Horizontal Eddy Viscosity Research Articles

Dynamics of the slope water off New England and its influence on the Gulf Stream as inferred from satellite IR data

The movements of warm core rings and the Gulf Stream front were studied in the New England slope water region by using both satellite IR data and available in situ sea surface temperature data. It is found that: (1) the rotating speed of the rings exhibits a marked seasonal variation with the maximum speed occurring in the winter and the minimum in the late fall; (2) the surface area of the rings increases monotonically with time; and (3) the Gulf Stream front at the north wall migrates seasonally and attains its northernmost position in the late fall. Based on these findings we calculated the horizontal eddy viscosity for the rings as 4 × 10 8 cm 2 5 . Furthermore, by comparing the seasonal variations of the rotational velocity of the rings and the total mass transport of the Gulf Stream, we suggest that the seasonal variation of the slope water hydrography through local air-sea interactions could play an important role in the dynamics of the Gulf Stream system.

Wind and melt driven circulation in a marginal sea ice edge frontal system: a numerical model

A seasonal ice edge zone is a unique frontal system with an air-ice-sea interface. This paper is a report on the numerical results from a quasi-three dimensional, time dependent, non-linear numerical model of circulation at a continental shelf-seasonal ice edge zone. The purpose of the experiments is to model the hydrography and circulation, including upwelling, baroclinic geostrophic flow, and inertial oscillations, at the ice edge with emphasis on examining the driving forces of wind and melting ice. It is suggested that the non-linear acceleration terms and vertical density diffusion terms are negligible and that the horizontal density diffusion terms are of secondary importance within the time and space scales of the experiments. The vertical eddy viscosity terms are important in a spin-up time scale and for Ekman transport and a bottom Ekman layer. The effects of the horizontal eddy viscosity terms are observable (a long-ice jet is diffused away from the ice edge) by the end (72 h) of the model runs. Model results are compared with available oceanographic and meteorological data for verification. The observed and modeled features of melt water induced water column stability, frontal structure, and ice edge upwelling are briefly discussed relative to observed ice edge primary production. Because the model is relatively general in nature, it is readily applicable to other seasonal or marginal ice edge zones in either hemisphere.

Observations of the Horizontal Interactions between the Internal Wave Field and the Mesoscale Flow

Abstract Momentum and energy transfers from the mesoscale horizontal velocity shear to the internal wave field have been deduced from an analysis of a closely spaced, 25 km, moored current-meter array. The correlation between the low-frequency horizontal shear and internal-wave-field continuum effective stress implies a significant horizontal eddy viscosity of O (106 cm2 s−1), somewhat larger than predicted by Muller (1976). A simple steady-state energy balance for the internal wave field using the observed correlation between the internal wave kinetic energy and the square of the low-frequency shear implies a 10-day relaxation time for the internal-wave Acid and a combined vertical viscosity and horizontal diffusivity not significantly different from zero. These estimates are within the experimental uncertainty of previous observational analyses.

Rotating rayleigh-taylor instability as a model of sinking events in the ocean

Abstract The development of small disturbances in a statically unstable two-layer rotating fluid is investigated. With horizontal eddy viscosity included, there is a maximum growth rate [sgrave]max at some preferred wavelength L max. This scale selection mechanism may determine the characteristic size of deep convection events in the ocean following a sudden cooling of the surface. Horizontal scales of 500m or less are estimated. Unless growth rates are small ([sgrave]max≪ f) the effect of the Earth's rotation on [sgrave]max and L max is minor.

