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The quasi-geostrophic theory leads to a single nonlinear partial differential equation for a streamfunction giving geostrophic velocity fields presumed to resemble the synoptic scales of atmospheric motion. This article is concerned with demonstrating that the quasi-geostrophic problem is well-posed mathematically, in the sense that solutions exist, and that they are continuously dependent on the initial data. The model studied is comprised of the quasi-geostrophic equation subject to the severe boundary condition that an isentrope coincides with the earth's surface. The main technique is the use of the eigenfunctions of an elliptic operator appearing within the quasi-geostrophic equation. These eigenfunctions provide the basis for a spectral model, which can be truncated to include a finite number of scales. The convergence properties of the solutions to the truncated model allow the existence of solutions to the entire model to be inferred with the methods of functional analysis. Thus, the conc...

Nondimensional parameters characteristic of the outer part of the planetary boundary layer have been determined by fitting a simple, Ekman-type theory to a number of averaged, observed velocity distributions, using the Wangara data of Clarke et al. (1971). The theoretical model is based on constant eddy viscosity in the outer layer and a linear variation of the geostrophic wind with height. At the lower boundary of the outer layer, the condition is applied that stress and velocity are parallel. This yields an equation for the cross-isobar angle as a function of drag coefficient, depth coefficient and nondimensional thermal wind. The data could be sorted into three well-defined, distinct groups, each characterized by a more or less constant value of the depth coefficient. The group with the lowest value of this parameter contains most of the nighttime data, the middle group the remaining nighttime data and most of the daytime ones, and the group with the largest depth, daytime data with cold air advection. The difference between the lowest and highest depth coefficients found here is about a factor of three. Within each group separately, the theoretically derived cross-isobar angle agrees remarkably well with the observed one, as a function of thermal wind.

Diagrams of silicate versus potential temperature from two years of data at a hydrographic station on the southern Puerto Rican insular slope 190 km west-southwest of Jungfern Passage sill indicate the presence of minor amounts of North Atlantic Deep Water (NADW) below 1600 m. Time-dependent sections of silicate indicate that this water is present only sporadically. Timedependent sections of salinity display no variation below 1600 m. These observations are consistent with sporadic overflow of NADW into the Venezuela Basin over Jungfern sill, accompanied by mixing and geostrophic spreading at intermediate depths westward along the Puerto Rico-St. Croix ridge.

A Simple slab model of the planetary boundary layer is extended to include vertical shear of the geostrophic wind. The layer depth is assumed to be determined by a Richardson number criterion. The cross-isobar angle for the surface wind is given in terms of the drag coefficient, the Froude number of the layer, and the angle between the thermal wind and the surface isobars. The theoretical results resemble the observations rather well.

Three subjective and two objective analysis techniques were employed to prepare height data for a synoptic case over North America. These heights were used as input to a ten-level quasi-geostrophic model for computation of omegas (vertical velocities in pressure coordinate system) over a grid 2° latitude square. Using a subjective analysis prepared by F. Sanders as a standard, all the techniques gave synoptically acceptable patterns locating the 500-mb low center within one grid square with the intensity within ±20 m. The average height differences at each level for each technique never exceeded 11 m. One subjective and one objective technique did slightly better than the others for the height field. Comparison of the quasi-geostrophic omegas with the Sanders' values shows good agreement for all analysis techniques for the broad scale rising and sinking centers and zero lines. The 500-mb centers are located within one grid interval with the main central values generally within 10–20% of the Sande...

Theoretical reasons for considering the temporal, zonal and time-and-zone root mean square values of meteorological parameters as their normal values in the extratropics are given. Using geostrophic relative vorticity (ζ g ) data for the Northern Hemisphere Extratropics for four months typical of the four seasons it is shown that the time averages of the zonal root mean square values of ζ g — calledK — are useful in determining the normal locations and other properties of extratropical jet streams.

Geostrophic drag coefficients are obtained from direct measurements of the momentum flux and from an objective analysis of the synoptic pressure field by the method of least squares. At a site in the Kiel Bight, a mean geostrophic drag coefficient cg = 0.0223 was obtained with near neutral/ slightly unstable conditions and a surface Rossby Number of 1.2 × 109.

