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Dec 1975

Volume 18, Issue 12, pp. 1595-1826

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Nonlinear behavior of line vortices

Matilde Macagno and Enzo Macagno

Phys. Fluids 18, 1595 (1975); http://dx.doi.org/10.1063/1.861075 (9 pages) | Cited 3 times

Online Publication Date: 3 September 2008

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A simple model is formulated following classical lines and used to investigate the decay of line vortices in which both self‐induced and ambient turbulence may exist. The effect of molecular viscosity is included, since it may be important at certain times and in certain regions of the vortices. Circulation overshoot, predicted for the first time by Saffman, is also predicted by this model when self‐induced turbulence is taken into account. This occurs even if no trace of overshoot is present initially. Experimental data of Focke, which exhibit overshoot, are used to test the model. The presence of a logarithmic portion in the circulation profiles of turbulent vortices is discussed critically.
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47.15.-x Laminar flows

Motion of a vortex pair approaching an opening in a boundary

Michael Karweit

Phys. Fluids 18, 1604 (1975); http://dx.doi.org/10.1063/1.861076 (3 pages) | Cited 5 times

Online Publication Date: 3 September 2008

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The flow properties of liquid helium are such that the motions of vortex rings and line vortices can be calculated from hydrodynamic potential theory. The two‐dimensional (line vortex) case allows a relatively simple method of calculating vortex motion: The ’’Routh path function’’ yields a solution through conformal mapping. Using this method, the case of a rectilinear vortex pair approaching a slit in a wall is analyzed. Criteria for their just passing through are found.
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67.25.dk Vortices and turbulence

Experimental results for oscillating flow in a curved pipe

B. R. Munson

Phys. Fluids 18, 1607 (1975); http://dx.doi.org/10.1063/1.861077 (3 pages) | Cited 13 times

Online Publication Date: 3 September 2008

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Experimental results are presented describing the character of the secondary flow developed within a curved pipe for which the primary (axial) flow is driven by an oscillating pressure gradient. The comparison between previous theory and experiment is excellent.
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47.15.-x Laminar flows

Burgers’ model with a renormalized Wiener–Hermite representation

W. C. Meecham, P. Iyer, and W. C. Clever

Phys. Fluids 18, 1610 (1975); http://dx.doi.org/10.1063/1.861078 (7 pages) | Cited 4 times

Online Publication Date: 3 September 2008

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The use of the Wiener–Hermite expansion for the turbulence problem is reviewed. The expansion is known to give good results for lower Reynolds’ number flows, up to a fluctuation Reynolds’ number of 20 using more recent time‐dependent bases. The use and meaning of these bases is discussed. A new development in Wiener–Hermite expansion is used to calculate Burgers’ model: It is known that these expansions are not unique; by taking advantage of this arbitrariness, a renormalization is presented and used to improve the convergence of the expansion. The procedure involves calculation for turbulence using a time‐dependent base for a short time. The calculation is then stopped and the resulting functions are adjusted in such a way as to minimize the non‐Gaussian part of the energy, at the same time preserving the last value of the transfer function. Following this, the calculation is resumed using these new, adjusted values for the functions; then the process is repeated. For the first time it was possible to obtain the equilibrium form k−2, of the energy spectrum for Burgers’ model of turbulence. The calculation proceeds, holding the non‐Gaussian part of the energy to but a few percent. Good results are obtained up to fluctuation Reynolds’ numbers of 100.
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47.27.Gs Isotropic turbulence; homogeneous turbulence

Investigation into the anomalous behavior of Pitot tubes in dilute polymer solutions

N. A. Halliwell and A. K. Lewkowicz

Phys. Fluids 18, 1617 (1975); http://dx.doi.org/10.1063/1.861079 (9 pages) | Cited 5 times

Online Publication Date: 3 September 2008

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The anomalous behavior of Pitot tubes in the turbluent pipe flow of a dilute polymer solution is investigated and discussed. Pitot tube error is measured using an alternative optical means of local velocity measurement. Unlike previous research in this field a strict control is kept on the state of degradation of the solutions and Pitot tube error is measured directly whilst the tube is under test. Errors which occurred in total head pressure measurement were found to be negative and in some cases were as high as 40%. Factors which influence the Pitot tube accuracy are identified and a dynamical similarity argument is used to express the error as a function of these. Pitot tube error is attributed to the polymer solutions storing energy viscoelastically at the tube tip. A critical review of explanations previously put forward for Pitot error with reference to the results is included in the discussion. The polymer used was polyox WSR‐301.
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47.15.-x Laminar flows

