Cutoffs in submarine channels

A paper on sinuous submarine channels that is based on work I have done with Jake Covault (at the Quantitative Clastics Laboratory, Bureau of Economic Geology, UT Austin) has been recently published in the journal Geology (officially it will be published in the October issue, but it has been online for a few weeks now).

The main idea of the paper is simple: many submarine channels become highly sinuous and cutoff bends, similar to those observed in the case of rivers, must be fairly common. The more high-quality data becomes available from these systems, the more evidence there is that this is indeed the case. I think this observation is fairly interesting in itself, as cutoffs will add significant complexity to the three-dimensional structure of the preserved deposits. We have addressed that complexity to some degree in another paper. However, previous work on submarine channels has not dealt with the implications of how the along-channel slope changes as cutoff bends form.

So, inspired by a model of incising subaerial bedrock rivers, we have put together a numerical model that not only mimics the plan-view evolution of meandering channels, by keeping track of the x and y coordinates of the channel centerline, but it also records the elevations (z-values) at each control point along this line. The plan-view evolution is based on the simplest possible model of meandering, described in a fantastic paper by Alan Howard and Thomas Knutson published in 1984 and titled “Sufficient conditions for river meandering: A simulation approach“. The key idea is that the rate of channel migration is not only a function of local curvature (in the sense of larger curvatures resulting in larger rates of outer bank erosion and channel migration), but it also depends on the channel curvature upstream from the point of interest; and the strength of this dependence declines exponentially as you move upstream. Without this ‘trick’ there is no stable meandering. The model starts out with an (almost) straight line that has some added noise; meanders develop through time as a dominant wavelength gets established. The z-coordinates are assigned initially so that they describe a constant along-channel slope; as the sinuosity increases, the overall slope decreases and local variability appears as well, depending of the local sinuosity of the channel. More discrete changes take place at the time and location of a cutoff: this is essentially a shortcut that introduces a much steeper segment into the centerline. The animations below (same as two of the supplementary files that were published with the paper) show how the cutoffs form and how the along-channel slope profiles change through time.


Animation of incising submarine channels with early meander cutoffs.



Evolution of along-channel gradient (top) and elevation profile (bottom) through time.

Meandering lowland rivers (like the Mississippi) have very low slopes to start with, so cutoff-related elevation drops are not as impressive as in the case of steep (with up to 40-50 m/km gradients) submarine channels (see diagram below). As a result, knickpoints that form this way in submarine channels are likely to be the locations of erosion and contribute to the stratigraphic complexity of the channel deposits. Previous work has shown that avulsion- and deformation-driven knickpoints are important in submarine channels; our results suggest that the meandering process itself can generate significant morphologic and stratigraphic along-channel variability.


Knickpoint gradient plotted against the overall slope. Dashed lines show trends for different cutoff distances (beta is the ratio between cutoff distance and channel width).

Finally I should mention that, apart from the seismic interpretation itself, we have done all the modeling, analysis, visualization, and plotting with Jupyter notebooks and Mayavi (for 3D visualization), using the Anaconda Python distribution. I cannot say enough good things about – and enough thanks for – these tools.

Stratigraphic patterns in slope minibasins

A few months ago some former colleagues and myself have published a paper on the stratigraphic evolution of minibasins that are present on many continental slopes (Sylvester, Z., Cantelli, A., and Pirmez, C., 2015, Stratigraphic evolution of intraslope minibasins: Insights from surface-based model: AAPG Bulletin, v. 99, no. 6, p. 1099–1129, doi: 10.1306/01081514082.) We used a simple model that investigates the interplay between subsidence and sedimentation and helps in the understanding of how stratal termination patterns relate to variations in sediment input and basin subsidence. Conventional sequence stratigraphy focuses on what is happening on the shelf and the shelf edge; and the origin of stratal patterns that characterize the typical ‘slug diagram’ are not trivial to relate to the three main parameters that influence continental-margin stratigraphy: changes in sea level, sediment supply, and subsidence. Our minibasin model is simpler because (1) it only has two parameters: sediment supply and subsidence (the impact of sea level – if any – is included in the sediment supply curve); and (2) sedimentary layers are deposited horizontally during each time step. The basic idea is to investigate the geometries of a system that consists of a depression that deepens through time and sediment is deposited in it with a certain rate. The figures that follow are color versions of the model cross sections that were published in the paper. The originals are actually interactive (you can zoom in and pan if you want to, which is useful if you want to check some details, especially along the basin margins). However, I could not get WordPress to display these in an interactive format; you can check out the interactive version of this post on Github. I have used the excellent, d3.js-based – and recently open-sourced – Plotly to create the interactive plots. A few brief comments follow so that the plots make some sense even if you don’t read the paper.

There are two types of layers in the model: (1) the ones that correspond to the sediment supply that comes into the basin from updip, mostly in the form of turbidity currents – these are shown in yellow color in the plots below; and (2) thinner ones that reflect an overall finer-grained background sedimentation – these are shown in brown color. The latter have a constant thickness across the basin. It is important to note that, while it is true that all the sand is supposed to be in the yellow units, not all of the yellow units are coarse-grained. In fact, slope minibsains have overall a much lower sand content than what a ‘yellow = sand’ assumption would imply in these plots. The blue dots are pinchout points, locations where the yellow layers terminate. In the paper we adopted the idea that ‘onlap’ refers to the case when these termination points move toward the basin edge, and ‘offlap’ corresponds to stratigraphic intervals where the pinchout locations migrate toward the center of the basin.

Sedimentary fills of slope minibasins commonly show strata that either converge toward or onlap onto the basin margins (or show a combination of the two). The first two cross sections are examples of ‘pure’ convergence and well-defined onlap. In the first case (convergence), both sediment input rate and subsidence are constant and have similar orders of magnitude, and, as a result, the locations where sedimentary layers terminate are stationary. The basin never gets very deep, as sedimentation keeps up with subsidence.

Screen Shot 2015-12-19 at 4.40.33 PM

Obviously animations are much better at conveying the idea of sedimentation taking place while the basin is subsiding (the background sedimentation is set to zero in the animation; the lower panel is a chronostratigraphic – or Wheeler – diagram):


For the formation of onlap, a relatively deep basin and high sediment input are needed. The model below has the same overall sediment volume as the previous one, but it was deposited in half the time. Subsidence was kept constant, just like in the convergence model.

Screen Shot 2015-12-19 at 4.42.09 PM

And here is the animation for the onlap scenario:


The next two plots illustrate what happens if the sediment input varies following a sinusoidal curve with three cycles, while subsidence is constant. In the first model, there is enough accommodation in the basin to keep all the sediment; in the second model, the sediment input exceeds the space available in the basin and each cycle has an upper part during which some of the sediment bypasses the basin. If you zoom in, you can see that both onlap and offlap are present in both cases.

Screen Shot 2015-12-19 at 4.44.14 PM

Screen Shot 2015-12-19 at 4.46.05 PM

And the animation for the no-bypass scenario:


Next we looked at what happens if sediment input is kept constant, but subsidence varies (following a sinusoidal function). It is tempting to think that the result must be similar to the case of variable sediment input and constant subsidence, but that is *not* the case. In  previous models onlap surfaces overlie condensed sections across the whole basin. In this new model, onlap surfaces are restricted to the basin margin and they correlate to sections with high sedimentation rate at the basin center.

Screen Shot 2015-12-19 at 4.46.51 PM

It is a cool idea that stratal patterns might tell you whether variability in sediment supply or deformation is more important. However, if both parameters are changing through time, with similar magnitudes, it is practically impossible to tell what is going on:

Screen Shot 2015-12-19 at 4.47.44 PM

This animation gives you an idea how can these two cases transition into each other, as sediment input goes from constant to sinusoidal and out-of-phase with subsidence:


So far all the simulations were based on sinusoidally varying parameters; what happens though if sediment supply is closer to an ‘on-off’ function? You can see the result below: each sedimentary cycle only shows onlap, the offlapping upper part is missing.

Screen Shot 2015-12-19 at 4.48.35 PM

The next two plots are attempts to reproduce the stratal patterns seen in two well-studied minibasins in the Gulf of Mexico: Brazos-Trinity Basin 4 and the Auger Basin.

