We study a sedimentary delta prograding over a fixed adversely sloping bathymetry, asking whether a perturbation to the advancing shoreline will grow (unstable) or decay (stable) through time. To start, we use a geometric model to identify the condition for acceleration of the shoreline advance (auto-acceleration). We then model the growth of a delta on to a fixed adverse bathymetry, solving for the speed of the shoreline as a function of the water depth, foreset repose angle, fluvial top set slope, and shoreline curvature. Through a linearization of this model, we arrive at a stability criterion for a delta shoreline, indicating that auto-acceleration is a necessary condition for unstable growth. This is the first time such a shoreline instability has been identified and analyzed. We use the derived stability criterion to identify a characteristic lateral length scale for the shoreline morphology resulting from an unstable growth. On considering experimental and field conditions, we observe that this length scale is typically larger than other geomorphic features in the system, e.g., channel spacings and dimensions, suggesting that the signal of the shoreline growth instability in the landscape might be “shredded” by other surface building processes, e.g., channel avulsions and alongshore transport.

Shorelines are the moving boundary between land and sea, and their evolution is of great importance to the estimated 10% of the global population that live in their proximity

Here our interest will focus on a new mechanism that might drive the instability of an advancing delta shoreline. Our motivation is the recent works from

In exploring the possible instability associated with auto-acceleration, we will appeal to the analogy between solid and liquid phase change processes and delta shoreline advance

Principally, we are interested in answering a number of key questions.

Under what conditions would an unstable shoreline growth arise and how would it evolve over time?

What, if any, is the connection between auto-acceleration an unstable shoreline growth?

What would the characteristic length scale of the instability be and how does this scale compare to other geomorphic length scales in deltaic shoreline settings, e.g., channel spacings?

To set the stage for our study, we adopt the delta geometry used in the

The one-dimensional model recently presented in

As we noted above, while meeting the auto-acceleration condition,

Schematics of sediment delta cross sections depositing on to a fixed basement with an adverse slope

The key ingredient in the analogy (see

In the case of a planar shoreline (curvature

The starting point for our stability analysis is to introduce a small perturbation of the planar front with the form

Schematic diagram of a perturbed shoreline.

At this point we need to emphasize three possible limitations of our analysis. In the first place while

Now that we have established that the condition of auto-acceleration can lead to unstable growth of a delta shoreline, we need to consider two issues. How, under a given set of conditions, will a shoreline instability evolve? What length scales (wavelengths) will the resulting instability exhibit?

In our analysis of the instability the obvious place to start is to explore the shape of the stability region and develop an understanding of how unstable shoreline perturbations might evolve with time. To provide a physical context that enables us to analyze our stability criterion under conditions that are consistent with realizable experimental systems, we consider the XES10 experiment reported in

To illustrate the shoreline stability region, under XES10 conditions, we use Eq. (

In our study of the evolution of an unstable shoreline we will consider the advance of a shoreline on the XES10 final basement profile. Here we will set the initial shore line position to

We can also use the above conditions to test the validity of the linear theory used in the derivation of the stability criterion; Eq. (

Following the typical approach of a morphological instability analysis (see

In the case of the initial perturbation exhibiting a number of modes (

Perhaps a better length scale to characterize the nature of unstable shoreline growth is the neutral wavelength. On appropriate rearrangement, this wavelength can be calculated by the substitution of the wave number definition

Our contention is that, determining the possible values of the neutral wavelength in experimental and field systems will inform us regarding the expected length scales of the instability in delta shoreline growth along adverse basement slopes.

As an example, let us again consider the end-point conditions found in the XES10,

Atchafalaya 1935 bathymetry data. Panel

As for the determining values of neutral wavelengths that could be characteristic of field settings, we consider predictions from Eq. (

As a larger-scale field example, we consider the Torok formation in the Colville Basin, as reported by

In this work we have used a geometric model and a linear stability analysis to investigate conditions under which the progradation of a planar sedimentary delta shoreline could become unstable, i.e., a condition where a perturbation of the shoreline will grow faster than the shoreline advance. Under the conditions of a constant unit discharge and a non-subsiding basement, we find the following.

A geometric model provides a simple condition for determining the onset of auto-acceleration, the positive seaward acceleration of the shoreline. This model shows that a necessary condition of auto-acceleration is an adverse basement slope with an absolute value exceeding the value of the top set (fluvial) slope; the amount of excess required to trigger auto-acceleration increases as the absolute value of the ratio of basement to foreset slope increases.

A linear stability analysis shows that, in an auto-acceleration condition, the growth of a delta shoreline prograding on a fixed adverse slope will become unstable; i.e., lateral perturbations on the shoreline, greater than a particular neutral wavelength, will grow faster than its bulk advance.

The analysis indicates that the fastest (dominant) growth perturbation wavelengths are at the lateral size of the system under consideration.

In experiment and field systems the neutral wavelength of the perturbations (the wavelength at which there is no growth or decay) is expected to be large, in excess of the widths of experimental systems and well beyond delimiting field length scales such as distributary channel spacings.

Thus, while we have clearly provided a positive answer to the question of this paper (“Can the growth of a deltaic shoreline be unstable?”), we can also conclude that observing clear signals of unstable growth in typical experimental and field delta systems would be unlikely. In other words, while delta building along an adverse basement slope is unstable, the resulting signal of the shoreline growth instability in the landscape will probably be “shredded” by other surface building processes, e.g., channel avulsions and alongshore transport.

The data for the Atchafalaya Bay field comparison was extracted from the
NCEI Estuarine Bathymetric Digitial Elevation Models web page (

MZ led the development of the details of the stability analysis and build numerical codes for the nonlinear model. GS led the development of the geometric model and comparisons of the stability analysis in field and laboratory settings. SL led the development of the concepts for the stability and nonlinear analysis. VV led the development of the calculations and interpretations of the length scales associated with the stability analysis.

The authors declare that they have no conflict of interest.

Vaughan R. Voller acknowledges additional support through the James L. Record Professorship. The authors are also grateful for discussions with Chris Paola, Liz Hajek, and Karen Kleinspehen and for the insightful and helpful comments from Andrew Ashton and referees John Shaw and Jorge Lorenzo-Trueba. The data supporting the conclusions of this work are self-contained in the mathematical analysis presented.

This research has been supported by the National Science Foundation USA (grant nos. DMS-1720420, ECCS-1307625, GRF 00039202).

This paper was edited by Orencio Duran Vinent and reviewed by John Shaw and Jorge Lorenzo-Trueba.