Magnetohydrodynamic Equilibrium Research Articles

Plasma beta dependence of ion temperature gradient driven turbulence influenced by Shafranov shift

Plasma β dependence of ion temperature gradient (ITG) driven turbulence is investigated using gyrokinetic simulations, where β is the normalized pressure. In our β scan, self-consistent magnetohydrodynamic (MHD) equilibrium state is numerically calculated for each value of β. It is found that the influence of the Shafranov shift cancels out the electromagnetic stabilizing effect on the ITG mode, and the growth rate of the ITG mode is accordingly unchanged as β increases. As a result, the turbulent energy transport does not decrease with β as suggested by the model (Ishizawa et al 2019 Phys. Rev. Lett. 123 025003). A significant difference from the model is the increase of the energy transport with β. It is also found that the critical onset β value for the kinetic ballooning mode is significantly increased by the influence of the Shafranov shift. The cancellation of the electromagnetic stabilization by the Shafranov shift is explained by the decrease of magnetic drift frequency in the dispersion relation of electromagnetic ITG modes obtained by using a fluid approximation.

Structure of pressure-gradient-driven current singularity in ideal magnetohydrodynamic equilibrium

Singular currents typically appear on rational surfaces in non-axisymmetric ideal magnetohydrodynamic (MHD) equilibria with a continuum of nested flux surfaces and a continuous rotational transition. These currents have two components: a surface current (Dirac δ-function in flux surface labeling) that prevents the formation of magnetic islands, and an algebraically divergent Pfirsch–Schlüter current density when a pressure gradient is present across the rational surface. On flux surfaces adjacent to the rational surface, the traditional treatment gives the Pfirsch–Schlüter current density scaling as , where is the difference of the rotational transform relative to the rational surface. If the distance s between flux surfaces is proportional to , the scaling relation will lead to a paradox that the Pfirsch–Schlüter current is not integrable. In this work, we investigate this issue by considering the pressure-gradient-driven singular current in the Hahm–Kulsrud–Taylor problem, which is a prototype for singular currents arising from resonant magnetic perturbations. We show that not only the Pfirsch–Schlüter current density but also the diamagnetic current density are divergent as . However, due to the formation of a Dirac δ-function current sheet at the rational surface, the neighboring flux surfaces are strongly packed with . Consequently, the singular current density , making the total current finite, thus resolving the paradox. Furthermore, the strong packing of flux surfaces causes a steepening of the pressure gradient near the rational surface, with . In general non-axisymmetric MHD equilibrium, contrary to Grad’s conjecture that the pressure profile is flat around densely distributed rational surfaces, our result suggests a pressure profile that densely steepens around them.

Effect of discreteness and misalignment on magnetic field and charged particle confinement in CFQS quasi-axisymmetric stellarator

The Chinese first quasi-axisymmetric stellarator (CFQS), which will be the first quasi-axisymmetric (QA) stellarator in the world, is now under construction. The primary task of the CFQS project is to realize a QA configuration and to examine its physical properties. Based on this task, two important issues were investigated in this work in order to estimate the robustness of the CFQS design from a physical perspective. One was the toroidal field (TF) ripple due to the discreteness of modular coils (MCs) which could potentially degrade the charged particle confinement in the CFQS configuration. The other was a possible MC misalignment in the assembly that would affect the magnetic field and charged particle confinement in the CFQS. Moreover, since the stellarator symmetry might be broken by the MC misalignment, such a case was also investigated in this work. By performing a magnetic field line tracing and charged particle orbit tracing calculation, it was found that the TF ripple does not affect the confinement property significantly and the magnetohydrodynamics equilibrium was robust against possible MC misalignments. These results are helpful in defining the reasonable tolerance of assembly accuracy.

