Shallow Fluid Layer Research Articles

Internal Solitary Waves

The expansion procedure introduced by Benney (1966) for weakly nonlinear, planar shallow‐water waves is used to provide an alternative derivation of the more general results of Benjamin (1966) for shallow fluid layers possessing arbitrary vertical stratification and horizontal shear. New solutions that include the effects of both shear and stratification are presented. The evolution equation for slowly varying cylindrical solitary waves traveling in a density‐stratified fluid is found using two‐timing techniques. Not surprisingly, one obtains the same coefficients for the nonlinear and dispersive terms as in the planar case. In the limit for uniform density it is shown that the free‐surface evolution equation of Miles (1978) for axisymmetric Boussinesq waves is recovered.

Transcritical, shallow-water flow over compact topography

The flow of a shallow layer of fluid over isolated topography is considered when the velocity is near the critical, long-wave speed. The response of the free surface is obtained when the length-scale of the topographic variation is comparable to the fluid depth (compact topography). The weakly nonlinear response is described by a forced Korteweg-de Vries equation. When the topography is compact, an inner and outer matching is required to obtain a uniform description of the free-surface deformation for all streamwise positions. The important classes of compact forcing are defined and general results for the shape of the free surface directly over the topographic forcing are obtained. A variety of stationary, outer solutions are presented including solitary and lee-wave responses.

Stability of some stationary solutions for the forced KdV equation

The forced KdV (fKdV) equation has been established by recent studies as a simple mathematical model capable of describing the physics of a shallow layer of fluid in response to external forcing. For a particular one-parameter family of forcings characterized by a wave amplitude parameter for a certain supercritical forcing distribution, the exact stationary solutions are known. We study the stability of these solutions as the parameter varies. The linear stability analysis is first carried out by exploring the structure of the spectrum of the associated eigenvalue problem using a perturbation technique applied to a neighbourhood of isolated parameter values, where eigenfunctions corresponding to eigenvalue zero can be expressed in closed form. The results identify a set of intervals in the parameter space corresponding to different types of manifestation of instability. In the region of the parameter space where the linear stability analysis fails to provide an answer, we discuss a nonlinear analysis that provides a sufficient condition for stability.

The propagation of two-dimensional and axisymmetric viscous gravity currents at a fluid interface

Viscous gravity currents resulting from the introduction of fluid between an upper layer of fluid of lesser density and a lower layer of greater density are analysed. The nonlinear equations governing the spread and shape of the intrusion are formulated for the cases of intrusion at low Reynolds number between deep ambient layers and of flow over a shallow layer of viscous fluid with a rigid lower boundary. Similarity solutions of these equations are obtained in both two-dimensional and axisymmetric geometries, under the assumption that the volume of intruding fluid increases with time like tα. The theoretical predictions are shown to be in reasonable agreement with experimental observations of the spreading of glucose syrups and of viscous hydrocarbons between fluid layers of differing densities. Scaling arguments are used to derive many new results for the rates of spread of intrusions in a wide variety of further situations. A compendium of spreading relations, including some previously isolated results, is derived within a coherent framework and tabulated.

Hadley Circulations on a Nonuniformly Heated Rotating Plate

Abstract Hadley circulations in a shallow fluid layer driven by a radial temperature gradient in the bottom plate have been investigated experimentally. The temperature on top of the fluid was uniform, and the vertical temperature gradient in the fluid was usually stable over the entire layer. The apparatus was either rotating or nonrotating. In the cast of stable stratification, axisymmetric one-loop circulations have been observed in the nonrotating as well as in the rotating case, in agreement with the classical concept of the Hadley circulation. In the nonrotating case with partially unstable stratification the flow was striated with jet of fluid moving rapidly to the center. In the corresponding rotating case the striations were spirals. In the parameter range of the experiments no example of azimuthal waves has been observed.

Onset of surface-tension-driven Bénard convection

An experimental investigation of the onset of convection in shallow fluid layers heated uniformly from below and cooled from above by an air layer has been made. If the depth of the silicone layer is smaller than 2 mm the onset of convection takes place in two stages. There is first a weak pattern, which is characterized by its appearance at ever smaller temperature gradients as the depth of the fluid is decreased. When the temperature difference across the fluid is increased a second strong pattern forms near the predicted critical Marangoni number. The cells in this pattern are hexagonal and seem to be what one has always referred to as Benard cells. The temperature gradient at which this pattern appears increases with decreased depth. The heat transfer through the fluid has been measured. The critical temperature gradient for the formation of the hexagonal pattern has been determined from the break of the heat transfer curve.

