Energy spectra associated with long's model for steady finite-amplitude flow over orography

Abstract

The process of wave steepening in Long's model of steady, two-dimensional stably stratified flow over orography is examined. Under conditions of the long-wave approximation, and constant values of the background static stability and basic flow, Long's equation is cast into the form of a nonlinear advection equation. Spectral properties of this latter equation, which could be useful for the interpretation of data analyses under mountain wave conditions, are presented. The principal features, that apply at the onset of convective instability (density constant with height), are:

i) a power spectrum for available potential energy that exhibits a minus eight-thirds decay, in terms of the vertical wavenumber k _z ^-;

ii) a rate of energy transfer across the spectrum that is inversely proportional to the wavenumber for large k _z ^-;

iii) an equipartition between the kinetic energy of the horizontal motion and the available potential energy, under the longwave approximation, although all the disturbance energy is kinetic at the point where convective instability is initiated. It is also shown that features i) and ii) apply to more general conditions that are appropriate to Long's model, not just the long-wave approximation. Application to fully turbulent flow or to conditions at the onset of shearing instability are not considered to be warranted, since the development only applies to conditions at the onset of convective instability.     

Understanding flow dynamics and density currents in a river-reservoir system under upstream reservoir releases

ABSTRACT

A three-dimensional flow and temperature model was applied for a 124 km river-reservoir system from Lewis Smith Dam tailrace to Bankhead Lock & Dam, Alabama. The model was calibrated against measured water levels, temperatures, velocities and flow rates from 4 May to 3 September 2011 under small constant release (2.83 m³/s) and large intermittent releases (~140 m³/s) from an upstream reservoir. Distributions of simulated flow and temperatures and particle tracking at various locations were analyzed which revealed the complex interactions of density currents, dynamic surface waves and solar heating. Flows in the surface and bottom layers moved in both upstream and downstream directions. If there was small constant release only from Smith Dam, simulated bottom temperatures at Cordova were on average 4.8°C higher than temperatures under actual releases. The momentum generated from large releases pushed bottom density currents downstream, but the released water took several days to reach Cordova.

Editor D. Koutsoyiannis; Associate editor B. Dewals     

Hydrological characteristics of vegetated river flows: a laboratory flume study

Abstract

Laboratory flume experiments were undertaken to measure the vertical profiles of mean flow velocity for three different flow discharges and four different stem densities of Hydrilla verticillata. The data were used to calculate three parameters, namely Manning's roughness coefficient, the Reynolds number and the Froude number. In addition, empirical equations were obtained for the vertical distribution of measured flow velocity within the transitional zone and above the plant canopy. The results show that: (a) the vertical distribution of measured flow velocity exhibits three zone profiles; (b) Manning's roughness coefficient decreases with increasing depth-averaged flow velocity; (c) the relationship between Manning's roughness coefficient and the depth-averaged flow velocity is within the smooth left inverse curve; (d) Manning's roughness coefficient significantly changes with increasing density of Hydrilla; (e) the Froude number is independent of the density of Hydrilla; and (f) both the Reynolds number and the Froude number increase with increasing depth-averaged flow velocity.

Citation Shi, J.Z., Li, Y.-H., Hughes, J.M.R., and Zhao, M., 2013. Hydrological characteristics of vegetated river flows: a laboratory flume study. Hydrological Sciences Journal, 58 (5), 1047–1058.

Editor Z.W. Kundzewicz     

Three-dimensional primary instabilities of a stratified,dissipative, parallel flow

Abstract

We demonstrate the existence of a class of dissipative, stratified, parallel shear flows which, as a consequence of linear supercritical instability, evolve directly into three-dimensional flows without the requirement for an intermediate two-dimensional finite-amplitude state. This represents a counter-example to a common misinterpretation of Squire's theorm, namely that the fastest-growing unstable mode of a dissipative parallel shear flow must be two-dimensional.     

