Re: Problems understanding irrational q values in Tokamaks

"Bruce Scott TOK" <[EMAIL PROTECTED]> wrote in
message news:[EMAIL PROTECTED]
> Oliver Keating asked:
>
> [sorry for not a timelier answer...  we had a conference...]
>
> |> I have started research into plasma instabilities in Tokamaks, but I
have
> |> hit a physics problem that I cannot get my head around.
> |>
> |> Let me set the scene:
>
> [need a poloidal as well as toroidal magnetic field to have closed flux
> surfaces, thereby confining the plasma...  basic Freidberg stuff (cf his
> book Ideal Magnetohydrodynamics, or Plasma Confinement by Hazeltine and
> Meiss)]
>
> |> This means that the actual magnetic field looks rather like a helix
> |> travelling around the tourus.
> |>
> |> At this point it is useful to define something called a "q" value,
which
> |> refers to the number of times the magnetic field goes around poloidally
for
> |> each toriodal turn.
> |>
> |> I had always thought that q *must* be rational, i.e. q = m/n where m
and n
> |> are integers.
>
> q is simply the ratio of the contravariant components of B, averaged
> over a surface.  The more useful magnetic coordinates are defined such
> that this ratio is constant on the surface, and therefore the magnetic
> field lines are straight on the coordinate plane covering the flux
> surface.
>
> Note that since B is a local vector field, and the only constraint is
> that div B = 0, the q ratio (also called the "pitch parameter" or
> because of some ideal MHD results, the "safety factor") can take
> arbitrary values.
>
> Magnetic shear is defined as the logarithmic derivative of q with
> respect to the minor radius.  Therefore, a sheared magnetic field will
> always have irrational values of q, except on a set of measure zero of
> surfaces, called "rational surfaces" which have q = m/n with m and n
> integers.
>
> |> The reason for this is that otherwise the field line cannot reconnect
on
> |> itself. For example, if the q value is 2, that means the magnetic field
goes
> |> around the tokamak 2 times before reconnecting on itself.
>
> There is no reason magnetic field lines, which are artificial
> constructs, are required to be closed.  The thing that is closed is the
> flux surface.  Again, since this is merely a surface upon which the
> value of a scalar field known as the poloidal flux is constant, there is
> nothing mysterious about closed, toroidal flux surfaces.  In fact, such
> closed surfaces are necessary to have a confined MHD equilibrium.
>
> |> Now, when I started looking at instabilities - growth of instabilities
> |> happens on "rational q surfaces". I find this extraordinary as it
implies
> |> there are an infnite set of irrational q surfaces. I asked about this
and it
> |> turns out to be true.
> |>
> |> My problem is that with an irrational q number, the magnetic field line
> |> spins around the tourus forever - it never manages to reconnect back on
> |> itself as it spirals around.
>
> No problem.  The vector field, not the field line, is the physical
> quantity.
>
> |> I don't understand this at all. I thought one of the consequences of
> |> "divB=0" is that all magnetic field lines must connect back on
themselves?
>
> No...  [note that div B = 0 is a local constraint]
>
> ..that is only really true in 2D (in the poloidal plane projection,
> the magnetic field does indeed close on itself...  the surfaces of
> revolution described by these curves and the symmetry axis are in fact
> the toroidal flux surfaces).
>
> |> Additionally, how can you have an infinite number of field lines but a
> |> finite magnetic field?
> |>
> |> As yet no one has managed to provide a answer that I understand!
>
> If you can deal with vectors and differential equations (which even the
> most basic stuff uses), it is easy...
>
> You might be blown away by the detail in the Callen et al text, which is
> the way it is because they also describe stellarator equilibria...
>
> A simple lecture on tokamak equilibrium is here:
>
>     http://www.rzg.mpg.de/~bds/write-ups/tokamak/
>
> A short papers on describing the flux tube geometry with globally
> consistent boundary conditions (required to address your questions) is
> here:
>
>     B Scott, Phys Plasmas 5 (1998) 2334-2339
>
> That is all for tokamaks, which are 2D because of the axisymmetry
> (vectors have three components, but depend on only two coordinates)...
>
> Three dimensional (i.e. non axisymmetric) equilibria are really, really
> complicated.  The coils are easy to build, but the plasma is hard to
> understand :-)

Depending on how you simplify it.

Due to the gaussian velocity distribution of the plasma, there are always
some particles losing the grip of the magnetic field, drifting away from the
confinement.

Meaning that there is always a certain percentage of density leak in the
plasma.

Another way to put it is, confined hot plasma will never work.

I don't think it is so hard to understand.

That's why it never worked and it never will.