globals [step time dt L var xbar scale truescale xnewmax xnewmin xnewbar histscale axmin axmax histbars Kx_theory Kx_actual]
patches-own [vel]
turtles-own 
[
  xc         ; unwrapped xcor 
  yc         ; unwrapped ycor
  delx
  dely
  xnew
  Gauss
  Gx
  localvel   ; local velocity
]


to setup
    ca
    crt 1000
    ask turtles [
        set color yellow
        set heading 90
        set xcor -1 * screen-edge-x
        set xnew -1.0 * screen-edge-x
        ]
    ;ask turtle 10 [set color green]
    
    if line
        [ ask turtles with [ who <= 500 ] [ set ycor random (screen-edge-y - 0.5) ]  
          ask turtles with [ who > 500 ] [ set ycor random -1.0 * (screen-edge-y - 0.5)] 
        ]
    ask turtles [
        set yc ycor
        ]
    set scale 1
    set truescale 1
    set axmin 0
    set axmax 2 * screen-edge-x
    set dt 0.5 ;L ^ 2 / 200
    ;set U_o 1
    set L (screen-edge-y - 0.5)
    set Kx_theory 8 / 945 * U_o ^ 2 * L ^ 2 / D_y
    grid-setup
end

to grid-setup
    ask patches
        [ 
        set vel U_o * (1 - (pycor / (screen-edge-y - 0.5) ) ^ 2)
        set pcolor scale-color blue vel (1.4 * U_o) 0 
        ]
    ask patches with [abs pycor = screen-edge-y]
        [set pcolor red]
end

to go
    move-turtles
    
    set step step + 1
    set time time + dt
    set var variance values-from turtles [xc]
    set xbar mean values-from turtles [xc]
    set Kx_actual 1 / 2 * var / time
    
    set-current-plot "Growth of Tracer Variance"
    set-current-plot-pen "theory" plotxy time 2 * 8 / 945 * U_o ^ 2 * L ^ 2 / D_y * (step * dt)
    set-current-plot-pen "actual" plotxy time var
    
    if (step mod 30 = 1) [
    set-current-plot "Histogram" clear-plot
    set histscale (max values-from turtles [xc] - min values-from turtles [xc]) / (max values-from turtles [xcor] - min values-from turtles [xcor])
    set axmin min values-from turtles [xc] - (min values-from turtles [xcor] + screen-edge-x) * histscale
    set axmin round axmin
    set axmax max values-from turtles [xc] + (screen-edge-x - max values-from turtles [xcor]) * histscale
    set axmax round axmax
    set-plot-x-range axmin axmax
    set-histogram-num-bars 50
    histogram turtles [xc]
    
    set-current-plot "Tracer Concentration Profile vs. Gaussian" clear-plot
    set-plot-x-range axmin axmax
    set-current-plot-pen "C(x)"  
    set histbars 70
    set-histogram-num-bars histbars
    histogram turtles [xc]
    set-current-plot-pen "Gauss"
    ask turtles [
        set Gx axmin + who * (axmax - axmin) / 999
        set Gauss 1000 * (axmax - axmin) / histbars / sqrt (2 * pi * var) * exp ( -1 * (Gx - xbar) ^ 2 / (2 * var) )
        plotxy Gx Gauss 
        ]
    ]
end

    
    
to move-turtles
    ask turtles
        [set localvel vel-of patch-here
         set delx localvel * dt
         set dely sqrt( 2 * D_y * dt) * (random-normal 0 1)
         bound-reflect
         set xc xc + delx
         set yc yc + dely 
         set ycor yc
         set xnew xnew + delx * truescale 
         ]
         
         set xnewmax max values-from turtles [xnew]
         set xnewmin min values-from turtles [xnew]
         set xnewbar mean values-from turtles [xnew]
             
         if (xnewmax > screen-edge-x)
             [ set scale min list 1 (0.67 * screen-edge-x / ( xnewmax - xnewmin ))
               ask turtles [ set xnew scale * xnew - screen-edge-x - scale * xnewmin ]
               set truescale truescale * scale
             ]
         ask turtles [set xcor xnew]
         
end

to bound-reflect
    if ( pcolor-of (patch-at delx dely) = red)  
        [ set dely dely - 2 * ( yc + dely - ( L * abs pycor-of patch-at delx dely / pycor-of patch-at delx dely ) ) ]
end          