Relationship between drift kinetics, gyrokinetics and magnetohydrodynamics in the long-wavelength limit

Gyrokinetic theories, even ‘global’ models, typically rely on a separation in scale (in perpendicular wavelength) between the fluctuations and the system scale. In such models direct simulation of system-scale dynamics like magnetohydrodynamic (MHD) motion is formally not consistent. Drift-kinetic theory, on the other hand, may be directly applied to model system-scale MHD-ordered behaviour. I review the long-wavelength limit of standard gyrokinetics and drift kinetics to present the relationships between these theories in an elementary fashion. This provides a pathway to global gyrokinetic modelling, resulting in an approach that is structurally similar to kinetic MHD, and I present dynamical equations for solving global field evolution in this framework. Departures from certain earlier global gyrokinetic theories include the appearance of magnetosonic (fast) modes, and the cross-coupling of the parallel and perpendicular currents with perpendicular and parallel magnetic field components. A periodic two-dimensional testcase is outlined as a benchmarking and implementation target, to help clarify practical aspects of these theories, with minimal complexity in terms of boundary conditions, and a proof-of-principle implementation of a field-solver is exhibited. To motivate this work, I first illustrate certain limitations of existing global gyrokinetic frameworks and directly identify how scale separation approximations lead to certain ‘missing’ system-scale field terms in global gyrokinetics, largely as a result of simplifications associated with the field representation in terms of $A_{\|}$ and $B_{\|}$ . As a result, the currents in the gyrokinetic Ampère's law resulting from a gyrokinetic equilibrium distribution do not match the currents implied by Ampère's law in a force-balance MHD equilibrium. I present a simple choice of equilibrium distribution function whose drift-kinetic currents are consistent with MHD currents; a specific Grad–Shafranov equilibrium is used to illustrate the size of the components of the parallel currents.

Analytical and Statistical Modelling of a Fast Ion Source Formed by Injection of a Neutral Beam into Magnetically Confined Plasma

Mathematical modelling of heating and current drive as well as yields and distributions of fusion products in a magnetically confined plasma subject to neutral beam injection requires, in turn, modelling of distributions of fast ions, which is a complex task including calculations of the source of suprathermal particles, i.e., the number of fast ions occurring in unit volume during unit time owing to the injection of fast atoms. The knowledge of the magnetohydrodynamic equilibrium, beam injection geometry and spatial distribution of the magnetic field are the necessary prerequisites. Explicit general analytical formulae for the source of fast ions have been obtained by two different methods. In addition, a method of statistical modelling is presented. Calculations of spatial and angular distributions of the fast ion source for a tokamak and verifications of the obtained results have been performed by a number of methods.

Non-axisymmetric magnetohydrodynamic equilibrium and stability in an axisymmetric toroidal device

This paper is dedicated to studying the radial broadening of helical states embedded in three-dimensional nonlinear magnetohydrodynamic (MHD) equilibria in an axisymmetric toroidal device. The stability of these helical equilibria is estimated by analyzing global MHD stability. An equilibrium bifurcation can develop a non-axisymmetric configuration with a helical core encompassed by an axis-symmetric plasma edge when the amplitude of an innermost magnetic helicity exceeds that of the primary axisymmetric magnetic component. It is found that internal negative magnetic shear drives the increase of the helicity and the decrease of the axisymmetric component via triple-mode nonlinear interactions, which induces the helical core to broaden radially. The helical structure mainly rests within the domain where the helicity is larger than the axisymmetric mode. Meanwhile, chaotic fields are generated outside the region predicted by the multi-region relaxed MHD model. Furthermore, the global stability analysis reveals that the helical equilibria are stabilized with flat or weak negative magnetic shear at the plasma core. As the core negative magnetic shear becomes significant, the helical equilibria become destabilized by kink modes even though the helical structure extends radially further out. This work is of importance and interest for reversed-field pinch plasma to reach steady quasi-single-helicity states.

A Data-Assimilation Based Method for Equilibrium Reconstruction of Magnetic Fusion Plasma: Solution by Adjoint Method

Accurately determining the magnetohydrodynamic (MHD) equilibrium is fundamentally important in controlling and stabilizing fusion plasmas. Development of reliable methods for equilibrium reconstruction is therefore crucial to advancing the fusion plasma research. However, since the dynamics of plasma phenomena are highly nonlinear and highly complicated dynamics, it is difficult to develop their rigorous mathematical model. We already proposed a method of equilibrium reconstruction for fusion plasmas based on data assimilation. In the method the problem is formulated as a parameter optimization and a solution by the sensitivity equation method is proposed. In this paper, we propose a solution of equilibrium reconstruction for fusion plasmas by the adjoint method for the purpose of reducing computation time. In order to ensure the applicability of the adjoint method to a wide class of fusion plasmas, the problem formulation and derivation of the reconstruction algorithm based on the adjoint method are performed in generic form. Furthermore, we implement the proposed method as equilibrium reconstruction algorithms applicable to axisymmetric toroidal plasmas, typical examples of which are tokamaks and reversed field pinches (RFPs). Finally, the proposed method is applied to a real RFP experiment, and it is shown that the proposed method has the capability of reconstructing the equilibrium with sufficient accuracy and dramatic reduction of computation time compared with the existing methods.