M2Baroclinic Tides in Johnstone Strait, British Columbia

Tide, current and water property data collected in the western basin of Johnstone Strait, British Columbia, are compared with analytical models for M2 semidiurnal motions in a stratified, rotating channel of uniform depth. We show that, for a rectangular basin in which there is a depth-independent mean flow, the tidal fluctuations are readily expressible in terms of a landward (eastward) propagating Kelvin wave and exponentially damped, seaward (westward) propagating baroclinic Kelvin waves. The internal M2 tide appears to be generated through interaction of the surface tide with the shallow sill at the eastern end of the basin and is dominated by a first-mode baroclinic wave whose amplitude undergoes attenuation over an e-folding distance of one wavelength (∼26 km). Near the sill the baroclinic current speed may exceed 50% of the barotropic flow speed. The along-channel energy flux (power) of the barotropic tide is comparatively uniform at ∼1.4×109 W throughout the western portion of the Strait but decreases by a factor of 2 across the sill. Less than 0.3% of this power loss is needed to account for the baroclinic energy flux of ∼2×106 W near the sill; dissipation due to bottom friction is the main sink for the barotropic tidal energy. Attenuation of the baroclinic waves appears to result from a combination of turbulent horizontal friction and bottom boundary layer friction. Based on the first mode amplitudes, we find a horizontal eddy viscosity of ∼4×106 cm2 s−1 and a vertical eddy viscosity of ∼103 cm2 s−1. Our analysis further indicates that higher modes may be subject to attenuation by critical-layer absorption in the presence of the mean estuarine circulation in the basin. The M2 tides and currents are shown to possess considerable variability over periods of weeks and to have seasonal trends that, qualitatively at least, are related to changes in the mean density structure.

Contributions to the dynamics of the Antarctic Circumpolar Current (A. C. C.) II. Integral relationship

Non-dimensional equations of motion are derived for the A.C.C. of the barotropic mode, including the bottom friction and the horizontal eddy viscosity. Integration of the vorticity equation along a streamline leads to the zeroth order stream function which is dependent only on depth divided by Coriolis parameter. Integration of the momentum equation along a streamline yields the relation between the momentum input by wind stress and its dissipation by the bottom friction and by the horizontal eddy viscosity. This relation determines the magnitude of the stream function. It explains differences in the total transport of the A.C.C. obtained byBryan andCox (1972), though it gives only one third of the total transport obtained byKamenkovich (1972) with his vertical eddy viscosity of 102cm2 s−1. With 1 cm2 s−1 of this viscosity,Bryan andCox obtained the transport of about 650 or less than 32×106m3s−1 for constant or variable depth models, respectively. The higher transport is mainly due to broadening of the width of the A.C.C., whereas the lower value is due to its narrowing and meandering which in turn make the horizontal eddy viscosity more effective (by exercising friction on both sides of the A.C.C.) and the wind stress input smaller than the almost zonal streamlines for constant depth. In the Appendix dynamics of the bottom boundary layer is treated to give rational estimates of the bottom stress in terms of the geostrophic flow and is compared to the recent observations of the benthic boundary current in the Straits of Florida and off San Diego.

Cabbeling at thermohaline fronts

A linear model with cabbeling dynamics is developed for a nonrotating, unstratified ocean and applied to observations of three medium‐ to large‐scale oceanic thermohaline (density balanced) fronts. The ability of the predicted cabbeling sinking and surface convergence rates to maintain a steady front against diffusion depends critically on the assumed values of horizontal and vertical eddy viscosities. There is good theoretical evidence for the existence of small‐scale (1 km), weak (surface currents 0.1 cm s−1) cabbeling fronts; however, the intensity and stability of such fronts will depend critically on the scale dependent behavior of dissipation processes.

A three-dimensional model of Hamilton Harbour incorporating spatial distribution of transient surface drag

This paper discusses the formulation of surface boundary conditions for a three-dimensional transport model for shallow lakes, specifically for Hamilton Harbour. The same hydrodynamic equations that describe the circulation of the ocean and the Great Lakes were used in this study. However, the boundary conditions (bed topography, shoreline configuration, and surface and bottom shear stress fields) have bigger effects on circulation in shallow enclosed lakes.In this study the flow is assumed to be incompressible and in hydrostatic equilibrium. A layered system is used in which the lake is considered to consist of a number of unequal layers in the vertical. The hydrodynamic equations are integrated vertically over each layer, and both vertical and horizontal eddy viscosities are introduced.The over-water wind stress is determined using the logarithmic wind velocity distribution and Von Karman's integral equation for turbulent flow over a rough movable surface of variable roughness, in conjunction with equations for wind–wave generation. Thus the wind drag coefficient is determined as a function of wind and wave characteristics, and is time- and space-dependent.

