

_{0}, and zero for

*y*< 0 and

*y*>

*y*

_{0}, but let it now have a cross-shore structure given by a function

*C*(

*x*), so that

*C*(

*x*) is taken to vanish at the coastal boundary

*x*=

*x*

_{c}without loss of generality because in this linear framework, a cross-shore-uniform stress component equal to any nonzero stress at the coastal boundary may be removed and treated by the previous methods. The condition on

*dC*/

*dx*at

*x*=

*x*

_{c}is a technical convenience that allows inviscid satisfaction of a no-normal-flow condition at the coastal boundary; its influence can be confined arbitrarily closely to the coastal boundary because

*C*(

*x*) is a specified function. For simplicity,

*x*

_{a}>

*x*

_{0}is also assumed here, so that the region of curl is confined to the shelf and slope.

*f*/

*H*contours driven by the curl of

*y*= 0) = Ψ(

*x*=

*x*

_{c}) = 0, this may be solved to obtain

*C*will typically decrease monotonically toward the coast, with all other terms in (25) being positive, this alongshore transport will be equatorward under most conditions. An exception can occur if

*d*

^{2}

*C*/

*dx*

^{2}> −

*adC*/

*dx*, but negative curvature of the mean cross-shore wind stress profile

*d*

^{2}

*C*/

*dx*

^{2}> 0) seems likely to occur under upwelling conditions only very close to the coastal boundary, if at all.

As the fast time response thus directly drives steady, barotropic, equatorward flow in the region of positive wind stress curl, it is not effective at producing poleward flow. Thus, as found in related numerical solutions by McCreary et al. (1987), wind stress curl is an unlikely cause of the poleward undercurrent during the upwelling season. It is interesting to note, however, that this mechanism does directly support the existence of a geostrophically balanced alongshore pressure gradient over the outer shelf. This gradient balances the forced cross-shore barotropic transport

Wind stress curl over the outer shelf and upper slope is therefore potentially a mechanism for the generation of geostrophically balanced cross-shore flow, a response that seems otherwise difficult to develop through coastal-trapped wave dynamics. In this context, it is worth noting that interior alongshore pressure gradients balanced by geostrophic onshore flow under upwelling conditions have been identified in California Current models and observations, for example, by Connolly et al. (2014) and Todd et al. (2011).

## 9. Discussion

The coupling of fast, coastal-trapped dynamics with slow, interior dynamics through the low-frequency outer-shelf sea surface height anomaly yields, as outlined in sections 3–6, an appealing model of the generation of subsurface poleward flows along eastern boundaries by upwelling winds. Solutions are obtained in a simplified setting in which the alongshore wind forcing is applied over a limited meridional range. A subsurface poleward flow response on the slow, planetary wave time scale is found over an extended alongshore region poleward of the wind forcing, where the slow-time-scale boundary conditions and response are understood to be uniform alongshore.

This simplified geometric setting does not directly represent observed conditions along eastern boundaries, in which alongshore wind forcing and subsurface poleward flow are both distributed over broad, overlapping alongshore regions. However, because the model is linear, the simplified setting easily generalizes to a superposition of multiple overlapping forcing regions, each with their respective poleward flow responses that may in turn overlap in space and time with other forcing or poleward-flow regions. This extension is not pursued quantitatively here but is straightforward from both a theoretical and a computational point of view. This generalized conceptual setting intuitively yields a much closer match between the implied model response and the general character of observed eastern-boundary conditions. Even with this generalization, however, the theoretical outline falls short of a rigorous asymptotic analysis. Thus, the results presented here are probably best interpreted as the exploration and illustration of an extended hypothesis regarding poleward undercurrents.

A critical aspect of the theory that is supported on general grounds but lacks rigorous justification is the assertion that a sustained, geostrophically balanced, outer-shelf sea surface height anomaly (3) develops in response to seasonal upwelling winds. This anomaly was computed from the fast, coastal-trapped solution in section 3, but under the artificial restriction that only the gravest continental-shelf wave mode was allowed to propagate. Note, however, in this context that the first three coastal-trapped wave modes computed by Chapman (1987) and the stochastically forced coastal-trapped wave model of Brink et al. (1987), both for a northern California shelf and slope profile, show nearly barotropic structures over the outer shelf that have the same sign as the inner-shelf and midshelf structure. Results for pressure gain *R*_{pT} from the latter model might be interpreted to estimate an outer-shelf sea level response to low-frequency wind on the order of Δ*ζ*_{b} ≈ (*gρ*_{0})^{−1}*R*_{pT}*τ*_{0} = 0.5 × 10^{−2} m at frequency 1.08 × 10^{−5} s^{−1}, and probably larger for lower frequencies, with Δ*ζ*_{b} < 0 for equatorward wind (Brink et al. 1987, their Figs. 8, 9, 16, 17). Kurapov et al. (2017, their Fig. 11) have recently analyzed the coherence of coastal sea level with sea surface height over the shelf in a coastal ocean model, and they find significant coherence specifically over the outer shelf that extends 1800 km poleward of the coastal reference point at frequencies approaching seasonal time scales; in general, these findings support the basic hypothesis (3) that the low-frequency sea surface height over the outer shelf responds coherently to coastal-trapped processes and has sufficient persistence to induce an interior response.