Strong toroidal magnetic fields sustained by the elastic crust in a neutron star

ABSTRACT We investigate new solutions for magnetized neutron stars with a barotropic core in magnetohydrodynamic (MHD) equilibrium and a magnetoelastic crust, which was neglected by previous studies concerning stars in MHD equilibrium. The Lorentz force of the barotropic star is purely irrotational and the structures of magnetic fields are constrained. By contrast, a solenoidal component of the Lorentz force exists in the elastic crust and the structures of the magnetic fields are less restricted. We find that the minor solenoidal component in the elastic crust is important for sustaining the strong magnetic field in the core. Unlike previous studies, the toroidal magnetic field exists in the entire region of the core, and we obtain equilibrium states with large toroidal magnetic fields, where the toroidal magnetic energy is larger than the poloidal magnetic energy. The elastic force of the crust sustains an order of 1015 G toroidal magnetic field in the core, and the maximum strength of the toroidal magnetic field is approximately proportional to the crust thickness.

Mapping the space of quasisymmetric stellarators using optimized near-axis expansion

A method is demonstrated to rapidly calculate the shapes and properties of quasi-axisymmetric and quasi-helically symmetric stellarators. In this approach, optimization is applied to the equations of magnetohydrodynamic equilibrium and quasisymmetry, expanded in the small distance from the magnetic axis, as formulated by Garren & Boozer [Phys. Fluids B, vol. 3, 1991, p. 2805]. Due to the reduction of the equations by the expansion, the computational cost is significantly reduced, to times of the order of 1 cpu second, enabling wide and high-resolution scans over parameter space. In contrast to traditional stellarator optimization, here, the cost function serves to maximize the volume in which the expansion is accurate. A key term in the cost function is $\| \boldsymbol {\nabla }\boldsymbol B \|$ , the norm of the magnetic field gradient, to maximize scale lengths in the field. Using this method, a database of $5\times 10^5$ optimized configurations is calculated and presented. Quasisymmetric configurations are observed to exist in continuous bands, varying in the ratio of the magnetic axis length to average major radius. Several qualitatively new types of configuration are found, including quasi-helically symmetric fields in which the number of field periods is two or more than six.

Direct construction of stellarator-symmetric quasi-isodynamic magnetic configurations

We develop the formalism of the first-order near-axis expansion of the magnetohydrodynamic equilibrium equations described by Garren & Boozer (Phys. Fluids B, vol. 3, issue 10, 1991, pp. 2805–2821) and Plunk et al. (J. Plasma Phys., vol. 85, issue 6, 2019; J. Plasma Phys., vol. 87, issue 6, 2021) for the case of a quasi-isodynamic, $N$ -field-period, stellarator-symmetric, single-well magnetic field equilibrium. The importance of the magnetic axis shape is investigated, and we conclude that control of the curvature and torsion is crucial to obtain omnigenous configurations with finite aspect ratio and low effective ripple, especially for a higher number of field periods. For this reason a method is derived to construct classes of axis shapes with favourable curvature and torsion. Solutions are presented, including a three-field-period configuration constructed at an aspect ratio of $A=20$ , with a maximum elongation of $e=3.2$ and an effective ripple under $1\,\%$ , which demonstrates that high elongation is not a necessary feature of quasi-isodynamic stellarators.

