Volcanoes in Orbit!

What we're concerned with here is the eruption column and subsequent plume arising from the volcanic eruption. This is a shield volcano, so it's not going to erupt a la Mount St. Helens; rather, it will slowly spew out material. Granted, for a volcano this large that's still going to be quite the eruption. First, let's figure out some basic properties and quantities:

• Height: 50 miles, or about 80 kilometers, as you stated. Obviously, no mountain should be this tall, but that really doesn't matter.
• Radius: Given an average slope of about $5^{\circ}$, the volcano should have a radius of $\sim80\text{ km}\tan(5^{\circ})\simeq914\text{ km}$. This means that the volcano should be more than one half as wide as the United States, and about three times as wide as Olympus Mons.
• Rate of mass ejection: The Volcano Explosivity Index (VEI) ranks how powerful a volcanic explosion is. The top level is VEI 8, ejecting over $1000\text{ km}^3$ of material. Let's assume that this volcano ejects $4000\text{ km}^3$ of material over a period of one week. Assuming typical ash densities of perhaps $1500\text{ kg m}^{-3}$, this gives us $Q\sim10^{10}\text{ kg s}^{-1}$. Carazzo et al. (2008) lists values for other eruptions (see Table 1 and Table 2). Our $Q$ is two orders of magnitude greater than most Plinian (think high, tall plumes) eruptions, and about the same as many explosive eruptions producing pyroclastic flows. Ours simply lasts a lot longer.

Let's start with a simplified model, namely, a Gaussian plume. This means that the concentration of the material, $C$, is $$C(x,y,z)=\frac{Q}{2\pi U\sigma_y\sigma_z}\exp\left(-\frac{y^2}{2\sigma_y^2}\right)\left[\exp\left(-\frac{(z-H)^2}{2\sigma_z^2}\right)+\exp\left(-\frac{(z+H)^2}{2\sigma_z^2}\right)\right]$$ where $$\sigma_y=\sqrt{2Dy\frac{x}{U}},\quad\sigma_z=\sqrt{2Dz\frac{x}{U}}$$ and

• The wind is blowing in the $x$-direction with speed $U$, which might be $25\text{ m/s}$, or $56\text{ mph}$.
• $y$ is the $y$-coordinate and $z$ is the $z$-coordinate.
• $H$ is the reference height - in our case, $80000\text{ m}$.
• $D$ is the diffusion coefficient, which is probably $\sim10^3\text{ m}^2\text{ s}^{-1}$.

There are some assumptions the model makes:

• The opening is roughly point-like.
• In our case, we assume that the diffusion coefficient is isotropic, i.e. it is not directionally dependent.
• This is a "steady state" solution, meaning that it is approximately constant. This works well for a week-long event.
• The atmosphere (and gravity) are homogeneous and don't affect the plume too much. On large scales, this assumption doesn't always work.

I wrote some Mathematica code to look at the concentration of the plume as a function of distance. This code currently outputs a contour graph in the plane $y=0\text{ m}$. It is cut off at $x=25\text{ m}$, because otherwise the densities near the opening are too large and make it hard to see the rest of the contours.

U = 25;
H = 80000;
Diff = 1000;
Q = 10000000000;
Sigmay[x_, y_] := Sqrt[2*Diff*Sqrt[y^2]*x/U];
Sigmaz[x_, z_] := Sqrt[2*Diff*Sqrt[z^2]*x/U];
Conc[x_, y_, z_] := Q/(2*Pi*U*Sigmay[x, y]*Sigmaz[x, z])*
   Exp[-y^2/(2*(Sigmay[x,y)^2)]*(Exp[-(z - H)^2/(2*(Sigmaz[x, z])^2)] + 
     Exp[-(z + H)^2/(2*(Sigmaz[x, z])^2)]);
TopView[x_, y_] := Conc[x, y, H];
SideView[x_, z_] := Conc[x, 0.001, z];
ContourPlot[SideView[x, z], {x, 25, 10000}, {z, 78000, 88000}, 
 PlotRange -> {{25, 10000}, {78000, 88000}, All}, Exclusions -> None,
  PlotLegends -> Automatic, Contours -> 50, ContourLines -> False, 
 RegionFunction -> Function[{x, y, z}, y - 80000 + 0.0875*x > 0]]

This outputs the following plot:

Notice that even two kilometers away from the summit, the plume is still incredibly dense. Its density doesn't become negligible until it's hundreds and hundreds of kilometers away from the summit. If you were to raise the upper limit on $z$, you would see that the plume rises to a height of over $160\text{ km}$! That's way beyond the upper limit of the mesosphere. And to be honest, I think I've been conservative in my estimate for $Q$.

Is this realistic?

First, let's consider what kind of eruption we're dealing with. It's a shield volcano, so most of the material that comes out will not be in the plume. However, as I said before, I likely underestimated $Q$, and so such a plume isn't that unrealistic.

That final height of the plume - $\sim80\text{ km}$ - seems a bit much. However, it's really not that far-fetched. In general, the relationship between plume height $H$ and $Q$ is $$Q\propto H^4$$ Kaminski et al. use the equation $$Q=aH^4+b$$ where, for $12\text{ km}\leq17\text{ km}$, $a=258\text{ kg s}^{-1}\text{ km}^{-4}$ and $b=-4.6\times10^6\text{ kg s}^{-1}$. We can assume that $H$ is going to be even greater than $17\text{ km}$, so we should expect the result to be close but not totally accurate. Rearranging, plugging in for $Q$ and then solving yields a total plume height of $79\text{ km}$ - which is very close to what Mathematica told us. Perhaps that result isn't too ridiculous after all.

One thing you'll need to consider is column collapse. It might be the most important factor behind the eruption's evolution. An eruption column will collapse when the bulk density of the material (a combination of its density and the density of the air inside the column) becomes too dense compared with the air around it. Looking at some standard atmospheric tables, we can see that the mesosphere is not at all dense, meaning that collapse is highly likely. The plume will form, certainly, and material will travel hundreds of kilometers (during which the Gaussian plume model is a good fit), but it won't last for that long.

So, what happens then? Well, the material now coming out of the volcano will form pyroclastic flows, which scare the living daylights out of me. They are fast, dense and hot, and destroy anything in their path. And they will begin by rushing down the mountain at fairly high speeds, and probabaly will not stop quickly. Assuming these flows reach high speeds ($300$-$600\text{ mph}$), they'll reach the base of the volcano in a few hours, after ruining anything on its slopes. An area the size of a medium-sized country will be decimated.

It's unclear what will happen when the flow reaches an atmospheric layer with a density high enough to support an eruption column. I'll have to get back to you on that. It's possible that the gas will form clouds of ash and soot at that altitude, which would then spread out even more. That said, that's currently just a conjecture on my part.

Here's the bottom line:

• Ash, gas and dust will be spread for hundreds of kilometers away from the volcano - possibly up to $1000\text{ km}$ or more.
• A plume will rise into the mesosphere and possibly low thermosphere before collapsing; I don't know how high it will actually go.
• Pyroclastic flows will then descend from the mountain, destroying anything on it. Any further atmospheric affects will be from them.

posted almost 7 years ago

HDE 226868‭

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