What are the different types of stellarators?
How does 3-dimensional shaping produce rotational transform?
What are flux surfaces and why are they important?
A stellarator is a type of magnetic confinement device used to confine hot plasma. The ultimate goal of magnetic confinement is to confine high temperature plasmas at sufficiently high densities and long enough confinement times so as to be applicable to fusion power production. Stellarators are a particlular type of "magnetic bottle" which use 3-dimensional shaping to try and improve the plasmas confinement and stability properties. A picture of one type of stellarator is shown below. The blue-magneta surface represents the outer surface of the plasma; the colors represent the strength of the magnetic field on this surface (magenta = high field, light blue = low field). The white lines are the magnetic field vector which wraps around the plasma surface. This property of the magnetic field to traverse or wrap around the plasma surface, rather than simply go around as a set of individual disconnected lines is known as rotational transform and is helpful in confining the plasma. In the stellarator, rotational transform is predominately produced by the 3-dimensional shaping of the plasma surface; although, in more recent designs (known as hybrids), a low level of plasma current flowing around the torus may also be relied upon for rotational transform.
A tokamak is also a toroidal (donut-shaped) plasma magnetic confinement device. However, the tokamak is axially symmetric (see the picture below) and relies only on 2-dimensional shaping (in the cross section you would get by cutting through the donut). The tokamak also has field lines that wrap around the plasma surface (show below as white lines). However, the tokamak achieves its rotational transform through internal plasma currents that flow around the torus. This reliance on plasma currents implies that the tokamak is not a steady state system, but will require some level of current drive. Also, the presence of current in the plasma can drive instabilities and turbulence in the plasma. However, the tokamak can offer very good plasma confinement due to its high degree of symmetry. The stellarator requires careful optimization to achieve similar levels of plasma confinement as the tokamak.
The main types of stellarators that are of current interest are the torsatron, and several forms of optimized systems. The latter group can be characterized first of all by what form of symmetry they try to achieve in the magnetic field variation. The three types of magnetic field symmetry are helical (the magnetic field isobars form a helical pattern on the plasma surface), toroidal (the magnetic field isobars form bands running the long way around the plasma surface), and poloidal (the magnetic field isobars form bands running the short way around the plasma surface). The torsatron is characterized by its magnetic coil structure which consists of helical twisting coils that wrap around the plasma. The optimized systems generally use modular coils which wrap the short way around the torus. These coils will genrally have a 3-dimensional shape (see light blue objects in example below).
and must be numerically determined by an optimization process. Some examples of the different plasma shapes of stellarators are at the link: world stellarators. In addition to the above types of stellarators, these devices can be either large or small aspect ratio. Aspect ratio is the ratio of the large outside radius of the torus to the small radius (i.e., through the plasma). Small aspect ratio stellarators have very fat, compact donut shapes; on the other hand, large aspect ratio have thinner bicycle tire-like shapes.
Rotational transform is the motion of the magnetic field lines the short way around the torus while they are also wrapping the long way around the torus. That is in the above 2 figures, the magnetic field lines do not in general simply come back to the same point and close on themselves, but they are displaced by some amount so that over many times around the torus they will completely cover it. This must be qualified by the fact that there are always some "rational surfaces" where the field lines do eventually (after multiple circuits around the torus) close on themselves.
This is a somewhat subtle aspect of stellarators. However, there is an analogy to water flowing around a donut-shaped pipe which has been used. Both the magnetic field and the flowing water are incompressible vector fields with a particular direction at every point so they will share some of the same properties. If you then took the toroidal pipe and either twisted it, or placed appropriate dents and bulges around it, one can imagine that the flow of water will become re-directed so that it is no longer moving only around the donut (toroidal direction). In particular the water may tend to develop swirls or velocity components the short way around the donut. This is equivalent to the rotational transform in the magnetic field that stellarators produce by 3-dimensional shaping.
Magnetic flux surfaces are the surfaces formed by the magnetic field as it wraps many times around the torus. The confined plasma tends to form isobars or constant pressure surfaces that coincide with these magnetic surfaces. If the magnetic surfaces are not closed or connect to the wall, then the hot plasma will not be well confined and escape rapidly to the wall and extinguish itself (also possibly damaging the wall). Stellarator designers must work hard to produce devices with good closed magnetic surfaces. Small asymmetries or manufacturing defects in the magnetic field coils can lead to open magnetic surfaces.