# Help understanding vectors in diagram from "Magnetohydrodynamic flow control during reentry" and the actual goal of this research

A link to the ESA Advanced Concepts Team page Magnetohydrodynamic flow control during reentry was recently shared under Re-entry Heat Shield Alternative. It says:

During the reentry phase, spacecraft experiences large aerodynamic and thermal loads that constraint the vehicle design. Entry vehicles have been traditionally equipped with passive thermal protection systems (TPS) which typically consist of ablative heat-shields and heat-resistant materials. Previous studies have shown that MHD flow control is a potential active TPS technology, since it is capable of reducing the thermal loads typically experienced during atmospheric entry [2]. Furthermore, MHD flow control could bring benefits in terms of vehicle controllability and re-usability.

In a short study within the Advanced Concepts Team, existing literature on this concept was combined to estimate the impact on the re-entry trajectory. In particular, the interest was to explore the maximum velocity achievable for direct entry. Using a simplified model of the entry dynamics, in combination with an analytical approximation of the MHD interaction, a trajectory analysis could be performed under varying levels of magnetic field strength.

References

H. Otsu, H. Shizuoka, A.Matsuda, T. Abe, D. Konigorski, Feasibility Study on the Flight Demonstration for a Reentry Vehicle with theMagnetic Flow Control System, AIAA 2006-3566, 2006. paywalled

T. Yoshino, T. Fujino, M. Ishikawa, Possibility of Thermal Protection in Earth Re-entry Flight by MHD Flow Control with Air-Core Circular Magnet, Institute of Electrical Engineers of Japan, 2009. paywalled

During a search I found Plasmas for High Speed Flow Control which seems to be related:

This paper presents experimental activities focused on supersonic flow control with plasma and MHD actuators. This work is carried out at ICARE, a laboratory located at the CNRS Campus in Orléans. The study of aerothermodynamic physics, one of the research fields of ICARE, is conducted with the experimental platform FAST consisting of three supersonic/hypersonic wind tunnels involved in aerothermodynamic testing for hypersonic flight and space technology.

The original link also contains the following diagram which I can't understand at all!

The flow of the plasma is indicated by the velocity vector $$\pmb{V}$$, but what causes the current $$\pmb{J}$$ (the orange "donut") and why does it appear to be perpendicular to all three other vectors $$\pmb{V}$$, $$\pmb{B}$$ and the Lorentz force $$\pmb{J} \times \pmb{B}$$?

I can't make heads or tails of this diagram.

Questions:

1. Why does the orange $$\pmb{J}$$ donut look perpendicular to three other mutually orthogonal vectors, which is not mathematically possible. Is there a simpler way to draw this? When it comes to 3D problems, extended prose just doesn't do it for me.
2. What is the point of this research? Does it keep the spacecraft cooler, or provide some against the plasma to change the attitude of the spacecraft?

• For donut shaped vector $J$, the direction would taken as the normal to the plane of $J$ (possibly counter clockwise here, hence normal comes out of the blunt shape) and then cross product with B points to the perpendicular direction from both of these two. So it is mathematically possible, just we are not able to see the directions clearly. $JXB$ is not coming towards us but is going away from us. Mar 28, 2021 at 10:14
• @OrangeDurito I don't know what "the direction of vector $\pmb{J}$ being normal to the plane of $\pmb{J}$" means, except to define what the phrase "the plane of" means, but that must be different than "the plane containing $\pmb{J}$". What is "the plane of $\pmb{J}$" such that $\pmb{J}$ would be normal to it? btw the title is already long. Since it's clear enough I elected not to make it even longer. I switched from \mathbf{J} to \pmb{J} (bold italic) but since I use MathJax so much I want to keep it consistent.
– uhoh
Mar 28, 2021 at 10:47
• First I've heard of a "donut shaped vector". Sounds like a trip to the pastry shop. Mar 28, 2021 at 13:23
• @OrganicMarble it's the direction that Homer Simpson is moving
– uhoh
Mar 28, 2021 at 13:24
• ObXKCD Magnetohydrodynamics combines the intuitive nature of Maxwell's equations with the easy solvability of the Navier-Stokes equations. It's so straightforward physicists add "relativistic" or "quantum" just to keep it from getting boring. Mar 30, 2021 at 9:42

Question 1 is a bit trickier. The current really is a circle, which means you have to visualize the trio of vectors as rotating around the body if you were to spin the circle like a wheel with those vectors attached just at the point they all emerge from, so at every point on the wheel, $$B$$ is pointing somewhere different.
Reading the second paper on the list, the one which is cited by the first, makes things a bit clearer, because they use diagrams that are actually much more complicated. The magnetic field is not pointing in just one direction, because magnetic field lines don't do that: they have to close, because there is no magnetic charge (Maxwell's equation $$\nabla\mathbf{\cdot} B=0$$). The model is quite small, and field is generated by a stack of three small (15 millimeter diameter) neodymium permanent magnets. This radiates from the sphere like hair, curving away from straight ahead and then back along the body to enter again at the back, forming a cylindrical sheath along the sides out of wild-looking "hair" at the "head", as in my feeble attempt to reconstruct their image:
The arrow is the direction in which the object is moving, the circle and the two rectangles are a side view of the three magnets (one spherical, two cylindrical), the curvy things are pieces of magnetic field lines, and the picture is rotationally symmetric about the axis of the arrow. Thus $$B$$ is not just in one direction: it points off of the hemispherical body nose in every direction, starting radialish but all sweeping back around to close behind (not shown).
The paths of charged particles in plasmas like this are generally spirals around magnetic field lines, and in this situation those can be thought of as adding up to create cylindrical currents like the orange torus labeled $$J$$. The interactions of those currents with the fields that created them, and with the additional fields and currents they create, always act to oppose what created them (Lenz's Law), and thus slow things down.