1. Introduction
Wellmixed (or nearly barotropic) outflows onto sloped bathymetry are common geophysical phenomena including rip currents, tidal inlet jets, flows through straits, and wavedriven jets through channels on coral reefs (Talbot and Bate 1987; Wolanski 1988; Symonds et al. 1995; Brun et al. 2020). These circulation features are characterized by steep gradients in water properties such as momentum and density relative to the ambient fluid and can play a key role in transporting scalars in the coastal ocean (Suanda and Feddersen 2015; Moulton et al. 2023). The physical scales of environmental jets determine which forces (inertia, bottom drag, rotation, etc.) affect their behavior. Smallscale outflows have been well studied in engineering literature (Jones et al. 2007), while largescale jets have been examined in geophysical fluid dynamics studies (Flexas et al. 2005). Wellmixed environmental outflows like reef pass jets and rip currents, however, can occur at intermediate transport and horizontal scales of
A key feature of barotropic and stratified outflows alike is that the governing dynamics tend to change along their course, typically understood as a transition from nearfield to farfield dynamics as the length and time scales develop (HornerDevine et al. 2015). In largescale jets/plumes, the near field can be inertia dominated (Ro > 1), while the far field tends to be rotationally dominated (Ro < 1). Planetary rotation can turn mid to low Ro outflows in the near field to eventually orient downcoast as a coastal current. It is not clear a priori how the Coriolis acceleration will affect the solution of intermediatescale outflows, but it is expected to depend on the relative impact of other coastal ocean dynamical processes such as inertia, topographic gradients, and bottom drag. Smallscale flows are typically inertia–pressure gradient dominated and are negligibly affected by rotation. Bottom friction can influence outflows at a range of scales depending on the jet/plume aspect ratio and bottom drag coefficient (Atkinson 1993; McCabe et al. 2009; HornerDevine et al. 2015).
Potential vorticity conservation has often been used to study and predict the behavior of outflows, especially in the far field (Whitehead 1985; An and McDonald 2004). The application of potential vorticity conservation can be used to obtain properties such as the width, depth, and transport of the farfield coastal current (Thomas and Linden 2007). Prior analytical solutions for the trajectory and transport of barotropic outflows based on vorticity conservation, however, are not wellbehaved on steep slopes, especially in the nearfield jet region (Beardsley and Hart 1978). In the near field, deflection and spreading cooccur, and examining individual vorticity components may clarify the kinematics of outflow jets there. Relative vorticity is comprised of shear and curvature constituents that arise in a natural coordinate system (Chew 1974). As an outflow jet deflects and spreads over the shelf slope, the curvature and shear vorticity, respectively, undergo transformation and potentially interchange (Chew 1975). Examining the dynamics of the individual shear and curvature relative vorticity constituents, therefore, can identify the physical processes governing the kinematics of a jet. This approach differs from using potential vorticity alone, where the shear and curvature vorticity constituents are combined.
Path equations for comparable geophysical flow features have been derived by considering the curvature vorticity constituent. Curvature vorticity has been used to derive path equations for the meandering of the Gulf Stream, approximated as a free jet over linear topography (Warren 1963). This theoretical model was later extended to describe the trajectory of the weakly nonlinear flow over arbitrary topography, capturing behaviors such as jet trapping, retroflection, and deflection (CushmanRoisin et al. 1997). Contour dynamics has also been used to predict the trajectory of outflows (Kubokawa 1991; Flierl 1999; Southwick et al. 2017). All of these models, however, assume potential vorticity conservation and do not account for bottom drag. For outflows onto a slope, significant vorticity can be generated and dissipated via bottom friction and topographic interaction, which should be accounted for.
Here, we derive a fully nonlinear dissipative streamwise curvature dynamics equation and analyze the kinematics (i.e., deflection and spreading) of a barotropic jet outflow onto a linear slope in shallow water. From the curvature dynamics equation and continuity equations, we then derive a simplified 1D streamwise system of ordinary differential equations (ODEs) valid for the domain interest—an idealized circular island domain with a linear slope. Then, we evaluate the validity of the 1D analytical model against idealized horizontally twodimensional depthintegrated (2DH) ocean circulation simulations across a range of bottom slopes, bottom friction, and Coriolis parameter values. In doing so, we are able to identify the influence of key physics on the fate of an outflow jet from the perspective of curvature dynamics, taking a particular interest in the poorly understood nonlinear nearfield region.
