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.     

Neutrally Stratified Turbulent Ekman Boundary Layer: Universal Similarity for a Transitional Rough Surface

The geostrophic Ekman boundary layer for large Rossby number (Ro) has been investigated by exploring the role played by the mesolayer (intermediate layer) lying between the traditional inner and outer layers. It is shown that the velocity and Reynolds shear stress components in the inner layer (including the overlap region) are universal relations, explicitly independent of surface roughness. This universality of predictions has been supported by observations from experiment, field and direct numerical simulation (DNS) data for fully smooth, transitionally rough and fully rough surfaces. The maxima of Reynolds shear stresses have been shown to be located in the mesolayer of the Ekman boundary layer, whose scale corresponds to the inverse square root of the friction Rossby number. The composite wall-wake universal relations for geostrophic velocity profiles have been proposed, and the two wake functions of the outer layer have been estimated by an eddy viscosity closure model. The geostrophic drag and cross-isobaric angle predictions yield universal relations, which are also supported by extensive field, laboratory and DNS data. The proposed predictions for the geostrophic drag and the cross-isobaric angle compare well with data for Rossby number Ro ≥ 10⁵. The data show low Rossby number effects for Ro < 10⁵ and higher-order effects due to the mesolayer compare well with the data for Ro ≥ 10³.     

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).     

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.     

On the transient adjustment of a mid-latitude abyssal ocean basin with realistic geometry: the constant depth limit

The early stages in the adjustment of a mid-latitude abyssal basin with realistic geometry are studied using an inverted one and one-half layer model of the Eastern Mediterranean Sea as a natural test basin. The model is forced with a localized sidewall mass source and a compensating distributed mass sink. A flat bottom basin is investigated for comparison with existing theories on abyssal gyral spin-up, and as a precursor to a study with realistic topography. As in existing theories, the early adjustment is dominated by sub-inertial Kelvin and Rossby waves. Obstacles and the varying coastal geometry do not impede the passage of the Kelvin wave, though the circuit time of the main Kelvin wave signal is reduced by an aggregate 6% for the abyssal Eastern Mediterranean basin. The scattering of the Kelvin wave due to small-scale variations in the coastline is also shown not to be significant to the adjustment. The relatively short period of time needed to reach a statistical steady state is attributed to western boundary current formation in response to local Kelvin wave dynamics. Upon cessation of the sidewall forcing, sub-inertial motion controls the spin-down adjustment with basin-scale Rossby waves becoming the most pronounced feature of the flow. Two dynamical issues of particular interest emerge in these simulations: the retardation of Kelvin wave propagation around the abyssal basin and the roles of detrainment and sidewall forcing in the interior vorticity balance. An idealized simulation using an elliptical basin is used to illustrate that the mechanism for Kelvin wave retardation is a geometrically induced dispersion due to large-scale variations in the coastline. A dynamical analysis of the interior circulation shows that detrainment alone does not develop a Sverdrup response. Both the localized sidewall injection and the detrainment are needed to describe the interior dynamics, with both poleward and equatorward flows developing during the adjustment.     

Meridional component of oceanic Rossby wave propagation

A three-dimensional spectral analysis of Topex altimeter data reveals a large meridional component k_y of the wavevector k for baroclinic Rossby waves of all timescales. Its existence necessitates some refinements in our estimates of certain basic properties of the Rossby wave field. In particular, by taking into account an actual off-zonal direction of k (often exceeding 70°), one finds that the wavelength, phase speed, and group velocity of mid-latitude Rossby waves (with periods less than 2 years) are much smaller than they appear to be on the assumption of a purely zonal wavenumber vector. Because of a shorter wavelength (yielding kL as high as 0.6, where L is the Rossby radius of deformation), these waves are essentially dispersive. Their group velocity vector may depart from zonal by more than 30°. An important intrinsic feature of the wave spectrum confirmed by our analysis is a broad-band distribution with respect to k_y. Some of the dynamical implications of the large k_y/k_x ratio are discussed.     

