## 1. Introduction

### a. Background

The Leeuwin Current system (LCS) off the coast of Western Australia consists of three main branches: the shelf currents, the Leeuwin Current (LC), and the Leeuwin Undercurrent (LUC). Its most energetic current, the LC, is directed poleward, and because it carries relatively warm tropical waters southward along the shelf break, it stands out as a narrow, warm band from the surrounding waters (Ridgway and Condie 2004). The LC thickens and its mean transport increases toward the south (Godfrey and Weaver 1991; Pattiaratchi and Woo 2009); at 32°S, Feng et al. (2003) estimated a mean transport of 3.4 Sverdrups (Sv; 1 Sv ≡ 10^{6} m^{3} s^{−1}), with 5 Sv in the Australian summer and 2 Sv in winter. The LCS, and in particular the offshore edge of the LC, is a region of high eddy energy compared to other eastern boundary current systems (Feng et al. 2005). Although most observational studies of the LCS have focused on the LC or the shelf currents, the LUC has been noticed on the offshore edge of the LC at several occasions. Its core was typically found at 200–600-m depth (Thompson 1984; Godfrey and Ridgway 1985; Smith et al. 1991). Thompson (1984) estimated an LUC transport of 5 Sv, slightly larger than that of the LC (4 Sv), but his findings have not been confirmed more recently. In regional model studies, the LUC is observed as a narrow, northward current at depths similar to those observed, its transport increasing northward from Cape Leeuwin, then weakening farther to the north upon losing mass by westward flow into the Subtropical Gyre (Domingues et al. 2007; Meuleners et al. 2007; Rennie et al. 2007).

A number of studies have investigated the dynamics of the LCS, and several mechanisms have been proposed that force the LC poleward against the winds and trap the currents along the eastern boundary. Kundu and McCreary (1986) and McCreary et al. (1986) explored the role of forcing by the Indonesian Throughflow (ITF) and the large-scale meridional density gradient using linear, continuously stratified models, in which Rossby waves are damped by vertical mixing (McCreary 1981). Their models simulated many features of the LCS, including coastal downwelling and an LUC. Because the currents were relatively weak when only one forcing mechanism was considered, they suggested that both the ITF and the buoyancy forcing might contribute to the LCS. To maintain strong eastern boundary currents, relatively large diapycnal mixing (≈10^{−3} m^{2} s^{−1}) was required in their models, casting doubt on whether mixing can be responsible for trapping the LCS to the eastern boundary. Idealized experiments utilizing eddy-resolving ocean general circulation models (OGCMs) have confirmed the key role of the meridional density forcing in driving the LC (e.g., Godfrey and Weaver 1991; Cessi and Wolfe 2013; Benthuysen et al. 2014). While currents similar to those in the LCS were found in flat bottom oceans, the strength of these currents increased to observed values in cases when a continental shelf was included (Godfrey and Weaver 1991; Benthuysen et al. 2014). Furthermore, the OGCM results suggested a relation between density advection by the LC and the LUC.

An instrumental role of the shelf in trapping the LCS has also been suggested by other modeling studies. The Csanady (1978, 1985) model has proven particularly useful for such investigations. It consists of a two-layer model in the open ocean matched to a uniformly stratified, linear, diffusive shelf model, where the density varies horizontally but is externally prescribed. Another simplifying assumption is that the Coriolis parameter *f* is constant. Weaver and Middleton (1989, 1990) applied the model to the LCS. In their solutions, the LC develops over the slope in response to the density forcing, suggesting that the interaction of the meridional density gradient with the continental slope [joint effect of baroclinicity and relief (JEBAR); Sarkisyan and Ivanov 1971] can be responsible for trapping the LCS. Because of the *f*-plane approximation, their model lacked a key mechanism allowing for Rossby waves to propagate off the shelf and adjust the interior ocean. Thus, the trapping effect of the shelf was conclusively proven by Furue et al. (2013), who recently found similar solutions on a *β* plane. These idealized models simulate many features of the LCS surprisingly well; they show a poleward intensification and deepening, that downwelling occurs in the LC region, and that meridional winds can be included to drive northward shelf currents. They do not, however, produce an LUC.

### b. Present research

In this manuscript, we explore the effect of an across-shelf density gradient on the LCS circulation, when the gradient takes the form of a density front on the offshore edge of the LC. With density increasing offshore, a vertical shear with a relative poleward flow near the surface and equatorward flow at depth, respectively, is expected due to thermal wind balance (e.g., Thompson 1987). But because the horizontal density gradient only determines the vertical shear, it does not necessarily drive an equatorward current. What processes then determine the total velocities and transports? How are these processes related to the properties of the LCS, and do they generate an LUC?