2. Theory
The path equations: A 1D ODE system
3. Methods
a. The idealized domain
b. Numerical model configuration
The depthintegrated equations of motion were solved numerically on this domain with the Regional Ocean Modeling System (ROMS; Shchepetkin and McWilliams 2005) with a uniform density field. A depthdependent quadratic bottom friction scheme was used corresponding to (4), where C_{D} is a constant bottom friction coefficient. The harmonic lateral viscosity was set to 0.2 m^{2} s^{−1}, consistent with other coastal oceanographic modeling studies (e.g., Kumar and Feddersen 2017). The simulations were integrated for 10 days with a time step (Δt = 0.25 s) in order to capture several inertial periods. The Coriolis parameter was set constant throughout the domain using the fplane approximation. The grid spacing was 15 m in the radial direction and 15–20 m in the azimuthal direction, giving a horizontal computational grid of dimensions (512 × 256). The inflow speed V_{0} was set to 0.125 m s^{−1} for all runs, which corresponds to an inflow rate of Q = 615 m^{3} s^{−1}, comparable to large reef pass jets (Herdman et al. 2017) and moderately sized rivers (Cole and Hetland 2016; Lemagie and Lerczak 2020). Radiation boundary conditions for momentum and the free surface were applied at the azimuthal and outer radial boundaries. Model simulations were carried out on the SuperMIC resource through LSU HPC, leveraging the parallelizability of ROMS. Each simulation required about 2500 nodehours, using 160 cores per run. The norm of the domainintegrated kinetic energy was computed to estimate convergence, and all runs were satisfactorily converged (δ_{KE}/ǁKEǁ < 0.01% h^{−1}) after 5 days, after which several inertial periods were captured. As the focus of this paper is on the steady behavior of the flow, model results shown here were timeaveraged over the final 3 h of the simulation, similar to the flow time scale given by τ = V_{0}/L_{J}, the jet initial velocity divided by the streamwise length scale of interest L_{J} ≈ 2 km, over which the jet typically has transitioned from nearfield to farfield dynamics.
c. Numerical experiments
The ODE system [(13)] was solved using the initial conditions from the circulation model. The initial values y_{0} were selected to minimize the least squared error between the path of the ODE jet and the path of the jet from the circulation model but only within 10% of y_{0}. SciPy’s bounded minimize routine was used (Virtanen et al. 2020). As the ODE system is nonlinear and sensitive to initial conditions, it is possible that there are useful and acceptable solutions in close proximity to nonphysical ones, so providing a narrow range of the initial values is appropriate. For typical parameters, the ODE system is unstable to perturbations about the initial state, as the real parts of the local Lyapunov exponents (the eigenvalues of the linearized Jacobian) are positive. ODE instability combined with the dimensionality of the phase space (n = 4) and the presence of nonlinearity suggest that the ODE system is likely to exhibit chaotic properties (Arfken et al. 2013); however, further investigation is beyond the scope of this study.
d. Center streamline identification
4. Results
a. Jet kinematics and trajectories
The set of numerical simulations captured a wide variety of jet behaviors as the model parameters were varied (Figs. 4 and 5). The influence of planetary rotation on the path of the jet was dramatic in simulations with steep bottom slope (Λ = 0.1). For cases more affected by planetary rotation [−15°, −30°], typical jet excursions (the crossshore distance traveled before rectifying to isobaths) were 1–2 km offshore. Despite the relatively high nearfield Rossby numbers of the jet (Ro ≈ 10), planetary rotation notably deflects the trajectory of the geophysically small jet flowing across a steep slope. The jet stayed coherent and narrow over a larger streamwise distance for low C_{D} cases, and noticeable spreading does not occur until after the jet is oriented downcoast. For high C_{D,} the jet spreads considerably more in the near field.
