Vortex evolution

Abstract

Friedmann's equation and the potential vorticity equation are generalised for turbulent motion. The generalised equations incorporate some new phenomena connected with turbulent transport of mass. It is proved that, if ?×[S×Ω+S(?·S)]≠0 where Ω is the absolute vorticity of the velocity and S is the turbulent density flux, then the Helmholtz-Kelvin theorem concerning the conservation of the velocity circulation around a closed path is violated and the potential vorticity is not a Lagrangian adiabatic invariant. The effects of this turbulent transport of mass on the creation or dissipation of vorticity discussed here is not equivalent to effects of baroclinicity or viscosity. Some possible implications of the new circulation theorem in geophysical and astrophysical fluid dynamics are discussed.     

Spin-up of a two-layer fluid in a rotating cylinder

Abstract

we report the results of experiments on the spin-up of two layers of immiscible fluid with a free upper surface in a rotating cylinder over a wide range of internal Froude numbers. Observations of the evolution of the velocity field by particle tracking indicates that spin-up of the azimuthal velocity in the upper layer take much longer than in a homogeneous fluid. Initially, spin-up occurs at a rate comparable to that of homogeneous fluid but, at high internal Froude number, a second phase follows in which the remaining lative motion decays much more slowly. Quantitative comparison of these measurements to the theory of Pedlosky (1967) shows good agreement.

Visualization of the interface displacement during spin-up detected the presence of transient azimuthal variations in the interface elevation over a wide range of Froude (F), Ekman (E), and Rossby (ε) number. nalysis of the occurrence of the asymmetric variations using the parameter space (Q, F), where Q = E ^1/2/ε, suggested by the baroclinic instability theory and experiments of Hart (1972), showed that the flow was stable for Q > 0.06 with no discernable dependence on F. This result, together with the prediction of Pedlosky's theory that radial gradient of potential vorticity in the two layers have opposite signs, suggests at the baroclinic instability mechanism was responsible for the asymmetries. The location and timing of these instabilities may account for the discrepancies between the observations and the Pedlosky (1967) theory.     

Residual vorticity induced by the action of tidal currents in combination with bottom topography in the Southern Bight of the North Sea

Abstract

Results are presented of calculations on the generation of residual vorticity by tidal currents over the bottom topography of the Southern Bight of the North Sea. A typical order of magnitude is 10^?6 to 10^?7 s ^?1. This is compared with current measurements on calm days, when similar magnitudes are found. At windspeeds less than about 5 m/s tidal generation of residual vorticity is important; at higher windspeeds wind effects begin to dominate. Our results are relevant in understanding the spatial variability of residual currents, because a non-zero vorticity implies the existence of horizontal gradients in the residual current field.     

Sediment-transport competence of rain-impacted interrill overland flow

Laboratory experiments to determine the maximum size of sediment transported in shallow, rain-impacted flow were conducted in a recirculating flume 4·80 m long and 0·50 m wide. Rainfall intensities were varied between 51 and 138 mm h⁻¹, flow was introduced from a header tank into the flume at rates ranging from 0 to 0·64 l s⁻¹, and experiments were conducted on gradients between 3·5 and 10°. The following equation was developed: ML = (REFE)^1·6363 in which M is particle mass, L is distance moved in unit time (cm min⁻¹), RE is rainfall energy (J m⁻² s⁻¹) and FE is flow energy (J m⁻² s⁻¹). This equation can be used to predict sediment-transport competence of interrill overland flow. The equation is limited in its utility insofar as it has been developed using quartz grains and takes no account of variations in absorption of rain energy by natural ground surfaces. © 1998 John Wiley & Sons, Ltd.     

