A note on circulation in differentially heated rotating stratified fluid and other elliptic problems

This note relates to the paper Circulation and boundary layers in differentially heated rotating stratified fluid by Whitehead and Pedlosky [Whitehead, J.A., Pedlosky, J., 2000. Circulation and boundary layers in differentially heated rotating stratified fluid. Dyn. Atmos. Ocean. 31, 1–21]. Here, we describe an alternative method of solution for the theoretical model developed therein, and provide a comparison with the original method used in the paper.     

Circulation and boundary layers in differentially heated rotating stratified fluid

Recent laboratory experiments with rotating stratified water in a cylinder have revealed many of the predictions of linearized, analytic theory. Earlier measurements of the velocity field generated in a cylinder by top heating compared well with theory. Large stratification clearly suppressed Ekman pumping so that the interior velocity field (primarily azimuthal) responded by satisfying no-slip top and bottom boundary conditions without the need for Ekman layers. This interior flow also occupied a boundary layer of greater thickness than the Ekman layer under some conditions. Theory and experiments have now been conducted for sidewall heating. As before, experiment and theory agree well over some parameter ranges. But for some parameters, the flow is unstable. The exact nature of the instability remains poorly understood. The size of one combination of both vertical and horizontal boundary layers is governed by the Rossby radius of deformation multiplied by the square root of the Prandtl number. Sidewall boundary layers and their scales will be reviewed with the present results in mind.     

On the interactions between planetary geostrophy and mesoscale eddies

Multiscale asymptotics are used to derive three systems of equations connecting the planetary geostrophic (PG) equations for gyre-scale flow to a quasigeostrophic (QG) equation set for mesoscale eddies. Pedlosky (1984), following similar analysis, found eddy buoyancy fluxes to have only a small effect on the large-scale flow; however, numerical simulations disagree. While the impact of eddies is relatively small in most regions, in keeping with Pedlosky’s result, eddies have a significant effect on the mean flow in the vicinity of strong, narrow currents.First, the multiple-scales analysis of Pedlosky is reviewed and amplified. Novel results of this analysis include new multiple-scales models connecting large-scale PG equations to sets of QG eddy equations. However, only introducing anisotropic scaling of the large-scale coordinates allows us to derive a model with strong two-way coupling between the QG eddies and the PG mean flow. This finding reconciles the analysis with simulations, viz. that strong two-way coupling is observed in the vicinity of anisotropic features of the mean flow like boundary currents and jets. The relevant coupling terms are shown to be eddy buoyancy fluxes. Using the Gent-McWilliams parameterization to approximate these fluxes allows solution of the PG equations with closed tracer fluxes in a closed domain, which is not possible without mesoscale eddy (or other small-scale) effects. The boundary layer width is comparable to an eddy mixing length when the typical eddy velocity is taken to be the long Rossby wave phase speed, which is the same result found by Fox-Kemper and Ferrari (2009) in a reduced gravity layer.     

The Ekman boundary-layer stress due to flow over a regular array of hills

The flow of water over a regular array of hills in a rotating laboratory experiment is studied as an analogue of planetary boundary layers. Gaussian-shaped hills of heighth = 1 cm andh = 1/3 cm covered the floor of a 228 cm diam rotating tank. The characteristic depth of the Ekman layer varied fromD = 0.1 cm toD = 0.3 cm and the Reynolds number for the Ekman layer varied up toRe = 150. The integrated boundary-layer stress as a function ofRe andD/h was determined by the rate of spin-up of the water after a small increase of the rotation rate.Selected functional relations were fitted to the empirical data by the method of least squares for the laminar and turbulent regimes of flow. These results indicate maximum non-linear effects of the hills and a minimum transition Reynolds number nearD/h = 0.2. The patterns of flow as indicated by dye tracers are briefly discussed. Peculiar circular bands of dye appear to be caused by the selective ejection of dye from the boundary layer when the circular flow is at a particular angle of incidence with respect to the triangular pattern of hills.The research reported herein was supported in part by the National Science Foundation under Grant NSF GA 4388.     

