Fishbone Mode Research Articles

Resistive wall mode and fishbone mode in ITER steady state scenario: roles of fusion-born alphas and plasma flow

Drift-kinetic effects of fusion-born alpha particles on the n= 1 (n is the toroidal mode number) resistive wall mode (RWM) is numerically investigated for a recent design of the ITER 10 MA steady state plasma scenario, utilizing a magneto-hydrodynamic (MHD)-kinetic hybrid toroidal model. While the fluid theory predicts unstable RWM as the normalized plasma pressure β N exceeds the no-wall Troyon limit and with the mode growth rate monotonically increasing with β N, inclusion of the drift-kinetic contribution of trapped alphas qualitatively modifies the behavior by stabilizing the mode at high β N. In fact, a complete stabilization of the n= 1 RWM up to the ideal-wall Troyon limit is found. On the other hand, another unstable branch—the alpha-driven n = 1 fishbone mode (FB)—is identified in the high-β N regime, with the mode frequency matching that of the toroidal precession frequency of trapped alphas. Fast plasma toroidal flow however helps mitigate the FB instability. Kinetic stabilization of the RWM and flow stabilization of the (alpha-triggered) FB result in an enhancement of β N from the design value of 3.22–3.52 for the ITER scenario considered, while still maintaining stable plasma operation against the aforementioned MHD instabilities.

Observation of Doppler shift modulated by the internal kink mode using conventional reflectometry in the EAST tokamak

In this paper we present a new experimental observation using a conventional reflectometry technique, poloidal correlation reflectometry (PCR), in the Experimental Advanced Superconducting Tokamak (EAST). The turbulence spectrum detected by the PCR system exhibits an asymmetry and induced Doppler shift during the internal kink mode (IKM) rotation phase. This Doppler shift is the target measurement of Doppler reflectometry, but captured by conventional reflectometry. Results show that the Doppler shift is modulated by the periodic changes in the effective angle between the probing wave and cutoff layer normal, but not by plasma turbulence. The fishbone mode and saturated long-lived mode are typical IKMs, and this modulation phenomenon is observed in both cases. Moreover, the value of the Doppler shift is positively correlated with the amplitude of the IKM, even when the latter is small. However, the positive and negative frequency components of the Doppler shift can be asymmetric, which is related to the plasma configuration. A simulated analysis is performed by ray tracing to verify these observations. These results establish a clear link between and IKM rotation, and are helpful for studying the characteristics of IKM and related physical phenomena.

Identification of MHD modes on EAST using a deep learning framework

The improvement of plasma parameters is severely limited by magnetohydrodynamic (MHD) instabilities. The identification of MHD modes is crucial for the study and control of MHD instabilities. In this study, an MHD mode identifier is developed based on a temporal convolutional network and long short-term memory (LSTM) network. The identifier is trained and tested on a small dataset containing 33 shots. Firstly, the temporal convolutional network encodes 27 diagnostic signals and then decodes them using LSTM network with different parameters to obtain the MHD modes and their frequency and intensity. The identifier exhibits an accuracy of approximately 98.38% on the test set and can accurately calculate the frequency and intensity of the MHD modes. To further examine the performance of the identifier, seven shots outside the dataset are used for shot-by-shot testing. The identifier can accurately identify the time period of tearing modes, and the identification accuracies of 2/1 and 3/2 tearing modes are 92.7% and 100%, respectively. The identification accuracy of the fishbone mode is slightly worse, only 82.1%. This is because the fishbone mode occurs intermittently. The frequent switching between the fishbone mode and no MHD behavior affects the identification of the fishbone mode. Overall, through the training of the small datasets, the identifier exhibits a good identification performance for the MHD modes. The proposed data-driven identifier can serve as a reference for establishing a large MHD mode database of EAST as well as a real-time MHD identification and control algorithm.

Investigation of performance enhancement by balanced double-null shaping in KSTAR

We report experimental observations on the effect of plasma boundary shaping towards balanced double-null (DN) configuration on the plasma performance in KSTAR. The transition from a single-null to a DN configuration resulted in improved plasma performance, manifested through changes in the pedestal region, decreased density, and core MHD activity variation. Specifically, the DN transition led to a wider and higher pedestal structure, accompanied by grassy edge-localized modes (ELMs) characteristics. The density decrease was a prerequisite for performance enhancement during DN shaping, increasing fast ion confinement. Optimizing the plasma near the core region was associated with the suppression of sawtooth instabilities and the occurrence of fishbone modes during the DN transition. Integrated modeling demonstrated that secondary effects of the DN shaping could increase core thermal energy confinement.