Point-source inertial particle dispersion

The dispersion of inertial particles continuously emitted from a point source is analytically investigated in the limit of small but finite inertia. Our focus is on the evolution equation of the particle joint probability density function p(x,?v,?t), x and v being the particle position and velocity, respectively. For arbitrary inertia, position and velocity variables are coupled, with the result that p(x,?v,?t) can be determined by solving a partial differential equation in a 2d-dimensional space, d being the physical-space dimensionality. For small (but nevertheless finite) inertia, (x,?v)-variables decouple and the determination of p(x,?v,?t) is reduced to solve a system of two standard forced advection–diffusion equations in the space variables x. The latter equations are derived here from first principles, i.e., from the well-known Lagrangian evolution equations for position and particle velocity.     

The destabilising nature of differential rotation

Abstract

In a rapidly rotating, electrically conducting fluid we investigate the thermal stability of the fluid in the presence of an imposed toroidal magnetic field and an imposed toroidal differential rotation. We choose a magnetic field profile that is stable. The familiar role of differential rotation is a stabilising one. We wish to examine the less well known destabilising effect that it can have. In a plane layer model (for which we are restricted to Roberts number q = 0) with differential rotation, U = sΩ(z)1 _?, no choice of Ω(z) led to a destabilising effect. However, in a cylindrical geometry (for which our model permits all values of q) we found that differential rotations U = sΩ(s)1 _? which include a substantial proportion of negative gradient (dΩ/ds ≤ 0) give a destabilising effect which is largest when the magnetic Reynolds number R _m = O(10); the critical Rayleigh number, Ra _c, is about 7% smaller at minimum than at R_m = 0 for q = 10⁶. We also find that as q is reduced, the destabilising effect is diminished and at q = 10^?6, which may be more appropriate to the Earth's core, the effect causes a dip in the critical Rayleigh number of only about 0.001%. This suggests that we see no dip in the plane layer results because of the q = 0 condition. In the above results, the Elsasser number A = 1 but the effect of differential rotation is also dependent on A. Earlier work has shown a smooth transition from thermal to differential rotation driven instability at high A [A = O(100)]. We find, at intermediate A [A = O(10)], a dip in the Ra_c vs. R_m curve similar to the A = 1 case. However, it has Ra_c ≤ 0 at its minimum and unlike the results for high A, larger values of R_m result in a restabilisation.     

Chaotic mixing of tracer and vorticity by modulated travelling Rossby waves

Abstract

We consider the mixing of passive tracers and vorticity by temporally fluctuating large scale flows in two dimensions. In analyzing this problem, we employ modern developments stemming from properties of Hamiltonian chaos in the particle trajectories; these developments generally come under the heading “chaotic advection” or “Lagrangian turbulence.” A review of the salient properties of this kind of mixing, and the mathematics used to analyze it, is presented in the context of passive tracer mixing by a vacillating barotropic Rossby wave. We then take up the characterization of subtler aspects of the mixing. It is shown the chaotic advection produces very nonlocal mixing which cannot be represented by eddy diffusivity. Also, the power spectrum of the tracer field is found to be k ^{? l} at shortwaves—precisely as for mixing by homogeneous, isotropic two dimensional turbulence,—even though the physics of the present case is very different. We have produced two independent arguments accounting for this behavior.

We then examine integrations of the unforced barotropic vorticity equation with initial conditions chosen to give a large scale streamline geometry similar to that analyzed in the passive case. It is found that vorticity mixing proceeds along lines similar to passive tracer mixing. Broad regions of homogenized vorticity ultimately surround the separatrices of the large scale streamline pattern, with vorticity gradients limited to nonchaotic regions (regions of tori) in the corresponding passive problem.

Vorticity in the chaotic zone takes the form of an arrangement of strands which become progressively finer in scale and progressively more densely packed; this process transfers enstrophy to small scales. Although the enstrophy cascade is entirely controlled by the large scale wave, the shortwave enstrophy spectrum ultimately takes on the classical k ^{? l} form. If one accepts that the enstrophy cascade is indeed mediated by chaotic advection, this is the expected behavior. The extreme form of nonlocality (in wavenumber space) manifest in this example casts some doubt on the traditional picture of enstrophy cascade in the Atmosphere, which is based on homogeneous two dimensional turbulence theory. We advance the conjecture that these transfers are in large measure attributable to large scale, low frequency, planetary waves.

