# Maxwell stress contribution to $\nabla \cdot \mathbf{P}$ in the Navier-Stokes equation for fluid in stars

I was reading through the following extract outlining how the Maxwell stress contributes to the $$\nabla \cdot \mathbf{P}$$ term of the Navier-Stokes equation for fluids in a star. Here $$\mathbf{P}$$ is a general symmetric stress tensor. I am struggling to see how the estimate $$\vert \nabla\cdot \mathbf{P}\vert/\rho \approx 3\times 10^{-3}$$ ms$$^{-2}$$ was derived using the information given in the passage. Can someone please provide a quick derivation of this result?

$$\nabla B^2 \sim B^2/l,$$ where $$l$$ is a length scale on which $$B^2$$ varies.

In SI units, $$B^2 \sim 4\times 10^{-8}$$ T$$^2$$ and $$\mu_0 = 4\pi \times 10^{-7}$$, so an order of magnitude for $$\nabla \cdot P_B$$ is $$4\times 10^{-8}/(8\pi \times 10^{-7}\times 10^{6}) \sim 10^{-8}$$ Pa/m.

The density of the solar photosphere (at optical depth unity in the visible spectrum) is around $$\rho \sim 10^{-4}$$ kg/m$$^3$$.

Thus $$\nabla \cdot P_B/\rho \sim 10^{-4}$$ m/s$$^2$$.

So no, I can't quite reproduce your number.

• Yes, I got something like that too. I think its the modulus that's maybe causing some difficulty... – user29126 Aug 24 '19 at 22:22
• I should also point out that the density specified earlier in the book is $\rho = 1$ g cm$^{-3}$. – user29126 Aug 24 '19 at 23:58
• @user29126 I doubt that this is the density of the photosphere, that is more like the average density of the entire Sun. – Rob Jeffries Aug 25 '19 at 14:04