Waveform distortion of off-axis fishbone instability in the nonlinear magnetohydrodynamic evolution in tokamak plasmas

We have investigated the waveform distortion of energetic particle driven off-axis fishbone mode (OFM) in tokamak plasmas with kinetic magnetohydrodynamic (MHD) hybrid simulations. We extended our previous simulations (Li et al 2022 Nucl. Fusion 62 026013) by considering higher-n harmonics in the MHD fluid, where n is toroidal mode number. The waveform distortion is successfully reproduced in the simulation for both magnetic fluctuations and temperature fluctuations. It is clarified that the waveform distortion arises from the superposition of the n = 2 harmonics on the fundamental n = 1 harmonics of OFM, where the n = 2 harmonics are generated by the MHD nonlinearity from the n = 1 OFM. Two types of waveform distortion can occur depending on the phase relationship between the n = 1 and n = 2 harmonics and the relative amplitude of the n = 2 harmonics to the n = 1 harmonics. Lissajous curve analyses indicate that the wave couplings between the n = 1 and n = 2 harmonics with phase-lock and lead to ‘rising distortion’ and ‘falling distortion’, respectively. The two types of waveform distortion can be attributed to the strong shearing profile of radial MHD velocity with n = 2 around the q = 2 magnetic flux surface. The dependence of waveform distortion on viscosity is investigated. It is found that the viscosity which is needed to reproduce the waveform distortion is larger than that in the experiment.

Internal Kink Instabilities Driven by Poloidal Flow via Non-Resonant Excitation in Tokamaks

Non-resonant excitation due to plasma inertia may be dominant in inducing internal kink (IK) instabilities. Poloidal rotation can effectively modify plasma inertia and cause non-resonant excitation to occur. An extended dispersion relation including poloidal rotation is established to study the IK mode and the fishbone (FB) mode. It is found that in rotating plasmas, even for a stable IK mode (i.e., the perturbed potential energy of background plasma δWc is positive) and in the absence of energetic particles (EPs), poloidal rotation can drive the IK mode via non-resonant excitation. Moreover, the IK mode is easy to be driven by poloidal rotation in weak magnetic shear plasmas. Similar to toroidal rotation, when poloidal rotation frequency exceeds a threshold, the FB mode can transform into a branch of a non-resonant mode. The real frequency of the mode, being independent of the precessional frequency of EPs, is just equal to the poloidal rotation frequency. Thus, the non-resonant mode is characterized by the long-lived mode (LLM) observed in toroidal rotating plasmas. The critical gradient of the poloidal rotation profile plays a crucial role in causing the resonant mode to evolve into a non-resonant one; for instance, only for a very peaked poloidal rotation profile can the FB mode transform into the LLM. In addition, the diamagnetic drift frequency of thermal ions can stabilize the FB and the IK modes.

Non-monotonic radial structures of fluctuating temperatures and densities associated with fishbone activities in KSTAR

General characteristics of a fishbone mode in KSTAR are investigated. Fishbone activities are observed with a Mirnov coil, an electron cyclotron emission radiometer (from the core to the edge of plasmas) and an beam emission spectroscopy system (core or edge plasmas) which are measuring fluctuations of poloidal magnetic fields, electron temperatures, and densities, respectively. Temporal evolutions of these fluctuations are similar to the observations from other tokamaks. An interesting and notable feature found in KSTAR fishbone modes is that radial coherence structures of electron temperature and density with respect to magnetic fluctuations are non-monotonic that they have a local minimum at r/a∼0.7 and a maximum at r/a∼0.9 in addition to the usual global peak near the q = 1 surface, where r/a is the normalized minor radius and q is the safety factor. Furthermore, the associated temporal evolution of the electron temperatures in slow-time scale, i.e., less than 1 kHz, with the fishbone activities show that there exist a drop in temperature or increase in temperature depending on inside or outside the q = 1 surface, respectively, from the core to the edge plasmas except that there are almost no temperature changes in the intermediate region which seems to be correlated with the non-monotonic coherence profile. Such a non-monotonic structure and the slow temporal evolution of temperatures are explained with barely trapped resonating fast ions with the banana orbit widths of the order of the minor radius, so that they transit the core and the edge regions simultaneously without trespassing the mid-plane intermediate region.

