Field Ripple Research Articles

Analyzing Fast-ions Trajectories in a Nuclear Fusion Reactor through Its Poincaré-Island Size and Ripple Resonance

Fast-ions confinement is a prominent subject in developing nuclear fusion reactors due to its importance in sustaining the burning plasma and keeping energy production. However, confining them has proven to be difficult until now, and one of the reasons is that the inherent discrete magnetic field produces a magnetic ripple. A better understanding of fast-ions transport using appropriate numerical calculation tools needs to be developed to overcome such a challenge in the engineering aspect. This study revisited data collection of fast ion transport simulated under the ripple presence in a nuclear fusion device. The ion trajectories were followed using two orbit-following equation schemes, and the ripple-resonance island size in the Poincaré section was compared. The result showed that the island size obtained by each scheme was different when the particle resonates with a stronger ripple field and, proportionally, the diffusion coefficients are different. The physical meaning and consequence behind this discovery were discussed in this paper.

Alien Aeolian Bedforms: A Comparative Sedimentary Analysis of the Dingo Gap Bedform and Hidden Valley Ripple Traverses, Gale Crater, Mars

AbstractAt the end of January 2014 (Sols 528–540), the Mars Science Laboratory rover Curiosity approached the valley opening known as Dingo Gap. Spanning this gap was an unusual bedform with a maximum height of 1.34 m and a maximum width of ∼8 m. Curiosity encountered, imaged, and successfully traversed this atypical bedform. In August of 2014, Curiosity began to traverse a field of ripples in Hidden Valley, but was forced to abort the attempt due to high wheel slip. The Hidden Valley ripples are ∼1/6 of the height of the Dingo Gap bedform, and yet posed a serious hazard to the rover. Here, we present a sedimentological analysis of the Dingo Gap bedform and the Hidden Valley ripples and discuss how their differences may have impacted rover traversability. The Dingo Gap and Hidden Valley bedforms are morphologically similar to many commonly observed bedforms on Mars that have few clear analogs on Earth, and Curiosity's encounters provide a framework for studying these bedforms in detail. Furthermore, the rover tracks across the bedforms have persisted for >5 Earth years, even through a global dust storm. The longevity of the tracks demonstrates that on multi‐year timescales (a) sediment transport on some Martian bedforms is inactive or slow enough that areas disturbed by the rover are not resurfaced, and (b) the rate of dust deposition on the Martian surface, even after a global dust storm, is too low to create an optically thick layer.

Simulations of NBI fast ion loss in the presence of toroidal field ripple on EAST

NBI fast ion losses in the presence of the toroidal field ripple on EAST have been investigated by using the orbit code GYCAVA and the NBI code TGCO. The ripple effect was included in the upgraded version of the GYCAVA code. It is found that loss regions of NBI fast ions are mainly on the low field side near the edge in the presence of ripple. For co-current NBIs, the synergy effect of ripple and Coulomb collision on fast ion losses is dominant, and fast trapped ions located on the low field side are easily lost. The ripple well loss and the ripple stochastic loss of fast ions have been identified from the heat loads of co-current NBI fast ions. The ripple stochastic loss and the collision-induced loss are much larger than the ripple well loss. Heat loads of lost fast ions are mainly localized on the right side of the radio frequency wave antennas from the inside view toward the first wall. For counter-current NBIs, the first orbit loss due to the magnetic drift is the dominant loss channel. In addition, fast ion loss fraction with ripple and collision for each NBI linearly increases with the effective charge number, which is related to the pitch angle scattering effect.

A comparison of the temporal evolution of hydrodynamics and inlet morphology during Tropical Storm Fay (2020)