The role of polydispersity in the mechanism of drag reduction

Donald L. Huntson and Michael M. Reischman

Phys. Fluids 18, 1626 (1975); http://dx.doi.org/10.1063/1.861080 (4 pages) | Cited 7 times

Online Publication Date: 3 September 2008

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The effects of polydispersity in drag reduction were studied by testing two narrow molecular weight distribution polystyrene samples individually and in mixtures. The results indicate that the largest molecules in the distribution determine the value of the onset point and that this value decreases with increasing molecular weight. Above the onset point the magnitude of the drag reduction effect is shown to be a complex function of both molecular weight and polydispersity.
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47.15.-x Laminar flows
46.35.+z Viscoelasticity, plasticity, viscoplasticity

Series solution for the planar asymmetric blunt‐body problem

Leonard W. Schwartz

Phys. Fluids 18, 1630 (1975); http://dx.doi.org/10.1063/1.861081 (9 pages) | Cited 1 time

Online Publication Date: 3 September 2008

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Accurate semi‐analytic solutions to the inverse blunt‐body problem have been obtained using a method of series expansion. Rational fractions are used for series summation and analytic continuation. Angles of incidence up to 30° and Mach numbers as low as 2 have been considered. The maximum entropy streamline will not wet the body surface in asymmetric flow. It may pass either above or below the stagnation streamline. Limit lines appear in the supersonic portion of the flow field, both in the shock layer and in its upstream analytic continuation.
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47.40.Ki Supersonic and hypersonic flows

Finite‐element analysis of compressible and incompressible viscous flow and heat transfer problems

Trifon E. Laskaris

Phys. Fluids 18, 1639 (1975); http://dx.doi.org/10.1063/1.861082 (10 pages) | Cited 6 times

Online Publication Date: 3 September 2008

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There are numerous problems of two‐dimensional viscous, compressible or incompressible steady‐state flows where the governing hydrodynamic equations are difficult to solve even numerically due to their elliptic‐hyperbolic character and the complex geometry of the flow configuration. For such problems, a finite‐element numerical technique has been developed whereby the steady‐state hydrodynamic equations and the associated boundary conditions are solved taking into full account the nonlinear convective terms, viscous terms, heat conduction terms, and variable fluid properties. The numerical technique is based upon a general formulation for the system of hydrodynamic equations making use of the method of weighted residuals, applied over discrete, distorted finite elements of the flow domain where the unknown fluid variables are expressed continuously in terms of polynomial approximating functions and nodal parameters. This process results in a set of nonlinear algebraic equations for the nodal parameters which are solved iteratively by using a multi‐dimensional Newton–Raphson scheme. To assess its accuracy, the method is applied to problems of compressible and incompressible viscous flow and heat transfer in diverging channels with plane walls, and also to a problem of normal shock wave in one‐dimensional flow. The results in all cases are in satisfactory agreement with existing analytical solutions and experimental data. Furthermore, the numerical scheme appears to be stable.
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47.15.-x Laminar flows
47.27.-i Turbulent flows

Variable viscosity effects on the onset of convection in porous media

D. R. Kassoy and A. Zebib

Phys. Fluids 18, 1649 (1975); http://dx.doi.org/10.1063/1.861083 (3 pages) | Cited 32 times

Online Publication Date: 3 September 2008

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The onset of convection in a horizontal, isotropic, water‐saturated porous medium is considered. The temperature difference between the top and bottom is as large as 250 °C. The effects of an eightfold variation in kinematic viscosity are included. The critical Rayleigh number is found to be substantially reduced from the classical value although the associated wavenumber is nearly the same. Neutral mode streamline and isotherm patterns are considerably distorted in the vertical direction in distinction to the symmetric patterns found in the constant viscosity classical calculation.
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47.15.-x Laminar flows
47.56.+r Flows through porous media

Cross‐hatching: A consequence of differential deformation of a viscous solid

Hans W. Stock

Phys. Fluids 18, 1652 (1975); http://dx.doi.org/10.1063/1.861084 (8 pages)

