The ponderomotive force is an important concept in plasma physics and, in particular, plays an important role in many aspects of the theory of laser plasma interactions including current concerns like wakefield acceleration and Raman amplification. The most familiar form of this gives a force on a charged particle that is proportional to the slowly varying gradient of the intensity of a high frequency electromagnetic field and directed down the intensity gradiant. For a field amplitude simply oscillating in time there is a simple derivation of this formula, but in the more general case of a travelling wave the problem is more difficult. Over the years there has been much work on this using Hamiltonian or Lagrangian averaging techniques, but little or no investigation of how well these theories work. Here we look at the very basic problem of a particle entering a region with a monotonically increasing electrostatic field amplitude and being reflected. We show that the equation of motion derived from a widely quoted ponderomotive potential only agrees with the numerically computed orbit within a restricted parameter range and that outside this range it shows features which are inconsistent with any ponderomotive potential quadratic in the field amplitude. Since the ponderomotive force plays a fundamental role in a variety of problems in plasma physics we think that it is important to point out that even in the simplest of configurations standard theories may not be accurate.
Abstract. We consider the action of the ponderomotive force of low-frequency Alfv´en waves on the distribution of the background plasma. It is assumed that the ponderomotive force for traveling waves arises as a result of the background inhomogeneity of medium under study. Expressions for the ponderomotive force obtained in this paper differ from pre- vious analogous results. The induced magnetic moment of medium is taken into account. It is shown that the well- known Pitayevsky’s formula for the magnetic moment is not complete. The role of the induced nonlinear thermal pressure in the evolution of the background plasma is considered. We give estimations for plasma displacement due to the long- and short-acting nonlinear wave perturbations. Some discus- sion of the ponderomotive action of standing waves is pro- vided.
is damped, this motion is temporary and as soon as the ampli- tude goes below the critical amplitude, the rain eventually falls towards the surface. We sought to identify these characteristics in several observational studies of coronal rain in the presence of transverse oscillations. We found that up-and-down motions of coronal rain is indeed seen. In one study we found that though we can explain the observed rain speeds, the amplitude neces- sary to explain the long periodicity is about ten times larger than what is observed. It is not likely that this discrepancy can be un- derstood by a line-of-sight effect of the plane of polarisation of the oscillation. We investigated two further observational studies where sub-ballistic rain is seen in the presence of oscillations. We found that the required amplitude also tends to be about an order of magnitude higher than what is observed. Therefore, the ponderomotive force is not likely to be the primary physical rea- son for sub-ballistic fall of coronal rain in most cases. But this is not surprising since sub-ballistic motion has been observed in the absence of any oscillations. This study, however, allows us to evaluate the contribution from the ponderomotive force on coronal rain kinematic.
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of P x calculated by the second-order backward Lie trans- formation. From Eq. (37), we see that θ, the normalized period to the phase η , increases as the curvature of the field and/or the laser field amplitude increases. Remarkably, the unbounded secular motion originating from the first-order ponderomotive force given in Eq. (30) is changed to the bounded solution, Eqs. (35) and (36), by taking into ac- count the second order curvature terms. This motion cor- responds to a betatron oscillation by which the particle is confined to a finite radial region.
Clearly there are approximations involved in the calculation of the average potential and kinetic energy, the assumed particle displacement and velocity not being exact, so the question arises as to how well this formula works. Despite the fact that it is a simple matter to integrate the equation for the exact particle orbit in the field we describe, little or no work has, so far as we know, been done to verify the accuracy of the various expressions for the ponderomotive force that have appeared in the literature. We shall do this for the very basic case of a particle moving in an electrostatic travelling wave with a monotonic intensity profile and show that in certain parameter ranges the motion is not described by the above equation and, indeed, cannot be predicted from any averaged Lagrangian with a potential that is proportional to the square of the field intensity. This suggests that it is necessary, in these regimes, to consider ponderomotive potentials to higher order in the field amplitude, a problem we do not believe to have been considered in the previous literature.