Screen Shot 2015-12-19 at 4.49.19 PM

Screen Shot 2015-12-19 at 4.50.08 PM

Finally, we looked at the common scenario of multiple linked minibasins, with sediment bypassing the updip basin serving as the sediment input for the basin downdip. The classic idea about this depositional setting is called ‘fill and spill’: the updip basin has to be filled first before any sediment gets into or spills the second basin. A ‘static’ version of this idea, with pre-formed basins and no subsidence during sedimentation, is illustrated in the plot below, with three sediment input cycles:

Screen Shot 2015-12-19 at 4.50.46 PM

The animated version looks like this:


However, this static fill and spill is probably the exception rather than the rule: most minibasins keep subsiding while sedimentation is taking place. So, in contrast with the previous model, the large-scale fills of the basins are time-equivalent, while at the scale of individual sediment input cycles the fill-and-spill model still works.

Screen Shot 2015-12-19 at 4.50.58 PM

Now let’s animate this:


For more details on these models, see the paper in AAPG Bulletin. Ten related animations are open access and available on the AAPG website. And, again, the plots are a lot more interesting if you can zoom, pan and rescale, so you should really check out the intended format of this post.

Forest cover change in the Carpathians since 1985

There has been a lot of talk in recent months about how bad the deforestation problem is in Romania, especially related to the expansion of the Austrian company Schweighofer. Although there is a lot of chit-chat in the media and on the internet on the subject, data and facts are not that easy to find. Not that easy — unless you make a bit of an effort: it turns out that a massive dataset on forest cover change in Eastern Europe is available for download, thanks to scientists at the University of Maryland who have published a paper on the dataset in the journal Remote Sensing of Environment.

The dataset is based on Landsat imagery collected between 1985 and 2012 and has a pixel size of 30×30 meters. It can be downloaded from this website. I am especially interested in what’s going on in parts of the Carpathians, because that’s where I grew up and went on lots of hikes and through lots of adventures in the 1980s and early 1990s. What follows are a few screenshots that give an idea about what is possible to see with this data.

This first image shows the whole Carpathian – intra-Carpathian area. There are five colors that correspond to different histories of forest cover: black pixels are places without forest during the time of study; green is stable forest; blue is forest gain; red is forest loss; and purple is forest loss followed by forest gain [there are two additional categories in the data, but they are not very common in this area].


The good news is that there is a lot of green in this map, which means that about 28.5% of the Carpathian area was continuously covered by forest since 1985. An additional 3.4% was without tree cover in 1985, but has gained forest cover since then. The forest is gone from 1.5% of the area; and 1.7% has lost and then regained tree cover.

If we zoom in to the ‘bend’ area, which is essentially the southeastern corner of Transylvania, Romania, it becomes more obvious that, although the big picture doesn’t look very bad, some places have been affected fairly significantly over the last three decades:


The black, forest-free patches in the middle are young intramontane basins with no forest. Deforestation looks to be more of a problem in the the Ciuc/Csík and Gheorgheni/Gyergyó basins. Let’s have a closer look at the Csík Basin:


A further zoom-in shows one of the areas with the most forest loss, the western side of the Harghita Mountains:


I am going to stop here; but this dataset has a lot more to offer than I showed in this post. The conclusion from this certainly shouldn’t be that everything is fine; often the issue is not so much the quantity of the forest being cut, but *where* it is being cut. Protected areas and national parks should clearly be green and stay green on these maps; and the Romanian Carpathians could use a few more protected lands, as many of these forests have never been cut (unlike a lot of forests in Western Europe).

I have used IPython Notebook with the GDAL package to create these images. The notebook can be viewed and downloaded over here.


P.V. Potapov, S.A. Turubanova, A. Tyukavina, A.M. Krylov, J.L. McCarty, V.C. Radeloff, M.C. Hansen, Eastern Europe’s forest cover dynamics from 1985 to 2012 quantified from the full Landsat archive, Remote Sensing of Environment, Volume 159, 15 March 2015, Pages 28-43,

Exploring the diffusion equation with Python

Ever since I became interested in science, I started to have a vague idea that calculus, matrix algebra, partial differential equations, and numerical methods are all fundamental to the physical sciences and engineering and they are linked in some way to each other. The emphasis here is on the word vague; I have to admit that I had no clear, detailed understanding of how these links actually work. It seems like my formal education both in math and physics stopped just short of where everything would have nicely come together. Papers that are really important in geomorphology, sedimentology or stratigraphy seemed impossible to read as soon as they started assuming that I knew quite a bit about convective acceleration, numerical schemes, boundary conditions, and Cholesky factorization. Because I didn’t.

So I have decided a few months ago that I had to do something about this. This blog post documents the initial – and admittedly difficult – steps of my learning; the purpose is to go through the process of discretizing a partial differential equation, setting up a numerical scheme, and solving the resulting system of equations in Python and IPython notebook. I am learning this as I am doing it, so it may seem pedestrian and slow-moving to a lot of people but I am sure there are others who will find it useful. Most of what follows, except the Python code and the bit on fault scarps, is based on and inspired by Slingerland and Kump (2011): Mathematical Modeling of Earth’s Dynamical Systems (strongly recommended). You can view and download the IPython Notebook version of this post from Github.

Estimating the derivatives in the diffusion equation using the Taylor expansion

This is the one-dimensional diffusion equation:

\frac{\partial T}{\partial t} - D\frac{\partial^2 T}{\partial x^2} = 0

The Taylor expansion of value of a function u at a point \Delta x ahead of the point x where the function is known can be written as:

u(x+\Delta x) = u(x) + \Delta x \frac{\partial u}{\partial x} + \frac{\Delta x^2}{2} \frac{\partial^2 u}{\partial x^2} + \frac{\Delta x^3}{6} \frac{\partial^3 u}{\partial x^3} + O(\Delta x^4)

Taylor expansion of value of the function u at a point one space step behind:

u(x-\Delta x) = u(x) - \Delta x \frac{\partial u}{\partial x} + \frac{\Delta x^2}{2} \frac{\partial^2 u}{\partial x^2} - \frac{\Delta x^3}{6} \frac{\partial^3 u}{\partial x^3} + O(\Delta x^4)

Solving the first Taylor expansion above for \frac{\partial u}{\partial x} and dropping all higher-order terms yields the forward difference operator:

\frac{\partial u}{\partial x} = \frac{u(x+\Delta x)-u(x)}{\Delta x} + O(\Delta x)

Similarly, the second equation yields the backward difference operator:

\frac{\partial u}{\partial x} = \frac{u(x)-u(x-\Delta x)}{\Delta x} + O(\Delta x)

Subtracting the second equation from the first one gives the centered difference operator:

\frac{\partial u}{\partial x} = \frac{u(x+\Delta x)-u(x-\Delta x)}{2\Delta x} + O(\Delta x^2)

The centered difference operator is more accurate than the other two.

Finally, if the two Taylor expansions are added, we get an estimate of the second order partial derivative:

\frac{\partial^2 u}{\partial x^2} = \frac{u(x+\Delta x)-2u(x)+u(x-\Delta x)}{\Delta x^2} + O(\Delta x^2)

Next we use the forward difference operator to estimate the first term in the diffusion equation:

\frac{\partial T}{\partial t} = \frac{T(t+\Delta t)-T(t)}{\Delta t}

The second term is expressed using the estimation of the second order partial derivative:

\frac{\partial^2 T}{\partial x^2} = \frac{T(x+\Delta x)-2T(x)+T(x-\Delta x)}{\Delta x^2}

Now the diffusion equation can be written as

\frac{T(t+\Delta t)-T(t)}{\Delta t} - D \frac{T(x+\Delta x)-2T(x)+T(x-\Delta x)}{\Delta x^2} = 0

This is equivalent to:

T(t+\Delta t) - T(t) - \frac{D\Delta t}{\Delta x^2}(T(x+\Delta x)-2T(x)+T(x-\Delta x)) = 0

The expression D\frac{\Delta t}{\Delta x^2} is called the diffusion number, denoted here with s:

s = D\frac{\Delta t}{\Delta x^2}

FTCS explicit scheme and analytic solution

If we use n to refer to indices in time and j to refer to indices in space, the above equation can be written as

T[n+1,j] = T[n,j] + s(T[n,j+1]-2T[n,j]+T[n,j-1])

This is called a forward-in-time, centered-in-space (FTCS) scheme. Its ‘footprint’ looks like this:

import numpy as np
import matplotlib.pyplot as plt
%matplotlib inline
%config InlineBackend.figure_format = 'svg' # display plots in SVG format

plt.title('FTCS explicit scheme',fontsize=12)

Screen Shot 2015-02-02 at 9.03.19 PM

Now we are ready to write the code that is the solution for exercise 2 in Chapter 2 of Slingerland and Kump (2011). This is an example where the one-dimensional diffusion equation is applied to viscous flow of a Newtonian fluid adjacent to a solid wall. If the wall starts moving with a velocity of 10 m/s, and the flow is assumed to be laminar, the velocity profile of the fluid is described by the equation

\frac{\partial V}{\partial t} - \nu \frac{\partial^2 V}{\partial y^2} = 0

where \nu is the kinematic viscosity of the fluid. We want to figure out how the velocity will change through time as a function of distance from the wall. [Note that I have changed the original 40 m/s to 10 m/s — the former seems like an unnaturally large velocity to me].

