A fluid experiment of large-scale topography effect on baroclinic wave flows

The effects of topography on baroclinic wave flows are studied experimentally in a thermally driven rotating annulus of fluid.Fourier analysis and complex principal component (CPC) analysis of the experimental data show that, due to topographic forcing, the flow is bimodal rather than a single mode. Under suitable imposed experimental parameters, near thermal Rossby number ROT = 0.1 and Taylor number Ta = 2.2 × 107, the large-scale topography produces low-frequency oscillation in the flow and rather long-lived flow pattern resembling blocking in the atmospheric cir-culation. The ‘blocking’ phenomenon is caused by the resonance of travelling waves and the quasi-stationary waves forced by topography.The large-scale topography transforms wavenumber-homogeneous flows into wavenumber-dispersed flows, and the dispersed flows possess lower wavenumbers.     

The effect of idealized water waves on the turbulence structure and kinetic energy budgets in the overlying airflow

The influence of an idealized moving wavy surface on the overlying airflow is investigated using direct numerical simulations (DNS). In the present simulations, the bulk Reynolds number is Re = 8000 (; where U₀ is the forcing velocity of the flow, h the height of the domain and v the kinematic viscosity) and the phase speed of the imposed waves relative to the friction velocity, i.e., the wave age varies from very slow to fast waves. The wave signal is clearly present in the airflow up to at least 0.15λ (where λ is the wave length) and is present up to higher levels for faster waves. In the kinetic energy budgets, pressure transport is mainly of importance for slow waves. For fast waves, viscous transport and turbulent transport dominate near the surface. Kinetic energy budgets for the wave and turbulent perturbations show a non-negligible transport of turbulent kinetic energy directed from turbulence to the wave perturbation in the airflow. The wave-turbulent energy transport depends on the size, tilt, and phase of the wave-induced part of the turbulent Reynolds stresses.According to the DNS data, slow waves are more efficient in generating isotropic turbulence than fast waves.Despite the differences in wave-shape as well as in Reynolds number between the idealized direct numerical simulations and the atmosphere, there are intriguing similarities in the turbulence structure. Important information about the turbulence above waves in the atmosphere can be obtained from DNS—the data must, however, be interpreted with care.     

Topographic generation of intermediate-scale motions by wind-driven currents — A model for the circulation in the vicinity of the Indian-Antarctic ridge

When a broad ocean current encounters a large-scale topographic feature, standing Rossby wave patterns can be generated. Short Rossby waves with a scale L_i = √ Q/β (Q is the speed of the approaching flow; β is the meridional gradient of f) are generated east of the topography. If the zonal scale of the topography, L, is planetary, long standing Rossby waves can be generated west of the topography, when the current has a meridional component. The long waves focus the disturbance zonally and produce alternating regions of intensified or reduced zonal flow. The meridional scale that characterizes these zonal bands is the intermediate scales, L = L_i^2/3L^1/3. When the meridional topographic scale is comparable to L, the amplitude of the long-wave disturbance is dominant. Using multiple-scale methods to exploit the scale gap between the planetary, intermediate and Rossby wave scales, the topographically induced pressure and velocity fields due to a zonal ridge are obtained. When the planetary-scale flow field is directed poleward, a westward counterflow can occur along the poleward flank of the ridge. The meridional scales of these topographically induced flows are comparable to those observed along the Indian-Antarctic Ridge by Callahan (1971).     

Upstream stagnation points in stratified flow past obstacles

An experimental study has been made of stagnation points and flow splitting on the upstream side of obstacles in uniformly stratified flow. A range from small to large values of Nh/U (where N is the buoyancy frequency, h_m is the maximum obstacle height and U is the undisturbed fluid velocity) has been covered, for three obstacle shapes which are, respectively, axisymmetric, and elongated in the across-stream and in the downstream directions. Upstream stagnation for the first two of these models does not occur until Nh_m/U > 1.05, where it occurs at z ≈ h_m/2. On the central line below this point the flow descends and diverges, and we term this ‘flow splitting’. For the third model (elongated in the downstream direction), stagnation upstream first occurs at Nh_m/U ≈ 1.43, at z ≈ 0. Results for this obstacle are not consistent with the ‘Sheppard criterion’, and this upstream flow stagnation is not apparently related to lee wave overturning, in contrast to flow over two-dimensional obstacles.     

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.     