Effect of deposition location of electron cyclotron resonance heating on active control of fishbone modes in the HL-2A tokamak

Experiment on suppressing fishbone activities is carried out in HL-2A tokamak by electron cyclotron resonance heating (ECRH). To achieve multiple deposition locations of ECRH, the magnetic field is in a range of 1.22–1.4 T from shot to shot. It is found that the fishbone modes exhibit different characteristics at different radial deposition locations. With the same injected power, the effect of off-axis ECRH is much better than that of on-axis heating. The fishbone modes can be completely suppressed when ECRH is deposited nearby the <i>q</i> = 1 rational surface, but would only mitigate in other cases. Further analysis indicate that injection of high power ECRH leads the electron temperature to increase, then the pressure gradient and plasma current density to change, finally safety factor to change and the minimum safety factor to reach a value larger than 1. Meanwhile, M3D-K simulation results show that the growth rate of fishbone mode declines with the increase of <i>q</i><sub>min</sub>. In other words, the growth of safety factor and disappearance of <i>q</i> = 1 rational surface induced by ECRH contribute to the suppression of fishbone activities. The experimental results reported here may not only help to better understand complex effects of ECRH on magnetohydrodynamic instability, but also provide a physics basis for actively controlling the energetic particle driven modes in the future magnetic confined fusion devices.

Multiple interactions between fishbone instabilities and internal transport barriers in EAST plasmas

Fishbone instabilities and internal transport barriers (ITBs) are frequently and sequentially observed in tokamak plasmas. Recently, the relationship between fishbone instabilities and ITBs was numerically studied, mainly on the basis of experimental results (Liu et al 2020 Nucl. Fusion 60 122001). It was identified that a radial electric field can be generated by the fishbone itself, which may act as a trigger for ITB formation. To gain a deeper understanding of this subject, in this work we further demonstrate the multiple interactions between fishbone instability and ITBs in Experimental Advanced Superconducting Tokamak (EAST) experiments (discharge #56933) using the hybrid kinetic-MHD code M3D-K. In multiple-n simulations, it is found that a zonal electric field can be induced in the nonlinear fishbone stage, leading to a relatively large E × B zonal flow that is sufficient to suppress the dominant microinstability before ITB formation; this should account for ITB triggering. After the ITB is triggered, the equilibrium pressure gradient increases and fast ions from the neutral beam injection accumulate in the ITB region. Linear simulations are performed to analyze the effect of ITB formation on fishbone instability. It is shown that due to the change of the pressure gradient during ITB expansion, the change in the bootstrap current density profile modifies the q-profile and then stabilizes the fishbone mode. Additionally, the accumulation of the fast ions leads to a broadening of fast ion distribution around the ITB region, which also has a stabilizing effect on the fishbone mode.

Thermal ion kinetic effects and Landau damping in fishbone modes

The kinetic–magnetohydrodynamic (MHD) hybrid simulation approach for macroscopic instabilities in plasmas can be extended to include the kinetic effects of both thermal ions and energetic ions. The new coupling scheme includes synchronization of the density and parallel velocity between thermal ions and MHD, in addition to pressure coupling, to ensure the quasineutrality condition and avoid numerical errors. The new approach has been implemented in the kinetic-MHD code M3D-C1-K, and was used to study the thermal ion kinetic effects and Landau damping in fishbone modes in both DIII-D and NSTX. It is found that the thermal ion kinetic effects can cause an increase of the frequencies of the non-resonant $n=1$ fishbone modes driven by energetic particles for $q_\mathrm {min}>1$ , and Landau damping can provide additional stabilization effects. A nonlinear simulation for $n=1$ fishbone mode in NSTX is also performed, and the perturbation on magnetic flux surfaces and the transport of energetic particles are calculated.

