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Can I get an explanation of what magnetic pressure is exactly in the context of MHD? From my simple understand, it's essentially just the force that magnetic field lines exert back onto a plasma that has some incident velocity (energy) perpendicular to the field lines. An example would be the solar wind interacting with the Earth's magnetic field. The field blocks the solar wind, but there is so perpendicular pressure being exerted onto the field from the incident plasma. Is my interpretation correct?

Second question (If my interpretation is correct), is this pressure directly related to the Larmor radii trapped particles will have? Since the plasma has a Maxwell distribution, then there will be a Maxwell distribution of Larmor radii and so those would then be correlated to the pressure term? So if I wanted to know then what is the momentum threshold for this distribution of energies to break through the field entirely, how would I solve for that given the equation (Just from what I found on Wikipedia):

enter image description here

Say I would simplify this extremely by just looking at the 1D version of this. The $B$ term would be the 1D gradient of a dipole field whose center is aligned with the $z$-axis, (So it's center is at $x = y = 0$), while the incident plasma will be traveling along $x$ or $y$. I assume I would change the $B$-terms to $dB_x$ and do an integration of the $B$ gradient over the $x$-space of interest to find the total energy/current required by say an ion to penetrate through the whole field? That is my best guess for this.

https://en.wikipedia.org/wiki/Magnetic_pressure

Mauricio
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Sophia
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  • Please do not use images for equations when you can type them instead. This would avoid broken links in the future. – Mauricio Mar 24 '23 at 02:37
  • @Mauricio Ok will do – Sophia Mar 24 '23 at 02:43
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    Pressure is just a form of energy density or energy per unit volume. So magnetic pressure is just the energy per unit volume contained within the magnetic field alone. – honeste_vivere Mar 24 '23 at 13:17
  • @honeste_vivere Oh so is that J term the current density that is producing the B field itself and not the plasma? – Sophia Mar 24 '23 at 19:09
  • If that J is the plasma: I know that the plasma pressure will alter the field geometry since it's a force acting on the field in a perpendicular manner, thus "squishing" it. Is this not what is being described by the above equation? If J is the current inducing the B field, what would be the proper set of equations that describe the altering of field geometries based on internal/external plasma pressures? – Sophia Mar 24 '23 at 19:14
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    $\mathbf{j}$ is indeed the current density, which is a source of magnetic fields, yes. Currents are carried by drifting, charged particles in a plasma (i.e., some subpopulation of the plasma will carry the current in some reference frame since currents are not frame-invariant). What you have shown above is just the expansion of a single term in generalized Ohm's law, where one assumes $\mathbf{j}$ = $\nabla \times \mathbf{B} / \mu_{o}$. – honeste_vivere Mar 24 '23 at 20:13
  • I see, so what equations address how magnetic field geometries (Magnetic field densities) change (Compress/Expand) based on incident charged particle fluxes? @honeste_vivere – Sophia Mar 24 '23 at 21:21
  • Maxwell's/Heaviside's equations plus a kinetic equation like a modified Boltzmann equation? – honeste_vivere Mar 24 '23 at 22:11
  • Yeah I was thinking about those too but it seems like a more complicated problem than I wish to pursue tbh haha. Thank you for the help! – Sophia Mar 24 '23 at 22:25
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    I briefly discuss magnetic pressure in this answer of mine wherein I show that this pressure term straightens a flow along a field line. – Kyle Kanos Mar 30 '23 at 17:57
  • @KyleKanos thanks I'll take a look! – Sophia Mar 31 '23 at 06:47

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