The simple map for a single-null divertor tokamak

1996 ◽  
Vol 56 (3) ◽  
pp. 569-603 ◽  
Author(s):  
Alkes Punjabi ◽  
Arun Verma ◽  
Allen Boozer
Keyword(s):  
Safety Factor ◽  
Single Degree Of Freedom ◽  
Divertor Plate ◽  
Divertor Tokamak ◽  
Magnetic Perturbations ◽  
Stochastic Layer ◽  
Field Lines ◽  
Basic Features ◽  
Area Preserving

We present the simple map for a single-null divertor tokamak. The simple map is an area-preserving map based on the idea that magnetic field lines are a single-degree- of-freedom time-dependent Hamiltonian system, and that the basic features of such systems near the X-point are generic. We obtain the properties of this map and the resulting footprints of field lines on the divertor plate. These include the width of the stochastic layer, the edge safety factor, the area of the footprint and the amount of magnetic flux diverted. We give the safety factor profile, the average and median values of strike angles, lengths and the Liapunov exponents. We describe how the effects of magnetic perturbations can be included in the simple map. We show how the map can be applied to the problem of the determination of heat flux on the divertor plate in tokamaks.

scholarly journals Determination of the stochastic layer properties induced by magnetic perturbations via heat pulse experiments at ASDEX upgrade

2017 ◽  
Vol 12 ◽  
pp. 831-837 ◽  
Author(s):  
D. Brida ◽  
T. Lunt ◽  
M. Wischmeier ◽  
G. Birkenmeier ◽  
P. Cahyna ◽  
...  
Keyword(s):  
Heat Pulse ◽  
Magnetic Perturbations ◽  
Stochastic Layer

Stochastic layer scaling in the two-wire model for divertor tokamaks

2009 ◽  
Vol 75 (3) ◽  
pp. 303-318 ◽  
Author(s):  
HALIMA ALI ◽  
ALKESH PUNJABI ◽  
ALLEN BOOZER
Keyword(s):  
Hamiltonian Structure ◽  
Mode Number ◽  
Arnold Diffusion ◽  
Step Size ◽  
Layer Width ◽  
Area Preserving Maps ◽  
Stochastic Layer ◽  
Mathematical Techniques ◽  
Field Lines ◽  
Area Preserving

AbstractThe question of magnetic field structure in the vicinity of the separatrix in divertor tokamaks is studied. The authors have investigated this problem earlier in a series of papers, using various mathematical techniques. In the present paper, the two-wire model (TWM) [Reiman, A. 1996 Phys. Plasmas3, 906] is considered. It is noted that, in the TWM, it is useful to consider an extra equation expressing magnetic flux conservation. This equation does not add any more information to the TWM, since the equation is derived from the TWM. This equation is useful for controlling the step size in the numerical integration of the TWM equations. The TWM with the extra equation is called the flux-preserving TWM. Nevertheless, the technique is apparently still plagued by numerical inaccuracies when the perturbation level is low, resulting in an incorrect scaling of the stochastic layer width. The stochastic broadening of the separatrix in the flux-preserving TWM is compared with that in the low mn (poloidal mode number m and toroidal mode number n) map (LMN) [Ali, H., Punjabi, A., Boozer, A. and Evans, T. 2004 Phys. Plasmas11, 1908]. The flux-preserving TWM and LMN both give Boozer–Rechester 0.5 power scaling of the stochastic layer width with the amplitude of magnetic perturbation when the perturbation is sufficiently large [Boozer, A. and Rechester, A. 1978, Phys. Fluids21, 682]. The flux-preserving TWM gives a larger stochastic layer width when the perturbation is low, while the LMN gives correct scaling in the low perturbation region. Area-preserving maps such as the LMN respect the Hamiltonian structure of field line trajectories, and have the added advantage of computational efficiency. Also, for a $1\frac12$ degree of freedom Hamiltonian system such as field lines, maps do not give Arnold diffusion.

