Simulation of START shots with the canonical profile transport model

The canonical profile transport model, which has been benchmarked previously for tokamaks with a conventional aspect ratio, is applied to simulations of the spherical tokamak START. A set of Ohmic shots is used to modify the model so that it is appropriate for the specific conditions of the spherical tokamak plasma. The application of the model as a tool to analyze neutral beam-heated START shots allows the estimation of the neutral beam-injection power absorbed by the plasma, P _NB ^abs , which is experimentally uncertain. The modeling shows that both P _NB ^abs and the energy confinement time increase with increasing the average density. Finally, the modified model is used to simulate the performance of the new megaampere spherical tokamak MAST at Culham.     

Application of the canonical profile theory to the problems of heat transport in tokamaks

Abstrac A transport model for describing electron and ion plasmatemperatures is developed on the basis of the canonical profile theory for a tokamak with an arbitrary cross section. A comparison with the data from experiments on eight different tokamaks shows that the model is capable of adequately simulating plasma discharges. A scaling for the behavior of the relative temperature gradient at half the plasma minor radius is constructed based on both an analysis of the experimental data and the results of numerical calculations. __________ Translated from Fizika Plazmy, Vol. 30, No. 1, 2004, pp. 3–10. Original Russian Text Copyright ? 2004 by Dnestrovskij, Dnestrovskij, Lysenko, Cherkasov, Walsh.     

Simulation of internal transport barriers by means of the canonical profile transport model

Models with critical gradients are widely used to describe energy balance in L-mode discharges. The so-called first critical gradient can be found from the canonical temperature profile. Here, it is suggested that discharge regimes with transport barriers can be described based on the idea of the second critical gradient. If, in a certain plasma region, the pressure gradient exceeds the second critical gradient, then the plasma bifurcates into a new state and a transport barrier forms in this region. This idea was implemented in a modified canonical profile transport model that makes it possible to describe the energy and particle balance in tokamak plasmas with arbitrary cross sections and aspect ratios. The magnitude of the second critical gradient was chosen by comparing the results calculated for several tokamak discharges with the experimental data. It is found that the second critical gradient is related to the magnetic shear s. The criterion of the transport barrier formation has the form (a ²/r)d/drln(p/p _c ) > z ₀ (r), where r is the radial coordinate, a is the plasma minor radius, p is the plasma pressure, p _c is the canonical pressure profile, and the dimensionless function z _O(r) = C _O + C ₁ s (with C _0i ～1, C _0e ～3, and C _1i,e ～2) describes the difference between the first and second critical gradients. Simulations show that this criterion is close to that obtained experimentally in JET. The model constructed here is used to simulate internal transport barriers in the JET, TFTR, DIII-D, and MAST tokamaks. The possible dependence of the second critical gradient on the plasma parameters is discussed.     

Variational problems for the canonical profiles

A noncontradictory formulation of the variational problem for a canonical profile is proposed that refines the problem posed by B.B. Kadomtsev for a circular plasma cylinder. The results are generalized to a toroidal plasma with an arbitrary cross section. For the problem in toroidal geometry, boundary conditions are proposed with which to single out the Kadomtsev-like solution (the canonical profile) from the solutions to the Euler equation. Canonical profiles for the L-and H-modes are constructed. For a number of interesting examples, it is numerically shown that the second variation of the magnetic energy functional is positive. The canonical profile transport model is outlined, and the relationship between the canonical, numerical, and experimental profiles in tokamaks is briefly discussed.     

Modification of the canonical profile transport model on the basis of new DIII-D experiments

Recent DIII-D experiments have shown that the stiffness of the ion temperature profile κ _i ^PC in the region 0.4 < ρ < 0.7 increases by one order of magnitude with increasing radius. At ρ < 0.4, the stiffness is low and the ion temperature profile is “soft.” The stiffness of the temperature profile also increases with decreasing the toroidal rotation velocity. The approximation of the experimental stiffness profiles allows one to modify the canonical profile transport model. The heat conductivity κ _i ⁰ in the plasma core is determined by minimizing the r.m.s. deviations of the calculated ion temperature from the measured one. This procedure also makes it possible to determine how κ _i ⁰ depends on the central ion temperature.     