Pattern Formation and Wave-Number Selection by Rayleigh–Bénard Convection in a Cylindrical Container

We describe an apparatus and procedures for simultaneous heat transport measurements and computer enhanced shadowgraph flow-pattern imaging in a shallow horizontal layer of fluid heated from below. The heat transport measurements have a resolution of better than 0.1%, and the shadowgraph technique can detect the flow field for ϵ ≡ (R – Rc)/Rc as small as 10-2 (Rc is the critical value of the Rayleigh number R for onset of convection).The apparatus and procedures were used to study pattern and wave-number evolution in a cylindrical layer of water with radius-to-height ratio L = 7.5 and Prandtl number σ = 6.1. We found that dynamic sidewall forcing during the early thermal transients after a change in the heat current from a subcritical to a supercritical value establishes a cylindrical flow pattern. Once created, this pattern is stable in our apparatus over the wide range 0.16 ≲ ϵ ≲ 8 even after the transients have decayed.With changing ϵ, adjustment in the wave number k takes place discontinuously by hysteretic changes at the cell center in the number of convection roll pairs. When ϵ is increased, the discontinuous changes at the cell center are towards smaller k and are preceded by a continuous loss of cylindrical symmetry (the middle roll pair moves off center). The selected wave numbers coincide neither with the zig-zag instability of the infinite system, as once suggested, nor with a linear extrapolation to ϵ = 0(1) of the recent prediction to lowest order in ϵ of Manneville and Piquemal and of Cross. Comparison of the selected k with measurements by others reveals no dependence upon L and σ.For ϵ < 0.16, the cylindrical pattern is unstable and decays on a time scale much longer than a horizontal diffusion time to patterns of rolls which tend to be perpendicular to the sidewalls and which contain defects. Once formed, these latter patterns will persist at large values of ϵ. These patterns also undergo a wave-number adjustment process with hysteretic changes mediated mostly by focus singularities near the walls. In these cases, larger values of ϵ also tend to produce smaller values of k.

Geostrophic Regimes on a Sphere and a Beta Plane

A general geostrophic equation is derived for a shallow layer of fluid on a sphere. This equation encompasses the planetary, intermediate, and quasi-forms of geostrophy and produces their equations directly when the appropriate parametric ordering relationships are chosen. The three regimes have proven useful for defining and describing oceanic and Jovian eddies and currents on the planetary, intermediate and synoptic scales respectively. The general geostrophic equation may be most useful in describing the interactions among these three different regimes of motion and between motions in high and low latitudes. The accuracy of the β-plane version of these equations is also examined in detail.

Pattern Formation and Wave-Number Selection by Rayleigh–Bénard Convection in a Cylindrical Container

We describe an apparatus and procedures for simultaneous heat transport measurements and computer enhanced shadowgraph flow-pattern imaging in a shallow horizontal layer of fluid heated from below. The heat transport measurements have a resolution of better than 0.1%, and the shadowgraph technique can detect the flow field for ϵ ≡ (R-Rc)/Rc as small as 10-2 (Rc is the critical value of the Rayleigh number R for onset of convection).The apparatus and procedures were used to study pattern and wave-number evolution in a cylindrical layer of water with radius-to-height ratio L = 7.5 and Prandtl number σ = 6.1. We found that dynamic sidewall forcing during the early thermal transients after a change in the heat current from a subcritical to a supercritical value establishes a cylindrical flow pattern. Once created, this pattern is stable in our apparatus over the wide range 0.16 ≲ ϵ ≲ 8 even after the transients have decayed.With changing ϵ, adjustment in the wave number k takes place discontinuously by hysteretic changes at the cell center in the number of convection roll pairs. When ϵ is increased, the discontinuous changes at the cell center are towards smaller k and are preceded by a continuous loss of cylindrical symmetry (the middle roll pair moves off center). The selected wave numbers coincide neither with the zig-zag instability of the infinite system, as once suggested, nor with a linear extrapolation to ϵ = O(1) of the recent prediction to lowest order in ϵ of Manneville and Piquemal and of Cross. Comparison of the selected k with measurements by others reveals no dependence upon L and σ.For ϵ < 0.16, the cylindrical pattern is unstable and decays on a time scale much longer than a horizontal diffusion time to patterns of rolls which tend to be perpendicular to the sidewalls and which contain defects. Once formed, these latter patterns will persist at large values of ϵ. These patterns also undergo a wave-number adjustment process with hysteretic changes mediated mostly by focus singularities near the walls. In these cases, larger values of ϵ also tend to produce smaller values of k.

