The concept of topographic steady state has substantially
informed our understanding of the relationships between landscapes,
tectonics, climate, and lithology. In topographic steady state, erosion rates
are equal everywhere, and steepness adjusts to enable equal erosion rates in
rocks of different strengths. This conceptual model makes an implicit
assumption of vertical contacts between different rock types. Here we
hypothesize that landscapes in layered rocks will be driven toward a state of
erosional continuity, where retreat rates on either side of a contact are
equal in a direction parallel to the contact rather than in the vertical
direction. For vertical contacts, erosional continuity is the same as
topographic steady state, whereas for horizontal contacts it is equivalent to
equal rates of horizontal retreat on either side of a rock contact. Using
analytical solutions and numerical simulations, we show that erosional
continuity predicts the form of flux steady-state landscapes that develop in
simulations with horizontally layered rocks. For stream power erosion, the
nature of continuity steady state depends on the exponent,

The formation of landscapes is driven by tectonics and climate, and often
profoundly influenced by lithology, the substrate on which tectonic and
climate forces act to sculpt Earth's surface. Much of our interpretation of
landscapes, and their relationship to climatic and tectonic forces, employs
concepts of landscape equilibrium, or steady state. Though there are a
variety of types of landscape steady state

Topographic equilibrium in layered rocks.

Bedrock channels are of particular geomorphic interest because they span most
of the topographic relief of mountainous terrains

The elevation profiles of bedrock channels enable analysis of landscapes for
evidence of transience, contrasts in rates of tectonic uplift, or the
influence of climate

Erosional continuity.

Topographic steady state has also been used to explain channel response to
substrate resistance, generally leading to a conclusion that channels are
steeper within more erosion-resistant bedrock and less steep within more
erodible rocks

Conceptual models of land surface response to changing rock type typically
employ the concept of topographic steady state, which makes an implicit
assumption of vertical contacts between the different rock types. In
topographic steady state, vertical incision rates are matched in the two rock
types (Fig.

Physical reasoning supports the idea that landscapes in layered rocks would
tend toward erosional continuity. If the upper layer retreats slower than the
lower layer in the direction of the contact, this produces a steep, or
possibly overhanging, land surface at the contact (Fig.

There are cases in natural systems where continuity is not maintained at all
times. For example, caprock waterfalls are similar to the case in
Fig.

Using the constraint of erosional continuity, one can write a very
general relationship between surface erosion rates and slopes at a
contact between two rock types,

The implications of the relationship in Eq. (

Parameters used in the 1-D model runs.

Channel slope response at a subhorizontal contact
from an assumption of continuity. The ratio of slope within the weaker rock
(

For the subhorizontal limit, where channel slopes are much greater than the
slope of the contact (

Channel profiles in subhorizontally layered rocks with
high uplift (2.5

Channel profiles in subhorizontally layered rocks
with low uplift (0.25

The slope ratio (

The channel continuity relations above apply to channels within the neighborhood of a contact. Though there are clear long-term constraints on the relative retreat rates of any two contacts, these are not sufficient to determine an entire profile. However, we hypothesize that the continuity relation applies along entire profiles, and therefore that it can be used to describe a type of equilibrium state that develops in layered rocks. If this is correct then there is a one-to-one relationship between erodibility and steepness that is predicted by the continuity relations. Here we test this hypothesis using simulations of channel and landscape evolution in horizontally layered rock.

We solve the stream power model using a first-order explicit upwind finite-difference method. This method is conditionally stable, and the time step was
adjusted to produce a stable Courant–Friedrich–Lax number of

Simulation results are most easily visualized in

For simulations where

For

The continuity relation (Eq.

In continuity steady state the slopes in both rock types are
different, in general, than the slopes that would be predicted by
topographic steady state. Combining Eqs. (

An example case of the ratio of slopes predicted
by continuity and topographic steady states. This example assumes a choice of
equal rock thicknesses in both rock types and a weak rock erodibility that is
twice that of the strong rock. Contrasts are in general strongest for

The influence of relative layer thickness on slopes in continuity steady state. If the relative thickness of the strong and weak layers is changed, the slopes that are far from base level in both rocks adjust correspondingly (solid lines), as predicted by continuity steady state. Grey bands depict the locations of weak rocks in the differing thickness model. The dashed lines depict channel profiles for simulations with equal layer thickness but the same erosional parameters. Increasing the weak layer percentage reduces topography overall.

Continuity steady state predicts that the ratios of slopes in the weak and
strong layers are independent of layer thickness (Eq.

