The heat flux is one of the key theoretical concepts used to quantify and understand transport in fusion devices. In this paper, a new method is introduced to calculate the heat flux including its confidence with high accuracy based on perturbed measurements such as the electron temperature. The new method is based on ideal filtering to optimally reduce the noise contributions on the measurements and piece-wise polynomial approximations to calculate the time derivative. Both methods are necessary to arrive at a heat flux and effective diffusion coefficient with high accuracy. The new methodology is applied to a measurement example using ECRH block-wave modulation at the Large Helical Device showing the merit of the newly developed methodology.

}, doi = {10.1088/1741-4326/aad13e}, author = {van Berkel, M. and Kobayashi, T. and Vandersteen, G. and Zwart, H. J. and Igami, H. and Kubo, S. and Tamura, N. and Tsuchiya, H. and M. R. de Baar and LHD Experiment Group} } @article {4083, title = {A systematic approach to optimize excitations for perturbative transport experiments}, journal = {Physics of Plasmas}, volume = {25}, year = {2018}, pages = {082510}, abstract = {In this paper, techniques for optimal input design are used to optimize the waveforms of perturbative experiments in modern fusion devices. The main focus of this paper is to find the modulation frequency for which the accuracy of the estimated diffusion coefficient is maximal. Mathematically, this problem can be formulated as an optimization problem in which the Fisher information matrix is maximized. First, this optimization problem is solved for a simplified diffusion model, while assuming a slab geometry and a semi-infinite domain. Later, the optimization is repeated under more general conditions such as a cylindrical geometry, finite domain, and simultaneous estimation of multiple transport coefficients. Based on the results of these optimizations, guidelines are offered to select the modulation frequency and to determine the optimality of the corresponding experiment. {\textcopyright} 2018 EURATOM

}, doi = {10.1063/1.5010325}, author = {van Berkel, M. and De Cock, A. and Ravensbergen, T. and Hogeweij, G. M. D. and Zwart, H. J. and Vandersteen, G.} } @article {4108, title = {Separation of transport in slow and fast time-scales using modulated heat pulse experiments (hysteresis in flux explained)}, journal = {Nuclear Fusion}, volume = {58}, year = {2018}, pages = {106042}, abstract = {Old and recent experiments show that there is a direct response to the heating power of transport observed in modulated ECH experiments both in tokamaks and stellarators. This is most apparent for modulated experiments in the Large Helical Device (LHD) and in Wendelstein 7 advanced stellarator (W7-AS). In this paper we show that: 1) This power dependence can be reproduced by linear models and as such hysteresis (in flux) has no relationship to hysteresis as defined in the literature; 2) Observations of "hysteresis" (in flux) and a direct response to power can be perfectly reproduced by introducing an error in the estimated deposition profile as long as the errors redistribute the heat over a large radius; 3) Non-local models depending directly on the heating power can also explain the experimentally observed Lissajous curves (hysteresis); 4) How non-locality and deposition errors can be recognized in experiments and how they affect estimates of transport coefficients; 5) That non-linear-non-local transport models offer a path in discerning deposition errors from non-local fast transport components otherwise experimentally indistinguishable. To show all this, transport needs to be analyzed by separating the transport in a slow (diffusive) time-scale and a fast (heating/non-local) time-scale, which can only be done in the presence of perturbations. (DOI dataset, OA: 10.4121/uuid:5fcf4247-da0e-4119-adcd-fc90b85b7f03)}, doi = {10.1088/1741-4326/aadc17}, author = {van Berkel, M. and Vandersteen, G. and Zwart, H. J. and Hogeweij, G. M. D. and Citrin, J. and Westerhof, E. and Peumans, D. and M. R. de Baar} } @article {3761, title = {Technical note on the linearity and power dependence of the diffusion coefficient in W7-AS}, journal = {Plasma Physics and Controlled Fusion}, volume = {59}, year = {2017}, pages = {062001}, abstract = {Transient electron temperature measurements of a step power experiment at W7-AS are reassessed by direct comparison of the up- and downward responses of the electron temperature. The analysis shows that the response at some distance to the center behaves linearly and the model predicted responses based on a power-dependent diffusion coefficient that vary from the measured step responses.

