On the stability of stratified plane couette flow

Abstract

Unbounded stratified plane Couette flow is shown to be stable against small amplitude disturbances. The Brunt-Väisälä frequency is assumed to be constant. Both viscosity and thermal diffusion are included, and shown to be stabilizing.     

The nonlinear development of disturbances in a zonal shear flow

Abstract

A study is made of the nonlinear stability of a weakly supercritical zonal shear flow in the β-plane approximation. The dynamics of initially small disturbances are examined. The main nonlinear effects are associated with the rearrangement of the critical layer. It is shown that as the wave grows in amplitude, linear regimes of the critical layer (viscous and nonstationary) change over to a nonlinear regime while the exponential law of disturbance growth becomes a power-law.     

On magneto-hydrodynamic disturbances in a fluid in contact with a fixed plane

Summary The Laplace-transform techniques have been made use of in investigating the disturbances in a conducting fluid in contact with a fixed plane.     

On the stability analysis of sharply stratified shear flows

When the stability of a sharply stratified shear flow is studied, the density profile is usually taken stepwise and a weak stratification between pycnoclines is neglected. As a consequence, in the instability domain of the flow two-sided neutral curves appear such that the waves corresponding to them are neutrally stable, whereas the neighboring waves on either side of the curve are unstable, in contrast with the classical result of Miles (J Fluid Mech 16:209–227, 1963) who proved that in stratified flows unstable oscillations can be only on one side of the neutral curve. In the paper, the contradiction is resolved and changes in the flow stability pattern under transition from a model stepwise to a continuous density profile are analyzed. On this basis, a simple self-consistent algorithm is proposed for studying the stability of sharply stratified shear flows with a continuous density variation and an arbitrary monotonic velocity profile without inflection points. Because our calculations and the algorithm are both based on the method of stability analysis (Churilov J Fluid Mech 539:25–55, 2005; ibid, 617, 301–326, 2008), which differs essentially from usually used, the paper starts with a brief review of the method and results obtained with it.     

On ionospheric disturbances in relation to geomagnetic disturbances

Summary Disturbances in the critical frequency of theF2-region of the ionosphere at Watheroo on international magnetically disturbed days are analyzed together with simultaneous geomagnetic data at the same station. The results show that the daily average disturbanceDm (foF2) becomes negative or positive according as the maximum of the disturbance daily variationSD (foF2) takes place in night time or in daytime respectively. This fact may show that bothDm (foF2) andSD (foF2) are due to the condition that the daytime increase in the daily variation of theF2-region is reduced or enhanced owing to vertical drift of the ionosphere caused by electric currents responsible for geomagneticSD.     

On the stability properties of the nongeostrophic disturbances in a barotropic zonal current

Summary The stability properties of the nongeostrophic disturbances are studied in a barotropic zonal current. The growth rate of the nongeostrophic disturbances is increased and the instability is generally shifted towards shorter wavelengths. Even in the structure of the nongeostrophic disturbances large differences are found as compared to geostrophic disturbances.On leave from Physical Research Laboratory, Ahmedabad, India.     

Ionosphere F2-region under the influence of the evolutional atmospheric gravity waves in horizontal shear flow

The ambipolar diffusion equation for the height distribution of electron density in the ionospheric F2-layer is solved in the presence of neutral horizontal shear flow. By using this nonstationary solution the reaction of the F2-region electron density on the evolution of atmospheric acoustic–gravity waves (AGW) is investigated. The evolution of the AGW and the corresponding behaviour of the height distribution of the F2-region electron density are described by the characteristic time, t_a, of transient development of shear waves in the horizontal shear flow. For long times t > t_a, the gravity wave frequency tends to the isothermal Brunt–Väisälä frequency, which appears in the F2-layer as wavelike behaviour of h_mF2 and N_mF2 with periods close to 16–20 min, when the scale height of the neutral gas is H = 60 km. The shear wave, which is due to the presence of horizontal shear flow, gives sufficient changes of the height profile of electron density for times of t ⩽ t_a.     