The record-setting North Atlantic hurricane season of 2020 had 30 named storms and reinforced the need for high-resolution, small-scale data collected in the nearshore zone during storm events to characterize storm impacts on coastal settings. To address these needs, hydrodynamic and morphologic data were collected during the 2020 Atlantic hurricane season, capturing fair weather conditions and the passage of Tropical Storm Fay (July 2020) near Chincoteague Inlet, Virginia. A sector-scanning rotary sonar captured high-resolution imagery of bedform evolution and data were analyzed to relate the migration of bedforms to the concomitant hydrodynamic conditions during the storm event. During the peak of the storm on 10 July 2020, significant wave height and period in Chincoteague Inlet were 0.96 m at 9.6 s arriving from the SSW (201°). The ripple field evolved during the storm in a manner consistent with that found in Hay and Mudge (2005): irregular ripples (O 20 cm wavelength) dominated during fair weather conditions, which developed into a washed-out, flat bed state as the storm arrived. During the peak of the storm, lunate megaripples (O 1 m wavelength) formed and migrated shoreward. A substantial outflow of freshwater from Chincoteague Bay occurred for up to seven days post-storm, and sediment transported by this outflow could serve as a yet-unidentified sediment source for the rapid growth of southern Assateague Island. This outflow of freshwater dampened waves and hindered ripple field recovery for up to seven days post-storm. These extreme event datasets are critical to inform coastal flood models and management decisions, as this work recognizes an increased risk of flooding for the town of Chincoteague from even the offshore passage of tropical storms.

PTC: Full and Drift Particle Orbit Tracing Code for α Particles in Tokamak PlasmasSupported by the National Natural Science Foundation of China (Grant Nos. 11975068 and 11925501), and the National KeyResearch and Development Program of China (Grant No. 2017YFE0300501).

Fusion born α particle confinement is one of the most important issues in burning plasmas, such as ITER and CFETR. However, it is extremely complex due to the nonequilibrium characteristics, and multiple temporal and spatial scales coupling with background plasma. A numerical code using particle orbit tracing method (PTC) has been developed to study energetic particle confinement in tokamak plasmas. Both full orbit and drift orbit solvers are implemented to analyze the Larmor radius effects on α particle confinement. The elastic collisions between alpha particles and thermal plasma are calculated by a Monte Carlo method. A triangle mesh in poloidal section is generated for electromagnetic fields expression. Benchmark between PTC and ORBIT has been accomplished for verification. For CFETR burning plasmas, PTC code is used for α particle source and slowing down process calculation in 2D equilibrium. In future work, 3D field like toroidal field ripples, Alfvén and magnetohydrodynamics instabilities perturbation inducing α particle transport will be analyzed.

Design and thermal-structural analysis of a high power ICRH travelling wave array antennas

Travelling wave array (TWA) antennas have been proposed for the ICRH (Ion Cyclotron Resonance Heating) antennas of fusion reactors in view to decreasing the antenna voltage and associated electric field. This paper reports the progress of the design and structural analysis of an actively cooled high power TWA antenna for WEST. First, the design of a non-cooled mock-up that will be tested in the TITAN facility is presented. The main objective is to assess the voltage stand-off of the antenna at power and electric field levels relevant for future nominal operation in a fusion device. The main characteristics of the mock-up are detailed and the results from thermal and structural analysis of the mock-up tested in TITAN are presented. In the second part, the compatibility with WEST environment for an actively-cooled TWA antenna is assessed. Mechanical, electromagnetic, thermal and hydraulics constrains (.i.e VDE, plasma radiation, toroidal magnetic field ripple…) are listed in a system loads specification. A Virtual Reality analysis using IRFM tools has been performed to check compliance of the design in term of assembly inside the WEST vacuum vessel.

Tomographic reconstruction of the runaway distribution function in TCV using multispectral synchrotron images

Synchrotron radiation observed in a quiescent Tokamak à Configuration Variable (TCV) runaway discharge is studied using filtered camera images targeting three distinct wavelength intervals. Through the tomographic simultaneous algebraic reconstruction technique (SART) procedure the high momentum, high pitch angle part of the spatial and momentum distribution of these relativistic particles is reconstructed. Experimental estimates of the distribution are important for verification and refinement of formation-, decay- and transport-models underlying runaway avoidance and mitigation strategy design. Using a test distribution it is demonstrated that the inversion procedure provides estimates accurate to within a few tens of percent in the region of phase-space contributing most to the synchrotron image. We find that combining images filtered around different parts of the emission spectrum widens the probed part of momentum-space and reduces reconstruction errors. Next, the SART algorithm is used to obtain information on the spatiotemporal runaway momentum distribution in a selected TCV discharge. The momentum distribution is found to relax towards an avalanche-like exponentially decaying profile. Anomalously high pitch angles and a radial profile increasing towards the edge are found for the most strongly emitting particles in the distribution.Pitch angle scattering by toroidal magnetic field ripple is consistent with this picture. An alternative explanation is the presence of high frequency instabilities in combination with the formation of a runaway shell at the edge of the plasma.