Oscillations of surface and interface elevations and velocity have been calculated numerically in using a two layer model of constant depth with 25×62 square meshes of 1 km intervals. In that model, the mean depth of the interface is 17.5m and the density difference between the upper and lower layers is 2.5×10-3g/cm3 corresponding to the temperature difference of 13°C.Linear hydrodynamic equations for the equivalent volume transport and the equivalent elevations were integrated by means of an implicit method, taking into account of wind stress, bottom stress and CORIOLIS force.In the present numerical experiments, the four cases of duration time were treated, 450 minutes (NW-(1)) and 1200 minutes (NW-(2)) for NW-wind and 450 minutes for SW-wind and ESS-wind, to which the constant wind speed of 5m/sec was applied.From the results of FOURIER spectrum analysis for the calculated time series of elevation and velocity at various sampling positions in the model lake, it was found that the long period internal KELVIN waves progressing along the lake shore in the anticlockwise direction exist with the same period as that in the case of no rotation of the earth, and that the mean lake currents consisted of two or three gyres, are essentially affected by geostrophic condition, as usually observed in real Lake Biwa.

The evolution of the structure of the atmospheric boundary layer during the cooling phase of the diurnal cycle is studied with the help of a one-dimensional numerical model. The model uses a turbulent energy equation for the determination of the eddy exchange coefficients, which also involve the local value of the Monin-Obukhov length. The surface temperature is prescribed as a function of time and the geostrophic wind is held constant. Results are presented in nondimensional form and cover a wide range of possible values for the external parameters. After several hours of cooling the inversion layer is found to reach a maximum depth, which depends strongly upon the magnitude of the geostrophic wind and, to a lesser extent, on the cooling rate of the ground surface.

From ship-borne radiosonde and radar wind observations during the Atlantic Trade Wind Experiment 1969 (ATEX) the relation between actual and geostrophic wind in the subcloud layer is determined. The frictional force and the divergence of vertical transport of momentum between the sea surface and 500-m height are computed by the ageostrophic method. Along the surface wind direction, balance between pressure gradient and frictional force is closely established. Perpendicular to the surface wind the expected balance of pressure gradient and Coriolis force is not obtained but a substantial frictional force results from this component. The boundary condition that the stress vanishes at the lowest wind maximum leads to an underestimation of surface stress by about 35 per cent. It is suggested that convective transport of momentum must take place in the subcloud layer.

Results of tests carried out during the period from December 1972 through January 1973 to determine the short-term forecasting accuracy of a multilevel numerical primitive-equation (PE) model of the global atmosphere developed at the Goddard Institute for Space Studies (GISS). Six 48-hr forecasts were carried out with the aid of this model, using nine vertical levels and a horizontal grid spacing of 4 deg in latitude and 5 deg in longitude for an effective grid point separation averaging slightly more than 400 km. Verification of forecast sea-level pressures, 1000-mb heights, and 500-mb heights, as well as 1000-mb and 500-mb vector geostrophic winds, shows that the model has forecast skill comparable to that of operational PE models. Based on the 36-hr evolution of 18 extratropical cyclones, the model forecasts exhibit a tendency toward underestimating their propagation speeds and overestimating their central pressures. Both deficiencies are attributed to inadequate horizontal grid resolution. Quantitative verification of forecast surface temperatures over the eastern United States shows a forecast skill equal to that achieved by combined dynamical-statistical procedures.

Two sets of initial conditions are used to integrate a three-level quasi-geostrophic model in spectral form. After a maximum perturbation kinetic energy is reached, a barotropic exchange in established between the zonal flow and the perturbation with an apparent periodicity from 2 to 4 days. The initial state includes a finite amplitude baroclinic mode which is highly unstable in the linear sense. This mode exhibits a negative growth rate for about the first two days of the integration due to barotropic exchanges with other modes. Spectra of kinetic and available potential energies, enstrophy, and omega2 are presented. The kinetic and available energy display a –3 slope for intermediate, scales for the initial integration period when perturbation kinetic energy is actively growing at the expense of the mean flow available energy. The effect of the variation of stratification with height, dissipation, heating, mountains, and truncation of the spectral system is discussed.