In previous analytical Csanady (1985)-type models, density over the shelf varies horizontally, but it cannot be internally determined. This simplification is okay for studying the general effects of the large-scale, alongshore coastal density field. The strong across-shelf density gradient in the LCS, however, results from density advection in the LC (e.g., Smith et al. 1991), and its position is strongly linked to the position of the LC itself. To explore that process, we extend the Furue et al. (2013) model by prescribing different densities in the coastal and offshore regions; the grounding line, then, is also the position of a density gradient (front), and its position is determined by a frontal equation (e.g., Dewar 1991) rather than by matching the layer thickness in both regions.

Although extending previous models, the model presented is still limited in several ways. Processes in the frontal region that determine the width of the LUC are not resolved, and because of the limited vertical resolution, the LC and the LUC are both contained in the surface layer (the LUC is depth intensified, however, due to the thermal wind shear). Because of these issues, the LUC cannot extend deeper than the LC. All these simplifications are expected to affect the results in a quantitative way. On the other hand, by improving the representation of several LCS features compared to previous models without a front, the model may be a useful tool to better understand the dynamical processes controlling eastern boundary current systems.

The manuscript is organized as follows. In section 2, we introduce our ocean model and discuss its configuration. Solutions and parameter sensitivities are discussed in section 3, and we close with a summary and conclusions (section 4).

## 2. Model

### a. Configuration

The model used in this study is a variable density, two-layer system in which the upper-layer density *ρ*_{1}(*x*, *y*) varies horizontally, but the deep-layer density *ρ*_{2} remains constant (e.g., Schloesser et al. 2012, 2014; Furue et al. 2013). Solutions are obtained in a semi-infinite basin with an eastern boundary at *x* = *x*_{e} = 0. Meridionally, the basin extends from *y* = *y*_{s} in the south to *y* = *y*_{n} in the north. We assume a midlatitudinal *β* plane, such that the Coriolis parameter has the form *f* = *f*_{0} + *β*(*y* − *y*_{n}), with *f*_{0} = −3 × 10^{−5} s^{−1} and *β* = 2 × 10^{−11} s^{−1} m^{−1}. The model ocean includes a continental shelf along the eastern boundary. For simplicity, the ocean depth *D*(*x*) varies only in the zonal direction; it monotonically increases away from the eastern boundary, and *D*(*x*_{e}) = 0. Solutions are invariant to the form of *D*(*x*), thus, a specification is not necessary for deriving the general solutions [*D*(*x*) = −1000*x* m (°)^{−1} is assumed for illustrating solutions in Figs. 3 and 6, described in greater detail below]. Because the surface layer has a finite depth, the model domain is zonally divided into two dynamically different regimes: a coastal region *R*_{c}, where the upper layer extends to the bottom, and an offshore region *R*_{o}, where the ocean is stratified (see Fig. 1). The two regions are separated by the “grounding line” of the upper layer *x*_{g}(*y*), which is internally determined by the model.

*x*≥

*x*

_{g}) and offshore regions (

*x*≤

*x*

_{g}). According to (1),

*ρ*

_{1}(

*y*

_{n}) =

*ρ*

_{n}is constant along the northern boundary and then (linearly) increases to

*ρ*

_{os}at the southern boundary in the offshore and

*ρ*

_{cs}in the coastal region, respectively. Densities in the offshore and onshore regions differ because in reality they result from different balances of processes. In the offshore region,

*ρ*

_{1}=

*ρ*

_{o}(

*y*) represents a mean, interior-ocean meridional density profile, with the density increasing toward the pole due to buoyancy forcing. In the coastal region, strong, poleward advection of relatively warm and low-salinity waters by the LC is balanced by surface buoyancy fluxes, “entrainment” of offshore water into the LC, and mixing of offshore and coastal waters by eddies (Ridgway and Condie 2004; Domingues et al. 2006; Pattiaratchi and Woo 2009). This complex balance (as well as resulting zonal, vertical, and temporal density variations within each of the regions) cannot be resolved by our model; however, its result is partly included by assuming lower densities in the coastal than in the offshore region [i.e., we consider cases

*ρ*

_{os}≥

*ρ*

_{cs}in (1)]. Note that the across-shelf density gradient is then modeled as a density front along

*x*

_{g}(

*y*), the dynamics of which are discussed in section 2e. Consequently, by internally adjusting the position of that front,

*x*

_{g}(

*y*), the model also controls density [i.e., it determines whether

*ρ*

_{1}(

*x*,

*y*) =

*ρ*

_{o}(

*y*) or

*ρ*

_{1}(

*x*,

*y*) =

*ρ*

_{c}(

*y*) at a given point (

*x*,

*y*)]. Wind forcing can be included in the model as discussed by Furue et al. (2013), but is not considered in the present manuscript.