Roll-pattern evolution in finite-amplitude Rayleigh–Beńard convection in a two-dimensional fluid layer bounded by distant sidewalls

This paper considers the temporal evolution of two-dimensional Rayleigh–Bénard convection in a shallow fluid layer of aspect ratio 2L ([Gt ] 1) confined laterally by rigid sidewalls. Recent studies by Cross et al. (1980, 1983) have shown that for Rayleigh numbers in the range R = R0 + O(L−1) (where R0 is the critical Rayleigh number for the corresponding infinite layer) there exists a class of finite-amplitude steady-state ‘phase-winding’ solutions which correspond physically to the possibility of an adjustment in the number of rolls in the container as the local value of the Rayleigh number is varied. It has been shown (Daniels 1981) that in the temporal evolution of the system the final lateral positioning of the rolls occurs on the long timescale t = O(L2) when the phase function which determines the number of rolls in the system satisfies a one-dimensional diffusion equation but with novel boundary conditions that represent the effect of the sidewalls. In the present paper this system is solved numerically in order to determine the precise way in which the roll pattern adjusts after a change in the Rayleigh number of the system. There is an interesting balance between, on the one hand, a tendency for the number of rolls to change by the least number possible and, on the other, a tendency for the even or odd nature of the initial configuration to be preserved during the transition. In some cases this second property renders the natural evolution susceptible to arbitrarily small external disturbances, which then dictate the form of the final roll pattern.The complete transition involves an analysis of the motion on three timescales, a conductive scale t = O(1), a convective growth scale t = O(L) and a convective diffusion scale t = O(L2).

The Resonant Interaction between a Long Internal Gravity Wave and a Surface Gravity Wave Packet

The long-short wave interactions between an internal gravity wave and a surface gravity wave packet in a two-layer fluid are investigated. When the phase speed of the internal wave coincides with the group velocity of the surface wave packet, a strong interaction occurs. The equations describing this interaction are derived both for a shallow (in comparison with a length scale of the long wave) fluid layer and for a deep one, and are studied numerically.

The effect of distant side-walls on the evolution and stability of finite-amplitude Rayleigh-Bénard convection

A recent study by Cross et al . (1980) has described a class of finite-amplitude phase-winding solutions of the problem of two-dimensional Rayleigh-Bénard convection in a shallow fluid layer of aspect ratio 2 L (≫ 1) confined laterally by rigid side-walls. These solutions arise at Rayleigh numbers R = R 0 + O ( L -1 ) where R 0 is the critical Rayleigh number for the corresponding infinite layer. Nonlinear solutions of constant phase exist for Rayleigh numbers R = R 0 + O ( L -2 ) but of these only the two that bifurcate at the lowest value of R are stable to two-dimensional linearized disturbances in this range (Daniels 1978). In the present paper one set of the class of phase-winding solutions is found to be stable to two-dimensional disturbances. For certain values of the Prandtl number of the fluid and for stress-free horizontal boundaries the results predict that to preserve stability there must be a continual readjustment of the roll pattern as the Rayleigh number is raised, with a corresponding increase in wavelength proportional to R - R 0 . These solutions also exhibit hysteresis as the Rayleigh number is raised and lowered. For other values of the Prandtl number the number of rolls remains unchanged as the Rayleigh number is raised, and the wavelength remains close to its critical value. It is proposed that the complete evolution of the flow pattern from a static state must take place on a number of different time scales of which t = O(( R - R 0 ) -1 ) and t = O(( R - R 0 ) -2 ) are the most significant. When t = O(( R - R 0 ) -1 ) the amplitude of convection rises from zero to its steady-state value, but the final lateral positioning of the rolls is only completed on the much longer time scale t = O(( R - R 0 ) -2 ).

Nonlinear Thermal Convection

The topic of this. review article is thermal convection in thin horizontal fluid layers, uniformly heated from below and/or cooled from above. If the temperature difference between the two horizontal boundaries is sufficiently small, the heat will be transferred through the fluid by conduction alone. For greater temperature differences the conduction state becomes unstable and a convective motion is set up. The first experimental investigations on thermal convection date back to Thomson (1881) and Benard ( 1900). The experiments by Benard in particular have attracted great attention and are today considered classical in fluid mechanics. This is essentially due to the surprising and fascinating pattern of very regular hexagonal cells obtained for large values of time in his experiments. Benard studied convective motion in a very shallow layer of viscous fluid (molten spermaceti) and made the motion visible by graphite or aluminum powder. The theory of thermal convection was initiated by Rayleigh ( 1916). He assumed that the amplitude of the motion was infinitesimal such that the equations could be linearized. He thus derived the critical temperature gradient (or in modern language, the critical Rayleigh number) for the onset of convection together with the wavenumber for the marginal stable mode. The resemblance of the cell pattern in the Rayleigh theory to certain cloud forms was early noticed by meteorologists. For example, it was pointed out that often the observed ratio between the vertical height and the lateral extent in cumulus clouds is the same as found in the theory for hexagonal cells. The theory on cloud formation is perhaps best applied to the convective motion set up in an altostratus layer when it breaks up into altocumulus clouds because of radiative cooling at the top of the layer. This application is, however, obscured by the effect of the release of latent heat, which is not accounted for in the theory. The theory on thermal convection has become very important in the study of motion in the earth's interior (see Turcotte & Oxburgh 1972) and also in some branches of astrophysics (Spiegel 1971 , 1972). The great interest shown to the theory for the last 10--15 years is partly due to these and other applications. The main