The trajectory of the numerically modeled jets compares favorably to predictions of the ODE system [(13)] given similar initial conditions (Fig. 4). It is apparent that the quality of the ODE solutions is best in the near field and departs from the corresponding 2DH solutions with increasing streamwise distance. This is to be expected because key simplifications (i.e., ϒ ≫ 1 and VK ≫ −∂V/∂n) are inevitably violated when the jet rectifies downcoast along isobaths. The ODE2DH model agreement was reasonably robust to parameter combinations testing its other core assumptions: parallel streamlines, no velocity shear at the centerline, and steadiness. For larger values of C_{D}, the jet exhibits radially spreading behavior; streamlines diverge from another and the centerline shear vorticity magnitude minimum is not particularly distinct. This implies that the shear vorticity is not weaker than the curvature vorticity, which will be verified later. The ODE system, nonetheless, reasonably recovers the path of the jet in the near field in most cases. 1D ODE solutions on shallow topographic slopes do not perform as well as on steep slopes for most combinations of C_{D} and f, as radial spreading due to friction more readily dominates over topographic divergence in the near field. This allows terms not accounted for in the reduced physics model, such as confluence veering and speed torque, to potentially influence the solution. Thus, the 1D ODE model is most useful under high inertia/low drag conditions.
Having demonstrated the ability of the reduced physics model to predict jet trajectory, we now examine the validity of the assumptions made in its derivation by examining the vorticity, divergence, and curvature budgets of the numerical model output. We select a representative case from the suite of runs to examine in detail, where the influences of the curvature terms in (10) on the jet are clearly illustrated and consistent with the simplifying assumptions: a steep slope, strong planetary rotation, and relatively weak bottom friction [ϕ = −30°, Λ = 0.1, C_{D} = 0.125] (Fig. 6). Over a path distance of 2 km, the jet orients as a coastal current downcoast after a ∼1km crossshelf excursion. A return flow is present on both sides of the jet as a result of the inflow boundary condition, which seems to contribute to its spreading.
b. Vorticity and divergence
Recall that the vorticity of any flow can be decomposed into shear (−∂V/∂n) and curvature (VK) components [(7)]. In the jet, shear vorticity of opposite signs is present on opposing sides of the center streamline (Fig. 6e), while the curvature vorticity of the feature indicates the sign and intensity of its deflection (Fig. 6b). The shear vorticity weakens as the jet spreads and the spanwise length scale increases, with clear minima along the centerline. The smoothly varying and consistently positive sign of curvature vorticity is consistent with the jet coherently deflecting anticlockwise, relaxing as it transitions into a farfield coastal current. The alongcenterline curvature vorticity begins in inertial balance (VK ≈ f) and dominates over the shear vorticity throughout the near field of the trajectory for ∼1.3 km. As the flow orients to isobaths, the curvature vorticity approaches the topographic curvature vorticity set by the island geometry (Fig. 7b). As long as the curvature vorticity dominates over the shear vorticity, the simplifications required arrive at the simplified curvature dynamics equation [(11)] and are reasonable.
The other key simplification in deriving (11) is that the volume dynamics are predominantly controlled by streamwise topographic change rather than diffluent streamlines. The rapid deceleration of the jet as it encounters the slope is clearly shown by the topographic divergence (Fig. 6c), while the weak, but steady spreading of the jet can be attributed to diffluence divergence (Fig. 6f). The topographic component dominates for about 1 km along the center streamline as the jet aligns toward isobaths until it no longer experiences a streamwise depth gradient (Fig. 7a). Thus, the assumption ϒ ≫ 1 is most valid in the jet near field. It is apparent that the assumptions about the divergence and vorticity components requisite for the simplified model are most valid in the near field of the jet. This is consistent with the adherence of the ODE system trajectories to the 2D numerical model in the near field of the jet and less congruence in the far field (Figs. 4 and 5). In summary, the terms involving curvature/shear vorticity and topographic/diffluence divergence dominate in the near field and far field of the jet, respectively (Fig. 7).
