The influence of the geostrophic wind advection approximation on a well-mixed layer

Numerical results indicate that advection of momentum in the boundary layer may significantly alter both the structure of the planetary boundary layer and its influence on the overlying free atmosphere. However, due to the nonlinearity of the inertial terms, it is always difficult to obtain the analytical solution of the boundary-layer model that retains the flow acceleration. In order to overcome this difficulty, the geostrophic momentum (hereafter GM) approximation has been introduced into boundary-layer models. By replacing the advected momentum with the geostrophic wind, the effect of the flow acceleration is partially considered and the original nonlinear partial differential equation set is converted to ordinary differential equations, the solutions of which can be obtained easily with standard techniques. However, the model employing GM fails to capture the features of the boundary layer when the spatio-temporal variation of the boundary-layer flow cannot be properly approximated by the geostrophic wind. In the present work, a modified boundary-layer model with the inertial acceleration in a different approximate form is proposed, in which the advecting wind instead of the advected momentum is approximated by the geostrophic wind (hereafter GAM).Comparing the horizontal velocity and boundary-layer pumping obtained from the classical Ekman theory, and the model incorporating (i) GM and (ii) GAM, it is found that the model with GAM describes most facets of the steady well-mixed layer beneath a north-westerly flow with embedded mesoscale perturbations that is considered in the present work. Inspection of the solution of the model with GAM shows that, within the limit of the validation of the model (i.e., the Rossby number R_O is not very large and the drag coefficient C_D is not too small), the horizontal convergence (divergence) is strengthened by the effect of the inertial acceleration in the region of maximum positive (negative) geostrophic vorticity. Consequently, the boundary-layer pumping there is intensified. It is found that the intensification is firstly strengthened and then weakened as R_O or C_D increases.     

The influence of geostrophic shear on the cross-isobar angle of the surface wind

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.     

A baroclinic planetary boundary-layer model,and its application to the Wangara data

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.     

Aircraft estimates of the geostrophic drag coefficient and the rossby similarity functions A and B over the sea

Estimates of the geostrophic drag coefficient and the Rossby similarity functions, A and B obtained from data collected by an instrumented aircraft over the sea are presented. The average value of the geostrophic drag coefficient is 0.027 and is independent of the geostrophic windspeed. The dependence of the similarity functions A and B on boundary-layer parameters is investigated. The function A is found to depend on baroclinicity parameters, while B depends on the parameter u _*/fh (where u _* is the surface friction velocity, f is the Coriolis parameter, and h is the boundary-layer depth). Using the geostrophic drag coefficient found here and the results of surface drag coefficient studies, a relationship between geostrophic windspeed and surface windspeed is obtained which shows good agreement with empirical data.     

Steady state hodographs in a baroclinic boundary layer

If the steady-state geostrophic wind vector varies exponentially with height in the planetary boundary layer, calculated hodographs of the 24-hour mean wind resemble the classic Ekman spiral distorted by thermal effects. For such an assumed distribution, Lettau's (1967) conclusion that the effects of thermal influence on the steady-state boundary-layer winds can be linearly superimposed on the effects of internal friction is justified.The minimum value of the cross-isobar angle of the surface wind for a given magnitude of the thermal wind vector occurs when that vector points about 345° to the right of the surface geostrophic wind vector and the maximum value occurs when it points at an azimuth of about 120° relative to the surface geostrophic wind vector. The range of values of the cross-isobar angle is almost directly proportional to the magnitude of the thermal wind vector.Hodographs resulting from the assumed variation of the geostrophic wind have approximately the same shape as 24-hour mean hodographs at two locations over the Great Plains.This work is part of a thesis submitted to the University of Wisconsin in partial fulfillment of the requirements for the Ph.D. Degree, written under the supervision of Professor H. Lettau, Department of Meteorology.     