Numerical investigation of alpha particle confinement under the perturbation of neoclassical tearing modes and toroidal field ripple in CFETR

The confinement of alpha particles in burning plasma is a key issue in fusion reactor design, including particle interaction with instabilities. This paper includes two topics: the effect of neoclassical tearing modes (NTMs) and toroidal field ripple on alpha particle loss, and the assessment of particle redistribution under an NTM with a reduced model. We consider Chinese fusion engineering test reactor parameters, the alpha particle distribution given by TRANSP/NUBEAM and the NTM perturbation function given by the initial value code TM1. We show that the synergistic effect of the NTM and ripple is negligible; the particle loss fraction does not change with increasing NTM amplitude. Only passing particles are affected by the mode particle resonance, producing profile flattening but no increased loss because only trapped particles are influenced by ripple. To study alpha particle profile flattening, the work adopts an innovative method of phase vector rotation to determine regions of good and broken Kolmogorov–Arnold–Moser surfaces and equilibrates the particle density according to local stochasticity.

Electromagnetic properties of curved HTS trapped field stacks under high-frequency cross fields for high-speed rotating machines

Superconducting electric propulsion systems, characterized by high power densities and efficiencies, provide a possibility to zero carbon emission for future aviation. Stacks of high temperature superconducting (HTS) coated conductors (CCs) have become an alternative for high field magnets applied to superconducting machines, given their excellent field trapping ability and thermal stability. High-frequency ripple fields always exist in high-speed electric machines. Most research work regarding HTS trapped field stacks (TFSs) was focused on their magnetization methods and amplitude of trapped flux density; however, their performance in the high-frequency environment remains unclear. Despite several numerical models established for flat HTS TFSs, a comprehensive analysis of curved ones is still lacking, which possess geometrical applicability for cylindrical rotating shafts. Aimed at exploring the electromagnetic properties of curved HTS TFSs applied to high-speed rotating machines, a 3D numerical model considering both the multilayer structure and the Jc (B) dependence of HTS CCs has been built. Current and magnetic flux density distributions, as well as loss properties of a curved HTS TFS have been studied in detail, under perpendicular and cross fields with varying frequencies ranging from 50 Hz to 20 kHz. Results have shown that, the widely adopted two-dimensional-axisymmetric models are inapplicable to study the electromagnetic distributions of TFSs because of the emergence of the electromagnetic criss-cross defined in this paper. High-frequency ripple fields can drive induced current towards the periphery of the HTS TFS due to the skin effect, leading to a fast rise of AC loss and even an irreversible demagnetization of the stack. This paper has qualified and quantified the high-frequency electromagnetic behaviours of curved HTS TFSs, providing a useful reference for their loss controlling and anti-demagnetization design in high-speed propulsion machines.

Numerical simulation of synergistic effect of neoclassical tearing mode and toroidal field ripple on alpha particle loss in China Fusion Engineering Testing Reactor