Finite-amplitude, neutral baroclinic eddies and mean flows in an internally heated, rotating fluid: 2. Effects of spatially varying N

The analytical model of finite-amplitude, quasi-geostrophic ‘free mode’ baroclinic eddies and mean zonal flows in a Cartesian channel, presented recently by Read, is extended to take account of vertical variations in the buoyancy frequency N. A series of exact solutions is presented to illustrate the effect of monotonically varying static stability on the structure and properties of the flow. The analytical solutions are then compared with a corresponding series of numerical simulations of steady wave flows in a rotating fluid annulus subject to internal heating and sidewall cooling. By suitable choices of internal heating distributions and boundary conditions, several different forms of N² profile could be obtained in the simulated flows, in which N² was concentrated to a greater or lesser extent towards the upper boundary. The resulting steady flows exhibited strong qualitative similarities in their structure and dependence upon the form of N²(z) to that of the analytical solutions when realistic profiles of N² were included in the latter, especially when an equivalent-barotropic component was included, although the latter component is unable to satisfy the simplest (internal jet) form of horizontal boundary condition as usually applied to Rossby waves.The relatively weak, though crucially important, forcing and dissipation processes in the annulus are examined using approximate quasi-geostrophic diagnostics of the major terms in the budget of potential enstrophy for the numerical simulations. Internal heating is found to be the major source of potential enstrophy for the mean zonal flow, solely by virtue of the variation of N² with height, but has only a minor direct effect upon the eddy flow component. Because of the presence of critical layers in the flow, all non-linear terms (including the third-order potential enstrophy flux divergence) are found to be significant in certain regions. Some implications for the value and applicability of EP flux diagnostics are discussed. Potential enstrophy budgets for horizontal regions enclosed by geostrophic streamlines are used to shed further insight into the maintenance of the flow against ‘friction’, and on the form of the potential vorticity-streamfunction relationship. Some implications of the results for other systems of geophysical interest are also discussed.     

Steady flow between two reservoirs of fluid with different densities and levels through a rectangular channel section of varying depth and width

The steady hydrostatic flow through a channel of rectangular cross section connecting reservoirs of infinite width and depth and containing inviscid fluids of different densities and levels is studied. The main goal is the determination of the discharges of the lighter and denser fluids in terms of the external conditions (reservoir levels, fluid densities and variation of width and depth along a channel). It is shown that the key parameter is δ, which is the ratio of relative reservoir level difference, γ, to relative density difference, ε. If δ<0 then the denser fluid plunges under the stationary lighter layer. If δ>δ_* (1<δ_*<1.5) then the lighter fluid runs up on a wedge of stationary heavier fluid. Here δ_* depends on the geometry of the constriction. The solutions describing these regimes are stated. If 0<δ<δ_* then both layers are in motion. A qualitative analysis of the solution for arbitrary bottom shape and channel width and arbitrary ε is presented and the problem is reduced to a system of two equations which can be easily solved numerically for any particular channel profile. We give detailed analyses for the following two cases: 1) the narrowest width of the channel is on the side of the heavier fluid and the top of the sill is on the side of lighter fluid; 2) the minima in channel depth and width coincide. In the second case the discharges for one class of geometries in the Boussinesq approximation are calculated and discussed.     

Estimates of water vapor flux and canopy conductance of Scots pine at the tree level utilizing different xylem sap flow methods

