Linear momentum of surface gravity waves changes with time during refraction by a horizontally variable current, as is predicted by ray theory; the momentum change per unit time requires a force by the current on the waves. According to Newton's third law, the waves apply an equal but opposite force back on the current. The wave force of linear waves on the current is calculated for a steady horizontal shear current and it is found to be directly proportional to the wave momentum times the shear in the current. For a current like the Gulf Stream it is theoretically possible for the wave force on the current to be as large as the Coriolis force on the current to the depth of wave influence; the effect on equatorial surface currents is likely to be even more significant. Considering the reasonable conjecture that the orbital angular momentum of the waves cannot be exchanged with the current, the growth or decay of the wave amplitude in the shear current is computed as well. An exponential growth or decay of the amplitude is obtained with the e-folding scale being proportional to the current shear. A comparison between the calculated wave force and the Coriolis force for reported data describing the reflection of waves by the Gulf Stream is presented. The potential effects of the wave force on the surface extent of such currents and their observations by remote sensing, including possible bias in estimation of their transport capacity, are discussed. Instances of potential positive and negative feedback acting during the interaction between the waves and the current are outlined.

Einstein's surface gravity wave model [Naturwissenschaften 4, 509 (1916)] is modified in order to investigate the properties of internal gravity waves of infinitesimal amplitude at the boundary between two layers of differing constant and parallel velocities and constant densities, where surface tension and viscosity are neglected. We find that there is one and only one (stable) wavelength that satisfies the balance of pressure forces between crest and trough in the steady reference frame, when the lower layer has the greater density, no matter what the sizes of the interior fluid speeds (relative to the wave) are in the two layers. But if it is just required that the velocity difference between layers remains constant, and the layer densities are also constant, then there are many stable wavelengths. In contrast to this result, the Kelvin-Helmholtz interpretation of the identical problem, only viewed in the fixed reference frame, is that the motion is unconditionally unstable, since mathematically a sufficiently small wavelength can always be found which makes the phase speed imaginary. However, in our opinion, the available observational evidence can support the existence of a single stable wave in many cases. Therefore, we present Einstein's physical approach that predicts a stable wave for any given set of environmental parameters, a prediction that is qualitatively consistent with some geophysical phenomena. We compare the results, and their interpretations, of the modified Einstein and Kelvin-Helmholtz models. An English translation of Einstein's very short German paper, upon which the present development builds, is provided in the Appendix.

When small-amplitude surface gravity waves progress in deep water, all the fluid particles are observed to orbit in circles within the depth of wave influence. Each fluid particle therefore has angular momentum with respect to the center of its orbit, and the angular momentum vectors are directed parallel to the crests and troughs or perpendicular to the wave number. The angular momentum per unit volume is calculated for each fluid particle and then, by vertical integration, the angular momentum per unit horizontal area is computed. The total energy and the magnitude of the angular momentum, both per unit area or both per unit volume, are found to be proportional, the factor of proportionality being the wave frequency; specifically, angular momentum magnitude equals energy divided by frequency. Wave action, which has become an increasingly popular quantity for interpreting surface gravity wave problems and is defined as wave energy divided by frequency, is the same thing as the magnitude of the angular momentum. A general equation for the conservation of angular momentum along wave rays is given, and it contains on the right-hand side two torque terms; one changes the magnitude and the other changes the direction of the angular momentum. In applying this equation to the wave-current refraction problem there is only one torque, which changes the direction of the angular momentum, and it is explicitly determined asa function of the horizontal shear in the current. The magnitude of the angular momentum, and therefore also the wave action, will be conserved along the rays if there are no torques that could alter it, as in pure wave-current refraction. Our conservation equation for angular momentum is easily adapted to describing wave generation and dissipation by including the appropriate torques on the right side, and this may prove to be helpful for calculations of wave evolution.