Global gyro-kinetic study of magnetic shaping effects on linear trapped electron mode instability in normal/reversed magnetic shear plasmas

A linear δf version of the gyro-kinetic Vlasov code GKNET (Gyro-Kinetic Numerical Experiment of Tokamak) is extended to the global cylindrical coordinates (R, φ, Z) and includes the kinetic electron response, to study the shaping effect on linear drift-wave instabilities in Tokamak plasmas. Cross-verifications are performed between three GKNET versions that use different electron models, i.e. the adiabatic electron model, hybrid electron model (only trapped electrons are calculated kinetically) and full-kinetic electron model (all electrons are calculated kinetically). A destabilizing effect of non-adiabatic passing electrons is observed in the full-kinetic case for the ion temperature gradient (ITG) mode in the short wavelength region and the trapped electron mode (TEM). The ion-to-electron mass ratio and the electron–electron collisionality have weak impacts on the destabilization of ITG/TEM. Utilizing magnetohydrodynamic equilibria designed with normal and reversed magnetic shear (NS and RS), the characteristics of linear TEMs and the corresponding impact of plasma shaping are studied based on the GKNET code with full-kinetic electrons. Due to the change in temperature/density gradient and magnetic shear either measured locally in the low field side (LFS) or averaged over the flux surface, plasma shaping shows different impacts on linear TEMs in NS and RS configurations. For the elongation κ, the increase in κ always stabilizes linear TEMs due to the reduction in the effective profile gradient over the flux surface. For the triangularity δ, in the NS case, the change in δ shows weak dependence on TEM linear growth rates due to the balance between variations of profile gradients and magnetic shear in the LFS. In the RS case, local magnetic shear in the peak gradient region is nearly zero. Thus, the variation in local profile gradient plays a dominant role on the linear growth in a plasma shaping scan. Consequently, the negative triangularity (δ < 0) has a destabilizing effect on the linear TEMs in the RS configuration mainly due to the upshift in local profile gradient in the LFS.

Development of controller for fast plasma position control coils with ISO-FLUX scheme on JT-60SA

Two types of poloidal magnetic field coils, superconducting poloidal field (SCPF) coils and in-vessel coils called fast plasma position control (FPPC) coils, will be installed in JT-60SA. We presented the different roles of SCPF and FPPC coils. The SCPF coils control plasma position and shape (P/S) and plasma current (I p), whereas the FPPC coils stabilize the perturbation of the n= 0 mode of magnetohydrodynamic (MHD) instability, such as vertical instability. This study developed a controller that outputs a coil voltage command for the power supply connected to each coil based on an ISO-FLUX scheme using an equilibrium control simulation code, MHD equilibrium control simulator (MECS). This controller stabilizes the horizontal and vertical plasma displacements using FPPC coils. FPPC coils have the advantage of FPPC due to fast coil current response; however, the induced current is also driven in FPPC coils. Thus, we proposed a control logic to mitigate the induced currents, particularly when the induced voltage is large. The difference in coil current response for SCPF and FPPC coils causes the coupling problem. Thus, decoupling between the SCPF and FPPC coils was established by employing the derivative treatment on the ISO-FLUX scheme in the FPPC control. To investigate the effectiveness of the FPPC control, using MECS we evaluated the allowable I p disruption intensity, which causes the plasma horizontal displacement, in the high elongation plasma, which relates to the plasma vertical displacement. Higher I p disruption intensity and elongation were allowed by adding the FPPC control. We investigated the controllability in the plasma ramp-up and flat-top operations. The support of FPPC control for SCPF control expands the plasma operation region which contributes to achieving the planned plasma operation in JT-60SA.

Two types of helical-core equilibrium states in tokamak plasmas

Helical distortion in the core region and its formation mechanism in weakly reversed magnetic shear tokamak plasmas are investigated using three-dimensional magnetohydrodynamic (MHD) equilibrium calculations and MHD stability analysis. It is found that there are two different types of helical equilibrium states: one is a rigid-body shift in the core region on a poloidal cross-section, while the other is a local advection only around the magnetic axis. Linear stability analysis of the corresponding axisymmetric equilibrium is carried to understand the origin of these two types of helical equilibrium states. The analysis shows that the former is related to the current-driven internal-kink instability, while the latter is linked to the pressure-driven quasi-interchange instability. The kink-mode-driven helical equilibrium state appears when the minimum value of the safety factor is close to unity, while the quasi-interchange-driven helical equilibrium state appears when is below unity. The appearance of the quasi-interchange state is attributed to the weak magnetic shear at the inner rational surface and finite β.