Summary During the Hartheim Experiment (HartX) 1992 conducted in the upper Rhine Valley, Germany, three different methods were used to measure sap flow in Scots pine trees via heating of water transported in the xylem: (1) constant heating applied radially in the sapwood (Granier-system-G), (2) constant heating of a stem segment (ermák-system-C), and (3) regulated variable heating of a stem segment that locally maintains a constant temperature gradient in the trunk (ermák/Schulze-system-CS). While the constant heating methods utilize changes in the induced temperature gradient to quantify sap flux, the CS-system estimates water flow from the variable power requirement to maintain a 2 or 3 degree Kelvin temperature gradient over a short distance between inserted electrodes and reference point. The C- and CS-systems assume that all transported water is encompassed and equally heated by the electrodes. In this case, flux rate is determined from temperature difference or energy input and the heat capacity of water. Active sapwood area need not be determined exactly. In contrast, the G-system requires an empirical calibration of the sensors that allows conversion of temperature difference into sap flow density. Estimates of sapwood area are used to calculate the total flux. All three methods assume that the natural fluctuation in temperature of the trunk near the point of insertion of heating and sensing elements is the same as that where reference thermocouples are inserted.Using all three systems, 24 trees were simultaneously monitored during the HartX campaign. Tree size within the stand ranged between 18 and 61 cm circumference at breast height, while sample trees ranged between 24 and 55 cm circumference. The smallest trees could only be measured by utilizing the G-system. Sap flow rates of individual trees measured at breast height increased rapidly in the morning along with increases in irradiance and vapor pressure deficit (D), decreased slowly during the course of the afternoon with continued increase inD, and decreased more slowly during the night.Ignoring potential effects introduced by the different methods, maximum flow rates of individual trees ranged between 0.5 and 2.5 kg H₂O h^–1 tree^–1 or 0.3 and 0.6 mm h^–1 related to projected crown area of trees and daily sums of sap flow for individual trees varied between 4.4 and 24 kg H₂O tree^–1 d^–1 or 1.1 and 6.0 mm d^–1. Maximum sap flow rates per sapwood area of trees varied least for the G-system (11–17 g cm^–2 h^–1) and was of similar magnitude as the C- (8–21 g cm^–2 h^–1) and CS-system (4–14 g cm^–2 h^–1).Regressions of total tree conductance (g _t ) derived from sap flow estimates demonstrated the same linear increase of conductance with increasing irradiance, however decrease of conductance with increasingD under non-limiting light conditions was different for the three systems with strongest reduction ofg _t measured with the CS-system followed by the C- and G-system. This led to different estimates of daily sap flow rates especially during the second part of the measurement period.Variation in sap flow rates is explained on the basis of variation in leaf area index of individual trees, heterogeneity in soil conditions, and methodological differences in sap flow measurements. Despite the highly uniform plantation forest at the scale of hectares, the heterogeneity in tree size and soil depth at the scale of square meters still make it difficult to appropriately and efficiently select sample trees and to scale-up water flux from individual trees to the stand level.With 5 Figures     

Stratified flow over three-dimensional topography

In order to investigate flows over topography in an atmospheric context, we have studied experimentally the wake structure of axi-symmetric Gaussian obstacles towed through a linearly stratified fluid. Three dimensionless parameters govern the flow dynamics: F, the Froude number based on the topography height h; Re, the Reynolds number and the aspect ratio r = h/L, where L is the topography horizontal scale. Two-dimensional (2-D), saturated lee wave (SLW) and three-dimensional (3-D) regimes, as defined in Chomaz et al. (1993), are found to be functions of F and r only (Fig. 1) as soon as Re is larger than Re_c ≈ 2000. For F < 0.7 the flow goes around the obstacle and the motion in the wake is quasi-two-dimensional. This 2-D layer is topped by a region affected by lee wave motions with amplitude increasing with r and F. For 0.7 < F < 1/r, the flow is entirely dominated by a lee wave of saturated amplitude which suppresses the separation of the boundary layer from the obstacle. Above the critical value 1/r, the lee wave amplitude decreases with F and a recirculating zone appears behind the obstacle. Simultaneously, coherent large-scale vortices start to be shed periodically from the wake at a Strouhal number which decreases as 1/F until it reaches its neutral asymptotic value.     

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.     

A laboratory simulation of mesoscale flow interaction with the Alps

A series of laboratory experiments, aimed at the simulation of some aspects of Alpine lee cyclogenesis has been carried out in the rotating tank of the Coriolis Laboratory of LEGI-IMG in Grenoble. Dynamic and thermodynamic processes, typical of baroclinic development triggered by the orography, were simulated. The background flow simulating the basic state of the atmosphere consisted of a stream of intermediate density fluid introduced at the interface between two fluid layers. The structure of the intermediate current was established by mixing fluid obtained from the upper layer of fresh water with fluid removed from the heavier salty layer below.The dynamical similarity parameters are the Rossby (Ro), Burger (Bu) and Ekman (Ek) numbers, although this last, owing to its small values, need not be matched between model and prototype, since viscous effects are not important for small time scales. The flow in both the prototype and laboratory simulation is characterized by hydrostatics; this requires (Ro²δ²/Bu)1 (where δ=H/L is the aspect ratio of the obstacle) which is clearly satisfied, in the atmosphere and oceans, and for the laboratory experiment.A range of experiments for various Rossby and Burger numbers were conducted which delimited the region of parameter space for which background flows akin to that found to the northwest of the Alps prior to baroclinic cyclogenesis events, were observed.One such experiment was carried out by placing a model of the Alps at the appropriate place in the flow field. The subsequent motion in the laboratory was observed and dye tracer motions were used to obtain the approximate particle trajectories. The density field was also analyzed to provide the geopotential field of the simulated atmosphere. Using standard transformations from the similarity analysis, the laboratory observations were related to the prototype atmosphere. The flow and the geopotential fields gave results compatible with the particular atmospheric event presented.     

