Spiral eddies were first seen in the sunglitter on the Apollo Mission 30 years ago; they have since been recorded on synthetic aperture radar (SAR) images and in the infrared. We present a small sample of images. The spirals are broadly distributed over the world's oceans, 10–25 km in size and overwhelmingly cyclonic. Under light winds favourable to visualization, linear surface features with high surfactant density and low surface roughness are of common occurrence. The linear features are wound into spirals in vortices associated with horizontal shear instability, modified by rotation, in regions where the shear is comparable with the Coriolis frequency. Two models for concentrating shear are presented: a softened version of the classical sharp Margules front, and the time–dependent Lagrangian model of Hoskins & Bretherton. Horizontal shear instabilities and both frontal models favour cyclonic shear and cyclonic spirals, but for different reasons.

Lattice results and dual QCD results for all heavy quark potentials through order (quark mass)-2 are exhibited and compared. The agreement on the whole is quite good, confirming the validity of dual QCD.

We review the assumptions leading to the description of long distance QCD by a Lagrangian density expressed in terms of dual potentials. We find the color field distribution surrounding a quark anti-quark pair to first order in their velocities. Using these distributions we eliminate the dual potentials from the Lagrangian density and obtain an effective interaction Lagrangian $L_I ( \vec x_1 \, , \vec x_2 \, ; \vec v_1 \, , \vec v_2 )$ depending only upon the quark and anti-quark coordinates and velocities, valid to second order in their velocities. We propose $L_I$ as the Lagrangian describing the long distance interaction between constituent quarks. Elsewhere we have determined the two free parameters in $L_I$, $\alpha_s$ and the string tension $\sigma$, by fitting the 17 known levels of $b \bar b$ and $c \bar c$ systems. Here we use $L_I$ at the classical level to calculate the leading Regge trajectory. We obtain a trajectory which becomes linear at large $M^2$ with a slope $\alpha' \simeq .74 \, \hbox{GeV}^{-1}$, and for small $M^2$ the trajectory bends so that there are no tachyons. For a constituent quark mass between 100 and 150 MeV this trajectory passes through the two known Regge recurrences of the $\pi$ meson. In this paper, for simplicity of presentation, we have treated the quarks as spin-zero particles.

We use Dual QCD to derive an effective potential to order (quark mass${)}^{\mathrm{\ensuremath{-}}2}$ for a constituent quark and antiquark. This is done by expanding the dual QCD Lagrangian to second order in the qq\ifmmode\bar\else\textasciimacron\fi{} spins and velocities around the static central potential, in which the quarks are both spinless and stationary. The field equations are then used to eliminate the dual gluon fields and the Higgs fields of dual QCD in favor of quark variables for an arbitrary but slowly moving qq\ifmmode\bar\else\textasciimacron\fi{} pair with a Dirac string of arbitrary shape connecting them. The result is a Lagrangian, and therefore, a potential, which depends only on the qq\ifmmode\bar\else\textasciimacron\fi{} positions, velocities, and spins. Dual QCD contains only three parameters, which can be determined from the vacuum energy density, the string tension, and the strength of the Coulomb singularity of the central potential. The only free parameters in the spin- and velocity-dependent part of the effective potential are, therefore, the masses of the c and b quarks. When inserted into a Schr\"odinger equation these potentials provide a complete effective constituent quark theory which can be used to calculate qq\ifmmode\bar\else\textasciimacron\fi{} energy levels in terms of the masses and the masses can thereby be fixed (agreement with experiment is excellent). The various potential, spin-spin, spin-orbit, and spinless velocity dependent, can also in principle be compared to lattice calculations of the same quantities. For the spin-orbit case, for example, the agreement is good, although lattice results are not yet precise enough for a real comparison to be made. For the potentials proportional to the velocity squared lattice results do not yet exist. We also attempt to extend the use of these potentials to heavy-light quark-antiquark systems through use of the Salpeter equation and the Dirac equation. The results of this effort are described in two Appendices.

We present Dirac's method for using dual potentials to solve classical electrodynamics for an oppositely charged pair of particles, with a view to extending these techniques to non-Abelian gauge theories.

