Nonlinear evolution of two fast-particle-driven modes near the linear stability threshold

A system of two coupled integro-differential equations is derived and solved for the non-linear evolution of two waves excited by the resonant interaction with fast ions just above the linear instability threshold. The effects of a resonant particle source and classical relaxation processes represented by the Krook, diffusion, and dynamical friction collision operators are included in the model, which exhibits different nonlinear evolution regimes, mainly depending on the type of relaxation process that restores the unstable distribution function of fast ions. When the Krook collisions or diffusion dominate, the wave amplitude evolution is characterized by modulation and saturation. However, when the dynamical friction dominates, the wave amplitude is in the explosive regime. In addition, it is found that the finite separation in the phase velocities of the two modes weakens the interaction strength between the modes

Gaussian Beam Diffraction and Self-Focusing in Weakly Anisotropic and Dissipative Nonlinear Plasma

The paper presents a simple and effective method to calculate polarization and diffraction of the Gaussian beam in nonlinear and weakly dissipative plasma. The presented approach is a combination of quasi-isotropic approximation of geometric optics with complex geometrical optics. Quasi-isotropic approximation describes the evolution of polarization vector reducing the Maxwell equations to coupled ordinary differential equations of the first order for the transverse components of the electromagnetic field. Complex geometrical optics describes the Gaussian beam diffraction and self-focusing and deals with ordinary differential equations for Gaussian beam width, wave front curvature, and amplitude evolution. As a result, the quasi-isotropic approximation + complex geometrical optics combination reduces the problem of diffraction and polarization evolution of an electromagnetic beam to the solution of the ordinary differential equations, which enable to prepare fast and effective numerical algorithms. Using combined complex geometrical optics/quasi-isotropic approximation for weakly anisotropic plasma, we assume that nonlinearity of anisotropy tensor is small and we restrict ourselves to considering only isotropic nonlinearity. The quasi-isotropic approximation effectively describes the evolution of microwave and IR electromagnetic beams in polarimetric and interferometric measurements in thermonuclear reactors and the complex geometrical optics method can be applied for modeling of electron cyclotron absorption and current drive in tokamaks

Gaussian Beam Diffraction and Self-Focusing in Weakly Anisotropic and Dissipative Nonlinear Plasma

The paper presents a simple and effective method to calculate polarization and diffraction of the Gaussian beam in nonlinear and weakly dissipative plasma. The presented approach is a combination of quasi-isotropic approximation of geometric optics with complex geometrical optics. Quasi-isotropic approximation describes the evolution of polarization vector reducing the Maxwell equations to coupled ordinary differential equations of the first order for the transverse components of the electromagnetic field. Complex geometrical optics describes the Gaussian beam diffraction and self-focusing and deals with ordinary differential equations for Gaussian beam width, wave front curvature, and amplitude evolution. As a result, the quasi-isotropic approximation + complex geometrical optics combination reduces the problem of diffraction and polarization evolution of an electromagnetic beam to the solution of the ordinary differential equations, which enable to prepare fast and effective numerical algorithms. Using combined complex geometrical optics/quasi-isotropic approximation for weakly anisotropic plasma, we assume that nonlinearity of anisotropy tensor is small and we restrict ourselves to considering only isotropic nonlinearity. The quasi-isotropic approximation effectively describes the evolution of microwave and IR electromagnetic beams in polarimetric and interferometric measurements in thermonuclear reactors and the complex geometrical optics method can be applied for modeling of electron cyclotron absorption and current drive in tokamaks

Gaussian beam evolution in inhomogeneous nonlinear media with absorption

The method of complex geometrical optics (CGO) is presented, which describes Gaussian beam (GB) diffraction and self−fo− cusing in smoothly inhomogeneous and nonlinear Kerr type and saturable fibres. CGO reduces the problem of Gaussian beam evolution in inhomogeneous and nonlinear media to the system of the first order ordinary differential equations for the complex curvature of the wave front and for GB amplitude, which can be readily solved both analytically and numerically. As a result, CGO radically simplifies the description of Gaussian beam diffraction and self−focusing effects as compared to other methods of nonlinear optics such as variational method approach, method of moments and beam propagation method. The power of CGO method is presented on the example of Gaussian beam propagation in saturable fibres with either focu− sing and defocusing refractive profiles. Besides, the influence of initial curvature of the wave front, phenomenon of weak ab− sorption and effect of either transverse and longitudinal inhomogeneity on GB propagation in nonlinear fibres is discussed in this paper