Upscale energy transfers amplifying the large scale wave do indeed occur in the course of the above-described process. However, the energy transfer is complete long before vorticity mixing has gotten very far, and therefore has little to do with chaotic advection. In this sense, the vorticity involved in the enstrophy cascade is “fossil vorticity,” which has already given up its energy to the large scale.

We conclude with some speculations concerning statistical mechanics of two dimensional flow, prompted by our finding that flows with identical initial energy and enstrophy can culminate in very different final states. We also outline prospects for further applications of chaotic mixing in atmospheric problems.     

Lateral friction in the oceans as an effect of potential vorticity mixing

Abstract

Applying a mixing-length calculation to potential vorticity rather than to momentum a new type of lateral friction appears in the oceanic mass transport equations. This friction is evaluated for the special case of horizontally homogeneous, quasi-geostrophic turbulence. The main effect is a westward force arising from the so-called β-term. This produces an additional southward interior transport and a strengthening of the western boundary current. A turbulent exchange coefficient K^H = 10⁸ em²s^?1 is sufficient to give a Gulf Stream transport twice that obtained by the classical Sverdrup model.     

Echo intensity data as a directional antenna for observing processes above sloping ocean bottoms

Relative ‘echo intensity’ data (dI) from a bottom-mounted four-beam 300 kHz acoustic Doppler current profiler (ADCP) are used to infer propagation of vigorous processes above a continental slope. The 3- to 60-m horizontal beam spread and the 2-Hz sampling allow the distinction of different arrival times t _i , i = 1,..., 4, at different distances in the acoustic beams from sharp changes in dI-content associated with frontal non-linear and turbulent bores or ‘waves’. The changes in dI are partially due to variations in amounts of resuspended material carried by the near-bottom turbulence and partially due to the fast variations in density stratification (‘stratified turbulence’), as inferred from 1-Hz sampled thermistor string data above the ADCP. Such bores are observed to pass the mooring up to 80 m above the bottom, having typical propagation speeds c = 0.15–0.5 m s⁻¹, as determined from dI(t _i ). Particle speeds in the immediate environment of a bore amount to |u|_env=c ± 0.05 m s⁻¹, the equality being a necessary condition for kinematic instability, whilst the maximum particle speeds amount |u|_max = 1.2–2c. The dI-determined directions of up-, down- and alongslope processes are all to within ±10° of the ADCP’s beam-spread averaged current (particle velocity) data.     

Decreasing trend of sediment transfer function of the Upper Yellow River,China, in response to human activity and climate change

Abstract

An index (F_s) for sediment transfer function is introduced, based on the sediment budget at the channel scale. The purpose of this study is two-fold: to gain a deeper insight into how F_s is influenced by natural and human factors, and to provide some new knowledge for decision making in the management of the Upper Yellow River, China. Since 1960, the F_s of the Lanzhou to Toudaoguai reach of the Upper Yellow River shows a decreasing trend. At the drainage basin level, the decreased F_s can be explained by changes in precipitation and air temperature, as well as by a number of variables describing human activity, such as reservoir regulation, water diversion, and soil and water conservation. The higher temperature reduces the transfer function, while the larger runoff coefficient increases it. At the channel level, the decreased F_s can be explained by a number of variables of flow and sediment input. Three countermeasures for restoration of the F_s are suggested.

Editor Z.W. Kundzewicz     

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.     

Hydromagnetic waves in a differentially rotating annulus. II. Resistive instabilities

Abstract

In part I of this study (Fearn, 1983b), instabilities of a conducting fluid driven by a toroidal magnetic field B were investigated. As well as confirming the results of a local stability analysis by Acheson (1983), a new resistive mode of instability was found. Here we investigate this mode in more detail and show that instability exists when B(s) has a zero at some radius s=s _c. Then (in the limit of small resistivity) the instability is concentrated in a critical layer centered on s _c . The importance of the region where B is small casts some doubt on the validity of the simplifications made to the momentum equation in I. Calculations were therefore repeated using the full momentum equation. These demonstrate that the neglect of viscous and inertial terms when the mean field is strong does not lead to spurious results even when there are regions where B is small.     