On the theory of the East African Low Level Jet Stream

Several theoretical models for the East African Low Level Jet Stream are described. They all share the notion that the northward advection of planetary vorticity across the equator, coupled with the presence of a north—south mountain barrier, leads to the formation of a low-level western boundary current (akin to the Gulf-Stream) along the equatorial east coast of Africa. They differ in the manner in which the planetary vorticity advection is balanced to obtain a quasi-steady state. A purely inertial model predicts the correct cross-stream scale of the jet, but does not reproduce the observed inner shear layer which reduces the jet velocity to zero inland near the highlands. The lateral friction model can produce a realistic jet profile if the horizontal eddy viscosity (appearing as a free parameter) is chosen appropriately. However this solution shows a recirculation, i.e., northerly flow, off the coast that has not yet been observed. Finally, a model that includes bottom friction over variable topography also can give realistic jet profiles. If one accepts that the mountains, the Beta effect, and some form of inertial or frictional acceleration act together to produce the cross-equatorial low level jet stream, then one can formulate the types of observations needed to distinguish between the various theories.

On the scale of the turbulent small masses of sea water

The relation between the initial spatial scale and the life time, of turbulent small masses of sea water produced at the strait or the interface between two water masses, is investigated analytically. In the analysis, their initial shape is assumed to be expressed as the Rankine vortex, and the horizontal eddy viscosity coefficient to obey then-th power-law with respect to scale. Particularly in the case ofn=4/3 (the 4/3 power-law), it is obtained that the life time is proportional to the 2/3 power of the initial scale. The proportional coefficient can be determined in the present analysis except a parameter related to the energy dissipation rate of each sea area.

LATERAL AND BOTTOM FORCES ON LONGSHORE CURRENTS

A two-component electromagnetic flowmeter has been used on a natural beach of slope 0.01 to measure mean longshore currents and the horizontal fluctuating velocities in the frequency range 0-1 Hz. The measurements extend up to 120 m offshore and span about one third of a wide surf-zone. The cross product of the fluctuating horizontal velocities, assumed to contain the combined effects of radiation and Reynold's stresses, is plotted as a function of distance from the shoreline. The on/offshore gradient of the cross-product is then equated with a bottom friction term either in the form used by Bowen (1969a) or in a form similar to that used by Longuet-Higgins (1970). The apparent values of bottom friction coefficient obtained in this way are at least a factor of two smaller than expected for Reynolds numbers and bottom roughness appropriate to the beach. Attempts to separate the radiation stress and the Reynold's stress contributions to the total stress term using cospectra fail to show distinguishable Reynold's stress contributions. Although this may be construed as being consistent with Battjes' (1975) beach slope dependent form for horizontal eddy viscosity rather than Longuet-Higgins' (1970) form, it is argued that, in fact, the significant horizontal turbulence was not measured at all but was confined to a surface layer above the flowmeter. This leads to the hypothesis that lateral friction, as a surface boundary layer, and the bottom friction act on a less turbulent central layer, and that the small measured friction coefficient in the present experiment is the result of the combined effects of these layers.