scholarly journals Prediction of the sliding type and critical factor of safety in homogeneous finite slopes

2019 ◽  
Vol 9 (7) ◽  
Author(s):  
Farzin Salmasi ◽  
Biswajeet Pradhan ◽  
Bahram Nourani
Keyword(s):  
Safety Factor ◽  
Factor Of Safety ◽  
Critical Factor ◽  
Specific Weight ◽  
Regression Equations ◽  
Plane Failure ◽  
Angle Of Internal Friction ◽  
Modes Of Failure ◽  
Constant Slope

Abstract In this paper, the effect of soil material parameters including soil specific weight (γ), cohesion (C), angle of internal friction ($$\emptyset$$ ∅ ), and geometric parameter of slope including angle with the horizontal (β) for a constant slope height (H) on factor of safety (Fs) was investigated. Fs was considered in two scenarios: (1) slope with dry condition, and (2) with steady-state saturated condition that comprises water level drawdown circumstances. In addition, the type of slip circle was also investigated. For this purpose, the SLOPE/W software as a subgroup of Geo-Studio software was implemented. Results showed that decreasing of water table level and omitting the hydrostatic pressure on the slope consequently would result in safety factor decrement. Comparison of the plane and circular failure surfaces showed that plane failure method produced good results for near-vertical slopes only. Determination of slip type showed that for state (30° < β < 45°), the three types of failure circles (toe, slope or midpoint circle) may occur. For state (45° < β < 60°), two modes of failure may occur: midpoint circle and toe circle. For state (β > 60°), the mode of failure circle is only toe circle. Linear and nonlinear regression equations were obtained for estimation of slope safety factor.

Maximum Amplification of Blade Response in Mistuned Multi Stage Assemblies

2010 ◽  
Author(s):  
Javier Avalos ◽  
Marc P. Mignolet
Keyword(s):  
Mass Matrix ◽  
Optimization Strategy ◽  
Single Degree Of Freedom ◽  
Reduced Order ◽  
Multi Stage ◽  
Maximum Amplification ◽  
Blade Model ◽  
Computational Validation ◽  
Theoretical Developments

This paper focuses on the determination of the maximum amplification of blade response due to mistuning in multi stage assemblies. The modal optimization strategy developed earlier in connection with single stage models is extended here to multi stage configurations. Theoretical developments are carried out first and lead to the new upper limit of (1+N1+N2(g2/g1)+…)/2, where Ni are the number of blades on the stages and gi = FiTMi−1Fi with Fi the force vector applied on a sector of stage i and Mi its mass matrix. For identical stages, this maximum equals the Whitehead limit observed with single stages but with the number of blades equal to sum of the numbers of blades of the coupled stages. A computational validation of the theoretical results is achieved next on both a single degree of freedom per blade model and a reduced order model of a blisk. These numerical optimization efforts confirm the theoretical developments and demonstrate that such high amplification factors can indeed be achieved with small levels of mistuning. The effects of the number of blades on the different stages, damping in the system, stage coupling strength, etc are discussed in details.

scholarly journals Cluster observations of ELF/VLF signals generated by modulated heating of the lower ionosphere with the HAARP HF transmitter

2004 ◽  
Vol 22 (7) ◽  
pp. 2643-2653 ◽  
Author(s):  
M. Platino ◽  
U. S. Inan ◽  
T. F. Bell ◽  
J. Pickett ◽  
E. J. Kennedy ◽  
...  
Keyword(s):  
High Frequency ◽  
Lower Ionosphere ◽  
Spectral Signature ◽  
Closed Field ◽  
Ionospheric Heating ◽  
Higher Power ◽  
Power Operation ◽  
Field Lines ◽  
Vlf Waves