We can compare the numerical results with the analytic solution, which is known for this problem:

V = V_0 \Big\{ \sum\limits_{n=0}^\infty erfc\big(2n\eta_1+\eta\big) - \sum\limits_{n=0}^\infty erfc\big(2(n+1)\eta_1+\eta\big) \Big\}


\eta_1 = \frac{h}{2\sqrt{\nu t}}


\eta = \frac{y}{2\sqrt{\nu t}}

dt = 0.0005 # grid size for time (s)
dy = 0.0005 # grid size for space (m)
viscosity = 2*10**(-4) # kinematic viscosity of oil (m2/s)
y_max = 0.04 # in m
t_max = 1 # total time in s
V0 = 10 # velocity in m/s

# function to calculate velocity profiles based on a 
# finite difference approximation to the 1D diffusion 
# equation and the FTCS scheme:
def diffusion_FTCS(dt,dy,t_max,y_max,viscosity,V0):
    # diffusion number (has to be less than 0.5 for the 
    # solution to be stable):
    s = viscosity*dt/dy**2 
    y = np.arange(0,y_max+dy,dy) 
    t = np.arange(0,t_max+dt,dt)
    r = len(t)
    c = len(y)
    V = np.zeros([r,c])
    V[:,0] = V0
    for n in range(0,r-1): # time
        for j in range(1,c-1): # space
            V[n+1,j] = V[n,j] + s*(V[n,j-1] - 
                2*V[n,j] + V[n,j+1]) 
    return y,V,r,s
# note that this can be written without the for-loop 
# in space, but it is easier to read it this way

from scipy.special import erfc

# function to calculate velocity profiles 
# using the analytic solution:
def diffusion_analytic(t,h,V0,dy,viscosity):
    y = np.arange(0,h+dy,dy)
    eta1 = h/(2*(t*viscosity)**0.5)
    eta = y/(2*(t*viscosity)**0.5)
    sum1 = 0
    sum2 = 0
    for n in range(0,1000):
        sum1 = sum1 + erfc(2*n*eta1+eta)
        sum2 = sum2 + erfc(2*(n+1)*eta1-eta)
    V_analytic = V0*(sum1-sum2)
    return V_analytic

y,V,r,s = diffusion_FTCS(dt,dy,t_max,y_max,viscosity,V0)

# plotting:
plot_times = np.arange(0.2,1.0,0.1)
for t in plot_times:
    V_analytic = diffusion_analytic(t,0.04,40,dy,viscosity)
    if t==0.2:
plt.xlabel('distance from wall (m)',fontsize=12)
plt.ylabel('velocity (m/s)',fontsize=12)
plt.title('comparison between explicit numerical 
    \n(FTCS scheme) and analytic solutions');

diffusion with FTCS scheme

The dots (analytic solution) overlap pretty well with the lines (numerical solution). However, this would not be the case if we changed the discretization so that the diffusion number was larger. Let’s look at the stability of the FTCS numerical scheme, by computing the solution with different diffusion numbers. It turns out that the diffusion number s has to be less than 0.5 for the FTCS scheme to remain stable. What follows is a reproduction of Figure 2.7 in Slingerland and Kump (2011):

dt = 0.0005 # grid size for time (m)
dy = 0.0005 # grid size for space (s)
y,V,r,s = diffusion_FTCS(dt,dy,t_max,y_max,viscosity,V0)
V_analytic = diffusion_analytic(0.5,0.04,V0,dy,viscosity)
plt.plot(y,V_analytic-V[0.5/dt],'--k',label='small s')

dy = 0.0010
dt = 0.00254
y,V,r,s = diffusion_FTCS(dt,dy,t_max,y_max,viscosity,V0)
V_analytic = diffusion_analytic(0.5,0.04,V0,dy,viscosity)
V_numeric = V[r/2-1,:]

plt.plot(y,V_analytic-V_numeric,'k',label='large s')
plt.xlabel('distance from wall (m)',fontsize=12)
plt.ylabel('velocity difference (m/s)',fontsize=12)
plt.title('difference between numerical and analytic 
   \n solutions for different \'s\' values',fontsize=14)

difference between analytic and numerical solutions

Laasonen implicit scheme

plt.title('Laasonen scheme',fontsize=12)

Laasonen scheme

Instead of estimating the velocity at time step n+1 with the curvature calculated at time step n, as it is done in the FTCS explicit scheme, we can also estimate the curvature at time step n+1, using the velocity change from time step n to time step n+1:

s\big(T(x+\Delta x)-2T(x)+T(x-\Delta x)\big) = T(t+\Delta t)-T(t)

Written in matrix notation, this is equiavlent to

s\big(T[n+1,j+1]-2T[n+1,j]+T[n+1,j-1]\big) = T[n+1,j]-T[n,j]

After some reshuffling we get

-sT[n+1,j+1] + (1+2s)T[n+1,j] - sT[n+1,j-1] = T[n,j]

This is the Laasonen fully implicit scheme. Unlike the FTCS scheme, the Laasonen scheme is unconditionally stable. Let’s try to write some Python code that implements this scheme. First it is useful for me to go through the logic of constructing the system of equations that needs to be solved. Let’s consider a grid that only consists of 5 nodes in space and we are going to estimate the values of T at the locations marked by the red dots in the figure below. Black dots mark the locations where we already know the values of T (from the initial and boundary conditions).

for i in range(0,4):
for i in range(0,5):
plt.title('first two time steps on a 1D grid of five points',fontsize=12)
first two time steps

First we write the equations using the Laasonen scheme centered on the three points of unknown velocity (or temperature) — these are the red dots in the figure above:

\begin{array}{rrrrrcl}   -sT[1,0]&+(1+2s)T[1,1]&-sT[1,2]&+0T[1,3]&+0T[1,4]&=&T[0,1] \\  0T[1,0]&-sT[1,1]&+(1+2s)T[1,2]&-sT[1,3]&+0T[1,4]&=&T[0,2] \\  0T[1,0]&+0T[1,1]&-sT[1,2]&+(1+2s)T[1,3]&-sT[1,4]&=&T[0,3]  \end{array}

It may seem like we have five unknowns and only three equations but T[1,0] and T[1,4] are on the boundaries and they are known. Let’s rearrange the equation system so that the left hand side has ony the unknowns:

\begin{array}{rrrrcrr}   (1+2s)T[1,1]&-sT[1,2]&+0T[1,3]&=&T[0,1]&+sT[1,0] \\  -sT[1,1]&+(1+2s)T[1,2]&-sT[1,3]&=&T[0,2]& \\  0T[1,1]&-sT[1,2]&+(1+2s)T[1,3]&=&T[0,3]&+sT[1,4]  \end{array}

In matrix form this is equivalent to

\begin{bmatrix} 1+2s & -s & 0 \\  -s & 1+2s & -s \\  0 & -s & 1+2s \end{bmatrix} \times   \left[ \begin{array}{c} T[1,1] \\ T[1,2] \\ T[1,3] \end{array} \right]  = \left[ \begin{array}{c} T[0,1]+sT[1,0] \\ T[0,2] \\ T[0,3]+sT[1,4] \end{array} \right]

This of course can be extended to larger dimensions than shown here.
Now we are ready to write the code for the Laasonen scheme. One important difference relative to what I did in the explicit scheme example is that in this case we only keep the last two versions of the velocity distribution in memory, as opposed to preallocating the full array of nt x ny size as we did before. This difference is not a significant time saver for simple problems like this but once you start dealing with more complicated tasks and code it is not possible and/or practical to keep the results of all time steps in memory.