Optimization of quasi-symmetric stellarators with self-consistent bootstrap current and energetic particle confinement

Quasi-symmetry can greatly improve the confinement of energetic particles and thermal plasma in a stellarator. The magnetic field of a quasi-symmetric stellarator at high plasma pressure is significantly affected by the bootstrap current, but the computational cost of accurate stellarator bootstrap calculations has precluded use inside optimization. Here, a new efficient method is demonstrated for optimization of quasi-symmetric stellarator configurations such that the bootstrap current profile is consistent with the geometry. The approach is based on the fact that all neoclassical phenomena in quasi-symmetry are isomorphic to those in axisymmetry. Therefore, accurate formulas for the bootstrap current in tokamaks, which can be evaluated rapidly, can be applied also in stellarators. The deviation between this predicted parallel current and the actual parallel current in the magnetohydrodynamic equilibrium is penalized in the objective function, and the current profile of the equilibrium is included in the parameter space. Quasi-symmetric configurations with significant pressure are thereby obtained with self-consistent bootstrap current and excellent confinement. In a comparison of fusion-produced alpha particle confinement across many stellarators, the new configurations have significantly lower alpha energy losses than many previous designs.

ATEQ: Adaptive toroidal equilibrium code

A radially adaptive numerical scheme is developed to solve the Grad–Shafranov equation for axisymmetric magnetohydrodynamic equilibrium. A decomposition with independent solutions is employed in the radial direction, and Fourier decomposition is used in the poloidal direction. The independent solutions are then obtained using an adaptive shooting scheme together with the multi-region matching technique in the radial direction. Accordingly, the adaptive toroidal equilibrium (ATEQ) code is constructed for axisymmetric equilibrium studies. The adaptive numerical scheme in the radial direction improves considerably the accuracy of the equilibrium solution. The decomposition with independent solutions effectively reduces the matrix size in solving the magnetohydrodynamic equilibrium problem. The reduction of the matrix size is about an order of magnitude as compared with the conventional radially grid-based numerical schemes. Also, in this ATEQ numerical scheme, no matter how accuracy in the radial direction is imposed, the size of matrices basically does not change. The small matrix size scheme gives ATEQ more flexibility to address the requirement of the number of Fourier components in the poloidal direction in tough equilibrium problems. These two unique features, the adaptive shooting and small matrix size, make ATEQ useful to improve tokamak equilibrium solutions.

Equilibrium reconstruction of DIII-D plasmas using predictive modeling of the pressure profile

New workflows have been developed for predictive modeling of magnetohydrodynamic (MHD) equilibrium in tokamak plasmas. The goal of this work is to predict the MHD equilibrium in tokamak discharges without having measurements of the kinetic profiles. The workflows include a cold start tool, which constructs all the profiles and power flows needed by transport codes; a Grad–Shafranov equilibrium solver; and various codes for the sources and sinks. For validation purposes, a database of DIII-D tokamak discharges has been constructed that is comprised of scans in the plasma current, toroidal magnetic field, and triangularity. Initial efforts focused on developing a workflow utilizing an empirically derived pressure model tuned to DIII-D discharges with monotonic safety factor profiles. This workflow shows good agreement with experimental kinetic equilibrium calculations, but is limited in that it is a single fluid (equal ion and electron temperatures) model and lacks H-mode pedestal predictions. The best agreement with the H-mode database is obtained using a theory-based workflow utilizing pressure profile predictions from a coupled TGLF turbulent transport and EPED pedestal models together with external magnetics and Motional Stark Effect (MSE) data to construct the equilibrium. Here, we obtain an average root mean square error of 5.1% in the safety factor profile when comparing the predicted and experimental kinetic equilibrium. We also find good agreement with the plasma stored energy, internal inductance, and pressure profiles. Including MSE data in the theory-based workflow results in noticeably improved agreement with the q-profiles in high triangularity discharges in comparison with the results obtained with magnetic data only. The predictive equilibrium workflow is expected to have wide applications in experimental planning, between-shot analysis, and reactor studies.

A Simple proto nebula constituted of plasma under magneto hydrodynamics conditions

Using simplifying assumptions, a region or time in the universe is conceived, which allows for a state constituted solely by a gaseous plasma subject to the current laws of gravity and electromagnetism to exist. Under these conditions, it is proposed that the presence of a portion of that medium, organized in a highly symmetric magnetic field and plasma structure, possesses the symmetry of a torus. We resort to the fact that magneto hydrodynamics (MHD) is the common state of this medium and that it has the property of attaching to it the mass. (I.e. in the language of MHD the mass of the torus is assumed to be magnetized.) The solution to this MHD equilibrium of the matter of the torus, which prevents it from coalescing through the gravitational pool, is presented. Further, analysis shows that a gravitational field generated by the torus may be capable of attracting the other, non-magnetized matter, under the influence of the torus’ gravitational pool. In this way it is shown that the torus shaped MHD structure, under the discussed conditions, satisfies essential properties, like having large regions where the Keplerian mass motion cannot be present. A scenario consistent with a few key properties of proto-nebulae in equilibrium is presented and considered.

