Reply To: Peak Oil from Truth Out

[Di Vita]

“Thank you for responding to my post, M. Di Vita.”

Thank you for your interest. Please, let me play devil’s advocate. For a hint of possible future developments, see the last lines of my post.

“Although to my knowledge […] Chaotic systems.”

Indeed, the so-called ‘edge localized modes’ (also known as ‘ELMs’, i.e., coherent oscillations observed in the boundary layer of tokamak plasmas) are currently described as ‘limit cycles’, i.e. as stable, spontaneously occurring, oscillating solutions of a system of simplified, non linear ordinary differential equations. The latter provide us with approximate desciption of mass, momentum ad energy transport in the plasma boundary layer. This approximate description works fine because of the relatively simple (axisymmetric) geometry of tokamak plasmas. (Unfortunately, simple geometries seem to play no role in the DPF). ELMs play a crucial role in the helium control strategy at ITER (if ELMs do not remove helium properly, duterium-tritium fusion keeps on storing helium inside the plasma, leading to unacceptable dilution of deuterium and tritium and stopping further nuclear fusion reactions). The mathematical theory of limit cycles is the same theory of chaotic systems (see e.g. Kutznezov’s textbook).

“I understand your point regarding a few versus many degrees of freedom. However, in the presence of an emergent strong magnetic field, those degrees of freedom do essentially contract to just a few.”

If only it were true… 🙂 Strong magnetic fields may justify simplification from fully three-dimensional geometry to two-dimensional geometry in a restricted class of problems. (Even so, most essential physics is lost: think e.g. of a magnetar). But such contraction implies no chaos at all. For example, in the Seventies Taylor was able to explain Bohm’s diffusion coefficient in a turbulent (non-chaotic) two-dimensional plasma only.

“The evidence of this is the behavior of the plasma itself in the DPF, in the way the “instabilities” form. This is order arising from chaos, a hallmark of Chaotic systems.”

Indeed, it is a hallmark. But not of Chaotic systems; just of what plasma physicists call ‘relaxation’, i.e. ‘spontaneous evolution towards a stable, organised configuration’. Relaxed states are often described with the help of variational principles, i.e. with wordings like: ‘as its own relaxed state, the plasma selects the state which minimises magnetic energy among all possible configurations with the same twistedness of magnetic field lines’. This is the so-called Taylor’s principle, formulated in 1974. In DPF pinch, hot spots seem to be realisations of Taylors’ principle. Numerically, it has been shown that relaxation to states described by Taylors’ principle is the outcome of the competition between heat transport processes on one side and magnetohydrodynamic turbulence on the other side. Turbulence, not chaos.

“Another characteristic of Chaotic systems is some mechanism of feedback. […] themselves.

Spontaneous filamentation, a common feature of many turbulent plasmas both in the lab and in space, is a well-known example of such self-organisation. Again, it is not necessarily due to Chaos. In the DPF, filamentary plasmas have been described with the help of the so-called Turner’s principle, a generalisation of Taylor’s principle.

“I am certainly […] will not work.”

This is certainly true. Even in conventional fluid dynamics (I mean, with zero magnetic field), nobody has yet achieved full suppression of turbulence in high-speed flows. (As you may know, sharks seem to be equipped with a particularly wrinkled skin which lowers turbulence-related drag -my dear friend Professor Bottaro at Genoa University in Italy is a renowned expert in the field. But reduction is no suppression). A fortiori, as far as I know nobody is able to tame magnetohydrodynamic turbulence today. (Admittedly, it is possible to raise the shock front angle in hypersonic plasma flow with the help of a magnetic field, as recently observed in a laboratory at Pisa, not too far from the Leaning Tower). In my opinon, the appeal of DPF lies precisely in the fact that it requires no turbulence suppression.

“What I would really love to see is someone who is well versed in Chaos take a crack at characterizing plasma in the DPF with the view in mind that it just might be Chaotic in nature. If it is and someone could identify an equivalent of a Feigenbaum constant, that could save many headaches and a whole lot of money. It’s worth taking a look.”

You are perfectly right. Indeed, I wonder if a simplified system of equations like the system quoted above in the discussion of ELMs may hold also for a DPF. Once the system has been written down ‘on the back of the enevelope’, then standard mathematical techniques (including e.g. the so-called ‘continuation analysis’, a tool available even in MATLAB) may find both ‘an equivalent of a a Feigenbaum constant’ and all islands of stability, provided that they exist. In turn, simplification forces us to give up any complete description of the system. Then, it requires knowledge of both a) the equations of motion of a plasma and b) the relevant quantities to be computed (such quantity should differ in ELMs and PDF, as both temperature, chemical composition and geometry are different). Since a) are well-known, the difficulty is to find out b). Of course, once the answer is known, it can be tested also on other plasmas, not just on the DPF. Then, the problem affects both the quest for nuclear fusion and the fundamental research.