from scipy.sparse import diags
def diffusion_Laasonen(dt,dy,t_max,y_max,viscosity,V0,V1):
    s = viscosity*dt/dy**2  # diffusion number
    y = np.arange(0,y_max+dy,dy) 
    t = np.arange(0,t_max+dt,dt)
    nt = len(t) # number of time steps
    ny = len(y) # number of dy steps
    V = np.zeros((ny,)) # initial condition
    V[0] = V0 # boundary condition on left side
    V[-1] = V1 # boundary condition on right side
    # create coefficient matrix:
    A = diags([-s, 1+2*s, -s], [-1, 0, 1], shape=(ny-2, ny-2)).toarray() 
    for n in range(nt): # time is going from second time step to last
        Vn = V #.copy()
        B = Vn[1:-1] # create matrix of knowns on the RHS of the equation
        B[0] = B[0]+s*V0
        B[-1] = B[-1]+s*V1
        V[1:-1] = np.linalg.solve(A,B) # solve the equation using numpy
    return y,t,V,s

Because this is a stable scheme, it is possible to get reasonable solutions with relatively large time steps (which was not possible with the FTCS scheme):

dt = 0.01 # grid size for time (s)
dy = 0.0005 # grid size for space (m)
viscosity = 2*10**(-4) # kinematic viscosity of oil (m2/s)
y_max = 0.04 # in m
V0 = 10.0 # velocity in m/s
V1 = 0.0 # velocity in m/s

for time in np.linspace(0,1.0,10):
    y,t,V,s = diffusion_Laasonen(dt,dy,time,y_max,viscosity,V0,V1)
plt.xlabel('distance from wall (m)',fontsize=12)
plt.ylabel('velocity (m/s)',fontsize=12)
plt.title('Laasonen implicit scheme',fontsize=14);

Diffusion with Laasonen scheme

Just for fun, let’s see what happens if we set in motion the right side of the domain as well; that is, set V1 to a non-zero value:

dt = 0.01 # grid size for time (s)
dy = 0.0005 # grid size for space (m)
viscosity = 2*10**(-4) # kinematic viscosity of oil (m2/s)
y_max = 0.04 # in m
V0 = 10.0 # velocity in m/s
V1 = 5.0 # velocity in m/s

for time in np.linspace(0,1.0,10):
    y,t,V,s = diffusion_Laasonen(dt,dy,time,y_max,viscosity,V0,V1)
plt.xlabel('distance from wall (m)',fontsize=12)
plt.ylabel('velocity (m/s)',fontsize=12)
plt.title('Laasonen implicit scheme',fontsize=14);

Case when velocities are nonzero on both sides

Crank-Nicolson scheme

The Crank-Nicholson scheme is based on the idea that the forward-in-time approximation of the time derivative is estimating the derivative at the halfway point between times n and n+1, therefore the curvature of space should be estimated there as well. The ‘footprint’ of the scheme looks like this:

plt.title('Crank-Nicolson scheme',fontsize=12)

Crank-Nicolson scheme

The curvature at the halfway point can be estimated through averaging the curvatures that are calculated at n and n+1:

0.5s\big(T[n+1,j+1]-2T[n+1,j]+T[n+1,j-1]\big) + 0.5s\big(T[n,j+1]-2T[n,j]+T[n,j-1]\big) = T[n+1,j]-T[n,j]

This can be rearranged so that terms at n+1 are on the left hand side:

-0.5sT[n+1,j-1]+(1+s)T[n+1,j]-0.5sT[n+1,j+1] = 0.5sT[n,j-1]+(1-s)T[n,j]+0.5sT[n,j+1]

Just like we did for the Laasonen scheme, we can write the equations for the first two time steps:

\begin{array}{rrrrrcl}   -0.5sT[1,0] & +(1+s)T[1,1] & -0.5sT[1,2] & = & 0.5sT[0,0] & +(1-s)T[0,1] & +0.5sT[0,2] \\  -0.5sT[1,1] & +(1+s)T[1,2] & -0.5sT[1,3] & = & 0.5sT[0,1] & +(1-s)T[0,2] & +0.5sT[0,3] \\  -0.5sT[1,2] & +(1+s)T[1,3] & -0.5sT[1,4] & = & 0.5sT[0,2] & +(1-s)T[0,3] & +0.5sT[0,4]  \end{array}

Writing this in matrix form, with all the unknowns on the LHS:

\begin{bmatrix} 1+s & -0.5s & 0 \\ -0.5s & 1+s & -0.5s \\ 0 & -0.5s & 1+s \end{bmatrix} \times   \left[ \begin{array}{c} T[1,1] \\ T[1,2] \\ T[1,3] \end{array} \right]  = \begin{bmatrix} 1-s & 0.5s & 0 \\ 0.5s & 1-s & 0.5s \\ 0 & 0.5s & 1-s \end{bmatrix} \times  \left[ \begin{array}{c} T[0,1] \\ T[0,2] \\ T[0,3] \end{array} \right] +  \left[ \begin{array}{c} 0.5sT[1,0]+0.5sT[0,0] \\ 0 \\ 0.5sT[1,4]+0.5sT[0,4] \end{array} \right]

Now we can write the code for the Crank-Nicolson scheme. We will use a new input parameter called ntout that determines how many time steps we want to write out to memory. This way you don’t have to re-run the code if you want to plot multiple time steps.

def diffusion_Crank_Nicolson(dy,ny,dt,nt,D,V,ntout):
    Vout = [] # list for storing V arrays at certain time steps
    V0 = V[0] # boundary condition on left side
    V1 = V[-1] # boundary condition on right side
    s = D*dt/dy**2  # diffusion number
    # create coefficient matrix:
    A = diags([-0.5*s, 1+s, -0.5*s], [-1, 0, 1], 
          shape=(ny-2, ny-2)).toarray() 
    B1 = diags([0.5*s, 1-s, 0.5*s],[-1, 0, 1], shape=(ny-2, ny-2)).toarray()
    for n in range(1,nt): # time is going from second time step to last
        Vn = V
        B =[1:-1],B1) 
        B[0] = B[0]+0.5*s*(V0+V0)
        B[-1] = B[-1]+0.5*s*(V1+V1)
        V[1:-1] = np.linalg.solve(A,B)
        if n % int(nt/float(ntout)) == 0 or n==nt-1:
            Vout.append(V.copy()) # numpy arrays are mutable, 
            #so we need to write out a copy of V, not V itself
    return Vout,s

dt = 0.001 # grid size for time (s)
dy = 0.001 # grid size for space (m)
viscosity = 2*10**(-4) # kinematic viscosity of oil (m2/s)
y_max = 0.04 # in m
y = np.arange(0,y_max+dy,dy) 
ny = len(y)
nt = 1000
V = np.zeros((ny,)) # initial condition
V[0] = 10
Vout,s = diffusion_Crank_Nicolson(dy,ny,dt,nt,viscosity,V,10)

for V in Vout:
plt.xlabel('distance from wall (m)',fontsize=12)
plt.ylabel('velocity (m/s)',fontsize=12)
plt.title('Crank-Nicolson scheme',fontsize=14);

Diffusion with Crank-Nicolson scheme

Fault scarp diffusion

So far we have been using a somewhat artificial (but simple) example to explore numerical methods that can be used to solve the diffusion equation. Next we look at a geomorphologic application: the evolution of a fault scarp through time. Although the idea that convex hillslopes are the result of diffusive processes go back to G. K. Gilbert, it was Culling (1960, in the paper Analytical Theory of Erosion) who first applied the mathematics of the heat equation – that was already well known to physicists at that time – to geomorphology.