Gaussian Beam Diffraction in Inhomogeneous and Nonlinear Saturable Media

The method of complex geometrical optics is presented, which describes Gaussian beam diffraction and self-focusing in smoothly inhomogeneous and nonlinear saturable media of cylindrical symmetry. Complex geometrical optics reduces the problem of Gaussian beam diffraction and self-focusing in inhomogeneous and nonlinear media to the system of the first order ordinary differential equations for the complex curvature of the wave front and for Gaussian beam amplitude, which can be readily solved both analytically and numerically. As a result, complex geometrical optics radically simplifies the description of Gaussian beam diffraction and self-focusing effects as compared to the other methods of nonlinear optics such as: variational method approach, method of moments, and beam propagation method. The power of complex geometrical optics method is presented on the example of Gaussian beam width evolution in saturable fibre with either focusing and defocusing refractive profiles. Besides, the influence of initial curvature of the wave front on Gaussian beam evolution in nonlinear saturable medium is discussed in this paper

On the wave amplitude blow-up in the Berk–Breizman model for nonlinear evolution of a plasma wave driven resonantly by fast ions

In this paper the Berk–Breizman (BB) model of plasma wave instability arising on the stability threshold is considered. An interesting although physically unacceptable feature of the model is the explosive behaviour occurring in the regime of small values of the collision frequency parameter. We present an analytical description of the explosive solution, based on a fitting to the numerical solution of the BB equation with the collision parameter equal to zero. We find that the chaotic behaviour taking place for small but non-zero values of the collision parameter is absent in this case; therefore, chaotic behaviour seems to be an independent phenomenon not directly related to the blow-up regime. The time and the velocity dependence of the distribution function are found numerically and plotted to better understand what actually happens in the model. It allows us to obtain a good qualitative understanding of the time evolution of the mode amplitude including the linear growth of the amplitude, reaching its maximum and then decreasing towards the zero value. Nevertheless, we have no satisfactory physical explanation of the amplitude evolution when the amplitude vanishes at some time and then revives but with an opposite phase

Simplified models for the nonlinear evolution of two fast-particle-driven modes near the linear stability threshold

An analytical model that is based on purely differential equations of the nonlinear dynamics of two plasma modes driven resonantly by high-energy ions near the instability threshold is presented here. The well-known integro-differential model of Berk and Breizman (BB) extended to the case of two plasma modes is simplified here to a system of two coupled nonlinear differential equations of fifth order. The effects of the Krook, diffusion and dynamical friction (drag) relaxation processes are considered, whereas shifts in frequency and wavenumber between the modes are neglected. In spite of these simplifications the main features of the dynamics of the two plasma modes are retained. The numerical solutions to the model equations show competition between the two modes for survival, oscillations, chaotic regimes and \u27blow-up\u27 behavior, similar to the BB model

Efficacy of fungicides and essential oils against bacterial diseases of fruit trees

In the framework of the performed studies, the antibacterial activity of the following fungicides was evaluated: Miedzian 50 WG (active substance - a.s. 50% copper oxychloride), Ridomil MZ Gold 68 WG (a.s. 3.8% metalaxyl-M and 64%, mancozeb), Euparen Multi 50 WG (a.s. 50% tolylfluanid), Captan 80 WG [a.s. 80% N-(captan)], Dithane Neotec 75 WG (a.s. 75% mancozeb). The evaluation also concerned the essential oils: lavender, sage, lemon balm, clove, and a preparation based on thyme oil (BioZell). Each preparation and compound was tested against the following bacterial pathogens: Erwinia amylovora, Xanthomonas arboricola pv. corylina, X. arboricola pv. juglandis, Pseudomonas syringae pv. syringae, Agrobacterium tumefaciens (presently Rhizobium radiobacter). Each preparation and compound was tested at a concentration of 1,000 ppm of active substance. Copper oxychloride was also tested at a concentration of 1,500 ppm. Among the tested fungicides, metalaxyl-M with mancozeb, mancozeb alone, and copper oxychloride inhibited all of the tested strains of pathogenic bacteria. Tolylfluanid did not inhibit any of the bacteria used. Out of the investigated essential oils, the strongest inhibitors of bacteria were: sage, cloves, and BioZell. The protective activity of the above mentioned fungicides was also evaluated in vivo. They were assessed against fire blight on apple blossoms and pear fruitlets, against bacterial canker on sweet cherry fruitlets, and against crown gall on sunflower seedlings (the test plant). All fungicides were applied at the same concentrations as those in the in vitro tests. Only copper oxychloride was found to show protective activity against the studied diseases. This result indicates that the antibacterial properties of the other fungicides did not correspond with their activity on the plant organs used in the in vivo experiment