02 May 2014

Should I run straight or in zigzag up steep mountain ascents?

Zigzag path in the Aosta Valley at Tor des Geants
I have written a lot about hill running in the last few posts. However, running up the hills around Uppsala is clearly different from going up a real mountain. I was clearly made aware of this when I read a mail from the race director of Petite Trotte à Léon (PTL) Jean-Claude Marmier earlier in the week. In his mail to all PTL 2014 runners he emphasizes the importance of following the rigourous regulations for safety reasons and also announces some new measures:

“After the Col du Petit Saint Bernard at KM 243, you will have to cross the Col de l’Argueray 2853 meters, a superb passage between France and Italy. This pass was on the 2012 PTL® itinerary. Unfortunately, its crossing had to be cancelled due to particularly unfavorable conditions.  This pass is protected on its north face by a miniscule glacier (free of crevasses) with a slope of 25°. Covered in snow this crossing is not a problem. At the end of the season, it could be sheet ice and become a sure obstacle for the trail-runner. This year, putting the odds in our favor, we have decided to add an anti-slip system to the obligatory equipment which allows safe progression on gentle snow-packed or icy slopes”

The icy slope through a glacier at Col de l'Argueray at 2853 meters
It will certainly be a formidable challenge to cross Col de l’Argueray and it will now be, as Col Malatra was during Tor des Géants, my mantra during the race as I think it might be the last really difficult pass before the finish line (although I think I will have to “eat my own words” [not sure if this is also an English expression – quite frequently used in Swedish] many times during the last 50 kilometers after this pass). I have never run a race in crampons, and neither trained in them either, and the first time will probably be during PTL if there is ice. Nevertheless, I become encouraged watching Anton Krupicka in his latest film “In the High Country”,  where he puts on crampons and effortlessly run up Long’s Peak covered in ice and snow.
 
Anton Krupicka putting on crampons in the film In the High Country 
There appears also be some really light crampons specifically designed for trail running from for instance Hillsound, Snowline and Kahtoola.
 
Crampons on trail running shoes
The light weight of the crampons will most likely not be so important for it enables fast running, I will walk and not run up Col de l’Argueray and its 25° slope, but as it adds more weight to the backpack during this long race. I am not sure of the paths up the pass either, but I expect it not to be a straight line but involving some zigzagging and traversing the slope. When I previously wrote about comparisons between different ultramarathon races the gradient of the slopes along the path was not specifically considered in the equation and an interesting mathematical, physical and physiological problem is at what gradient it becomes more advantageous to run in zigzag rather than straight up the slope.  This critical angle at which this transition should take place is of course, as so much else in mountain ultrarunning and life in general, something subjective and influenced by a number of factors and I do not think one should think so much about it but rather follow the path and when there is a choice feel what is right at that particular moment, much like what Sam Robson so eloquently says in his latest blog post. However, I could anyway not stop thinking about this problem and looked into what was written in the literature about it. I will only discuss going uphill and come back to downhill running in a separate blog post later on.

An often referred study is by Llobera and Sluckin from a couple of years back (Llobera & Sluckin “Zigzagging: theoretical insights on climbing strategies” J Theor Biol 2007; 249: 206-217). In this study they build a semi-quantitative theoretical model of the behavior of humans walking on terrain with relief by minimizing the metabolic energy cost per unit of distance and the structure is according to the authors resembling the Landau Theory of Phase Transitions. They find that hairpin bends (switchbacks) and shortcuts appear as efficient strategies for downhill walkers, and switchbacks are retained also in uphill walking. For weakly inclined slopes, the best strategy is to walk/run directly downhill and uphill, while for sufficiently steep slopes, depending on the terrain, the best strategy would be to transition to a “broken symmetry solution corresponding to the switchback trail patterns typical of rugged environments” i.e to zigzag down- or uphill. The gradient at which this transition should occur in this model is at an angle of 16° (28.7% slope).

In classical studies based on volunteers running on treadmills with various slopes, Margaria and colleagues (Margaria et al “Energy cost of running” J Appl Physiol 1963; 18: 367-370) and Davey and colleagues (Davey et al “Running uphill: an experimental result and its applications” J Opl Res Soc 1994; 45: 25-29 and Davey et al “Speed, gradient and workrate in uphill running” J Operational Res 1995; 46: 43-49) determine the angle to be around 20° (36.4 % slope) when it is more efficient to walk or run in zigzag rather than run straight up the hill from an energetic perspective. Data from Minetti and colleagues in their treadmill tests of 10 elite athletes practicing endurance mountain racing rather than normal healthy controls indicate critical gradient for transition of around 15.5° (27.6 % slope) (Minetti et al “Energy cost of walking and running at extreme uphill and downhill slopes” J Appl Physiol 2002; 93: 1039-46).

Antohony Kay at the Department of Mathematical Sciences at Loughborough reviews these studies in greater detail in his paper “Route choice in hilly terrain” which is freely available on the web at http://www.academia.edu/2876712/Route_choice_in_hilly_terrain. Another freely available paper discussing this is Ulrich Leuthäusser’s “About walking uphill: time required, energy consumption and the zigzag transition” where he recalculates the critical gradient according to the formula below and find it to be 13.8° (24.6% slope). Surprisingly it is in his model very robust between 13° - 16° regardless of power (i.e. how fast one is moving uphill). His paper is available at http://sigmadewe.com/fileadmin/user_upload/pdf-Dateien/Bergaufgehen_engl.pdf .

The formula for the critical gradient in Leuthäusser's paper
Quite interestingly, looking at different terrains the paths appear to both go straight up at slopes greater than this, but also in many cases to go in zigzag at slopes lower than this. For safety reasons, and in respect of the nature, I think that in most cases one should anyway follow the path present and not take shortcuts even though it might not be the fastest way up from a race pace perspective. When running (or walking when it comes to most uphill sections) PTL I expect the path to frequently be non-existing and, at least with crampons, the route choice will most likely be straight even though the slope might be higher than mathematical critical gradient of around 15° (26.7%) as outlined above.

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