Mississippi Embayment Project: Bradshaw

From GeoMod

Contents 1 Determine model domain 2 Governing equations 3 Discretized equations 4 Model parameters 5 Initial conditions 6 Boundary conditions 7 Programming 8 Model verification and validations 9 Results of model simulations 10 Discussion 11 Conclusion

Determine model domain

What spatial and temporal areas are going to be covered by the model. What is the scope of the model.

• This 2-D model will be used to simulate the erosion and subsidence of the Earth's crust to form the Mississippi Embayment.

Governing equations

Discretized equations

• 2D Diffusion Equation

Where:

Z = elevation of land surface

K = diffusion coefficient

dt = time step

dx = change in distance in x-direction

dy = change in distance in y-direction

S=K*dt/dx^2

and

Model parameters

• nrows = 41
• ncols = 41
• dx = 100000.0 m
• dt = 1000 yr
• K = 500000 m/yr
• S = K*dt/dx*dx

Initial conditions

• Initial elevation = 34000 m
• Average erosion rate = 89 m/m.y.

Boundary conditions

• Northern Boundary: no flow
• Southern Boundary: no flow
• Eastern Boundary: no flow
• Western Boundary: no flow

Programming

from visual import *
from raster_map import *


''' set dimensions of grid'''
nrows = 41
ncols = 41


'''set parameters'''
dx = 100000.0 #meters
dz = 32000.0 #meters
dt = 1000 #year
K = 500000 #meters per year
S = K * dt / (dx*dx)

'''Determine if S exceeds the stability threshold'''
#if S > 0.0:
    #exit


'''Initial conditions: create elevation array'''
Z = raster_map(dz*ones((nrows,ncols)), dx=dx) #elevation array
color_scale = color_map(32000.0, 33000.0, 100.0,  colmin=vector(1,0,0), colmax=vector(0,1,0))
Z.data[20,20]=dz+999.0 #high elevation near center

'''Boundary conditions'''
Z.data[:,0]=dz #South
Z.data[:,nrows-1]=dz #North
Z.data[0,:]=dz #West
Z.data[ncols-1,:]=dz #East


'''Draw map of elevation'''

print Z.data
Z.line_3d(color_map=color_scale,scale=100.0)

nsteps = 1000

for nt in range(1, nsteps+1):
    Z_new = Z.data[1:nrows-1, 1:ncols-1] + S * (Z.data[2:nrows, 1:ncols-1] - 4 * Z.data[1:nrows-1, 1:ncols-1] +
                                              Z.data[0:nrows-2, 1:ncols-1] + Z.data[1:nrows-1, 2:ncols] +
                                              Z.data[1:nrows-1, 0:ncols-2])
    
    '''update'''
    Z.data[1:nrows-1,1:ncols-1] = Z_new
    Z.update_line_3d()
    print nt, Z_new,

Model verification and validations

• Assumptions

• Uniform topography
• Original crust topography was flat
• Uniform erosion rate for the entire area
• No effect from water runoff or streams on erosion
• All boundaries are no flow

Results of model simulations

Erosion was simulated for a high point of elevation located in the center of the model. The model seemed to work, but would need to be combined with the thermal expansion model to simulate any actual erosion or subsidence of the Earth's crust.

Discussion

Since the equation used to simulate erosion does work, the next step is to combine the erosion model with the model used to simulate thermal expansion of the crust from an underlying hot spot.

Conclusion