The wave velocity c, wavelength lambda, and the surface elevation zeta of solitary waves of elevation and depression are calculated as functions of the wave height H for a given undisturbed depth h. For infinitesimal height (H much-less-than h), c = (gh)1/2 and lambda = 2pih/3(1/2) for both the elevation and the depression, where g is the acceleration of gravity. At finite height, c and lambda increase for the elevation and decrease for the depression as H increases. The calculation involves balancing static and dynamic pressure differences all along the surface and conserving mass through all cross sections below the surface in the reference frame moving with the wave velocity. The horizontal component of the fluid velocity is taken independent of depth below the surface, but the irrotational assumption is not used. The results obtained are compared with the corresponding ones given by the standard theory based on the nonlinear equation of Korteweg and de Vries, and the qualitative differences between the two sets of results are noted.

A combination and modification of two existing methods, which involves balancing static and dynamic pressure differences between points along the surface and conserving mass through cross sections below the surface in the reference frame moving with the phase velocity, is applied to surface gravity waves of arbitrary amplitude in water of finite depth. For a given still water depth and wave height the method determines in closed form the phase velocity, wavelength, and wave profile of the stable wave. The main assumption is that the horizontal component of the fluid velocity be independent of depth. The motion is not assumed to be irrotational. The wavelength of the stable wave is found to be about 3.6 times the still water depth for infinitesimal amplitude, and at finite amplitude the wavelength decreases as the amplitude increases. Therefore, shallow water waves are concluded to be unstable even at infinitesimal amplitude, for which the assumption is accurate. Previously it has been argued that only at finite amplitude will shallow water waves change form as they propagate. The wave profile is found to be sinusoidal for infinitesimal amplitude and to be asymmetric at finite amplitude, the crests being higher and narrower and the troughs shallower and broader. These results are consistent with well-known theoretical work and laboratory measurements.

Observations of ocean features and their interaction near Pt Conception, California, are reported for a two-month period during the early spring of 1983 at the time of the El Nino. The development of a body of warm water into a clockwise rotating eddy is described and the observed increase in the spatial scale of the eddy is found to compare favourably with the value calculated rrom a model of geostrophic adjustment. The subsequent interaction of the warm core eddy with a cold coastal jet directed offshore is discussed in relation to the approximate one-month lifetime observed for the eddy. The interaction of the eddy with incoming swell, and the resulting gradients in wave energy and wind stress are discussed in an accompanying paper in this issue (Sheres and Kenyon 1989). The measurements in this report are based mainly on NOAA-7 infrared images, but in situ current and wind data from moorings and temperature and salinity data from a California Cooperative Fisheries Investigation (CALCOFI) cruise are used as well. Water velocities up to 55cm/s (averaged over 12-24 hours) were calculated from tracing temperature features in successive infrared images; they compare well with CALCOFI and moored near-surface current data, when available.

The interaction between an ocean eddy and incoming swell is investigated numerically producing characteristic refraction patterns of wave number and energy. The velocity distribution in the model eddy follows observations off Pt. Conception that are discussed in Sheres and Kenyon in this issue. The energy refraction patterns exhibit focus regions (caustics) where wave energy is concentrated; adjacent regions have a reduced wave energy concentration. Due to the eddy-induced refraction of incoming Pacific swell frequently observed in the area, approximately 20 km of coastline near Pt. Conception and about 5 km of coastline south of Morro Bay (California) would have relatively high waves. The Santa Maria Basin in between, as well as part of the entrance to the Santa Barbara Channel, would have relatively low waves. These gradients of wave energy depend on the velocity distribution of the eddy and the group velocity of the waves. Change in both the velocities while maintaining their ratio constant produces the same refraction patterns in the limited number of cases that were tested. During the 30 days residency of the eddy off Pt. Conception, the gradients of wave energy were likely to be significant. The potential effect of such gradients on the momentum exchange between wind and water (wind stress) and between waves and surface flow are discussed briefly.

