Nuclear Fusion: What is the cause of transport? (Plasma leakage) Context: I've been reading about The Hairy Ball Theorem which shows that the ideal shape for magnetic confinement fusion has to be a torus. Given that tokamaks use toroidal magnetic fields, I assumed that plasma leakage then wouldn't be a problem.
What is the cause of plasma leakage?
(If I misunderstood anything, feel free to correct me)
 A: 
What is the cause of plasma leakage?

It's no one cause, there are dozens of reasons.
First off there is a natural leakage inherent to any real-world fluid due to a pure random-walk process. The plasma particles are orbiting the long axis of the torus while spinning around the "lines of force", thereby tracing out helical paths. The helical diameter is larger than the inter-orbital spacing, meaning any single particle will overlap the paths of others during its motion, and this means there will be multiple chances for scattering as they orbit and collide. This causes the particles to undergo a random-walk process that eventually takes them outside the boundary of the confinement field, and/or into the walls of the reactor.
Basic math suggests this rate, now known as "classical diffusion" is low enough that a reactor would work. There is a dependency on the square of the field strength, so it seemed that even low-power machines would be useful testing systems because as long as they worked a little one could then build a machine that would work completely by scaling up the magnets for production. So in the 1950s you see many small-scale tabletop devices being built.
When they did, they found the actual confinement time was dramatically lower than classical diffusion suggested, and increasing the magnet power had no effect. This was determined to be due to natural instabilities in the plasma itself.
To illustrate a simple example, consider a plasma torus where, purely at random, one section of the plasma is slightly higher density. When a current is run through the plasma, as it is in the pinch machines, the current creates a field that pulls the plasma down into a filament. However, as one section has slightly higher density, the field in this region is higher, so it collapses faster, which increases the density, which increases the field...
This instability, the "sausage", is inherent to the plasma. Similar examples include the kink, the flute (aka interchange) and various higher-order MHD modes where standing waves in the plasma cause "pump out".
It took about 15 years to come up with ways to solve these issues, which were first demonstrated with convincing effect in the T-3 tokamak in 1968. The key was to use more external magnetic field compared to the field from the internal current, which causes the overall long-axis path to be more "spirally" and thus smoothes out the instabilities before they can build up.
As new toks came online, it was soon noticed that yet new instabilities were being seen. A key one among these is now known as the banana orbit. Consider a single particle orbiting the reactor; when it is at the outside of the torus the magnetic field is lower than it is when it moves toward the inside of the curve - simply due to geometry, the magnets are closer together on the smaller radius. If the particle has a velocity below a threshold value, it will reflect off the increasing field in the same fashion as in a magnetic mirror. Now you have low-energy particles bouncing back and forth within limited regions of the reactor, which from above look like the shape of a banana. The higher-energy ions, the ones you need for fusion, keep scattering off of these low-energy ones.
So then we added more complexity. One is to "scrape off" ions near the outside of the confinement area, another is to divert them into a cooler, typically liquid lithium in modern designs, while other fields and heaters can be used to control the action of these ions and use them constructively.
Today we have yet more instabilities to deal with, and these are truly destructive. There are conditions that form that cause electrons to bunch up and create channels that accelerate the electrons to relativistic speeds. These "disruptions" are extremely annoying, in one case burning a hole into the vacuum chamber. Controlling these is a major ongoing area of research in the field.
And on top of all of this, you still have that random walk going on.