On the exact shape of the horizontal profile of a topographically rectified tidal flow

Abstract

Starting from the nonlinear shallow water equations of a homogeneous rotating fluid we derive the equation describing the evolution of vorticity by a fluctuating bottom topography of small amplitude, using a multiple scale expansion in a small parameter, which is the topographic length scale relative to the tidal wave length. The exact response functions of residual vorticity for a sinusoidal bottom topography are compared with those obtained by a primitive perturbation series and by harmonic truncation, showing the former to be invalid for small topographic length scales and the latter to be only a fair approximation for vorticity produced by planetary vortex stretching. In deriving the exact shape of the horizontal residual velocity profile at a step-like break in the bottom topography, it is shown that the Lagrangian profile only exists in a strip having the width of the amplitude of the tidal excursion at both sides of the break, and that it vanishes outside that interval. Moreover, in the limit of small amplitude topography at least, it vanishes altogether for the generation mechanism by means of planetary vortex stretching. The Eulerian profile is shown to extend over twice the interval of the Lagrangian profile both for production by vortex stretching and by differential bottom friction. These finite intervals over which the residual velocity profiles exist for a step-like topography are not reproduced by harmonic truncation of the basic equation. This method gives exponentially decaying profiles, indicating spurious horizontal diffusion of vorticity. In terms of orders of magnitude, the method of harmonic truncation is reliable for residual velocity produced by vortex stretching but it overestimates the residual velocity produced by differential bottom friction by a factor 2.     

Statistical dynamics of internal gravity waves-turbulence

Abstract

Numerical simulations of internal gravity waves-turbulence are carried out for the inviscid, viscous and forced-dissipative two-dimensional primitive equations using the spectral method. Some of the results are compared with the predictions of the eddy damped quasi-normal Markovian (EDQNM) closure for internal waves of Carnevale and Frederiksen, generalized for periodic boundary conditions and possible random forcing and dissipation. The EDQNM reduces to the Boltzman equation of resonant interaction theory in the continuum space limit and as the forcing and dissipation vanish. However, the limit is singular in the sense that as well as conserving total energy, E, and total cross-correlation between the vorticity and buoyancy fields, C, an additional conservation law, viz. z-momentum, P_z , occurs in the limit. This means that the resonant interaction equilibrium (RIE) solution of the Boltzmann equation differs from the statistical mechanical equilibrium (SME) solution of the EDQNM closure.

The statistical stability of the SME and RIE spectra for the primitive equations is tested by integrating the inviscid equations using initial realizations of these spectra with random phases. It is found that E and C are accurately conserved while P_z undergoes large amplitude variations. The approach to equilibrium of initial disequilibrium spectra is monitored by examining the evolution of the entropy. The increase and asymptotic approach to a constant value corresponding to complete chaos is consistent with the behaviour predicted by the EDQNM closure.

For the viscous decay and forced-dissipative experiments, the behaviour of the entropy is also consistent with that predicted by the EDQNM closure. There is approximate equipartition of potential and total kinetic energies throughout the integrations from initial conditions having equal potential and total kinetic energies and as well equal vertical and horizontal energies, but as expected, the ratio of horizontal to vertical kinetic energy increases with time to a value greater than unity.

With Laplacian viscous dissipation and thermal diffusivity, the statistical steady states produced in the forced-dissipative experiments have k^?3 power laws for k≧7. A comparison with the power laws for kinetic energy and passive scalar variance produced in a numerical simulation of the two-dimensional passive scalar problem is also presented.     