The Dynamic Structure of the Frontal Zone in the Coastal Upwelling Region off Oregon

We studied the frontal zone of the coastal upwelling region off Oregon, from observations made in two successive years. The measurements were made between July and September in 1965 and 1966. The alongshore flow field was determined by combining direct measurements and geostrophic calculations. A near-surface southward jet and a subsurface northward undercurrent existed in the frontal zone. They were separated by an inclined frontal layer (permanent pycnocline). The frontal layer tended to intersect the sea surface about 10 km offshore, where a surface front was formed. Through a combination of direct current measurement and water mass analysis, the cross-stream flow was estimated to be seaward near the surface, shoreward at the top of the inclined frontal layer, but seaward at the bottom of the inclined frontal layer and shoreward below that. During a 25 h anchor station, a high degree of correlation existed between the vertical structure of the alongshore and cross-stream flows. An anomalously warm water mass occurred at the base of the frontal layer. We believe it was formed near the surface front and that it sank and flowed seaward along the base of the inclined frontal layer. Vertical shears in the horizontal velocity were caused by the mean baroclinic flow and the tidal and longer period baroclinic oscillations. A zone of low dynamic stability was produced near the base of the inclined frontal layer, coincident with the warm anomaly, providing a mixing mechanism for the erosion of the warm anomaly and the broadening of the frontal layer offshore. Estimates of temporal and spatial scales and of horizontal eddy viscosity coefficients are given. Internal tidal motions provided an energy flux to the mean motion. A conceptual model is presented for the mean state (averaged over a fortnight or, equivalently, over one or more upwelling “wind event cycles”) of coastal upwelling.

Short-Period Variations in a Great Lakes Coastal Current by Aerial Photogrammetry

Measurements of the surface velocity structure off the Keweenaw Peninsula of Lake Superior were obtained in 1971 and 1972, using aerial photography to track surface drift cards. Variations in the current structure are described at 9-min intervals, over a 45-min period of one experiment, using streamlines and isotachs extending across the entire coastal region. Speed contour irregularities and eddies of about 100 m diameter can be traced in some of the aerial sequences. Speed fluctuations of 25% of the mean flow occur frequently. The horizontal divergence and relative vorticity structure for each sequence is also calculated; magnitudes of each are up to three times that of the local Coriolis parameter. Both inshore and offshore countercurrents are observed. The region of anticyclonic shear is typically twice as wide as the cyclonic shear region. Cross-stream velocity gradients are about three times larger than those measured in the Gulf Stream. Rossby numbers range from 0.5 to 0.8, and inertial accelerations appear to be larger than local accelerations at least 25% of the time. Horizontal eddy viscosity coefficients range from ±103 to ±105 cm2 s−1. Geostrophic calculations based on bathythermograph sections, airborne radiometer flights and meteorological data are also discussed.

On the attenuation of sea waves by rain

Abstract Some conflicting evidence on Reynolds' (1900) hypothesis that rain should attenuate any wave motion on the sea surface is discussed. It is concluded that rain drops ought to produce vortex rings in the sea which mix the water to a sufficient depth to affect most waves, as asserted by Reynolds. By introducing vertical and horizontal eddy viscosities to model the mixing process, an estimate is found for the rate of attenuation of wave energy by the rain. Consequently, the net effect on the wave field of attenuation by rain and generation by wind is calculated.

Dynamics of rip currents

The dynamics of rip currents is investigated using a set of shallow water equations with a horizontal eddy viscosity term. A boundary layer analysis is introduced based on the observed fact that rip currents are rather narrow. Similarity solutions of the model equations are found which seem to give reasonable representations of the velocity profile and other characteristics of rip currents. The mechanisms that are believed to be responsible for the formation of rip heads are studied. The turning of rip currents due to the longshore momentum carried by the entrained fluid is analyzed.

Lake currents associated with the thermal bar

A time-dependent theoretical study of the thermal bar is presented. The temperature and velocity distributions are obtained for a system consisting of water contained in a rotating annulus with a horizontally varying heat flux applied at the free upper surface. An equation of state is used that shows a parabolic dependence of density on temperature, the maximum density occurring at 4°C. The analysis assumes small Rossby and Ekman numbers and allows for different values of vertical and horizontal eddy viscosities and conductivities. The temperature field is found from the time-dependent heat conduction equation. An interior flow field is found utilizing the ‘thermal-wind’ balance and is matched to Ekman layers. Side boundary layers are found, the form of which depends on the relative sizes of the Ekman number and the ratio of eddy viscosities or conductivities, but their contributions are small compared to the interior. Both of the lines of zero meridional velocity and stream function proceed from the shore to the center of the lake, marking a change in direction of all velocity components as they go. As time elapses, the magnitudes of the velocity components get larger in the shoreward water and smaller in the seaward water. Eventually the shoreward regime encompasses the entire lake.

