Constant Density Gradient Research Articles

Gravity anomalies of irregularly shaped two‐dimensional bodies with constant horizontal density gradient

An equation to compute the gravity anomalies of two‐dimensional (2-D) bodies with density contrast varying with depth (z axis) was developed by Murthy and Rao (1979). I develop an equation for computing the gravity anomalies of 2-D bodies with constant horizontal density gradient. By combining this equation with the equation of Murthy and Rao, I estimate the depth of the sedimentary basin which is adjacent to the master fault associated with the Rio Grande rift in New Mexico, where the density is assumed to decrease basinward from the fault (Cordell, 1979).

Stably stratified shear flow over low hills

AbstractAn analysis is presented for the mean flow over hills with low slopes, for various kinds of stable stratification in the approach flow. (The Froude number F based on the height is always greater than unity.) The analysis is based on dividing up the flow into different regions with different dynamics.We concentrate on the flow in the inviscid outer region of the flow. It is shown that in the lower part of the outer region, the middle layer, where the shear is large, the upwind velocity profile and the shape of the hill, rather than the upwind density profile, largely determine the vertical perturbation velocity. In the ‘upper layer’ the structure of the flow depends largely on the form of the upwind density profile. For example, this largely determines the location of maximum surface wind. New solutions for stable conditions are obtained when there is a constant density gradient, when (N/UB)2 ∞ z−2 and when there is an elevated inversion layer. In the first and third of these cases, we compare the results with the hydrostatic approximation, and also obtain new general formulae for the near‐surface winds. An approximate formula is developed for surface winds at arbitrary values of F(>1) for uniform stable stratification.

The nature of upstream blocking in uniformly stratified flow over long obstacles

The general method described in Baines 1988 has been applied to stratified flows of finite depth over long obstacles where the flow initially has uniform horizontal velocity. The fluid consists of a finite number of homogeneous layers of equal thickness and with equal density increments. This represents the state of continuous stratification with constant density gradient as closely as possible, for a given number of layers. Two-, three-, four- and sixty-four-layered models are studied in detail. The results are expressed in terms of the initial Froude number F0 (F0 = U/ĉ1 where U is the fluid speed and ĉ1 is the speed of the fastest long internal wave mode in the fluid at rest) and the obstacle height. In general, introduction of an obstacle into the flow causes disturbances to propagate upstream (columnar disturbance modes) which alter the velocity and density profiles there. These may accumulate to cause upstream blocking of some of the fluid layers if F0 is sufficiently small. As the number of fluid layers increases, so does the range of F0 for which this upstream blocked flow occurs. There are no upstream disturbances for F0 > 1, and for F0 < 1 the upstream disturbances are of the rarefaction type if upstream blocking does not occur. The results for three and four layers show how several coexisting modes may interact to affect the upstream profiles. The results for sixty-four-layers provide theoretical support for the observational criterion (Baines 1979b) that blocking in initially uniformly stratified flow occurs when Nhm/U > 2 (N is the Brunt—Väisälä frequency and hm the obstacle height), provided that more than two modes are present. In some situations, layered models are found to be inadequate as a representation of continuous stratification when one or more layers thicken to the extent that their discreteness is significant.

The formation of longitudinal fronts in a coastal plain estuary

Longitudinal estuarine fronts, aligned parallel to the axis of the estuary, are characteristic of the York River. Their time scale for genesis and decay is of the order of hours; they are usually located at the inner edge of the shoals or over the main channel. Field measurements have shown marked intratidal differences in density and current speeds across this estuary. These can be correlated with changes in depth and the strength of the advective processes. Differential advection between the channel and shoal regions, when acting upon a constant longitudinal density gradient, is shown to be of sufficient magnitude to generate strong lateral density gradients, and thus fronts, at particular times within the tidal cycle.

