## 1. Introduction

[2] Depth- and time-averaged surf zone currents, initiated by the transfer of momentum due to wave breaking, are often classified in the literature as alongshore-uniform (1DH) or nonuniform (2DH). 1DH flow corresponds to an alongshore-uniform shore-parallel jet [*Bowen* 1969a; *Longuet-Higgins*, 1970a; *Thornton and Guza*, 1986; many others], and occurs on long straight beaches with oblique wave angles. 2DH flows contain across-shore-directed currents, with an extreme case being a rip current system [*MacMahan et al*., 2006 and references therein]; these typically occur on alongshore-nonuniform beaches with small (nearly shore-normal) wave angles.

[3] Between these two extremes, there is a continuum of flows typified by meandering or separation of the alongshore current. In these cases, an alongshore current is propagating over spatially varying bathymetry or through a spatially varying wave field, and hence exhibits somewhat-2DH behavior, although perhaps less pronounced than a rip current. These, too, have been well documented [*Sonu*, 1972; *Slinn et al*., 2000], but have received less theoretical treatment than the extreme cases described above. One exception is the numerical study by *Wu and Liu* [1984], who simulated steady 2DH flows on undulating bathymetry with oblique wave angles, and found that the results did not correspond to a linear superposition of simple 1DH and 2DH end-members. Instead, nonlinear advection was shown to play a leading-order role in determining the magnitude and alongshore position of flow undulations. When the strength of the background alongshore current was increased, flow undulations became increasingly weaker, and were pushed downstream, illustrating the fact that advection acts to bypass minor variations in forcing. This effect can be important for the prediction of surf zone circulation in the field. For example, *Long and Özkan-Haller* [2005] present a realistic case where a linearized model predicted the presence of a rip current extending past the surf zone, whereas a nonlinear model did not. They explain that the inertia of the background flow was acting to “sweep away” smaller rips in the nonlinear model. Hence, an understanding of what constitutes a “strong” background flow, or a “minor/short” variation in forcing, is important for bridging the gap between 1DH and 2DH regimes.

[4] Our approach is to study the role of nonlinear advective terms in a weakly nonlinear setting, using a combination of numerical experiments and linearized analysis. In section 2, we investigate advection by selectively including/excluding it in a numerical model (equations (1)-(3), in which the advective terms are in square brackets). The difference between the alongshore current with and without advection is interpreted in terms of an effective transfer function, analyzed using spectral methods, which provides a quantified description of the length-dependent shifting/dampening of flow perturbations described above. In other words, we hypothesize that advection can be viewed as a spatial filter, which we seek to characterize using a transfer function. Then, in section 3, we analyze simplified equations of motion to better understand the numerical results. We find that the numerical results can be largely explained using a single nondimensional parameter, which quantifies the strength of the advective terms. We also derive an asymptotic approximation for the transfer function. This approximation is only valid under specific simplifying conditions, which we describe, but may represent useful progress toward a more general understanding.