Confinement of fusion born alpha particles in tokamak is the key issue to burning plasma. Apart from toroidal field ripple, instabilities can induce energetic particles to lose and be redistributed. Based on the parameters of China Fusion Engineering Testing Reactor (CFETT) hybrid scenario, alpha particle distribution and neoclassical tearing mode structure, the alpha particle loss induced under perturbation of ripple and neoclassical tearing mode (NTM) is calculated with the guiding center code ORBIT. The inputs have the initial distribution of alpha particles which is obtained with the TRANSP/NUBEAM code, the static NTM perturbation with different amplitudes which is obtained from TM1 code, and the ripple field from engineering design. The results show that the heat load on last closed flux surface is about 0.1 MW/m<sup>2</sup>, with ripple and collision included. The collisionless stochastic ripple diffusion is the main loss channel of initial alpha particle distribution in the CFETR, and the ripple perturbation has no influence on passing particles. The loss fraction does not increase with the NTM perturbation amplitude increasing, the synergistic effect is negligible. The scanning of ripple amplitude shows that the synergistic effect is slight. The monoenergetic initial distribution of alpha particles can give different types of orbits in the plane of (<inline-formula><tex-math id="M1">\begin{document}$ {P_\zeta },\mu $\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="11-20201972_M1.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="11-20201972_M1.png"/></alternatives></inline-formula>), such as the domains of trapped particle and passing particle, lost particle and confined particle. The trapped fraction of initial alpha particles is about 27%, ripple loss region in phase space is narrow and away from the main trapped particle distribution. The increasing of ripple perturbation in simulation does enlarge the ripple loss domain in the phase space (<inline-formula><tex-math id="M2">\begin{document}$ {P_\zeta },\mu $\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="11-20201972_M2.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="11-20201972_M2.png"/></alternatives></inline-formula>), which is corresponding to a lager ripple loss fraction and has more trapped-passing boundaries. The NTM perturbation does enlarge the orbit excursions of trapped particles, and thus increasing the trapped passing transition near the boundary. The slight synergistic effect in calculation with larger ripple amplitude is explained by ripple loss region having more trapped-passing boundaries, not by the profile flattening of trapped particles. The NTM perturbation and finite collision can transit the passing particle to trapped particle near the boundary. With the help of kinetic Poincare plot, neither direct particle loss nor profile flattening of trapped particles is observed. The loss fraction enhancement can happen only when the profile flattening of trapped particles takes place within the ripple loss region, which is not the case in CFETR. The conclusion of this work contributes a lot to the design of CFETR and the study of alpha particle physics.

Neutron rate estimates in MAST based on gyro-orbit modelling of fast ions

A discrepancy between predicted and measured neutron rates on MAST using TRANSP/NUBEAM has previously been observed and a correction factor of about 0.6 was needed to match the two: this correction factor could not be accounted for by the experimental uncertainties in the plasma kinetic profiles nor in the NBI energy and power (Cecconello et al 2019 Nucl. Fusion 59 016006). Further causes of this discrepancy are here studied by means of TRANSP/NUBEAM and ASCOT/BBNBI simulations. Different equilibria, toroidal field ripples, uncertainties on the NBI divergence value and gyro-orbit effects were studied and simulations were performed with both transport codes. It was found that the first three effects accounted for only a 5% variation in the fast ion density. On the other hand, full gyro-orbit simulations of the fast ions dynamics carried out in ASCOT/BBNBI resulted in an approximately 20% reduction of the fast ion population compared to TRANSP/NUBEAM. A detailed analysis of the fast ion distributions showed how the drop occurred regardless of the energy at pitch values ⩽−0.4. The DRESS code was then used to calculate the neutron rate at the neutron camera detector’s location showing that the discrepancy is considerably reduced when the full gyro-orbit fast ion distribution is used, with now the correction factor, used to match experimental and predicted neutron rates, being around 0.9.

Verification of effect of secondary conductor bar on starting characteristics in line‐start type self‐excited wound‐field synchronous motor

This study clarifies experimentally the effect of the secondary conductor bar on line-start characteristics in a line-start type self-excited wound-field synchronous motor. This motor is based on a brushless wound-field synchronous motor having a rotor winding connected to a diode rectifier circuit. And it has a secondary conductor bar that is independently connected at a pitch of one pole, without end-ring. First, the operation principle of the proposed motor and its magnetic circuit design are described, and then the details of the prototype for principle verification are clarified. Next, when the combination of the connecting of the secondary conductor bar is changed, an experiment of line-starting characteristics and measured rotor field current is demonstrated. Consequently, it was clarified that the influence on the synchronous pull-in, speed ripple, and rotor field current.

Modeling cross-field demagnetization of superconducting stacks and bulks for up to 100 tapes and 2 million cycles

Superconducting stacks and bulks can act as very strong magnets (more than 17 T), but they lose their magnetization in the presence of alternating (or ripple) transverse magnetic fields, due to the dynamic magneto-resistance. This demagnetization is a major concern for applications requiring high run times, such as motors and generators, where ripple fields are of high amplitude and frequency. We have developed a numerical model based on dynamic magneto-resistance that is much faster than the conventional Power-Law-resistivity model, enabling us to simulate high number of cycles with the same accuracy. We simulate demagnetization behavior of superconducting stacks made of 10–100 tapes for up to 2 million cycles of applied ripple field. We found that for high number of cycles, the trapped field reaches non-zero stationary values for both superconducting bulks and stacks; as long as the ripple field amplitudes are below the parallel penetration field, being determined by the penetration field for a single tape in stacks. Bulks keep substantial stationary values for much higher ripple field amplitudes than the stacks, being relevant for high number of cycles. However, for low number of cycles, stacks lose much less magnetization as compared to bulks.