Geostrophic drag coefficients

Data on the relationship of the surface wind to the geostrophic wind at Porton Down, Salisbury Plain, are presented for various stability conditions and analysed in the light of the Rossbynumber similarity theory. For near-neutral conditions, the geostrophic drag coefficients for geostrophic wind speeds 5 to 15 m s^-1 are close to those found by other workers but at higher speeds the values are low. Comparisons of geostrophic and radar wind speeds for ⋍900-m height, suggest that undetectably small mean cyclonic curvatures of the trajectories of the air are responsible for this departure. A value of the geostrophic drag coefficient for the open sea at wind speeds around 8 m s^-1 (neutral conditions) is deduced from recent observations of the drag in relation to the surface wind, combined with the ratios of 900-mb radar wind to surface wind obtained from the North Atlantic weather ship data tabulations of Findlater et al. (1966).     

Interaction between the atmospheric and oceanic boundary layers

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, and , 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 C _d, the geostrophic wind coefficient C _f, and the angles , and , of the turbulent flow at the sea-air interface are functions of a dimensionless number, mfG/kg, with S ₁ and S 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.Formerly Visiting Research Associate, Applied Physics Branch, Earth Observations Division, NASA-Manned Spacecraft Center, Houston, Texas.     

One-Equation Turbulence Modelling for Atmospheric and Engineering Applications

A new algebraic turbulent length scale model is developed, based on previous one-equation turbulence modelling experience in atmospheric flow and dispersion calculations. The model is applied to the neutral Ekman layer, as well as to fully-developed pipe and channel flows. For the pipe and channel flows examined the present model results can be considered as nearly equivalent to the results obtained using the standard k– model. For the neutral Ekman layer, the model predicts satisfactorily the near-neutral Cabauw friction velocities and a dependence of the drag coefficient versus Rossby number very close to that derived from published (G. N. Coleman) direct numerical simulations. The model underestimates the Cabauw cross-isobaric angles, but to a less degree than the cross-isobar angle versus Rossby dependence derived from the Coleman simulation. Finally, for the Cabauw data, with a geostrophic wind magnitude of 10 ms^–1, the model predicts an eddy diffusivity distribution in good agreement with semi-empirical distributions used in current operational practice.     

étude expérimentale de la couche limite de montagne

The characteristics of the boundary layer over complex terrain (Lannemezan - lat.: 43.7° N and, long.: 0.7 ° E) are analyzed for various scales, using measurements obtained during the COCAGNE Experiment. In this first part, the dynamic characteristics of the flow are studied with respect to atmospheric stability and the relief at small (~20 km) and medium scales (~100 km). These relief scales depend on the topographical profile of the Lannemezan Plateau along the dominant axis of the wind (E-W) and the Pyrénées Mountains located at the south of the experimental site. The terrain heterogeneities have a standard deviation of ~48 m and a wavelength of ~2 km.The averaged vertical profiles of wind speed and direction over the heterogeneous terrain are analyzed. The decrease of wind speed within the boundary layer is greater than over flat terrain (WANGARA Experiment). However, a comparison between ETTEX (complex terrain) and COCAGNE vertical wind speed profiles shows good agreement during unstable conditions. In contrast, during neutral conditions a more rapid increase with normalized height is found with COCAGNE than with ETTEX and WANGARA data. The vertical profiles of wind direction reveal an influence of the Pyrénées Mountains on the wind flow. The wind rotation in the BL is determined by the geostrophic wind direction-Pyrénées axis angle (negative deviation) as the geostrophic wind is connected with the Mountain axis.When the geostrophic wind does not interact with the Pyrénées axis, the mean and turbulent wind flow characteristics (drag coefficient C _D, friction velocity u _*) depend on the topography of the plateau. When the wind speed is strong (>6 m s ^-1), an internal boundary layer is generated from the leading edge of the Plateau.     

