GLY Geomorphology Notes

Transcription

1 GLY Geomorpholoy Notes Dr. Peter N. Adams Revised: Oct Hillslope Processes Associated Readins: Anderson and Anderson (2010), pp To examine processes operatin on hillslopes, we can look first at the physics and eomorphic results of individual events (e.. what happens when a raindrop hits a pile of loose sediment), then we can look at how individual events, of different manitudes, stack up in time. We can differentiate deterministic events, those that are predictable once we know the conditions (e.. the trajectory of an ejected rain from a rain drops assault) from stochastic events, that we don t know what spatial distribution or temporal order of events (the distribution of rain drops and the timin of storms) Rainsplash This is an example of a particular process that transports material downslope in a diffusive-manner. Raindrops blast loose sediment into the air, and if that sediment is on a slope, the trajectories of rains travelin downslope are loner than the trajectories of rains travelin upslope - so there s a net downslope transport. [schematic of sediment ejecta on a slope when hit by raindrop] Althouh rainsplash may not be operatin everywhere, by pickin apart the details of this process, we illustrate an approach by which we can look at other processes. 1

2 By delvin into the details of rainsplash, we will see how it depends upon climate and type of hillslope material. We divide this problem into three parts: 1. Determine the number of rains ejected per impact and impacted rain velocity as a function of rain drop size 2. Determine the resultin trajectories of ejected rains 3. Determine the probability distribution of drop sizes and total number of raindrops per unit area per unit time durin storm Mass and Velocity of Raindrop The number and velocity of disloded rains is proportional to the kinetic enery of the raindrop, hence we need to know the raindrop flux and the kinetic enery of each impact. Mass of Raindrop E k = 1 2 mv2 (1) The mass of a raindrop is easy to calculate. We know raindrops are spherical so their volume is V = 4 3 πr3 = πd3 (2) 6 and their mass is M = ρ wπd 3 (3) 6 Note that the mass has a cubic dependence on particle diameter. Velocity of Raindrop - Force Balance The velocity of impact is a little trickier to compute. We must start with the idea of a fall velocity or terminal velocity or constant terminal settlin velocity (CTSV). Imaine a skydiver: he doesn t accelerate all the way to the round, but rather he reaches a CTSV, has a thrill, opens his parachute (thereby increasin his rain size or particle diameter), and slows to a new CTSV. 2

3 Raindrops do the same thin and its a ood thin they do, otherwise we d be takin our life in our hands oin outside in the rain. (Diaram of raindrop fallin, the downward-directed ravitational force and the upward-directed dra force) CTSV is reached when the downward force of ravity is equaled by the upward force of dra on the particle. At that time, the forces are balanced and there is no net force, so no acceleration, hence the velocity is unchanin. The dra force is F dra = 1 2 ρ airc d πd 2 4 w2 (4) the density, fall velocity, and diameter are all in this equation as well as an intriuin coefficient obsessed over by automotive desin enineers called the dra coeffiecient. Velocity of Raindrop - Reynolds Number and Dra Coefficient Understandin the dra coefficient in detail requires a course in fluid mechanics, but know the followin. The dra coefficient has the behavior shown in a plot of C d vs. Re This requires us to understand the concepts of laminar and turbulent flow. These are flow reimes in a fluid that have very different behavior. Laminar flow is smooth and predictable whereas turbulent flow is chaotic and hopelessly inconsistent. The Reynolds number (Re) is a dimensionless value representin the relative importance of intertial to viscous forces within a flowin medium and it distinuishes these two flow reimes. Hih Re means turbulent, where low Re means laminar. This plot shows the behavior of the dra coefficient with respect to the flow reime. Its constant for hih Re (turbulent) and inversely proportional to Re for laminar flow. The force balance is therefore dw dt = F + F dra = ρ wπd ρ πd 2 airc d 4 w2 (5) We can set the left side equal to zero, because theres no acceleration, so the velocity is not chanin thru time. And we can solve for the fall velocity, w, which has two outcomes, based on which flow reime it is in: 3

4 for low Re: for hih Re: w = D2 ρ w 18µ (6) Dρ w w = (7) 0.3ρ air We now asses the dependence of E k of a rain drop on its size. Drops reater than a mm in diameter fall with enouh velocity to be in the turbulent reime and therefore w D 1/2, so, E k = 1 2 mv2 D 3 [D 1/2 ] 2 = D 4 (8) This result, the fact that the enery has such a stron dependence upon rain drop size, tells us what kind of storms are most effective in transportin material downslope by this process; that is convective storms with hih columns within which raindrops row to very lare diameters. So the occasional convective summer storm of the Colorado front rane can do much more work on the landscape than the many wimpy little winter drizzles of Seattle. Now we can asses the total flux of kinetic enery, simply by summin the individual kinetic eneries of all the drops. We know how to calculate the E k of each drop size, so we just need to know the distribution of drop sizes and interate the product of KE and drop size distribution function with respect to drop size. E k = N E ki = N i=1 0 E k (D)p(D)dD (9) Trajectory of Ejected Grains At this stae, we must connect the meteoroloical information to eomorphic results. In other words, we need to address the question what is the fate of ejected rains? Here is the microphysics view of the impact: (Diaram of rain drop impact and ejecta trajectory). 4