An analytic approach to developing transport threshold models of neoclassical tearing modes in tokamaks

Transport threshold models of neoclassical tearing modes in tokamaks are investigated analytically. An analysis is made of the competition between strong transverse heat transport, on the one hand, and longitudinal heat transport, longitudinal heat convection, longitudinal inertial transport, and rotational transport, on the other hand, which leads to the establishment of the perturbed temperature profile in magnetic islands. It is shown that, in all these cases, the temperature profile can be found analytically by using rigorous solutions to the heat conduction equation in the near and far regions of a chain of magnetic islands and then by matching these solutions. Analytic expressions for the temperature profile are used to calculate the contribution of the bootstrap current to the generalized Rutherford equation for the island width evolution with the aim of constructing particular transport threshold models of neoclassical tearing modes. Four transport threshold models, differing in the underlying competing mechanisms, are analyzed: collisional, convective, inertial, and rotational models. The collisional model constructed analytically is shown to coincide exactly with that calculated numerically; the reason is that the analytical temperature profile turns out to be the same as the numerical profile. The results obtained can be useful in developing the next generation of general threshold models. The first steps toward such models have already been made.     

Review of mixing length estimates and effects of toroidicity in a fluid model for turbulent transport in tokamaks

Basic aspects of turbulent transport in toroidal magnetized plasmas are discussed. In particular the fluid closure has strong effects on zonal flows which are needed to create an absorbing boundary for long wave lengths and also to obtain the Dimits nonlinear upshift. The fluid resonance in the energy equation is found to be instrumental for generating the L–H transition, the spin-up of poloidal rotation in internal transport barriers, as well as the nonlinear Dimits upshift. The difference between the linearly fastest growing mode number and the corresponding longer nonlinear correlation length is also addressed. It is found that the Kadomtsev mixing length result is consistent with the non-Markovian diagonal limit of the transport at the nonlinearly obtained correlation length.     

Pellet injection into H-mode ITER plasma with the presence of internal transport barriers

The impacts of pellet injection into ITER type-1 ELMy H-mode plasma with the presence of internal transport barriers (ITBs) are investigated using self-consistent core-edge simulations of 1.5D BALDUR integrated predictive modeling code. In these simulations, the plasma core transport is predicted using a combination of a semi-empirical Mixed B/gB anomalous transport model, which can self-consistently predict the formation of ITBs, and the NCLASS neoclassical model. For simplicity, it is assumed that toroidal velocity for ω _E×B calculation is proportional to local ion temperature. In addition, the boundary conditions are predicted using the pedestal temperature model based on magnetic and flow shear stabilization width scaling; while the density of each plasma species, including both hydrogenic and impurity species, at the boundary are assumed to be a large fraction of its line averaged density. For the pellet’s behaviors in the hot plasma, the Neutral Gas Shielding (NGS) model by Milora-Foster is used. It was found that the injection of pellet could result in further improvement of fusion performance from that of the formation of ITB. However, the impact of pellet injection is quite complicated. It is also found that the pellets cannot penetrate into a deep core of the plasma. The injection of the pellet results in a formation of density peak in the region close to the plasma edge. The injection of pellet can result in an improved nuclear fusion performance depending on the properties of pellet (i.e., increase up to 5% with a speed of 1 km/s and radius of 2 mm). A sensitivity analysis is carried out to determine the impact of pellet parameters, which are: the pellet radius, the pellet velocity, and the frequency of injection. The increase in the pellet radius and frequency were found to greatly improve the performance and effectiveness of fuelling. However, changing the velocity is observed to exert small impact.     

Formation of an internal transport barrier in the ohmic H-mode in the TUMAN-3M tokamak

In experiments on studying the ohmic H-mode in the TUMAN-3M tokamak, it is found that, in high-current (I _p～120–170 kA) discharges, a region with high electron-temperature and density gradients is formed in the plasma core. In this case, the energy confinement time τ _E attains 9–18 ms, which is nearly twice as large as that predicted by the ELM-free ITER-93H scaling. This is evidence that the internal transport barrier in a plasma can exist without auxiliary heating. Calculations of the effective thermal diffusivity by the ASTRA transport code demonstrate a strong suppression of heat transport in the region where the temperature and density gradients are high.     

