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.     

Richardson number profiles in laboratory experiments applied to shallow seas

Abstract

A unified analysis is given of the critical conditions for the onset of stratification due to either a vertical or a horizontal buoyancy flux, with tidal or wind stirring.

The critical conditions for the onset of stratification with a horizontal buoyancy flux are found to be of the form of ratios of the tidal slope, or wind setup, to the equivalent surface slope due to the lateral density gradient. These ratios, which are easily determined from sea data, indicate that the profiles of critical flux Richardson Number, averaged over the stirring cycle, are similar to those inferred from the laboratory experiments of Hopfinger and Linden (1982) in which there is zero mean shear turbulence with a stabilising buoyancy flux, and also that the efficiency for the conversion of kinetic energy to potential energy for tidal stirring is similar to that for wind stirring.

The observed much greater efficiency for wind stirring, compared with tidal stirring with a vertical buoyancy flux, is also consistent with the existence of flux Richardson Number profiles in the sea similar to those occurring in the corresponding laboratory experiments. Using the solution of the turbulent kinetic energy equation for the water column, the relative importance of the production of turbulent kinetic energy, and its diffusion by turbulence are assessed, and the critical conditions for the onset of stratification with a vertical buoyancy flux are shown to reduce the classical Simpson—Hunter form.     

Steady flow past an island with applications to Bermuda

Abstract

Temperature and salinity data from the vicinity of Bermuda reveal large vertical displacements of the isopycnals of over 100 m close to the island. A model based on the steady flow of an inviscid, stratified ocean past a circularily symmetric island on a rotating plane gives good qualitative agreement. The effects of island slope and nonlinearities are accounted for in a perturbation procedure.

In an anomalous area over the left slope of the island (looking downstream) large steps were observed in the temperature and salinity profiles. The theoretical flow is shown to have a minimum Richardson number in this region. In a quasi-empirical manner it is possible to compute a Richardson number profile from the observed density data. This procedure gives values very close to that needed for instability to be possible suggesting that instabilities promote mixing and the development of the observed layers.     

Robust flood statistics: comparison of peak over threshold approaches based on monthly maxima and TL-moments

ABSTRACT

Flood quantile estimation based on partial duration series (peak over threshold, POT) represents a noteworthy alternative to the classical annual maximum approach since it enlarges the available information spectrum. Here the POT approach is discussed with reference to its benefits in increasing the robustness of flood quantile estimations. The classical POT approach is based on a Poisson distribution for the annual number of exceedences, although this can be questionable in some cases. Therefore, the Poisson distribution is compared with two other distributions (binomial and Gumbel-Schelling). The results show that only rarely is there a difference from the Poisson distribution. In the second part we investigate the robustness of flood quantiles derived from different approaches in the sense of their temporal stability against the occurrence of extreme events. Besides the classical approach using annual maxima series (AMS) with the generalized extreme value distribution and different parameter estimation methods, two different applications of POT are tested. Both are based on monthly maxima above a threshold, but one also uses trimmed L-moments (TL-moments). It is shown how quantile estimations based on this “robust” POT approach (rPOT) become more robust than AMS-based methods, even in the case of occasional extraordinary extreme events.

Editor M.C. Acreman Associate editor A. Viglione     

INTERNATIONALE GLAZIOLOGISCHE GRÖNLANDEXPEDITION, 1957–1959

Abstract

Equations are derived for vapor transport within a constant flux layer of the atmosphere. The physical model is based on the Reynolds analogy for fully turbulent flow, so that vapor and momentum flux are assumed similar. The shear stress is deduced from analysis of the wind profile by means of equations available for non-neutral conditions. Empirical expressions are proposed using the friction velocity rather than the wind velocity at a given level in order to take account of the stability of the air.     

Stable and unstable barotropic shelf waves in a coastal current

Abstract

We consider the linearized stability of a barotropic coastal current flowing parallel to a straight coastline over a continental shelf and slope whose depth varies monotonically with distance from the coast. Some necessary conditions for stability and various semi-circle theorems are reviewed for general current profiles and bottom topography. A criterion for topography to be a destabilizing influence is derived. Some general results for stable waves are also described. Analytic solutions are obtained for a piece-wise linear current profile and the exponential depth profile (Buchwald and Adams, 1968). Dispersion diagrams are obtained for a monotonic current profile, where it is shown that the effect of topography is destabilizing, and for a triangular current profile. The dispersion diagrams generally contain a finite number (usually one or two) of unstable waves, and a set of stable waves, which may be infinite in number. The results are applied to some specific coastal regimes.     

