In his letter in the February
2006 issue of PHYSICS TODAY (page 10), David Montgomery expresses concern about the theoretical
basis for predicting the performance of ITER, the planned international magnetic fusion energy
experiment. He focuses specifically on the use of ideal (nondissipative) magnetohydrodynamics
(MHD) as the zeroth-order description of equilibrium in magnetically confined plasmas. He correctly
points out that such an ordering is not generally possible in the mechanics of neutral fluids.
Although in principle
that concern could be legitimate, in practice a wide range of experiments has confirmed that the
ideal MHD model excellently describes the equilibrium of high-temperature magnetically confined
plasmas. Global and detailed local measurements of plasma equilibrium show excellent agreement
with ideal theory. Fast instabilities are also well described by ideal theory; slower, dissipative
instabilities require more elaborate description, but under the usual operating conditions,
the basic ordering that starts from ideal equilibrium is not violated. Ideal MHD calculations
are routinely used in major magnetic confinement experiments to assemble and interpret measurements
of magnetic fields, pressure gradients, and flow speeds, and to predict stability boundaries.
Those MHD codes track experimental behavior consistently and in detail.
Why does ideal MHD provide
a legitimate zeroth-order description of magnetically confined plasmas? The answer is tied to
researchers' broad success in using magnetic fields to confine plasmas in relatively quiescent
states free of large-scale MHD instability. By contrast, the evolution of globally turbulent
neutral fluids depends on a dynamic balance between ideal forces, acceleration, and dissipation.
In magnetically confined plasmas, the ideal equilibrium forces—j × B,
∇p,
and to a lesser degree ρ(v · ∇)v—are
measured to balance each other and dominate strongly over acceleration and nonideal dissipation
terms, confirming experimentally the ordering procedure that Montgomery questions. Violation
of that ordering corresponds to global acceleration on an ideal MHD time scale, and so to rapid plasma
loss, which is observed only under circumstances in which ideal MHD stability is compromised.
Even Montgomery's model calculations support the use of an ordering that starts from ideal equilibrium
MHD, since the results he displays show a very low flow speed, in the range of 1 m/s.
One of the most exciting
developments in fusion-energy research over the past decade has been the increasing agreement
between magnetized plasma theory and confinement experiments. Recent progress in achieving
predictive understanding of confined plasma behavior seems to us, an experimentalist and a theorist,
truly remarkable. Theory—including MHD theory, more elaborate fluid models, and fully
kinetic plasma descriptions—does not play a "decorative" role today in high-temperature
plasma physics; such tools provide an essential guide in the design and execution of experiments.
Theory and advanced computing, particularly through the US Department of Energy's Scientific
Discovery Through Advanced Computing and Leadership-Class Computing initiatives, are key elements
of US preparations for ITER operation.
Montgomery replies:
It isn't easy to reply to citation-free claims about physics projects that assert "broad successes,"
but that is apparently what has to be done when tweaking the controlled fusion program. A first point
to be made is that fundamental properties of differential equations are not trivially changed
by "ordering" the terms, particularly when the ordering involves dropping those terms with the
highest numbers of derivatives. Then even the boundary conditions necessary to determine a solution
change. One way to see this would be to try to re-derive the results of our nonideal toroidal steady-state
calculation1 from a perturbation series that starts with ideal MHD.
As far as the alleged differences
between MHD and fluid dynamics go, the mechanics of rotating fluids (as in oceans and atmospheres)
are actually not that much different. The role of MHD Lorentz forces gets taken over to a large degree
by the Coriolis force. The principal results that emerge in geostrophic flow, Ekman circulation,
thermal convection, and so on would all be impossible without the inclusion of dissipation coefficients
at crucial places in the essential nonideal aspects of the phenomena.2
I would say that whether
flows of meters per second amount to a serious threat for ITER likely depends on the geometry of the
flow in a cross section of the toroid. A simple poloidal rotation of that magnitude might be no problem;
but a dipolar flow pattern whose streamlines connect the hot geometrical center with the cool perimeter
could be a disaster. In our article for the Journal of Plasma Physics, Leon Kamp and I found
a transition from the second behavior to the first as the shape of the cross-sectional boundary
and the Hartmann number were varied.1 Is that result in dispute?
2.See, for example, H. P. Greenspan, The Theory of Rotating Fluids, Cambridge U. Press, Cambridge, UK (1968); J. Pedlosky, Geophysical Fluid Dynamics, Springer-Verlag, New York (1979).
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