Online Publication Date: 3 September 2008

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The hypothesis, proposed by Probstein and Gold, that cross‐hatching results from the differential deformation of an inelastic material at its surface, has been used in this analysis. Wall pressure and wall shear stress perturbations leading to the deformation were calculated with the linearized small perturbation theory. The supersonic boundary layer was composed of an inviscid outer layer and a viscous sublayer. Both laminar and turbulent mean velocity and Mach number profiles were used in the calculation. In contrast to the work of Probstein and Gold the viscous sublayer was included and wall pressure perturbations were also considered. The essential result is the prediction of the pattern cant angle, which was not possible with the inviscid analysis. The theory qualitatively confirms the experimentally observed features of cross‐hatching, similarly seen by Probstein and Gold. For instance, for cross‐hatching to occur, the boundary layer must be supersonic and turbulent, whilst the inelastic body material must be of the Maxwell type. Theory and experiment also show that the pattern is charcterized by a cant angle equal to the local Mach angle, by a wavelength inversely proportional to the static pressure and directly proportional to the viscosity of the material.
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47.15.-x Laminar flows
47.40.Ki Supersonic and hypersonic flows

Solution of Maxwell’s moment equations by the method of rational truncation and coordinate straining

D. Baganoff and J.P. Elliott

Phys. Fluids 18, 1660 (1975); http://dx.doi.org/10.1063/1.861085 (6 pages)

Online Publication Date: 3 September 2008

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A method is developed for truncating and closing any valid system of moment equations of the Boltzmann equation. The method is based on the idea that a rational truncation procedure for any system of moment equations must be developed from an orthonormal expansion for the distribution function, and that the convergence of the expansion can be accelerated if the coordinates in velocity space are scaled in accordance with any asymmetry in the distribution function that may be apparent from the physical situation. The procedure allows the determination of closure relations among any set of moments that may be convenient for a particular problem. Application of the method to the problem of shock‐wave structure gives, for the first time, results at only the 13‐moment level of approximation that are comparable to the Mott–Smith solution.
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47.10.-g General theory in fluid dynamics

Velocity profile in the Knudsen layer for the Kramer’s problem

S. K. Loyalka

Phys. Fluids 18, 1666 (1975); http://dx.doi.org/10.1063/1.861086 (4 pages) | Cited 18 times

Online Publication Date: 3 September 2008

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Since the BGK model is based on the assumption of constant collision frequency, and since this model has been found inadequate in describing some experimental data, the numerical study of a variable collision frequency model proposed earlier by Cercignani and Loyalka and Ferziger is described. Specifically, the Kramer’s problem for this model is solved, and it is found that the ’’velocity defect’’ in the Knudsen layer is quite sensitive to the velocity dependence of the collision frequency. In fact, for the hard sphere collision frequency, the present results agree reasonably well with the recent experimental data of Reynolds, Smolderen, and Wendt.
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51.20.+d Viscosity, diffusion, and thermal conductivity
51.10.+y Kinetic and transport theory of gases

Criteria for acoustic instability in a gas with ambient vibrational and radiative nonequilibrium

J. Srinivasan and W. G. Vincenti

Phys. Fluids 18, 1670 (1975); http://dx.doi.org/10.1063/1.861072 (8 pages) | Cited 7 times

Online Publication Date: 3 September 2008

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When acoustic waves propagate in a gas, nonequilibrium phenomena can manifest themselves in either (or both) the ambient and perturbed states of the gas. To study the relatively unexamined effects of ambient nonequilibrium, the present paper uses a macroscopic, gas‐dynamic approach to investigate the influence of such nonequilibrium, primarily as regards vibration and secondarily as regards associated radiation. In situations where the vibrational temperature exceeds the translational temperature, ambient vibrational nonequilibrium, from whatever source, is shown to cause instability (i.e., wave amplification) provided the vibrational relaxation time is a descreasing function of temperature (or increases at less than a specified rate) and the degree of ambient nonequilibrium is sufficiently high. Ambient radiative nonequilibrium will cause instability, in cases where the external‐source temperature exceeds the gas temperature, provided the absorption coefficient is an increasing function of temperature and the rate of radiant energy input is sufficiently large.
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51.40.+p Acoustical properties
43.25.-x Nonlinear acoustics

Hydromagnetic boundary layers in a rotating cylindrical container

Somaraju Vempaty and David E. Loper

Phys. Fluids 18, 1678 (1975); http://dx.doi.org/10.1063/1.861073 (9 pages) | Cited 7 times