Flooding and drying mechanisms of the seasonal Sudd flood plains along the Bahr el Jebel in southern Sudan

Abstract

The flooding and drying mechanisms of the seasonal flood plains of the Sudd swamps in southern Sudan are, while dependent on the river levels, influenced by a complex interaction between soil, vegetation, topography and seasonal trends in rainfall and evapotranspiration. Based on field measurements, these components have been assessed in detail and evaluated regarding their function in the seasonal cycle of flooding and drying. A detailed analysis of soil and evapotranspiration conditions, as well as the interaction with vegetation and meteorological conditions, has been conducted using field and laboratory experiments. Sources, processes, flow directions and the fate of the floodwaters on both the river-fed seasonal flood plains and the rain-fed grasslands have been established. The results show that river spill is responsible for flooding these areas while no return flow occurs, and drying is caused by evapotranspiration. Rainfall can only cause temporary flooding in extreme events.

Citation Petersen, G. & Fohrer, N. (2010) Flooding and drying mechanisms of the seasonal Sudd flood plains along the Bahr el Jebel in southern Sudan. Hydrol. Sci. J. 55(1), 4–16.     

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.     

Nonlinear stability of baroclinic flows over topography

Abstract

A new nonlinear stability criterion is derived for baroclinic flows over topography in spherical geometry. The stability of a wide class of exact three-dimensional nonlinear steady state solutions subject to arbitrary disturbances is established. The resonance condition, at the highest total wavenumber, for the steady state solutions and the stability criteria for baroclinic flow in the absence of topography provide the boundaries of the regions of stability in the presence of topography. The analogous results for flow on periodic or infinite beta planes incorporating non-orthogonal function large scale flows are also discussed.     

Large-scale vortices in helical turbulence of incompressible fluid

Abstract

The interaction of a mean flow with a random fluctuation field is considered. This interaction is described by the averaged Navier-Stokes equation in which terms nonlinear in the fluctuation field are expressed in terms of the mean flow and the statistical properties of the fluctuation field, which is assumed to be homogeneous, isotropic, and helical. Averaged equations are derived using a functional technique. These equations are solved for a mean background flow that depends linearly on the position vector. The solutions show that large-scale vortices may arise in this system.     

Field theoretical techniques in statistical fluid dynamics: With application to nonlinear wave dynamics

Abstract

A derivation of two-point Markovian closure is presented in classical statistical field theory formalism. It is emphasized that the procedures used in this derivation are equivalent to those employed in the quantum statistical field theory derivation of the Boltzmann equation. Application of these techniques to the study of two-dimensional flow on a β-plane yields a quasi-homogeneous, quasi-stationary transport equation and a renormalized dispersion relation for Rossby waves     

Critical-level crossing of rossby-gravity wave-packets in a linear-shear,stratified flow

Abstract

The propagation of Rossby-gravity waves, which are of astronomical interest near the critical level (or corotation point), is considered in a linear-shear, exponentially stratified flow by means of a ray tracing method. It is shown, using the analytic solutions for the stream function, that for a Richardson number J > 1/4 a wave-packet can cross the critical level in a finite time. In the unstable stratified case (J < 0), it cannot cross the critical level for large |J| (J < 0), but may do so for some intermediate |J| values.

Based on the above results, the possible existence of the regular normal mode with a discrete point eigenvalue in the continuous spectrum is discussed for bounded systems.     

Solitary Rossby Waves in Zonally Varying Jet Flows

The dynamics of solitary Rossby waves (SRWs) embedded in a meridionally sheared, zonally varying background flow are examined using a non-divergent barotropic model centered on a midlatitude g -plane. The zonally varying background flow, which is produced by an external potential vorticity (PV) forcing, yields a modified Korteweg-de Vries (K-dV) equation that governs the spatial-temporal evolution of a disturbance field that contains both Rossby wave packets and SRWs. The modified K-dV equation differs from the classical equation in that the zonally varying background flow, which varies on the same scale as the disturbance field, directly affects the disturbance linear translation speed and linear growth characteristics. In the limit of a locally parallel background flow, equations governing the amplitude and propagation characteristics of SRWs are derived analytically. These equations show, for example, that a sufficiently large (small) translation speed and/or a sufficiently weak (strong) background zonal shear favor transmission (reflection) of the SRW through (from) the jet. Conservation equations are derived showing that time changes in the domain averaged amplitude ("mass") or squared amplitude ("momentum") are due to zonal variation in both the linear, long-wave phase speed and linear growth; dispersion and nonlinearity do not affect the "mass" or "momentum". Provided (1) the background PV forcing is sufficiently small, or (2) the background PV forcing is meridionally symmetric and the disturbance is a SRW, the dynamics of the disturbance field is Hamiltonian and mass and energy are thus conserved. Numerical solutions of the K-dV equation show that the zonally varying background flow yields three general classes of behavior: reflection, transmission, or trapping. Within each class there exists SRWs and Rossby wave packets. SRWs that become trapped within the zonally localized jet region may exhibit the following behaviors: (1) an oscillatory decay to a steady state at the jet center, (2) the creation of additional SRWs within the jet region, or (3) a steady-state wherein the solution has a smoothed step-like structure located downstream along the jet axis.     