Generation and atmospheric heat exchange of coastal polynyas in the Weddell Sea

The forcing mechanisms for Antarctic coastal polynyas and the thermodynamic effects of existing polynyas are studied by means of an air-sea-ice interaction experiment in the Weddell Sea in October and November 1986.Coastal polynyas develop in close relationship to the ice motion and form most rapidly with offshore ice motion. Narrow polynyas occur frequently on the lee side of headlands and with strong curvature of the coastline. From the momentum balance of drifting sea ice, a forcing diagram is constructed, which relates ice motion to the surface-layer wind vector v _z and to the geostrophic ocean current vector c _g . In agreement with the data, wind forcing dominates when the wind speed at a height of 3 m exceeds the geostrophic current velocity by a factor of at least 33. This condition within the ocean regime of the Antarctic coastal current usually is fulfilled for wind speeds above 5 m/s at a height of 3 m.Based on a nonlinear parameter estimation technique, optimum parameters for free ice drift are calculated. Including a drift dependent geostrophic current in the ice/water drag yields a maximum of explained variance (91%) of ice velocity.The turbulent heat exchange between sea ice and polynya surfaces is derived from surface-layer wind and temperature data, from temperature changes of the air mass along its trajectory and from an application of the resistance laws for the atmospheric PBL. The turbulent heat flux averaged over all randomly distributed observations in coastal polynyas is 143 W/m². This value is significantly different over pack ice and shelf ice surfaces, where downward fluxes prevail. The large variances of turbulent fluxes can be explained by variable wind speeds and air temperatures. The heat fluxes are also affected by cloud feedback processes and vary in time due to the formation of new ice at the polynya surface.Maximum turbulent fluxes of more than 400 W/m² result from strong winds and low air temperatures. The heat exchange is similarly intense in a narrow zone close to the ice front, when under weak wind conditions, a local circulation develops and cold air associated with strong surface inversions over the shelf ice is heated above the open water.     

Sea surface winds derived from cloud motion vectors over the tropical Atlantic

The relationship between satellite-derived low-level cloud motion, surface wind and geostrophic wind vectors is examined using GATE data. In the trades, surface wind speeds can be derived from cloud motion vectors by the linear relation: V = 0.62 V _s + 1.9 m s^–1 with a mean scatter of ±1.3 m s^–1. The correlation coefficient between surface and satellite wind speed is 0.25. Considering baroclinicity, i.e., the influence of the thermal wind, the correlation coefficient does not increase, because of the uncertainty of the thermal wind vectors. The ratios of surface to geostrophic wind speed and surface to satellite wind speed are 0.7 and 0.8, respectively, with a statistical uncertainty of ±0.3. Calculations of the ratio of surface to geostrophic wind speed on the basis of the resistance law yield V/V _g = 0.8 ± 0.2, in agreement with experimental results. The mean angle difference between the surface and the satellite wind vectors amounts to - 18 °, taking into account baroclinicity. This value is in good agreement with the mean ageostrophic angle - 25 °.     

Estimation of the Exchange Coefficient of Heat During Low Wind Convective Conditions

For the first time, the exchange coefficient of heat C_H has been estimated from eddy correlation of velocity and virtual temperature fluctuations using sonic anemometer measurements made at low wind speeds over the monsoon land atJodhpur (26°18' N, 73°04' E), a semi arid station. It shows strong dependence on wind speed, increasing rapidly with decreasing wind speed, and scales according to a power law C_H = 0.025U₁₀ ^-0.7 (where U₁₀ is the mean wind speed at 10-m height). A similar but more rapid increase in the drag coefficient C_Dhas already been reported in an earlier study. Low winds (<4 m s^-1) are associated with both near neutral and strong unstable situations. It is noted that C_H increases with increasing instability. The present observations best describe a low wind convective regime as revealed in the scaling behaviour of drag, sensible heat flux and the non-dimensional temperature gradient. Neutral drag and heat cofficients,corrected using Monin–Obukhov (M–O) theory, show a more uniform behaviour at low wind speeds in convective conditions, when compared with the observed coefficients discussed in a coming paper.At low wind convective conditions, M-O theory is unable to capture the observed linear dependence of drag on wind speed, unlike during forced convections. The non-dimensional shear inferred from the present data shows noticeable deviations from Businger's formulation, a forced convection similarity. Heat flux is insensitive to drag associated with weak winds superposed on true free convection. With heat flux as the primary variable, definition of new velocity scales leads to a new drag parameterization scheme at low wind speeds during convective conditionsdiscussed in a coming paper.     