Quasi-geostrophic stratified flow over isolated finite amplitude topography

The quasi-geostrophic response of a stratified stream incident upon isolated finite amplitude topography on a f-plane is examined in the limit of a Boussinesq, incompressible, inviscid fluid. Compact solutions are derived subject to the following stipulations: uniform upstream velocity and stratification, a circular obstacle and an entirely isentropic/isopycnic lower surface.It is shown that for a semi-infinite flow domain the criterion for Taylor cap formation (i.e., a region of closed streamlines) is . However, for the isentropic lower boundary condition the solutions exist (i.e., have physical validity) only if R₀F⁻¹ < 0.5. (Here R₀ and F refer to the Rossby and Froude numbers defined respectively in terms of the mountain half-width and height.) Also considered are the modifications both to the flow response and to the foregoing existence criterion that are induced by the introduction of an upstream profile comprising two layers of uniform but different stratification. In addition, the relationship of the derived solutions to the results obtained in previous studies is explored, and in particular an outline is given of the impact of adopting the ‘traditional’ simplified lower boundary condition.     

Turbulence Structure of Stable Boundary Layers with a Near-Linear Temperature Profile

By using a thermally stratified wind tunnel, we have successfullysimulated stably stratified boundary layers (SBL), in which the meantemperature increases upward almost linearly. We have investigated the flow structure and the effects of near-linearstable stratification on the transfer of momentum and heat. Thevertical profiles of turbulence quantities exhibit different behaviour in two distinct stability regimes of the SBLflows with weak and strong stability. For weak stability cases, theturbulent transfer of momentum and heat is basically similar to that for neutral turbulent boundary layers, although it is weakenedwith increasing stability. For strong stability cases, on the other hand,the time-mean transfer is almost zero over the whole boundary-layer depth.However, the instantaneous turbulent transfer frequently occurs in bothgradient and counter-gradient directions in the lower part of the boundary layer. This is due to the Kelvin–Helmholtz (K–H) shear instability and therolling up and breaking of K–H waves. Moreover, the internal gravity wavesare observed in the middle and upper parts of all stable boundary layers.     

一个耦合气候模式中的副极地涡旋指数与北大西洋经向翻转流

The subpolar gyre index (SPG), derived from the analysis of sea surface height (SSH), is proposed to be a potential indicator for the North Atlantic Meridional Overturning Circulation (AMOC) based on observation as well as the Ocean General Circulation Model (OGCM). We investigated the correspondence between the SPG and the AMOC in a coupled climate model. Our results confirm that the SPG can be used as an early indicator for the AMOC in the subtropical North Atlantic. Changes in the SPG are closely related to variations in the air-sea heat exchange in the Labrador Sea, and variations in deep water formation and southward dense water transport with the deep western boundary current (DWBC) in the North Atlantic. Citation: Gao, Y. Q., and L. Yu, 2008: Subpolar gyre index and the North Atlantic meridional overturning circulation in a coupled climate model, Atmos. Oceanic Sci. Lett., 1, 29-32     

The lower symmetric regime and its transition to waves in rotating annulus convection

Measurements of the temperature and zonal velocity fields which develop in a rotating annulus of fluid with an upper surface, differentially heated from the inner to outer cylinder, are described for the lower symmetric regime (small radial temperature differences). The temperature field is essentially conductive for moderate to large rotation rates, Ω (>1.0 sec⁻¹). The zonal velocity field is poorly approximated by the thermal wind equation.Measurements of the transition to waves from the lower symmetric regime at very large rotation rates are presented for positive and negative radial temperature differences. They suggest that the centrifugal buoyancy force and the free surface curvature may be important factors for the lower symmetric-wave transition at large Ω. By varying the stratification of the fluid over a range of 10³ independently of the radial temperature difference, Δr_wT, it is conclusively shown that several theories are correct in predicting that the lower symmetric transition is independent of the stratification at small Δr_wT > 0 for large enough Ω.     