The temporal evolution of a geostrophic flow in a rotating stratified basin

A laboratory study in a rotating stratified basin examines the instability and long time evolution of the geostrophic double gyre introduced by the baroclinic adjustment to an initial basin-scale step height discontinuity in the density interface of a two-layer fluid. The dimensionless parameters that are important in determining the observed response are the Burger number S=R/R₀ (where R is the baroclinic Rossby radius of deformation and R₀ is the basin radius) and the initial forcing amplitude (H₁ is the upper layer depth). Experimental observations and a numerical approach, using contour dynamics, are used to identify the mechanisms that result in the dominance of nonlinear behaviour in the long time evolution, τ>2⁻¹ (where τ is time scaled by the inertial period T_I=2π/f). When the influence of rotation is moderate (0.25≤S≤1), the instability mechanism is associated with the finite amplitude potential vorticity (PV) perturbation introduced when the double gyre is established. On the other hand, when the influence of rotation is strong (S≤0.1), baroclinic instability contributes to the nonlinear behaviour. Regardless of the mechanism, nonlinearity acts to transfer energy from the geostrophic double gyre to smaller scales associated with an eddy field. In the lower layer, Ekman damping is pronounced, resulting in the dissipation of the eddy field after only 40T_I. In the upper layer, where dissipative effects are weak, the eddy field evolves until it reaches a symmetric distribution of potential vorticity within the domain consisting of cyclonic and anticyclonic eddy pairs, after approximately 100T_I. The functional dependence of the characteristic eddy lengthscale L_E on S is consistent with previous laboratory studies on continuously forced geostrophic turbulence. The cyclonic and anticyclonic eddy pairs are maintained until viscous effects eventually dissipate all motion in the upper layer after approximately 800T_I. The outcomes of this study are considered in terms of their contribution to the understanding of the energy pathways and transport processes associated with basin-scale motions in large stratified lakes.     

On the instability of a planetary boundary layer with Rossby-number similarity

The formation of longitudinal vortex rolls in the planetary boundary layer (PBL) is investigated by means of perturbation analysis. The method is the same as that used by previous authors who have investigated the instability of a laminar Ekman layer. To study the instability of the turbulent boundary layer of the atmosphere, vertical profiles are needed of the eddy viscosity and of the two components of the basic flow. These profiles have been obtained by a numerical PBL-model; they are universal for zz ₀. (However, the stability equations are not completely universal, i.e., independent of the external parameters). For each thermal stratification, expressed by the internal stratification parameter , one has a set of three consistent profiles.The numerical solution of the stability equations gives the critical values and the perturbation growth rates as functions of thermal stratification and of the surface Rossby number Ro₀. This is in contrast to the case of a laminar Ekman layer, where the instability depends on a Reynolds number only, which makes atmospheric applications difficult. The most unstable perturbations are found for Ro₀ = 1 × 10⁶ approximately, which is in the range of surface Rossby numbers observed in the atmosphere. However, considering vortex rolls oriented in the direction of the surface stress, the instability is found to be universal, i.e., independent of the external parameters combined in the surface Rossby number. With respect to thermal stratification, the results show that the instability of the perturbations increases with increasing static stability.     

On the effects of vertical eddy viscosity on rossby and eady waves in the ocean

The effects of vertical eddy viscosity on simple mesoscale waves in the ocean are studied. The decay of Rossby waves is investigated by one-dimensional depth-dependent linear stability problems which are derived for the interior non-viscous or viscous quasigeostrophic flow using parameterizations of the top and bottom boundary layers corresponding to Ekman suction, no-stress and bottom-stress boundary conditions.The non-slip condition at the bottom yielding an O(E_v^1/2)-Ekman layer causes very short damping times for the 0th Rossby mode. This suggests that this boundary condition is not suitable for mesoscale wave studies, because a Rossby wave fit for the MODE eddy can be done satisfactorily without any damping. Reasonable results for damping times of Rossby waves are obtained by prescribing the bottom stress, resulting from the constant-stress layer at the bottom, and the free-slip condition at the surface. The growth rates of Eady waves are reexamined using this bottom-stress condition.Vertical viscosity in the interior of the ocean, e.g. internal wave induced viscosity, may have a significant influence on the dynamics of the mesoscale motions, comparable to that of the boundary layers in some cases. The results are compatible with the sparse observations available.     