Online Publication Date: 3 September 2008

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Linear, steady, axisymmetric flow of an electrically conducting homogeneous fluid confined within a rigid, rotating electrically insulated cylinder is analyzed. The fluid motions are driven by differential rotation of horizontal boundaries. The applied magnetic field and the rotation vector are aligned normal to the horizontal boundaries. The magnetic Reynolds number, the Rossby number, and the Ekman number are assumed to be much less than unity. The dynamics of the inviscid interior and vertical boundary layers are investigated as functions of the rotational magnetic interaction parameter α2(=σB2/2ρΩ) which measures the ratio between the magnetic force and the Coriolis force. If α2 ≪1, the flow behaves as a nonmagnetic rotational flow to dominant order with Stewartson’s E1/3 and E1/4 double layer structures and the electric currents pumped by the Ekman–Hartmann layer return through the interior. At the other extreme, if α2≫E−1/3 the flow behaves as a strongly magnetic nonrotational flow with a single vertical layer (≈E1/4 α−1/2) of parabolic structure. The intermediate range 1≪α2≪E−1/3, which is a transition from weakly magnetic to the strongly magnetic flows, is characterized by a quadruple vertical boundary layer structure: (1) a nonmagnetic E1/3 layer, (2) a nonmagnetic E1/4α1/2 layer, (3) a magnetic α−2 layer to transport the electric current vertically, and (4) a magnetic E1/4α−1/2 layer with axial scale E1/4α3/2 to feed to feed the current from the Ekman–Hartmann layers into the α−2 layer.
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47.65.-d Magnetohydrodynamics and electrohydrodynamics
47.15.-x Laminar flows

Laser target model

D. G. Colombant, K. G. Whitney, D. A. Tidman, N. K. Winsor, and J. Davis

Phys. Fluids 18, 1687 (1975); http://dx.doi.org/10.1063/1.861074 (11 pages) | Cited 50 times

Online Publication Date: 3 September 2008

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A numerical simulation model is described for the dense hot plasma produced in the interaction of a focused laser pulse with a target of moderate atomic weight. Specific results are given for the line and continuum x‐ray emission, together with other parameters of the plasma, for the case of an aluminum target.
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52.50.Jm Plasma production and heating by laser beams (laser-foil, laser-cluster, etc.)
52.65.-y Plasma simulation

Computational study of the first stage of hypersonic ion beam neutralization: The cross neutralization stage

C. Pomot and J. M. Dolique

Phys. Fluids 18, 1698 (1975); http://dx.doi.org/10.1063/1.861087 (6 pages) | Cited 1 time

Online Publication Date: 3 September 2008

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A study is made of the first stage of evolution of a hypersonic ion beam in which thermoelectrons are emitted by a heated grid, known as the neutralizer. Downstream from the neutralizer there appears successively as a sheath a range of periodic and quasi‐stationary electric field and a front where the electric field oscillates with the plasma frequency. The sheath is self‐regulated. Some electrons are trapped in the periodic and stationary electric field. The characteristics of the periodic, quasi‐stationary range correspond to those of both an experimental study and one‐dimensional time‐independent macroscopic theory. This quasi‐stationary regime builds up in a time smaller than ωpe−1 and is studied for a few periods ωPe−1. The subsequent evolution of this state of nonequilibrium is not investigated. The experimental study has shown that, as for the neutralization of a subsonic ion beam, it leads to a field‐free, homogeneous medium: a ’’synthesized plasma.’’ The importance of the first stage described herein, which may be called the gross neutralization stage, is due to the properties of mean neutrality in the current and in the charge insured by the regulating sheath, properties which will be preseved downstream.
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47.40.Ki Supersonic and hypersonic flows
41.75.Ak Positive-ion beams
41.75.Cn Negative-ion beams

Motion of a plasma column in a perturbing magnetic field

G. Miller

Phys. Fluids 18, 1704 (1975); http://dx.doi.org/10.1063/1.861088 (6 pages) | Cited 4 times

Online Publication Date: 3 September 2008

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The motion of a plasma column acted upon by a small external magnetic field is calculated using the sharp‐boundary ideal magnetohydrodynamic model. The natural oscillation frequencies of the system are obtained. For the theta pinch, simple approximate formulas useful in considering feedback and dynamic stabilization schemes are derived.
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52.30.-q Plasma dynamics and flow