5 When the impact occurs, a rain should travel a distance dictated by its horizontal velocity, u 0, multiplied by twice the time it takes to et to the top of its trajectory. L = 2tu 0 (10) The vertical velocity is w = w 0 + t (11) When w = 0, the particle is at the top of its path, so the time it takes to et to the top of its path is: t = w 0 (12) Subbin back into the equation for distance, we et: L = 2u 0w 0 v 0, the initial launch speed is related to the components by So the final expression for travel distance becomes (13) w 0 = v 0 sin θ (14) u 0 = v 0 cos θ (15) L = 2v2 0 cos θ sin θ On a slope, the downhill directed rains will be propelled further than the uphill directed rains, and the averae of uphill and downhill transport distances determines the net rain splash transport rate. The mismatch in uphill and downhill distances is proportional to local slope, so L net dz (17) dx We can solve crudely for the discrepancy between a rain travelin down a slopin surface and a rain travelin down a flat surface, and we et the net difference in lenth is: L = 2v2 0 cos 2 θ tan α (18) The sum of the discrepancies between upslope path shortenin and downslope path lenthenin is rouhly twice this. tan α is rouhly the local slope, so we et the followin for net transport lenth L net = 4v2 0 cos 2 θ dz dx (16) (19) 5

6 For a flat surface, dz/dx = 0, and we et a net transport of zero. If the mean transport rate increases as the slope increases, this suests that we miht be dealin with a diffusive process. We are close to havin a rule for rainsplash transport, we just need a couple more thins: Q = m p nl net N (20) where m p is the mass of the particle, n is the number of ejected rains per impact, L net is the net transport distance, N is the raindrop flux. The upshot is that we have a flux linearly related to local slope and the transport constant in front of the slope is related to the meteoroloically important variables, and hence, rainsplash should diffuse the hillslope Probability Distribution of Drop Size and Areal Density Lastly, we measure drop size distribution with a simple contraption: a pan filled with flour. Raindrops become balled-up in a layer of flour dust, the dust balls are dried and measured knowin that the volume of the pellet corresponds to the surface area of the drop Frost Heave and Creep The slow, downslope mass movement of material in response to ravity on hillslopes is iven the eneral term Creep. Solifluction results from frozen soils attainin excess water durin the freezin process by the rowth of ice lenses. This aids downslope movement by supersaturatin near-surface soil upon thawin. In reions that freeze and thaw reularly (i.e. perilacial landscapes), material moves seasonally down a hillslope, by flexin upward durin freezin and collapsin upon thaw. Expansion is normal to the surface, but collapse is vertical, so the total cycle of motion is a downslope rachetin. The displacement profile is iven by [ ] Di z V i = H i tan θ D i (21) 6

7 where D i is the frost penetration depth, and the heiht of the maximum heave is H i = βd i The mass dischare for the i-th event is obtained by interatin the displacement profile q i = Di 0 βd i ρ b z x [ Di z D i ] dz = ρ bβ z 2 x D2 i (22) The total mass dischare is obtained by interatin the individual mass dischare events throuh the probability distribution of events in time Q = f 0 q i p(d)dd = f 0 ρ b β 2 z x D2 i p(d)dd (23) The total downslope movement is set by the heiht of the heave and the local slope. To et the total displacement downslope over a period of time, one needs to know the timin and manitude of freeze-thaw events, and round water content. The profile appears exponential, because the frequency of shallow freeze events is so much reater than the deep freeze events. It is the product of dischare per event and the probability distribution of thaw depths that must be interated. This process has both a deterministic and stochastic component to the total calculation of downslope material transport. The deterministic component is heave profile associated with a sinle event. The stochastic component is manitude and timin of freeze-thaw events, which depends on the weather. Note that this relationship is much more informative than the simple statement Q = ks 24.3 Gelifluction There appears to mechanism operatin that promotes flow durin the thawin process resultin in much reater downslope displacement than calculated from freezethaw alone Bioenic Processes In veetated landscapes, rainsplash and sheet wash may not be very effective, but other mechanisms are available. 7

8 Rodents Tree Throw References Anderson, R. S., and S. P. Anderson (2010), Geomorpholoy: The Mechanics and Chemistry of Landscapes, Cambride University Press. 8