The structure of separated flow past a circular cylinder in a rotating frame

Abstract

This paper examines the detailed E ^1/4-layer structure of separated flow past a circular cylinder in a low-Rossby-number rotating fluid as the Ekman number E tends to zero. This structure is based on an initial proposal by Page (1987) but with some modifications in response to further evidence, outlined both in this paper and elsewhere, on the behaviour of E ^1/4-layer flows in this context. Numerical calculations for flow in an E ^1/4 shear layer along the separated free streamline are described and the mass flux from this layer is then used to calculate the higher-order flow within the separation bubble. The flow structure is found to have two forms, depending on the value of the O(1) parameter λ, and these are compared with results from published “Navier-Stokes” type calculations for the flow at small but finite values of E.     

Three‐dimensional non‐geostrophic disturbances in a baroclinic zonal flow

Abstract

A study is made to determine the stability properties of a baroclinic zonal current on which small amplitude three‐dimensional non‐geostrophic disturbances are superimposed. The flow is assumed to be bounded to the north and south by rigid vertical walls and the Rossby number Ro is taken to be small compared to unity. It is then shown that if the perturbation quantities are expanded in power series in Ro the leading or zero order terms in the series correspond to the quasi‐geostrophic solution obtained by Eady (1949) and that the higher order terms represent the “non‐geostrophic” effects neglected by the latter.

It is shown that to the second order in Ro the non‐geostrophic effects decrease the growth rates of those disturbances which are found to be unstable according to Eady's analysis but do not alter their speed of propagation. The results indicate, on the other hand, that to the same order of approximation the stable waves travel at a speed which is different from that given by Eady's solution. The modification of the perturbation wave structure by the non‐geostrophic effects is also investigated. It is found in particular that to the first order in Ro the latter produce a northward tilt with height in the ridge (or trough) lines of the meridional and vertical particle velocity fields away from the lateral boundaries.     

Wave hierarchies in alluvial river flows

Abstract

The long wave equations governing the flow in alluvial rivers and channels are considered. The linearized equations are re-cast in the form of a single equation of wave hierarchy type as discussed by Whitham (1974). The dynamic and kinematic waves are of third and second order respectively. Behaviour at the wave fronts is considered and a roll-wave type instability is revealed.

For stable flow, the theory is used to make both qualitative and quantitative predictions in the areas of short and long term floods, tidal waves and channel dredging.

The non-uniformity in the quasi-steady theory on bedform development [see, for example, Reynolds (1985)] as the Froude number, F, approaches unity is also discussed, and appropriate scalings are obtained to derive a theory which remains valid when F ～ 1.     

Topographic instability and multiple equilibria on an f-plane

Abstract

The flow of a two-layer flow in a rotating channel on an f-plane over topography with sinusoidal variation of height in a direction parallel to the flow is investigated. When the two layers flow in opposite directions a resonance is found when the topographic scale matches the free mode of the system. We examine the stability of the forced mode in the vicinity of this resonance by means of a perturbation expansion of the topographic height. Both subresonant and super-resonant instabilities are found and their equilibration is examined. For small values of the dissipation multiple equilibria are found. The topographic drag releases potential energy even when the flow is baroclinically stable.     

On the hyperbolic nature of the equations of alluvial river hydraulics and the equivalence of stable and energy dissipating shocks

Abstract

The depth-averaged hydraulic equations augmented with a suitable bed-load sediment transport function form a closed system which governs the one-dimensional flow in an alluvial river or channel. In this paper, it is shown that this system is hyperbolic and yields three families of shock-wave solutions. These are determined to be temporally stable in restricted regions of the (H, F ₀)-plane, via the Lax shock inequalities. Further, it is demonstrated that this criterion is equivalent to the energy dissipation criterion developed by Needham and Hey (1991).     

Asymptotic aspects of the Boussinesq approximation for gases and liquids

Abstract

The asymptotic formulation of the Boussinesq approximation relates the pressure of the fluid to a thermodynamical quantity involving the heat capacity c_Po . In this paper we examine the implications of such a scaling, in particular: (i) the singular degeneracy of the equation of state ρ = ρ* (1 ?α* (T?;T*)) of a liquid: this equation of state is valid only for small values of the coefficient α T*; (ii) in which manner the scaling introduces the Mach number of the flow as a small parameter e for a compressible fluid. The equations at order zero with respect to ? are the same equations for gases and for liquids only if the thermodynamics of the medium is described by using the Brunt-Väisälä frequency instead of the temperature.     