Hamiltonian description of almost geostrophic flow

Abstract

Geostrophic flow in the theory of a shallow rotating fluid is exactly analogous to the drift approximation in a strongly magnetized electrostatic plasma. This analogy is developed and exhibited in detailed to derive equations for the slow nearly geostrophic motion. The key ingredient in the theory is the isolation, to whatever order in Rossby number desired, of the fast motion near the inertial frequency. One of the remaining degrees of freedom represents a new approximate constant of the motion for nearly geostrophic flow. This is the analogue of the familiar magnetic moment adiabatic invariant in the plasma problem.

The procedure is a Rossby number expansion of the Hamiltonian for the fluid expressed in Lagrangian, rather than Eulerian variables. The fundamental Poisson brackets of the theory are not expanded so desirable properties such as energy conservation are maintained throughout.     

Impact of forest degradation on streamflow regime and runoff response to rainfall in the Garhwal Himalaya,Northwest India

Baseflows have declined for decades in the Lesser Himalaya but the causes are still debated. This paper compares variations in streamflow response over three years for two similar headwater catchments in northwest India with largely undisturbed (Arnigad) and highly degraded (Bansigad) oak forest. Hydrograph analysis suggested no catchment leakage, thereby allowing meaningful comparisons. The mean annual runoff coefficient for Arnigad was 54% (range 44–61%) against 62% (53–69%) at Bansigad. Despite greater total runoff Q_t (by 250 mm year^–1), baseflow at Bansigad ceased by March, but was perennial at Arnigad (making up 90% of Q_t vs. 51% at Bansigad). Arnigad storm flows, Q_s, were modest (8–11% of Q_t) and occurred mostly during monsoons (78–98%), while Q_s at Bansigad was 49% of Q_t and occurred also during post-monsoon seasons. Our results underscore the importance of maintaining soil water retention capacity after forest removal to maintain baseflow levels.

EDITOR D. Koutsoyiannis; ASSOCIATE EDITOR D. Gerten     

Spatial variability of the surface energy balance of Lake Kasumigaura and implications for flux measurements

ABSTRACT

The spatial variability of the lake surface energy balance and its causes are not well-understood. Energy balance maps (90 m resolution) of Lake Kasumigaura (172 km²), Japan, obtained by interpolating station data and bulk equations, allowed an investigation of these issues. Due to lake-scale variations in meteorological variables and small-scale fluctuations of surface temperature, T_s, surface heat fluxes differed horizontally at two distinct scales, while radiative fluxes were more uniform. As the key variable to surface flux T_s was only homogeneous for directions with a longer fetch or under calm wind conditions. Using these findings, the suitability of two flux station locations, one at the centre of the lake and another within a cove, was considered. Although both locations satisfied the fetch requirements, T_s was not always found to be homogeneous in the cove, making this location less suitable for flux measurements, an issue that, to date, has been overlooked.     

A steady state of the disc dynamo

Abstract

We discuss the steady states of the αω-dynamo in a thin disc which arise due to α-quenching. Two asymptotic regimes are considered, one for the dynamo numberD near the generation thresholdD ₀, and the other for |D| ? 1. Asymptotic solutions for |D—D ₀| ? |D ₀| have a rather universal character provided only that the bifurcation is supercritical. For |D| ? 1 the asymptotic solution crucially depends on whether or not the mean helicity α, as a function ofB, has a positive root (hereB is the mean magnetic field). When such a root exists, the field value in the major portion of the disc is O(l), while near the disc surface thin boundary layers appear where the field rapidly decreases to zero (if the disc is surrounded by vacuum). Otherwise, when α = O(|B|^?s) for |B| → ∞, we demonstrate that |B| = O(|D|^1/s ) and the solution is free of boundary layers. The results obtained here admit direct comparison with observations of magnetic fields in spiral galaxies, so that an appropriate model of nonlinear galactic dynamos hopefully could be specified.     