Radial density distribution of charged particles in an electronegative discharge plasma with space charges

P. Kocian

Phys. Fluids 18, 1710 (1975); http://dx.doi.org/10.1063/1.861089 (6 pages)

Online Publication Date: 3 September 2008

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The radial distribution of the electrons, as well as the positive and negative ions, respectively, has been determined theoretically in the positive column of the low‐pressure discharge in an electronegative gas. The problem has been solved using a system of hydrodynamic equations and Poisson’s equation. This latter equation must be considered because of the volume recombination of the negative and positive ions and because of the space charges produced in the positive column. The method has been applied to the special case of the conical equipotentials often occuring in electronegative gases.
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52.80.Dy Low-field and Townsend discharges

Resonance cones in a warm plasma for finite magnetic fields

Keith H. Burrell

Phys. Fluids 18, 1716 (1975); http://dx.doi.org/10.1063/1.861090 (18 pages) | Cited 23 times

Online Publication Date: 3 September 2008

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The resonance cone structure excited by a localized source is studied theoretically and experimentally for a warm plasma with a finite, uniform magnetic field. The electrostatic Green’s function is evaluated by asymptotically expanding it in the limit where the observation point is far from the source. Results are obtained that are valid for all angles and for all frequencies below the upper hybrid. The theory predicts the functional dependence of the angular location of the main resonance cone peak and the angular spacing between interference peaks on the physical parameters. Experimental parameter studies were made to check the predicted functional dependence; it was found to produce a good fit to the data. Both density and temperature can be found from these measurements; thus, resonance cone measurements are a useful diagnostic technique in any magnetoplasma in which antennas can be inserted.
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52.40.Db Electromagnetic (nonlaser) radiation interactions with plasma
52.25.Os Emission, absorption, and scattering of electromagnetic radiation

Nonlinear saturation of stimulated diffusion scattering

J. Weinstock, T. D. Rognlien, and Kai Fong Lee

Phys. Fluids 18, 1734 (1975); http://dx.doi.org/10.1063/1.861091 (6 pages) | Cited 2 times

Online Publication Date: 3 September 2008

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A nonlinear instability theory is presented for stimulated scattering of electromagnetic ordinary‐mode radiation from electrostatic ’’diffusion modes.’’ These ’’diffusion modes’’ are excited unstable by the incident radiation, and have their linear damping rates proportional to the diffusion tensor. Two diffusion modes are found; one corresponds to thermal diffusion and the other to particle diffusion. The thermal diffusion mode was previously calculated by Berger, Goldman, and DuBois for a two‐dimensional model. The present calculation is made for three dimensions to show that the radiation is scattered over a wide angle in the plane perpendicular to a uniform magnetic field. A three‐dimensional model is necessary to determine nonlienar effects. The saturated level of the electrostatic diffusion modes is calculated explicitly. For ionospheric modification experiments it is found that the electron density fluctuations are almost 1% of the ambient electron density.
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94.20.Qq Particle precipitation
94.20.Ss Electric fields; current system
52.40.Db Electromagnetic (nonlaser) radiation interactions with plasma

Collisions in multi‐dimensional plasma simulations

Y. Matsuda and H. Okuda

Phys. Fluids 18, 1740 (1975); http://dx.doi.org/10.1063/1.861092 (8 pages) | Cited 9 times

Online Publication Date: 3 September 2008

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Collisions in multi‐dimeniosnal plasma simulation models using finite‐size particles are studied numerically and analytically. Drag and diffusion of particles in velocity space are measured and are compared with the theoretical predictions for two‐, two‐and‐a‐half‐, and three‐dimensional models. For two‐ and three‐dimensional models, the measurement agrees reasonably well with the Balescu–Lenard theory. For the two‐and‐a‐half‐dimensional model, it is shown that the collisions strongly depend on the choice of the angle between magnetic field and wave vector due to the limited phase space available in the model. For a strongly magnetized plasma with Ωe≳ωpe, the effective collisions are significantly reduced for a simulation plasma when a≳ρe (where a is particle size, and ρe is the gyroradius) in contrast to a real plasma where the effective collisions depend weakly on the magnetic field.
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52.20.Hv Atomic, molecular, ion, and heavy-particle collisions
52.65.-y Plasma simulation