c. Curvature dynamics
Having justified the simplifications required to derive the reduced physics model in the near field of the jet, we can explicitly examine the curvature dynamics. By examining the leadingorder balances, we can gain physical intuition into the mechanisms by which the jet is deflected, with the terms readily interpreted as directing the curvature of the center streamline one way or another. Along the jet center streamline in the representative case, the leadingorder nearfield balance is between the curvature and Coriolis stretching terms, slope torque, and curvature dissipation (Fig. 8). In this region (s ≤ 500 m), the jet’s curvature increases after emerging in nearly inertial balance (Fig. 7). As the jet moves offshore, curvature stretching turns the jet anticlockwise in the same direction of its initial curvature. The Coriolis stretching term opposes the curvature stretching term, nudging the jet toward inertial balance. The two important effects of bottom drag in the near field are the slope torque and curvature dissipation. Slope torque arises because the jet exits at a slight angle relative to isobath normals and contributes to the anticlockwise deflection of the jet. Curvature dissipation relaxes the curvature of the jet and incidentally increases its offshore excursion scale. The net effect of the bottom drag terms, however, is not particularly substantial in this case. Note that the dominant terms in the near field are exactly the four terms identified by the reduced physics model [(11)], confirming the validity of the ODE system under these conditions.
On both flanks of the jet, shear divergence is primarily in balance with the speed torque and shear stretching terms (Fig. 9). The sign of the shear divergence is consistent with the jet spreading (i.e., attenuation of the spanwise shear). Speed torque acts to spread the jet, while shear stretching opposes it, especially in the near field when the jet is normally incident to isobaths. Confluence veering and shear divergence are strong at the jet–ambient interface, across which the velocity reverses direction and streamlines deflect (Fig. 9).
We now consider curvature dynamics variation across the parameter space. The curvature gradient flux [
As the spreading of the jet is enhanced over shallower slopes (Fig. 4), confluence veering becomes progressively more important (Fig. 10). Speed torque and shear divergence also become increasingly important in the near field of jets on shallower slopes, for which the centerline is not as distinct. Shear and spreading are neglected in the reduced physics model. Speed torque and shear divergence are often inversely proportional to another, but speed torque remains relevant in the far field while shear divergence does not. Instead, the Coriolis stretching and curvature dissipation balance against speed torque in the far field. This implies that the streamwise shear variation is related to the Coriolis acceleration and bottom slope and inversely to bottom drag. In other words, the vorticity acquisition due to friction and sloped topography occurs in the shear vorticity components rather than the curvature components for lower slope cases. Shear stretching, horizontal viscosity, and unsteadiness are negligible along the jet streamline.
5. Discussion
a. Jet behavior and classification
The curvature of the coastline (K_{i}) may also affect the behavior of the jet. Bathymetric curvature and planetary rotation set a shallow bound on the isobath upon which the jet can settle into a coastal current. On a circular island, the coastal current could be in cyclogeostrophic balance for a topographic Rossby number of unity VK_{i}/f ≤ 1. If we assume negligible spreading, (Q_{i} = V_{0}h_{0}w_{0} = V_{i}h_{i}w_{0}) and h_{i} ≈ Λ/K_{i}, then the topographic Rossby number of the coastal current is
b. Extension of curvature dynamics to buoyant flows
6. Summary
We derived the general form of the curvature dynamics equation and used it to study the deflection and spreading of an idealized barotropic jet outflow on an idealized coast. By integrating along the center streamline and assuming a bathymetric profile, we were able to use the curvature dynamics equation to develop a 1D ODE system along the center streamline coordinate that effectively predicted the nearfield trajectory of the jet when compared against 2D numerical solutions of the governing equations. The key assumptions made in deriving the reduced physics model, minimal shear vorticity, and weak diffluence relative to topographic divergence along the center streamline were verified in the near field of the jet but not in the far field after the jet had oriented itself along isobaths. Prior simplified kinematic models of barotropic outflow jets break down in the nearfield region (Beardsley and Hart 1978), where the ODE system developed here is especially accurate.