The simulation of moving rainwater droplets requires the determination of the ponderomotive force, and the concentration on the total force F is a reasonable simplification with respect to other influences like the weather. The computation of the electric fieldaround the droplet is a computationally expensive task, which is dealt with finite integration techniques in 5, 6 or by finite elements on an adaptive triangular grid in 4. Similarly, a related problem is solved in 7 in the investigation of ferromagnetic fluids.
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Initially, the loop top is displaced downwards by the mass of the condensation region, resulting in onset of a vertically po- larised transverse loop oscillation. For low condensation region densities (µ = 0.19 to µ = 0.54) the cool plasma is confined to the loop top by combination of the pressure from the underly- ing plasma and the ponderomotive force resulting from the loop oscillations directed towards the antinode of the oscillation, that is, the loop apex for the fundamental harmonic (Verwichte et al. 2017). The amplitude and period of the loop oscillation stays approximately constant for the duration of the simulation. For the high condensation region densities (µ = 0.62 to µ = 0.83) the cool plasma falls towards the loop footpoints along both sides of the loop similar to a real coronal rain shower scenario. This results in long period, large amplitude oscillations transi- tioning into persistent, smaller amplitude, shorter period oscil- lations once the condensations have fallen into the lower loop legs, where their mass has negligible e ff ect on the loop oscil- lation period. The downfalling plasma blobs elongate as they fall due to the di ff erential e ff ective gravity component acting along the finite length of the blob. The blobs develop elongated wings, leaving an evacuated region behind. The development of this v-shape is likely caused by the fact that the motion of the blob is subsonic in the background plasma but supersonic for its own temperature, leading to shock behaviour. The blobs are then gradually slowed down and ultimately rebound multiple times in the lower loop leg due to the combined e ff ect of the under- lying plasma pressure and magnetic tension force counteracting the e ff ective gravity (Mackay & Galsgaard 2001). This type of
The ponderomotive influence of ion cyclotron waves on the field-aligned distribution and motion of ions in the equatorial zone of the magnetosphere is examined. The hydrodynamic, quasi-hydrodynamic and “test-particle” approaches are used for the study of ponderomotive wave-particle interaction. Particular attention has been given to the challenging questions encountered in applying the general theory to the magnetospheric physics. The closed system of quasi-linear equations describing the ponderomotive effects is derived. Analytical investigation of the basic equations has demonstrated the diverse manifestations of the ponderomotive impact of ion cyclotron waves on the ion population in the magnetosphere. It is found that the redistribution of ion density under the action of ponderomotive force with increase in the wave amplitude follows the pattern of phase transition of the second kind. The density distribution changes qualitatively as the point in plane of the governing parameters of system crosses a demarcation line. It has been found that the magnetic equator is an attractor for heavy ion. The period of the finite (oscillatory) motion of a heavy ion, which is trapped in the potential trough in the vicinity of magnetic equator, depends on the wave frequency, wave amplitude, together with the energy of motion. In addition, the diffusion equilibrium of ions in a multicomponent plasma is considered, and the ponderomotive separation of ions in a binary mixture is demonstrated. It is shown that the heavy ions collect near the magnetic equator provided the waves are comparatively strong. It suggests that the ponderomotive effects play a part in formation of structure and dynamics of the magnetosphere.
