Effects of Magnetic Field on Two-Plasmon Decay Instability in Homogeneous Plasma Xinfeng Sun ( 孙新锋 ), Zhonghe Jiang ( 江中和 ), Xiwei Hu ( 胡希伟 ) School of Electrical and Electronic Engineering, HUST

Outline 2/18  Background;  Absolute instability;  Convective amplification;  Future works. Huazhong University of Science and Technology

Background 3/18  TEXTOR ECRH experiment, strong scattering phenomenon occurs when the rf wave passing the O point of a magnetic island[1], which leads to energy loss. [1] E. Westerhof, et al. PRL, (2009). [2] E.Z. Gusakov and A. Yu. Popov, PRL, (2010).  The threshold of two-plasmon decay instability of magnetized plasma is low( kW).  Parametric decay instabilities of X wave lead to the scattering, X-X+IB and X-UH+UH are two main channels[2]. Huazhong University of Science and Technology

Background 4/18 Incident frequency ： 140GHz Scattering frequencies ： 138.2GHz 、 137.5GHz Huazhong University of Science and Technology

Definitions 5/18 Parametric Decay ： A sideband mode is produced by the coupling of pump wave with the eigenmode of plasma. Absolute and Convective Instabilities ： Mode Coupling ： One mode is produced by the coupling of two branch mode. Huazhong University of Science and Technology

Coupling equations: B 0 =0 6/18 Coupling equations [3] (Two daughter wave) ： [3] R. Yan, A. V. Maximov and C. Ren, Phys. Plasmas 17, (2010). Fluid equations ： Huazhong University of Science and Technology

Dispersion relation of upper hybrid wave 7/18 (1) (2) (3) (4) (5) Dispersion relation after the correction of Electromagnetic wave ： Huazhong University of Science and Technology

8/18 Coupling equations of magnetized plasma ： Magnetic field ( L is inhomogeneous scale length ) ： Coupling equations of weak magnetic field Huazhong University of Science and Technology

9/18 Common formulation ： Intermediate variables ： Coupling equations of weak magnetic field Huazhong University of Science and Technology

Growth of absolute instability 10/18 Assuming the field amplitude vary with time as e Γ t and the spatial variation is ignored: Without the magnetic field [4] [4] K. Baumgartel, K. Sauer, Topics on Nonlinear Wave – Plasma Interaction (Birkhauser Verlag, Basel, 1987), Chap. 7, p.81. Huazhong University of Science and Technology

Comparison with the numerical results 11/18 [5] H. C. Barr, et al., Phys. Fluids (1984). First order approximation, 8 order matrix equation of dispersion, without analytic solution, the growth is given by numerical result. Both of the plasma density and magnetic field are uniform Huazhong University of Science and Technology

12/18 Critical density and frequency of pump wave: n c =2.84×10 27 m -3 and =3.0×10 15 rad/sec, plasma frequency, oscillation speed, thermal speed and the wavenumber of perpendicular direction:,V osc =V e =0.1c,k y =0.2,respectively. Huazhong University of Science and Technology Comparison with the numerical results

13/18 Comparison with the numerical results Huazhong University of Science and Technology Critical density and frequency of pump wave: n c =2.84×10 27 m -3 and =3.0×10 15 rad/sec, plasma frequency, oscillation speed, thermal speed and the wavenumber of perpendicular direction :, V osc =V e =0.1c,k y =0.4,respectively.

Convective Amplification 14/18 Laplace Transform where Eliminating, and multiplying the conjugate of on both sides Huazhong University of Science and Technology

15/18 Solution of the above equation: Solving A(x), and extending the interaction domains to the infinite According to the linear dispersion ： Convective Amplification Huazhong University of Science and Technology

Amplification factor and threshold of convection 16/18 Substituting into A(x) ， the amplification factor is Rosenbluth criterion[6] ， A(x)>1 [6]M.N. Rosenbluth, Phys. Rev. Lett (1972). Huazhong University of Science and Technology

Summary 17/18 Summary ： Piliya–Rosenbluth coefficient, the enhancement of absolute instability should be satisfied Threshold of the absolute instability  Analytical expressions of the absolute growth rate, convective growth and their threshold intensity have been obtained.  The behavior of the absolute growth rate of two-plasmon is more complex with the presence of magnetic.  The threshold of absolute instability decreases with the increasing of magnetic field and the inhomogeneity scale length of magnetic field.  The growth of convective instability decreases with the increasing of magnetic field, while the threshold is just opposite. Huazhong University of Science and Technology

Future works 18/18  According to the analytic solutions and the numerical results, the physics mechanism of the growth and saturation of two-palsmon decay instabilities will be further studied. And the analytic solution under the strong magnetic field will also be given.  The two-plasmon decay instability will be evaluated by numerical method according to the TEXTOR discharge parameters of ECRH experiment.  Kinetic or Fluid-Kinetic method will be used to study the effects of magnetic field on two-plasmon decay instability[7], which should be compared with the fluid model. [7] A. C. Machacek and J. S. Wark, Phys. Plasmas (2001). Huazhong University of Science and Technology

2015/03/25 Huazhong University of Science and Technology