How faithfully will the geostrophic currents represent the existing ocean currents?

It is widely recognized that the geostrophic flows computed by the dynamic method of Bjerknes and collaborators represent the actual currents pretty faithfully. However, what would be the reason that a geostrophic current derived by only retaining the terms of Coriolis and the pressure gradient forces in the hydrodynamical equations agrees so closely with the actual ocean current of the same area? In this attempt was assumed an imaginative ocean of homogeneous water and uniform depth on a rotating earth but with neither continent nor islands. The average wind distribution observed along several meridians over the Pacific Ocean was assumed to prevail in this sea throughout with no variation in east-west direction. Taking the curvature of the earth surface, rotation of the earth, Coriolis forces, pressure gradients and the horizontal and vertical eddy viscosity into account, the equations of motion were solved and velocity components were derived for all latitudes. A comparison of the east-west components thus obtained with the corresponding components of the geostrophic flows, reveals that they agree well in higher latitudes but there appears a remarkable disagreement in lower latitudes. This means that a special care must be taken in replacing the existing currents with the geostrophic flows at lower latitudes.

NUMERICAL EXPERIMENTS WITH A SLOWLY VARYING MODEL OF THE TROPICAL CYCLONE

Additional experiments with the slowly varying axisymmetric hurricane model described by Anthes are presented. Variable horizontal eddy viscosity coefficients are utilized to study circulations that result from an increased heating function and a heating function with a double maximum along the radial direction. Infrared radiative cooling is modeled for the clear tropical atmosphere following Sasamori. A representative cooling rate is used to study the effect of cooling in the hurricane environment on the dynamics and energetics of the slowly varying tropical cyclone.

Longshore currents generated by obliquely incident sea waves: 2

The profile of the longshore current, as a function of distance from the swash line, is calculated by using the concept of radiation stress (introduced in an earlier paper) together with a horizontal eddy viscosity μe of the form μe = ρNx(gh)1/2, where ρ is the density, x is the distance offshore, g is gravity, h is the local mean depth, and N is a numerical constant. This assumption gives rise to a family of current profiles whose form depends only on the nondimensional parameter P = (Π/2)(sN/αC), where s denotes the bottom slope, α is a constant characteristic of breaking waves (α ≑ 0.41), and C is the drag coefficient on the bottom. The current profiles are of simple analytic form, having a maximum in the surf zone and tending to zero at the swash line. Comparison with the laboratory experiments of Galvin and Eagleson (1965) shows remarkably good agreement if the drag coefficient C is taken as 0.010. The theoretical profiles are insensitive to the exact value of P, but the experimental results suggest that P never exceeds a critical value of 2/5.

Cell Method for Computing Lake Circulation

The equations of fluid motion are solved using finite differences for the three-dimensional, wind-driven circulation in a homogeneous lake of arbitrary bottom topography. The method allows arbitrary choices for the wind stress, the horizontal and vertical eddy viscosities, and the distribution of vertical eddy viscosity. For the examples investigated the nonlinear terms of the equations contributed little or nothing, indicating that lake circulation is well described by linear equations. The magnitudes of the horizontal and vertical eddy viscosities have a strong effect on the current velocity but only a minor effect on the circulation patterns. The distribution of the vertical eddy viscosity appears to play a minor role in the circulation pattern. Sample calculations indicate that some combinations of horizontal eddy viscosity, vertical eddy viscosity, and distribution of vertical eddy viscosity are nearly equivalent in that these combinations produce similar response curves, similar current velocities and circulation patterns having small differences.