Theoretical study of long-lived mode in tokamaks

Long-lived mode (LLM) in rotating plasmas is studied by using a alternative dispersion relation. This work offers a theoretical interpretation of LLM observed in experiments [Heidbrink, Phys. Rev. Lett. 57, 835 (1986)10.1103/PhysRevLett.57.835; Chapman, Nucl. Fusion 50, 045007 (2010)10.1088/0029-5515/50/4/045007]. It is found that in rotating plasmas, LLM can transform from the fishbone mode. The real frequency of LLM, independent of the characteristic frequency of energetic ions (EIs), is proportional to the on-axis rotation frequency of bulk plasma. The key conditions to cause transition between the fishbone mode and LLM are weak magnetic shear (for both monotonic and nonmonotonic safety factor profile) and plasma rotation. The fishbone mode can evolve into LLM only when rotating speed is bigger than a critical value in low shear plasmas. The critical beta of EIs to induce LLM is very low. LLM is nonresonantly excited by EIs, and the driving sources mainly come from plasma inertia.

Simulation study of energetic-particle driven off-axis fishbone instabilities in tokamak plasmas

Kinetic-magnetohydrodynamic hybrid simulations were performed to investigate the linear growth and the nonlinear evolution of off-axis fishbone mode (OFM) destabilized by trapped energetic ions in tokamak plasmas. The spatial profile of OFM is mainly composed of m/n = 2/1 mode inside the q = 2 magnetic flux surface while the m/n = 3/1 mode is predominant outside the q = 2 surface, where m and n are the poloidal and toroidal mode numbers, respectively, and q is the safety factor. The spatial profile of the OFM is a strongly shearing shape on the poloidal plane, suggesting the nonperturbative effect of the interaction with energetic ions. The frequency of the OFM in the linear growth phase is in good agreement with the precession drift frequency of trapped energetic ions, and the frequency chirps down in the nonlinear phase. Two types of resonance conditions between trapped energetic ions and OFM are found. For the first type of resonance, the precession drift frequency matches the OFM frequency, while for the second type, the sum of the precession drift frequency and the bounce frequency matches the OFM frequency. The first type of resonance is the primary resonance for the destabilization of OFM. The resonance frequency which is defined based on precession drift frequency and bounce frequency of the nonlinear orbit for each resonant particle is analyzed to understand the frequency chirping. The resonance frequency of the particles that transfer energy to the OFM chirps down, which may result in the chirping down of the OFM frequency. A detailed analysis of the energetic ion distribution function in phase space shows that the gradient of the distribution function along the E′ = const. line drives or stabilizes the instability, where E′ is a combination of energy and toroidal canonical momentum and conserved during the wave–particle interaction. The distribution function is flattened along the E′ = const. line in the nonlinear phase leading to the saturation of the instability.

Transition from fishbone mode to β-induced Alfvén eigenmode on HL-2A tokamak

In the presence of energetic particles (EPs) from auxiliary heating and burning plasmas, fishbone instability and Alfvén modes can be excited and their transition can take place in certain overlapping regimes. Using the hybrid kinetic-magnetohydrodynamic model in the NIMROD code, we have identified such a transition between the fishbone instability and the β-induced Alfvén eigenmode (BAE) for the NBI heated plasmas on HL-2A. When the safety factor at magnetic axis is well below one, typical kink-fishbone transition occurs as the EP fraction increases. When q 0 is raised to approaching one, the fishbone mode is replaced with BAE for sufficient amount of EPs. When q 0 is slightly above one, the toroidicity-induced Alfvén eigenmode dominates at lower EP pressure, whereas BAE dominates at higher EP pressure.

Chirping modes in hybrid tokamak discharges

In the so-called “hybrid” shots characterized by shear-free core with safety factor slightly above 1, an interesting yet unexplained chirping activity has been observed in tokamaks with a high fast ion content. Namely, fishbone modes with m=n=1 and downward frequency chirping are accompanied by frequency jumps of the tearing modes with m=n+1 and n > 1 [here m(n) is the poloidal (toroidal) mode number]. In the present work, these related phenomena are attributed to a combination of the two impulsive toroidal torques: (1) recoil torque applied to the shear-free core during fishbone burst and (2) electromagnetic torque applied to the tearing layer. We have developed a quasi-linear model in which fast transients of the plasma toroidal rotation induced by these torques qualitatively reproduce the observed chirping dynamics.

