Foundations of circulation modeling systems

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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
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Circulation models what is under the hood?
Correlation skill
Bottom salinity
0 psu
34 psu
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Variables

• Velocities u, v, w
• Water level ?
• Water density ?
• Salinity and temperatures S, T
• Basic variables 7
• Equations needed 7

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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
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Topology/bathymetry
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Discretization (space)
nodes 27416 elements
53314 ? levels 24 min
element area 942 m2 max element area
89834 m2
Refined grid (hi-res)
fDB16
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The consequences of grid resolution
Time step consideration?
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Grays River example of cascading grids
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Grays river detail
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Introduction to governing equations
Continuity
Salt and heat conservation
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Introduction to governing equations
Conservation of momentum (from Newtons 2nd law
fma)
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Introduction to governing equations

• Equation of state
• ?? ?? (s, T, p)
• Turbulence closure equations

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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
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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
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Conservation of mass - water
Rearranging,
For incompressible fluids, like water and,
thus
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Conservation of mass of a solute
Consider a 1D system with stationary fluid and a
solute that is diffusing
dz
dy
dx
Let concentration (mass /unit volume) of solute
inside the control volume C
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Conservation of mass of a solute - diffusion
Conservation of mass states that
Thus
or
or
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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

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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
Applying Ficks law to the 1D mass conservation
equation for a solute, we get
or