Effect of non-axisymmetric perturbations on the ambipolar Er and neoclassical particle flux inside the ITER pedestal region

The transport dynamics of impurities in the pedestal region of ITER plasmas is of crucial interest since this regulates the penetration of impurities from the edge into the core plasma, where an excessive accumulation of impurities can degrade their fusion performance. In the pedestal region of H-mode tokamak plasmas, anomalous transport is highly reduced and impurity transport is found to be well described by the neoclassical theory. Under these conditions, perturbations to the axisymmetric tokamak geometry can strongly affect both the radial electric field and particle transport. In this work, we describe the results of numerical studies performed to quantify the effects on the pedestal ambipolar electric field and radial particle fluxes of the non-axisymmetric fields, associated with both the intrinsic toroidal field ripple and extrinsic fields applied for ELM control, for ITER Q = 10 plasma conditions with emphasis on high Z impurity transport. It is found that the effect of the ITER toroidal field ripple on high Z impurity transport is negligible. On the contrary, extrinsic three-dimensional fields applied for ELM control cause a strong modification of the pedestal ambipolar electric (to less negative values) and the appearance of multi-valued solutions for the pedestal electric field, analogue to core stellarator transport significantly increasing the outward character of neoclassical pedestal transport for both the main plasma ions (D and T in ITER) and high Z impurities, suggesting a strong modification of the background plasma profiles. Finally, it is found that for the Z impurity, its quantitative evaluation has uncertainties (with important implications for the radial flow direction) associated with the high poloidal Mach number ~ 1, due to the high pedestal electric field and large ion mass, which renders the first order neoclassical theory questionable. Detailed ITER MHD 3D equilibria and a benchmark of SFINCS against NEO and NCLASS codes has also been obtained.

The Effect of Removing the High Power Pulse Transformer on the Microwave Pulse Duration in the Preionization Phase in Taban Tokamak

In this study, a simple, economical and efficient preionization system (2.45 GHz, 1 kW) was constructed to assist the current startup in Taban tokamak ( $B_{T}= 0.7$ T, $R = 0.45$ m, a = 0.15 m). Power pulse transformer is a part of the preionization circuit, and its removal from the system can have impact on the performance of microwave duration and plasma preionization. In the present system, the magnetron was connected directly to the capacitor using a triggered spark gap. This new preionization system allows studying the electron cyclotron emission (ECR) preionization phenomenon at different toroidal magnetic fields. The toroidal magnetic field ripple was also used to determine the entrance moment of the ECR layer into the chamber. As a result, the exact moment of sending microwave can be determined using the moment of the ECR layer formation inside the chamber for maximum absorption of wave power.

Experimental study on effect of the toroidal magnetic field ripple on the ECR layer in TABAN tokamak

Abstract The study of ECR assisted breakdown on TABAN tokamak (BT = 7000 G, R = 45 cm, a = 15 cm) has been directed to find the best conditions for ECR pre-ionization and plasma generation. The toroidal magnetic field ripple ( δ ) was investigated to obtain optimum pressure at pre-ionization phase. The internal chamber can be divided into two areas considering the δ value: underneath and between the coils.The toroidal magnetic field ripple was used to determine the moment of the Electron Cyclotron Resonance (ECR) layer entrance into the chamber, and consequently, calculate the accurate breakdown time and then optimum pressure range (3 × 10−2 Pa ∼ 5 × 10−2 Pa) was obtained at pre-ionization phase. In addition, the toroidal magnetic field and its ripple have been experimentally measured using magnetic probes and simulated by COMSOL software as well.