Rotating channel flow: Control and upstream currents

Abstract

Theory and experiments are presented for critically controlled flow of a layer of inviscid rotating fluid. Flow is controlled by a level passage. For a wide upstream channel of fixed depth (i.e. constant potential vorticity) the volume flux on the right-hand wall is unaffected by passage flow. This suggests that specifying Bernoulli potential on the right-hand passage wall produces a physically well-posed condition. The specification results in one less dimensionless number than was required by previous formulations to specify flow in the controlled passage. The upstream flow needs the same number as before, so that a range of upstream conditions produce exactly the same passage flow. A laboratory study is conducted using a thin layer of water under air. This is pumped in steadily at various locations in a deep rotating upstream basin, with fluid leaving through a level passage. All currents in the upstream basin cross to the left-hand wall as the current approaches the passage over a sloping bottom. The current crosses back to the right-hand wall within the passage. Velocity profiles of currents agree reasonably well with constant potential vorticity theory. To the right of the detached upstream current is a closed gyre that connects the upstream flows (that have different patterns depending on source location) with the unique passage flows. The results suggest that gyres upstream of critically controlling passages in the ocean might serve as adjustment regions between the relatively unconstrained upstream flows and the tightly controlled passage flows.     

Taylor columns in horizontally sheared flow

Abstract

Solutions of the steady, inviscid, non-linear equations for the conservation of potential vorticity are presented for linearly sheared geostrophic flow over a right circular cylinder. The indeterminancy introduced by the presence of closed streamline regions is removed by requiring that the steady flow retains above topography a given fraction of that fluid initially present there, assuming the flow to have been started from rest. Those solutions which retain the largest fraction in uniform and negatively sheared streams satisfy the Ingersoll (1969) criterion (that, in the limit of vanishingly small viscosity, closed streamline regions are stagnant) and so are unaffected by Ekman pumping. These flows are set up on the advection time scale. In positively sheared flows the maximum retention solutions do not satisfy the Ingersoll criterion and thus would be slowly spun down on the far longer viscous spin-up time.

For arbitrary isolated topography, both the partial retention and Ingersoll problems are reduced to a one-dimensional non-linear integral equation and the solution of the Ingersoll problem obtained in the limit of strong positive shear. The stagnant region is symmetric about the zero velocity line and extends to infinity in the streamwise direction. Its cross-stream width is proportional to the rotation rate and fractional height occupied by the obstacle and inversely proportional to the strength of the shear, decreasing inversely as the square of distance upstream and downstream.     

Wave patterns generated by disturbances travelling horizontally in rotating stratified fluids

The pattern and propagation of waves generated by steady or oscillatory disturbances travelling horizontally in a rotating, stratified fluid are studied following a technique developed by Lighthill. Both two‐ and three‐dimensional distrubances are investigated. The results show how rotation modifies internal wave patterns in a stratified fluid and how stratification modifies inertial wave patterns in a rotating fluid. The results are used to compute the effective diminution of Taylor column length due to the presence of density stratification. They also show that the appearance of wave crests upstream of a disturbance is possible only when the disturbance is unsteady and that observations of upstream blocking in a two‐dimensional stratified flow can be explained by the existence of a certain class of plane waves as modified by viscosity.     

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.     

A large class of non-zonal oceanic flows satisfying the arnold-blumen sufficient condition for stability

Abstract

Recently Andrews has discussed an example of a topographically-forced non-zonal now satisfying the Arnold-Blumen sufficient condition for stability. At large distances from the topographic centre this flow becomes purely zonal and westward. After underlining the richness of solutions of the Andrews model, the present paper goes on to show that Andrews' technique can be applied successfully to a model where the vertical profile of static stability resembles those found in the ocean. In particular we obtain a large class of hydrodynamic stable flows, forced by the bottom topography, for continuously stratified fluids (two layers each with uniform Brunt-Väisälä frequency).     

