Lab 1.1: Sedimentary Transport

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Course completed Spring 2025.

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Lab 1.1: Sedimentary Transport

Introduction

In this lab, you will develop a model of the profile of a prograding delta. You will then use your model to understand the conditions that explain variations in delta shape.

Bulk sediment transport: diffusion

The change of elevation over time is proportional to the second partial derivative of the topography with respect to space (the curvature). This equation is sometimes referred to as the hillslope application of the diffusion equation:

\[\dfrac{\partial h}{\partial t} = K \dfrac{\partial^2 h}{\partial x^2} \label{eq:diff}\tag{diffusion equation}\]

The diffusion equation is derived from combining the continuity equation, which expresses the conservation of mass:

\[\dfrac{\partial h}{\partial t} = - \dfrac{\partial S}{\partial x} \label{eq:cont}\tag{continuity equation}\]

with an expression of sediment transport rate, \(S\), as a diffusive flux. In this formulation, \(S\) is linearly proportional to slope:

\[S = - K \dfrac{\partial h}{\partial x}\tag{diffusive flux}\]

Kenyon and Turcotte 1985 solves the diffusion equation analytically by introducing a coordinate system that moves along with the landward edge of the delta front, which is steadily prograding at a rate, \(u_0\):

\[\xi = x - u_0t\]

\[t' = t \label{eq:mrf} \tag{moving reference frame}\]

and the associated partial derivatives:

\[\begin{split}\begin{aligned} \dfrac{\partial h}{\partial x} & =\dfrac{\partial h}{\partial \xi} \\ \dfrac{\partial h}{\partial t} & =\dfrac{\partial h}{\partial t'} - u_0\dfrac{\partial h}{\partial \xi} \\ \end{aligned}\end{split}\]

Now, we can write the diffusion equation within a moving reference frame as:

\[\begin{aligned} \dfrac{\partial h}{\partial t'} - u_0 \dfrac{\partial h}{\partial \xi} = K \dfrac{\partial^2 h}{\partial \xi^2} \label{eq:diffmrf} \end{aligned}\]

In this reference frame, the height of the delta does not change with respect to time, so \(\dfrac{\partial h}{\partial t'}=0\), and this form of the diffusion equation reduces to the ordinary differential equation:

\[\begin{aligned} 0 = \dfrac{d^2 h}{d \xi^2} + \dfrac{u_0}{K} \dfrac{d h}{d \xi} \end{aligned}\]

with the following solution:

\[\begin{aligned} h(\xi) = h_0 e^{\left(-\dfrac{u_0 \xi}{K}\right)} \end{aligned}\]