A double vortex was observed at the entrance to the Santa Barbara Channel, California, with sequential IR imagery from the NOAA 7 satellite during May 1984. While the wind blew strongly to the southeast, the double vortex moved westward with a velocity greater than that estimated for the mutual interaction of two idealized vortices. The effect of the double vortex on incoming Pacific swell is calculated by numerical integration of the ray equations. This calculation shows areas of focusing of the rays near the entrance to the channel, which suggests elevated wave energy there with reduced wave energy nearby. The pattern of wave refraction by mesoscale features, such as a double vortex, can help detect them and estimate their velocity distribution from remotely sensed surface wave imagery. The similarities and differences between refraction patterns produced by double vortices and by eddies is outlined. The double vortex and its generation by an episodic or tidal flow is investigated using Kashiwai's (1984) approach. An example of the related generation of a single vortex in the Alboran Sea by tidal flow in the Strait of Gibraltar is briefly discussed.

Modulation of the wavenumber and amplitude of swell by mesoscale current and bathymetry features leads in turn to patchiness of the short gravity capillary wave field due to the modulated swell. A selective review of these processes and their implication to air sea interaction and remote sensing is presented.

First, a general review is presented of wave-current interaction processes ( for horizontal shears) and their effect on radar backscatter and radar imagery ( SAR/ RAR ) Then numerical results on the refraction of wave energy trajectories by complex bottom topography (finite depth) and a linear shear current are presented. For deep water, the wave-energy trajectories are given for mesoscale currents ( e.g. eddies and double-vortex configurations). The focusing of wave energy by variable currents found here should have important influence on the spatial scale of wind stress over the ocean, and on optical and acoustic properties of the upper layer of the ocean.

Aerial photographs showing strong refraction of swell near the northern California coastline were obtained on September 8, 1982. The possibility that this refraction is due to a large horizontal current shear is discussed; a maximum shear value of the order of 10−3 s−1, with the velocity directed along a linear slick‐like front, is calculated from the wave data in the photo. In situ current data obtained at 90 m depth at a nearby midshelf site on two surface moorings separated by 390 m also show occasional periods of large horizontal velocity shear. On January 4, 1982, a maximum horizontal shear of order 10−3 s−1 was observed for several hours. These separate observations taken together indicate that large horizontal surface velocity shears of order 10−3 s−1 occur over the northern California continental shelf, and that large surface current shear can be usefully detected and measured by high flying high resolution sensors.

Ocean imagery obtained by satellite carried sensors in the visible, microwave and IR frequency bands contain synoptic data about the state of the ocean that is available in close to "real time" on a global scale. The information includes quantitative data about surface waves, temperatures, velocity, shear, chlorophyl content and bathymetry in shallow water. Here we will report on two examples of the use of remote sensing in ocean measurements: 1. The use of high resolution ocean images of swell to determine wavelength and direction of the dominant waves as well as to detect and measure surface currents and horizontal shears. 2. The use of infra red ocean imagery to determine water velocities and circulation patterns. Archived imagery of both types can be used for obtaining historic data on a global scale.

Spatial changes in the wavelength and direction of short‐wavelength surface gravity waves, caused by their interaction with much longer wavelength gravity waves, are calculated by applying geometrical optics to the short waves in the reference frame that reduces the long waves to a steady current, which is horizontally variable but unidirectional. The wave rays are very nearly parallel to the direction of the current. Along a ray the wavelength and the angle between the propagation direction of the short waves and the ray direction are both maximum at the long wave troughs (where the steady current speed is maximum) and minimum at the crests (where the current speed is minimum), whether the short waves travel with or against the long waves. The short waves can never be reflected by the long waves. The calculated magnitudes of the wavelength and direction changes of the short waves are valid for arbitrary surface slope of both short and long waves, assuming no wave breaking occurs. Therefore these results are more general than any known ones, but they agree with earlier work when all wave slopes are taken small compared to unity and the wave number directions of the short and long waves are parallel.

Synoptic surface current data in a lagoon have been obtained utilizing a new approach that evolved from the kinematics of wave-current interaction. Surface current at a position was determined from wavelength and direction of two monochromatic wavetrains with known frequency; these wave data were required only at the position of current determination. The data, collected by aerial photography, were processed optically by two-dimensional Fourier transforms. Possible extension to surface flow measurements in the open ocean is discussed.