Note on the flow of a stratified fluid over a stationary obstacle in a channel

Abstract

A new method is introduced to produce a uniform stratified flow over a stationary obstacle in an open channel. The flow is achieved by discharging the flow from the channel through a sink. The details of the sink are unimportant. The flow speed is limited only by the sink capacity. Selective withdrawal at lower densimetric Froude numbers is effectively eliminated through the use of a contraction. The standing, free-surface, long wave arising from the initiation of the flow is also eliminated by the contraction. Experiments are conducted for flow over a sphere for a range of Reynolds numbers from O(10²) to O(10³) and a range of Richardson numbers from O(10^?1) to O(10). Dye and neutrally buoyant droplets are used for quantitative analysis of the wake structure. The wake is also probed by a hot-film anemometer. The frequency of vortex shedding is obtained. Comparison with data from towed experiments is also presented.     

The symmetric baroclinic instability of an equatorial current

Abstract

The stability of a baroclinic zonal current to symmetric perturbations on an equatorial β-plane is considered. The fluid is assumed to be Boussinesq, inviseid, adiabatic, hydrostatic, and stably stratified. The solutions exhibit the same stability properties as those on an f-plane: instability occurs whenever Ri < 1/(1 + d), where Ri is the Richardson number and d is a measure of the horizontal shear of the current; the most unstable motions tend to parallel the isotherms of potential temperature; and they have infinitely small scales of variation perpendicular to the isotherms. The variation of Coriolis parameter leads to one important difference in the structure of the eigenfunctions: the rapidly growing modes are concentrated in high latitudes, and the slowly growing ones in low latitudes.

The suggestion that the symmetric cloud bands observed at low latitudes in Jupiter's atmosphere are caused by symmetric instabilities is re-examined in the light of these results. These cloud bands would have to be associated with the slowly-growing, low-latitude modes. These modes consist of small scale motions parallel to the isotherms, with the magnitude of the motions having a large scale modulation as a function of latitude. The time scales of these modes and the latitude scales of their modulation agree qualitatively with the observations of Jupiter's cloud bands, so long as Ri is not very close to zero or to its critical value.     

Hydrologic indicators of hydraulic conditions that drive flow–biota relationships

Abstract

This article tests the association between streamflow alteration and the alteration of ecologically significant hydraulic environments. There has been a recent shift in environmental flow assessments to develop rapid desktop-based approaches that are applicable in a regional context. Streamflow statistics (e.g. minimum monthly flow) are often chosen to predict the impact of streamflow alteration on aquatic ecosystems. The assumption that the flow–biota relationship will be obscured by the effect of how streamflow interacts with channel morphology is often acknowledged, but not quantified. In this study, streamflow statistics are derived for 19 reaches in four river systems in Victoria, Australia. Hydraulic metrics were used to quantify ecologically significant surface flow conditions (Froude number) and the area of bench inundation, shallow and deep water. Multivariate analysis was used to investigate the correlation between streamflow statistics altered with regulation and the hydraulic metrics. It was found that streamflow statistics have a weak correlation to surface flow condition and the area of shallow water under natural streamflow conditions. The results show that hydrologic statistics have limited utility in quantifying changes in hydraulic environments. A similar magnitude of flow alteration can produce diverse hydraulic results. The confounding influence of channel morphology prevents streamflow statistics being an adequate surrogate for the assessment of hydraulic alteration. Modelling flow–biota relationships in a regional context is limited by the inadequacy of streamflow statistics to model ecologically significant hydraulic function. Improving knowledge of ecohydraulically significant hydrologic statistics will improve the effectiveness of environmental flow planning to sustain instream habitat conditions. A probabilistic approach is required to enable a risk-based approach to desktop generalization of flow–biota relations.

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

Citation Turner, M. and Stewardson, M., 2014. Hydrologic indicators of hydraulic conditions that drive flow–biota relationships. Hydrological Sciences Journal, 59 (3–4), 659–672.     

