The governing equations for a shallow surfacewater system consist of conservation of mass, volume, and momentum. Leendertse and Gritton (1971) and Leendertse (1987) present the following governing equations, which were modified by Swain et al. (2004) to include aerially distributed sources and sinks, describing the (1) conservation of water volume, (2) conservation of momentum in the x-direction, (3) conservation of momentum in the y-direction, and (4) solute mass transport:

D

D

D

D

where h is water stage [L], d is water depth [L], v_x and v_y are vertically averaged velocities in the x- and y- directions [LT^-1], q_sg is a source/sink term representing the volumetric exchange between surface water and ground water per unit area [LT^-1], q_r is a rainfall source term representing the volumetric rate per unit area [LT^-1], q_et is an evapotranspiration sink term representing the volumetric rate per unit area [LT^-1], f is the Coriolis parameter [T^-1], g is gravitational acceleration [LT^-2], is water density [ML^-3], R is the bottom-stress coefficient [T^-1], C_d is the windstress coefficient [L⁰], _a is air density [ML^-3], W is wind speed [LT^-1], is the angle between wind direction and the positive y-axis [degrees], k is the horizontal momentum-exchange coefficient [L²T^-1], C is solute concentration for a conservative nonreactive constituent [ML^-3], D_x and D_y are the dispersion coefficients in the x- and y-directions [L²T^-1], C_sg is the leakage concentration between surface water and ground water [ML^-3], and C_r is the solute concentration of rainfall. In this paper, the source concentration for rainfall and the sink concentration for evapotranspiration are both assumed to be zero, because C represents salinity concentration, which is considered conservative and non-reacting. The transport equation (Eq. (4)) can easily be extended to represent reactive and decaying constituents. Fluid density is related to salinity, in practical salinity units (psu), using the following equation of state:

D

where _f is the reference fluid density (that is, the density of freshwater) [ML^-3], and /C is the slope of a linear relation between fluid density and salinity [L⁰]. For salinities ranging between freshwater and typical seawater, /C has an approximate value of 0.7. The effect of temperature on fluid density is not considered here, although it could be important for some applications. For the Everglades application, seasonal temperature variations can be substantial, but spatial variations are assumed to have a negligible effect on flow. Simultaneous solutions to Eq. (1), (2), (3), (4), (5) result in spatial distributions for h, C, , v_x, and v_y.