Here I used the Crank-Nicolson scheme to model a fault scarp with a vertical offset of 10 m. To compare the numerical results with the analytical solution (which comes from Culling, 1960), I created a function that was written using a Python package for symbolic math called sympy. One of the advantages of sympy is that you can quickly display equations in \LaTeX.

import sympy
from sympy import init_printing
x, t, Y1, a, K = sympy.symbols('x t Y1 a K')
y = (1/2.0)*Y1*(sympy.erf((a-x)/(2*sympy.sqrt(K*t))) + sympy.erf((a+x)/(2*sympy.sqrt(K*t))))

Screen Shot 2015-02-06 at 5.13.34 PM

The variables in this equation are x – horizontal coordinates, t – time, a – value of x where fault is located, K – diffusion coefficient, Y1 – height of fault scarp.

from sympy.utilities.lambdify import lambdify
# function for analytic solution:
f = lambdify((x, t, Y1, a, K), y)

dt = 2.5 # time step (years)
dy = 0.1 # grid size for space (m)
D = 50E-4 # diffusion coefficient in m2/yr 
# e.g., Fernandes and Dietrich, 1997
h = 10 # height of fault scarp in m
y_max = 20 # length of domain in m
t_max = 500 # total time in years
y = np.arange(0,y_max+dy,dy) 
ny = len(y)
nt = int(t_max/dt)
V = np.zeros((ny,)) # initial condition
V[:round(ny/2.0)] = h # initial condition

Vout,s = diffusion_Crank_Nicolson(dy,ny,dt,nt,D,V,20)

for V in Vout:

plt.xlabel('distance (m)',fontsize=12)
plt.ylabel('height (m)',fontsize=12)
plt.title('fault scarp diffusion',fontsize=14);
plt.plot(y,np.asarray([f(x0, t_max, h, y_max/2.0, D) 
   for x0 in y]),'r--',linewidth=2);

Fault scarp diffusion

The numerical and analytic solutions (dashed red line) are very similar in this case (total time = 500 years). Let’s see what happens if we let the fault scarp evolve for a longer time.

dt = 2.5 # time step (years)
dy = 0.1 # grid size for space (m)
D = 50E-4 # diffusion coefficient in m2/yr
h = 10 # height of fault scarp in m
y_max = 20 # length of domain in m
t_max = 5000 # total time in years
y = np.arange(0,y_max+dy,dy) 
ny = len(y)
nt = int(t_max/dt)
V = np.zeros((ny,)) # initial condition
V[:round(ny/2.0)] = h # initial condition

Vout,s = diffusion_Crank_Nicolson(dy,ny,dt,nt,D,V,20)

for V in Vout:

plt.xlabel('distance (m)',fontsize=12)
plt.ylabel('height (m)',fontsize=12)
plt.title('fault scarp diffusion',fontsize=14);
plt.plot(y,np.asarray([f(x0, t_max, h, y_max/2.0, D) 
   for x0 in y]),'r--',linewidth=2);

Fault scarp diffusion over long time

This doesn’t look very good, does it? The reason for the significant mismatch between the numerical and analytic solutions is the fixed nature of the boundary conditions: we keep the elevation at 10 m on the left side and at 0 m on the right side of the domain. There are two ways of getting a correct numerical solution: we either impose boundary conditions that approximate what the system is supposed to do if the elevations were not fixed; or we extend the space domain so that the boundary conditions can be kept fixed throughout the time of interest. Let’s do the latter; all the other parameters are the same as above.

dt = 2.5 # time step (years)
dy = 0.1 # grid size for space (m)
D = 50E-4 # diffusion coefficient in m2/yr
h = 10 # height of fault scarp in m
y_max = 40 # length of domain in m
t_max = 5000 # total time in years
y = np.arange(0,y_max+dy,dy) 
ny = len(y)
nt = int(t_max/dt)
V = np.zeros((ny,)) # initial condition
V[:round(ny/2.0)] = h # initial condition

Vout,s = diffusion_Crank_Nicolson(dy,ny,dt,nt,D,V,20)

for V in Vout:

plt.xlabel('distance (m)',fontsize=12)
plt.ylabel('height (m)',fontsize=12)
plt.title('fault scarp diffusion',fontsize=14);
plt.plot(y,np.asarray([f(x0, t_max, h, y_max/2.0, D) 
   for x0 in y]),'r--',linewidth=2);

Fault scarp diffusion, extended domain

Now we have a much better result. The vertical dashed lines show the extent of the domain in the previous experiment. We have also gained some insight into choosing boundary conditions and setting up the model domain. It is not uncommon that setting up the initial and boundary conditions is the most time-consuming and difficult part of running a numerical model.

Further reading

R. Slingerland and L. Kump (2011) Mathematical Modeling of Earth’s Dynamical Systems

W. E. H. Culling (1960) Analytical Theory of Erosion

L. Barba (2013) 12 steps to Navier-Stokes – an excellent introduction to computational fluid dynamics that uses IPython notebooks

I have blogged before about the geosciency aspects of the diffusion equation over here.

You can view and download the IPython Notebook version of this post from Github.

Questions or suggestions? Contact @zzsylvester

Rivers through time, as seen in Landsat images

Thanks to the Landsat program and Google Earth Engine, it is possible now to explore how the surface of the Earth has been changing through the last thirty years or so. Besides the obvious issues of interest, like changes in vegetation, the spread of cities, and the melting of glaciers, it is also possible to look at how rivers change their courses through time. You have probably already seen the images of the migrating Ucayali River in Peru, for example here. This river is changing its course with an impressive speed; many – probably most – other rivers don’t show much obvious change during the same 30-year period. What determines the meander migration rate of rivers is an interesting question in fluvial geomorphology.

The data that underlies Google Earth Engine is not accessible to everybody, but the Landsat data is available to anyone who creates a free account with Earth Explorer. It is not that difficult (but fairly time consuming) to download a set of images and create animations like this (click for higher resolution):


This scene also comes from the Ucayali River (you can view it in Google Earth Engine over here) and it is a nice example of how both neck cutoffs and chute cutoffs form. First a neck cutoff takes place that affects the tight bend in the right side of the image; this is followed by a chute cutoff immediately downstream of the neck cutoff location, as the new course of the river happens to align well with a pre-existing chute channel. The third bend in the upper left corner shows some well-developed counter-point-bar deposits. There is one frame in the movie for each year from 1985 to 2013, with a few years missing (due to low quality of the data).

Update (04/14/2016): If you want to use the animation, feel free to do so, as long as you (1) give credit to NASA/USGS Landsat, (2) give credit to me (= Zoltan Sylvester, geologist), and (3) link to this page. Note that you can see/download the high-resolution version if you click on the image.

Update (04/15/2016): Matteo Niccoli (who has a great blog called MyCarta) has created a slower version of the animation:


Memorable moments and photos from 2013

We are well into 2014 now, but it is not too late to look back at 2013 and pick some of the best moments (which means photos in my case) of the year that just passed.

We started out the year with a trip to Zion and Bryce Canyon National Parks. Although it was fairly cold (especially at Bryce Canyon NP — this park has a much higher elevation overall than Zion NP), we had lots of sunshine and did several day hikes. Visiting these parks in the winter is a great idea — they are a lot less crowded than in the summer, and obviously the landscapes and sights are quite different when they are covered with snow.

Zion National Park is paradise for a sedimentologist: there are endless, top-quality exposures of the Navajo Sandstone, showing all kinds of sedimentary structures characteristic of deposits of wind-blown sand. I have included two examples here; you can find more on my Smugmug site.

Sedimentologically, Bryce Canyon National Park is a bit less exciting than Zion, but this is counterbalanced by the fantastic geomorphology of this place. I haven’t seen Bryce Canyon in the summer, but I wouldn’t be surprised if it was more beautiful when it’s covered with snow.

In February, I went on a ‘business’ trip to Torres del Paine National Park in Southern Chile: I attended a field consortium meeting organized by Steve Hubbard’s group at the University of Calgary. I have been to this area several times before, as it has some of the best outcrops of turbidites (= deep-water sediments) in the world, but I was once again shocked how uniquely beautiful Chilean Patagonia can be.

At the end of the official trip, Zane Jobe (who is blogging at Off the Shelf Edge) and I did a bit of geo-turism: we went to see Glacier Grey and Lago Grey, and then did a day hike in the park to check out the actual Torres del Paine. The rest of the photos are here.

In July, my wife and I took a few days to do some hiking and running in Rocky Mountain National Park. I was struggling with a running injury at that time, but the mountains and the trails acted as efficient tranquilizers. More photos at Smugmug.

In September I attended a research conference on turbidity currents in Italy and Peter Talling showed us some of the classic outcrops of the Marnoso-Arenacea Formation. These rocks are very unique because they were deposited by huge submarine flows that covered the entire basin floor. Always wanted to see them and it was enlightening to get up close to them.


Turbidites of the Marnoso-Arenacea Formation, Italian Apennines. David Piper and Bill Arnott for scale

In October we spent a long weekend in Moab, Utah, to participate in our first trail races, but we also did some hiking. Running the Moab Trail Marathon was an amazing experience (I think I will have to do it again this year); unfortunately I didn’t take a camera with me, as I was trying to focus on running (and surviving the race).


Typical view in Canyonlands National Park

To continue with the theme of ‘national parks in winter’, some friends from California and the two of us wrapped up the year with a Christmas trip to Yellowstone and Grand Teton National Parks. More photos, of course, at Smugmug.