In recognition of his pioneer work and notably effective leadership in the experimental investigation of the atmosphere’s surface boundary layer, W. C. ‘Bill’ Swinbank has been called ‘ . . . the arch-apostle of the small, controlled field experiment’ (Priestley, 1974). That this appellation is well deserved is demonstrated by the published record of the research arising from the field programs he conducted in Australia during the 1960’s, under the auspices of the CSIRO Division of Meteorological Physics. Swinbank’s major papers on boundary-layer research were published in the decade before this journal came into existence, although his very last note on the subject (‘The geostrophic drag coefficient’, 1974) did appear in these pages. It therefore seems appropriate that this memorial volume include some more specific reference to his earlier work by quoting selections chosen as much for style as content in order to recall Bill Swinbank td those who knew him, and to introduce him to those who did not. In part because of the work summarized here, we have come to know that, given uniform conditions of weather and terrain, distributions of wind and temperature in the surface boundary layer can be sufficiently steady in the mean to enable micrometeorological measurements approaching laboratory quality to be obtained in the field. Exploiting the fact that such conditions frequently exist over the plains of southeastern Australia, Bill Swinbank organized and led a remarkable series of micrometeorological field expeditions whose measurements set a standard of precision and internal consistency seldom approached during the ensuing decade. His own descriptions of the design, conduct, and analysis of these studies are distributed among a number of papers and reports. Recollected, they provide an account of unique interest to the student of the atmospheric boundary layer. Should some few points made then now seem obvious to the more informed reader (who benefits from more than ten years’ hindsight), one must ask why similarly exacting standards are not universally observed in boundary-layer field work today.

The Northern Hemisphere 700 mb contour heights from 20 N to 70 N for the period 1947-1970 are studied in conjunction with 272 key days, where the daily increase of the Ci index equals or exceeds 1.0. The superposed epoch method is applied from 33 days before to 66 days after the key day for a variety of zonal and meridional indices. It is shown that the 700 mb height difference between 20 N and 55 N increases significantly in winter 4 days following geomagnetic disturbance (in summer a less prominent but statistically significant increase is found 2 days earlier). The effect is most clear in winter in the quadrant 90-175 W and corresponds to a 7% increase in the mean geostrophic westerly flow. The statistical significance of the results is established by applying Student's t-test to the difference of each daily mean from the continuum.

Analysis of hemispheric temperature variance data on five isobaric surfaces in terms of two-dimensional spectral decomposition shows that the available potential energy distributes with a slope in the neighborhood of −3 for the scale range 14≤n≤25. Although this slope varies with pressure, indications are that the observations substantiate the expectations of geostrophic turbulence theory. The noted deviations from −3 are discussed in terms of the distribution of energy in vertical modes which are not in the range in which −3 statistics should be expected. Vertical scales necessary for a three-dimensional spectral representation are considered with regard to the Brunt-Vaisala frequency distribution.

A new empirical assessment of the functions A and B of Rossby-number similarity theory is made based on the Wangara data. Variations of these functions with stability, baroclinicity and time of day are discussed. It is found that B is dependent on stability in agreement with older data but contradicting the prediction of Csanady (1972). Coefficients expressing the variation of A and B with the two components of baroclinicity have been derived from the data, and these are claimed to be correct in regard to sign and approximately in regard to magnitude. Longer period time changes, represented by the diurnal cycle, are shown to result in systematic differences in A and B between the case of increasing stability and that of decreasing stability, for the same value of the stability parameter. The first attempt, to our knowledge, to present the actual functional form of the wind departure components (based on field data) is made. As the surface layer is approached in near-neutral conditions, the departure component in the direction of the surface wind assumes the expected logarithmic form.

We consider here the flow induced by applying a wind stress at the surface of an initially quiescent lake. It is assumed that the Ekman number, based on an eddy viscosity, is small, and that the Rossby number is at most of the order of (Ekman number)½. Under these conditions, which are met in practice, a linear theory is applicable. The linear problem is solved using boundary layer methods. There are essentially five distinct regions: an outer region in which the horizontal velocity is independent of depth, Ekman layers at the upper and lower boundaries, a corner region at the edge of the lake at which the Ekman layers meet, and a shear layer adjacent to the corner region. Study of the Ekman layers provides the equations which hold in the outer and shear layer regions, and consideration of the corner region provides the boundary condition. The outer flow proves to he geostrophic and directed along curves of constant depth. The shear layer is needed to satisfy the boundary condition of zero net ou...

