Equivalence of turbulence in solid materials

The governing equations for a fluid and a solid are effectively the same and many times analysis can be done for a solid using the Navier-Stokes equations with the equation of state and/or the stress tensor computation modified. Because the equations are effectively the same, the phenomenon present in solutions are effectively the same.

What is the equivalence of turbulence in a solid? How do the scaling laws change?

• What kind of solid are you talking about? The governing equations for a solid are very different than those for a fluid, unless you're talking about some highly non-linear solid. – jasonlarkin Feb 14 '14 at 6:11

The onset of turbulence in fluids is determined by the Reynolds number $$\mathrm{Re} = \frac{vL}{\nu},$$ where $L$ is the characteristic length scale, $v$ the characteristic velocity, and $\nu$ the viscosity. The onset of turbulence in fluids occurs for $\mathrm{Re}$ greater than about 1000 or more, depending on geometry.

If we want to see the equivalent to turbulence in solids, we will have to figure out how to make the Reynolds number large enough. The trouble is that to the extent that solids can be treated as fluids, they're fluids with really, really high viscosities. So we have to make $L$ or $v$ or both really really big in order to counter this.

We could try to make $v$ high, by making one part of the solid move extremely rapidly relative to another. However, for all the solids I can think of, this will just break the solid.

So the alternative is to make the characteristic length scale really long. At first I thought the Earth's mantle might be an example where this happens, depending on how you look at it. The Earth's mantle is pretty much a solid, in that if you could take a part of it from very deep below the crust and put it in the lab while keeping it under high pressure, it would probably behave like a rock. (The mantle isn't made of magma, which forms only near the surface.) However, on very long time scales it flows. If you look at simulations of the mantle's dynamics (e.g. via a google image search), you will see that it has eddies at various spatial scales, which suggested to me that it might be at least slightly turbulent. However, as Michael Brown pointed out in a comment, the length scale is nowhere near large enough to counter the high viscosity and low speed of the flow, and the Reynolds number comes out to about $10^{-21}$ to $10^{-24}$, far too small for turbulence -- those multiply-sized eddies must be due to stratification effects and the non-Newtonian nature of the mantle fluid.

So in order to have a turbulent solid, you woud need to have something about $1000/10^{-21}=10^{24}$ times larger than the Earth (it would be a black hole), or moving $10^{24}$ times faster (this would be much faster than light). Since this is impossible, I think it is unlikely that turbulence can be observed in solids.

• +1: nice example. But I am confused now. I am not a geologist, but with some quick numbers I found on wikipedia (won't vouch for accuracy): $\mu\sim 10^{21\sim24} \mathrm{Pa\cdot s}$ (exponent covers a range of values), $v\sim20 \mathrm{mm/yr}$, a density $\rho\sim 5\times10^3 \mathrm{kg/m^3}$ and $L\sim 10^3 \mathrm{km}$. So I get $\mathrm{Re}\sim10^{-21\sim24}$. Obviously made a big mistake somewhere, but where? – Michael Brown Apr 9 '13 at 8:13
• @MichaelBrown I have to confess I didn't look at the numbers before posting. I did wonder whether the size would really be enough to counter the high viscosity and low speed, and your calculation suggests it isn't. On the other hand those simulation images do look somewhat like turbulent fluids, but perhaps there is another explanation. I will edit my post to be a bit more cautious about claiming that mantle convection is turbulent. – Nathaniel Apr 9 '13 at 8:28
• @MichaelBrown Actually, thinking about it, it isn't plausible that mantle convection could be turbulent after all. In order for it to be turbulence, inertial effects should be more important than forcing, and I can't really imagine that anything is moving inertially inside the Earth, except in the inner and outer core. I think the multiply-sized eddies must be due to stratification effects. – Nathaniel Apr 9 '13 at 8:30
• Thanks. That makes sense I guess. Is that the sound of geophysics suddenly getting interesting? ;) – Michael Brown Apr 9 '13 at 8:59
• "Geology is physics slowed with trees on top" - Terry Pratchet, me thinks. – mart Apr 9 '13 at 11:10