Delivering environmental flows in the Murray-Darling Basin (Australia)—legal and governance aspects

Abstract

The hydrology of water-dependent ecosystems around the world has been altered as a result of flow regulation and extraction for a variety of purposes including agricultural and urban water supply. The flow regime of the Murray-Darling Basin in Australia is no exception, with attendant impacts on the health of the environment. Restoration of parts of the flow regime is a key feature of environmental flow delivery. However, environmental flow delivery in a system that is managed primarily to provide a secure and stable supply for irrigation presents challenges for managers seeking to return more natural flow variability in line with ecosystem requirements. The institutional arrangements governing releases of water from storage can influence the ability of managers to respond to natural cues, such as naturally rising flows in a river. As such, the legal and governance aspects of environmental flow delivery are likely to be important influences on the outcomes achieved.

Editor Z.W. Kundzewicz; Guest editor M. Acreman

Citation Banks, S.A. and Docker, B.B., 2014. Delivering environmental flows in the Murray-Darling Basin (Australia)—legal and governance aspects. Hydrological Sciences Journal, 59 (3–4), 688–699.     

An index to assess hydromorphological quality of Estonian surface waters based on macroinvertebrate taxonomic composition

We developed an index (MESH – Macroinvertebrates in Estonia: Score of Hydromorphology) to assess hydromorphological quality of Estonian surface waters based on macroinvertebrate taxonomic composition. The MESH is an average score based on the affinities of selected indicator taxa to flow velocity and bottom type. As both parameters were highly correlated (r = 0.65) indicator response to both parameters were combined. The list of MESH indicators includes 394 freshwater macroinvertebrate taxa derived from 3282 samples collected from rivers and lakes during 1985–2009. The indicators were selected out of 690 taxa, by applying the information-theoretical Kullback–Leibler divergence. The individual scores of macroinvertebrates range from 0 to 3, the higher scores indicating faster flow and/or solid bottom substrate. For standing waters, flow velocity was always considered zero. Among the reference waterbodies, mean MESH was the highest for small streams followed by middle streams, large streams, and lakes. In lakes with medium water hardness (the prevailing type in Estonia), the MESH decreased gradually from stony to muddy bottom. The highest MESH values for standing waters were observed in the stony surf zone of very large lakes (area > 100 km²). The lowest values occurred for small lakes with exceptional hydrochemical characteristics (soft- and darkwater, and calcareous types). Similarly, MESH indicated stream degradation by damming. Mean MESH in reservoirs with a muddy bottom was significantly lower than in reservoirs with a hard bottom, or in unregulated stream sections.     

A numerical study of detached shear layers in a rotating sliced cylinder

Abstract

The low Rossby number flow in a rotating cylinder with an inclined bottom, of small slope, is examined when part of the lid of the container is rotating at a slightly different rate. The resulting flow is calculated numerically by solving the governing equations for the two-dimensional geostrophic motion which approximates the flow in most of the fluid including the inertially-modified E ^¼ -layers. The presence of ageostrophic regions, on the container walls and beneath the velocity discontinuity on the lid, is accounted for in the governing equations and their boundary conditions. This study supplements previous work on this configuration, in which the zero Rossby number flow was calculated and experimental results were presented, by enabling a direct comparison to be made between the results of the low Rossby number theory and the experiments. The numerical results for a range of Rossby and Ekman numbers compare well with those from the experiments despite a severe limitation on the size of the Rossby number arising from the analysis in the ageostrophic part of the detached shear layer.     

The equatorial dynamics of a deep homogeneous ocean

Abstract

The flow properties of an homogeneous fluid which is bounded by two concentric spheres and two meridional planes which intersect along a diameter of the spheres are investigated. The spheres rotate about this diameter with slightly different angular velocities. As in the axisymmetric case studied by Proudman (1956) and Stewartson (1966) the viscous terms in the equations of motion are important only in boundary layers on the spheres and on the cylinder C which circumscribes the inner sphere and which has generators parallel to the axis of rotation, provided the Ekman number E is small. In the inviscid region the velocities are independent of the coordinate measuring distance along the axis of rotation and are much weaker, by a factor 0(E ^½), than the velocities in the Ekman layer on the driving surface (outer sphere). (It is assumed that the reference frame is fixed in the slower rotating inner sphere.) If the separation of the spheres is small compared to their radii then the asymmetric circulation inside C is characterized by an intense jet along the western wall. Loss of fluid from this jet sustains the eastward and northward flow in the inviscid interior where motion is driven by the suction of the Ekman layer on the outer sphere. (Geophysical conventions have been adopted.) Outside C an intense current is present on the eastern, not western, wall while motion in the inviscid region is westward, and away from the axis of rotation. Though there is no transport across C in the inviscid region, the meridional transport of the Ekman layer on the outer sphere is continuous across C and increases, through suction, as the equator is approached until it drains into an eastward flowing equatorial current of width 0(E ^1/7). The eastern boundary current outside C and shear layers on C carry this fluid to the intersection of C and the western wall where it feeds the western boundary current inside C.