Experimental observation of low-frequency magnetohydrodynamic instabilities driven by energetic electrons in low hybrid current drive plasmas

Energetic electrons driving low-frequency magnetohydrodynamic instabilities, e.g., electron fishbone (eFB) modes and electron beta-induced Alfvén eigenmodes (eBAEs), are found in co- and counter-current drive low hybrid current drive (LHCD) plasma in HL-2A, respectively. The eBAEs are found in LHCD plasma for the first time. Two branches eFB modes are observed in the core of plasma, and they can transit from the high-frequency one to the low-frequency one continuously. The different mode structures and positions are obtained by tomography of soft x-ray arrays. The frequency jump phenomena of eFB modes are also found. Two eBAEs, with poloidal and toroidal mode numbers m/n = 3/1 and 5/2, are found in the edge of the plasma. The single m/n = 3/1 mode is found when the LHCD power (P LHCD) is 0.44 MW. The strong m/n = 5/2 mode coexists with the weak 3/1 mode when P LHCD = 0.52 MW. Although the current drive direction of LHCD is opposite to the plasma current, the two eBAEs also propagate in the electron diamagnetic drift direction poloidally, which is the same direction as the eFB modes in co-current drive LHCD plasma. With increasing P LHCD (or changed current profile), the value of the edge safety factor (q) decreases. The evolutions of the mode numbers of eBAEs may be related to changes in the edge q factors.

Progress of Experimental Studies in the HL-2A Tokamak

During the last several years, the HL-2A experiment has made significant progress in the following areas: (1) lower-hybrid wave (LHW) heating and current drive, (2) plasma confinement and turbulent transport, (3) magnetohydrodynamic (MHD) instabilities and energetic particle physics and (4) H-mode and edge localized mode (ELM) control. The results show that the LHW system working in the co-current mode can reach higher driving efficiency and full non-inductive lower-hybrid current drive (LHCD) has been achieved. The intrinsic poloidal torque characterized by the divergence of the residual stress is deduced from synthesis for the first time. The dynamics of spectral symmetry breaking in drift wave turbulence is in good agreement with the development of the poloidal torque to drive the edge poloidal flow. The influence of the cross-phase dynamics on turbulent stress was also investigated. The ion internal transport barrier has been observed in the NBI-heated plasma, and inside the barrier the ion thermal transport is reduced to the neoclassical level. Besides, micro-turbulence is modulated by the rotation frequency of the magnetic island, and this modulation effect is related to a critical island width. Strong E × B shear is found at the island boundary. Three kinds of axisymmetric modes, beta-induced Alfven eigenmode (BAE), toroidal Alfven eigenmode (TAE) and the ellipticity-induced Alfven eigenmode (EAE), are found to be driven unstable by nonlinear mode coupling between Alfven eigenmodes and tearing mode which is well explained by the nonlinear gyrokinetic theory. The fishbone and tearing modes were actively controlled by the electron cyclotron resonance heating (ECRH). The dynamics of the edge plasma flows and turbulence during the L–I–H transition have been dedicatedly investigated. The geodesic acoustic mode (GAM) and limit cycle oscillation (LCO) coexist for a short time and disappear in the H-mode plasma with the increasing of E × B shear flow before the I–H transition, which plays an important role in the turbulence suppression. Different techniques, such as LHW, ECRH, resonant magnetic perturbation (RMP), and impurity seeding by the laser blow-off (LBO) and supersonic molecular beam injection (SMBI), have been successfully applied to control the large ELMs. It has been found that pedestal turbulence enhancement might be responsible for the observed mitigation effect.

Dependence of fishbone cycle on energetic particle intensity in EAST low-magnetic-shear plasmas

The dependence of fishbone cycle on energetic particle intensity has been investigated in EAST low-magnetic-shear plasmas. It is observed that the fishbone mode growth rate, saturation amplitude as well as fishbone cycle frequency clearly increase with increasing neutral beam injection (NBI) power. Moreover, enhanced electron density and temperature perturbations as well as energetic particle loss were observed with greater injected NBI power. Simulation results using M3D-K code show that as the NBI power increases, the resonant frequency and the energy of the resonant particles become higher, and the saturation amplitude of the mode also changes, due to the non-perturbative energetic particle contribution. The relationship between the calculated energetic particle pressure ratio and fishbone cycle frequency is obtained as ${f_{\textrm{FC}}} = 2.2{(1000{\beta _{\textrm{ep,calc}}} - 0.1)^{5.9 \pm 0.5}}$ . Results consistent with the experimental observations have been achieved based on a predator–prey model.

