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Model Analysis:

Analysis Procedure:

  • Three consecutive digital images were selected to represent each of three distinct phases of glacier flow (for a total of 9 analyzed images) during the model run: valley flow (phase 1), valley-piedmont transition flow (phase 2), and piedmont flow (phase 3).  We define valley flow as Flubber flow constrained completely within the plexiglass valley.  Valley-piedmont transition flow is defined as the narrow time interval after the Flubber has begun flowing on the continental plain, but before it has reached the side walls on the plain.  Piedmont flow is defined as Flubber flow on the continental plain while in contact with the side walls (see characteristic images of each phase below)
  • Images were digitally reoriented to remove camera movements (see error analysis below).  Flow marker positions in each image were then digitized manually, manipulated (using the equations below), gridded and plotted in Surfer 8 (Golden Software) to produce contour and vector plots of velocity, strain, vorticity and dilatation for each flow phase.    

Equations:


For all model observations, axes are oriented with the z-axis perpendicular to the
valley towards vertical, the x-axis transverse to the valley axis in the valley plane,
and the y-axis parallel to the valley axis in the valley plane, as shown.
 
Velocity, Strain, Vorticity and Dilatation during Three Flow Phases:  
 
Click on the panels below to view analyses of each flow phase
   
  
Phase 1: Valley
Flow
 Phase 2: Valley- Piedmont Transition
 Phase 3: Piedmont
Flow



 Cross-Sections:
 







The figures to the left display y-z cross-sections of the piedmont Flubber after completion of the model run.  Slices in the Flubber were made using a pizza cutter.  Folding in the y-z plane is due to a y-z velocity gradient, which causes the Flubber to continuously fold over itself at the toe as observed in the movies.  Basal velocity approaches 0, and velocity is highest on the surface.  The lower figure shows that layers become extremely thin (~1 mm) during flow and do not mix vertically. Colored chalk markers are visible on the Flubber surface.




 




The figures to the left display x-z cross-sections of the piedmont (above) and valley (below) Flubber, revealing the nature of longitudinal bands below the surface.  Coupling between the Flubber and the substrate and walls produces overturning due to basal drag (vertical velocity gradient) that varies across the glacier (lateral velocity gradient), producing the observed pattern. The pattern is symmetric around the center longitudinal axis (y-axis) of the glacier, as would be expected from the geometry of the model. Variations in the width of Flubber bands are most likely related to variations in the initial thickness of each colored block during set-up.  



 Error Analysis:

There are several potential sources of error in the analyses described above that are difficult to quantify:  

  • Slight movement of the digital cameras between frames is evident in the two movies, even though tripods were used to minimize movement.  We used two procedures to correct for this error in the numerical analyses.  First, all images used in the analyses were digitally reoriented (a.k.a. co-registered) to remove as much relative camera movement as possible.  Second, two fixed points (not on the flowing Flubber) were digitized along with the flow markers for each analysis.  The average displacement of the fixed pints was then subtracted from the displacement of all of the flow markers. 
  • Marker digitization was completed manually, and thus errors were potentially introduced through inaccuracies in picking the same marker in consecutive images.  This is particularly a problem when digitizing the larger colored chalk pieces because the displacement of a large chalk marker between two consecutive images was often only 2-3 times larger than the radius of the marker itself.  Effort was made to digitize the markers at the same exact location on consecutive images, but residual errors were expected.  An estimate of this error can be obtained by determining the difference between the displacement of two fixed points; if there is no error in digitizing, then the displacement due to camera motion should be the same at all fixed points.  Analysis reveals that manual digitizing error averaged 3.4%.
  • Trigonometric errors due to changes in the camera look angle as the Flubber flows down-valley and the camera is shifted could produce errors in marker displacement measurements between consecutive images.  This error is assumed to be minimal because analyses were done on consecutive images and marker displacements were generally on the order of only 0-2 cm.
  • One of the most significant sources of error is the low spatial density of markers, particularly during the valley-piedmont transition.  Additional markers were placed on the Flubber during piedmont flow.  The density of data points must be considered when evaluating the velocity, strain, vorticity and dilatation plots shown above.  Knotted contours are generally indications of low data density, and should be disregarded.