Consideration of the outflow’s curvature dynamics particularly clarified the role of the Coriolis–topographic interaction in steering the jet. A barotropic flow stretching vertically as it flows across a steep slope will experience an alongstreamline curvature gradient flux. Thus, rather than following a steady inertial trajectory, the jet’s curvature is enhanced in the near field due to vortex stretching before being rectified to isobaths by Coriolis acceleration and bottom drag. The relaxation of curvature in the nearfield to farfield transition, requisite for the jet to rectify to isobaths as a coastal current, can therefore be attributed to the Coriolis stretching and curvature dissipation. In the far field, the jet reached a quasisteady state where the flow curvature approaches that of the topography and alongstream variation of flow properties is minimal.
Prior applications of curvature dynamics have generally been presented in a simplified form in oceanographic studies. Ochoa and Niiler (2007), for instance, consider a balance between curvature gradient flux and the Beta effect, finding a strong agreement with the path of the Agulhas Current. Niiler and Robinson (1967) consider a balance between advective, topographic, and rotational effects as a model for the Gulf Stream, and Chew (1974) additionally incorporates vortex tilting and shear vorticity effects on curvature to study drogue paths in the Gulf of Mexico Loop Current, but neither study accounted for bottom drag. To our knowledge, outflow jets have not yet been analyzed from a curvature dynamics perspective. In this application, topography, advection, drag, rotation, and shear terms can all be of leading order and must be considered to explain the behavior of the flow. The more general and inclusive dynamics captured in (10) can therefore be used to better understand more complex coastal circulation patterns.
It was found that even over relatively small
Acknowledgments.
Support for this work came from Duke University, National Science Foundation Physical Oceanography and LTER Programs (OCE1435133, OCE2123708, OCE1637396, and OCE2224354), National Center for Supercomputing Applications Blue Waters Fellowship, American Australian Association Graduate Fellowship, and the Honda Marine Science Foundation. Code development, numerical modeling, and computational resources were supported by the National Science Foundation XSEDE Program (TGOCE170004; LSU HPC allocation 1920), the Blue Waters sustainedpetascale computing project which is supported by the National Science Foundation (OCI0725070 and ACI1238993), the State of Illinois, and the National GeospatialIntelligence Agency. We acknowledge Dr. Rob Hetland and an anonymous reviewer for thoughtful comments that improved the manuscript.
Data availability statement.
Model input files and results for the simulations analyzed in this article are available in a Zenodo repository (https://10.5281/zenodo.8355246).
APPENDIX
Computing Curvature Dynamics Equation Terms in Cartesian Coordinates

Curvature$\frac{\partial \alpha}{\partial s}=\frac{1}{{V}^{2}}\left[{u}^{2}\frac{\partial \upsilon}{\partial x}{\upsilon}^{2}\frac{\partial u}{\partial x}u\upsilon \left(\frac{\partial u}{\partial x}\frac{\partial \upsilon}{\partial y}\right)\right].$

Confluence/diffluence$\frac{\partial \alpha}{\partial n}=\frac{1}{{V}^{3}}\left[{u}^{2}\frac{\partial \upsilon}{\partial y}+{\upsilon}^{2}\frac{\partial u}{\partial x}u\upsilon \left(\frac{\partial \upsilon}{\partial x}+\frac{\partial u}{\partial y}\right)\right].$

Divergence$\frac{\partial V}{\partial s}=\frac{1}{{V}^{2}}\left[{u}^{2}\frac{\partial u}{\partial x}+{\upsilon}^{2}\frac{\partial \upsilon}{\partial y}+u\upsilon \left(\frac{\partial \upsilon}{\partial x}+\frac{\partial u}{\partial y}\right)\right].$

Streamnormal shear$\frac{\partial V}{\partial n}=\frac{1}{{V}^{2}}\left[{u}^{2}\frac{\partial u}{\partial y}+{\upsilon}^{2}\frac{\partial \upsilon}{\partial x}u\upsilon \left(\frac{\partial u}{\partial x}\frac{\partial \upsilon}{\partial y}\right)\right].$
Equations (A3)–(A6) can be used to compute the vorticity and divergence components as in (7) and (8). After additionally computing the first and secondorder velocity derivatives, it is possible to directly compute the streamwise curvature gradient and shear divergence in Cartesian coordinates.