Free-boundary plasma equilibria with toroidal plasma flows

Magnetohydrodynamic equilibrium schemes with toroidal plasma flows and scrape-off layer are developed for the ‘divertor-type’ and ‘limiter-type’ free boundaries in the tokamak cylindrical coordinate. With a toroidal plasma flow, the flux functions are considerably different under the isentropic and isothermal assumptions. The effects of the toroidal flow on the magnetic axis shift are investigated. In a high beta plasma, the magnetic shifts due to the toroidal flow are almost the same for both the isentropic and isothermal cases and are about 0.04a 0 (a 0 is the minor radius) for M 0 = 0.2 (the toroidal Alfvén Mach number on the magnetic axis). In addition, the X-point is slightly shifted upward by 0.0125a 0. But the magnetic axis and the X-point shift due to the toroidal flow may be neglected because M 0 is usually less than 0.05 in a real tokamak. The effects of the toroidal flow on the plasma parameters are also investigated. The high toroidal flow shifts the plasma outward due to the centrifugal effect. Temperature profiles are noticeable different because the plasma temperature is a flux function in the isothermal case.

Overview of coordinated spherical tokamak research in Japan

Spherical tokamak (ST) research in Japan has produced many innovative results: (i) plasma start-up to I p > 70 kA was achieved by electron cyclotron wave (ECW) with N ∥ = 0.75, while electron heating to T e > 500 eV was achieved with N ∥ = 0.26 on QUEST. (ii) The radiofrequency (RF)-induced transport model was combined with the x-ray emission model, and extended magnetohydrodynamics equilibrium with kinetic electrons was developed to interpret fast-electron-dominated lower hybrid wave sustained plasmas on TST-2. (iii) Density as high as 30 times the cutoff density was achieved by electron Berstein wave current drive combined with electron beam injection on LATE. (iv) Multiple plasmoids formed by tearing instability in the elongated current sheet were observed, and flux closure and ion heating by plasmoid-mediated fast magnetic reconnection were observed on HIST. (v) Optimization of ECW-assisted inductive start-up with a vertical field with positive decay index was performed on TST-2. (vi) Stabilization of the vertical displacement event by a set of upper and lower helical field coils was demonstrated on TOKASTAR-2. (vii) A 6 h discharge was achieved by cool-down of the center stack cover on QUEST, where the plasma duration limit was consistent with the wall saturation time estimated by modeling. (viii) Extension of ion heating by plasma merging was achieved on TS-3U, TS-4U, UTST, MAST, and ST40.

Recent developments in engineering design for the quasi-axisymmetric stellarator CFQS

A quasi-axisymmetric stellarator, the CFQS, has been designed as a joint project of the National Institute for Fusion Science and Southwest Jiaotong University to prove intrinsic advantages of quasi-axisymmetry. Principal parameters of the CFQS are as follows: the major radius is 1 m, the magnetic field strength is 1 T, the aspect ratio is 4, and the toroidal periodic number is 2. The magnetic field configuration is designed based on that of the CHS-qa. Enhanced confinement properties within the context of neoclassical theory are achieved by its quasi-axisymmetric configuration. In the entire radial range, the magnetic well is retained to keep favourable stability features in the magnetohydrodynamic equilibrium. A magnetic field coil system was designed for the CFQS, which consists of 16 modular coils, 12 toroidal field coils, and 4 poloidal field coils. The supporting structure is designed to withstand strong electromagnetic force under 1 T operation, maintaining enough space for heating and diagnostic systems. The mock-up modular coil with the most complicated shape was constructed by Hefei Keye Electro Physical Equipment Manufacturing Co., Ltd. to check manufacturability and the achieved accuracy. A heat-run test was performed to check the temperature rise of conductors, and the capability of 1 T operation was confirmed. After various tests for the mock-up coil, construction of actual modular coils and the vacuum vessel has begun.