Abstract. It is now well known that amplitude modulated HF transmissions into the ionosphere can be used to generate ELF/VLF signals using the so-called "electrojet antenna". Although most observations of the generated ELF/VLF signals have been made on the ground, several low and high-altitude satellite observations have also been reported (James et al., 1990). One of the important unknowns in the physics of ELF/VLF wave generation by ionospheric heating is the volume of the magnetosphere illuminated by the ELF/VLF waves. In an attempt to investigate this question further, ground-satellite conjunction experiments have recently been conducted using the four Cluster satellites and the HF heater of the High-Frequency Active Auroral Research Program (HAARP) facility in Gakona, Alaska. Being located on largely closed field lines at L≈4.9, HAARP is currently also being used for ground-to-ground type of ELF/VLF wave-injection experiments, and will be increasingly used for this purpose as it is now being upgraded for higher power operation. In this paper, we describe the HAARP installation and present recent results of the HAARP-Cluster experiments. We give an overview of the detected ELF/VLF signals at Cluster, and a possible explanation of the spectral signature detected, as well as the determination of the location of the point of injection of the HAARP ELF/VLF signals into the magnetosphere using ray tracing.

Exact determination of critical damping in multiple exponential kernel-based viscoelastic single-degree-of-freedom systems

2019 ◽  
Vol 24 (12) ◽  
pp. 3843-3861 ◽  
Author(s):  
Mario Lázaro
Keyword(s):  
Past History ◽  
Degree Of Freedom ◽  
Single Degree Of Freedom ◽  
Single Degree ◽  
Critical Curves ◽  
History Of ◽  
Definition Of ◽  
Analytical Expressions ◽  
Critical Damping

In this paper, exact closed forms of critical damping manifolds for multiple-kernel-based nonviscous single-degree-of-freedom oscillators are derived. The dissipative forces are assumed to depend on the past history of the velocity response via hereditary exponential kernels. The damping model depends on several parameters, considered variables in the context of this paper. Those parameter combinations which establish thresholds between induced overdamped and underdamped motion are called critical damping manifolds. If such manifolds are represented on a coordinate plane of two damping parameters, then they are named critical curves, so that overdamped regions are bounded by them. Analytical expressions of critical curves are deduced in parametric form, considering certain local nondimensional parameters based on the Laplace variable in the frequency domain. The definition of the new parameter (called the critical parameter) is supported by several theoretical results. The proposed expressions are validated through numerical examples showing perfect fitting of the determined critical curves and overdamped regions.

scholarly journals Rotation of the magnetic field in Earth's magnetosheath by bulk magnetosheath plasma flow

2006 ◽  
Vol 24 (1) ◽  
pp. 339-354 ◽  
Author(s):  
M. Longmore ◽  
S. J. Schwartz ◽  
E. A. Lucek
Keyword(s):  
Magnetic Field ◽  
Field Line ◽  
Plasma Flow ◽  
Mean Value ◽  
Flow Coupling ◽  
Parker Spiral ◽  
Magnetic Field Model ◽  
Field Lines ◽  
The Magnetic Field

Abstract. Orientations of the observed magnetic field in Earth's dayside magnetosheath are compared with the predicted field line-draping pattern from the Kobel and Flückiger static magnetic field model. A rotation of the overall magnetosheath draping pattern with respect to the model prediction is observed. For an earthward Parker spiral, the sense of the rotation is typically clockwise for northward IMF and anticlockwise for southward IMF. The rotation is consistent with an interpretation which considers the twisting of the magnetic field lines by the bulk plasma flow in the magnetosheath. Histogram distributions describing the differences between the observed and model magnetic field clock angles in the magnetosheath confirm the existence and sense of the rotation. A statistically significant mean value of the IMF rotation in the range 5°-30° is observed in all regions of the magnetosheath, for all IMF directions, although the associated standard deviation implies large uncertainty in the determination of an accurate value for the rotation. We discuss the role of field-flow coupling effects and dayside merging on field line draping in the magnetosheath in view of the evidence presented here and that which has previously been reported by Kaymaz et al. (1992).

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