Alternatively, it might be hoped that a longer time limit of the multiple-mode fast time response might still be considered, in which the geostrophic flow partially approaches the collapsed limit; the boundary condition for the interior flow might then be imposed instead at a location over the midshelf or even farther onshore. However, planetary wave speeds computed for the shallow fluid depths over the shelf are so slow that conditions imposed at the inner-shelf boundary in linear models can excite only weak, interior geostrophic motions, and only on very long time scales. It is presumably for this reason that the undercurrent response in the linear shelf-topography model of McCreary et al. (1987) was found to be overly weak unless the depth of the shelf region was artificially increased to 300 m—twice the depth of the outer-shelf isobath at which the interior boundary condition (3) is imposed here. Consequently, in order to retain the simplicity of the linear, inviscid theoretical setting, the domain of the slow, interior dynamics was here restricted to the regime offshore of the outer continental shelf, where the linear, slow-mode dynamics retain greater relevance. More generally, the coastal-upwelling jet over the shelf, represented in the present calculations by the strong equatorward flow in the continental-shelf wave solution, is known rapidly to become nonlinear, with alongshore flows consistently reaching or exceeding 0.5 m s^{−1}. Under these conditions, the linear approximation can be supported over the shelf only for the fastest coastal-trapped modes, and a dynamical decomposition based on linear theory for the slow time scale cannot be reasonably supported over the shelf.

An additional aspect of the theory that merits further attention is the dependence of the slow, interior solution either on the unrealistic flat-seafloor geometry (2) or on the unusual, steep-topography boundary condition (9) over the continental slope. There is rational motivation for both of these alternatives, and both yield appealing and dynamically similar physical models of subsurface poleward flow development. Nonetheless, development and analysis of a similar model with generalized interior dynamics that would allow the imposition of a more standard boundary condition over a realistic slope geometry would be a useful extension of the present approach. Even in the quasigeostrophic setting, extension to a full-domain solution that includes the response in the wind-forcing region and is not subject to the long-wave approximation would likewise be of interest, although such a solution can probably be obtained only numerically.

## 10. Summary

The main physical hypothesis illustrated by this analysis and explored in the solutions is that poleward undercurrents on eastern boundaries may be understood as the consequence of planetary wave radiation that is forced by large-scale, low-frequency, geostrophic sea surface height disturbances over the outer continental shelf. The separation of time scales between the fast, coastal-trapped dynamics and the slow, geostrophic, interior dynamics supports a conceptual decomposition of the wind-driven response into these two components. In this linear framework, the fast, coastal-trapped, *f*-plane response determines the alongshore structure and amplitude of the geostrophically balanced sea surface height disturbance over the outer continental shelf. The planetary waves excited by these slowly varying, mean sea surface height disturbances then radiate offshore from the outer shelf on the slow, *β*-plane time scale. These geostrophic waves penetrate downward as they propagate and induce geostrophic disturbances at depths below the level of the outer shelf, which are compelled to adjust to an effective no-normal-flow condition as the upper slope is approached. It is the cross-shore pressure gradients in this region of adjustment of deep interior pressure disturbances to the slope boundary that support the model poleward undercurrent. This model undercurrent is geostrophic; alongshore pressure gradients do not exist in the model in the alongshore-uniform limit considered here, and the poleward flow is balanced instead by cross-shore pressure gradients. These results extend, and suggest reinterpretation of, several previous linear models, including those of McCreary (1981) and McCreary et al. (1987, 1992), that connect poleward undercurrent generation to planetary wave propagation in a generally similar way.

The uniform alongshore structure of this model undercurrent emphasizes the tendency of the dynamics to support coherent undercurrent flows over large alongshore scales, but is, of course, unrealistic in its perfect symmetry. Numerous factors can be anticipated to break up this structure in the coastal ocean, including alongshore variations in winds and topography as well as nonlinear and other linear interactions with shelf and upper-slope flows. For example, it is well known that offshore excursions of the coastal jet from the shelf onto the slope are frequent events, especially late in the upwelling season, and these events can include dramatic interactions with the poleward undercurrent (Barth et al. 2000). The linear decomposition also breaks down when this occurs, as nonlinear motions impinge on the geostrophic interior. The linear, long-wave theory for the interior motions may nonetheless give some insight even under these conditions. Recent altimeter-based studies of open-ocean sea surface height variability are consistent with the extension of nondispersive, long-wave dynamics into short-wave regimes that would be dispersive in linear quasigeostrophic theory (Chelton et al. 2011; Samelson et al. 2016). The long-wave theory [(5)] may therefore prove to be a better model of the propagation characteristics of general interior geostrophic motions than might be expected from the formal linearization and long-wave approximation.

The basic viability of the theory proposed here could be tested by suitable analysis of numerical simulations conducted with the current generation of nonlinear, primitive-equation, coastal-circulation models, which have been shown to reproduce many elements of observed flow characteristics over the continental shelf and slope (e.g., Springer et al. 2009; Durski et al. 2015). A natural starting point for this analysis would be to separate the transports over the outer continental shelf—in suitable spatial and temporal means—into slowly varying geostrophic, Ekman, and other ageostrophic components, and then to examine the possible relationship of these components to model representations of poleward undercurrents. Such an effort is in progress (R. Duran and R. M. Samelson, 2017, unpublished manuscripts), and will be reported on elsewhere.

## Acknowledgments

This research was supported the National Aeronautics and Space Administration (NASA) Ocean Vector Winds Science Team, NASA Grant NNX14AM66G. I am grateful for conversations with J. McCreary and for comments from two anonymous reviewers that stimulated clarifications of the formulation, including the schematic Fig. 1.

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