Stability of the Rossby-Haurwitz wave

A spectral method is used to integrate the primitive equations for the motion on a sphere of a shallow layer of fluid with a free surface. A simple Rossby-Haurwitz wave of zonal wavenumber 4 is found to change its form little over 24 days, whilst one of wavenumber 8 breaks down completely in 5 days. Inspired by the integrations, the barotropic stability of a wave to a perturbation composed of another wave and a zonal flow is considered. The theory suggests that waves of zonal wavenumber greater than 5 may be unstable because the feedback of the perturbation on itself is positive. Those of wavenumber 5 and less are stable, the feedback being negative. The theory and the general barotropic instability of Rossby waves is tested by integrations of the non-divergent barotropic model, in which the wave is perturbed by a zonal flow. Some agreement with theory is found. Waves of zonal wavenumber 8 rapidly break down to produce jets in the same direction as the perturbation zonal flow. The stability of these jets is also examined.

Stability of the Rossby‐Haurwitz wave

AbstractA spectral method is used to integrate the primitive equations for the motion on a sphere of a shallow layer of fluid with a free surface. A simple Rossby‐Haurwitz wave of zonal wavenumber 4 is found to change its form little over 24 days, whilst one of wavenumber 8 breaks down completely in 5 days. Inspired by the integrations, the barotropic stability of a wave to a perturbation composed of another wave and a zonal flow is considered. The theory suggests that waves of zonal wavenumber greater than 5 may be unstable because the feedback of the perturbation on itself is positive. Those of wavenumber 5 and less are stable, the feedback being negative. The theory and the general barotropic instability of Rossby waves is tested by integrations of the non‐divergent barotropic model, in which the wave is perturbed by a zonal flow. Some agreement with theory is found. Waves of zonal wavenumber 8 rapidly break down to produce jets in the same direction as the perturbation zonal flow. The stability of these jets is also examined.

Friction Term Formulation and Convective Instability in a Shallow Atmosphere

Abstract The form of the friction terms for a shallow layer of fluid on a sphere is discussed for isotropic and transversely-isotropic fluids. We then examine the nature of convection in a transversely-isotropic fluid and find that long flat convection cells with a width to height ratio of 2(νH/νV)½ are produced, where νH, νV are the horizontal and vertical diffusion coefficients. The critical Rayleigh number is given by Rc=4π4νH/νV, but another Rayleigh number P=βgΔTd5/(νVκVL), with a constant critical value Rc=4π4 is shown to be a more relevant parameter. Results for convection in a rotating system are also given.

Convection experiments with electrolytically heated fluid layers

Convection experiments described by Tritton &amp; Zarraga (1967) with electrolytically heated fluid layers were renewed in order to investigate the reported phenomena, which were hitherto unknown and which contradicted a corresponding theory of Roberts. While the apparatus was essentially unchanged, provisions were incorporated to study the possible influence of several flow and equipment parameters on the convection pattern. With the exception of the temperature dependence of the electric conductivity, the new experiments displayed no essential effects of the convection parameters. Experiments with shallow fluid layers revealed a clear co-orientation of the convection flows with the electric current and a strong time dependence of the hexagonal patterns. Experiments with deeper fluid layers exhibited a considerably diminished time and direction dependence of the convection flow, and a significant reduction of the dilation of the cells. Based on these observations, it is concluded that no drastic differences between theory and experiments, and between internal and external heating, exist, provided the heating is sufficiently uniform.

Distant side-walls cause slow amplitude modulation of cellular convection

The effect of vertical boundaries on convection in a shallow layer of fluid heated from below is considered. By means of a multiple-scale perturbation analysis, results for horizontally unbounded layers are modified so that they ‘fit in’ to a rectangular region. The critical Rayleigh number and critical wave-number are determined. Motion is predicted to have the form of finite ‘rolls’ whose axes are parallel to the shorter sides of the dish. Aspects of the non-linear development and stability of this motion are studied. The general question of convective pattern selection in a bounded layer is discussed in the light of available theoretical and experimental results.