The relation between this study and the experiments of Baker and Robinson (1970) is discussed.     

Internal hydraulic control in rotating fluids—applications to oceans

Abstract

Laboratory experiments and analysis of shallow water equations in a rotating fluid show that channel flow is governed by the ratio of the width of the channel to the Rossby radius of deformation R= √[g&Delta;ρH/ρf ²]. Flows through narrow ocean openings exhibit blocking and clear evidence of hydraulic control. These imply that formulae can be derived for width, volume flux, and velocity scales of the currents. A new version of the constant potential vorticity problem is solved, and it is shown to predict volume flux within 22% of the zero potential vorticity results. Next a systematic method of predicting volume flux through ocean passages is described. Some examples are given from the Denmark Straits overflow and the flow of Antarctic Bottom Water into the western Atlantic Ocean. Two-layer flows and counter-flows with rotation in a narrow passage, the so-called lock exchange flow problem, duplicate flows at a number of important straits and openings to bays. A potential vorticity formulation is reviewed. The flows in the mouths of various bays such as Funka Bay in Hokkaido, Japan, Spencer Gulf in South Australia, and Chesapeake Bay in the United States has R < width of the mouth, and the two currents are separated by a front. The width of the front and the density difference can be predicted with good results.     

Magnetic Fluctuations for Small Magnetic Prandtl Numbers

Numerous studies of magnetic fluctuations with a zero mean-field for small magnetic Prandtl numbers (Pr _m 1) show that magnetic fluctuations cannot be generated by turbulent fluid flow with the Kolmogorov energy spectrum. In addition, the generation of magnetic fluctuations with a zero mean-field for Pr _m 1 were not observed in numerical simulations. However, in astrophysical plasmas the magnetic Prandtl numbers are small and magnetic fluctuations are observed. Thus a mechanism of generation of magnetic fluctuations for Pr _m 1 still remains poorly understood. On the other hand, in astrophysical applications (e.g., solar and stellar convection zones, galaxies, accretion disks) the turbulent velocity field cannot be considered as a divergence-free. The generation of magnetic fluctuations by turbulent flow of conducting fluid with a zero mean magnetic field for Pr _m 1 is studied by means of linear and nonlinear analysis. The turbulent fluid velocity field is assumed to be homogeneous and isotropic with a power law energy spectrum ( k ^–p ) and with a very short scale-dependent correlation time. It is found that magnetic fluctuations can be generated when the exponent p > 3/2. It is shown also that the growth rates of the higher moments of the magnetic field are larger than those of the lower moments, i.e., the spatial distribution of the magnetic field is intermittent. In addition, the effect of compressibility (i.e., u 0) of the low-Mach-number turbulent fluid flow u is studied. It is demonstrated that the threshold for the generation of magnetic fluctuations by turbulent fluid flow with u 0 is higher than that for incompressible fluid. This implies that the compressibility impairs the generation of magnetic fluctuations. Nonlinear effects result in saturation of growth of the magnetic fluctuations. Asymptotic properties of the steady state solution for the second moment of the magnetic field in the case of the Hall nonlinearity for the low-Mach-number compressible flow are studied.     

Nonlinear dynamics of a stellar dynamo in a spherical shell

Abstract

A simple mean-field model of a nonlinear stellar dynamo is considered, in which dynamo action is supposed to occur in a spherical shell, and where the only nonlinearity retained is the influence of the Lorentz forces on the zonal flow field. The equations are simplified by truncating in the radial direction, while full latitudinal dependence is retained. The resulting nonlinear p.d.e.'s in latitude and time are solved numerically, and it is found that while regular dynamo wave type solutions are stable when the dynamo number D is sufficiently close to its critical value, there is a wide variety of stable solutions at larger values of D. Furthermore, two different types of dynamo can coexist at the same parameter values. Implications for fields in late-type stars are discussed.     

On the growth of disturbances to forced and dissipated barotropic flows

Abstract

We consider the growth of disturbances to large-scale zonally-asymmetric steady states in a truncated spectral model for forced and dissipated barotropic flow. A variant of the energy method is developed to optimize the instantaneous disturbance energy growth rate. The method involves solving a matrix eigenvalue problem amenable to standard numerical techniques. Two applications are discussed. (1) The global stability of a family of steady states is assessed in terms of the Ekman damping coefficient r. It is shown that monotonic global stability (i.e., every disturbances energy monotonically decays to zero) prevails when r≥r_c . (2) Initially fastest-growing disturbances are constructed in the r<r_c regime. Particular attention is paid to a subregion of the r<r_c regime where initially-growing disturbances exist despite stability with respect to normal modes. Nonlinear time-dependent simulations are performed in order to appraise the time evolution of various disturbances.     