Application of a Large-Eddy Simulation Database to Optimisation of First-Order Closures for Neutral and Stably Stratified Boundary Layers

Large-eddy simulation (LES) is a well-established numerical technique, resolving the most energetic turbulent fluctuations in the planetary boundary layer. By averaging these fluctuations, high-quality profiles of mean quantities and turbulence statistics can be obtained in experiments with well-defined initial and boundary conditions. Hence, LES data can be beneficial for assessment and optimisation of turbulence closure schemes. A database of 80 LES runs (DATABASE64) for neutral and stably stratified planetary boundary layers (PBLs) is applied in this study to optimize first-order turbulence closure (FOC). Approximations for the mixing length scale and stability correction functions have been made to minimise a relative root-mean-square error over the entire database. New stability functions have correct asymptotes describing regimes of strong and weak mixing found in theoretical approaches, atmospheric observations and LES. The correct asymptotes exclude the need for a critical Richardson number in the FOC formulation. Further, we analysed the FOC quality as functions of the integral PBL stability and the vertical model resolution. We show that the FOC is never perfect because the turbulence in the upper half of the PBL is not generated by the local vertical gradients. Accordingly, the parameterised and LES-based fluxes decorrelate in the upper PBL. With this imperfection in mind, we show that there is no systematic quality deterioration of the FOC in the strongly stable PBL provided that the vertical model resolution is better than 10 levels within the PBL. In agreement with previous studies, we found that the quality improves slowly with the vertical resolution refinement, though it is generally wise not to overstretch the mesh in the lowest 500 m of the atmosphere where the observed, simulated and theoretically predicted stably stratified PBL is mostly located. The submission to a special issue of the “Boundary-Layer Meteorology” devoted to the NATO advanced research workshop “Atmospheric Boundary Layers: Modelling and Applications for Environmental Security”.     

Cyclonic/anticyclonic gyre asymmetries: laboratory and intermediate-model experiments

Laboratory models of rapidly rotating geophysical flows often show significant asymmetries with respect to the sign of the gyre forcing. In this paper we focus on the instability of separated boundary currents and the resulting transition to time-dependent motion in a slightly sliced cylinder driven by a differentially rotating lid. This transition occurs more readily for cyclonic (co-rotating) gyre forcing, when compared with that observed for anticyclonic forcing, even though the system Rossby number is very small. Quasi-geostrophic models are invariant to changes in the sign of the forcing, so a more accurate theoretical framework must be used to capture the observed asymmetries. An intermediate model, which includes a second-order nonlinear Ekman suction relation, is proposed and integrated numerically. The results are in significantly better agreement with the laboratory observations, and simple diagnostics illustrate which of the higher-order physical effects are responsible for the enhanced instability of cyclonically forced gyres.     

TURBULENCE STRUCTURE IN A STRATIFIED BOUNDARY LAYER UNDER STABLE CONDITIONS

Turbulence structure in stably stratified boundary layers isexperimentally investigated by using a thermally stratified wind tunnel. Astably stratified flow is created by heating the wind tunnel airflow to atemperature of about 50 °C and by cooling the test-section floor to asurface temperature of about 3 °C. In order to study the effect ofbuoyancy on turbulent boundary layers for a wide range of stability, thevelocity and temperature fluctuations are measured simultaneously at adownwind position of 23.5 m from the tunnel entrance, where the boundarylayer is fully developed. The Reynolds number, Re, ranges from 3.14× 104 to 1.27 × 105, and the bulk Richardson number, Ri,ranges from 0 to 1.33. Stable stratification rapidly suppresses thefluctuations of streamwise velocity and temperature as well as the verticalvelocity fluctuation. Momentum and heat fluxes are also significantlydecreased with increasing stability and become nearly zero in the lowest partof the boundary layer with strong stability. The vertical profiles ofturbulence quantities exhibit different behaviour in three distinct stabilityregimes, the neutral flows, the stratified flows with weak stability(Ri = 0.12, 0.20) and those with strong stability (Ri= 0.39,0.47, 1.33). Of these, the two regimes of stratified flows clearly showdifferent vertical profiles of the local gradient Richardson number Ri,separated by the critical Richardson number Ri cr of about 0.25. Moreover,turbulence quantities in stable conditions are well correlated with Ri.     

Intermittent Bursting of Turbulence in a Stable Boundary Layer with Low-level Jet

The atmospheric stable boundary layer (SBL) with a low-level jet is simulated experimentally using a thermally stratified wind tunnel. The turbulence structure and flow characteristics are investigated by simultaneous measurements of velocity and temperature fluctuations and by flow visualization. Attention is focused on the effect of strong wind shear due to a low-level jet on stratified boundary layers with strong stability. Occasional bursting of turbulence in the lower portion of the boundary layer can be found in the SBL with strong stability. This bursting originates aloft away from the surface and transports fluid with relatively low velocity and temperature upward and fluid with relatively high velocity and temperature downward. Furthermore, the relationship between the occurrence of turbulence bursting and the local gradient Richardson number (Ri) is investigated. The Ri becomes larger than the critical Ri, Ri_cr = 0.25, in quiescent periods. On the other hand, the Ri number becomes smaller than Ri_cr during bursting events.     