### b. Model equations

*x*

_{g}(see section 2e below). The set of equations considered in each layer is thenwhere

*i*= 1, 2 is the layer index (layer two is only considered in the offshore region),

**V**

_{i}= (

*U*

_{i},

*V*

_{i}) are the layer depth–integrated horizontal transports per unit width, and

**∇**= (∂

_{x}, ∂

_{y}) denotes the horizontal gradient operator, and

**f**×

**V**

_{i}= (−

*fV*

_{i},

*fU*

_{i}). Friction

*h*

_{1}and

*h*

_{2}are the layer thicknesses,

*η*is the sea surface height (SSH),

*ϕ*

_{1}≡

*gρ*

_{1}/

*ρ*

_{2}, and

*g*is the acceleration due to gravity. Analogously, we define

*ϕ*

_{2}≡

*g*and the reduced gravity

*ϕ*′ ≡

*ϕ*

_{2}−

*ϕ*

_{1}. To distinguish values of

*ϕ*

_{1}in the coastal and offshore regions, we also use

*ϕ*

_{o}(

*y*) =

*ϕ*

_{1}at

*x*<

*x*

_{g}and

*ϕ*

_{c}(

*y*) =

*ϕ*

_{1}at

*x*>

*x*

_{g};

*ϕ*′ =

*ϕ*

_{2}−

*ϕ*

_{o}is only used in context of offshore densities. Because (2) and (3) are essentially the same as in Furue et al. (2013), our readers are referred to that paper for a more detailed derivation and discussion.

### c. Vertical shear

*υ*_{1}(

*z*) is not necessary for obtaining the depth-integrated solution and circulation; however, it is useful for providing a better picture of the three-dimensional flow. Assuming again a layered density structure and geostrophy, vertical integration of the hydrostatic equation givesAccording to (4), velocities are proportional to the SSH gradient at the surface, and the vertical shear is determined by the thermal wind. Note that by vertically integrating the rhs of (4) over the upper layer, we directly regain the depth-integrated pressure gradient (3a).

### d. Boundary conditions

*y*=

*y*

_{n}in the Southern Hemisphere). For simplicity and comparability, we choose the same conditions as Furue et al. (2013), that is,andWith

*H*

_{n}≡

*D*[

*x*

_{g}(

*y*

_{n})] and density being constant at

*y*=

*y*

_{n}, there is no flow across the northern boundary according to (5). No eastern boundary condition needs to be specified, because information propagates along potential vorticity characteristics, and, because

*h*

_{1}→ 0 as

*x*→

*x*

_{e}, no characteristics originate from the eastern boundary [for a discussion of solutions with

*h*

_{1}(

*x*

_{e}) ≠ 0, see Furue et al. (2013)]. Moreover, because of the direction of Rossby wave propagation, solutions can extend indefinitely in western and poleward directions.

### e. Frontal equation

To complete solutions, coastal and offshore solutions have to be matched across the grounding line *x*_{g}(*y*). In the Furue et al. (2013) model, this is accomplished by matching the coastal and offshore layer thickness. In case of different densities in the coastal and offshore regions considered here, however, solutions cannot be matched by that condition because the continuity is then violated across the front (section 3c). A mechanism responsible for frontogenesis in similar models (e.g., Dewar 1991; Dewar et al. 2005; Schloesser et al. 2014) is related to baroclinic Rossby wave characteristics intersecting in the interior ocean, carrying different information from the basin boundaries; because, in general, variables do not match across the intersection, fronts exist there. In the limit of infinitesimally small mixing, these fronts are infinitesimally narrow, and their position is determined by requiring that volume is conserved across the front.

*x*

_{g}(

*y*) (Fig. 1) giveswhere

*y*

^{±}=

*y*± Δ

*y*/2, with Δ

*y*being the meridional length of the integration area; |

*x*

_{g}(

*y*

^{+}) −

*x*

_{g}(

*y*

^{−})| is its zonal extent;

*U*

_{o}and

*V*

_{o}are the horizontal transports per unit width in the offshore region;

*U*

_{c}and

*V*

_{c}are the corresponding transports in the coastal region; and

_{LUC}is the transport of the Leeuwin Undercurrent driven by the pressure gradient across the front. To obtain an expression for

_{LUC}, we will employ the typical boundary layer assumption that the current is geostrophic along

*x*

_{g}. As in similar applications (e.g., Dewar 1991; Dewar et al. 2005; Schloesser et al. 2014), (eddy) processes not resolved by the model are assumed to generate continuous transitions of variables across the front. Furthermore, because the width of the front is infinitesimally small, gradients of external parameters across the front (e.g., depth

*D*) are neglected. Taking the limit Δ

*y*→ 0 in (6) then yieldswhich is solved for the grounding line in section 3c.

