First approximation sea-level dependent mathematical model for volume eroded and submarine profile development in a semienclosed sea: Kiel Bay, Western Baltic

Abstract

Given a marine basin of near homogeneous lithology, a known sea level curve, and known submarine abrasion rates, a model is developed to estimate the volume of material eroded by marine action. Assumptions of the model are that erosion is effected solely by submarine abrasion, which is assumed known and uniform through time, and that the volume eroded is small relative to the total volume of the basin. The basis of the model is that the volume eroded V, between time limits t₁ andt ₂,is essentially a function of the perimeter length l of the basin, which in turn is time-dependent on the sea level curve, so that

$$V = k\int_{t1}^{t2} {l(t)dt} $$

where k is an abrasion rate constant. The model was tested on Kiel Bay, Western Baltic, a shallow semienclosed, essentially nontidal sea, for which considerable data is available. Critical for numerical integration of the model is the k value, representing the volume eroded from the shore normal profile per unit length of shoreline per year. A number of possible k values were utilized, the most likely realizing a total volume eroded over the past 9000 y, since the sea first entered Kiel Bay, of 2.60×10⁹ m ³. From this model, long-term average vertical submarine abrasion rates for Kiel Bay can be deduced as being between 0.001 and 0.0004 m/y.

An extension to the model analyzes the effect of sea level transgression rate on whether cliffs develop and predicts the theoretical form of the submarine profile based on varying abrasion rates summarized as

$$y_B (x) = \left\{ {\begin{array}{*{20}c} {h({x \mathord{\left/ {\vphantom {x {V_c }}} \right. \kern-\nulldelimiterspace} {V_c }}) - a_T (x) for x > 0;} \\ {x\tan \theta - a_T (x) for x > 0.} \\ \end{array} } \right.$$

Here the origin x=0, y_B=0 is chosen horizontally at the position where cliff formation first occurs, and vertically at the sea level at that time. The coordinate x measures distance inshore from the origin, y_B(x) is the vertical position of the sea floor, a_T(x) is the total depth abraded, tanθ is the original land surface slope, V_c the rate of cliff retreat, and h(t) the sea-level at time t. The synthetic profiles are compared to actual erosional profiles from representative sectors of Kiel Bay. The model predicts cliff development began at about 5800 yB.P., resulting in submarine abrasion profile lengths of about 1740 m and cliff heights of about 17.4 m for original land surface slopes of 0.01. This agrees to within about 10% of mean values obtained from bathymetric and topographic maps.