Models for quasi‐linear equations

Barbara Abraham‐Shrauner

Phys. Fluids 18, 1748 (1975); http://dx.doi.org/10.1063/1.861093 (8 pages) | Cited 1 time

Online Publication Date: 3 September 2008

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The quasi‐linear equation for a weakly turbulent plasma is treated by specifying the time dependence of the imaginary part of the frequency of a mode γk(t) for several models. For unstable plasmas an oscillatory diffusion coefficient, D, is found and several methods are discussed for insuring that D is positive. For wave modes that stabilize at high electric field fluctuation levels, it is found that the one‐particle distribution function must be treated as an operator if the diffusion time is equal to or greater than growth time. Our results confirm the picture given by Burns and Knorr within the limits of our models.
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52.35.Mw Nonlinear phenomena: waves, wave propagation, and other interactions (including parametric effects, mode coupling, ponderomotive effects, etc.)

Space‐time structure of ion beam‐plasma turbulence

F. Doveil and D. Grésillon

Phys. Fluids 18, 1756 (1975); http://dx.doi.org/10.1063/1.861094 (6 pages) | Cited 11 times

Online Publication Date: 3 September 2008

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The turbulence excited by an ion beam penetrating an unmagnetized plasma is investigated. One‐dimensional linear kinetic theory predicts the ion acoustic mode to be driven unstable when the beam velocity vb ranges from cs to 2cs. Numerical calculations are reported for the three dimensional problem, in which attention is given to unstable waves. The direction of propagation, the phase velocities, and the growth rate contours in the (k∥, k⊥) plane, are given as a function of vb. This instability is created in a double plasma device, and investigated by means of space and time correlations. Turbulence is excited, consisting of modes propagating at a phase velocity which agrees with theory. As vb increases, an abrupt transition from one‐dimensional to three‐dimensional structure occurs as expected.
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52.35.Ra Plasma turbulence

Anomalous electron transport and lower‐hybrid wave damping

C. Chu, J. M. Dawson, and H. Okuda

Phys. Fluids 18, 1762 (1975); http://dx.doi.org/10.1063/1.861095 (7 pages) | Cited 18 times

Online Publication Date: 3 September 2008

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The transport of electrons across magnetic fields was investigated by two‐dimensional computer experiments. Under the modest condition ωce≈ωpe, which occurs in many laboratory plasmas and fusion devices, it is found that the electron heat and momentum transport rates are much enhanced above the classical collisional values even for plasma very close thermal equilibrium. The enhancement is due to electron turbulent motion in the fields of thermally excited lower hybrid waves. Nonthermal plasmas could show an even larger degree of enhanced transport. Employing a fluid model which includes turbulent viscous terms in the electron momentum equation, linearized theory gives predictions which are in reasonably good agreement with the observed wave damping. The computer results throw some light on the theory of strong turbulence, comparison with such theories points up some discrepancies and indicates the proper method for including the turbulence effects. Estimates of the electron heat transport in three‐dimensional thermonuclear plasmas due to this mechanism gives values comparable to those predicted by neclassical theory from thermal fluctuations alone; enhancement of the fluctuations above the thermal level will given higher transport rates. These results suggest that nonclassical effects can be very important in plasma transport and could be important for fusion plasmas.
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52.25.Fi Transport properties
52.35.-g Waves, oscillations, and instabilities in plasmas and intense beams
FREE

Nonlinear stabilization of beam plasma interactions by parametric effects

K. Papadopoulos

Phys. Fluids 18, 1769 (1975); http://dx.doi.org/10.1063/1.861096 (9 pages) | Cited 77 times

Online Publication Date: 3 September 2008

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The nonlinear stabilization of the kinetic stage of electron beam plasma instabilities by parametric effects is investigated. It is found that within a definite range of plasma parameters parametric instatilities induced by the beam generated waves can stabilize the system at a level of wave energy density substantially lower than expected by quasi‐linear theory. This occurs because at a certain level of beam‐generated plasma waves, the transfer rate of wave energy outside the spectral region in resonance with the beam exceeds the beam plasma instability growth rate. A model of a quasi‐steady state for the case of continuous beam injection is proposed. The possibility of utilizing ultrarelativistic electron beams for achieving ignition temperatures in a tokamak is discussed.
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52.35.Mw Nonlinear phenomena: waves, wave propagation, and other interactions (including parametric effects, mode coupling, ponderomotive effects, etc.)
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