}, doi = {10.1088/1361-6587/aa6a2b}, author = {van Berkel, M. and Zwart, H. J. and Hogeweij, G. M. D. and M. R. de Baar} } @article {3869, title = {New evidence and impact of electron transport non-linearities based on new perturbative inter-modulation analysis}, journal = {Nuclear Fusion}, volume = {57}, year = {2017}, pages = {126036}, abstract = {A new methodology to analyze non-linear components in perturbative transport experiments is introduced. The methodology has been experimentally validated in the Large Helical Device for the electron heat transport channel. Electron cyclotron resonance heating with different modulation frequencies by two gyrotrons has been used to directly quantify the amplitude of the non-linear component at the inter-modulation frequencies. The measurements show significant quadratic non-linear contributions and also the absence of cubic and higher order components. The non-linear component is analyzed using the Volterra series, which is the non-linear generalization of transfer functions. This allows us to study the radial distribution of the non-linearity of the plasma and to reconstruct linear profiles where the measurements were not distorted by non-linearities. The reconstructed linear profiles are significantly different from the measured profiles, demonstrating the significant impact that non-linearity can have.

}, doi = {10.1088/1741-4326/aa827a}, author = {van Berkel, M. and Kobayashi, T. and Igami, H. and Vandersteen, G. and Hogeweij, G. M. D. and Tanaka, K. and Tamura, N. and Zwart, H. J. and Kubo, S. and Ito, S. and Tsuchiya, H. and M. R. de Baar and LHD Experiment Group} } @article {3701, title = {LPMLE3: A novel 1-D approach to study water flow in streambeds using heat as a tracer}, journal = {Water Resources Research}, volume = {52}, year = {2016}, pages = {6596-6610}, keywords = {frequency domain, groundwater-surface water interaction, Groundwater/surface water interaction, heat tracer, hyporheic zone, maximum-likelihood estimator, Modeling, Time series analysis}, doi = {10.1002/2015WR017453}, author = {Schneidewind, U. and van Berkel, M. and Anibas, C. and Vandersteen, G. and Schmidt, C. and Joris, I. and Seuntjens, P. and Batelaan, O. and Zwart, H. J.} } @article {2658, title = {Estimation of the thermal diffusion coefficient in fusion plasmas taking frequency measurement uncertainties into account}, journal = {Plasma Physics and Controlled Fusion}, volume = {56}, year = {2014}, pages = {105004}, abstract = {In this paper, the estimation of the thermal diffusivity from perturbative experiments in fusion plasmas is discussed. The measurements used to estimate the thermal diffusivity suffer from stochastic noise. Accurate estimation of the thermal diffusivity should take this into account. It will be shown that formulas found in the literature often result in a thermal diffusivity that has a bias (a difference between the estimated value and the actual value that remains even if more measurements are added) or have an unnecessarily large uncertainty. This will be shown by modeling a plasma using only diffusion as heat transport mechanism and measurement noise based on ASDEX Upgrade measurements. The Fourier coefficients of a temperature perturbation will exhibit noise from the circular complex normal distribution (CCND). Based on Fourier coefficients distributed according to a CCND, it is shown that the resulting probability density function of the thermal diffusivity is an inverse non-central chi-squared distribution. The thermal diffusivity that is found by sampling this distribution will always be biased, and averaging of multiple estimated diffusivities will not necessarily improve the estimation. Confidence bounds are constructed to illustrate the uncertainty in the diffusivity using several formulas that are equivalent in the noiseless case. Finally, a different method of averaging, that reduces the uncertainty significantly, is suggested. The methodology is also extended to the case where damping is included, and it is explained how to include the cylindrical geometry.

}, doi = {10.1088/0741-3335/56/10/105004}, author = {van Berkel, M. and Zwart, H. J. and Hogeweij, G. M. D. and Vandersteen, G. and van den Brand, H. and M.R. de Baar and ASDEX Upgrade Team} } @article {2660, title = {Explicit approximations to estimate the perturbative diffusivity in the presence of convectivity and damping. I. Semi-infinite slab approximations}, journal = {Physics of Plasmas}, volume = {21}, year = {2014}, pages = {112507}, abstract = {In this paper, a number of new approximations are introduced to estimate the perturbative diffusivity (χ), convectivity (V), and damping (τ) in cylindrical geometry. For this purpose, the harmonic components of heat waves induced by localized deposition of modulated power are used. The approximations are based on semi-infinite slab approximations of the heat equation. The main result is the approximation of χ under the influence of V and τ based on the phase of two harmonics making the estimate less sensitive to calibration errors. To understand why the slab approximations can estimate χ well in cylindrical geometry, the relationships between heat transport models in slab and cylindrical geometry are studied. In addition, the relationship between amplitude and phase with respect to their derivatives, used to estimate χ, is discussed. The results are presented in terms of the relative error for the different derived approximations for different values of frequency, transport coefficients, and dimensionless radius. The approximations show a significant region in which χ, V, and τ can be estimated well, but also regions in which the error is large. Also, it is shown that some compensation is necessary to estimate V and τ in a cylindrical geometry. On the other hand, errors resulting from the simplified assumptions are also discussed showing that estimating realistic values for V and τ based on infinite domains will be difficult in practice. This paper is the first part (Part I) of a series of three papers. In Part II and Part III, cylindrical approximations based directly on semi-infinite cylindrical domain (outward propagating heat pulses) and inward propagating heat pulses in a cylindrical domain, respectively, will be treated.