On the use of horizontal acoustic Doppler profilers for continuous bed shear stress monitoring

Continuous monitoring of bed shear stress in large river systems may serve to better estimate alluvial sediment transport to the coastal ocean.Here we explore the possibility of using a horizontally deployed acoustic Doppler current profiler(ADCP) to monitor bed shear stress,applying a prescribed boundary layer model,previously used for discharge estimation.The model parameters include the local roughness length and a dip correction factor to account for sidewall effects.Both these parameters depend on river stage and on the position in the cross-section, and were estimated from shipborne ADCP data.We applied the calibrated boundary layer model to obtain bed shear stress estimates over the measuring range of the HADCP.To validate the results,co-located coupled ADCPs were used to infer bed shear stress,both from Reynolds stress profiles and from mean velocity profiles. From HADCP data collected over a period of 1.5 years,a time series of width profiles of bed shear stress was obtained for a tidal reach of the Mahakam River,East Kalimantan,Indonesia.A smaller dataset covering 25 hours was used for comparison with results from the coupled ADCPs.The bed shear stress estimates derived from Reynolds stress profiles appeared to be strongly affected by local effects causing upflow and downflow,which are not included in the boundary layer model used to derive bed shear stress with the horizontal ADCP.Bed shear stresses from the coupled ADCP are representative of a much more localized flow,while those derived with the horizontal ADCP resemble the net effect of the flow over larger scales.Bed shear stresses obtained from mean velocity profiles from the coupled ADCPs show a good agreement between the two methods,and highlight the robustness of the method to uncertainty in the estimates of the roughness length.     

The stability of dissipative magnetohydrodynamic shear flow in a parallel magnetic field

Abstract

Strong decay bounds are obtained for linearized perturbations to an unbounded, plane Couette flow in a parallel magnetic field. Finite conductivity and molecular viscosity are found to be stabilizing. Those modes decaying most slowly have the form of rolls aligned with the shear flow. The non-aligned rolls decay at an enhanced rate. Stability bounds at finite amplitude are obtained for flows bounded in one direction using energy methods.     

On the mathematical stability of stratified flow models with local turbulence closure schemes

Occasionally, numerical simulations using local turbulence closure schemes to estimate vertical turbulent fluxes exhibit small-scale oscillations in space, causing the eddy coefficients to vary over several orders of magnitude on short distances. Theoretical developments suggest that these spurious oscillations are essentially due to the way the eddy coefficients depend on the vertical gradient of the model’s variables. An instability criterion is derived based on the assumptions that the artefacts under study are due to the development of small-amplitude, small time- and space-scale perturbations of a smooth solution. The relevance of this criterion is demonstrated by applying it to a series a closure schemes, ranging from the Pacanowski–Philander formulas to the Mellor–Yamada level 2.5 model.     

Unconfined linear flow to a horizontal well

The validity of a previously proposed but untested modification to equations for flow to a horizontal well is assessed using a specially developed finite-difference model. This modification extends confined flow equations to allow the head in the well and the saturated depth at the well to be estimated in unconfined conditions. The study is limited to the case of two-dimensional flow with no flow in the direction parallel to the line of the well. The results show that the modified equations for both a finite unconfined aquifer and, by inference, an infinite unconfined aquifer are adequately accurate for practical application.     

Hydromagnetic flow due to a transient plane

Summary The electromagnetic equations of Maxwell as well as the equations of fluid dynamics have been used to investigate the hydromagnetic flow due to the transient motion of a plane. The expressions for the velocity and skin-friction have been derived for small values of time by using Laplace transform techniques.     

Instability of plane parallel shear flow (toward a mechanistic picture of how it works)

This paper reviews work done over the last twelve years on wave overreflection in shear flows, and on the development of a mechanical picture of how overreflection and instability work. It is argued that the wave geometry of a flow configuration is the primary determinant of its stability. It is also argued that the primary mechanism for wave amplification is the Orr mechanism.     