Assuming that the width of the frontal region is zero in our model poses a notable simplification, and resolving the frontal processes in a more complete model will likely affect the results in a quantitative way. Yet, by qualitatively capturing the effect of fronts on the large-scale circulation, we demonstrate that the model’s capability to reproduce the LCS circulation can be improved in comparison to models where this effect is not considered at all.

## 3. Solutions

In this section, we present our model solutions. We start by deriving the response in the coastal and offshore regions, and the solution is completed by finding the grounding line *x*_{g}(*y*). Then, we discuss a base solution for the model, as well as sensitivity to several parameters.

### a. Coastal region

*x*≥

*x*

_{g}), the surface layer extends to the bottom of the ocean (

*h*

_{1}=

*η*+

*D*and

*h*

_{2}= 0; see Fig. 2), such that the model reduces to a one-layer system. Starting with (2b) for layer one, substituting the cross-differentiated (2a) with (3a), and using

*ϕ*

_{1x}=

*D*

_{y}= 0, it is straightforward to show thatis conserved along characteristics given bywhere the characteristic velocities areNote that

*c*

^{x}is proportional to the planetary

*β*, and

*c*

^{y}to the topographic

*β*. The slope of characteristic curves is given byEquation (10) can be integrated using boundary conditions, such that the coastal solution is essentially known along characteristics. For the simple set of boundary conditions used in the present study [(5)], the solution can be inverted to

*γ*(

*x*,

*y*); let

*γ*along

*x*,

*y*) in the coastal region that is intersected by a northern boundary characteristic, and all variables are known except for the value of

*x*,

*y*) in the coastal region. Using (8) to solve (12) for the layer thickness, we getSubstitution of (13) into (3a) and (2a) gives the horizontal transportsto complete the coastal solution.

### b. Interior-ocean solution

The solution in the offshore region is found following the same procedure as in Luyten et al. (1983). First, we demonstrate that the deep layer is at rest, then we use that property to solve for the surface-layer response.

_{2x}=

*V*

_{2}, Ψ

_{2y}= −

*U*

_{2}) and the cross-differentiated (2a) with (3b), it is straightforward to show that (2b) for layer two can be written asa statement that transports in layer two are aligned with

*h*

_{2}/

*f*characteristics. Because boundary conditions (5) imply that transports vanish across the boundary

*y*=

*y*

_{n}, the steady-state, deep-layer transports vanish along characteristics extending across that boundary. To proceed, we obtain a relation between the gradients of sea surface height and upper-layer thickness, assuming that all characteristics extend across

*y*

_{n}, so that the deep layer is at rest entirely (that assumption is easily verified after the solution is completed). Because the deep-layer transports are geostrophic, we set the deep pressure term (3b) to zero, which givesNext, (16) is integrated using (5), which yieldsrelating the sea surface elevation to the upper-layer thickness and the northern boundary condition.

*ϕ*

_{1x}= 0 in the offshore region, it follows that the steady-state upper-layer thickness is constant in the zonal direction [from (17), it then also follows that

*η*does not vary with longitude], and no upper-layer flow occurs in the meridional direction:

### c. Frontal solution

_{LUC}(

*y*), assuming that its alongfront component is in geostrophic balance. Integration of the pressure term (3a) across the front giveswhere subscripts

*c*and

*o*denote the coastal and offshore values of variables

*ϕ*

_{1}and

*h*

_{1}, respectively. After substitution of (14), (19), and (21) into (7), several terms cancel, and using

*β*.

*D*

_{gy}=

*D*

_{x}

*dx*

_{g}/

*dy*, we get after some reorganizationwhich is a nonlinear ordinary differential equation (ODE) for

*D*

_{g}(

*y*). We have found closed form solutions only in special cases (section 3i); however, it can be easily integrated numerically, and the grounding line

*x*

_{g}(

*y*) is then obtained by inverting

*D*(

*x*

_{g}) =

*D*

_{g}. With the grounding line determined, the solution is complete.