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period. Case (i) corresponds to dominant relativistic nonlinearity while case (ii) refers to the situation when the ponderomotive  and relativistic  nonlinearities are simultaneously operative. In such a case the nonlinearity in the dielectric function occurs is caused by the electron mass variation due to large laser irradiance and the change in electron density as a consequence of the ponderomotive force. Very few studies on self focusing [35, 36] and cross focusing  of the laser beams have been made, incorporating the combined effect of relativistic and ponderomotive nonlinearities. Further the effect of an ultra intense laser pulse on the propagation of an electron plasma wave has been analyzed by Kumar et al.  in the relativistic-ponderomotive regime. Most of the analyses on self focusing of the laser beams are devoted to the Gaussian beams [3–5, 33–43]; nevertheless a few studies have also been published on the self focusing of super Gaussian beams [44– 46], self trapping of degenerate modes of laser beams , and self trapping of Bessel beams  by considering the nature of different irradiance distributions of the beams. Apart from these investigations, recently the optical beams with central shadow, usually known as dark hollow beams (DHB) have received much attention from the physics community because of their wide and attractive applications in the field of modern optics, atomic optics and plasmas [49–52]; numerous experimental techniques [53–55] have also been developed for the production of the DHBs. Further for the explanation of the dynamics and other propagation characteristics, several theoretical models for DHBs [the beam with zero central intensity] like the TEM 01 mode doughnut beam, some higher order Bessel beams, superposition of off-axis Gaussian beams and dark-hollow Gaussian beams etc. have been introduced [56–59]. In relatively recent studies the propagation of DHBs in paraxial optical systems  and turbulent atmospheres  has been discussed in detail.
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Abstract. We discuss two different physical processes that create localized high β plasma regions. One is nonlinear wave-steepening, generating magnetic decreases (MDs) by a ponderomotive force. The other is the mirror instability generating alternating high and low β plasma regions. It is demonstrated that MDs and mirror modes are observa- tionally quite different structures. MDs spatially occur in interplanetary space and mirror modes primarily in plane- tary magnetosheaths. MDs are characterized by: 1) variable (exponentially decreasing number with increasing) angular changes, 2) variable (exponentially decreasing) thicknesses, and 3) no characteristic inter-event spacings. In sharp con- trast, mirror modes are characterized by: 1) little or no an- gular changes across the structures, 2) a characteristic scale size, and 3) are quasiperiodic in nature.
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In this article, we demonstrate the prominent impact of the RR effect on the collective dynamics of radiating elec- trons, which tends to contain them in the highest field areas (in opposition to the ponderomotive force which expels them) during the interaction of an ultraintense laser pulse with a deuterium target. This can produce a compact high energy synchrotron radiation source of the order of tens of MeV over the laser pulse duration. We introduce an analytical model based on a kinetic ap- proach, which confirms the main characteristics of the radiating electron source that are observed in Particle- In-Cell (PIC) simulations. This builds upon the previous studies of the RR effect on electron beams during the in- teractions with a laser pulse [17–19], by extending to a laser-plasma interaction.
Photon acceleration also plays a role in laser modulational instability [45–47]. This instability occurs when the energy of a long laser pulse is bunched longitudinally by a co-propagating plasma wave. The underlying process is periodic acceleration and deceleration of the laser’s photons by the density fluctuations of the plasma wave. In turn, the laser’s ponderomotive force helps to enhance the plasma wave, creating a positive feedback loop (see also Ref.  for the relationship between the ponderomotive force and the photon dispersion in plasma). The laser amplitude modulation due to this instability can be seen in Fig. 1, frames (c) and (d).
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In this paper, we discuss the ion acceleration due to the PA combined with the ICR mechanism. We have investi- gated whether or not the PA/ICR acceleration scheme can configure an eﬃcient electric thruster compared with the VASIMR type thruster. Dodin et al.  have theoretically analyzed that the ponderomotive eﬀect can be optimized by minimizing the eﬀect of the transverse heating. How- ever, their study has focused on single-particle dynamics. The analysis relevant to the application to electric thrusters under realistic parameters has not been performed. Em- sellem  has proposed to apply the ponderomotive force to an electric thruster by using a fluid simulation. However, he did not discuss the resultant thrust at all. In this paper, we statistically evaluate the energy gain by the PA / ICR scheme, and classify the results in terms of the adiabatic parameter, Λ = L B Ω 0 /v 0|| , where L B is the axial divergent
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As in any empirical analysis, we first discuss availability and quality of relevant data. We use the original source reporting the estimates of labor force participation rate in the U.S. - the BLS (2008). As clear from Figure 1, the original growth rate is characterized by a high volatility induced by relatively low accuracy of corresponding measurements. Therefore, we use MA(5) in our comparisons of observed and predicted time series. One also has to bear in mind that the difference between the LFP measured according to the Census Bureau's definition and some true LFP, which represents the variable for a valid quantitative analysis. Due to numerous revisions to the CB's definition over time this difference is also time dependent and introduces significant noise in the analysis.