Application of a fiber-matrix model to transport in renal tubules

The effects of tight junction structure on water and solute fluxes across proximal tubular epithelium were examined with fiber-matrix equations previously derived by Curry and Michel (1980. Microvascular Research. 20:96-99). Using plausible estimates of tight junction fiber length and width the model predicts solute (Ps) and water permeability (Lp) coefficients that agree with the measured values. When fiber-matrix and pore models were compared for physiologically relevant ranges of matrix void fraction (80-98%) and pore radii (0-20 A), the fiber-matrix model predicted a 10-fold higher Lp/Ps ratio. Lp/Ps was most sensitive to small changes in tight junction structure when void fractions exceeded 90%. Void fractions of 96.5% and 97.1% predicted previously measured values for Lp and solute permeabilities in rat and rabbit proximal tubules. These values are consistent with void fractions and permeabilities of artificial membranes. The fiber-matrix tight junction model was incorporated into a model of reabsorption from the rat proximal tubule developed by Weinstein (1984). American Journal of Physiology. 247:F848-F862.) A void fraction of 98% predicted the experimental results for isosmotic reabsorption driven by active transport. Changing void fraction over the range of 97-99% produced a 50-75% change in predicted volume reabsorption with active transport. According to the fiber-matrix model: (a) solute permeabilities alone cannot be used to predict Lp, (b) previously measured solute permeabilities in the proximal tubule are compatible with significant water reabsorption through a water-permeable tight junction, and (c) hydraulic and solute permeabilities may be sensitive to small changes in tight junction fiber length and diameter or ionic strength within the tight junction.     

On the minimum electron transport coefficients in tokamaks in the range of low collision frequencies

There are two close empirical scalings, namely, the T-11 and neo-Alcator ones, that provide correct estimates for the energy confinement time in tokamaks in ohmic heating regimes in the linear part of the dependence τ _E (\(\bar n_e \)) in the range of low values of \(\bar n_e \) and 〈ν _e ^* 〉 ≤ 1. The similar character of electron energy confinement in this range, which expands with increasing magnetic field B ₀, has stimulated the search for dimensionless parameters and simple physical models that would explain the experimentally observed dependences χ _e ～ 1/n _e and τ _Ee ～ \(\bar n_e \). In 1987, T. Okhawa showed that the experimental data were satisfactorily described by the formula χ_e⊥ = (c ²/ω _pe ² )ν _e /qR, in deriving of which the random spatial leap along the radius r on the electron trajectory was assumed to be the same as that in the coefficient of the poloidal field diffusion, while the repetition rate of these leaps was assumed to be ν _e /qR. In 2004, J. Callen took into account the decrease in the fraction of transient electrons with increasing toroidal ratio ? = r/R and corrected the coefficient c ²/ω _pe ² in Okhawa equation by the factor σ _‖ ^Sp /σ _‖ ^neo . If one takes into account this correction and assumes that the frequency of the stochastic process is equal to the reciprocal of the half-period of rotation of a trapped electron along its banana trajectory, then the resulting expression for χ_e⊥ will coincide with the T-11 scaling: χ _e ^an ∞ ?^1.75(T _e /A _i )^0.5/(n _e qR) at A i = 1. If the same stochastic process also involves ions, it may result in the opening of the orbit of a trapped ion at the distance ～(c/ω _pe )(m _i /m _e )^1/4. In this case, the calculated coefficient of electron and ion diffusion D is close to D ^an ≈ χ _e ^an /2.     

Accuracy of reconstruction of the transport coefficients and ECRH power profile in tokamaks from solutions to inverse problems

Experiments on ECR plasma heating provide a powerful method for studying electron transport in tokamak plasmas and reconstructing the transport coefficients. An analysis of such experiments requires, however, reliable knowledge of the profile of the ECRH power and the region where it is deposited. In the present paper, the transport coefficients and ECRH power profile are reconstructed from the solutions to inverse problems by analyzing the transient process after ECR heating is switched on or off. In order to calculate the plasma parameters with this approach, it is necessary to know how the accuracy of the solutions to inverse problems depends on the errors in the input parameters. The accuracy of reconstruction of the ECRH power profile and transport coefficients in model problems is investigated as a function of errors in measuring the electron temperature. The results of this investigation are used to analyze experiments with ECR plasma heating in the T-10 tokamak and to estimate the accuracy of the results obtained.     

