A note on the geostrophic flow past a crater in a rotating fluid

Abstract

Laboratory experiments are described on the flow past a solid obstacle in a rotating, homogeneous fluid. Specifically, the obstacle has the form of a walled crater specially constructed so that the volume of the depression is identically equal to the volume of the walls. The results show that closed streamlines occur rather more easily above such topography than above other obstacle types of the same scale but that the conditions for closure are determined essentially by the detailed geometry of the crater, the value of the Rossby number, and the depth of the fluid. The observed flow patterns are analysed and classified and attempts to quantify the most common flow type are made.     

The interaction of topography and shear flow in a homogeneous rotating fluid

Abstract

Laboratory experiments are described in which a cylindrical obstacle is moved azimuthally through a rotating cylindrical tank of fluid in which a basic azimuthal flow inversely dependent upon the tank radius is generated by means of a source-sink arrangement. A technique is described whereby the flow can be adjusted until, relative to the obstacle, it is forward near the centre of the tank and reversed near the rim, with a monotonic variation between these extremes. The sense of this shear, relative to the obstacle, can be altered so that it either opposes or coincides with the sense of the basic rotation. Both cases are investigated in the experiments. The results of both qualitative and quantitative measurements are presented, and some comparison with related theoretical work is attempted.     

A note on the use of zero potential vorticity models

Abstract

The acceptability of zero potential vorticity models as approximations for natural systems of small, but finite, potential vorticity is studied for bounded frontal flows of arbitrary profile. It is demonstrated that all (infinitely) long-wave solutions of the zero potential vorticity front are asymptotic limits for some (not necessarily long-wave) solutions of the small potential vorticity front. In contrast, for downstream-varying solutions there is no simple way of demonstrating this property. These findings suggest that the use of zero potential vorticity models should be carefully examined in other, non-frontal, problems as well. Finally we show that the longwave solutions of the zero potential vorticity flow are at most neutral (quasi-stable).     

Potential vorticity and the stirring of a rotating fluid

Summary The concept of potential vorticity is employed to demonstrate the quintessential interplay of an embient rotation with the forced or free stirring of a fluid.     

The effect of random recharge on uniform steady free-surface flow in heterogeneous porous formations

The effect of parametric uncertainty in recharge rate and spatial variability of hydraulic conductivity upon free-surface flow is investigated in a stochastic framework. We examine the three-dimensional free-surface gravitational flow problem for sloped mean uniform flow in a randomly heterogeneous porous medium under the influence of random recharge. We develop analytic solutions for the variance of free-surface position, head, and specific discharge on the free surface. Additionally, we obtain semi-analytic solutions for the statistical moments of head and specific discharge beneath the free-surface. Statistical moments are derived using a first-order approximation and then compared with their parallel in an unbounded medium. The effect of recharge mean and variability on the statistical moments is analyzed. Results can be applied to more complex flows, slowly varying in the mean.     

Integrated surface–groundwater flow modeling: A free-surface overland flow boundary condition in a parallel groundwater flow model

Interactions between surface and groundwater are a key component of the hydrologic budget on the watershed scale. Models that honor these interactions are commonly based on the conductance concept that presumes a distinct interface at the land surface, separating the surface from the subsurface domain. These types of models link the subsurface and surface domains via an exchange flux that depends upon the magnitude and direction of the hydraulic gradient across the interface and a proportionality constant (a measure of the hydraulic connectivity). Because experimental evidence of such a distinct interface is often lacking in field systems, there is a need for a more general coupled modeling approach.     

A note on the heat transfer by the axisymmetric thermal convection in a rotating fluid annulus

Abstract

Some new measurements are presented of the axisymmetric heat transport in a differentially heated rotating fluid annulus. Both rigid and free upper surface cases are studied, for Prandtl numbers of 7 and 45, from low to high rotation rates. The rigid lid case is extended to high rotation rates by suppressing the baroclinic waves, that would normally develop at some intermediate rotation rate, with the use of sloping endwalls.

