The Insomniac Notes on Landscape Evolution
Gary Parker
03:30 – 06:00, February 26, 2004
Introduction
These notes are motivated by a lunchtime discussion with Efi Foufoula on Friday, February 13, 2004 in Minneapolis, USA and several extended discussions with Andrea Rinaldo during the period February 16-20, 2004, in Alto Zolto, Italy. The author greatly appreciated these discussions.
A proposed equation for landscape evolution
Let denote surface elevation, where is a horizontal (2D) spatial vector and t is time. A general equation of landscape evolution can be written in the form
(1)
where vI denotes an incision rate, k is a coefficient of hillslope diffusion and u is an uplift rate. Recently the following form has been proposed for vI (Banavar et al. 2001);
(2)
where A is the “drainage area” upstream of a point, is identical to slope S and b is a constant. Equation (2) is a specific evaluation of an empirical form which has been in the literature for many years;
(3)
Here A is an integral property of the variable h itself. It is commonly interpreted as the drainage area upstream of a point, and is a surrogate for the flow discharge at the point.
Banavar et al. (2001) have added a random “noise” term n to (1), so that their complete field equation for landscape evolution is
(4)
Equation (5) should prove to be a useful tool for the study landscape evolution. The author has three comments that may be relevant to the problem.
The area term
Equation (5) is a continuous field equation rather than a discretized form. It seems to this writer than A is not the right variable to use in a field formulation. The correct variable would seem to a differential quantity that can be defined as follows. Let ds denote an arc length along a line of constant elevation. Now consider the area dA bounded by dn, orthogonals to the end points of dn and the drainage divide, and define
(5)
It is the parameter A rather than A that is used in the work of Izumi and Parker (1995, 2000). In a pixellated formulation it may be sufficient to use A instead of A.
The slope-area relation for incision rate
This writer believes that the “incisional” form of (1) is preferable to the “transportational” form of the original (and very inspiring) work of Willgoose et al (1991). Recent developments, and in particular the work of Sklar and Dietrich (1998), however, suggest that major advances can be made in the incision relation. This is an area that the author is pursuing with a graduate student, Phairot Chatanantavet, as part of his effort for the National Center for Earth-surface Dynamics. The analysis assumes that all the incision is concentrated in well-defined bedrock channels, so that drainage area A becomes well-defined (drainage area defined by dn = channel width, orthogonals to the channel banks and the divide). Space does not allow for a complete exposition here, but the form of the relation can be written as
(6)
Here b is an abrasion coefficient (1/m), qsw is the volume transport rate per unit width of sediment of a size sufficient to induce wear (m2/s), po denotes the fraction of the bedrock surface that is not covered by alluvium (dimensionless), no is an exponent and qswc denotes the capacity volume transport rate per unit width of wear-inducing sediment. To this relation is added a) a differential equation governing sediment yield which defines a relation between qsw and dA/ds, where s is now a down-channel coordinate, b) a standard sediment transport relation for qswc and c) a relation for boundary shear stress that specifically brings channel slope S and rainfall rate I upstream (or equivalently A) into the formulation.
This is not meant to imply that the form of (3) is of no value for the study of landscape evolution. The formulation of (4), however, should be considered to be a model equation that is subject to further development.
Fluctuations in time and space
Existing models of landscape evolution all approach a steady-state balance between uplift and erosion in which spatial fluctuations are present but temporal fluctuations are entirely absent. This author has been led to believe that a) (4) has this property in the absence of noise and b) noise must be added to maintain temporal fluctuations in (4). This author believes that intrinsic processes associated with landscape evolution are perfectly capable of generating and maintaining temporal fluctuations, even at steady state in the absence of noise, that the temporal fluctuations are somehow linked to the spatial fluctuations, and that they play an important role in the form that landscapes take.