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Bite forces were obtained from nine male (1.36±0.28 g) and 10 female (0.55±0.10 g) beetles with an isometric Kistler force transducer (type 9203, Kistler Inc., Winterthur, Switzerland) connected to a Kistler charge amplifier (type 5058A). The force transducer was mounted in a setup with two small plates that were grasped by the mandibles, thus enabling us to measure bite forces. Spacing between the bite plates could be adjusted [for a detailed description of the experimental setup, see Herrel et al. (Herrel et al., 1999)]. The analog signals were A/D converted (USB-6009, 14-bit, National Instruments, Austin, TX, USA) and recorded at a sample frequency of 200 Hz using a purpose-written LabVIEW routine (LabVIEW 9.0.1f2, 32- bit version, National Instruments). When a defensive response was provoked, males bit the bite plates with the large protrusions positioned halfway along the mandibles (P; see Fig. 5). In real fights, males also regularly (but not exclusively) use the mandibular protrusions. These were selected as the experimental bite point because: (1) they provide point contact with the bite plates, (2) they guarantee standardized experimental conditions within and between individuals (i.e. similar location of force application) and (3) males were eager to bite with them. Females bit with the tip of their mandibles, the only possibility given their jaw size and morphology. The minimal distance between the bite plates (imposed by the constraints of the setup) was 3 mm for both sexes. The maximal possible bite plate distance (behaviorally imposed: when the spacing became too large, specimens refused to bite) was 5 mm for females and 12 mm for males. We varied the distance between the bite plates between these boundaries with intervals of 1 mm. The sequence was determined randomly. For every plate distance (and hence mandibular angle) three recordings of 30 s were made, and the highest force observed in these 90 s recording was retained as the maximal bite force for that individual.
The analytical model is based on the simple assumption that minimum curvature of a flight trajectory is constrained by the fly’s maximum locomotor capacity and thus by the limits of total aerodynamic force production. We further assume that the production of moments around the three body axes requires only negligible aerodynamic force and that total force balance can thus be reduced to the vector sum of upward force, thrust and side slip. This assumption is fostered by results obtained during object orientation behaviour of tethered flying fruit flies that modulate their yaw moments within a range of approximately ±1.0 nN m peak to peak (Heisenberg and Wolf, 1984), using graduated alterations in the bilateral difference in wing stroke amplitude (Götz, 1983). We approximated the mean length of the moment arm for yaw turning between the fly’s centre of gravity and the aerodynamic force vector to be 2.1 mm, assuming that the chord-wise aerodynamic circulation is at maximum close to a span-wise location of 65% wing length [wing length, 2.5 mm (Lehmann, 1994; Birch and Dickinson, 2001; Lehmann and Pick, 2007; Ramamurti and Sandberg, 2001)]. Consequently, the production of a yaw turning moment of 1.0 nN m would require an aerodynamic force of approximately 0.47 μ N, which is only approximately 3% of the flight force Drosophila produces at maximum locomotor performance (Lehmann, 2004; Lehmann and Dickinson, 1997). The numerical model also requires that the fly produces lateral forces during turning due either to banking (roll turn) towards the inner curve side or to modification of wing kinematics without large changes in body posture, similar to those observed in hover flies (Collett 1980a; Collett, 1980b). We moreover ignored body inertia during translational acceleration and the mass moment of inertia during rotational accelerations of the animal because a recent study showed that yaw turning is dominated by drag on the wings and not body inertia (Hesselberg and Lehmann, 2007).