Experiments with density currents on a sloping bottom in a rotating fluid

The behaviour of a density current on a sloping bottom in a rotating system is investigated by laboratory experiments. The main result is that the dense bottom outflow induces cyclonic vortices in the upper fluid layer, which are formed periodically and move to the west parallel to coast. Two regimes of vortex formation have been identified. For strong density currents and weak rotation, vortices are formed by stretching of the upper layer near the source as found also in the experiments by Lane-Serff and Baines (1998) [Lane-Serff, G.F., Baines, P.G., 1998. Eddy formation by dense flows on slopes in a rotating fluid. J. Fluid Mech. 363, 229–253]. For weak density currents and strong rotation vortices are due to instability of the bottom plume itself as found in the numerical simulations of Jiang and Garwood (1996) [Jiang, L., Garwood, W. Jr., 1996. Three-dimensional simulations of overflows on continental slopes. J. Phys. Oceanogr. 26, 1224–1233].     

Turbulence in stably stratified fluids: A review of laboratory experiments

This is a review of laboratory studies of mixing in stably stratified fluids away from the direct intluence of boundary layers, and was written to introduce the session on laboratory experiments at the IUCRM Colloquium.Internal waves (Section 2) can lead to turbulence by creating regions of unstable density gradients through their mutual interactions, by individually breaking by forming rotors, and by reducing the local Richardson number until Kelvin-Helmholtz instability results. They may be important in radiating energy from turbulent or spreading regions. Critical-layer absorption of internal waves is not found to be a direct cause of turbulence in experiments with 15 Richardson number 5, although the modification of the vertical density profile may be significant.Turbulent-laminar interfaces without mean shear (Section 3) and with mean shear (Section 4) are described. The source of turbulence in these experiments is partly external, either generated by an oscillating grid or by a (relatively) moving boundary. The development of turbulence generated entirelywithin a stratified layer by Kelvin-Helmholtz instability, is described in detail in Section 5, and the results are compared with measurements in the ocean and in the atmosphere.     

On evaluation of the oceanic surface drift current speed and direction

The surface drift current speed and direction are defined using the resistance laws for turbulent Ekman boundary layers. Stratification conditions that coincide in the atmospheric and oceanic boundary layers yield approximate formulas whereby the drift current and geostrophic wind directions coincide and the geostrophic wind factor k is equal to the square root of the air and water density ratio. The theoretical estimates of k are compared with available experimental data.     

Large-Eddy Simulations on the Effects of Surface Geometry of Building Arrays on Turbulent Organized Structures

Turbulent organized structures (TOS) above building arrays were investigated using a large-eddy simulation (LES) model for a city (LES-CITY). Square and staggered building arrays produced contrasting behaviour in terms of turbulence that roughly corresponded to the conventional classification of ‘D-type’ and ‘K-type’ roughness, respectively: (1) The drag coefficients (referred to the building height) for staggered arrays were sensitive to building area density, but those for square arrays were not. (2) The relative contributions of ejections to sweeps (S₂/S₄) at the building height for square arrays were sensitive to building area density and nearly equalled or exceeded 1.0 (ejection dominant), but those for staggered arrays were insensitive to building area density and were mostly below 1.0 (sweep dominant). (3) Streaky patterns of longitudinal low speed regions (i.e., low speed streaks) existed in all flows regardless of array type. Height variations of the buildings in the square array drastically increased the drag coefficient and modified the turbulent flow structures. The mechanism of D-type and K-type urban-like roughness flows and the difference from vegetation flows are discussed. Although urban-like roughness flows exhibited mixed properties of mixing layers and flat-wall boundary layers as far as S₂/S₄ was concerned, the turbulent organized structures of urban-like roughness flows resembled those of flat-wall boundary layers.     