Exploring grain settling with Python

Grain settling is one of the most important problems in sedimentology (and therefore sedimentary geology), as neither sediment transport nor deposition can be understood and modeled without knowing what is the settling velocity of a particle of a certain grain size. Very small grains, when submerged in water, have a mass small enough that they reach a terminal velocity before any turbulence develops. This is true for clay- and silt-sized particles settling in water, and for these grain size classes Stokes’ Law can be used to calculate the settling velocity:

Screen Shot 2013-08-09 at 5.54.33 PM where R = specific submerged gravity (the density difference between the particle and fluid, normalized by fluid density), g = gravitational acceleration, D is the particle diameter, C1 is a constant with a theoretical value of 18, and the greek letter nu is the kinematic viscosity.

For grain sizes coarser than silt, a category that clearly includes a lot of sediment and rock types of great interest to geologists, things get more complicated. The reason for this is the development of a separation wake behind the falling grain; the appearance of this wake results in turbulence and large pressure differences between the front and back of the particle. For large grains – pebbles, cobbles – this effect is so strong that viscous forces become small compared to pressure forces and turbulent drag dominates; the settling velocity can be estimated using the empirical equation

Screen Shot 2013-08-09 at 5.54.40 PMThe important point is that, for larger grains, the settling velocity increases more slowly, with the square root of the grain size, as opposed to the square of particle diameter, as in Stokes’ Law.

Sand grains are small enough that viscous forces still play an important role in their subaqueous settling behavior, but large enough that the departure from Stokes’ Law is significant and wake turbulence cannot be ignored. There are several empirical – and fairly complicated – equations that try to bridge this gap; here I focus on the simplest one, published in 2004 in the Journal of Sedimentary Research (Ferguson and Church, 2004):

Screen Shot 2013-08-09 at 5.54.47 PM

At small values of D, the left term in the denominator is much larger than the one containing the third power of D, and the equation is equivalent of Stokes’ Law. At large values of D, the second term dominates and the settling velocity converges to the solution of the turbulent drag equation.

But the point of this blog post is not to give a summary of the Ferguson and Church paper; what I am interested in is to write some simple code and plot settling velocity against grain size to better understand these relationships through exploring them graphically. So what follows is a series of Python code snippets, directly followed by the plots that you can generate if you run the code yourself. I have done this using the IPyhton notebook, a very nice tool that allows and promotes note taking, coding, and plotting within one document. I am not going to get into details of Python programming and the usage of IPyhton notebook, but you can check them out here.

First we have to implement the three equations as Python functions:

import numpy as np
import matplotlib.pyplot as plt
rop = 2650.0 # density of particle in kg/m3
rof = 1000.0 # density of water in kg/m3
visc = 1.002*1E-3 # dynamic viscosity in Pa*s at 20 C
C1 = 18 # constant in Ferguson-Church equation
C2 = 1 # constant in Ferguson-Church equation
def v_stokes(rop,rof,d,visc,C1):
        R = (rop-rof)/rof # submerged specific gravity
        w = R*9.81*(d**2)/(C1*visc/rof)
        return w
def v_turbulent(rop,rof,d,visc,C2):
        R = (rop-rof)/rof
        w = (4*R*9.81*d/(3*C2))**0.5
        return w
def v_ferg(rop,rof,d,visc,C1,C2):
        R = (rop-rof)/rof
        w = ((R*9.81*d**2)/(C1*visc/rof+
        return w

Let’s plot these equations for a range of particle diameters:

d = np.arange(0,0.0005,0.000001)
ws = v_stokes(rop,rof,d,visc,C1)
wt = v_turbulent(rop,rof,d,visc,C2)
wf = v_ferg(rop,rof,d,visc,C1,C2)
plot([0.25, 0.25],[0, 0.15],'k--')
plot([0.25/2, 0.25/2],[0, 0.15],'k--')
plot([0.25/4, 0.25/4],[0, 0.15],'k--')
text(0.36, 0.11, 'medium sand', fontsize=13)
text(0.16, 0.11, 'fine sand', fontsize=13)
text(0.075, 0.11, 'v. fine', fontsize=13)
text(0.08, 0.105, 'sand', fontsize=13)
text(0.01, 0.11, 'silt and', fontsize=13)
text(0.019, 0.105, 'clay', fontsize=13)
xlabel('grain diameter (mm)',fontsize=15)
ylabel('settling velocity (m/s)',fontsize=15)
D = [0.068, 0.081, 0.096, 0.115, 0.136, 0.273,
    0.386, 0.55, 0.77, 1.09, 2.18, 4.36]
w = [0.00425, 0.0060, 0.0075, 0.0110, 0.0139, 0.0388,
    0.0551, 0.0729, 0.0930, 0.141, 0.209, 0.307]


The black dots are data points from settling experiments performed with natural river sands (Table 2 in Ferguson and Church, 2004). It is obvious that the departure from Stokes’ Law is already significant for very fine sand and Stokes settling is completely inadequate for describing the settling of medium sand.

This plot only captures particle sizes finer than medium sand; let’s see what happens as we move to coarser sediment. A log-log plot is much better for this purpose.

d = np.arange(0,0.01,0.00001)
ws = v_stokes(rop,rof,d,visc,C1)
wt = v_turbulent(rop,rof,d,visc,C2)
wf = v_ferg(rop,rof,d,visc,C1,C2)
plot([1.0/64, 1.0/64],[0.00001, 10],'k--')
text(0.012, 0.0007, 'fine silt', fontsize=13,
plot([1.0/32, 1.0/32],[0.00001, 10],'k--')
text(0.17/8, 0.0007, 'medium silt', fontsize=13,
plot([1.0/16, 1.0/16],[0.00001, 10],'k--')
text(0.17/4, 0.0007, 'coarse silt', fontsize=13,
plot([1.0/8, 1.0/8],[0.00001, 10],'k--')
text(0.17/2, 0.001, 'very fine sand', fontsize=13,
plot([0.25, 0.25],[0.00001, 10],'k--')
text(0.17, 0.001, 'fine sand', fontsize=13,
plot([0.5, 0.5],[0.00001, 10],'k--')
text(0.33, 0.001, 'medium sand', fontsize=13,
plot([1, 1],[0.00001, 10],'k--')
text(0.7, 0.001, 'coarse sand', fontsize=13,
plot([2, 2],[0.00001, 10],'k--')
text(1.3, 0.001, 'very coarse sand', fontsize=13,
plot([4, 4],[0.00001, 10],'k--')
text(2.7, 0.001, 'granules', fontsize=13,
text(6, 0.001, 'pebbles', fontsize=13,
xlabel('grain diameter (mm)', fontsize=15)
ylabel('settling velocity (m/s)', fontsize=15)


This plot shows how neither Stokes’ Law, nor the velocity based on turbulent drag are valid for calculating settling velocities of sand-size grains in water, whereas the Ferguson-Church equation provides a good fit for natural river sand.

Grain settling is a special case of the more general problem of flow past a sphere. The analysis and plots above are all dimensional, that is, you can quickly check by looking at the plots what is the approximate settling velocity of very fine sand. That is great, but you would have to generate a new plot – and potentially do a new experiment – if you wanted to look at the behavior of particles in some other fluid than water. A more general treatment of the problem involves dimensionless variables; in this case these variables are the Reynolds number and the drag coefficient. The classic diagram for flow past a sphere is a plot of the drag coefficient against the Reynolds number. I will try to reproduce this plot, using settling velocities that come from the three equations above.

At terminal settling velocity, the drag force equals the gravitational force acting on the grain:

Screen Shot 2013-08-09 at 5.49.31 PM

We also know that the gravitational force is given by the submerged weight of the grain:

Screen Shot 2013-08-09 at 5.34.42 PM

The drag coefficient is essentially a dimensionless version of the drag force:

Screen Shot 2013-08-09 at 5.49.08 PM

At terminal settling velocity, the particle Reynolds number is

Screen Shot 2013-08-09 at 5.59.14 PM

Using these relationships it is possible to generate the plot of drag coefficient vs. Reynolds number:

d = np.arange(0.000001,0.3,0.00001)
C2 = 0.4 # this constant is 0.4 for spheres, 1 for natural grains
ws = v_stokes(rop,rof,d,visc,C1)
wt = v_turbulent(rop,rof,d,visc,C2)
wf = v_ferg(rop,rof,d,visc,C1,C2)
Fd = (rop-rof)*4/3*pi*((d/2)**3)*9.81 # drag force
Cds = Fd/(rof*ws**2*pi*(d**2)/8) # drag coefficient
Cdt = Fd/(rof*wt**2*pi*(d**2)/8)
Cdf = Fd/(rof*wf**2*pi*(d**2)/8)
Res = rof*ws*d/visc # particle Reynolds number
Ret = rof*wt*d/visc
Ref = rof*wf*d/visc
loglog(Res,Cds,linewidth=3, label='Stokes')
loglog(Ret,Cdt,linewidth=3, label='Turbulent')
loglog(Ref,Cdf,linewidth=3, label='Ferguson-Church')
# data digitized from Southard textbook, figure 2-2:
Re_exp = [0.04857,0.10055,0.12383,0.15332,0.25681,0.3343,0.62599,0.77049,0.94788,1.05956,
Cd_exp = [479.30811,247.18175,199.24072,170.60068,112.62481,80.21341,45.37168,39.89885,34.56996,
loglog(Re_exp, Cd_exp, 'o', markerfacecolor = [0.6, 0.6, 0.6], markersize=8)