Formation of Antarctic Bottom Water takes place by the mixing of dense shelf water with warmer fresher open-ocean water along the edge of the continental shelves in such areas as the Weddell and Ross seas. The amount formed depends critically on the structure of the outflow from the shelf; this outflow is produced geostrophically by the strong east-west salinity and density gradient on the shelf. Since the effective freezing of sea ice produces a surface salinity influx which varies only in the north-south direction, it is necessary to seek a mechanism for the east-west density gradient. Such a mechanism is found in the geostrophically-induced upwelling which occurs on the western boundary of the shelf: the mixing of denser subsurface water with lighter surface water induces a net east-west density gradient. This corresponds to a geostrophic outflux of dense subsurface water in a boundary layer which grows in thickness as time increases. Downwelling occurs on the southern and eastern shelf boundaries. The predicted density distribution agress well with observations; only the northward fluxes are to small. Hence it is probable that several years of forcing are necessary before a limit cycle is achieved.

Systematic stratifications and analyses of low-level radiosonde data are performed for portions of the eastern half of the United States. The procedures are designed to specify changes in the planetary boundary layer wind profile resulting from variations in baroclinicity. The angle between the winds and isobars, the ageostrophic wind components, the surface stress, and the surface wind speeds are all shown to be functions of the orientation of the thermal wind vector relative to the surface geostrophic wind. These variations are consistent with a mixing-length model of the additional turbulent momentum transport initiated by the vertical shear of the geostrophic wind.

The two-layer system of an atmosphere over water bodies is reduced to a single-layer problem. Values of the interfacial quantities, such as the friction velocity, the surface velocity, the angles, alpha and beta, between the surface shear stress and the geostrophic wind velocity and the surface wind velocity, respectively, and the surface roughness, all of which depend upon external parameters, such as the geostrophic wind and stratifications, are obtained. The geostrophic drag coefficient, the geostrophic wind coefficient, and the angles alpha, and beta, of the turbulent flow at the sea-air interface are functions of a dimensionless number, mfG/kg, with S sub 1 and S sub 2 as two free stratification parameters. The surface roughness is uniquely determined from the geostrophic wind rather than from the wind profile in the boundary layer.

The strength and latitudinal position of the principal transport in the Pacific North Equatorial Countercurrent has been developed theoretically by Sverdrup (1947) and by Reid (1948) to be the result of a mass redistribution in response to Ekman divergence on the β-plane. More recently, the principal transport of the Countercurrent was found in the surface layer above the thermocline (Wyrtki and Kendall, 1967) and a good approximation for the Countercurrent transport in the upper layer (< 150 m) can be obtained by considering only the slope of the thermocline in a fashion after the Margules equation. Independently, Miropol'skiy (1970) developed a two-layer thermocline model from considerations of the conservation of heat and turbulent kinetic energy that prescribes the thickness of the upper layer in terms of the net heat flux and the flux of momentum across the sea surface. Therefore, it is possible that meridional variations in these fluxes could produce a meridional slope in the thermocline (a redistribution of mass induced by heating and mixing) that would be in geostrophic equilibrium with the flow in the upper layer. To determine whether this thermal/mixing process could maintain the Pacific North Equatorial Countercurrent, the long-term mean meridional profile of the heat and momentum fluxes at the sea surface were used to calculate the ‘thermal/mixing’ transport from 2·5°N to 10°N along 122·5°W. The direction of the ‘thermal/mixing’ transport is calculated to be eastward between 5 and 10°N in agreement with the long-term mean averaged direction of the North Equatorial Countercurrent (Kendall, 1970), and in agreement with the results of Sverdrup's theory. In addition, the magnitude of the ‘thermal/mixing’ transport is comparable to the observed geostrophic transport of the Countercurrent, corresponding to a maximum vertically averaged eastward speed of 35 cm s−1 near 7·5°N.