The influence of the coast on the dynamics of upwelling fronts: Part I. Laboratory experiments

We describe laboratory experiments on the instability and later evolution of a front in a two-layer rotating fluid. In particular, we focus on the influence of a nearby boundary on instability growth and eddy formation. The front is produced through the adjustment of a buoyant fluid initially confined within a bottomless cylinder. Typically a front in quasi-cyclostrophic balance establishes after two rotation periods, after which it becomes unstable. Measurements of the velocity and vorticity fields at the surface are made which provide detailed information on the evolution of the front as the instability grows to finite amplitude. We focus on the time evolution of the vorticity and distinguish between the cyclonic and anticyclonic components. The spatial averages of the cyclonic and anticyclonic vorticity first grow exponentially. This growth saturates when eddies form and are advected across the front. The growth rate depends upon two nondimensional parameters: the width W of the upwelling region in units of the internal radius of deformation and the depth ratio δ between the two layers. Measurements of the growth rates for the average of the cyclonic and anticyclonic vorticity are compared to the values inferred from a simplified model for baroclinic instability. A good agreement is obtained when the front develops far from the boundary (i.e. W1). However, the agreement is only qualitative when the front is near the boundary (i.e. W1). We find that, as W decreases, the growth of cyclonic eddies consisting of dense—“coastal”—water is enhanced compared to that of anticyclonic vorticity consisting of buoyant—“off-shore”—water. This crucial effect of the boundary with respect to the instability of the front has significant impact on exchanges across the front.     

Determination of the Atmospheric Boundary Layer Height from Radiosonde and Lidar Backscatter

The height of the atmospheric boundary layer is derived with the help of two different measuring systems and methods. From radiosoundings the boundary layer height is determined by the parcel method and by temperature and humidity gradients. From lidar backscatter measurements a combination of the averaging variance method and the high-resolution gradient method is used to determine boundary layer heights. In this paper lidar-derived boundary layer heights on a 10 min basis are presented. Datasets from four experiments – two over land and two over the sea – are used to compare boundary layer heights from both methods. Only the daytime boundary layer is investigated because the height of the nighttime stable boundary layer is below the range of the lidar. In many situations the boundary layer heights from both systems coincide within ±200 m. This corresponds to the standard deviation of lidar-derived 10-min values within a 1-h interval and is due to the time and space variability of the boundary layer height. Deviations appear for certain situations and depend on which radiosonde method is applied. The parcel method fails over land surfaces in the afternoon when the boundary layer stabilizes and over the ocean when the boundary layer is slightly stable. An automatic radiosonde gradient method sometimes fails when multiple layers are present, e.g. a residual layer above the growing convective boundary layer. The lidar method has the advantage of continuous tracing and thus avoids confusion with elevated layers. On the other hand, it mostly fails in situations with boundary layer clouds     

Elevated stratified layers observed with sodar during VTMX

Summary A combination of low frequency sodar, radar wind profiler and in-situ balloon-borne measurements of temperature and water vapor have been used to investigate the structure of elevated stratified layers within the transition layer above the nocturnal boundary layer during the Vertical Transport and Mixing Field Campaign in Salt Lake City Utah, during October, 2000. Elevated layers determined from sodar and radar vertical time sections were penetrated with a balloon-born instrument package to determine the fine scale temperature and moisture structure of the layers. As expected a potential temperature increase was found in the upper half of the layers; however the magnitude was considerably smaller than found above the daytime well-mixed layer and the vertical distance of the increase was quite variable. Mixing ratio, in the mean was found to have a relative maximum in the lower portion of the layers. It was found that the potential temperature within the layers decreased with time relative to background values, regardless of whether the layer descended or ascended.     

Experiments on steady buoyancy transfer through turbulent fluid layers separated by density interfaces

A laboratory experiment was performed to investigate mixing across a density interface which separates two turbulent fluid layers and coexists with a stabilizing buoyancy flux. It was found that the buoyancy flux (q₀) across the interface and through the turbulent layers (of depth D) becomes steady and constant in magnitude in the vertical direction, only when , where u is the horizontal r.m.s. velocity at the base of the mixed layers. The results suggest that mixing across the density interface is controlled by a dynamically important buoyancy gradient induced in the turbulent layers and that parameters such as the bulk Richardson number, , where Δb is the interfacial buoyancy jump, are of secondary importance. Measurements are used to infer the mixing mechanism at the interface, the mixing efficiency of stratified fluids and the entrainment law. Some geophysical applications of the results are also discussed.