Diffusive destabilisation of the baroclinic circular vortex

Abstract

It is shown that, even for vanishingly small diffusivities of momentum and heat, a rotating stratified zonal shear flow is more unstable to zonally symmetric disturbances than would be indicated by the classical inviscid adiabatic criterion, unless σ, the Prandtl number, = 1. Both monotonic instability, and growing oscillations ("overstability") are involved, the former determining the stability criterion and having the higher growth rates. The more σ differs from 1, the larger the region in parameter space for which the flow is stable by the classical criterion, but actually unstable.

If the baroclinity is sufficiently great for the classical criterion also to indicate instability, the corresponding inviscid adiabatic modes usually have the numerically highest growth rates. An exception is the case of small isotherm slope and small σ.

A single normal mode of the linearized theory is also, formally, a finite amplitude solution; however, no theoretical attempt is made to assess the effect of finite amplitude in general. But, in a following paper, viscous overturning (the mechanism giving rise to the sub‐classical monotonic instability when σ > 1) is shown to play an important role at finite amplitude in certain examples of nonlinear steady thermally‐driven axisymmetric flow of water in a rotating annulus. Irrespective of whether analogous mechanisms turn out to be identifiable and important in large‐scale nature, it appears then that a Prandtl‐type parameter should enter the discussion of any attempt to make laboratory or numerical models of zonally‐symmetric baroclinic geophysical or astrophysical flows.     

A heat pulse technique for the determination of small‐scale flow directions and flow velocities in the streambed of sand‐bed streams

The hyporheic zone (HZ) has the capability to eliminate and attenuate nutrients and contaminants in riverine systems. Biogeochemical reactions and the potential elimination of contaminants are strongly controlled by the flow paths and dynamics in the HZ. Nevertheless, an easily applicable method for the field determination of flow patterns in the HZ is still lacking. Therefore, a heat pulse technique, which traces the movement of a short heat pulse in the upper part of the HZ and other sand beds, was developed. Five rods are vertically driven into the sediment of the streambed; one rod with a heater as point source located in about 10‐cm sediment depth and four rods with four temperature sensors in 3 cm distance, arranged concentrically with 7 cm diameter around the heating rod. Subsequently, a heat pulse is applied and the resulting breakthrough curves are indicative of flow velocities and flow directions in the streambed. A rough data analysis procedure is also suggested. In addition, laboratory experiments were performed to test the heat pulse technique. These experiments were validated based on coupled numerical modelling of flow and heat transport. First field tests of the method prove that the method is easily applicable under field conditions. These first field tests showed highly complex flow patterns with flow velocities from 1·8 to 4·9 cm min^?1 and flow directions from parallel to surface flow to opposite to surface flow. This suggests the need for a robust method to quantify hyporheic flow patterns in situ. Copyright © 2011 John Wiley & Sons, Ltd.     

Mixing efficiency in stably-stratified decaying turbulence

Abstract

The behavior of the flux Richardson number R _f, as a function of the overall Richardson number Ri ₀, was investigated for a stably stratified, grid-generated, turbulent flow evolving in a closed-loop water channel. The turbulent dissipation rate ε, the buoyancy or vertical mass flux p wbar; and the rms density fluctuation ρ′ were obtained from simultaneous single-point measurements of the horizontal and vertical velocity components and density fluctuations. From these, R _f and Ri ₀ were calculated at each point in the spatially evolving flow. The resulting curves of R _f vs. Ri ₀ exhibit the full range of behavior found in the very different case studied by Linden (1980). The length scale arguments of Gibson (1980) and Stillinger et al. (1983b) provide an underlying mechanism which successfully accounts for the shape of the R _f vs. Ri ₀ curve.     

A local stability analysis of astronomical disks

Abstract

A fifth-order dispersion relation describing the local stability of a differentially rotating flow against small perturbations is derived. Finite viscosity and conductivity and both vertical (parallel to the rotation axis) and radial gradients in density, temperature and pressure are included. A general form is assumed for the equation of state, although this is not exploited in the paper. A number of special cases are studied: with negligible viscosity and conductivity, it is shown that modes can often be separated into two high frequency (modified acoustic), two intermediate frequency (combined inertial and internal waves) and a low frequency mode. In convectively unstable situations the intermediate frequency modes may be replaced by a damped/growing pair of instablities. Various criteria for mode excitation are given. It is shown that viscosity always inhibits instability at very short wavelengths, while non-zero conductivity may destabilize the flow. At intermediate wavelengths viscosity could also play a destabilizing role. A parameter study of the effects of fluctuations in the conductivity shows that it could cause mode excitation under certain circumstances.