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Comparison with models of pennate muscle architecture Our model of segmented musculature is similar to the most widely used model for relating muscle fiber strain and fiber force to tendon excursion and muscle force in pennate musculature (Benninghoff and Rollhäuser, 1952; Gans and Bock, 1965; Alexander, 1968). More sophisticated pennate muscle models have also been developed, in which curved muscle fibers and deformation of the aponeuroses have been modeled (Woittiez et al., 1984; Huijing and Woittiez, 1984; Zuurbier and Huijing, 1992; Van Leeuwen and Spoor, 1993). The basic pennate model assumes that the distance between the tendon sheets, usually drawn as the width of the muscle, does not change during muscle contraction (Otten, 1988). With this assumption, the pennate model is mathematically identical to our y =1 model of segmented musculature, rotated 90° such
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relatively upright posture of the limbs should increase normal force and decrease shear force because body mass is being accelerated in a horizontal direction (or more force is redirected through the limbs in a horizontal direction). Thus, this posture (and decrease in speed) should also decrease shear force when the cows move on relatively slippery surfaces. By contrast, Kapatkin et al. found no differences in peak forces or impulses (vertical and craniocaudal components of the SRF) generated by dogs trotting on linoleum versus carpet-surfaced trackways (Kapatkin et al., 2007). Therefore, dogs make no change in shear force, at least when running on carpet versus linoleum. Finally, Tokay geckos may increase shear force (up to twice the adhesive force, depending on the direction of shear force relative to the setae) within arrays of thousands of setae to cause powerful friction and adhesion forces that can prevent a gecko from slipping from smooth, dry glass (Autumn et al., 2006). My comparison among these three groups is simplistic, mainly because the data collected and the taxa are so varied. Nevertheless, it demonstrates astonishing diversity among tetrapods in how they cope with traveling on relatively smooth surfaces.
One of the reasons why surface forces are of interest is their importance in the interaction between colloid particles so it would be useful to be able to measure these forces directly. Until recently this has not been possible because of the difficulty in manipulating microscopic objects and in measuring the minute forces involved. Since the magnitude of surface forces depends on the radius of curvature of the substrate, the normalized surface force could be measured more accurately between surfaces with large radius. For example, in the SFA technique, the radius of substrates is about 2 cm, and this requires a resolution of about 1 |iN for typical double-layer experiments. For analogous measurements on a 2 pm particle a resolution of about 0.1 nN is required. However, the development of the Scanning Tunneling Microscope (STM) by Binnig and Rohrer in 198294 and its spectacular success in microscopic measurement have changed the scale on which scientists contemplate measurement. One implication of these developments for surface force measurement is that the technology now exists for measurement of forces on colloid-sized objects. Measurements of this kind are described in Chapters 5 and 6. This section contains a brief description of the STM and the derivative Atomic Force Microscope (AFM) which have enabled these measurements.
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Most basidiomycete fungi actively eject their spores. The process begins with the condensation of a water droplet at the base of the spore. The fusion of the droplet onto the spore creates a momentum that propels the spore forward. The use of surface tension for spore ejection offers a new paradigm to perform work at small length scales. However, this mechanism of force generation remains poorly understood. To elucidate how fungal spores make effective use of surface tension, we performed a detailed mechanical analysis of the three stages of spore ejection: the transfer of energy from the drop to the spore, the work of fracture required to release the spore from its supporting structure and the kinetic energy of the spore after ejection. High-speed video imaging of spore ejection in Auricularia auricula and Sporobolomyces yeasts revealed that drop coalescence takes place over a short distance (~5 μ m) and energy transfer is completed in less than 4 μ s. Based on these observations, we developed an explicit relation for the conversion of surface energy into kinetic energy during the coalescence process. The relation was validated with a simple artificial system and shown to predict the initial spore velocity accurately (predicted velocity: 1.2 m s –1 ;