# Reynolds number for golf ball:
rof_air = 1.2041 # density of air at 20 degrees C
u = 50 # velocity of golf ball (m/s)
d = 0.043 # diameter of golf ball (m)
visc_air = 1.983e-5 # dynamic viscosity of air at 20 degrees C
Re = rof_air*u*d/visc_air
loglog([Re, Re], [0.4, 2], 'k--')
text(3e4,2.5,'$Re$ for golf ball',fontsize=13)
xlabel('particle Reynolds number ($Re$)', fontsize=15)
ylabel('drag coefficient ($C_d$)', fontsize=15);


The grey dots are experimental data points digitized from the excellent textbook by John Southard, available through MIT Open Courseware. As turbulence becomes dominant at larger Reynolds numbers, the drag coefficient converges to a constant value (which is equal to C2 in the equations above). Note however the departure of the experimental data from this ideal horizontal line: at high Reynolds numbers there is a sudden drop in drag coefficient as the laminar boundary layer becomes turbulent and the flow separation around the particle is delayed, that is, pushed toward the back; the separation wake becomes smaller and the turbulent drag decreases. Golf balls are not big enough to reach this point without some additional ‘help’; this help comes from the dimples on the surface of the ball that make the boundary layer turbulent and reduce the wake.

You can view and download the IPython notebook version of this post from the IPython notebook viewer site.

Ferguson, R. and Church, M. (2004) A simple universal equation for grain settling velocity. Journal of Sedimentary Research 74, 933–937.

My job description, in simple words

I couldn’t refrain from playing with the ‘upgoerfive’ tool, an online text editor that only allows you to use the one thousand most common words of the English language. Here is my attempt to describe what I do.

I study how small pieces of rock get together, move along, and settle down to form beds. Water usually moves smaller pieces further than large ones but lots of small and big pieces together can move really fast and build thick beds on the water bottom. I also look at the form of these beds and think about how much space is left in between the small pieces. This space is filled with water and sometimes with other stuff that people like to burn in their cars. It is nice that such beds are often seen in beautiful places around the world that I like to visit.

Here is the link to the original upgoerfive post. You can try it out yourself over here.

The case for scales, rates, and numbers in geology

A long time ago I used to study geology in a nice city called Cluj, in the middle of that interesting part of Romania known as Transylvania. This was a place and time where and when I learned about quartz, feldspar, species of coral and foraminifera in great detail, heard about sequence stratigraphy and turbidites for the first time, and went on some great geological field trips. Not to mention the half-liter bottles of beer that would be significant components of any decent geological trip or spontaneous philosophical discussion in the evening. The less pleasant part was that many of the classes I took involved brute-force memorization of fossils, minerals, chronostratigraphic names, and formations. Although the geological vocabulary that I picked up was pretty broad and proved useful as a good set of words, terms, and definitions to play with, I forgot many of the details by now. If you asked me what was the difference between granite and granodiorite, I would have to check. I don’t remember at all what fossils are characteristic of the Late Jurassic. And, despite doing some fieldwork myself over there, I cannot remember the stratigraphic nomenclature in Transylvanian Basin; I would have to look it up (probably in this paper).

After college, it took me about one year to realize with convincing clarity that there was a lot left to learn. I went on to grad school on the other side of the planet, at a well-known university. Many of the classes I took over there were – unsurprisingly – quite different; a lot more focus on laws, processes and the connections between geological things than the ‘things’ themselves. It was also there that I started to see the links between geology and physics and math. I picked up quite a bit of math and physics during high school, but then quickly relegated them to the status of “stuff that is rarely used in geology”. At grad school, it dawned on me that numbers and mathematical laws are not only useful in geology, but are in fact necessary for doing good earth science. Maybe I am stating the obvious, but here it goes anyway: geology deals with enormous variations in scale, both in space and time; and it is not enough to say that the river was deep (how deep?), the tectonic deformation was fast (how fast?), the sea-level highstand lasted long (how long?), or the sediment gravity flows were high-energy flows (I am not even sure what that means). One of the most important things I learned was an appreciation for physical and quantitative insight in geology, that is, having at least an idea, a feel for what are the scales and rates involved in the formation of the rocks you are looking at. I cannot say it better than Chris Paola, one of the important and influential advocates of moving sedimentary geology closer to physics and math:

“For the ‘modal’ sedimentary-geology student, it is not sophisticated computational skills or training in advanced calculus that is lacking, but rather the routine application of basic quantitative reasoning. This means things like estimating scales and rates for key processes, knowing the magnitudes of basic physical properties, and being able to estimate the relative importance of various processes in a particular setting. Understanding scales, rates and relative magnitudes is to quantitative science what recognizing quartz and feldspar is to field geology. Neither requires years of sophisticated training, but both require repetition until they become habitual.”

Developing these skills is a lot easier if one is not afraid of tinkering with simple computer programs. Want to really understand what Stokes’ law is about? There is no better way than typing the equation into an Excel spreadsheet or a Matlab m-file and see how the plot of settling velocity against grain size looks like. What about settling in a fluid with different viscosity? Change the variable, and compare the result with the previous curve. High-level programming languages like Matlab or Python* are a lot easier to learn than languages closer to ‘computerese’ and farther from English, and they are great tools for these kinds of exercises and experiments. As somebody interested in stratigraphic architecture, I have become especially fond of creating surfaces that vaguely resemble real-world landscapes and then see how the evolution of these surfaces through time – deposition over here, erosion over there – creates stratigraphy. Complex three dimensional geometry is a lot easier to grasp if you can visualize and dissect it on the computer screen.

Of course, numbers, diagrams and images that come from computer programs are only useful if they demonstrably say something about the real world. Data collection in the field and the laboratory are equally important. But nowadays we often have more data than we wished for, and quantitative skills come handy for visualizing and analyzing large datasets – and comparing them to models.

Not everyone is excited about the growing number of earth scientists who tend to see equations ‘in the rocks’. The logo of the Sedimentology Research Group at the University of Minnesota features the Exner equation carved into a pebble, allegedly as a response to the exclamation “I haven’t seen yet an equation written on the rocks!” There is some concern that many geology graduates nowadays do not get to see, to map and to sample enough real rocks and sediments in the field. Although I think this unease is not entirely unsubstantiated, I wouldn’t want to sound as pessimistic as Emiliano Mutti – one of the founding fathers of deepwater sedimentology – does in the last phrase of a review article:

“This approach raises a problem, and not a small one: in connection with data collection in the field, how many field geologists are being produced in these times of increasingly computerized geology; and how good are they?”

As far as I know, geological field work is still an important part of the curriculum in many departments of geology – as it clearly should be. The number one reason I have become a geologist was that I loved mountains, hiking, and being outdoors in general, way before I started formally studying geology. And I still take every opportunity to go to the field. But I cannot see the growth of “computerized geology” – and of quantitative geology in general – as a bad thing. Does dry quantification take away the beauty and poetry of geology? I don’t think so. Unweaving the rainbow, unfolding a mountain, and reconstructing a turbidity current only add to our appreciation of the scale and grandeur of geology.

* I will let you know later whether this is true about Python…

** I have started writing this post for Accretionary Wedge #38, mostly because I found the call for posts quite inspiring, but haven’t finished it in time. Read all the good stuff at Highly Allochthonous.

Salt and sediment: A brief history of ideas

Salty weirdness
Salt is a weird kind of rock. At first sight, it behaves like most other rocks: if you pick up a piece, it is hard, it is heavy, and it breaks if hit with a hammer. But put it under stress for thousands of years, and salt will behave like a fluid: relatively small forces can cause it to flow toward less stressful surroundings. This often means it will try to find its way to the surface.