Modelling the influence of hydrogeological parameters on low flow in hilly and mountainous regions of Serbia / Modélisation de l'influence de paramètres hydrogéologiques sur les basses eaux dans des régions de collines et de montagnes de Serbie

Abstract

During dry weather periods of the year with long rainless intervals, streams slacken to what is generally termed “low flow”. This work presents an analysis of the influence of hydrogeology on low flows, using multiple linear regression, in natural medium and small streams in hilly and mountainous regions of Serbia. The study cases encompass 61 gauged catchments south of the rivers Sava and Danube. Characteristic relevant minimum mean 30-day flows of 80- or 95-percentile exceedence (Q _80%, Q _95%) are taken as dependent variables. Independent variables are the observable hydrogeological quantities: catchment area upstream of a gauging station; surface area of a hydrogeological soil category in a catchment; number of perennial springs of minimum flow higher than or equal to 1 L/s in a gauged catchment; number of perennial springs, each of minimum flow higher than or equal to 1 L/s, in a given hydrogeological soil category of the catchment; cumulative perennial spring flow of minimum single flow higher than or equal to 1 L/s in a catchment; and cumulative perennial spring flow of minimum single flow higher than or equal to 1 L/s in a hydrogeological soil category of a catchment. Through multiple linear regression between the characteristic relevant low flow and the hydrogeological elements, 16 models are developed and analysed, each based on a different combination of hydrogeological elements and characteristic low flow. The regional relationships developed for the minimum mean 30-day flows of 80- and 95-percentile exceedences are evaluated. The statistical tests of the representation quality of each multiple regression relationship show that the models justify the use of hydrogeological elements.     

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.     

A Correction on Coastal Heads for Groundwater Flow Models

We introduce a simple correction to coastal heads for constant‐density groundwater flow models that contain a coastal boundary, based on previous analytical solutions for interface flow. The results demonstrate that accurate discharge to the sea in confined aquifers can be obtained by direct application of Darcy's law (for constant‐density flow) if the coastal heads are corrected to ((α + 1)/α)h_s ? B/2α, in which h_s is the mean sea level above the aquifer base, B is the aquifer thickness, and α is the density factor. For unconfined aquifers, the coastal head should be assigned the value . The accuracy of using these corrections is demonstrated by consistency between constant‐density Darcy's solution and variable‐density flow numerical simulations. The errors introduced by adopting two previous approaches (i.e., no correction and using the equivalent fresh water head at the middle position of the aquifer to represent the hydraulic head at the coastal boundary) are evaluated. Sensitivity analysis shows that errors in discharge to the sea could be larger than 100% for typical coastal aquifer parameter ranges. The location of observation wells relative to the toe is a key factor controlling the estimation error, as it determines the relative aquifer length of constant‐density flow relative to variable‐density flow. The coastal head correction method introduced in this study facilitates the rapid and accurate estimation of the fresh water flux from a given hydraulic head measurement and allows for an improved representation of the coastal boundary condition in regional constant‐density groundwater flow models.     

Prediction of base flows from basin characteristics: a case study from Zimbabwe / Prévision de débits de base à partir de caractéristiques du bassin: une étude de cas au Zimbabwe

Abstract

Abstract Base flows make up the flows of most rivers in Zimbabwe during the dry season. Prediction of base flows from basin characteristics is necessary for water resources planning of ungauged basins. Linear regression and artificial neural networks were used to predict the base flow index (BFI) from basin characteristics for 52 basins in Zimbabwe. Base flow index was positively related to mean annual precipitation (r = 0.71), basin slope (r = 0.76), and drainage density (r = 0.29), and negatively related to mean annual evapotranspiration (r = –0.74), and proportion of a basin with grasslands and wooded grasslands (r = –0.53). Differences in lithology did not significantly affect BFI. Linear regression and artificial neural networks were both suitable for predicting BFI values. The predicted BFI was used in turn to derive flow duration curves of the 52 basins and with R ² being 0.89–0.99.     

