<p><span class="math-container">$\nabla B^2 \sim B^2/l,$</span> where <span class="math-container">$l$</span> is a length scale on which <span class="math-container">$B^2$</span> varies.</p>

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

<p>The density of the solar photosphere (at optical depth unity in the visible spectrum) is around <span class="math-container">$\rho \sim 10^{-4}$</span> kg/m<span class="math-container">$^3$</span>.</p>

<p>Thus <span class="math-container">$\nabla \cdot P_B/\rho \sim 10^{-4}$</span> m/s<span class="math-container">$^2$</span>.</p>

<p>So no, I can't quite reproduce your number.</p>

CC BY-SA 4.0
2019-08-24T22:22:56.437Z
Yes, I got something like that too. I think its the modulus that's maybe causing some difficulty...
user29126
CC BY-SA 4.0
2019-08-24T23:58:39.067Z
I should also point out that the density specified earlier in the book is $\rho = 1$ g cm$^{-3}$.
user29126
CC BY-SA 4.0
2019-08-25T14:04:05.980Z
@user29126 I doubt that this is the density of the photosphere, that is more like the average density of the entire Sun.
ProfRob
CC BY-SA 4.0
2019-08-24T20:30:58.877Z
2019-08-24T20:30:58.877Z
ProfRob
<p>I was reading through the following extract outlining how the Maxwell stress contributes to the <span class="math-container">$\nabla \cdot \mathbf{P}$</span> term of the Navier-Stokes equation for fluids in a star. Here <span class="math-container">$\mathbf{P}$</span> is a general symmetric stress tensor. </p>

<p><a href="https://i.sstatic.net/8p4JL.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/8p4JL.png" alt="enter image description here"></a></p>

<p>I am struggling to see how the estimate <span class="math-container">$\vert \nabla\cdot \mathbf{P}\vert/\rho \approx 3\times 10^{-3}$</span> ms<span class="math-container">$^{-2}$</span> was derived using the information given in the passage. Can someone please provide a quick derivation of this result?</p>

<star><the-sun><astrophysics><star-systems><stellar-structure>
Maxwell stress contribution to $\nabla \cdot \mathbf{P}$ in the Navier-Stokes equation for fluid in stars
user29126
# Maxwell stress contribution to $\nabla \cdot \mathbf{P}$ in the Navier-Stokes equation for fluid in stars

(Asked by: username_0 on 2019-08-24)

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.

[![enter image description here](https://i.sstatic.net/8p4JL.png)](https://i.sstatic.net/8p4JL.png)

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?

**Tags:** star, the-sun, astrophysics, star-systems, stellar-structure


---

## 1 answers

### Answer by username_1 on 2019-08-24. Score: 2

$\nabla B^2 \sim B^2/l,$ 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.

> **Comments:**
> **username_0**: Yes, I got something like that too. I think its the modulus that's maybe causing some difficulty...  
> **username_0**: I should also point out that the density specified earlier in the book is $\rho = 1$ g cm$^{-3}$.  
> **username_1**: @username_0 I doubt that this is the density of the photosphere, that is more like the average density of the entire Sun.  

---

