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.     

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.     

Principles for capturing the upstream effects of deep sills in low resolution ocean models

Principles for incorporating the upstream effects of deep sills into numerical ocean circulation models using nonlinear analytical hydraulic models are discussed within the context of reduced gravity flow. A method is developed allowing the upstream influence of a numerically unresolvable deep sill or width contraction to be reproduced. The method consists of placing an artificial boundary in the numerical model's overflowing layer at some distance upstream of the actual sill or width contraction of the deep strait. Given the model state at time t, the dependent flow variables are then predicted at the artificial boundary at time t + Δt by using the method of characteristics in combination with quasi-steady hydraulic laws. The calculation requires the use of Riemann invariants and examples are given for a simple nonrotating flow and for rotating channel flow with uniform potential vorticity. The computation is considerably simplified by linearizing the relevant equations in the vicinity of the artificial boundary, resulting in a linear wave reflection problem. The reflection coefficients for the two cases are calculated and these can be used directly to numerically satisfy the boundary condition in a straightforward way.     

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     

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].     

Particle dispersion and mixing induced by breaking internal gravity waves

The purpose of this paper is to analyze diapycnal mixing induced by the breaking of an internal gravity wave — the primary wave — either standing or propagating. To achieve this aim we apply two different methods. The first method consists of a direct estimate of vertical eddy diffusion from particle dispersion while the second method relies upon potential energy budgets [Winters, K.B., Lombard, P.N., Riley, J.J., D’Asaro, E.A., 1995. J. Fluid Mech. 289, 115–128; Winters, K.B., D’Asaro, E.A., 1996. J. Fluid Mech. 317, 179–193]. The primary wave we consider is of small amplitude and is statically stable, a case for which the breaking process involves two-dimensional instabilities. The dynamics of the waves have been previously analyzed by means of two-dimensional direct numerical simulations [Bouruet-Aubertot, P., Sommeria, J., Staquet, C., 1995. J. Fluid Mech. 285, 265–301; Bouruet-Aubertot, P., Sommeria, J., Staquet, C., 1996. Dyn. Atmos. Oceans 29, 41–63; Koudella, C., Staquet, C., 1998. In: Davis, P. (Ed.), Proceedings of the IMA Conference on Mixing and Dispersion on Stably-stratified Flows, Dundee, September 1996. IMA Publication]. High resolution three-dimensional calculations of the same wave are also reported here [Koudella, C., 1999].A local estimate of mixing is first inferred from the time evolution of sets of particles released in the flow during the breaking regime. We show that, after an early evolution dominated by shear effects, a diffusion law is reached and the dispersion coefficient is fairly independent of the initial seeding location of the particles in the flow.The eddy diffusion coefficient, K, is then estimated from the diapycnal diffusive flux. A good agreement with the value inferred from particle dispersion is obtained. This finding is of particular interest regarding the interpretation of in situ estimates of K inferred either from tracer dispersion or from microstructure measurements. Computation of the Cox number, equal to the ratio of eddy diffusivity to molecular diffusivity, shows that the Cox number varies within the interval [9, 262], which corresponds to the range of vertical eddy diffusivity measured in the interior of the ocean. The Cox number is found to depend on the turbulent Froude number squared.We show eventually that mixing results in a weak distortion of the initial density profile and we relate this result to observations made at small scale in the ocean.Comparisons between the analysis of the two-dimensional and high resolution (256³) three-dimensional direct numerical simulations of the primary wave were also conducted. We show that the energetics and the amount of mixing are very close when the primary wave is of small amplitude. This results from the fact that, for a statically stable wave, the dynamics of the initially two-dimensional primary wave remains mostly two-dimensional even after the onset of wavebreaking.     

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     

The characteristics of a laboratory produced turbulent Ekman layer

The characteristics of the atmospheric turbulent Ekman boundary layer have been qualitatively simulated in an annular rotating wind tunnel. Observed velocity spirals found to exist within the wind tunnel resembled qualitatively those found in the atmosphere in that a two-layer structure was evident, consisting of a log-linear portion topped by an outer spiral layer. The magnitude of the friction velocity u _* obtained from the log-linear profile agreed with that measured directly, i.e., that obtained from the relation: u _* = (u′w′)^1/2. Also, the effects of surface roughness on the characteristics of the boundary layer agreed with expected results. In cases where the parametric behaviour predicted by theory departed from the observed behaviour, the probable cause was the inherent size limitations of the wind tunnel. The ability to maintain dynamic similarity is constrained by the limited radius of curvature of the wind tunnel. The vertical distribution of turbulent intensity in the wind tunnel was found to agree qualitatively with an observed atmospheric distribution. Also, a vertical distribution of eddy diffusivity was calculated from tunnel data and found to give qualitatively what one might expect in the atmosphere.     

A suggested refinement for O'Brien's convective boundary layer Eddy exchange coefficient formulation

With observational data collected and interpreted by Crane et al. (1977), the adequacy of the O'Brien polynomial to represent the exchange profile of heat and pollution in a convective boundary layer is examined and a refinement suggested. Also, it is shown that the height of the surface layer, h=0.04 z _i , developed by Blackadar and Tennekes (1968) for a neutrally stratified boundary layer (with z _z =0.25u _*/f) appears to be equally valid for the convective boundary layer where z _i , defined as the top of the mixed layer, is used.     

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.     

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.     

Synoptic Controls on Boundary-Layer Characteristics

We report the characteristics of the three-dimensional, time evolving, atmospheric boundary layer that develops beneath an idealised, dry, baroclinic weather system. The boundary-layer structure is forced by thermal advection associated with the weather system. Large positive heat fluxes behind the cold front drive a vigorous convective boundary layer, whereas moderate negative heat fluxes in the warm sector between the cold and warm fronts generate shallow, stably stratified or neutral boundary layers. The forcing of the boundary-layer structure is quantified by forming an Eulerian mass budget integrated over the depth of the boundary layer. The mass budget indicates that tropospheric air is entrained into the boundary layer both in the vicinity of the high-pressure centre, and behind the cold front. It is then transported horizontally within the boundary layer and converges towards the cyclone’s warm sector, whence it is ventilated out into the troposphere. This cycling of air is likely to be important for the ventilation of pollution out of the boundary layer, and for the transformation of the properties of large-scale air masses.     

The influence of the earth's rotation on planetary boundary-layer turbulence

The planetary boundary layer (PBL) differs from other simple boundary layers in that it forms on the earth's rotating surface. While the effect of the earth's rotation on the mean wind vector of the PBL is well known, the rotational influence on PBL turbulence is not yet established. In the present work, the latter effect is investigated using numerical models that account for the influence of the earth's rotation on the turbulence. It is found that the earth's rotational influence on PBL turbulence is negligible, and therefore does not need to be included in turbulence models used to simulate PBL flows.