### d. Base solution

Here, we describe our base solution, which uses the parameters *ρ*_{n} = 1023 kg m^{−3}, *ρ*_{cs} = 1025 kg m^{−3}, *ρ*_{os} = 1026 kg m^{−3}, *ρ*_{2} = 1026.1 kg m^{−3}, and *H*_{n} = 100 m. The model domain extends meridionally from *y*_{n} = 10°S to *y*_{s} = 35°S. The surface density field given by (1) with these parameters is shown in Fig. 3 (top), the gray curve indicating the position of the grounding line and front *x*_{g}, which separates the relatively light water in the coastal region and the denser water offshore. Figure 3 (bottom) shows the upper-layer streamfunction and sea surface height. Note that because it is more intuitive to view solutions in a map, we have chosen a specific bathymetry profile for illustrating our solution, that is, *D* = −*x* × 1000 m (°)^{−1} (Fig. 3, top; contours). Because solutions are independent of a specific profile, however, the *x* axis can also be interpreted as ocean depth.

Starting from *H*_{n} = 100 m at the northern boundary, the grounding depth *D*_{g}(*y*) deepens markedly toward the south, reaching *D*_{g}(*y*_{s}) = 227 m at the southern boundary. White arrows indicate the direction of the depth-integrated flow. Transports are directed poleward and slightly onshore in the coastal region and proportional to the ocean depth [cf. (14)]. Note that because of JEBAR the coastal flow does not follow the contours of constant potential vorticity (|*f*| increases poleward and layer thickness decreases toward the coast). The frontal current along *x*_{g} is directed equatorward, and the depth-integrated interior flow is westward in the north of the dashed line at *y* < 20.6°S and eastward in the southern part of the domain. In the interior ocean, sea surface height varies only with latitude and reaches a maximum of 0.02 m at about 14.7°S, before reducing to −0.28 m at the southern boundary. Sea surface height decreases shoreward across the front at *x*_{g}(*y*) by a maximum of 0.1 m at *y* = 22.6°S and by 0.02 m at *y* = *y*_{s}. Onshore of *x*_{g}(*y*), *η* increases to *η* = 0 at the eastern boundary.

Because the density varies horizontally in the upper layer, the horizontal currents have a vertical shear related to the thermal wind (section 2c). At the surface, where velocities are proportional to the SSH gradient [(4)], interior-ocean surface currents are eastward over much of the domain and westward only at *y* > 14.7°S. The SSH difference across the frontal current drives an equatorward current across the entire water column. With density decreasing shoreward across the current, the flow is strongest at the bottom of the layer. The integrated transport of the frontal current reaches a maximum of 4.4 Sv at 22.6°N and reduces to 3.4 Sv at *y* = *y*_{s}. It is of similar strength as the poleward flow in the coastal region, which reaches 3.9 Sv at the southern boundary.

### e. Grounding depth

Although generally no closed form solution can be found for the grounding depth *D*_{g}(*y*), we can still show that it is proportional to *H*_{n}. We start by defining *μ*(*y*) ≡ *D*_{g}/*H*_{n}. Substitution into (26) easily verifies that *μ*_{y} is independent of *H*_{n}. Because *μ*(*y*_{n}) = 1 (i.e., also independent of *H*_{n}), it follows that *μ*(*y*) is independent of *H*_{n} and hence *D*_{g} = *μH*_{n} ∝ *H*_{n}. Thus, by setting the stratification along the northern boundary, the Indonesian Throughflow has a strong control over the grounding depth that extends over the full length of the LCS. This strong impact is affiliated with coastally trapped Rossby waves, which allow for information to propagate poleward along the continental shelf.

The grounding depth is also sensitive to the coastal density profile *ρ*_{c}(*y*). Figure 4 (top) shows meridional profiles of the grounding depth in solutions where *ρ*_{cs} is varied and all other parameters remain as in the base solution. It indicates that the LC extends to larger depths when the density difference across the front decreases and the alongshore density gradient increases. (That the alongshore density gradient and the across-front density difference both contribute to the deepening is verified by changing parameters such that only one or the other changes.) Consequently, the LC also deepens when *ρ*_{os} is reduced (not shown). Changing the deep stratification offshore has relatively little effect on the grounding depth. When the layer two density is increased to *ρ*_{2} = 1027 kg m^{−3} in the base solution, for example, *D*_{g}(*y*_{s}) increases from 227 to 243 m, a change that is relatively small compared to those caused by changing *ρ*_{cs} (Fig. 4).

### f. SSH difference across the LUC

A feature of our base solution that has not been reported in observations is the shoreward decrease in SSH across the LUC, Δ*η* ≡ *η*_{o} − *η*_{c}. In fact, observations typically show the poleward flow of the LC extending over the LUC, indicating that the offshore decrease in SSH extends across the LUC (Δ*η* < 0). Figure 4 (bottom) shows Δ*η* when the coastal density profile is varied. The SSH difference increases when the density difference across *x*_{g} decreases and vice versa. In the case with *ρ*_{cs} = 1024 kg m^{−3}, the Δ*η* is negative, that is, the SSH increases eastward across the LUC over the southern part of the domain, somewhat closer to observations.