When deposited, sand and mud have lots of pore space filled with water and have relatively low density. However, as they get buried by more sediment, much pore space is lost, both through compaction and cementation. Sediments turn into sedimentary rocks, become harder, and their density increases. In contrast, salt doesn’t have much pore space to begin with; its density will stay the same, regardless of depth of burial. As both salt and sediment are buried to greater depths, an unstable condition develops: lighter salt lying under denser material. In addition, the location of the salt layer in the sediment column is not entirely random: it is in the nature of sedimentary basins to initially place salt at the bottom of the sediment pile. Extensive salt layers usually form early in a basin’s lifetime, when seawaters invade for the first time shallow depressions on a continent that is about to split into two along a rift zone. The Dead Sea is an obvious example that comes to mind.

Layering salt and sediment in this unstable order is a recipe for a spectacular geological show. As salt is trying to find its way to the surface, it forms drop-shaped blobs called diapirs; but also ridges, walls, and salt sheets. Several sheets can connect laterally into a huge salt canopy, a new salt layer that is entirely out-of-place or allochtonous. Salt can also act as a lubricating layer at the base of a thick sequence of sedimentary rocks. But I am rushing ahead a little bit; salt tectonics is such a new – but rapidly growing – science that salt canopies, despite their widespread presence in the subsurface Gulf of Mexico, were not recognized and described until the 1980s.

Tectonics vs. buoyancy, Europe vs. America
Before the beginning of the twentieth century, even with the role that salt played in human history, little was known about how salt domes formed. This was an age of rampant speculation; surface data was scarce because salt does not last very long after exposed as it quickly gets dissolved and washed away by precipitation. Many geologists thought that formation of salt domes didn’t require any significant salt deformation or displacement. But things have changed dramatically in 1901, with the discovery of the Spindletop oil field on top of a salt dome in southeastern Texas. The recognition that oil is often found on top of and around salt domes created a much stronger interest in understanding how exactly salt formations are put in place.

European geologists thought that the main driving force was compression, the force that causes folding and thrusting and builds mountains. In Romania, where the Eastern Carpathians take a sharp turn toward the southwest, salt was found in the cores of oil-bearing anticlines. The contacts with the surrounding rocks were clearly discordant. These are the structures that prompted Ludovic Mrazec, professor of geology at University of Bucharest, to coin the term “diapir” in 1907.

Mrazec’s explanation of how salt diapirs form. From Barton (1925).

Salt in Germany and Poland also seemed to occur invariably in a compressional setting, in the cores of folds, next to folds that had no salt associated. It seemed obvious that salt was ‘pushed up’ by tectonic forces, and it appeared unlikely that the rise of salt itself was causing the folding.

But the discovery of a multitude of salt diapirs in the Gulf of Mexico made it clear that they can occur far away from any mountains and compressive tectonic forces. The much simpler setting and relative lack of deformation in the Gulf proved informative. “The Roumanian salt-dome geologist possibly may have more to learn from the American salt domes than the American salt-dome geologist has to learn from the Roumanian domes. The occurrence of the American domes in a region of tectonic quiescence suggests that tectonic thrust cannot have the importance postulated by Mrazec” – wrote Donald Barton in 1925.

This was also the time when the density difference between salt and sediment came into discussion. Gravity measurements in the Gulf of Mexico showed anomalies above salt domes that were due to the lower density of salt. It was increasingly recognized that density inversion must play an important role in diapirism, especially where compressive tectonic forces were absent. In addition, by the 1930s geologists have reached a consensus that salt diapirs must somehow punch through the overlying sediment. They seemed to ignore the fact that, as Wade (1931) put it, you cannot drive a putty nail through a wooden board. As mentioned before, salt does behave like a fluid over geological time scales. But how can it penetrate thick layers of hardened sedimentary rock?

A brilliant idea: downbuilding
The solution to this problem came in 1933, from the same Donald Barton who was discussing the differences between European and American salt domes in 1925. He suggested that diapirs can form without much piercement of the sediment above. Instead, once a small dome is initiated, it simply can stay in place, always at or close to the surface, while sediment is deposited around it and the source salt layer subsides: “it is the sediments which move, and not the salt core. The energy requirement (…) is very much less than if there were actual upward movement of the salt.”

The evolution of salt diapirs through ‘downbuilding’. Salt domes are always close to the surface and diapirism goes hand-in-hand with sedimentation. From Barton (1933).

This was a key insight: it got rid of the “room problem”, the need for moving huge volumes of hard rock out of the way of the rising salt. It also highlighted that salt movement can happen at the same time with sedimentation, a fact that became abundantly obvious later as high-quality seismic data became available. But the concept of ‘downbuilding’ was ignored for the next fifty years.

Animation showing how downbuilding works. Blue represents salt, yellow is sediment. To mimic mass balance for salt (-- what is lost from the source layer must go into the salt dome), the blue area is kept constant through the animation.

The beauty of instabilities

The main reason for conveniently forgetting Barton’s idea was that density inversion between two fluids could be nicely studied in the lab and described with elegant equations. In one of the papers that kicked off this fascination with Rayleigh-Taylor instabilities, Nettleton (1934) used corn syrup and less dense crude oil to visualize diapir-like blobs of fluid in a transparent cylinder and to show that gravity alone, without any help from contractional forces, was enough to generate structures similar to salt domes.

Less dense crude oil (black) forming diapir-like blobs as rising through higher-density corn syrup (yellow). Redrawn from Nettleton (1934).

One problem with this approach was that oil and syrup can be photographed during deformation, but the transient structures could not be carefully dissected and analyzed later. Materials of higher viscosity were needed for that; however, increasing the viscosity resulted in a density difference too small to get the fluids moving in the first place. The trick was to place the whole experiment in a centrifuge and use the centrifugal force to imitate a larger-than-normal gravitational force. This approach formed the basis of a productive line of research on gravity tectonics in the laboratory of the Norwegian-Swedish geologist Hans Ramberg. The results are probably more relevant to what is happening deeper in the Earth, at higher temperatures and pressures, where most rocks become more similar in behavior to salt.

Modern salt tectonics
By the late 1980s it has become quite obvious that kilometer-thick piles of sedimentary rock cannot be treated as fluids and salt-sediment interaction is more similar to placing and deforming slabs of brittle material on top of a viscous fluid. Seismic from salt-bearing sedimentary basins suggested that the history of salt movement and sedimentation were highly interconnected and Barton’s downbuilding concept was strongly relevant.

Three-dimensional seismic data also showed the variety and complexity of allochtonous salt bodies in salt-rich sedimentary basins. Sandbox experiments with more realistic material properties and ongoing sedimentation during deformation were performed and the results beautifully visualized. The behavior of turbidity currents flowing over complex salt-related submarine topography was investigated. Hundreds of scientific papers were written on salt tectonics, both by industry geoscientists and researchers in the academia.

N-S cross section in the Gulf of Mexico. Large volumes of the Jurassic Louann salt have been displaced and squeezed into a salt canopy surrounded by much younger sediments. From Pilcher et al., 2011

And there is quite a bit left to explore and understand.

References and further reading
Barton, D. C. (1926) The American Salt-Dome Problems in the Light of the Roumanian and German Salt Domes, AAPG Bulletin, v. 9, p. 1227–1268.

Barton, D. C. (1933) Mechanics of Formation of Salt Domes with Special Reference to Gulf Coast Salt Domes of Texas and Louisiana, AAPG Bulletin, v. 17, 1025–1083.

Hudec, M., & Jackson, M. (2007) Terra infirma: Understanding salt tectonics. Earth Science Reviews, 82(1-2), 1–28.

Jackson, M. (1996) Retrospective salt tectonics, in M.P.A. Jackson, D.G. Roberts, and S. Snelson, eds., Salt tectonics: a global perspective: AAPG Memoir 65, p. 1–28. [great summary of the history of salt tectonics]

Mrazec, L. (1907) Despre cute cu sȋmbure de străpungere [On folds with piercing cores]: Bul. Soc. Stiint., Romania, v. 16, p. 6–8.

Nettleton, L. L. (1934) Fluid Mechanics of Salt Domes, AAPG Bulletin, v. 18, p. 1–30.

Pilcher, R. S., Kilsdonk, B., & Trude, J. (2011) Primary basins and their boundaries in the deep-water northern Gulf of Mexico: Origin, trap types, and petroleum system implications. AAPG Bulletin, v. 95(2), p. 219–240.

Wade, A. (1931) Intrusive salt bodies in coastal Asir, south western Arabia: Institute of Petroleum Technologists Journal, v. 17, p. 321–330, 357–361.