We use the classical approximation to the dual QCD field equations to calculate the term in the heavy-quark potential that is proportional to angular momentum squared. This potential combined with the potentials obtained in our earlier work gives a result which is essentially the dual of the potential acting between a monopole-antimonopole pair carrying Dirac electric dipole moments and rotating in a relativistic superconductor. These potentials are used to fit the masses of the low-lying states of the ccBAR and bbBAR systems. The agreement, achieved with only four parameters, two of which are roughly determined in advance, is better than 1%. We also predict the masses of the lightest cbBAR states.

The dual magnetic mass of a hot quark-gluon plasma is computed in the lowest order of dual QCD, which predicts a well-defined (dual) gauge-invariant result for it. This is because, in dual QCD, electricity and magnetism are interchanged, so magnetic calculations in dual QCD are easy if the corresponding electric ones in ordinary QCD are easy, and vice versa. We obtain the (leading-order) numerical result m_(mag)=(7/12)^(1/2)gT for the dual magnetic mass, where g is the dual (color-magnetic) coupling constant.

We use the static potentials between heavy quarks derived from the classical approximation to the dual QCD field equations to calculate most of the low-lying states of the ccBAR and bbBAR systems. The agreement, achieved with only four parameters, two of which are roughly determined in advance, is better than 1%. The spin-orbit force cannot yet be calculated from dual QCD; if we use the Breit-Fermi form for it, all the ccBAR and bbBAR levels can be predicted. We also predict the masses of the lightest cbBAR states.

We use the classical approximation to dual QCD to compute the distribution of color fields around heavy quarks and the static potential between them. The resulting potential is in excellent agreement with phenomenologically obtained ones. As the distance between the quark-antiquark pair increases, the color field lines evolve from a squashed dipole distribution to a flux-tube distribution. It is noteworthy that the potential becomes linear well before the appearance of a fully developed flux tube. A comparison of these field distributions with lattice Monte Carlo calculations could test quantitatively whether dual superconductivity is the physical mechanism for confinement in QCD.

We review the attempts to use dual (electric) vector potentials rather than the standard magnetic vector potentials to describe QCD, particularly in the infrared regime. The use of dual potentials is motivated by the fact that in classical electrodynamics, in a medium with a dielectric constant vanishing at small momenta (as is believed to be the case in QCD), electric potentials provide a far more convenient language than do magnetic potentials. To begin with, we outline attempts to construct the QCD Lagrangian in terms of dual potentials and describe the various possibilities, their shortcomings and advantages, which so far exist. We then proceed to use the most attractive (albeit consistent as a field theory only at the tree leve l) of these Lagrangians in a number of applications. We show that it describes a non-Abelian dual superconductor (so that it automatically confines color), derive the static quark-antiquark potential, and various temperature dependent effects, such as deconfinement and chiral symmetry breaking.

We review the physical basis of dual QCD and solve the equations describing a QCD SU(3) flux tube. We use the solution to elucidate the physical connection between QCD and dual superconductivity.

We propose an explicit form for the long-range limit of the SU(N) Yang-Mills Lagrangian expressed as a function of the dual (color-electric) vector potentials. While we cannot rigorously derive the Lagrangian, it can be made plausible that it follows from conventional Yang-Mills theory. This dual long-distance QCD Lagrangian has many of the properties of a magnetic superconductor. It has classical solutions corresponding to confined tubes of quantized electric color flux which result from a dual Meissner effect. However, the confining pressure is not produced by a scalar Higgs field, as in ordinary superconductivity, but by a magnetic condensate field which arises naturally from the nonlocal form of the dual Lagrangian. Within the classical approximation, we find the explicit distribution of color fields surrounding a flux tube. Semiclassical quantization around this solution can be expected to yield the QCD string, and the semiclassical expansion parameter is 1/N, where N is the number of colors.

In the context of the formulation of QCD with dual potentials, we show that chiral-symmetry breaking occurs only in the confined state. Therefore, the transition temperature, beyond which chiral symmetry is restored, is the same as the deconfinement temperature. To carry out the calculation, it is necessary to couple quarks to dual gluons. We indicate how this is done (to lowest order in the magnetic coupling constant) and give the Feynman rules for quark--dual-gluon vertices.

The QCD Lagrangean expressed in terms of dual potentials exhibits, at zero temperature, spontaneous symmetry breaking giving rise to dual superconductivity and therefore to color confinement. At a finite temperature, the spontaneous symmetry breaking disappears and so does confinement. We estimate the deconfining transition temperature using a high-temperature expansion.