List of illustrations Some useful constants and units Introduction 1. The facts which call for explanation. A sketch of the surface distribution of the meteorological elements over the globe 2. Some statistical and thermal relationships 3. Water-vapour in the atmosphere. The ascent of damp air 4. Thermodynamics of the atmosphere 5. Radiation 6. Radiation in the troposphere 7. Radiative equilibrium and the stratosphere 8. The general equations of motion 9. Motion under balanced forces: the gradient wind 10. Surfaces of discontinuity 11. The general aspects of turbulence 12. Turbulence in the atmosphere: the eddies as diffusing agencies 13. The classification of winds 14. The transformations of energy in the atmosphere 15. The growth of cyclic circulations and of pressure inequalities in the atmosphere 16. The idea of air masses 17. The Polar front and its relation to the development of cyclones 18. Anticyclones 19. The general circulation of the atmosphere Appendix Index of names Index of subjects.

Physical and Dynamical Meteorolgy. By D. Brunt. Second edition. Pp. xxiv, 428. 25s. 1939. (Cambridge)

Three-dimensional (3D) turbulence has both energy and helicity as inviscid constants of motion. In contrast to two-dimensional (2D) turbulence, where a second inviscid invariant—the enstrophy—blocks the energy cascade to small scales, in 3D there is a joint cascade of both energy and helicity simultaneously to small scales. It has long been recognized that the crucial difference between 2D and 3D is that enstrophy is a nonnegative quantity whereas the helicity can have either sign. The basic cancellation mechanism which permits a joint cascade of energy and helicity is illuminated by means of the helical decomposition of the velocity into positively and negatively polarized waves. This decomposition is employed in the present study both theoretically and also in a numerical simulation of homogeneous and isotropic 3D turbulence. It is shown that the transfer of energy to small scales produces a tremendous growth of helicity separately in the + and − helical modes at high wave numbers, diverging in the limi...

The joint cascade of energy and helicity in three-dimensional turbulence

A recently developed formulation of the inverse source problem as a Fredholm integral equation of the first kind provides motivation for the development of analytical characterizations of the nonuniqueness in the inverse source problem. Nonradiating sources, i. e., sources for which the field is identically zero outside a finite region, are introduced. It is then shown that the null space of the Fredholm integral equation is exactly the class of nonradiating sources.

Non-uniqueness in the inverse source problem in acoustics and electromagnetics

Maxwell’s equations describing electric and magnetic fields limit the shapes field lines can take. But exotic solutions exist where the field lines are linked and knotted. A proposal now shows how such solutions could be realized experimentally. Maxwell’s equations allow for curious solutions characterized by the property that all electric and magnetic field lines are closed loops with any two electric (or magnetic) field lines linked. These little-known solutions, constructed by Ranada1, are based on the Hopf fibration. Here we analyse their physical properties to investigate how they can be experimentally realized. We study their time evolution and uncover, through a decomposition into a spectrum of spherical harmonics, a remarkably simple representation. Using this representation, first, a connection is established to the Chandrasekhar–Kendall curl eigenstates2, which are of broad importance in plasma physics and fluid dynamics. Second, we show how a new class of knotted beams of light can be derived, and third, we show that approximate knots of light may be generated using tightly focused circularly polarized laser beams. We predict theoretical extensions and potential applications, in fields ranging from fluid dynamics, topological optical solitons and particle trapping to cold atomic gases and plasma confinement.

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Linked and knotted beams of light

Publisher Summary This chapter describes the photon wave function, and explains the basic properties of a well-defined mathematical object—a six-component function of space-time variables—that describes the quantum state of the photon. The most essential property that does not hold for the photon wave function is that the argument of the wave function cannot be directly associated with the position operator of the photon. The position operator for the photon simply does not exist. However, one should remember that for massive particles also, the true position operator exists only in the nonrelativistic approximation. The concept of localization associated with the Newton-Wigner position operator is not relativistically invariant. The photon wave function is not restricted to the wave mechanics of photons. The same wave functions also appear as mode functions in the expansion of the electromagnetic field operators.