Application of a single-solute non-steady-state phloem model to the study of long-distance assimilate transport

A mass-balanced, finite-difference solution to Münch's osmotically generated pressure-flow hypothesis is developed for the study of non-steady-state sucrose transport in the phloem tissue of plants. Major improvements over previous modeling efforts are the inclusion of wall elasticity, nonlinear functions of viscosity and solute potential, an enhanced calculation of sieve pore resistance, and the introduction of a slope-limiting total variation diminishing method for determining the concentration of sucrose at node boundaries. The numerical properties of the model are discussed, as is the history of the modeling of pressure-driven phloem transport. Idealized results are presented for a sharp, fast-moving concentration front, and the effect of changing sieve tube length on the transport of sucrose in both the steady-state and non-steady-state cases is examined. Most of the resistance to transport is found to be axial, rather than radial (via membrane transport), and most of the axial resistance is due to the sieve plates. Because of the sieve plates, sieve tube elasticity does not provide a significant enhancement to conductivity at high pressure, as previously suspected. The transit time of sucrose through a sieve tube is found to be inversely proportional to the square of the sieve tube's length; following that observation, it is suggested that 20 1-m-long sieve tubes could transport sucrose 20 times faster than a single 20 m sieve tube. Short sieve tubes would be highly sensitive to differentials between loading and unloading rate, and would require close cooperation with adjacent companion cells for proper function.     

Application of experimental data to a new model of tissue oxygen transport

A recently published equation for oxygen uptake in cells (Hills, 1970) is corrected and applied to data gathered under appropriate experimental conditions. The result is found to support the hypothesis of facilitated intracellular oxygen transport.     

Study of Turbulence in the Globus-M Tokamak Plasma during the Transition to the ELM-free H-mode

Plasma Physics Reports - Here we report the results of the turbulence study in the high-confinement mode (H-mode) with and without edge localized modes (ELMs). The study was performed by the...     

Investigation of the H-mode during ECRH in the T-10 tokamak

An improved confinement regime with an external transport barrier (H-mode) is obtained during electron-cyclotron resonance heating of a plasma in the T-10 tokamak. A characteristic feature of this regime is a spontaneous density growth accompanied by a drop in the intensity of D_α line and an increase in β_p by a factor of ～1.6. The threshold power for the L-H transition is close to that predicted by the ITER scaling. The best characteristics of the H-mode are achieved with decreasing q _L to 2.2. It is shown that the external transport barrier arises for electrons, whereas the heat transport barrier insignificantly contributes to improved confinement.     

K-Na transport: the Brewer model replaces the active transport model

In the 1940's several experimental observations were made regarding the K+ and Na+ content of chilled and restored red blood cells. As a consequence, the concept of active transport was developed. Brewer, a physicist, developed a model for membrane transport based on the electrical properties of double bonds in the ground and excited states. Of particular importance is the membrane double bond P = O. This model was largely formulated from isotope concentration studies using mass spectroscopy, photospectrometry and the nature of malignant cells. In this study, it is shown that the Brewer model completely explains the experimental results which led to the concept of active transport. In addition, it also explains the results of some adjunct experiments.     

A canonical model of multistability and scale-invariance in biological systems

Multistability and scale-invariant fluctuations occur in a wide variety of biological organisms from bacteria to humans as well as financial, chemical and complex physical systems. Multistability refers to noise driven switches between multiple weakly stable states. Scale-invariant fluctuations arise when there is an approximately constant ratio between the mean and standard deviation of a system's fluctuations. Both are an important property of human perception, movement, decision making and computation and they occur together in the human alpha rhythm, imparting it with complex dynamical behavior. Here, we elucidate their fundamental dynamical mechanisms in a canonical model of nonlinear bifurcations under stochastic fluctuations. We find that the co-occurrence of multistability and scale-invariant fluctuations mandates two important dynamical properties: Multistability arises in the presence of a subcritical Hopf bifurcation, which generates co-existing attractors, whilst the introduction of multiplicative (state-dependent) noise ensures that as the system jumps between these attractors, fluctuations remain in constant proportion to their mean and their temporal statistics become long-tailed. The simple algebraic construction of this model affords a systematic analysis of the contribution of stochastic and nonlinear processes to cortical rhythms, complementing a recently proposed biophysical model. Similar dynamics also occur in a kinetic model of gene regulation, suggesting universality across a broad class of biological phenomena.