FORMULATION OF 2-D MARGIN MODEL

PRODUCING 3-D STRATIGRAPHY

Gary Parker

(Based on the Swenson-Paola-Voller Approach)

January 11, 1998

In thinking about the problem, I formulated two models: one consisting of a coastal plain and a slope of constant slope (Gilbert delta), and another with a coastal plain, a shelf and a slope of constant slope. Here I summarize the formulation for the latter; handwritten notes on both are enclosed.



DEFINITIONS

x = offshore coordinate

y = alongshore coordinate

t = time

s(y,t) = distance from sediment source to shoreline

b(y,t) = distance from sediment source to shelf-slope rollover

u(y,t) = distance from sediment source to base of slope

(x,y,t) = sediment surface elevation

(x,y,t) = basement elevation

 = subsidence rate

(x,y,t) =  -  = sediment deposit thickness

Z(t) = water surface elevation

qo(y,t) = volume sediment input rate/alongshelf distance at x = 0 in m2/s

Hb = depth from water surface to wave base (constant)

Ss = slope of slope (constant)

s(y,t) = angle shoreline makes to the y axis

b(y,t) = angle shelf-slope rollover boundary makes with the y axis

= unit outward vector normal to shoreline

= unit outward vector normal to shelf-slope rollover boundary

= time rate of normal migration of the shelf-slope rollover boundary (> 0 for outward

migration)

n = coordinate outward normal from the shelf-slope rollover boundary

rb = radius of curvature of the shelf-slope rollover position

nu = normal distance from the shelf-slope rollover position to the base of the slope

SETUP

The angle s of the shoreline to the y-axis is defined by the following relations

The unit normal vector outward to the shoreline is given by

Corresponding variables can be defined for the position of the shelf-slope rollover:

and

The radius of curvature of the shelf-slope rollover position rb is given by

Let denote the time rate of normal advance of the shelf-slope rollover position ( > 0 denotes outward advance), and let denote b/t. It follows from geometry that


Finally the subsidence rate  is given by

GOVERNING RELATIONS

Coastal plain. This is assumed to be purely diffusional. An interesting research problem is the determination of the along-coast diffusion coefficient cy.

Shelf. This is assumed to be diffusive-advective.

Slope. This maintains a constant slope Ss.

where n denotes an outward normal coordinate extending from the shelf-slope rollover.

BOUNDARY AND CONTINUITY CONDITIONS

The sediment infeed rate is specified at x = 0;

Here qo(y,t) can be specified as a line source, or a sum of point sources.

At the shoreline, the bed elevation must be that of the water and the sediment transport rate normal to the boundary must be continuous;

At the shelf-slope boundary, the bed elevation must be that of the elevation of the wave base and sediment continuity must be satisfied across the boundary. The first of these conditions yields

The second of these conditions requires no small amount of calculation. I obtained the following result:

where in the above equation n is an outward normal coordinate extending from the shelf-slope boundary, nu is the normal distance from the shelf-slope boundary to the bottom of the slope (where  = 0) and rb is the radius of curvature of the position of the shelf-slope break in the alongshelf direction.

The computation of nu must be performed iteratively. We wish to compute the distance u at some point y. Consider the vector illustrated below with length nu that has its origin along the shelf-slope boundary at point y + nu sinc and its end at point (u, y). Thus c is the angle of b to y at point y + nu sinc. It follows from geometric considerations that

Once nu is found, u is given as follows;


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