Groundwater composition and recharge origin in the shallow aquifer of the Djerid oases,southern Tunisia: implications of return flow

Abstract

Major ions and stable isotopes in groundwaters of the Plio-Quaternary shallow aquifer of the Djerid oases, southern Tunisia, were investigated to elucidate the origin of groundwater recharge and the mineralization processes. It has been demonstrated that the groundwater composition is mainly controlled by the water–rock interaction, the encroachment of brines from the Chotts and the return flow of irrigation waters. The isotopically depleted groundwater samples suggest that the recharge waters derive from an old palaeoclimatic origin. However, the enriched groundwater samples reflect the presence of evaporated recharge water. Furthermore, the large negative deuterium-excess values indicate the effect of secondary evaporation processes, probably related to the return flow of irrigation waters pumped from the underlying aquifer.

Editor D. Koutsoyiannis; Associate editor E. Custodio

Citation Tarki, M., Dassi, L. and Jedoui, Y., 2012. Groundwater composition and recharge origin in the shallow aquifer of the Djerid oases, southern Tunisia: implications of return flow. Hydrological Sciences Journal, 57 (4), 790–804.     

Upon boundary conditions imposed by a stratified fluid

Abstract

It is shown that in the limit of slow steady linearized flow and infinite stratification a stratified region lying above or below a layer of destabilized fluid produces zero velocity and zero stress boundary conditions. These novel boundary conditions constrain flow more fully than the more common rigid or free conditions, and arise because the penetrative flow in the stabilized region must pump only a finite amount of heat.     

ACCURACY OF KINEMATIC WAVE AND DIFFUSION WAVE APPROXIMATIONS FOR TIME-INDEPENDENT FLOW WITH MOMENTUM EXCHANGE INCLUDED

Errors in the kinematic wave and diffusion wave approximation for time-independent (or steady-state) cases of channel flow with momentum exchange included were derived for three types of boundary conditions: zero flow at the upstream end, and critical flow depth and zero depth gradient at the downstream end. The diffusion wave approximation was found to be in excellent agreement with the dynamic wave approximation, with errors of less than 1% for KF²₀≥7·5 and up to 12% for KF²₀≤0·75 for the upstream boundary condition of zero discharge and finite depth, where K is the kinematic wave number and F₀ is the Froude number. The kinematic wave approximation was reasonably accurate except at the channel boundaries and for small values of KF²₀ (≤1). The accuracy of these approximations was significantly influenced by the downstream boundary condition both in terms of the error magnitude and the segment of the channel reach for which these approximations would be applicable. © 1997 by John Wiley & Sons, Ltd.     

Non-linear hydrodynamic stability of oceanic flows

Abstract

In this paper an analytical method to study the hydrodynamic stability of simple barotropic, non-divergent flows is discussed. The method is based on the variational approach introduced by Arnold and derived from the Lyapunov stability criteria. In this context, the sufficient condition for the stability of a steady barotropic flow ψ(x,y) is obtained when dP(ψ)/dP_{ψ = ψ}, the derivative of the absolute vorticity P(ψ), is positive definite. In this case, we discuss the effect of higher derivatives dⁿP(ψ)/dψⁿψ_{ψ = ψ} on the non-linear stability. Then we show that some classical examples of oceanic non-divergent flows (i.e. lee waves downstream an Island, steady flows through a Strait, the Fofonoff gyre) are stable to finite-amplitude perturbations.     

The instability of barotropic circular vortices

Abstract

The linear, normal mode instability of barotropic circular vortices with zero circulation is examined in the f-plane quasigeostrophic equations. Equivalents of Rayleigh's and Fjortoft's criteria and the semicircle theorem for parallel shear flow are given, and the energy equation shows the instability to be barotropic. A new result is that the fastest growing perturbation is often an internal instability, having a finite vertical scale, but may also be an external instability, having no vertical structure. For parallel shear flow the fastest growing perturbation is always an external instability; this is Squire's theorem. Whether the fastest growing perturbation is internal or external depends upon the profile: for mean flow streamfunction profiles which monotonically decrease with radius, the instability is internal for less steep profiles with a broad velocity extremum and external for steep profiles with a narrow velocity extremum. Finite amplitude, numerical model calculations show that this linear instability analysis is not valid very far into the finite amplitude range, and that a barotropic vortex, whose fastest growing perturbation is internal, is vertically fragmented by the instability.