Introduction to modeling cinder cones in S. E. Idaho
Bro. David E. Brown, Dept. of Maths., BYU–Idaho
21 February 2011
The author wishes to thank Austin Blaser and his colleagues for introducing him to the delights of modeling cinder cones.
Introduction
This document is to serve as background reading for an extended assessment item (think “project” or
“sequence of exam questions”) or possibly as an introduction to the diffusion equation in some future
semester. Consequently, it’s supposed to have enough detail to help students find their way through the
modeling morass, but with some things left unaddressed.
Eruption
The following scenario has occurred in southern and southeastern Idaho multiple times: Magma—molten
rock from deep under the earth’s surface—erupts through a crack or hole in the ground (known as a vent).
Often, the magma around here has enough water in it to make for fairly runny lava. Water trapped in the
magma can flash into steam when the lava erupts. If the lava contains enough water (for steam) and gases
(such as carbon dioxide, hydrogen sulfide, and so on), the eruption can be quite dramatic, with lava exploding
into the air into showers of red-hot droplets of rock. These droplets, if small enough, bear a (mostly poetic)
resemblance to the cinders thrown out by a crackling fire. Red-hot at first, they cool in the air as they fall
to the ground, building up a cone-shaped mound around the vent, with a “throat” in it, through which the
lava continues to erupt. When the eruption is done, there’ll be a crater in the middle of the cone.
Now, lava droplets may cool enough while flying through the air to not be liquid when they hit the
ground. Even so, they may still be hot enough and soft enough to “weld” themselves to each other upon
contact. And they might not.1 In either case, a cinder cone is pretty much a one-shot volcano. The Menan
Buttes west of town (one of which is known as R Mountain), are cinder cones. If you see them from the air,
you can make out three separate cones, from three separate eruptions, through three separate vents.2
A freshly-erupted cinder cone is a beautiful thing. See Figure 1 for a picture of Capulin Mountain, a
cinder cone in northeastern New Mexico. At about 60,000 years old, it is quite young and still fresh. Also
see Figure 2 for a model of a freshly-erupted cinder cone.
Erosion
If the cinders don’t “weld” together very well as they fall, the rock making up the resulting cinder cone is
not very solid. (The technical way of saying this is that the rock is poorly consolidated.) It’s relatively easily
1 The truth, as usual, is more complicated, with some degree of sticking going on, even if not much real welding occurs.
If you want a close-up view of what a cinder cone looks like, go out to “R Mountain,” park somewhere safe, get out, and
look at the rocks and the soil around you. Unless I’m much mistaken, a fair amount of welding occurred when R Mountain was
formed, but this cinder cone is perhaps not as solidly welded as some.
Be sure not to trespass. And you know the rules: take only photographs and leave only footprints, etc.
2 Contrast this with, say, Mount Rainier, east and south of Seattle. That’s a volcano that has erupted many times out of
the same vent. Likewise its recently-more-dramatic neighbor Mount Saint Helens, farther south.
1
Figure 1: Capulin Mountain (foreground), a recently erupted cinder cone in northeastern New Mexico.
(Source: National Park Service website.)
Figure 2: An idealized freshly-erupted cinder cone.
eroded, say, by individual raindrops that make little splashes as they hit the cone. The side of the cone is
sloped, of course; gravity being what it is, any cinder or cinder dust kicked up by a raindrop is more likely
to go downhill than up. Likewise, as animals of various kinds (including people!) walk around on the cinder
cone, they disturb cinders and dust, which again are more likely to go downhill than up.
This slow erosion, occurring on such a small scale in comparison to the cone as a whole, qualifies as a
diffusion process.3 A few comments about this process are in order, in preparation for quantifying it. First,
eroded material moves downhill (on average) over time, but “downhill” isn’t always away from the geometric
center of the cone. The cone has a crater on top of it. Material outside the crater will move away from the
center of the mountain; material inside the crater will move toward the center. It follows that the crater
decreases in radius as it fills with eroded material.
Second, the crater’s concave-up initial shape slowly flattens out until the crater is full. For a brief moment
in time, the top of the cinder cone is approximately flat. Thereafter, the cone has a blunt sort of peak, which
slowly wears down until the entire mountain flattens out.
3 By way of contrast, if wind or running water rapidly and relentlessly carries away significant amounts of material from the
cone as a whole, erosion is not diffusive, but advective. (Actually, the diffusive processes would still be at work, so—depending
on the conditions—erosion may actually be diffusive-advective.) Or if a glacier comes along and plows into the cinder cone,
then wind and rain and critters scampering about may well have far less effect by comparison. In that case, I wouldn’t use
either diffusion or advection to model erosion with. I’d actually start from scratch, because I don’t have an appropriate model
memorized, nor do I know where to get one.
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Let’s look at this from another point of view: Pick a place on the mountainside and watch what happens
to the elevation there, over time. If the place you picked happens to be on the crater floor, the elevation
increases for a while, as the crater fills, then tops out, then decreases as the mountain erodes. If you happened
to pick a place on the cinder cone outside the crater, then the elevation can only decrease with time. If you
pick a place on the plain, near the cinder cone but not on it, the elevation will not change at all until eroded
material from the cinder cone reaches it, then elevation will increase with time.
If we’re going to quantify this erosive process, we’re going to have to set things up properly. Here’s where
the symbols start piling up around our throats, like so many cinders about the volcano’s throat. Hopefully
the symbols will weld together!