The response of vertically averaged boundary layer flow to specified time-dependent pressure gradients is examined. The adjustment rate of the boundary layer flow to the pressure gradient field is found to be proportional to the strength of the coupling between the flow and boundary stresses. The angle between the steady flow and geostrophic wind is also proportional to the strength of this coupling. The relationship of the transient component of the boundary layer wind to the “frictionless” transient wind component is found to be similar to the relationship of the steady boundary layer wind to the geostrophic wind. In flows characterized by transient Rossby number of order unity or greater, the production of cross-isobar mass transport is more sensitive to local accelerations than to typical variations of the surface drag coefficient. The transient portion of the vertical motion field, associated with the geostrophic vorticity tendency, is important when the time scale of the pressure gradient i...

The results of extended integrations of a two-level, quasi-geostrophic model with Newtonian heating and dissipation in terms of surface friction, internal friction, and lateral diffusion are described. The major emphasis is on an analysis of the integrations in wavenumber space, including the calculations of spectra for available potential energy, kinetic energy, enstrophy, energy generations, conversions and dissipation, as well as the nonlinear cascades of the first three quantities. It is found that the fluxes of available potential energy and kinetic energy through the wavenumber domain are very small above planetary wavenumber n = 10, while the enstrophy flux is large and positive for 6 ≤ n ≤ 10, but decreases rapidly for n > 10. The available potential energy, the kinetic energy and the enstrophy as a function of wavenumber vary approximately as n−5, n−3 and n−1, for 10 ≤ n ≤ 20. Dimensional considerations based on a balance between the convergence of the enstrophy flux and the dissipation ...

Simultaneous path and bottom velocity measurements made during the Transient Meander Experiment, reported in Part I, are analyzed in terms of a quasi-geostrophic thin jet model of the meandering Gulf Stream. The theory gives an explicit representation of the velocity field which may be used to decompose the observed velocities. This representation is shown to be consistent with the observations. The dynamics of this model provides an equation of the path of the Stream, a cross-sectional average of the vorticity equation. A linearized form of this equation is used to examine the relations between the space and time scales of the variability. The historical data on the space and time scales of the meandering are shown to be consistent with those implicit in the linearized form of the path equation. The contributions to the local vorticity balance are estimated from the observations reported in Part I. The data, although complicated by observational errors, suggest a balance between the local rate o...

Confirming a prediction of deep circulation theory, hydrographic sections along 3ats. 12°S and 23°S between Madagascara nd the Central Indian Ridge exhibit a western boundary current flowing northward close against Madagascar at depths greater than 3000–3500 m. It is 300–400 km wide at Lat. 12°S, and its volum transport is etiamted geostrophically to be 4–5 × 106 m3s−1. The current dervies from deep water int he Circumpolar Current, which moves northward into the western Crozet Basin and appears to pass into the Madagascar Basin near 27°S, 65°E. It is uncertain what fraction of the current continues northward into the Somali Basin. In the layer 2000–3000 m, property distributions in the Madagascar and Mascarenee basis are ambiguous concerning the sense of flow, and there are no boundary currents; geostrophic considerations suggest a net flow northwards of perhaps 1 × 106 m3 s−1 in this layer. Evidently very little of the high-salinity deep water in the Mozambique Basin passes northward into other parts of the Indian Ocean. Some general problematical aspects of these observations for deep circulation theory are noted briefly.

Vertical velocities at the 800, 600 and 400 mbar surfaces over India have been calculated, making use of a 3-level geostrophic baroclinic model. Further, the effects of non-adiabatic heating is included into the model and vertical velocity due to diabatic heating is obtained for the same period. A numerically obtained vertical velocity field due to baroclinicity and diabatic heating is seen to be in agreement with the observed weather patterns.

