Estuarine models: what is under the hood? Correlation skill 0 psu Bottom salinity 34 psu

Lin and Kuo 2003

Variables Sediment concentration: C Velocities: u, v, w Water level:  Water density:  Salinity and temperatures: S, T Basic variables: 7 (excludes sediment concentration) Equations needed: 7

Process Differential Equations Boundary conditions Topology/ bathymetry + + Numerical algorithm Discretization (grid) Algebraic equations Code & computer Post-processing Solution (variables are known at grid locations) Skill assessment

Topology/bathymetry

Discretization Refined grid (“hi-res”) fDB16 # nodes: 27416 # elements: 53314 #  levels 24 min element area: 942 m^2 max element area: 89834 m^2 Refined grid (“hi-res”) fDB16

Grays River: example of cascading grids

Grays river: detail

Introduction to governing equations Continuity Depth-averaged form: Salt and heat conservation

Introduction to governing equations Conservation of momentum (from Newton’s 2nd law: f=ma)

Introduction to governing equations Equation of state  =  (s, T, p) Turbulence closure equations

Conservation of mass - water Consider a control volume of infinitesimal size dz dy dx Let density = Let velocity = Mass inside volume = Mass flux into the control volume = Mass flux out of the control volume =

Conservation of mass-water Conservation of mass states that Rate of change of mass inside the system = Mass flux into of the system – Mass flux out the system Thus and, after differentiation by parts

Conservation of mass - water Rearranging, For incompressible fluids, like water and, thus

Conservation of mass of a solute Consider a 1D system with stationary fluid and a solute that is diffusing dz dy dx Let flux of mass per unit area entering the system = Let flux of mass per unit area leaving the system = Let concentration (mass /unit volume) of solute inside the control volume = C

Conservation of mass of a solute - diffusion Conservation of mass states that Mass flux of solute leaving the system – mass flux of solute entering the system = rate of change of solute in the system Thus or or

Conservation of mass of a solute - diffusion How do we quantify qD ? In a static fluid, flux of concentration (q), occurs due to random molecular motion It is not feasible to reproduce molecular motion on a large scale. Thus, we wish to represent the molecular motion by the macroscpoic property of the solute (its concentration, C) Also, from observation we know In a fluid of constant C (well mixed liquid), there is no net flux of concentration Solute moves from a region of high concentration to regions of low concentration Over some finite time scale, the solute does not show any preferential direction of motion

Conservation of mass of a solute - diffusion Based on these observations, Adolph Fick (1855) hypothesized that (molecular processes are represented by an empirical coefficient analogous to diffusivity) or in three dimensions Fick’s law Diffusion coefficient Applying Fick’s law to the 1D mass conservation equation for a solute, we get or