Minimal advective travel time along arbitrary streamlines of porous media flows: The Fermat–Leibnitz–Bernoulli problem revisited

Travel time of marked fluid particles along arbitrary streamlines in arbitrary porous streamtubes is estimated from below based on the Cauchy–Bunyakovskii (Schwartz) and Jensen inequalities. In homogeneous media the estimate is strict and expressed through the length of the streamline, hydraulic conductivity, porosity and the head fall. The minimum is attained at streamlines of unidirectional flow. The bounds for heterogeneous soils, non-Darcian flows and unsaturated media are also written. If such bounds are attained the corresponding trajectories become brachistochrones. For example, in a two-layered aquifer and seepage perpendicular to the layers there is a unique conductivity–porosity ratio which makes a broken streamline brachistocronic. Similarly, if conductivities of two layers are fixed there is a unique incident angle between flow in one medium and the interface which makes a refracted streamline brachistocronic.     

Nonlinear internal waves in ideal rotating basins

Abstract

Starting from Euler's equations of motion a nonlinear model for internal waves in fluids is developed by an appropriate scaling and a vertical integration over two layers of different but constant density. The model allows the barotropic and the first baroclinic mode to be calculated. In addition to the nonlinear advective terms dispersion and Coriolis force due to the Earth's rotation are taken into account. The model equations are solved numerically by an implicit finite difference scheme. In this paper we discuss the results for ideal basins: the effects of nonlinear terms, dispersion and Coriolis force, the mechanism of wind forcing, the evolution of Kelvin waves and the corresponding transport of particles and, finally, wave propagation over variable topography. First applications to Lake Constance are shown, but a detailed analysis is deferred to a second paper [Bauer et al. (1994)].     

Flow of a double-diffusive stratified fluid in a differentially-rotating cylinder

Abstract

A study has been made of a basic state of axisymmetric flow, at large rotational Reynolds numbers, in a double-diffusive stratified fluid contained in a vertically-mounted, differentially-rotating cylindrical cavity. The aim is to describe the qualitative characteristics of the flow of a fluid, the density of which is stratified by two diffusive effects, i.e., temperature and salinity gradients. Attention is confined to situations in which the temperature and salinity gradients make opposing contributions to the overall density profile, the undisturbed stratification being gravitationally stable. Finite difference numerical solutions of the governing Navier-Stokes equations have been obtained using the Boussinesq approximation. The results are presented in a way that illustrates the explicit effects of double-diffusivity when the cavity aspect ratio, height/radius, is O(1). The principal non-dimensional parameters characterizing the flow field are identified. In the interior core, the primary dynamic balance is between the horizontal density gradient and the vertical shear of the prevailing azimuthal velocity. The effective stratification is seen to decrease as the double-diffusivity increases, even if the overall stratification parameter, St, is held constant. The solute field contains a very thin boundary layer structure at large Lewis numbers. The effective stratification increases with the Prandtl number. Results have been derived for extreme values of the cavity aspect ratio. For small cavity aspect ratios, the dominant dynamic ingredients are viscous diffusion and rotation. For large aspect ratios, the bulk of the flow field is determined by the rotating sidewall. In this case, the direct influence of the double-diffusivity is minor.     

Riparian responses to reduced flood flows: comparing and contrasting narrowleaf and broadleaf cottonwoods

Abstract

To enable assessment of risks of water management to riparian ecosystems at a regional scale, we developed a quantile-regression model of abundance of broadleaf cottonwoods (Populus deltoides and P. fremontii) as a function of flood flow attenuation. To test whether this model was transferrable to narrowleaf cottonwood (Populus angustifolia), we measured narrowleaf abundance along 39 river reaches in northwestern Colorado, USA. The model performed well for narrowleaf in all 32 reaches where reservoir storage was <75% of mean annual flow. Field data did not fit the model at four of seven reaches where reservoir storage was >90% of mean annual flow. In these four reaches, narrowleaf was abundant despite peak flow attenuation of 45–61%. Poor model performance in these four reaches may be explained in part by a pulse of narrowleaf cottonwood expansion as a response to channel narrowing and in part by differences between narrowleaf and broadleaf cottonwood response to floods and drought.