The SSH difference is also sensitive to the layer two density in our model. When the density is increased to *ρ*_{2} = 1027 kg m^{−3}, the maximum of Δ*η* doubles (not shown). Note that in a more realistic model than the one considered here, where the offshore stratification continuously varies with depth, the effective *ρ*_{2} can be smaller near the northern boundary, where *D*_{g} is small. Our results suggest that this reduction in *ρ*_{2} could then also lead to a reduction in Δ*η* and hence a more realistic SSH structure.

### g. Transports

*V*

_{1}in (14) across the coastal region, which givesThe LUC transport is given by (21). To compare it to

_{LC}, however, it is useful to write it in terms of the grounding depth. Substitution of (13) and (20) into (21) givesWith

*D*

_{g}∝

*H*

_{n}, we note that both transports are proportional to

Meridional profiles of _{LC} (black curves) and _{LUC} (gray curves) are shown in Fig. 5. When the coastal density profile is varied (Fig. 5, top; solutions are the same as in Fig. 4), _{LC} and _{LUC} both significantly increase with the alongshore density gradient (i.e., when *ρ*_{cs} increases), consistent with JEBAR becoming stronger. A similar increase in LC transport is also observed in OGCMs (e.g., Benthuysen et al. 2014). In all solutions, the LUC is considerably stronger than the LC in the northern part of the domain, which is different from realistic models and observations. This discrepancy may be due to the no-flow northern boundary condition. In the southern part of the domain, on the other hand, transports of the LC and LUC are of the same order and in better agreement with observations. When the layer two density is varied (Fig. 5, bottom), we find that the LC transport remains relatively constant, consistent with the grounding depth not changing significantly (section 3e). The LUC transport increases significantly when *ρ*_{2} increases, however, consistent with Δ*η* increasing across the front (section 3f).

*ρ*

_{cs}= 1024 kg m

^{−3}(Fig. 5, top; dotted curves) is that

_{LUC}reverses its sign near

*y*=

*y*

_{s}. Because of Δ

*η*< 0 at

*y*< 22°S (Fig. 4, bottom), the near-surface part of the frontal current is poleward there and equatorward at depth due to the thermal wind shear (4), much like in observations. The depth where the current changes its direction

*z*

_{o}(

*y*) can be obtained by zonally integrating (4) across the front and solving for

*z*, which givesIntegrating (4) zonally across the front and vertically from

*z*

_{o}to the surface then gives the transport of the upper, poleward portion of the frontal current. Assuming |

*z*

_{o}| ≫ |

*η*|, that transport isA more sensible definition of the LC and LUC transports in the case Δ

*η*< 0 is then given, perhaps, by adding Δ

*ρ*

_{cs}= 1024 kg m

^{−3}, where

*z*

_{o}= 0 at

*y*= 22°S and

*z*

_{o}(

*y*

_{s}) = −89 m, the poleward flow increases to Δ

### h. Coastal buoyancy balance

Although densities are prescribed in our model, it is instructive to consider the buoyancy budget in the coastal region. The two processes resolved by our model are meridional advection by the LC, *Q*_{A} ∝ _{LC}*ϕ*_{cy}, and entrainment of offshore waters into the LC, *Q*_{E} ∝ _{LCy}(*ϕ*_{c} − *ϕ*_{o}), which constitute a buoyancy increase (heating) and loss (cooling), respectively. In the base solution, *Q*_{A} is about twice as large as *Q*_{E} at all latitudes, which means that all other unresolved processes must combine to buoyancy loss to close the balance. The main processes contributing to the buoyancy budget in addition to *Q*_{A} and *Q*_{E} are surface buoyancy forcing and eddy mixing across the front (Domingues et al. 2006), both effectively cooling the LC, consistent with our base solution.

The buoyancy budget differs significantly in the solutions with *ρ*_{cs} = 1024 and 1026 kg m^{3}. In the former solution, both *Q*_{A} and *Q*_{E} are reduced relative to the base solution due to the weakened current. Because *ϕ*_{cy} decreases, whereas (*ϕ*_{c} − *ϕ*_{o}) increases at the same time, *Q*_{A} and *Q*_{E} roughly balance each other in the *ρ*_{cs} = 1024 kg m^{3} case, such that (unrealistic) surface heating may be required to close the budget. In the *ρ*_{cs} = 1026 kg m^{−3} solution, on the other hand, *Q*_{A} is much stronger because of the increased LC transport and alongshore density gradient, whereas *Q*_{E} = 0 because of *ϕ*_{o} = *ϕ*_{c}, and hence the solution requires a very large surface buoyancy flux (cooling) to balance advection.