On one integral of equations of motion of relative horizontal flow

Резюме Бьул выведен интеграл уравнений двнженид Лагранжа, соответствующжа, соответствуюший чоступательниым, неустойчивым, тошоидальным волнам на границе раздела двух зональнух течений. Внд границы раздела в течение развития волны меняется, превращаясь из обычной трохоиды в циклоиду. Числовые оценки показывают, что рассматриваемые волны цхематичесски соответствуйт коротким воянам на квазистационарных атмосфернын фронтах.

---

---

Dedicated to Professor Bedřich Ŝalamon on his 85th Birthday     

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.     

Baroclinic energetics and zonal plane distribution of monsoon disturbances

The adiabatic, quasi-geostraphic, 25-layer, numerical, linear model with Ekman boundary layer friction is utilised to perform the baroclinic stability analysis of the mean monsoon zonal wind profile. It is shown thec _i is a function of the resultant wavenumber alone. This relation is able to explain the effects of the lateral walls on the unstable waves.The energetics and zonal plane distribution of the short and long preferred viscous waves are computed. The upward motion of the short wave together with the warm (cold) core lies to the west of the surface trough position above (below) 850 mb. Further, it is shown that the main source of kinetic energy for the wave lies in the middle layer (850–700 mb) which is transported to the lower and upper layers. Computed is found to be in good agreement with observed values.     

Deep heterogeneity of the stress state in the horizontal shear zones

The formation structures of brittle destruction in a rock layer above an active strike-slip fault in the crystalline basement is studied. The problem is analyzed from the standpoint of loading history, when after the stage of pure gravitational loading, an additional strain state of uniform horizontal shear of both the layer and underlying basement develops, which is further followed by a vertically nonuniform shear caused by the activation of the deep fault. For the studied object, irreversible fracture deformations on macro- and microlevels arise as early as the initial stage of loading under the action of gravitational stresses. These deformations continue evolving on the megascopic level in the course of horizontal shearing that is quasi-uniform both along the depth and laterally. The final formation of the structural ensemble occurs after a long stage of horizontal displacement of the blocks of the crystalline basement—the stage of localized shear. The theoretical analysis of the evolution of the stress state and morphology of the failure structures established the presence of numerous fractures with the normal dip-slip components in the intermediate-depth part of the rock mass, which are formed at the stages of uniform and localized horizontal shearing. The fractures with a strike-slip component mainly arise in the upper and near-axial deep parts of the section.     

Mathematical models of groundwater flow with a horizontal drain

An analytical review is given to drainage theory, the foundation of which was laid down by researchers in our country in the first half of the 20th century. Later the author of this paper developed and thoroughly studied a complex of mathematical models describing two-dimensional groundwater flow in drained soils based on the boundary problems of the theory of analytical functions. These models were used to provide a hydrodynamic substantiation for the regularities in subsoil water dynamics that were established by ameliorators in relation to the leaching of salts from soils; to identify the specific features of groundwater flow with drainage in the presence in the flow of one or two unknown free boundaries; and to develop approaches to the analysis of such flows in a direct formulation.     

Ground motions around a deep semielliptic canyon with a horizontal edge subjected to incident plane SH waves

We study scattering of antiplane shear waves induced by a deep semielliptic canyon with a horizontal edge. We employ the region-point-matching technique to cope with the problem considered. Through an auxiliary boundary, a part of the circumference of a semiellipse, the whole analyzed region is divided into two subregions. We express the displacement fields in terms of Mathieu functions. We unify two distinct elliptic coordinates via a simple coordinate transformation relation. Integration of the coordinate transformation relation into the region-point-matching technique simplifies the procedure for constructing simultaneous equations. Imposing the continuity conditions and traction-free ones, we obtain the expansion coefficients. Frequency-domain results demonstrate ground motion variability based on several key factors. Ground surface responses under seismic shaking are also simulated in the time domain.