The effect of vertical temperature and salinity gradients on double-diffusive interleaving

Theoretical and experimental work has been carried out to understand the almost horizontal oceanic intrusions that occur at density-compensating fronts, and in particular how the layer separation and slope of the intrusions depend on the vertical concentration gradients of the two diffusing components. In the experiments salt and sugar were used as analogues of ocean temperature and salinity, respectively. Matching stable linear density distributions were set up on the two sides of a vertical barrier, with an excess of sugar on one side and salt on the other. When the barrier was removed interleaving motions formed spontaneously. A similar experiment was performed by Ruddick and Turner (1979, Deep-Sea Research, 26, 903–913), but they assumed that the ensuing motion depended only on the total vertical density gradient and not on the relative contributions to this from the two components. In oceanic terms, the layer depth was found to be approximately independent of salinity gradient S z at constant total density gradient, except when S z was very small. For zero temperature gradient, T z = 0, h varied as S z to the power −0.7 ± 0.2. In all experiments the slope of intrusions was such that hot, salty water sank. A theoretical analysis is given of the dependence of the sense of layer slope on the ratio of the two vertical gradients, and suggestions are made for further experiments. Preliminary results of an optical investigation of the detailed structure of the layers are reported.

The stability of vortices in a rotating, stratified fluid

Axisymmetric flows with a two-layer density stratification are produced by releasing either a constant flux of fluid from a point source or a constant volume of fluid into a rotating environment with a different density. In both experiments the density interface intersects one horizontal boundary, forming a front. Transition to non-axisym-metric flow is observed and can be described by two parameters: θ, the square of the ratio of the internal Rossby radius of deformation to the horizontal length scale of the flow, and δ, the fraction of the total fluid depth occupied by the layer inside the front. For θ [Lt ] 1 and δ > 10−1 unstable disturbances obtain most of their energy from the potential energy of the flow, whilst for δ < 10−1 extraction of kinetic energy from the basic shear becomes the dominant driving mechanism. When the front intersects the free surface, n = 2 is the minimum azimuthal wavenumber for an unstable disturbance. At large amplitude of the growing waves, baroclinic and barotropic processes combine to form n vortex dipole structures which entrain buoyant fluid from the original vortex and propagate radially over the free surface. Vortices are also produced by the continuous release of fluid from a confined source at its own density level in a region of constant density gradient. As in the two-layer case the axisymmetric vortex grows to a critical size and then becomes unstable to a disturbance with wavenumber n = 2, producing, at large amplitude, two vortex pairs.

On the nonlinear reflexion of a gravity wave at a critical level. Part 1

In this paper we examine the nonlinear interaction of a forced internal gravity wave in a stratified fluid with its critical level. The representative Richardson number J is taken to be large and the undisturbed state consists of a hyperbolic-tangent velocity profile and an almost constant density gradient. It is assumed that at large values of a non-dimensional time t the flow outside the critical layer is steady, consisting of the mean shear together with a disturbance periodic in x that corresponds to the single harmonic of the incident wave of small amplitude e. The requirements of a match across the critical layer lead to a reflected wave and a transmitted wave both of whose amplitudes are O (e e −νπ ) when 1 [Lt ] t [Lt ] e −2/3 , where ν = ( J − ¼) ½ . For ν [Gt ] 1 the layer therefore acts as a wave absorber, and the purpose of this investigation is to ascertain whether this property persists on an even longer time scale. At times t = O (e −2/3 ) the layer has thickness O (e 2/3 ) and the first few terms of an expansion in powers of e 2/3 t show that higher harmonics are forced on the outer flow, and the reflexion and transmission coefficients develop with time. The leading-order correction to these coefficients is calculated explicitly; that to the transmission coefficient is again exponentially small in ν though that to the reflexion coefficient is O (ν −1 ). The reflexion coefficient is therefore increasing and the critical layer begins to restore wave energy to the outer flow. Owing to the complexity of the calculation higher-order corrections are not obtained here, but the results presented are in agreement with predictions of earlier workers that the layer acts as an absorber and a reflector but not as a transmitter.