V Photon Wave Function

In the present paper we introduce eigenfunctions of the curl operator. The expansion of vector fields in terms of these eigenfunctions leads to a decomposition of such fields into three modes, one of which corresponds to an irrotational vector field and two of which correspond to rotational circularly polarized vector fields of opposite signs of polarization. Under a rotation of coordinates, the three modes which are introduced in this fashion remain invariant. Hence we have introduced the Helmholtz decomposition of vector fields in an irreducible, rotationally invariant form.These expansions enable one to handle the curl and divergence operators simply. As illustrations of the use of the curl eigenfunctions, we solve four problems. The first problem that is solved is the initial value problem of electromagnetic theory with given time- and space-dependent sources and currents and we show that the radiation and longitudinal modes uncouple in a very simple way. In the second problem we show how fluid motion...

Eigenfunctions of the Curl Operator, Rotationally Invariant Helmholtz Theorem, and Applications to Electromagnetic Theory and Fluid Mechanics

After reviewing th eproperties of the photon considered as a quantized particle of zero mass, positive energy, and unit spin, the expansion of the unquantized and quantized electromagnetic fields and vector and scalar potentials in terms of the photon wave functions and creation and destruction operators in reviewed and extended. The most e use of the eigenfunctions of the curl operator. The dichotomy between the photon and wave picture of electromagnetic radiation is discussed and resolved. The results are applied to the calculation of the exact electromagnetic matrix elements and transition probabilities (i.e., with retardation taken into account exactly) for hydrogenic atoms. The exac matrix elements are very simple in form. The notion of multipole radiation of the usual treatments is irrelevant. However, it is shown how multipoles appear as an approximation. (auth)

Photon wave functions and the exact electromagnetic matrix elements for hydrogenic atoms

Previously it has been shown that all finite energy, causal solutions of the time-dependent three-dimensional acoustic equation and Maxwell’s equations have forms for large radius that, except for a factor $1/r$, represent one-dimensional wave motions along straight lines through the origin. The asymptotic region is called the wave zone. Conversely, it also has been shown how the exact finite energy causal solutions can be obtained from the asymptotic solutions in the wave zone through the use of a refined Radon transform. Thus a way of finding exact, causal three-dimensional solutions from the essentially one-dimensional solutions in the wave zone has been obtained.When the asymptotic solutions are confined to a finite radial interval within a cone, the exact solutions are termed “bullets” and asymptotically represent packets of acoustic and electromagnetic energy “shot” through the cone, which is a kind of “rifle.” No reflectors are needed. In the present paper explicit examples of such exact solutions ...

Acoustic and electromagnetic bullets: derivation of new exact solution of the acoustic and Maxwell's equations

The object of the time‐dependent inverse source problem of electromagnetic theory and acoustics is to find time‐dependent sources and currents, which are turned on at a given time and then off to give rise to prescribed radiation fields. In an early paper for the three‐dimensional electromagnetic case, the present writer showed that the sources and currents are not unique and gave conditions which make them so. The ideas of that paper are reformulated for the three‐dimensional electromagnetic case and extended to the acoustical three‐dimensional case and the one‐dimensional electromagnetic and acoustic cases. The one‐dimensional cases show very explicitly the nature of the ambiguity of the choice of sources and currents. This ambiguity is closely related to one which occurs in inverse scattering theory. The ambiguity in inverse scattering theory arises when one wishes to obtain the off‐shell elements of the T matrix from some of the on‐shell elements (i.e., from the corresponding elements of the scatterin...

Harry E. Moses

Papers

Eigenfunctions of the Curl Operator, Rotationally Invariant Helmholtz Theorem, and Applications to Electromagnetic Theory and Fluid Mechanics

Photon wave functions and the exact electromagnetic matrix elements for hydrogenic atoms

Acoustic and electromagnetic bullets: derivation of new exact solution of the acoustic and Maxwell's equations

The time‐dependent inverse source problem for the acoustic and electromagnetic equations in the one‐ and three‐dimensional cases

New Test of the Synchronization Procedure in Noninertial Systems