Note that the model has problems to reproduce a realistic heat budget offshore. One reason is that (5) do not allow for a flow of light water across the northern boundary. A second reason is that the model does not allow for a vertical density gradient within the upper layer. Because the sheared flow across the western boundary tends to be more eastward near the surface (where water is lighter in reality) and more westward at the bottom of the upper layer (where water is denser in reality), the model ocean effectively loses buoyancy. This buoyancy loss is compensated by a somewhat unrealistic buoyancy gain in the equatorward-flowing LUC.

### i. Case ϕ_{c} = ϕ_{o}

*ϕ*

_{c}=

*ϕ*

_{o}. Interestingly, two solutions formally exist in that limit: one solution with and one without a front. Thus, by comparing the two solutions, we can isolate the changes in circulation that are due to including frontal dynamics in the model. In both solutions, the grounding depth can be integrated analytically. We start by setting

*ϕ*

_{o}=

*ϕ*

_{c}=

*ϕ*

_{1}in (25), which gives

*h*

_{o}≠

*h*

_{c}) and divide (31) by

*h*

_{c}−

*h*

_{o}. After using

*η*≪

*D*to neglect terms of order

*η*relative to terms of order

*D*, we getNote that the rhs of (32) is the same as that of (10), and consequently,

*x*

_{g}(

*y*) is coincident with the characteristic curve intersecting [

*x*

_{g}(

*y*

_{n}),

*y*

_{n}]. On the coastal side of

*x*

_{g},

*γ*(

*x*

_{g}) =

*ϕ*

_{n}

*H*

_{n}/

*f*

_{n}is then conserved along the grounding line, and it follows from (9) that the ocean depth along

*x*

_{g}is given by

*ϕ*

_{c}=

*ϕ*

_{o}, because

*h*

_{c}=

*h*

_{o}then ensures that the lhs of (31) vanishes. The grounding line

*x*

_{g}(

*y*) and

*D*

_{g}(

*y*) are then found using the matching condition

*h*

_{c}=

*h*

_{o}. Substitution of (20) and (13) givesNote that (34) is the grounding depth obtained in the solutions of Furue et al. (2013); hence, we retain their solution as a special case. Moreover, this solution does not exist in the case

*ϕ*

_{c}≠

*ϕ*

_{o}, because then the continuity cannot be satisfied across

*x*

_{g}without introducing a difference in layer thickness [to check that statement substitute

*h*

_{c}=

*h*

_{o}into (25)].

Both solutions with *ϕ*_{c} = *ϕ*_{o}, and all other parameters as in the base solution, are illustrated in Fig. 6. In the solution with an LUC (Fig. 6, top), the LC transport and its southward deepening are much stronger (also see Figs. 4 and 5, top). Offshore, the solution without LUC (Fig. 6, bottom) has decreasing sea surface height and onshore flow throughout the domain, whereas the LUC solution displays a sea surface height maximum at *y* = 16°S, and the depth-integrated flow is westward in the northern part of the domain (*y* > 22.5°S).

Why do multiple solutions exist in case *ϕ*_{c} = *ϕ*_{o}? From a mathematical standpoint, these can exist because (31) with (5) does not satisfy the Lipschitz condition with respect to *x*_{g}, that is, *x*_{gy} is well defined and bounded through (31) only for *h*_{c} ≠ *h*_{o}. From a physical standpoint, we have argued that the frontal dynamics are strongly associated with the density difference across the front, so considering the limit where the density vanishes may be a somewhat ill-posed problem. This is also indicated by *x*_{g} becoming aligned with a characteristic of constant *γ* in the frontal solution with *ϕ*_{c} = *ϕ*_{o}, because characteristics intersecting with *x*_{g} may provide a key mechanism for maintaining strong gradients across *x*_{g}. Note that a similar situation with multiple solutions arises in the limit *D*_{x} → ∞, such that the model approaches the limit of a flat bottom and vertical wall along the eastern boundary; because our solutions are independent of *D*_{x}, they remain essentially unchanged, just the width of the coastal region becomes infinitesimally small. Thus, although the boundary conditions become the same in the limit *D*_{x} → ∞, the eastern boundary solutions with a shelf do not approach the flat bottom solutions discussed in Schloesser et al. (2012, 2014). In this situation, the ambiguity is resolved when the finite strength mixing is included in the model, which introduces an alternative (minimum) width scale for the boundary current.