The effect of ekman suction on a flow over a shallow topography in the beta plane

The effect of bottom Ekman layer suction on a homogeneous, constant depth, eastwards, low Rossby number flow over a shallow bottom topography in the beta plane is studied. The governing vorticity equation is obtained by expanding the velocities in the continuity and momentum equations in powers of the Rossby number, ?, and matching the vertical velocity with the vertical velocity at the outer edge of the bottom Ekman layer obtained from the Ekman layer solution. The suction effect is then linearized using an Oseen approxiamation and the resulting linear model is solved using Fourier transforms with the requirement that the solution behave like a vortex near the origin which is equivalent to the effect of an isolated bump, i.e., a Green's function solution is obtained. An analytical solution is thus, obtained in integral form and then numerically integrated. The effect of Ekman suction is found to be a damping of the downstream Rossby waves in a distance of order $\text{2√2U/f}{}_0\text{E}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}$, an increased upstream influence, and a counterclockwise rotation of the closed streamline region about the origin. It is pointed out that the vortex solutions can be superimposed in order to obtain the solution for flow over topographies of finite horizontal text. This technique was used to compute the flow over a right circular cylinder. The results agree favorably with the experimental results of McCartney (1975).     

On modelling geophysical flows having low rossby numbers

Abstract

New, fourth‐order “c” grid Coriolis term treatments are compared with widely used second‐order treatments. Their improved accuracy is demonstrated by a grid convergence study for a relevant linear problem. Such an accuracy improvement is relatively easy and costs little for low Rossby number flows compared with high Rossby number flows, because one must consider only the Coriolis and pressure gradient terms in low Rossby number flows. The “c” grid is favourable for the latter, but the Coriolis terms benefit greatly by the higher order treatments analysed herein.     

Direct numerical simulation of a vigorously heated low Reynolds number convective boundary layer

Computations of the buoyantly unstable Ekman layer are performed at low Reynolds number. The results are obtained by directly solving the three-dimensional time-dependen Navier-Stokes equations with the Boussinesq buoyancy approximation, resolving all relevant scales of motion (no turbulence closure is needed). The flow is capped by a stable temperature inversion and heated from below at a rate that produces an inversion-height to Obukhov-length ratio −z_i/L_* = 32. Temperature and velocity variance profiles are found to agree well with those from an earlier vigorously heated under-resolved computation at higher Reynolds number, and with experimental data of Deardorff and Willis (Boundary-Layer Meteorol., 32: 205–236, 1985). Significant helicity is found in the layer, and helical convection patterns of the scale of the inversion height are observed.     

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.     

Instabilities of the tidally induced bottom boundary layer in the rotating frame and their mixing effect

To investigate the stability of the bottom boundary layer induced by tidal flow (oscillating flow) in a rotating frame, numerical experiments have been carried out with a two-dimensional non-hydrostatic model. Under homogeneous conditions three types of instability are found depending on the temporal Rossby number Ro_t, the ratio of the inertial and tidal periods. When Ro_t < 0.9 (subinertial range), the Ekman type I instability occurs because the effect of rotation is dominant though the flow becomes more stable than the steady Ekman flow with increasing Ro_t. When Ro_t > 1.1 (superinertial range), the Stokes layer instability is excited as in the absence of rotation. When 0.9 < Ro_t < 1.1 (near-inertial range), the Ekman type I or type II instability appears as in the steady Ekman layer. Being much thickened (100 m), the boundary layer becomes unstable even if tidal flow is weak (5 cm/s). The large vertical scale enhances the contribution of the Coriolis effect to destabilization, so that the type II instability tends to appear when Ro_t > 1.0. However, when Ro_t < 1.0, the type I instability rather than the type II instability appears because the downward phase change of tidal flow acts to suppress the latter. To evaluate the mixing effect of these instabilities, some experiments have been executed under a weak stratification peculiar to polar oceans (the buoyancy frequency N²  10⁻⁶ s⁻²). Strong mixing occurs in the subinertial and near-inertial ranges such that tracer is well mixed in the boundary layer and an apparent diffusivity there is evaluated at 150–300 cm²/s. This suggests that effective mixing due to these instabilities may play an important role in determining the properties of dense shelf water in the polar regions.     

