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

Nonlinear Evolution of Plasma Modes Driven by Fast Particles in Tokamaks

In fusion plasmas, high-energy ions arising from plasma heating as wellas being generated in fusion reactions may lead to the occurrence ofwave micro-instabilities. The basic reason for these instabilities is thedeviation of the high-energy ions distribution function from the thermodynamicequilibrium. The presence of thermonuclear instabilities mayin turn cause anomalous losses of plasma energy and high-energy particlesand consequently may have direct impact on the operation scenariosand ignition conditions. Investigations of the initial phase of these instabilitiesare connected with an identification of the stability thresholdwith respect to wave excitation by fast ions as well as with the studyof nonlinear dynamics of the wave - fast ion system above the stabilitythreshold.The theory describing the nonlinear dynamics of a driven mode nearthe marginal stability threshold has been developed by H. Berk andB. Breizmann et al. in the 90’s and was also verefied to some extendin tokamak experiments. This theory is limited to the case of only asingle plasma mode with a fixed wave number. However, in practicemany plasma modes with different wave numbers may be excited in atokamak plasma.In the present thesis, the single mode theory is extended to the caseof two different, linearly unstable plasma modes driven by fast ions at thelinear stability threshold. Futhermore, based on analogy to mechanicalnonlinear systems, the model equations are reduced to a set of differentialequations of the nonlinear oscillator type. Numerical analysis of the twomode model reveals interesting features of the mode amplitude behavior,depending on the effect of classical relaxation processes represented bythe Krook, diffusive, and dynamical friction collision operators

Nonlinear Evolution of Plasma Modes Driven by Fast Particles in Tokamaks

In fusion plasmas, high-energy ions arising from plasma heating as wellas being generated in fusion reactions may lead to the occurrence ofwave micro-instabilities. The basic reason for these instabilities is thedeviation of the high-energy ions distribution function from the thermodynamicequilibrium. The presence of thermonuclear instabilities mayin turn cause anomalous losses of plasma energy and high-energy particlesand consequently may have direct impact on the operation scenariosand ignition conditions. Investigations of the initial phase of these instabilitiesare connected with an identification of the stability thresholdwith respect to wave excitation by fast ions as well as with the studyof nonlinear dynamics of the wave - fast ion system above the stabilitythreshold.The theory describing the nonlinear dynamics of a driven mode nearthe marginal stability threshold has been developed by H. Berk andB. Breizmann et al. in the 90’s and was also verefied to some extendin tokamak experiments. This theory is limited to the case of only asingle plasma mode with a fixed wave number. However, in practicemany plasma modes with different wave numbers may be excited in atokamak plasma.In the present thesis, the single mode theory is extended to the caseof two different, linearly unstable plasma modes driven by fast ions at thelinear stability threshold. Futhermore, based on analogy to mechanicalnonlinear systems, the model equations are reduced to a set of differentialequations of the nonlinear oscillator type. Numerical analysis of the twomode model reveals interesting features of the mode amplitude behavior,depending on the effect of classical relaxation processes represented bythe Krook, diffusive, and dynamical friction collision operators

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

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

Egzamin ósmoklasisty. Vademecum nauczyciela. Języki obce

Vademecum ósmoklasisty zawiera: a. podstawę programową do szkoły podstawowej wraz z komentarzem, b. założenia egzaminu ósmoklasisty wraz z przykładami zadań egzaminacyjnych, c. wybrane zagadnienia, ważne w procesie kształcenia oraz obecne w arkuszu egzaminacyjnym, wraz z propozycją rozwiązań metodycznych. Przygotowany materiał ma wspierać nauczycieli w pracy w zreformowanej szkole. Podpowiada rozwiązania metodyczne i mamy nadzieję okaże się ciekawym, inspirującym i pomocnym poradnikiem w pracy dydaktycznej

Evaluation of the rectal V30 parameter in patients diagnosed with postoperative endometrial cancer

Background: The present paper reports on analysis of 184 patients who were diagnosed with endometrial cancer. The main objective of this study was to address parameter Vrec(30Gy) which determines a volume of the rectum irradiated with a dose of 30 Gy during radiotherapy. Materials and methods: All patients were irradiated with an IMRT technique on linear accelerators. The planning target volume (PTV) contour was determined by a radiation oncologist. The clinical target volume (CTV) was drawn on CT images obtained in a prone position. For statistical analysis, appropriate tests (e.g. the Shapiro-Wilk, Wilcoxon) were used. Results and discussion: The performed analysis showed that the recommended condition for Vrec(30Gy) is met only in 3% of patients and the observed median value exceeds 90%. The obtained results were compared with the studies in which the Vrec(30Gy) values were related to various radiotherapy techniques. Conclusions: The analysis showed that the condition for Vrec(30Gy) is satisfied in the case of only 3% of patients. Due to the difficulty with meeting the condition, it should be reconsidered based on real results.

Egzamin ósmoklasisty. Ramowy program szkoleń nauczycieli. Język polski, matematyka, języki obce

Publikacja ta zawiera scenariusze zajęć wraz z kartami pracy, z których • pierwszy scenariusz przybliża założenia podstawy programowej z języka polskiego lub matematyki albo języków obcych nowożytnych, koncepcję egzaminu z danego przedmiotu, strukturę arkusza egzaminacyjnego oraz typy zadań • pozostałe scenariusze z danego przedmiotu odnoszą się do wybranych zagadnień ważnych z perspektywy podstawy programowej oraz egzaminu ósmoklasisty