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We use an idealized numerical model to investigate the dynamics and fate of a small river discharging into the surf zone. Our study reveals that the plume reaches a steady state, at which point the combined advective and diffusive freshwater fluxes from the surf zone to the inner shelf balance the river discharge. At a steady state, the surf zone is well-mixed vertically due to wave-enhanced vertical turbulent diffusion and has a strong cross-shore salinity gradient. The horizontal gradient drives a cross-shore buoyancy-driven circulation, directed offshore at the surface and onshore near the bottom, which opposes the wave-driven circulation. Using a scaling analysis based on momentum and freshwater budgets, we determine that the steady-state alongshore plume extent (Lp) and the fraction of river water trapped in the surf zone depend on the ratio of the near-field plume length to the surf zone width (Lnf/Lsz) across a wide range of discharge and wave conditions, and a limited set of tidal conditions. This scaling also allows us to predict the residence time and freshwater fraction (or dilution ratio) in the steady-state plume within the surf zone, which range from approximately 0.1 to 10 days and 0.1 to 0.3, respectively. Our findings establish the basic dynamics and scales of an idealized plume in the surf zone, as well as estimates of residence times and dilution rates that may provide guidance to coastal managers. # Data from: Dynamics and scaling of a small river discharging into the surf zone [https://doi.org/10.5061/dryad.2280gb608](https://doi.org/10.5061/dryad.2280gb608) The present dataset includes the [COAWST model](https://www.usgs.gov/centers/whcmsc/science/coawst-a-coupled-ocean-atmosphere-wave-sediment-transport-modeling-system) outputs used to describe the dynamics and scaling of a small river discharging into the surf zone. ## File structure The data are structured as follows: 1. plume_scale.mat - Data of plume scales of all the cases, where * Hs: significant wave height [m] * Q: river discharge [m^3 s^-1] * L_nf: near-field plume length [m] * L_p: alongshore plume extent [m] * h_sz: water depth at the surf zone edge [m] * x_sz: surf zone width [m] * S_in: inflow salinity [PSU] * g_p: reduced gravity at the river mouth [m s^-2] * g_p*_*0: reduced gravity at the river mouth calculated using the density difference between river inflow and ambient ocean water [m s^-2] * Eta_0: water surface elevation anomaly at the river mouth [m] * V_sz: total volume of freshwater trapped in the surf zone [m^3] * T: the time required for the plume to reach a steady state [day] * L_t: plume turning distance [m] * S_bar: averaged salinity in the plume [PSU] 2. DepthAveraged.mat - Depth-averaged flow fields. DepthAveraged_BaseCase.mat, DepthAveraged_Case1.mat, DepthAveraged_Case3.mat, DepthAveraged_Case4.mat, DepthAveraged_Case6.mat, DepthAveraged_Case7.mat, DepthAveraged_Case8.mat, DepthAveraged_Case9.mat, DepthAveraged_Case16.mat, DepthAveraged_Case17.mat, DepthAveraged_Case18.mat, DepthAveraged_Case19.mat includes the results of the base case, cases 1, 3, 4, 6-9, and 16-19, respectively. In these files: * Wetdry_mask: wet/dry mask on RHO-points [binary] * Wetdry_mask_u: wet/dry mask on U-points [binary] * Wetdry_mask_v: wet/dry mask on V-points [binary] * Z: free-surface [m] * S: surface salinity [PSU] * Hs: significant wave height [m] * U: vertically integrated u-momentum component [m s^-1] * U_st: vertically-integrated u-Stokes drift velocity [m s^-1] * V: vertically integrated v-momentum component [m s^-1] * V_st: vertically-integrated v-Stokes drift velocity [m s^-1] 3. FullField_BaseCase.mat - 3D flow fields for the base case, where * Z: free-surface [m] * S: salinity [PSU] * Hs: significant wave height [m] * Lw: mean wavelength [m] * U: u-momentum component [m s^-1] * U_st: u-Stokes drift velocity [m s^-1] * V: v-momentum component [m s^-1] * V_st: v-Stokes drift velocity [m s^-1] * W: w-momentum component [m s^-1] * W_st: w-Stokes drift velocity [m s^-1] * Aks: salinity vertical diffusion coefficient [m^2 s^-1] * Akv: vertical viscosity coefficient [m^2 s^-1] * Cs_r: S-coordinate stretching curves at RHO-points [-] * Cs_w: S-coordinate stretching curves at W-points [-] 4. FreshwaterTrace_BaseCase.mat - Time series of freshwater volume and fluxes for the base case, where * i_sz: XI-index of the location of the surf zone edge [-] * i_shore: XI-index of the location of the shoreline [-] * Vsz: volume of freshwater in the plume in the surf zone [m^3] * Vis: volume of freshwater in the plume in the inner shelf [m^3] * Vsz_total: total volume of freshwater in the surf zone [m^3] * Vis_total: total volume of freshwater in the inner shelf [m^3] * R2SZ_flux: freshwater flux discharging into the surf zone [m^3 s^-1] * Vchannel: volume of freshwater in the plume in the river channel [m^3] * Vchannel_total: volume of freshwater in the river channel [m^3] * SBoundary_flux_SZ: the freshwater fluxes through the southern domain boundaries of the surf zone [m^3 s^-1] * SBoundary_flux_IS: the freshwater fluxes through the southern domain boundaries