Dynamical aspects of animal grouping: Swarms, schools, flocks, and herds

The ascent and emplacement of basaltic magma on the earth and moon is modeled by the application of geological and physical observations and constraints. Relatively simple mathematical models of the motion of gas/liquid mixtures are shown to be adequate in the treatment of basaltic eruptions, provided that allowance is made for the coalescence of gas bubbles and that realistic geological and petrochemical constraints are applied to the numerical values of variables. Because gas exsolution from magmas on the earth and moon commonly occur at depths of less than 2 km, it is generally convenient to consider separately the rise of bubble-free magmatic liquid at depth in a planetary crust and the more complex motions occurring near the surface with gas exsolution.

http://www.planetary.brown.edu/pdfs/262.pdf

Ascent and eruption of basaltic magma on the earth and moon

Dust to planetesimals: Settling and coagulation in the solar nebula

We report the results of an extensive numerical study of the small-scale turbulent dynamo. The primary focus is on the case of large magnetic Prandtl numbers Prm, which is relevant for hot low-density astrophysical plasmas. A Prm parameter scan is given for the model case of viscosity-dominated (low Reynolds number) turbulence. We concentrate on three topics: magnetic energy spectra and saturation levels, the structure of the magnetic field lines, and intermittency of the field strength distribution. The main results are as follows: (1) the folded structure of the field (direction reversals at the resistive scale, field lines curved at the scale of the flow) persists from the kinematic to the nonlinear regime; (2) the field distribution is self-similar and appears to be lognormal during the kinematic regime and exponential in the saturated state; and (3) the bulk of the magnetic energy is at the resistive scale in the kinematic regime and remains there after saturation, although the magnetic energy spectrum becomes much shallower. We propose an analytical model of saturation based on the idea of partial two-dimensionalization of the velocity gradients with respect to the local direction of the magnetic folds. The model-predicted saturated spectra are in excellent agreement with numerical results. Comparisons with large-Re, moderate-Prm runs are carried out to confirm the relevance of these results and to test heuristic scenarios of dynamo saturation. New features at large Re are elongation of the folds in the nonlinear regime from the viscous scale to the box scale and the presence of an intermediate nonlinear stage of slower than exponential magnetic energy growth accompanied by an increase of the resistive scale and partial suppression of the kinetic energy spectrum in the inertial range. Numerical results for the saturated state do not support scale-by-scale equipartition between magnetic and kinetic energies, with a definite excess of magnetic energy at small scales. A physical picture of the saturated state is proposed. Subject heading gs: magnetic fields — methods: numerical — MHD — plasmas — turbulence

https://iopscience.iop.org/article/10.1086/422547/pdf

Simulations of the Small-Scale Turbulent Dynamo

For incompressible turbulent flow with large Reynolds and Peclet numbers, mechanisms for the spectral transfer of kinetic energy, heat, and mass at large wavenumbers are proposed. Based on these mechanisms, turbulent energy, temperature, and the concentration spectrum functions are deduced for the entire universal equilibrium range of wavenumbers. The resulting one‐dimensional energy, temperature, and concentration spectra are compared with measurements.

Structure of Turbulent Velocity and Scalar Fields at Large Wavenumbers

The equations describing the motion of a gas carrying small dust particles are given and the equations satisfied by small disturbances of a steady laminar flow are derived. The effect of the dust is described by two parameters; the concentration of dust and a relaxation time τ which measures the rate at which the velocity of a dust particle adjusts to changes in the gas velocity and depends upon the size of the individual particles. It is shown that if the dust is fine enough for τ to be small compared with a characteristic time scale associated with the flow, then the addition of dust destabilizes a gas flow; whereas if the dust is coarse so that τ is relatively large, then the dust has a stabilizing action.For plane parallel flow, it is shown that the stability characteristics for a dusty gas are still determined by solutions of the Orr-Sommerfeld equation, but with the basic velocity profile replaced by a modified profile which is in general complex. A simple, although unrealistic, example is used to illustrate some features of the action of dust. It is intended to describe the solution of the modified Orr-Sommerfeld equation for plane Poiseuille flow in a later paper.

