Calculations of the viscosity expressions appearing in the fluid approach to neoclassical transport of nonaxisymmetric toroidal plasmas are presented. By assuming the Chew–Goldberger–Low form for the viscosity tensor and a large-aspect-ratio plasma, the drift kinetic equation for the plateau regime is solved and expressions for the parallel and toroidal viscosity are calculated as functions of plasma and heat fluxes. The viscosity expressions found for the plateau regime are similar to the expressions valid in the collision-dominated regime.
In this paper, the fluid equation approach is used to analyze the time evolution of the plasma rotation and the ambipolar electric field in a nonsymmetric toroidal plasma subject to an external biasing voltage induced by a probe. Under consideration is a plasma with low rotation speed in the Pfirsch–Schlüter or the plateau regime that includes the effects of a background neutral gas. A time-dependent charge conservation equation is used to determine the ambipolar electric field as a function of time. It is found that, after the application of the biasing voltage, the electric field and the plasma rotation change quickly and reach steady-state after a time inversely proportional to the sum of the momentum damping rates due to parallel viscosity and ion–neutral collisions. The steady state is characterized by a radial electric field and a plasma rotation that are proportional to the electric current flowing through the biasing probe. The direction of the plasma flow is determined by the relative magnitude of the momentum damping rates on the flux surface. From the steady-state solution, an expression for the radial electric conductivity is obtained, which includes the effect of collisions with neutrals as well as viscosity. Axisymmetric systems without neutrals are also discussed, which is a special case since there is no momentum damping in the toroidal direction. Here, the toroidal velocity increases continuously in time with the bias and never reaches steady state. Finally, a model for nonsymmetric magnetic fields is presented and the viscous damping rate, the radial conductivity and the spin-up rate for a plasma in the Pfirsch–Schlüter regime are calculated. As examples, the cases of the rippled tokamak and the classical and helically symmetric stellarators are evaluated.
The objectives of this study were to analyze the injuries suffered during the previous year by amateur padel players according to the characteristics of the racket, their usual volume of practice and their experience in padel. A total of 950 amateur players (X age: 31.68 years; X weight: 70.84 kg; X height: 170.9 cm) participated voluntarily, completing an ad-hoc questionnaire. The results indicated that the appearance of the injuries and their location was different according to the sex of the amateur padel players. Men had a higher incidence of muscle and ligament injuries in the shoulder, and tendon injuries in the elbow. On the other hand, women had a greater probability of having muscle injuries in the shoulder and arm, ligament injuries in the elbow and bone injuries in the wrist and elbow. In general, tendon injuries were the most common injury in padel and the shoulder and elbow were the most affected areas. Moreover, men tend to use heavy (CSR = 6.0), fiberglass or carbon (CSR = 2.1), diamond-shaped rackets (CSR = 3.2), with a hard core (CSR = 4.4) and with two or more over grips (CSR = 2.7). Women usually use less heavy (CSR = 6.0), round-shaped rackets (CSR = 4.9), with a soft core (CSR = 4.4) and with one or no over grips (CSR = 2.7). In addition, men tend to play padel more often and have been practicing for longer. In conclusion, although the risk of injury depends on many factors, we identified that the characteristics of the racket, the volume of weekly practice, the experience of the player and the gender of the player are fundamental aspects to take into account for the prevention of injuries in amateur padel players.
The application of the well-known moment equation approach to neoclassical transport requires the use of Hamada coordinates in three-dimensional toroidal plasmas. The lack of analytical expressions and the difficulty associated with the numerical calculation of these coordinates has strongly limited the use of this approach. In this paper analytical calculations of Hamada coordinates for a large-aspect-ratio tokamak are presented. Knowledge of these coordinates for this relatively simple two-dimensional case will allow, for the first time, a general analytical application of the moment equation approach, and also a check of this approach against well-known results. In particular, it is shown here that the approach correctly reproduces the Pfirsch–Schlüter diffusion rate.
The fluid description of neoclassical transport theory is applied to a general nonaxisymmetric toroidal multispecies plasma with sources. From moment equations, expressions for plasma rotation, electric current, radial heat and particle transport fluxes, and ambipolar electric field are obtained. These expressions are valid in both the plateau and collision-dominated regimes. The effect of momentum and energy flux sources on particle and heat flows, electric current, radial transport fluxes, and on the ambipolar electric field is explicitly shown. In these calculations radial temperature gradients are taken into account.
In this paper the diffusion of guiding centers induced by stochastic magnetic and electric field fluctuations, with both time and space dependence, is analyzed for the case of tokamak plasmas. General experimental results on tokamak fluctuations are used to derive guidingcenter equations that properly describe the particle motion. These equations assume uniform average magnetic and electric fields with random stationary Gaussian fluctuations that constitute a homogeneous and cylindrically symmetric turbulence. By applying Novikov's theorem, a Fokker-Planck equation for the probability distribution function is derived and an expression for the guiding-center diffusion coefficient is obtained. This coefficient not only contains the standard terms due to the stochastic wandering of the magnetic lines and the stochastic electric drift, but also new terms due to the stochastic curvature and VB drifts. The form of these terms is shown explicitly in terms of the correlation functions of the fields.
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