Rectified flow along a vertical coastline

Laboratory experiments are conducted on a physical system in which an oscillatory, along-shore, free stream flow of a homogeneous fluid occurs in the vicinity of a long coastline with vertical slope; the model sea-floor is horizontal. Particular attention is given to the resulting rectified (mean) current which is along the coastline with the shore on the right, facing downstream. In the lateral far field region defined by (1), where y is the offshore coordinate and H is the depth of the fluid, the motion field is approximately independent of the lateral distance from the coast. The vertical structure of the cross-stream motion in this region consists of Ekman layers near the sea-floor and interior adjustment flows, both periodic in time. In the near field, defined by (1), the motion is strongly dependent on the cross-stream coordinate as well as time, and rectified currents are observed. The mechanism responsible for the rectification is a complex nonlinear coupling between laterally directed adjustment flows driven by the transport in the bottom Ekman layers, and the free stream motion field. The rectified current is found to be substantially wider than the Stewartson layer thickness but much narrower than the Rossby deformation radius. The characteristic width, δ_y, of the rectified current is shown to scale as , where Ro is the Rossby number Ro_t is the temporal Rossby number and E is the Ekman number. Experiments are presented which support this scaling.     

Experiments with viscous source flows in rotating systems

The dynamics of oceanic bottom currents are examined both theoretically and in the laboratory. A class of similarity solutions for steady flow indicates that the geostrophic current is drained by Ekman flux at its downslope edge and ultimately extinguished at a downstream distance of order $\text{(}\text{fQ}\text{s}{}^2\text{g}{}_{\mathrm{r}}\text{)}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}$ magnified by $\text{E}{}^{\mathrm{<mtext>?1</mtext><mtext>2</mtext>}}$.Laboratory source flows are found to be consistently wave-like. Nevertheless, certain gross features of the steady Ekman flux mechanism are observed. Instabilities are classified according to the magnitudes of Rossby (?) and Ekman (E) numbers for the steady flow scaling. Characteristic were forms include: (1) a meandering jet (E < ?, 10^?2); (2) transverse waves on broad contour current (? < E, 10^?2); and (3) transverse waves on a viscous flow (?, E > 10^?2). The dispersion relation for source flow waves resembles that of baroclinic instabilities for a uniform two-layer channel flow, and an empirically determined stability boundary is in rough agreement with the inviscid channel flow criterion.An interpretation of field measurements from the Denmark Strait Overflow in terms of the laboratory results is presented.     

The Rose Spit Eddy in Dixon Entrance: Evidence for its existence and underlying dynamics

Abstract

The existence and dynamics of the so‐called “Rose Spit Eddy” in Dixon Entrance, British Columbia, are investigated by (i) analysing published observations of low‐frequency Eulerian and Lagrangian currents in the region; (ii) interpreting tidal residuals produced by the Hecate Model (a non‐rotating hydraulic model of Hecate Strait and Dixon Entrance); and (iii) running a barotropic, non‐linear numerical tidal model over simplified topography to investigate residuals produced over the Rose Spit sill.

Observations have consistently revealed persistent basin‐wide, surface‐intensified cyclonic shears within central and eastern Dixon Entrance. The Hecate hydraulic tidal model also produced a tidal residual cyclonic gyre in central Dixon Entrance, but with velocities considerably larger than those observed. Barotropic numerical simulations of tidal streams flowing over a representation of the Rose Spit sill produced residual flows along the sill in reasonable agreement with observations and theory. A southward‐directed jet flow was produced off Cape Chacon. Elsewhere, tidal rectification was weak. Run without the Coriolis force, organized flow along the sill broke down, although the headland jet off Cape Chacon persisted.

We submit that the observed Rose Spit eddy results from interactions between buoyancy‐driven coastal currents and tidally rectified flows generated over the Rose Spit sill, and near Cape Chacon, and perhaps indirectly, over the western flank of Learmonth Bank (which although west of the Rose Spit eddy, contributes to the cross‐channel flow across the Entrance). These regions of localized tidal stress will each favour recirculation of a portion of the coastal current within the Entrance, helping to form the eddy.

We believe that the Hecate hydraulic model eddy was generated to a significant degree by phase errors introduced at the northern open boundary, where a rocking barrier was used to force currents. A second rocking barrier also produced a large cyclonic gyre, not supported by observations, near the model's southern boundary.     