Progress in the conceptual design of the CFETR toroidal field coil with rectangular conductors

A comprehensive research facility project was approved in December 2018 with funding of 345 million EUR, to support the research and development of the China Fusion Engineering Test Reactor (CFETR). As part of this project, a full-size CFETR toroidal field (TF) coil will be designed, manufactured and tested by the Institute of Plasma Physics Chinese Academy of Sciences. Two options are being explored in parallel for the TF coil design, using either circle-in-square or rectangular cable-in-conduit conductors (CICCs). The rectangular CICC has been reported to have some merits for a DEMO TF coil, as reported in designs of the EU-DEMO and K-DEMO. First this paper presents the progress in the conceptual design of the CFETR TF coil with rectangular CICCs, according to the most recent reference single-null configuration and radial build of CFETR. Then, electromagnetic analyses are performed to give the magnetic field distributions, toroidal field ripple, in-plane and out-of-plane Lorentz loads. Finally, 3D global and 2D local mechanical analyses are conducted, and the detailed mechanical behavior of the TF coil is illustrated and discussed. Our analyses indicate that the present TF design is reasonable, considering the ITER criteria, and provides valuable insight into the mechanical behavior of the CFETR TF coil system.

Effect of magnetic field ripple on parallel electron transport during microwave plasma heating in the Proto-MPEX linear plasma device

The Prototype Material Plasma Exposure eXperiment (Proto-MPEX) is a linear plasma device located at Oak Ridge National Laboratory to develop the plasma source concept for the Materials Plasma Exposure eXperiment (MPEX). Recent experiments have demonstrated the heating of electrons in a high-density deuterium plasma () via electron Bernstein waves and source plasma production via helicon waves in this linear configuration. Moreover, experimental observations suggest that the magnetic ripple adversely affects the parallel transport of the heated electrons toward the target where material samples are to be exposed to the plasma. To understand the transport process during microwave application, a test-particle Monte-Carlo (MC) code has been developed that incorporates the effects of Coulomb collisions, magnetic mirror adiabatic trapping and electron cyclotron interaction via a quasilinear RF heating operator. MC calculations indicate that the absence of 2nd turning points along the trajectory of cyclotron heated electrons significantly reduces adiabatic trapping and maximizes power transport to the Target surface. Thermalization of cyclotron heating electrons is analyzed with the MC code and it is found that for conditions relevant to Proto-MPEX, a significant fraction of the absorbed heating power is coupled to the Target surface via ballistic fast electrons; however, under certain conditions, up to 60% of the absorbed RF power can be dissipated via thermalization of the fast electrons on the background plasma. Results are compared with experimental measurements. Finally, the implications of the results are discussed in the context of the upcoming MPEX device.

Fusion α loss due to toroidal field ripple perturbation in CFETR

The key issue of burning plasma in tokamak is the loss and confinement of the energetic particles, and one of the loss channels is the perturbation due to discrete toroidal field coils. The work investigated the α ripple loss of CFETR hybrid scenario with plasma transportation code TRANSP/NUBEAM and guiding center code ORBIT, which have considered the initial α distribution, classical steady state slowing down distribution and other models. The results of the work can give the information of the lost particles, including the lost position, fraction and localized heat load on the wall. The study reveals the heat load on CFETR is about 0.13 MW/m2, which is below the safety threshold of ITER, 0.5 MW/m2, and the paper also gives the direction to reduce the α ripple loss.

Mud bedforms in a natural ice flume

An ephemeral, ice-based flume developed in Medway Creek (London, Ontario) during a February thaw when water at ~1oC flowed over the ice surface forming a < 10 cm deep, ~ 3 m wide channel. Eroded muddy bank sediment, composed of silt to medium-sand sized aggregates, formed linear streamers that revealed streaks in the boundary layer. In water 6-8 cm deep with a flow velocity of ~ 8-12 cm/s, mud aggregates were molded into lunate, transverse, and ovoid ripples a few mm high. Clear water allowed mud aggregates in streamers to be observed accreting to, and migrating over mud ripples. Downstream of larger ripples, mud streamers were swept clear of the bed, perhaps due to vortices shed by the ripple. Where flow exceeded ~ 12 cm/s, mud ripples were gradually replaced by mud aggregate streamers which in turn were washed out in an area of faster (undetermined rate) flow. The flow conditions and bedforms in this ephemeral, natural flume are closely comparable to those described from laboratory flumes at 25oC; however the increased viscosity of water at 1oC may alter the stability field of mud ripples.