Continuity steady state is perturbed near base level, because a constant rate
of base-level fall is imposed and continuity steady state requires vertical
incision at different rates in each rock type. Despite this discrepancy
between base-level topographic equilibrium and continuity steady state,
theoretical profiles produced using Eqs. (

In a horizontally layered rock sequence, a segment of stream profile with
erosion rate equal to uplift is continuously developing at base level. The
slope of this base-level segment in

The strong rock knickpoint begins with a head start equal to the

To generalize the damping behavior of the base-level perturbations it
is useful to analyze a dimensionless version of

Parameters used in the

The dimensionless damping length scale,

Simulations of knickpoint propagation and damping
from base level. Entire equilibrium profiles are depicted for cases where

Results of the

To illustrate this damping behavior, we run two simulations with somewhat
longer damping length scales. Both simulations have profile lengths of 500 km, uplift rates of 2.5

To determine whether continuity steady state is obtained within whole
landscape models, or whether addition of hillslope processes might eliminate
it,

The stream power model used in

Floating-point digital elevation models (DEMs) were produced for the final
time step for each

Topographic steady state is not attained within layered rocks with
non-vertical contacts since the spatial distribution of erodibility changes
in time

Our simulations and analysis support the conclusions of

If we compare the

As noted by

When contacts between rocks dip at slopes much greater than the channel
slope, then the vertical contact limit from Eq. (

For

During constant uplift, channels cannot attain continuity steady state at base level, because it requires different vertical incision rates in each rock type. However, the perturbations introduced by stream segments in topographic equilibrium at base level rapidly decay over a length scale that is primarily a function of the ratio of rock erodibilities, with larger erodibility contrasts resulting in shorter decay lengths. Practically speaking, for rocks that have erodibilities sufficiently different to have a strong effect on the profile, base-level perturbations of continuity steady state decay after a couple rock contacts are passed.

Though steepness ratios are a fixed function of rock erodibility in continuity steady state, absolute steepness values depend on rock layer thickness. Since natural systems will not generally have regular patterns of thickness or erodibility, this has implications for the ability of natural systems to approach continuity steady state. As new rock layers with different thicknesses or erodibilities are exposed at base level, the absolute steepness values that would represent continuity steady-state change. Therefore, continuity steady state may often represent a moving target, where the landscape is constantly adjusting toward it but never reaching it. The introduction of rock layers with varying thickness and erodibility can produce transience in landscapes that are experiencing otherwise stable tectonic and climate forcing. This only applies, however, for absolute steepness values. Steepness ratios, and their relationship to erodibility, would be expected to be relatively constant in time if sufficiently far from base level. Since the relationship between erodibility and steepness will change in both time and space as new layers are exposed at base level, this may confound attempts to identify erodibility values using channel profiles within steep channels in subhorizontal rocks. However, since steepness ratios do not depend on these dynamics, analysis of steepness ratios derived from profiles, rather than absolute steepness values, may enable quantification of the relative erodibility of layers.

We speculate that the simulated dynamics in subhorizontal rocks provide a
potential means to generate caprock waterfalls, a feature that has long
fascinated geologists

Though the focus of this work is on bedrock channel profiles in layered rocks, the concepts of continuity and flux steady state can be applied in general to any mathematical model for erosion. Much like topographic steady state, both continuity and flux steady state result from negative feedback within the uplift-erosion system that drives it toward steady state as uplift and erosion become balanced. Such feedback mechanisms are likely to be present within most erosional models. Though topographic steady state has been a powerful theoretical tool to understand landscapes, the generalized concept of erosional continuity may prove more useful in interpreting steep landscapes in subhorizontal rocks.

Topographic steady state has provided a powerful tool for understanding the
response of landscapes to climate, tectonics, and lithology. However, within
layered rocks, topographic steady state is only attained in the case of
vertical contacts. In topographic steady state, vertical erosion rates are
equal everywhere, and steepness adjusts with rock erodibility to produce
equal erosion. Here we generalize this idea using the concept of erosional
continuity, which is a state where retreat rates of the land surface on
either side of a rock contact are equal in the direction parallel to the
contact rather than in the vertical direction. Using a stream power erosion
model with

For continuity steady state, the relationships between rock erodibility and
landscape steepness differ most from topographic steady state when the rock
contacts are subhorizontal, that is, when contact dips are less than channel
slope. In the subhorizontal case, contrasts in steepness are larger than
predicted by topographic steady state. These contrasts are largest when

The simulation inputs, the code used to run the 1-D simulations, and the code to create the figures in the manuscript are
archived in a GitHub repository (

Here we detail how the constraint of channel continuity can be used to derive
the relationship given in Eq. (

Geometric relationships used to derive the
equation for continuity of the channel at a contact between two rock types.
Note that the slope of the contact plane (

In this section we use the subscript

Using the stream power model, erosion rates in two channel segments above and
below a contact are

Combining Eqs. (

The authors declare that they have no conflict of interest.

Matija Perne acknowledges the support of the Slovenian Research Agency through Research Programme P2-0001. Matthew D. Covington, Matija Perne, and Evan A. Thaler acknowledge support from the National Science Foundation under EAR 1226903. Any opinions, findings, conclusions or recommendations expressed in this material are those of the author and do not necessarily reflect the views of the National Science Foundation. The simulation outputs and code on which this work is based are available upon request.Edited by: G. Hancock Reviewed by: K. Whipple and two anonymous referees