Modeling erosion
Time to make some choices. I vote for keeping life simple, by ignoring the entire volcano except half of one
cross-section, as in Figure 3. What exactly are we going to model? Erosion? How do you quantify erosion?
There may be lots of ways, but it’s probably most straightforward to model the “height,” or elevation of the
cinder cone, at each point. You see, as the cone erodes, we can expect that (eventually, at least) the surface
of the cone gets closer and closer to the surrounding land surface. So let u stand for the unknown function
that gives the elevation (or height) of the cone above some reference elevation (say, that of a surrounding
plain, for example), at each point. So, if we’re using only a cross-section, elevation will be in the y-direction,
and we can use x to measure the distance from the center of the cinder cone. So u will be a function of
x and t, because the elevation depends on distance out from the center of the cone, and because elevation
changes with time—if not, there’s no erosion going on!
Figure 3: Half the cross-section of a freshly erupted cinder cone.
Let us here observe that eroded material finds its way slowly downhill, but there’s no way for it to simply
disappear. And without a fresh eruption, there’s no way for more cinders to mysteriously appear. So, the
volcano’s mass is conserved—it just gets spread out all over the place (by erosion, of course). We can get
some mathematical mileage out of this conservation law, if we can write the law down. For this, we need
symbols, including one for density. I’ve read very little about this, but geologists seem to be happy assuming
the density ρ of the rock involved is constant.4
4 Of course, we’re talking about dust as well as rock, and dust is likely to have different density than rock. But the dust
involved is only a thin layer on top of the rock.
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Conservation of mass
Now, if we just use a cross-section of the cinder cone, we have two dimensions, total. We can measure
the amount of mass of any part of this 2-d area by multiplying the area by density. So ρ has units of
mass/distance2 . Rock is heavy stuff, but perhaps kg/m2 will suffice. And while we’re in the neighborhood,
let’s use M to stand for mass.
So take a thin slice (say, from x to x + ∆x) of the half-cross-section of the cinder cone, and think about
material eroding off the top of it. Material enters the slice on the side where the elevation u is higher, and
leaves on the side where the elevation is lower. Over any given tiny interval ∆t of time, the change in mass
within the slice is “the amount of material entering less the amount leaving.” More precisely, let φ(x) stand
for “flux,” the time rate at which mass passes the point x from left to right, in units of mass/time. We have
∆M = φ(x) − φ(x + ∆x) ∆t.
(1)
Note that if mass enters the slice on the left side, φ(x) > 0; if mass exits there, then φ(x) < 0. If mass
enters the slice from the right side, φ(x + ∆x) < 0; if it exits there, then φ(x + ∆x) > 0. This sort of thing
determines the sign of ∆M , you see. Note also that if there’s no new eruption, and if mass doesn’t simply
disappear into the thin air, then mass can only appear in the slice or disappear from it by moving downhill,
crossing the boundaries of the slice as it goes. So Equation (1) accounts for all the mass that comes and
goes through our little slice.
Now, mass is density × area, as we noted before, and area is approximately elevation × ∆x. Therefore,
(1) can be rewritten as
ρ∆u∆x = (φ(x) − φ(x + ∆x))∆t,
or
φ(x + ∆x) − φ(x)
∆u
=−
.
∆t
∆x
Let both ∆t and ∆x go to zero. If u is differentiable with respect to t and if φ is differentiable with respect
to x, then we have
ρut = −φx ,
(2)
ρ
which expresses the conservation of mass for the erosion of cinder cones.
Flux
Now let’s analyze flux. We hold the following truths to be self-evident. Besides: They’ve been confirmed by
observation in the field or in the laboratory.
1. If the ground is literally flat (i.e., if u is constant) then our diffusive erosive mechanisms don’t really
move any mass around (on average).
2. Where elevation is higher in one place than another, gravity will pull eroded material down from the
higher location to the lower one.
3. The greater the change in elevation (over a given horizontal distance), the faster eroded material moves
downhill.
4. The rate at which the eroded material moves downhill will depend on things like how well cinders or
dust grains cling to each other. This, in turn, depends on (e.g.) how rough their surfaces are, which
depends, in turn, on the chemical composition of the lava from which the cinders were made, as well
as the temperature, rate of cooling, and so on.
Item 3 says flux ought to be proportional to the slope of the hillside. Item 1 serves as a reality check: where
the slope of the hillside is 0, there should be no flux. So, “proportional” makes sense here, where “inversely
proportional would not. Item 4 says the constant of proportionality K0 ought to depend on the material of
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which the cinders are made. Item 2 says that (as you go left to right) if the slope of the hillside is negative,
flux ought to be positive, and vice-versa. Putting these four ideas together, we get
φ(x) = −K0 ux .
(3)
That’s flux.
The diffusion equation
Substituting Equation (3) into the conservation law (1) gives us
ρut = K0 uxx .
Divide this equation through by ρ and define k = K0 /ρ, to arrive at
ut = kuxx ,
(4)
which is affectionately known as the diffusion equation. It doesn’t exactly quantify “erosion” for you, but it
does describe the function that gives the elevation at each point of the half-cross-section of the cinder cone.
I have it on good authority that values of k run anywhere from just under 5 to about 125, give or take,
depending on environmental conditions. For example, wetter climates induce more erosion by rainfall, and
therefore need higher values of k, than drier climates. But I’ll leave it up to you to decide what the units of
k ought to be.
Now, all we need is a way to solve this equation, so as to find out what happens to our cinder cone over
time, and how. . .
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