Seasonal climate patterns and their influence on calibration of the Hargreaves-Samani equation

Abstract

Annual patterns in climate parameters were studied to evaluate how these influence the quality of reference evapotranspiration (ETo) estimates obtained from the Hargreaves-Samani (HS) equation, since the method only uses the measured temperature directly. The work evaluates how these patterns can be used to improve the HS ETo estimates. Ten-year moving averages from a set of California Irrigation Management Information System (CIMIS) stations were used to evaluate the relationships between solar radiation (R_s), temperature (T) and ETo. The results indicate that T treads behind solar radiation and its value peaks some 25 days later. Thus, the main irrigation season in the Mediterranean climate (1 May–30 September) can be divided into three phases: increasing R_s and T; decreasing R_s with increasing T; and decreasing R_s and T. Non-univocal annual cycles were observed between R_s and T, ETo and R_s, and ETo and T. These annual patterns result in important seasonal changes in the ratio between the HS and Penman-Monteith (FAO PM) ETo estimates. The changes are particularly important during the irrigation season, where the FAO PM initially calculates greater ETo values than the HS methodology, and from the end of May to early September, where the HS equation overestimates the ETo values (by 17 mm, or 3%). These patterns obtained from 2000–2009 data were used to calibrate and improve HS ETo estimates at new sites for the 2010–2011 period. Calibration based on the proposed seasonal region-wide FAO PM/HS ETo ratios improved both the bias (decreased from 0.40 to 0.36 mm d^-1) and r² (increased from 0.67 to 0.87) of the ETo estimates for the irrigation season. The proposed methodology can be easily applied to other regions, even when the existing weather stations are sparse.

Editor Z.W. Kundzewicz     

Laboratory and numerical model studies of a negatively-buoyant jet discharged horizontally into a homogeneous rotating fluid

Abstract

The results of laboratory experiments and numerical model simulations are described in which the motion of a round, negatively-buoyant, turbulent jet discharged horizontally above a slope into a rotating homogeneous fluid has been investigated. For the laboratory study, flow visualisation data are presented to show the complex three-dimensional flow fields generated by the discharge. Analysis of the experimental data indicates that the spatial and temporal developments of the flow field are controlled primarily by the lateral and vertical discharge position of the jet (with respect to the bounding surfaces of the container of width W) and the specific momentum (M ₀) and buoyancy (B ₀) fluxes driving the jet. The flow is seen to be characterised by the formation of (i) a primary anticyclonic eddy (PCC) close to the source, (ii) an associated secondary cyclonic eddy (SCE) and (iii) a buoyancy-driven bottom boundary current along the right side boundary wall. For the parameter ranges studied, the size L _{p, s} and spatial location x _{p, s} of the PCC and SCE (and the nose velocity u _N of the boundary current) are shown to be only weakly-dependent upon the value of the mixed parameter M ₀Ω/B ₀, where Ω is the background rotation rate. Both L _p and x _p are shown to scale with the separation distance y?/W of the right side wall (y = 0) from the source (y = y?), both L _s and x _s scale satisfactorily with the length scale l _M (= M ₀ ^3/4/B ₀ ^½) and u _N is determined by the appropriate gravity current speed [(g']₀ H]^½ and the separation distance y?/W.

Numerical model results show good qualitative agreement with the laboratory data with regard to the generation of the PCC, SCE and boundary current as characteristic features of the flow in question. In addition, extension of the numerical model to

diagnose potential vorticity and plume thickness distributions for the laboratory cases allow the differences in momentum-and buoyancy-dominated flows to be clearly delineated. Specifically, the characteristic features of the SCE are shown to be strongly dependent upon the value of M ₀Ω/B ₀ for the buoyant jet flow; not least, the numerical model data are able to confirm the controlling role played by the boundary walls in the laboratory experiments. Quantitative agreement between the numerical and laboratory model data is fair; most significantly, the success of the former model in simulating the dominant flow features from the latter enables the reliable extension of the numerical model to be made to cases of direct oceanic interest.