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

Citation Wilding, T.K., Sanderson, J.S., Merritt, D.M., Rood, S.B., and Poff, N.L., 2014. Riparian responses to reduced flood flows: comparing and contrasting narrowleaf and broadleaf cottonwoods. Hydrological Sciences Journal, 59 (3–4), 605–617.     

Wave-induced topographic formstress in baroclinic channel flow

Large-scale zonal flow driven across submarine topography establishes standing Rossby waves. In the presence of stratification, the wave pattern can be represented by barotropic and baroclinic Rossby waves of mixed planetary topographic nature, which are locked to the topography. In the balance of momentum, the wave pattern manifests itself as topographic formstress. This wave-induced formstress has the net effect of braking the flow and reducing the zonal transport. Locally, it may lead to acceleration, and the parts induced by the barotropic and baroclinic waves may have opposing effects. This flow regime occurs in the circumpolar flow around Antarctica. The different roles that the wave-induced formstress plays in homogeneous and stratified flows through a zonal channel are analyzed with the BARBI (BARotropic-Baroclinic-Interaction ocean model, Olbers and Eden, J Phys Oceanogr 33:2719–2737, 2003) model. It is used in complete form and in a low-order version to clarify the different regimes. It is shown that the barotropic formstress arises by topographic locking due to viscous friction and the baroclinic one due to eddy-induced density advection. For the sinusoidal topography used in this study, the transport obeys a law in which friction and wave-induced formstress act as additive resistances, and windstress, the effect of Ekman pumping on the density stratification, and the buoyancy forcing (diapycnal mixing of the stratified water column) of the potential energy stored in the stratification act as additive forcing functions. The dependence of the resistance on the system parameters (lateral viscosity ε, lateral diffusivity κ of eddy density advection, Rossby radius λ, and topography height δ) as well as the dependence of transport on the forcing functions are determined. While the current intensity in a channel with homogeneous density decreases from the viscous flat bottom case in an inverse quadratic law ~δ ^–2 with increasing topography height and always depends on ε, a stratified system runs into a saturated state in which the transport becomes independent of δ and ε and is determined by the density diffusivity κ rather than the viscosity: κ/λ ² acts as a vertical eddy viscosity, and the transport is λ ²/κ times the applied forcing. Critical values for the topographic heights in these regimes are identified.     

On the transport of moisture in polythermal glaciers

Abstract

We reconsider the problem of formulation of a model for polythermal glaciers, focussing attention in particular on the temperate zone where ice and water can coexist at the melting temperature. The energy equation for the ice-water mixture in this zone introduces a moisture flux, and a constitutive law for this flux is required. By analogy with the flow through a porous medium, we use Darcy's law (i.e. the second momentum equation of a two-phase flow model with “porous” geometry), and then require a mechanical constitutive relation relating the water pressure p _w to the average ice pressure p _i . Experience in two phase flows suggests that p _w =p _i may be problematical, and experience in soil mechanics suggests it is inaccurate. A constitutive relation is therefore presented based on work of Nye (1976), and its effect on the well-posedness of the model is examined. Considerations of the sort presented here have clear relevance in the formulation of similar problems in other geophysical situations, notably mantle convection.     

Reflection of hydrostatic mountain waves from spatially nonuniform layers

Abstract

The steady, hydrostatic, inviscid, Boussinesq flow of a stably stratified fluid over a bell-shaped ridge is investigated within the framework of a linear model. The three layer model atmosphere introduced is such that the Brunt-Väisälä frequency is constant in each layer but the interfaces of the middle layer are allowed to vary gently in the cross-ridge direction. In essence, the model can be tuned in both vertical and horizontal directions. These cross-ridge variations can produce significant differences in both the cross-ridge surface wind and the surface drag compared to the response obtained by use of a horizontally uniform reflecting layer. These changes are sensitive to both the vertical location of the middle layer and to the slope of its lower interface at the ridge crest. Many of these features are explained by means of a conventional layered-model analysis.