    
; add model procedures here

@#$#@#$#@
GRAPHICS-WINDOW
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CC-WINDOW
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Command Center

BUTTON
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setup
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Growth of Tracer Variance
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Tracer Variance
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MONITOR
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MONITOR
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@#$#@#$#@
WHAT IS IT?
-----------
This is a model of shear dispersion.  It simulates the release of a passive tracer into a channel of width B with a longitudinal velocity profile U(y).  The individual tracer particles (think of them as molecules) advect with the current at the same time as they undergo a random diffusion process in the y-direction. The combination of transverse diffusion and "differential advection" gives rise to a longitudinal dispersion process that follows Fick's Law.  This means that the tracer concentration profile in the longitudinal direction is Gaussian and longitudinal variance grows linearly with time, after an initial transient period.  
  

HOW IT WORKS
------------
The user first initiates the trace molecules by pressing the "setup" button.  This action releases the molecules in a transverse line at x = 0. Pressing the "Go" button then starts the simulation.  Prior to starting the simulation, the user may use the slider buttons to vary the values of U, the longitudinal velocity along the channel centerline, and Dy, the trasverse diffusion constant.  The transverse-dependence of the velocity, U(y) is visually represented by the color variation of the channel (dark blue is the maximum velocity, near the channel centerline, while white corresponds to zero velocity at the edges).  The red lines are the solid channel boundaries, which tracer molecules are prohibited from crossing.  The triangles represent the tracer particles.  

Once the simulation has begun, the motion of the tracer can be monitored by watching the particles move within the colored channel.  In addition, the behavior of the tracer is shown by the three plots entitles "Histogram", "Tracer Concentration Profile vs. Gaussian", and "Growth of Tracer Variance".  The histogram plot shows a histogram of the locations of all tracer particles with time.  Note that the x axis changes with time, as the tracer becomes more and more spread out.  The next plot compares the tracer concentration profile to an ideal Gaussian curve. When the Fickian dispersion regime has been reached, the tracer curve should closely resemble a Gaussian.  The last curve shows the growth of the tracer variance with time, and compares it with the prediction based on Fickian dispersion theory.  Note that when the slope of the line from the simulation matches the slope of the ideal linear curve, the Fickian regime has been reached.      


THINGS TO NOTICE
----------------
Notice that the higher the value of Dy, the faster the particles move in the transverse direction.  
Note that the time necessary to reach the Fickian dispersion regime depends on the values of U and Dy.  See if you can determine this dependence through some combination of scaling and simulation.  Try varying the values of Dy and for each run, note the time at which you think Fickian dispersion has begun (Gaussian concentration profile and/or linear growth of tracer variance).



THINGS TO TRY
-------------
This section could give some ideas of things for the user to try to do (move sliders, switches, etc.) with the model.


EXTENDING THE MODEL
-------------------
This section could give some ideas of things to add or change in the procedures tab to make the model more complicated, detailed, accurate, etc.


NETLOGO FEATURES
----------------
This section could point out any especially interesting or unusual features of NetLogo that the model makes use of, particularly in the Procedures tab.  It might also point out places where workarounds were needed because of missing features.


RELATED MODELS
--------------
This section could give the names of models in the NetLogo Models Library or elsewhere which are of related interest.


CREDITS AND REFERENCES
----------------------
This section could contain a reference to the model's URL on the web if it has one, as well as any other necessary credits or references.
@#$#@#$#@
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@#$#@#$#@
NetLogo 1.3.1
@#$#@#$#@
@#$#@#$#@
@#$#@#$#@