Nonlinear dynamics of the fishbone-induced alpha transport on ITER

The fishbone-induced transport of alpha particles is computed for the ITER 15 MA baseline scenario (ITER Physics Expert Group on Energy Drive 1999 Nucl. Fusion 39 2471) using the non-linear hybrid Kinetic-MHD code XTOR-K. Two limit cases have been studied, in order to analyse the characteristic regimes of the fishbone instability: the weak kinetic drive limit (Berk et al 1999 Phys. Plasmas 6 3102) and the strong kinetic drive limit (Zonca et al 2015 New J. Phys. 17 013052). In both those regimes, characteristic features of the n = m = 1 fishbone instability are recovered, such as a strong up/down-chirping of the mode frequency, associated with a resonant transport of trapped and passing alpha particles. The effects of the n = m = 0 sheared poloidal and toroidal plasma rotation are taken into account in the simulations. The shear is not negligible, which implies that the fishbone mode frequency has a radial dependency, impacting the wave-particle resonance condition. Phase space hole and clump structures are observed in both non-linear regimes, centered around the precessional and passing resonances. These structures remains attached to the resonances as the different mode frequencies chirp up and down. In the non-linear phase, the transport of individual resonant trapped particles is identified to be linked to mode-particle synchronization. On this basis, a partial mechanism for the non-linear coupling between particle transport and mode dominant down-chirping is proposed. The overall transport of alpha particles through the q = 1 surface is of order 2–5\\% of the initial population between the simulations. The loss of alpha power is found to be directly equal to the loss of alpha particles.

Experimental progress of hybrid operational scenario on EAST tokamak

Extensive experiments of advanced scenario development, which contribute to the ITER hybrid operational scenario have been carried out on experimental advanced superconducting tokamak (EAST) tokamak recently with the ITER-like tungsten divertor. The in this operational scenario is intermediate up to 2.1 (EAST#78987, ∼ 2.1, I p ∼ 0.45 MA, q 95 = 3.7, ∼ 1.5 T, 3 MW neutron beam injection and 1 MW 4.6 GHz lower hybrid wave). In these hybrid H-mode plasmas, the internal transport barrier (ITB) has been frequently observed with central flat q profile and it is found that the fishbone mode (m/n = 1/1) can be beneficial to sustain the central flat (q(0) ∼ 1) q profile, thus a stable ITB can be obtained. In this case, better plasma performance is achieved. The formation of the ITB of the electron density is related to the fishbone activities. Energy transport analysis shows that the fishbone instabilities have a suppression on electron turbulent energy transport, while the ITB of ion temperature is due to the suppression of high-k modes (electron temperature gradient). The mechanism of turbulence suppression from fishbone instabilities in the EAST tokamak is not clear and needs more investigation. It is also observed that the power threshold for ITB formation is 3.5 MW, which is consistent with the scaling law for other tokamaks. The dimensionless parameter () obtained in the EAST reaches 0.3, but is still lower than the ITER hybrid scenario design (G ≥ 0.4) and needs more extension. Further investigation of extending the operational regimes, such as expanding the ITB foot outwards, would be important for the development of the hybrid and steady-state scenarios for next-step fusion devices like ITER and CFETR.

Synergistic effect of Coriolis and centrifugal forces from poloidal flow on internal kink and fishbone modes in tokamak plasmas

The synergy of Coriolis force and centrifugal force is proposed to study the influence of poloidal plasma rotation on internal kink and fishbone modes. A new dispersion relation is established by making use of energy principle when Coriolis and centrifugal forces are taken into account in the momentum equation. The significant discovery is that the destabilizing (stabilizing) effect of poloidal flow on internal kink (fishbone) mode is greatly increased due to the synergy of Coriolis and centrifugal forces. Poloidal flow can neither effectively destabilize internal kink mode nor stabilize fishbone mode with any one of both centrifugal force and Coriolis force being excluded. It is most interesting that the internal kink mode, being stable with positive δWc (perturbed potential energy of bulk plasma), is unstable when poloidal rotation frequency exceeds a threshold. It is difficult for poloidal flow without shear to destabilize the internal kink mode with δWc > 0. The physical mechanism of poloidal flow destabilization of internal kink mode mainly comes from the modification of plasma inertial due to Coriolis and centrifugal forces.