This document discusses slope-area relations on random topography. It begins by introducing the concept of slope-area relations and noting they are conventionally thought to arise from fluvial erosion. It then presents analytical results showing that for Gaussian random surfaces, plan curvature correlates inversely with slope. Specifically, gentle slopes typically have high convergence and steep slopes have low convergence. This geometric correlation between plan curvature and slope results in a slope-area relation on random surfaces without erosion. The document concludes that slope-area relations can arise from how rivers choose their paths on random topography, not just erosion, and that Gaussian surfaces provide a model for this behavior.

1. Acausal Slope-Area Relations:
Drainage Basins on Random Topography
Norbert Sch¨orghofer (MIT)
in collaboration with Daniel Rothman (MIT)
2002

2. Drainage Basin
0
100
200
300
400
(km)
0 100 200 300 400 500
(km)
-1000 0 1000 2000
Topography (m)
Drainage basin of
Ma’adim Vallis, Mars
(Aharonson et al.,
PNAS, 2002)
(but this talk is about
terrestrial landscapes)

3. Slope-Area Relation
s ... (local) slope, A...drainage area
Slope-area relation: s = f(A)
Often s ≈ KA−θ
θ is usually 0.3...0.6 on Earth
The slope-area law is conventionally thought to be a consequence
of fluvial erosion. (J.J. Flint, A.D. Howard, W. Dietrich, K.X
Whipple, ...)

4. Flow Convergence on Surfaces
h(x,y)
convergent
divergent
HIGH
LOW
Δ
plan-curvature κ =
|t| − |t′|
|t|
·
1
∆
≈ “focusing strength”
More intuitively, 1/κ is the radius of curvature of the contour
line.

5. ... now let the contour spacing ∆ go to zero ...
κ =
(∂yh)2∂xxh − 2(∂xh)(∂yh)∂xyh + (∂xh)2∂yyh
(∂xh)2 + (∂yh)2
3/2
slope s = (∂xh)2 + (∂yh)2 = |∇h|
... rewritten ...
κ =
(∇h)t ∂yyh −∂xyh
−∂xyh ∂xxh
(∇h)
|∇h|3
Unless a special cancellation occurs, one expects κ ∼ 1/|∇h|

6. Gaussian Surfaces
Definition:
h(r) =
k
a(k)e−i(k·r+ϕ(k))
A surface is called Gaussian if ϕ is random and uniformly dis-
tributed in [0, 2π[.
The amplitudes a(k) are arbitrary.
Properties of Gaussian surfaces:
• h, hx, hy, hxx, hxy, ... are distributed Gaussian
• h and hx are uncorrelated, h and hxx are correlated, hx and
hxx are uncorrelated, ...
Longuet-Higgins (1957); Adler (1981)

7. Analytical Results for Gaussian Surfaces
κ|s denotes κ spatially averaged over all locations with slope s,
i.e. a conditional average.
κ|s = 0 (due to up-down symmetry)
κ2
|s = ......................... = C/s2
C = 3
8
(∂xxh)2 + 1
2
(∂xyh)2 + 3
8
(∂yyh)2 + 1
4
(∂xxh)∂yyh
Hence, for Gaussian surfaces, plan-curvature correlates with slope
indeed as
|κ| ∼ 1/s

8. Summary of Curvature-Slope Relation
naive expectation: κ ∼ 1/s
κ ∼ 1/s for ALL Gaussian surfaces, independent of their spatial
correlations.
Gentle slopes typically have high convergence and steep slopes
typically have low convergence.
A strong correlation between κ and s results from a basic geo-
metric tendency.
Actually, it is difficult to imagine landscapes without such a cor-
relation.

9. Parabolic surfaces with height contours (dotted lines) and stream-
lines (solid lines). The surface to the left is less steeply sloped
than the surface to the right and therefore collects water into the
main stream more rapidly. The shading of the surface represents
the plan-curvature.

10. Plan view of the surfaces shown on the previous slide. The
shaded area is the contributing area of a contour segment with
the same length in both figures. This illustrates the connection
between local slope, plan-curvature, and contributing area.

11. 10
0
10
1
10
2
10
3
10
4
10
5
2
5
10
20
Area
Slope
triangles = Gaussian surface
stars = fracture surface (Lopez & Schmittbuhl 1998)
circles = Juan River basin, California
The drainage area is in units of grid cells; slopes are multiplied
by an arbitrary prefactor.

12. 0 0.2 0.4 0.6 0.8 1
0
0.1
0.2
0.3
0.4
Hurst exponent H
θ
Concavity exponent θ of the slope-area relation for Gaussian
surfaces (circles). The triangle corresponds to the Juan River
basin, California. The star indicates a fracture surface (Lopez &
Schmittbuhl 1998). With the exception of the Juan River basin,
none of these surfaces was shaped by erosion.

13. Conclusions
• A slope-area relation can arise not only from fluvial erosion,
but because of the way rivers choose their path.
• The convergence of flow lines correlates inversely with slope,
so that lower slopes are associated with stronger conver-
gence.
• Gaussian surfaces serve as a model of random topography:
– Plan-curvature correlates inversely with slope (analytically).
– Concavity exponent is 0...0.3, depending on Hurst expo-
nent (numerically).
– Slope-area relations with slopes decaying slower than the
cube root of drainage area are insensitive to the processes
that have shaped the landscape.

14. Publications
N. Schorghofer & D.H. Rothman. Basins of attraction on ran-
dom topography. Phys. Rev. E 63, 026112 (2001)
N. Schorghofer & D.H. Rothman. Acausal relations between
topographic slope and drainage area. Geophys. Res. Lett. 29,
1633 (2002)