Model studies suggest that sudden formation of supraglacial debris cover might cause glaciers to advance, by inhibiting ice melt in the lower reaches and altering the driving stresses if the debris deposit is massive enough (basically the weight of the rockfall deposit forces the ice to flow faster). For example, Vacco and others (2010) used a numerical glacier flow-line model with superimposed rock debris to show that a glacier advance caused by deposition of a rock avalanche on the ice will be followed by stagnation of the advanced ice lobe, producing distributed, hummocky deposits quite different from the single moraine ridges typically dated in paleoclimatic reconstructions. This type of rapid advance is different to periodic fast and slow flow that is characteristic of true ‘surge-type’ glaciers.

The cool thing is though that the surface debris cover shows really nice evidence of former surges at the surface of the glacier. For example, look at this photo of the Susitna Glacier in Alaksa:

I found this image on wikiversity, but its credited to Brian John (The image appears on a website entitled, “Stonehenge and the Ice Age” at http://brian-mountainman.blogspot.com/2011_06_01_archive.html), though I suspect it might come from the USGS archives originally.

In that photo every splurge forward of ice from one of the tributary glaciers has deformed the main line of the medial moraines and formed a loop, storing the record of weird anomalous flow for posterity. This way in which surface debris cover tracks irregular speedups of ice flow has been exploited as a means to map the flow behavior of glaciers throughout Alaska range in a clever study by Herreid and Truffer (2016), in which they use satellite data from LANDSAT to map the motion of debris features on the glacier surface and see if their displacement indicates a localized speed-up of ice flow.

Another awesome example, maybe not of real surging but switches of dominant flow between two tributaries – I’m not sure – can be seen on the Walsh Glacier which descends from within the Kluane National Park in Canada to flow across border into the USAs Wrangell-St Elias National Park:This glacier surface really looks like turbulent fluid flow frozen in time. These parks are full of amazing (debris-covered) glaciers, faultlines, and geomorphological processes. I really hope I get to go there one day!

References:

Vacco, D. A., Alley, R. B. and Pollard, D.: Glacial advance and stagnation caused by rock avalanches, Earth Planet. Sci. Lett., 294(1–2), 123–130, doi:10.1016/j.epsl.2010.03.019, 2010

Herreid, S. and Truffer, M.: Automated detection of unstable glacier flow and a spectrum of speedup behavior in the Alaska Range, J. Geophys. Res. F Earth Surf., 121(1), 64–81, doi:10.1002/2015JF003502, 2016.