## 4. Summary and discussion

Relatively simple dynamical models with a uniform “coastal” region over the continental shelf and a two-layer structure farther offshore have been shown to simulate many observed features of eastern boundary current systems such as the Leeuwin Current system off the coast of Western Australia (Weaver and Middleton 1989, 1990; Furue et al. 2013). In particular, they suggest that the poleward Leeuwin Current (LC) is “driven” by the alongshore density gradient interacting with the bottom slope (JEBAR; Sarkisyan and Ivanov 1971). Perhaps the most significant feature of the current system not explained by these kinds of models is the equatorward Leeuwin Undercurrent (LUC). In the present study, we extend these previous models by allowing different surface densities to be prescribed in the offshore and coastal regions and introducing a more general (frontal) matching condition to describe the resulting density front. Thus, our model implicitly takes into account poleward density advection by the LC and, to some extent, determines the surface density internally by determining the grounding line. The inclusion of the front changes the circulation considerably.

Our model circulation now includes a Leeuwin Undercurrent. It flows along the boundary (front) between the coastal and offshore regions located just offshore of the LC and is forced by differences (jumps) in SSH and density across the front. In a base solution (section 3d), the model LUC is consistent with observations (e.g., Thompson 1984; Pattiaratchi and Woo 2009) and more realistic models (Domingues et al. 2007; Meuleners et al. 2007; Rennie et al. 2007) in that it has a reasonable maximum strength, it reaches a maximum before weakening toward Cape Leeuwin, and because the density increases shorewards, the LUC is strongest near the bottom. On the other hand, SSH decreases shorewards across the LUC, and hence the equatorward flow extends up to the surface, whereas in reality the flow typically reverses its direction and is poleward above the LUC. We have shown that these features can be modeled more realistically when the across-shore density difference increases or the offshore stratification decreases. Yet, it appears that the density structure of the model and the no-flow northern boundary condition used are still too simplistic to realistically simulate all features of the LUC at the same time.

The Leeuwin Current in our model strengthens and deepens toward the south. Although we impose a no-flow condition at the northern boundary, it reaches a transport of about 4 Sv at the southern boundary, which compares reasonably well with the observed mean current strength (e.g., Thompson 1984; Pattiaratchi and Woo 2009). A poleward strengthening and deepening of the LC is also observed in solutions without a density front and LUC; however, by including a front these features become significantly more pronounced (section 3i). The deepening of the LC also has implications for the energetics of the LCS. The associated downwelling of relatively light waters generates considerable available potential energy that can then be released through instabilities (such processes are not resolved by our model). Thus, the very pronounced deepening of the surface layer is somewhat consistent with the high eddy kinetic energy observed in the LC (Feng et al. 2005).

The divergences of the LC and LUC drive a zonal offshore circulation in the surface layer. Depth-integrated transports are westward in the northern part of the domain and eastward farther to the south. The sea surface height reaches a maximum in the northern part of the domain, and, because it is proportional to the sea surface slope, the near-surface flow is eastward at most latitudes.

Our model is restricted in several ways. In particular, it has very limited vertical resolution and does not resolve the processes determining the width of the across-shelf density gradient and hence that of the LUC. Resolving the LUC width may be the key to modeling a more realistic LCS, for example, by affecting the frontal mass balance (7) and by allowing the LUC to extend to greater depths. On the other hand, we note that the processes ultimately determining the LUC width may also not be realistically represented in OGCMs used to study the LCS (e.g., Domingues et al. 2007; Meuleners et al. 2007; Rennie et al. 2007; Benthuysen et al. 2014). Although eddy resolving, such models still have to parameterize submesoscale processes like mixed layer instabilities, which may be important given the sensitivity of the LC to the mixed layer depth. While the importance of eastern boundary current eddies and instabilities has been recognized, and is an active field of research (e.g., Spall 2010), our simple model illustrates how processes of different scales dynamically interact in a boundary current system and may be useful for interpreting more complex systems.

In conclusion, we have extended a Csanady (1985) type of coastal model to allow for alongshore fronts. Including such dynamics has strong implications for the LCS circulation, which now includes an LUC. Moreover, because the LCS stratification serves as boundary condition for the stratification offshore, LCS dynamics also affect the circulation in the interior ocean.

Helpful comments by Jay McCreary, Ryo Furue, Lew Rothstein, and Peter Cornillon and three anonymous reviewers are gratefully acknowledged. My work is supported through NASA Grant NNX11AF23G and NSF Grant OCE-1060397.

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