Presentation of the paper: An unstructured grid, three- dimensional model based on the shallow water equations Vincenzo Casulli and Roy A. Walters Presentation by Charles Seaton All figures taken from the paper unless otherwise specified

UnTrim is an extension of the Trim family of models to an unstructured grid. The model is 1)Semi-implicit 2)Finite difference for the momentum equations 3)Finite volume for the continuity equation The version of the model given in the paper is: 1)Barotropic (only handles constant density) 2)Handles wetting and drying 3)Ambiguous on the use of vertical discretization 4)Ambiguous as to turbulence closure method 5)Flexible in its handling of advection, vertical viscosity, and coriolis

Governing Equations Barotropic momentum equations Continuity equation Depth integrated continuity equation

Orthogonal Unstructured Grid Centers defined by a set of line segments which intersect the element sides perpendicularly. Elements can conceptually be any convex polygon For model implementation, elements can be triangular or quadrilateral. U eta Elevation is defined at the element center Velocity is defined at the intersection of the dividing line and the element side Height is defined at the element sides h

Delaunay triangulation and Voronoi regions Voronoi regions: regions of space that are closest to a set of points) Delaunay triangulation: no triangles circumcircle contains any points not in the triangle image from wikipediaimage generated in matlab

Discretization of momentum Velocity discretized by theta method (semi-implicit) Surface and vertical friction terms discretized implicitly Velocity rotated from x-y components to cross and along side components Advection, coriolis and horizontal friction discretized explicitly (discretization of these terms is flexible within this model, but given in E-L form in paper) Discretization of cross-side velocity E-L discretization of coriolis, advection and horizontal friction

Discretization of continuity Semi-implicit Finite volume Dependent on U n+1

Solution Algorithm System of the momentum equation and continuity equation is decomposed into two systems of equations to simplify solution method Momentum Continuity

Details of continuity Velocity Level widths Meat of the continuity equation Friction terms

Method of solution Convert momentum to equation for U Substitute into continuity equation Gives system of equations to solve for eta n+1 that is explicit for all terms except eta With a solution for eta n+1 can solve for U n+1 With a solution for U n+1 can solve for w n+1 (vertical velocity) Solution for depths are taken from eta n+1 for adjacent elements

Properties Mass conservative (finite volume) both globally and locally Reduces to trim if polygons are uniform rectangles Accuracy is second order in space on uniform grid (U defined at side centers), second order in time for theta = 0.5 Accuracy drops to first order if as grid becomes non-uniform Formally stable on uniform grid, empirically stable for non- uniform grid, stability dependent on advection and horizontal friction term discretization Reduces to 2d trim if only one vertical layer specified Finite element version has worse accuracy on uniform grid, but equivalent on non-uniform grid

Simulation with analytical solution 1 dimensional bay with very small tides and no advection, friction or coriolis effects Regular grid Confirms 2 nd order convergence for regular grid

Big Lost River flood Stability test under very difficult conditions Sharp wave front moving through a constriction and inundating flats Nearly critical flows Model is stable, and shows typical hydrodynamic features (in a highly qualitative sense)

Jade-Weser Estuary Complex estuary: many channels, strong tides, significant freshwater inputs Neglecting density, neglecting wind (?) 300 s timestep, multiple vertical discretizations Increase from 1 to 60 vertical layers only slows model run by a factor of 3 Successfully produces tide lag and amplification