Streamwise curvature gradient$\frac{\partial K}{\partial s}=\frac{u}{{V}^{4}}\left[{u}^{2}\frac{{\partial}^{2}\upsilon}{\partial {x}^{2}}+2u\frac{\partial u}{\partial x}\frac{\partial \upsilon}{\partial x}{\upsilon}^{2}\frac{{\partial}^{2}u}{\partial x\partial y}2\upsilon \frac{\partial \upsilon}{\partial x}\frac{\partial u}{\partial y}\left(u\frac{\partial \upsilon}{\partial x}+\upsilon \frac{\partial u}{\partial x}\right)\left(\frac{\partial u}{\partial x}\frac{\partial \upsilon}{\partial y}\right)u\upsilon \left(\frac{{\partial}^{2}u}{\partial {x}^{2}}\frac{{\partial}^{2}\upsilon}{\partial x\partial y}\right)\right]+\frac{\upsilon}{{V}^{4}}\left[{u}^{2}\frac{{\partial}^{2}\upsilon}{\partial x\partial y}+2u\frac{\partial u}{\partial y}\frac{\partial \upsilon}{\partial x}{\upsilon}^{2}\frac{{\partial}^{2}u}{\partial {y}^{2}}2\upsilon \frac{\partial u}{\partial y}\frac{\partial \upsilon}{\partial y}\left(u\frac{\partial \upsilon}{\partial y}+\upsilon \frac{\partial u}{\partial x}\right)\left(\frac{\partial u}{\partial x}\frac{\partial \upsilon}{\partial y}\right)u\upsilon \left(\frac{{\partial}^{2}u}{\partial x\partial y}\frac{{\partial}^{2}\upsilon}{\partial {y}^{2}}\right)\right]3K\left(\frac{u}{{V}^{2}}\frac{\partial V}{\partial x}+\frac{\upsilon}{{V}^{2}}\frac{\partial V}{\partial y}\right).$

Shear divergence$\frac{\partial}{\partial s}\frac{\partial V}{\partial n}=\frac{u}{{V}^{3}}\left[{u}^{2}\frac{{\partial}^{2}u}{\partial x\partial y}+2u\frac{\partial u}{\partial x}\frac{\partial u}{\partial y}{\upsilon}^{2}\frac{{\partial}^{2}\upsilon}{\partial {x}^{2}}2\upsilon {\left(\frac{\partial \upsilon}{\partial x}\right)}^{2}u\upsilon \left(\frac{{\partial}^{2}u}{\partial {x}^{2}}\upsilon \frac{{\partial}^{2}\upsilon}{\partial x\partial y}\right)\left(u\frac{\partial \upsilon}{\partial x}+\upsilon \frac{\partial u}{\partial x}\right)\left(\frac{\partial u}{\partial x}\frac{\partial \upsilon}{\partial y}\right)\right]+\frac{\upsilon}{{V}^{3}}\left[{u}^{2}\frac{{\partial}^{2}u}{\partial {y}^{2}}+2u{\left(\frac{\partial u}{\partial y}\right)}^{2}{\upsilon}^{2}\frac{{\partial}^{2}\upsilon}{\partial x\partial y}2\upsilon \frac{\partial \upsilon}{\partial x}\frac{\partial \upsilon}{\partial y}u\upsilon \left(\frac{{\partial}^{2}u}{\partial x\partial y}\upsilon \frac{{\partial}^{2}\upsilon}{\partial {y}^{2}}\right)\left(u\frac{\partial \upsilon}{\partial y}+\upsilon \frac{\partial u}{\partial y}\right)\left(\frac{\partial u}{\partial x}\frac{\partial \upsilon}{\partial y}\right)\right]2\frac{\partial V}{\partial n}\left(\frac{u}{{V}^{2}}\frac{\partial V}{\partial x}+\frac{\upsilon}{{V}^{2}}\frac{\partial V}{\partial y}\right).$
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