A parameter P is defined as the square of the ratio of the (non-rotating) thermal sidewall layer thickness to the Ekman layer thickness. For small P the heat transport remains unaffected by the rotation, but as P increases to order unity the Ekman layer becomes thin enough to inhibit the radial mass transport, and hence the heat flux. No explicit Prandtl number dependence is observed. Also this scaling allows the identification of the region in which the azimuthal velocity reaches its maximum. Direct comparisons are drawn with previous experimental and numerical results, which show what can be interpreted as an inhibiting effect of increasing curvature on the heat transport.     

Comment on the effect of a vorticity centre on a frontal boundary

Zusammenfassung Es wird der Einfluss des sogenannten Vorticity-Zentrums auf die Grenzfläche zweier Luftmassen der mittleren geographischen Breiten studiert. Zwecks Aufklärung der Rolle des Bewegungs- und Druckfeldes in diesem Prozesse wurde ein einfaches Modell konstruiert und numerisch gelöst. Die sich entwickelnde frontale Welle kann beziehungsweise als ein Anfangswert beim Lösen der mit der Stabilität der Wellen verbundenen Probleme mittels der Gleichungen, linearisiert durch die Methode der kleinen Störungen, dienen. Bei der Konstruktion des Modells wird ausser anderem von der Geometrie der Bewölkung ausgegangen, wie sie auf den Aufnahmen der meteorologischen Satelliten erfasst ist.     

A note on the effect of the ground on the convective temperature field in the atmospheric surface layer

Summary The paper presents the solution of the equation of heat conduction with density of heat sources given generally. For two special cases the computed central temperature excesses of model[5] are compared with the results of some authors[1, 6] who deal with convection in the surface layer.     

A note on multiple flow equilibria

A set of ordinary differential equations describing a mechanical system subject to forcing and dissipation is considered. A topological argument is employed to show that if all time-dependent solutions of the governing equations are bounded, the equations admitN steady solutions, whereN is a positive odd integer and where at least (N–1)/2 of the steady solutions are unstable. The results are discussed in the context of atmospheric flows, and it is shown that truncated forms of the quasigeostrophic equations of dynamic meteorology and of Budyko-Sellers climate models satisfy the hypotheses of the theorem.     

Axially symmetric flow round a non-conducting obstacle in the shape of a hemisphere by a homogeneous electrical current field

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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.     

Shear flow instabilities in a rotating system

Abstract

The stability of a plane parallel shear flow with the profile U(z) = tanh z is considered in a rotating system with the axis of rotation in the z-direction. The establishment of the basic flow requires a baroclinic state, but baroclinic effects are suppressed in the stability analysis by assuming a limit of high thermal conductivity. It is shown that the strongest growing disturbance changes from a purely transverse form in the limit of vanishing rotation rate to a nearly longitudinal form as the angular velocity of rotation increases. An analytical solution of the stability equation is obtained for vanishing growth rates of the transverse form of the instability. But, in general, the solution of the problem requires numerical integrations which demonstrate that the preferred direction of the wave vector of the instability is towards the left of the direction of the mean flow.     

A note on the secular variation of the geomagnetic field

Geomagnetic variations with (apparent) periods from 13 to 30 y have been reported by Alldredge (1977) who has argued that the origin of these signals must be found in the core of the Earth rather than outside the Earth. It is shown in this paper that a portion of these geomagnetic variations (perhaps most of the variations) might well be due to geomagnetic signals of much longer period, originating in the core, that appear to have periods of 13–30 y because of an artifact of the data processing. Much of the remainder of these variations could well be of external origin. A method for processing these data is suggested that is superior to Alldredge's method in minimizing the generation of spurious waves by long-period secular variation.     