In 2000 the author gave a presentation at the fall American Geophysical Union meeting concerning (1). A copy of the presentation is attached. The basis of the talk is the hypothesis that elevation h constantly fluctuates in time and space, even at steady state, due to self-generated internal (and thus inherently nonlinear) processes rather than random noise. The work was inspired by the experiments of Hasbargen and Paola (reference to be added) and discussions with Peter Haff and Brad Murray. The work also used turbulence as a metaphor.
Consider the Navier-Stokes and continuity equations;
(7)
Shear flows create their own turbulence, the flow field of which eventually reaches a macroscopic steady state with fluctuations in both time and space. Writing
(8)
where the overbar denotes averaging and the prime denotes fluctuations, (7) can be manipulated into a form describing the evolution of the mean kinetic energy of the turbulence per unit mass k, where
(9)
This relation takes a standard form;
(10)
where P is a production term, D is a dissipation term and T are transfer terms that neither create nor destroy fluctuations. The forms for P and D are
(11a,b)
In a steady-state turbulent flow, ¶k/¶t drops to zero and the spatial integral of P balances the corresponding integral of D (in the absence of import or export of turbulent energy).
A form analogous to (10) can be derived for landscapes. Consider the following decomposition;
(12)
where again the prime denotes a fluctuating quantity. It is assumed that both temporal and spatial fluctuations persist even at steady state and even in the absence of noise. The uplift rate is taken to be constant for simplicity. The following balance equation can be derived from (1);
(13)
where
(14a,b)
Here P denotes a production rate of elevation fluctuations by incision and D denotes a destruction rate by hillslope diffusion.
It is easy to demonstrate that P as defined by (14a) is a positive quantity. If incision is concentrated in the channels, incision in the absence of hillslope diffusion will make the channels deeper without changing hillslope evolution, thus increasing .
The condition for a steady-state landscape is now one for which the integral of P equals the integral of D, a condition that is maintained by both temporal and spatial fluctuations. The author used considerations of this type to derive a scale for the characteristic hillslope length in the attached AGU presentation.
Challenge for the future
If the above arguments are right, the noise term in (4) is not necessary in order to obtain fluctuations at steady state. The term is hardly meaningless; there are always imposed fluctuations. This notwithstanding, however, a system that creates its own fluctuations seems intrinsically more interesting than one that needs to have them imposed.
Again, an analogy to turbulence seems to be in order. In the case of grid turbulence, the intrinsic production terms P of (11a) is zero. The fluctuations are maintained by the imposed oscillation of the grid. (But the motion of the grid need not be random in order to generate a random response.) In the case of shear turbulence, the flow creates its own fluctuations.
So to this author the challenge seems to be the determination of a form for vI which when installed in (1) generates its own fluctuations. One can envision any number of physical processes that might cause this to happen (lateral channel migration, cyclic steps, incision that increases nonlinearly with increasing adjacent slope of the hillslope). The elucidation of even one such form should lead to some fascinating new progress in the area of landscape evolution.
References
Banavar, J., Colaiori, F., Flammini, A., Maritan, A. and Rinaldo, A., 2001, Scaling, optimality and landscape evolution, Journal of Statistical Physics, 104(1/2).
Hasbargen, L: and Paola, C. Reference to be added.
Izumi, N. and Parker, G, 1995, Inception of channelization and drainage basin formation: upstream driven theory, Journal of Fluid Mechanics, 283, 341-363.
Izumi, N. and Parker, G, 1995, Linear stability analysis of channel inception: downstream-driven theory, Journal of Fluid Mechanics, 419, 239-262.
Sklar, L., and W. E. Dietrich, 1998, River longitudinal profiles and bedrock incision models: Stream power and the influence of sediment supply, in Rivers Over Rock: Fluvial Processes in Bedrock Channels, Geophys. Monogr. Ser., vol. 107, edited by K. J. Tinkler and E. E. Wohl, pp. 237–260, AGU, Washington, D. C.
Willgoose, G., Bras, R. L. & Rodriguez-Iturbe, I., 1991, A coupled channel network growth and hillslope evolution model. 1. Theory, Wat. Resour. Res. 27, 1671-1684.