1 Foundations of circulation modeling systems EBS566: Estuary and Ocean Systems II – Lecture 3, Winter 2010 Instructors: T. Peterson, M. Haygood, A. Baptista Division of Environmental and Biomolecular Systems, Oregon Health & Science University

2 Circulation models: what is under the hood? Correlation skill Bottom salinity 0 psu34 psu

3 Variables  Velocities: u, v, w  Water level:   Water density:   Salinity and temperatures: S, T  Basic variables: 7  Equations needed: 7

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

5 Topology/bathymetry

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

7 The consequences of grid resolution Time step consideration?

8 Grays River: example of cascading grids

9 Grays river: detail

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

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

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

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

14 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 Conservation of mass-water and, after differentiation by parts

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

16 Consider a 1D system with stationary fluid and a solute that is diffusing 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 dx dy dz

17 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 Conservation of mass of a solute - diffusion or

18 How do we quantify q D ? 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

19 Based on these observations, Adolph Fick (1855) hypothesized that 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 (molecular processes are represented by an empirical coefficient analogous to diffusivity) Conservation of mass of a solute - diffusion