A note on the approximate determination of the dispersion of Love waves in a laterally nonhomogeneous layer lying over a homogeneous half-space

Rayleigh's principle and the concept of the local wave number have been utilised for the approximate determination of the dispersion of Love waves propagating in a laterally heterogeneous layer lying over a homogeneous half-space. The shear wave velocity and the rigidity in the surface layer have been assumed to decrease with the increase of the lateral distance from the origin. The range of validity of the dispersion equation obtained by this method has been examined critically. It was found that: (a) for existence of Love waves the minimum value of shear wave velocity in the layer must be less than that in the matter below, and (b) the phase velocity of Love waves decreases with the increase of the lateral distance from the origin.     

A note on disturbances in a certain type of visco-elastic solid deformed by a magnetic field

Summary The present note is an attempt to study the propagation of disturbances in a semiinfinite visco-elastic rod which is of three-parameter type, and which is subjected to a static magnetic field.     

A note on the Aitken nuclei and ion flow measurements in the lower stratosphere

In the light of new measurements of small aerosol particles in the lower stratosphere, some of the old investigations—which were only published in part in scientific journals—are reviewed and compared. The discussion focuses on whether the Aitken nuclei (AN) size distribution up to 20 km can be described by Junge’s or log-normal density functions and under what circumstances one can find a bimodal distribution of these particles. The ion flow in correlated to the stratospheric pollution (intercept with the jet aircraft, volcanic activity) and is, in mean, directly proportional to the aircraft altitude in the undisturbed lower stratosphere. Note: The GCCPR, Univ. Missouri at Rolla, reports quoted in the article can be obtained from the author.     

A nonlinear,resistive boundary layer in rotating hydromagnetic flow

A new nonlinear boundary layer in rotating hydromagnetic flows is presented. The purpose of this layer is to provide a smooth transition between the magnetic field at a rigid, electrically insulating boundary and that far from the boundary. The boundary layer problem is solved in a cylindrical geometry, assuming that the variables obey the hydromagnetic analog of the von Karman similarity. The primary momentum balance within this boundary layer is between pressure, Coriolis and hydromagnetic forces; the viscous and nonlinear inertia forces are unimportant. Diffusion of the magnetic field is balanced by advection where the advecting fluid flow is induced by the action of the hydromagnetic body forces within the boundary layer. The structure of the layer depends upon a single parameter which measures the strength of the normal magnetic field. Steady solutions are presented and their uniqueness and temporal stability are analyzed. The relevance of this layer to the core of the Earth is discussed. It is estimated that this boundary layer would be at least as thick as one tenth of the core radius if it exists within the core.     

The effects of alongshore variation in bottom topography on a boundary current—(topographically induced upwelling)

A theory which describes the constant f-plane flow of a steady inviscid baroclinic boundary current over a continental margin with a bathymetry that varies slowly in the alongshore but rapidly in the offshore directions is developed in the parameter regime (L_D/L)² ≤ Ro 1, where L_D is the internal deformation radius, L the horizontal length scale, and Ro the Rossby number. To lowest order in the Rossby number the flow is along isobaths with speed q_o = V_u(h,z)|Vh|/α, where V_u(h,z) is the upstream speed, α the upstream bottom slope at depth h, and Vh the bottom slope downstream at depth h. The lowest order flow produces a variation in the vertical component of relative vorticity along the isobath as the magnitude and direction of Vh vary in the downstream direction. The variation of vorticity requires a vertical as well as a cross-isobath flow at first order in the Rossby number. The first order vertical velocity is computed from the vorticity equation in terms of upstream conditions and downstream variations of the bathymetry. The density, pressure, and cross-isobath flow at first order in the Rossby number are then calculated. It is shown that in the cyclonic region of current (d/dh(V_u/α) > 0), if the isobaths diverge in the downstream direction ((∂/∂s)|Vh| < 0), then upwelling and onshore flow occur. The theory is applied to the northeastern Florida shelf to explain bottom temperature observations.