Equilibrium theory of the planetary boundary layer with an inversion lid

The maintenance of an elevated inversion in steady flow above a cold, rotating surface is shown to be possible for a certain range of the buoyancy number bfV _g, where b is the buoyant acceleration appropriate to the density deficiency of the fluid above the inversion, f is Coriolis parameter and V _gis geostrophic velocity (so that fV _gis also horizontal pressure gradient in kinematic units). The height of the inversion lid is determined by a balance of surface stress and buoyancy, in a way which may be deduced from laboratory experiments. With the aid of such empirical evidence a theory is constructed for the layer below the inversion lid. The cross-isobar angle of ground-level stress is found to increase with the buoyancy number, to a limiting value of 90, by which time the inversion descends to the ground. Under typical conditions, a temperature difference of order 10C is necessary to eliminate the possibility of an equilibrium, elevated inversion lid and reduce ground level wind stress to a vanishingly small value.Woods Hole Oceanographic Institution Contribution #3011On leave from the University of Waterloo, Ontario     

On the drag coefficient over shallow water

It is shown that if the wind-wave spectrum in shallow water is approximately independent of wind speed due to the combined effects of white-capping and bottom friction, then the wave-induced drag coefficient has a maximum value when the wind speed is twice the maximum wave-speed; as the wind speed increases further, the drag coefficient slowly decreases. This result is consistent with the observations of Hicks et al. (1974).     

Vertical profiles of the structure parameter of temperature in the stable,nocturnal boundary layer

Vertical profiles of the structure parameter of temperature C _infT ^sup2 in the stable, nocturnal boundary layer (NBL) have been obtained with the analytic models described by Nieuwstadt (1984, 1985) and Sorbjan (1986) and the numerical model of Duynkerke and Driedonks (1987). These theoretical profiles are compared with observed profiles from the meteorological mast at Cabauw, The Netherlands. From the observations, it is found that C _infT ^sup2 is large in the surface layer and small at the top of the NBL. Observations during nights with moderate geostrophic winds or during the first few hours of nights with a high geostrophic wind show a continuous decrease of C _infT ^sup2 from the surface layer to the top of the NBL. Observations made later on nights with a high geostrophic wind show the development of a maximum of C _infT ^sup2 at about three quarters of the NBL. From the comparison with the models, we conclude that the observed profiles are most satisfactorily described by the model of Duynkerke and Driedonks.     

An ice breeze mechanism for boundary-layer jets

The existence of a low-level (z=~1000 m) jet adjacent to a sea-ice boundary is investigated with a two-dimensional numerical model. A thermally-direct ice breeze circulation is induced by specifying an ice-sea surface temperature gradient, with the mean geostrophic wind parallel to the ice edge. Pressure changes associated with over-water mixed-layer development create an increase in geostrophic velocity that accounts for most of the increase in wind speed. A change in initial geostrophic wind direction has significant effects on location and intensity of the low-level jet; geostrophic winds parallel to the ice edge result in stronger jets than occur with cross-ice geostrophic winds. An inertial oscillation simulated by the model in 1-D makes a negligible contribution to the low-level jet.     

Models for estimating offshore winds from onshore meteorological measurements

To understand and estimate wind speed differences across the coastal zone, two models, one theoretical and another semi-empirical, have been developed and verified by available data sets. Assuming that: (1) mean horizontal motion exists across the coastal zone; and (2) the geostrophic wind does not change appreciably at the top of the planetary boundary layer (PBL), the equation of motion in the direction of the wind can be reduced so that 341-01, where U, H, and C _D are wind speed, height of PBL, and drag coefficient over the sea and land, respectively. For practice, C _D SEA has been modified from a formula with U _LAND as the only input. H _SEA may be estimated routinely from known H _{D LAND LAND} and the temperature difference between land and sea, which can be provided by such means as remote sensing from meteorological satellites. For a given coast, Cmay be estimated also. This formula is recommended for weather forecasters. The semiempirical formula is based mainly on the power law wind distribution with height in the PBL. The formula states that 341-02. Simultaneous offshore and onshore wind measurements made at stations ranging from Somalia, near the equator, to the Gulf of Alaska indicated that values of a and b are 2.98 and 0.34 with a correlation coefficient of -0.95. For oceanographic applications, a simplified equation, i.e., 341-03, is also proposed.     