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.     

低纬斜压两层模式大气半地转适应过程

利用斜压两层模式研究了赤道平面近似下的低纬热带大气适应过程。指出低纬斜压大气适应过程主要受重力惯性内波控制。通过重力惯性内波对初始非地转能量的频散，使纬向运动达到地转平衡，而经向维持非地转运动，正压模式下称为半地转平衡，斜压模式下称为半热成风平衡。通过对垂直运动方程的求解，可知，垂直运动只与重力惯性内波相联系，其产生与初始斜压位涡度无关，而只与初始时刻的垂直运动和垂直运动倾向有关，半地转适应使运动趋向水平运动。讨论了半热成风平衡的建立及其物理机制，指出由于重力惯性内波激发出垂直运动，与垂直运动相联系的水平辐合辐散调整流场和温度场之间的关系，使温压场最终达到半热成风平衡。通过对适应过程终态的分析，指出平均温度场和切变流场之间的适应方向决定于初始非半地转扰动的尺度与斜压Ｒｏｓｓｂｙ变形半径有关的特征尺度的比值，当比值大于１时，切变流场向平均温度场适应；当比值小于１时，平均温度场向切变流场适应     

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.     

A new class of steady solutions of the barotropic vorticity equation

Early studies of the wind-driven ocean problem were concerned with finding steady solutions of the equations of motion, first by analytic, and later by numerical methods. The problem is characterized by an Ekman number (?) and a Rossby number (α). Prior to this study, steady solutions were confined to the ranges α, ? < < 1.In this work the problem is considered in a circular domain with lateral friction and a partial-slip boundary condition. Steady solutions are found analytically in the new regime α > 1. Furthermore, for large enough values of ?, steady solutions are determined numerically in the intermediate range between this high-α regime and the classical low-α regime. That is, for some values of ?, steady solutions can be found through all values of α. Numerical evidence suggests that below a critical value of ?, steady solutions in this intermediate range exist, but cease to be stable. Similar results have been obtained with bottom friction and with different boundary conditions.     

地形与Ekman边界层中的气流

利用σ坐标讨论地形与边界层气流是有很多方便的地方,因为,在此坐标中,下边界条件较为简单。在本工作中,首先将混合长理论加以推广并将它用于σ坐标,于是导得了用以描述地形上空边界层气流的控制方程,对边界层气流的特征,特别是对于Ekman抽吸作用进行了详细分析。指出有三种因子影响边界层顶部的垂直运动,第一种因子是边界层内涡度分布,这是与边界层中由于摩擦作用所引起的辐合辐散有直接联系;第二种因子是由于边界层顶部的气流爬坡运动所引起的;第三种是由于边界层中跨越等压线的分量爬坡所引起的,它出现于当等压线与地形等高线相平行时,或地转风呈现绕流情况时,这一作用最为明显。     

Observations of the barotropic Ekman layer over an urban terrain

Data from low-level soundings over Cambridge, U.S.A. were selected on the basis of an Ekman-like variation of the wind vector with altitude combined with evidence of a barotropic atmosphere. The method of geostrophic departure was used to determine the shear-stress distribution. The analysis yields the dimensionless properties of the barotropic Ekman layer under neutral and stable stratification. Some important results include: the geostrophic drag coefficient displays no dependence on the degree of static stability; the dimensionless height of the boundary layer decreases with increasing stability in agreement with the prediction of Zilitinkevich; the properties of the urban surface layer, where the roughness elements are multistory buildings, show no dependence on atmospheric stability under the moderate wind conditions which display the Ekman-like wind profile; and the directions of the horizontal shear stress and the vertical derivative of the velocity vector usually tend to be parallel only near the surface layer. Values of the two constants of the Rossby number similarity theory are found for the neutral barotropic Ekman layer at a surface Rossby number equal to 2 × 10⁵. The implications of the work with respect to wind-tunnel simulation of the flow over models of urban areas are discussed.