of the inner shelf [m^3 s^-1] * NBoundary_flux_SZ: the freshwater fluxes through the northern domain boundaries of the surf zone [m^3 s^-1] * NBoundary_flux_IS: the freshwater fluxes through the northern domain boundaries of the inner shelf [m^3 s^-1] * WBoundary_flux: the freshwater fluxes through the westhern domain boundary [m^3 s^-1] 5. DepthAveraged_XDiagnostic.mat - Depth-averaged diagnostic output of cross-shore momentum terms. DepthAveraged_XDiagnostic_BaseCase.mat includes the results of the base case at the steady state, and DepthAveraged_XDiagnostic_0day_1mWave.mat includes those at the start of river flow. In these files: * ubar_xadv: time-averaged 2D u-momentum, horizontal XI-advection term [m s^-2] * ubar_yadv: time-averaged 2D u-momentum, horizontal ETA-advection term [m s^-2] * ubar_xvisc: time-averaged 2D u-momentum, horizontal XI-viscosity term [m s^-2] * ubar_yvisc: time-averaged 2D u-momentum, horizontal ETA-viscosity term [m s^-2] * ubar_prsgrd: time-averaged 2D u-momentum, pressure gradient term [m s^-2] * ubar_zqsp: time-averaged 2D u-momentum, quasi-static pressure [m s^-2] * ubar_zbeh: time-averaged 2D u-momentum, Bernoulli head [m s^-2] * ubar_bstr: time-averaged 2D u-momentum, bottom stress term [m s^-2] * ubar_wbrk: time-averaged 2D u-momentum, wave breaking term [m s^-2] 6. DepthAveraged_YDiagnostic_BaseCase.mat - Depth-averaged diagnostic output of alongshore momentum terms, where * vbar_xadv: time-averaged 2D v-momentum, horizontal XI-advection term [m s^-2] * vbar_yadv: time-averaged 2D v-momentum, horizontal ETA-advection term [m s^-2] * vbar_xvisc: time-averaged 2D v-momentum, horizontal XI-viscosity term [m s^-2] * vbar_yvisc: time-averaged 2D v-momentum, horizontal ETA-viscosity term [m s^-2] * vbar_prsgrd: time-averaged 2D v-momentum, pressure gradient term [m s^-2] * vbar_zqsp: time-averaged 2D v-momentum, quasi-static pressure [m s^-2] * vbar_zbeh: time-averaged 2D v-momentum, Bernoulli head [m s^-2] * vbar_bstr: time-averaged 2D v-momentum, bottom stress term [m s^-2] * vbar_wbrk: time-averaged 2D v-momentum, wave breaking term [m s^-2] 7. grid.zip - Model grid file. * This grid file is designed for use with [ROMS](https://www.myroms.org/index.php), the hydrodynamic module of the COAWST modeling system. A diagram illustrating how the variables are placed on the grid and where the boundaries lie relative to the grid is available on [WikiROMS](https://www.myroms.org/wiki/Grid_Generation). * This grid file is in NetCDF format, which can be opened and used by a wide range of application software such as MATLAB, Python, and Panoply. For more detailed information, please refer to its [official website](https://www.unidata.ucar.edu/software/netcdf/). ## Code/Software All the post-processing scripts and data are prepared by MATLAB. 

Abstract We use an idealized numerical model to investigate the dynamics and fate of a small river discharging into the surf zone. Our study reveals that the plume reaches a steady state, at which point the combined advective and diffusive freshwater fluxes from the surf zone to the inner shelf balance the river discharge. At a steady state, the surf zone is well mixed vertically due to wave-enhanced vertical turbulent diffusion and has a strong cross-shore salinity gradient. The horizontal gradient drives a cross-shore buoyancy-driven circulation, directed offshore at the surface and onshore near the bottom, which opposes the wave-driven circulation. Using a scaling analysis based on momentum and freshwater budgets, we determine that the steady-state alongshore plume extent (L_p) and the fraction of river water trapped in the surf zone depend on the ratio of the near-field plume length to the surf-zone width (L_nf/L_sz) across a wide range of discharge and wave conditions and a limited set of tidal conditions. This scaling also allows us to predict the residence time and freshwater fraction (or dilution ratio) in the steady-state plume within the surf zone, which ranges from approximately 0.1 to 10 days and from 0.1 to 0.3, respectively. Our findings establish the basic dynamics and scales of an idealized plume in the surf zone, as well as estimates of residence times and dilution rates that may provide guidance to coastal managers. Significance StatementSmall rivers and estuaries often carry pollutants, sediments, and larvae into the coastal ocean, where wave action in the surf zone can trap them near the shore. This process can play an important role in the flux of material into and out of the nearshore ecosystem and presents a potential risk to swimmers when materials are harmful. The present study uses a numerical model to investigate the fate of freshwater discharged from small rivers into the surf zone and the processes through which trapped riverine freshwater escapes from the surf zone. These results establish a basis for predicting the fate of river-borne materials from coastal rivers and understanding the exchange between the surf zone and the inner shelf. Additionally, this work provides a theoretical framework for predicting the residence time and concentration of river-borne material trapped in the surf zone.