Coastal-trapped waves with finite bottom friction

Coastal-trapped waves with finite-amplitude bottom friction are explored. “Finite-amplitude” in this context means that the bottom stresses are large enough to change the wave modal structure. The importance of bottom friction is measured by the nondimensional number r/(ωh), where r is a bottom resistance coefficient, ω the wave frequency and h the water depth. Increasing bottom drag causes free wave modes to adjust by having their amplitude maxima for alongshore current translate offshore to the point that, with relatively large bottom stress, the alongshore current variance is trapped entirely on the slope, even though pressure variations remain substantial right up to the coast. In conjunction with these adjustments, wave frequency, hence propagation speed, varies and the wave damping is usually less than would be expected based on a weak-friction perturbation calculation. Stronger density stratification increases wave damping, all else being the same. A mean alongshore flow can strongly affect modal structure and wave damping, although general trends are difficult to discern. Results suggest that bottom friction may cause an observed tendency for lower frequency alongshore current fluctuations to become relatively more important with distance offshore.     

The cross‐channel flow at the entrance of Lancaster Sound

Abstract

A westward current flows along the northern side of Lancaster Sound and an eastward current flows along the southern side. A cross‐channel flow is commonly observed to link them near the eastern entrance of Lancaster Sound; this flow is modelled assuming inviscid flow and conservation of potential vorticity. It is shown that the westward decrease in depth is sufficient to cause a cross‐channel flow that couples the inflow to the outflow. The modelled cross‐channel flow takes place at a distance inside the entrance that is less than that observed for the surface current. Obviously stratification reduces the coupling of the surface current to the bathymetry. A more realistic result is obtained with the barotropic model if the bottom slope is halved.

An inviscid mean barotropic flow out of the channel is also modelled and found to be concentrated on the southern bank in order to conserve potential vorticity. It seems that barotropic instability and friction would limit the narrowing of the flow.     

An extended Miles' theory for wave generation by wind

Miles' inviscid theory of surface wave generation by wind is (a) modified by replacing the logarithmic shear velocity profile with one which applies right down to the wave surface and which exhibits an explicit dependence on the roughness of the surface, and (b) extended to include the effects of the interaction of wave with air flow turbulence by considering the wave-modified mean flow as the mean of the actual turbulent air flow over water waves and using this in a mixing-length model.The surface pressure is shown to depend significantly on the flow conditions being aerodynamically smooth or rough. Its component in phase with the surface elevation is practically unaffected by the wave-turbulence interaction. However, such interaction tends to increase the rate of energy input ß from wind to waves travelling in the same direction, e.g., the increase is 2gk ² for aerodynamically rough flow, where gk is the Von Karman constant. It also provides damping of waves in an adverse wind which can be about 10% of the growth rate in a favourable wind.     

On Integral Measures Of The Neutral Barotropic Planetary Boundary Layer

Two types of neutral planetary boundary layer (PBL) are distinguished:truly neutral – developed against a neutrally stratified free flow, and conventionally neutral – developed against a background stable stratification. Atmospheric PBLs treated asneutral are almost always conventionally neutral. Theoretical reasoning and results from large-eddy simulation (LES) show that A and B coefficients of the Rossby-number similarity theory are not constants. The same is true for thecoefficient C_h in the Rossby–Montgomery formula for the neutral boundary-layer depth h = C_hu*/|f|, where u_* is the friction velocity. Contrary to classical ideas, A, B and C_h depend on the ratio _N N/|f| of the free-flow Brunt–V*auml;isäl ä frequency N to the absolute value of the Coriolis parameter |f|. This new development can explain why atmospheric and LES estimates of A, B and C_h appear inconsistent. It results from neglecting the fact that atmospheric data for _N 10² were compared with LES data for _N = 0, violating an obvious requirement of similarity with respect to _N.     

Prediction Errors Associated with Sparse Grid Estimates of Flows over Hills

Numerical simulations of geophysical flows have to be done on very sparse grids. Nevertheless, flows over moderately sloped hills can be predicted quite accurately as long as the near ground vertical resolution is reasonably dense. Recirculation flows behind steeper hills are associated with slow convergence towards grid independent integrations, but even then moderately stratified flows of this type can be predicted usefully accurately. For better horizontal grids than about half the hill-height Δx ₁/H ≈ 0.5 or so, separation and recirculating domains are predicted with an error factor comparable to 0.3. The characteristic wavelength of lee waves is predicted more accurately while the lee wave amplitude and the maximum turbulence intensity in recirculating domains are underestimated by factors comparable to 0.3. Strongly stratified flows may be associated with hydraulic transitions and even this is predicted on quite coarse grids, up to say Δx ₁/H ≈ 0.5. However, the details of such flows turn out to be predicted with considerable errors also on high-resolution grids. Inaccurate modelling of stratified turbulence is a main contributor to this error.