In the western boundary current of Bryan and Cox's model ocean, a regularly spaced series of eddies grow, move northwards and decay. A qualitative dynamical explanation of this phenomenon runs as follows. There is a longshore pressure gradient throughout the upper layers of the western boundary region, which is not geostrophically balanced at the shore (water cannot flow out of the wall); several related phenomena occur as a result. First, the outflow needed to balance the gradient geostrophically offshore is largely provided by vigorous upwelling near the wall. Second, the geostrophic imbalance in the nearshore region is used to accelerate the boundary current northwards, and to overcome lateral friction. The net effect is to lift warm subtropical water up and over the colder water further north, thereby releasing potential energy. The upwelling and northwards movement also generate anticyclonic vortivity in the upper layers, not alo of which is lost to friction. The longshore pressure gradient, and all the associated phenomena,intensify spontaneously at certain locations; the flow pattern is such a region is similar to a northwards-moving warm front in the atmosphere. Much of the vorticity production in these places escapes the effect of friction, so the current spins seaward in an anticyclonic arc, to foom the northern edge of an eddy. It has not been showm conclusively that the spontaneous eddy growth ocurs through baroclinic instability. Richardson numbers are high and the flow is quasigeostrophic; eddy wavelengths are comparable to those expected of baroclinic waves, though eddy growth rates are rather slow; and the energy source for the complete western boundary flow is baroclinic. However, it is not possible with available model data to make a meaningful distinction between mean and eddy quantities, so no firm conclusions can be reached concerning the role of the eddies in the energy balance. There is some evidence for shear instability, however. After the eddies have developed in the baroclinic component of flow, they begin to show prominently in the depth-averaged flow. The reason is that the nonlinear accelerations experienced by the baroclinic flow act like a stress on the depth-averaged flow; this stress is large compred to the wind stress on the nearshore side of an eddy, and has a strong anticyclonic curl. Consequently, vorticity and kinetic energy are transferred in significant amounts to the depth-averaged flow. It is possible that closer examination of the model might reveal non-physical effects, arising from the use of a finite-difference scheme and of artificial ‘edddy diffusion’: but these effects are not expected to be serious in the western boundary region.

The system of the time dependent continuity and movement equations, describing the essential processes that determined the behaviour of oxygen, was solved numerically. The influence of wind on distribution of atomic and molecular oxygen was estimated. The redistribution of density of oxygen under the influence of neutral wind affects the direction and velocity of wind. So, there is the mutual action of the wind field and density distribution. In the movement equations we take into account the electrodynamic force, arising because of the motion of plasma in the geomagnetic field and deflexion of the wind from geostrophic one. The boundary condition, the density of nitrogen and temperature were taken from the models by Groves and Jacchia for winter in the northern hemisphere. As a result the velocity was found to be about 10–120 m/sec depending on height, latitude and local time. The oxygen density increases in the winter hemisphere and decreases in the summer one in comparison with the zero velocity case.

It is shown that although Coriolis forces cause large production rates of stress in a convective planetary boundary layer, there is a control mechanism, involving mean wind shear which prevents stress levels from becoming large. Higher-order-closure model calculations are presented which show that the stress profiles are essentially linear, regardless of wind direction, providing the geostrophic wind shear vanishes and the wind speed jump across the capping inversion is negligible. It is shown that it will he very difficult to verify these predicted stress profiles experimentally because of averaging time problems. A simple two-layer model is developed which leads to geostrophic drag and heat transfer expressions in fairly good agreement with Wangara data.

The seasonal variations of the dynamic topography relative to 500 decibars in the western equatorial Pacific, using data from 6900 hydrographic stations, are compared with the seasonal fluctuations of sea level observed at eleven islands. It is shown that the changes in the meridional profile of sea level correlate with changes in the strength of the major ocean currents measured by their speed or by geostrophic transports. The strength of the North Equatorial Current and that of the Countercurrent vary synchronously, both being strong in fall and weak in spring. The South Equatorial Current varies exactly out of phase with the two Northern Hemisphere currents, but is apparently in phase with the Undercurrent. Investigation of the particular example indicates that large anomalies seem to behave in the same fashion as the seasonal fluctuations of these currents.

The Ekman-Taylor problem for the planetary boundary layer is solved in the case of a thermal wind which varies linearly with height. The upper boundary condition is a vanishing a- geostrophic wind, while the lower boundary condition is continuity of the stress vector across the interface between the planetary boundary layer and the surface layer. The latter condition is used to determine the magnitude and the direction of the wind at the bottom of the Ekman layer. Theoretical hodographs are compared with observed hodographs based on five years of ohserva- tions from Ship N in the Pacific, giving fair agreement. The divergence, the vorticity, and the vertical velocity are calculated through the Ekman layer with emphasis on differences between the classical barotropic and the baroclinic cases; these differences are significant, especially in the vertical velocities as compared to the standard approximation. An extension of the present study to include thermal stratification is desirable.