/pdf/on-the-stability-of-laminar-flow-of-a-dusty-gas-ea2rtxcrgx.pdf

On the stability of laminar flow of a dusty gas

This paper is a contribution to the study of statistically homogeneous, dynamically passive vector fields convected by a turbulent fluid and subject to a molecular diffusivity λ that is small compared with the kinematic viscosity v . Two types are considered: the first, denoted by F , has the property that the total flux across a material surface moving with the fluid is conserved if λ = 0 (e.g. magnetic field); and the second, denoted by G , is the gradient of a conserved scalar quantity θ (e.g. temperature gradient). Attention is focused on small-scale variations with length-scale less than $(v^3|\epsilon)^{\frac {1}{4}}$ . A theory of Batchelor's in terms of Eulerian correlations for the distribution of θ for the case when λ [Lt ] v is extended and applied to the vector fields, thereby giving equations for the covariance tensors of F and G appropriate for separations less than $(v^3|\epsilon)^{\frac {1}{4}}$ . According to these equations, the effect of convection on small-scale components of the fields is to amplify and also to distort by reducing the scale; the ratio of these two effects is measured by a parameter σ. It is shown that if $\sigma \textless {\frac{5}{2}$ , the small-scale structure is stable against perturbations however small λ/ v may be, the amplification being eventually balanced by the dissipation which is increased by the reduction of scale. In the case of the quantity G , σ = 1. The value of σ for the case of F is not known, but reasons are given for believing that it is less than one, and it is concluded that the behaviour of $\overline{\bf F^2}$ and $\overline{\bf G^2}$ in a field of homogeneous turbulence is qualitatively the same. In particular, $\overline{\bf F^2}$ does not grow indefinitely with time as predicted by previous arguments. The correlation functions for small separations and the corresponding spectrum functions for a statistically steady state are obtained. The relation between this analysis and that for random vector fields in a uniform straining motion of infinite extent is considered in detail, for Pearson has shown that, if the strain is an irrotational distortion, then $\overline{\bf F^2} \rightarrow \infty$ with time. It is shown that this divergence is due to the amplification of components with very small wave-numbers or, equivalently, of very large scale, and it is therefore not considered relevant to a study of homogeneous turbulence. The particular case of the magnetic field in a good conductor is considered. If the Lorentz forces are unimportant, it is estimated that the magnetic energy of a weak seed field will be in general amplified by the turbulence by a factor lying somewhere between the Reynolds and magnetic Reynolds numbers of the turbulence before ohmic dissipation as increased by the reduction of scale limits the growth, and it is suggested further that the magnetic field will eventually decay to zero in the absence of external electromotive forces. In an appendix, the theory is applied tentatively to the turbulent vorticity (which satisfies the same equation as F if λ = v ) and an expression for the energy spectrum function for very large wave-numbers is deduced. This is compared with an expression given by Townsend, and is found to have a similar qualitative behaviour but gives values about one-half as large.

/pdf/on-the-fine-scale-structure-of-vector-fields-convected-by-a-4fgks8acd4.pdf

On the fine-scale structure of vector fields convected by a turbulent fluid

The existence of steady, one-dimensional, finite-amplitude waves in a cold collison-free plasma is investigated for the case in which the plasma and the magnetic field are uniform far ahead of the wave. It is supposed that the magnetic field at infinity is inclined at an angle β to the direction normal to the plane of the waves (0 ≤ β ≤ ½π). It is shown that the problem is equivalent to determining the orbits of a particle in a uniformly rotating field of force. 

Two types of waves are found to exist for β ≠ 0: solitary waves and quasishocks. In a quasi-shock, the plasma and magnetic field oscillate irregularly behind the wave front about mean values, which are different from the values ahead of the wave. The width of the wave front is of the order of the ion gyroradius. The quasi-shocks resemble oblique magnetohydrodynamic shock-waves. For the case β = 0, only solitary waves exist, which have already been described (Saffman 1961). 

The stability of the waves is considered, and it is concluded that the quasishocks are stable but that the solitary waves for β ≠ 0 are unstable.

/pdf/on-hydromagnetic-waves-of-finite-amplitude-in-a-cold-plasma-5fvrq87y9e.pdf

On hydromagnetic waves of finite amplitude in a cold plasma

This paper considers the question of how large the magnetic energy can be in stationary homogeneous turbulence at large Reynolds number of an incompressible conducting fluid for which the magnetic diffusivity λ is much less than the kinematic viscosity ν. An approximate equation, that takes account of the effect of the Lorentz forces on the turbulence, is proposed for the spectrum function of the magnetic energy for wave-numbers lying in the equilibrium range. This equation is used to determine the magnetic spectrum function and the level of magnetic energy for the case when a statistically steady magnetic field is maintained by a relatively small input of magnetic energy, by a weak applied field say, on the scale of the energy-containing eddies; it being supposed as suggested by earlier work that the magnetic energy would eventually die away in the absence of external electromotive forces. The results are complicated, there being essentially four different regimes depending in a fairly involved way on the relative values of λ/ν, the turbulent Reynolds number, and the ratio of the energy of the applied field to the energy of the turbulence. The main conclusions about the amplification factor are shown diagrammatically in figure 4. The wavenumber at which the magnetic spectrum has its maximum tends to decrease, as the applied field is increased, from the conduction cut-off wave-number (e | νλ^2)^(1/2) to values lying in the inertial subrange, much less than (e | ν^3)^(1/4). 

The case of a turbulent dynamo is also examined, and it is concluded that, if it exists, the equilibrium magnetic energy would be given by Batchelor's criterion of equipartition of energy between the magnetic field and the small, energy-dissipating eddies. The magnetic energy in the dynamo is found to lie mainly in Fourier components of wave-number about (e | νλ^2)^(1/2).

/pdf/the-amplification-of-a-weak-magnetic-field-by-turbulent-2zw14rme7g.pdf

The amplification of a weak magnetic field by turbulent motion of a fluid of large conductivity

A one-dimensional steady solution of the equations of motion of a cold plasma in a magnetic field is obtained. The plasma is of semi-infinite extent, bounded by a plane interface which separates it from a vacuum or medium at rest. The particles approach from infinity, are reflected at the front, and return to infinity in the opposite direction. At infinity, the magnetic field is parallel and anti-parallel to the plasma streams, and is inclined at an angle to the normal to the interface. The front is a current sheet across which the lines of force are bent, with the component of the magnetic field in the plane of the front changing direction. The inertia of the electrons is neglected, and the characteristic frequency associated with the front is the ion gyro-frequency.

/pdf/the-structure-of-a-hydromagnetic-front-in-a-cold-collision-vn1fc2tgtn.pdf

P. G. Saffman

Papers

On the stability of laminar flow of a dusty gas

On the fine-scale structure of vector fields convected by a turbulent fluid

On hydromagnetic waves of finite amplitude in a cold plasma

The amplification of a weak magnetic field by turbulent motion of a fluid of large conductivity

The structure of a hydromagnetic front in a cold collision-free plasma