Dependence of geostrophic drag on intensity of convection,baroclinicity, and acceleration

This paper examines the practical importance of stability, baroclinicity, and acceleration in the bulk ABL similarity formulations, in light of the random errors inherent in field measurements. This is done by propagating the measurement uncertainties through a theoretical model for the bulk ABL similarity functionsA ₀ andB ₀, under a range of assumed (but always unstable) conditions. It is shown that random measurement errors and acceleration effects may overwhelm most effects of baroclinicity and stability, once conditions are at least slightly unstable. Because of this, it is hard to discern a clear functional dependence ofA ₀ andB ₀ on degree of instability. Thus, for a given value ofh _i/z₀, whereh _i is the inversion height andz ₀ is the surface roughness, the geostrophic drag coefficient, which depends onA ₀ andB ₀, and weakly onh _i/z₀, may also be taken to be nearly independent of degree of instability.     

On the extension of the wind profile over homogeneous terrain beyond the surface boundary layer

Analysis of profiles of meteorological measurements from a 160 m high mast at the National Test Site for wind turbines at Høvsøre (Denmark) and at a 250 m high TV tower at Hamburg (Germany) shows that the wind profile based on surface-layer theory and Monin-Obukhov scaling is valid up to a height of 50–80 m. At higher levels deviations from the measurements progressively occur. For applied use an extension to the wind profile in the surface layer is formulated for the entire boundary layer, with emphasis on the lowest 200–300 m and considering only wind speeds above 3 m s^?1 at 10 m height. The friction velocity is taken to decrease linearly through the boundary layer. The wind profile length scale is composed of three component length scales. In the surface layer the first length scale is taken to increase linearly with height with a stability correction following Monin-Obukhov similarity. Above the surface layer the second length scale (L _MBL ) becomes independent of height but not of stability, and at the top of the boundary layer the third length scale is assumed to be negligible. A simple model for the combined length scale that controls the wind profile and its stability dependence is formulated by inverse summation. Based on these assumptions the wind profile for the entire boundary layer is derived. A parameterization of L _MBL is formulated using the geostrophic drag law, which relates friction velocity and geostrophic wind. The empirical parameterization of the resistance law functions A and B in the geostrophic drag law is uncertain, making it impractical. Therefore an expression for the length scale, L _MBL , for applied use is suggested, based on measurements from the two sites.     

An Analytical Study of the Diurnal Variations of Wind in a Semi-geostrophic Ekman Boundary Layer Model

A time-dependent semi-geostrophic Ekman boundary-layer model based on the geostrophic momentum approximation is used to study the diurnal wind variation in the planetary boundary layer (PBL) and the evolution of the low-level nocturnal jet (LLJ). The coefficient of eddy viscosity varies periodically with time, varies linearly with height in the surface layer and is constant above the surface layer. The influence of horizontal advection of momentum on the diurnal wind variation in the PBL, the development of inertial oscillations (IOs) and the formation of the LLJ are examined.In comparison with the Ekman solutions, the diurnal wind variation in semi-geostrophic Ekman boundary-layer dynamics has the following features: (1) the phase angle of the diurnal wind wave shifts with height, the rate of shifting is increased in anticyclonic regions and decreased in cyclonic regions, (2) the time of occurrence of the low-level maximum wind speed is later in anticyclonic regions and earlier in cyclonic regions, (3) the height of occurrence of the maximum wind speed is higher in the anticyclonic and lower in cyclonic regions, (4) the wind speed maximum and the amplitude of the diurnal wind variation are larger in anticyclonic and smaller in cyclonic regions, (5) the period of IOs is larger in anticyclonic regions and smaller in cyclonic regions, (6) anticyclonic vorticity is conducive to the generation of LLJ in the PBL. These features are interpreted by means of the physical properties of semi-geostrophic Ekman boundary-layer dynamics and inertial oscillation dynamics.