}, keywords = {approximation theory, convection, damping, diffusion, plasma simulation, plasma temperature, plasma waves}, doi = {10.1063/1.4901309}, author = {van Berkel, M. and Zwart, H. J. and Tamura, N. and Hogeweij, G. M. D. and Inagaki, S. and M.R. de Baar and Ida, K.} } @article {2661, title = {Explicit approximations to estimate the perturbative diffusivity in the presence of convectivity and damping. II. Semi-infinite cylindrical approximations}, journal = {Physics of Plasmas}, volume = {21}, year = {2014}, pages = {112508}, abstract = {In this paper, a number of new explicit approximations are introduced to estimate the perturbative diffusivity (χ), convectivity (V), and damping (τ) in a cylindrical geometry. For this purpose, the harmonic components of heat waves induced by localized deposition of modulated power are used. The approximations are based upon the heat equation in a semi-infinite cylindrical domain. The approximations are based upon continued fractions, asymptotic expansions, and multiple harmonics. The relative error for the different derived approximations is presented for different values of frequency, transport coefficients, and dimensionless radius. Moreover, it is shown how combinations of different explicit formulas can yield good approximations over a wide parameter space for different cases, such as no convection and damping, only damping, and both convection and damping. This paper is the second part (Part II) of a series of three papers. In Part I, the semi-infinite slab approximations have been treated. In Part III, cylindrical approximations are treated for heat waves traveling towards the center of the plasma.

}, keywords = {approximation theory, convection, damping, diffusion, heat transfer, plasma transport processes}, doi = {10.1063/1.4901310}, author = {van Berkel, M. and Hogeweij, G. M. D. and Tamura, N. and Zwart, H. J. and Inagaki, S. and M.R. de Baar and Ida, K.} } @article {2662, title = {Explicit approximations to estimate the perturbative diffusivity in the presence of convectivity and damping. III. Cylindrical approximations for heat waves traveling inwards}, journal = {Physics of Plasmas}, volume = {21}, year = {2014}, pages = {112509}, abstract = {In this paper, a number of new explicit approximations are introduced to estimate the perturbative diffusivity (χ), convectivity (V), and damping (τ) in cylindrical geometry. For this purpose, the harmonic components of heat waves induced by localized deposition of modulated power are used. The approximations are based on the heat equation in cylindrical geometry using the symmetry (Neumann) boundary condition at the plasma center. This means that the approximations derived here should be used only to estimate transport coefficients between the plasma center and the off-axis perturbative source. If the effect of cylindrical geometry is small, it is also possible to use semi-infinite domain approximations presented in Part I and Part II of this series. A number of new approximations are derived in this part, Part III, based upon continued fractions of the modified Bessel function of the first kind and the confluent hypergeometric function of the first kind. These approximations together with the approximations based on semi-infinite domains are compared for heat waves traveling towards the center. The relative error for the different derived approximations is presented for different values of the frequency, transport coefficients, and dimensionless radius. Moreover, it is shown how combinations of different explicit formulas can be used to estimate the transport coefficients over a large parameter range for cases without convection and damping, cases with damping only, and cases with convection and damping. The relative error between the approximation and its underlying model is below 2\% for the case, where only diffusivity and damping are considered. If also convectivity is considered, the diffusivity can be estimated well in a large region, but there is also a large region in which no suitable approximation is found. This paper is the third part (Part III) of a series of three papers. In Part I, the semi-infinite slab approximations have been treated. In Part II, cylindrical approximations are treated for heat waves traveling towards the plasma edge assuming a semi-infinite domain.

}, keywords = {approximation theory, Bessel functions, boundary-value problems, convection, damping, plasma transport processes}, doi = {10.1063/1.4901311}, author = {van Berkel, M. and Tamura, N. and Hogeweij, G. M. D. and Zwart, H. J. and Inagaki, S. and M.R. de Baar and Ida, K.} }