7th Asia-Paciﬁc Conference on Plasma Physics, 12-17 Nov. 2023 at Port Messe Nagoya

Change of relaxation path by inclusion of Hamiltonian dynamics to simulated
annealing of reduced magnetohydrodynamics
M. Furukawa1 and P. J. Morrison2
1
Faculty of Engineering, Tottori University, Japan,
2
Department of Physics and Institute for Fusion Studies, University of Texas at Austin, Texas,USAe-mail (speaker): furukawa@tottori-u.ac.jp
In simulated annealing (SA), we solve an initial-value
problem of artiﬁcial dynamics derived from an original
Hamiltonian system so that the energy of the systemmonotonically changes while the Casimir invariants arepreserved. The energy minimum (extremum) is an equilibrium. We have applied the double bracket formulationof SA[1, 2] to reduced magnetohydrodynamics (MHD)
for equilibrium calculations as well as stability analysis.
In [3], we have developed a method to accelerate the relaxation of SA since equilibrium and stability analyses
using relaxation are time consuming. In this study, weexamine whether inclusion of the original Hamiltonian
dynamics can accelerate the relaxation compared to thatby SA only.
The original Hamiltonian dynamics adopted here is
the low-beta reduced MHD[4]. The evolution equations
for the vorticity U and the magnetic ﬂux function ψ incylindrical geometry are, after normalization,∂U∂J= [U, φ] + [ψ, J] − ε∂t∂ζ∂ψ∂φ= [ψ, φ] − ε∂t∂ζ= f 1,(1)= f 2,(2)where φ is the stream function and J is the current density, ε is the inverse aspect ratio, [f, g] is the Poissonbracket, and ζ is the toroidal angle coordinate.
The evolution equations of SA adopted here are˜
∂U˜ − ε ∂J
= [U, φ̃] + [ψ, J]∂t∂ζ∂ψ∂ φ̃
= [ψ, φ̃] − ε∂t∂ζ= f˜1 ,(3)= f˜2 ,(4)where the artiﬁcial advection ﬁelds are deﬁned by∫φ̃(x, t) := −α11 (t)d3 x′ g(x, x′ )f 1 (x′ , t),D∫
˜J(x, t) := −α22 (t)d3 x′ g(x, x′ )f 2 (x′ , t).

(5)
(6)

D

Here, g is a Green’s function in three dimensions. The coefﬁcients α11 and α22 may depend on time[3], although
they should be both positive for monotonic decrease ofthe energy. Equations (3) and (4) have the same form
as Eqs. (1) and (2), but the advection ﬁelds φ and J are
˜replaced by the artiﬁcial ones φ̃ and J.
In this study, we solved∂U= f˜1 + cf 1 ,∂t∂ψ= f˜2 + cf 2 ,∂twhere c is a parameter expressing how much the original
dynamics is included.

Figure 1: Time evolutions of total energy for various values of parameter c expressing the ratio of the original
Hamiltonian dynamics to the SA. Inclusion of the original dynamics decelerated for the tested cases.
Figure 1 shows an example of time evolutions of total energy by Eqs. (7) and (8). The initial condition wastaken to be a cylindrically symmetric equilibrium plus
small-amplitude helical perturbations. The cylindricallysymmetric equilibrium has no ﬂow and is spectrally stable. The helical perturbations were taken to be dynamically accessible. The coefﬁcients α11 and α22 were ﬁxed
as 100. The relaxation of energy was decelerated for thetested cases with either c > 0 or c < 0.
Acknowledgement
M.F. was supported by JSPS KAKENHI Grand Number JP21K03507, while P.J.M. was supported by the U.S.
Department of Energy (Contract No. DE-FG05–80ET53088) and the Humboldt Foundation. The authors bothwould like to acknowledge the JIFT program for the support of MF to visit IFS in the Spring of 2019 when a portion of this work was carried out. A comment by Dr.
Toshinori Kouchi was useful.
References[1] G. R. Flierl, P. J. Morrison, Physica D 240, 212(2011).
[2] P. J. Morrison, Phys. Plasmas 24, 055502 (2017).

(7)
(8)

[3] M. Furukawa and P. J. Morrison, Phys. Plasmas 29,102504 (2022).
[4] H. R. Strauss, Phys. Fluids 19, 134 (1976).