# How long for released dye to uniformly saturate the oceans?

Suppose a supertanker filled with purple dye breaks open at a particular point in the middle of the Pacific Ocean and spills an enormous volume of purple dye into the ocean all at once.

1. Imagine that there are no ocean currents and that all of the world's oceans are perfectly still. How long would it take for the purple dye to diffuse and uniformly saturate the world's oceans? (You can assume that the world is completely covered with water, if that makes it simpler.)

2. Now imagine that there are ocean currents. How long would it take for the purple dye to uniformly saturate the world's oceans? (Potentially helpful reference here.)

3. Is it possible that #1 could actually be faster then #2? Could the dye be caught in a closed-loop current and stay there indefinitely, never reaching the rest of the world? My understanding is that when currents flow past each other in opposite directions, almost no material can cross the interface between them.

• I think Fukushima is doing this experiment right now. – JEB Jan 13 at 17:08
• Your understanding on item 3 is incorrect. Any amount of convection will enhance the rate of spreading of the dye, particularly if high concentration and low concentration streams are making contact with one another. This greatly enhances the rate of mass transport. – Chet Miller Jan 13 at 23:45

That is a very interesting question. I really like guesstimates :D.

The first part I would try to solve with Brownian motion. According to Einstein we have

$$x^2=t\frac{k_BT}{3r\pi\eta}$$

with

• $$x$$: average distance from the source
• $$t$$: time since the spill
• $$k_B$$: Boltzmann constant
• $$T$$: temperature
• $$r$$: the radius of the particle
• $$\eta$$: the viscosity of the dye

I would further assume that the dye and the water are two distinct phases, so they do not mix, the dye builds an upper layer, on top of the water.

In addition, I would assume that the Earth is a perfect sphere with diameter $$d=12'000$$ km, so $$x=\frac{d\cdot\pi}{2}\approx18\cdot10^6$$ m when the globe is completely covered. (It only has to "travel" half across the globe, because it is travelling in both directions simultaneously. Both "streams" will meet on the other side.)

The average global temperature is about 15°C if I am not mistaken. So $$T=288$$ K.

The radius of a particle: Let us consider indigo, with a molar mass of $$262$$ g/mol and a density of $$1$$g/cm$$^3$$, which yields and average particle volume of $$262$$ cm$$^3$$/mol $$= 4\cdot10^{-22}$$ cm$$^3$$, which roughly corresponds to a particle radius of $$8\cdot10^{-8}$$ cm (under the assumption of cubic particles, but spheres and cubes are not that much different...). This radius seems reasonable compared to the size of an atom which is of the same order. For simplicity, let us just assume $$r=10^{-9}$$ m.

The viscosity of dye should be somewhere between water and honey, I assume, maybe in the range of olive oil or slightly less. I will assume $$\eta=40$$ mPa s.

Putting everything together we find something like

\begin{align} t&=\frac{3r\pi\eta x^2}{k_BT}\\ &=\frac{3\cdot10^{-9}\text{ m}\cdot\pi\cdot0.04\text{ Pa s}\cdot 3\cdot10^{14}\text{ m^2}}{1\cdot10^{-23}\text{ J/K}\cdot288\text{ K}}\\ &\approx 4\cdot10^{25} \text{ s} \end{align}

where I used J = Pa m$$^3$$ from the Boltzmann constant.

So it takes roughly $$10^{18}$$ years to completely colourise our planet just by Brownian motion, longer than the age of our universe. So if there is somewhere a huge supertanker leaking, we would not have noticed yet! Stay tuned for our colourful world!