On internal gravity waves in an accelerating shear flow

The investigation of the effects which a changing mean flow has on a uniform train of internal gravity waves (Thorpe 1978a) is continued by considering waves in a uniformly accelerating stratified plane Couette flow with constant density gradient. Experiments reveal a change in the mode structure and phase distribution of the waves, and their eventual breaking near the boundary where the mean flow is greatest, the phase speed of the waves being positive. A linear numerical model is devised which accurately describes the waves up to the onset of their breaking, and this is used to investigate their energetics. The working of the Reynolds stress against the mean velocity gradient results in a very rapid transfer of energy from the waves to the mean flow, so that by the time breaking occurs only a small fraction of their initial energy remains for possible transfer into potential energy of the fluid.The consequences have important applications in oceanography and meteorology, to flow stability and flow generation, and explain some earlier laboratory observations.

Repeated-cascade theory of strong turbulence in a magnetized plasma

A two-dimensional Navier-Stokes equation of vorticity in fluid turbulence is used to model drift turbulence in a plasma with a strong constant magnetic field and a constant mean density gradient. The nonlinear eddy diffusivity is described by a time-integrated Lagrangian correlation of velocities, and the repeated-cascade method is employed to choose the rank accounting for nearest-neighbor interactions, to calculate the Lagrangian correlation, and to close the correlation hierarchy. As a result, the diffusivity becomes dependent on the plasma's induced diffusion and is represented by a memory chain that is cut off by similarity and inertial randomization. Spectral laws relating the kinetic-energy spectrum to the -5, -5/2, -3, and -11 powers of wavenumber are derived for the velocity subranges of production, approach to inertia, inertia, and dissipation, respectively. It is found that the diffusivity is proportional to some inverse power of the magnetic field, that power being 1, 2/3, 5/6, and 2, respectively, for the four velocity subranges.

The deepening of a mixed layer in a stratified fluid

In this paper two aspects of the deepening of a mixed layer in a stratified fluid are examined in the laboratory. The first is the deepening of a layer into a region of constant density gradient. Turbulence is produced by an oscillating grid which generates a horizontally homogeneous field of motion with no significant mean flow. It is found that the rate at which the potential energy of the basic stratification is increased by the mixing does not bear a simple relationship to the rate of energy input by the grid. On the other hand, when allowance is made for the decay of turbulent energy away from the grid and only that portion to reach the bottom of the mixed layer is considered, the rate of potential energy increase is found to be proportional to this available energy. The second aspect to be discussed is the effect of energy radiation by internal waves in the region below the mixed layer. Estimates are made of the possible loss of energy to these waves, which reduces the amount available to deepen the layer. An experimental demonstration of up to 50 % reduction in the mixing rate due to the presence of internal waves is given. Finally, the implications of these results are discussed in the light of current theoretical models of the deepening process.

On standing internal gravity waves of finite amplitude

Two-dimensional internal gravity waves in a rectangular container are examined theoretically and experimentally in (a) fluids which contain a single density discontinuity and (b) fluids in which the density gradient is everywhere continuous. The fractional density difference between the top and bottom of the fluid is small.Good agreement is found between the observed and calculated wave profiles in case (a). Unlike surface standing waves, which tend to sharpen at their crests as the wave amplitude increases, and which eventually break at the crests when fluid accelerations become equal to that of gravity, internal wave crests are found to be flat and exhibit no instability. In the case (a) breaking is found to occur at the nodes of the interfacial wave, where the current shear, generated by the wave itself, is greatest. For sufficiently large wave amplitudes, a disturbance with the form of a vortex but with direction of rotation reversing twice every cycle, grows at the wave node and causes mixing. This instability is found to be followed by the generation of cross-waves, of which two different forms are observed.Several modes of oscillation can be generated and are observed in a fluid with constant density gradient. The wave frequencies and shape are well predicted by theory. The experiments failed to establish any limitation of the possible wave amplitudes.

A Simple Device for Making Constant Density Gradients

Abstract The possibility of adapting the well-known mixing device shown in Fig. 1 for production of density gradients with retained hydrostatic equilibrium is investigated. It is shown that a specific form of the unstirred container can be found that gives a strictly linear density course. Instead of specifically shaped containers, specific plungers can be used in cylindrical containers. All pertinent equations arc given. The modified arrangement is useful for falling as well as for rising densities by use of exchangeable plungers. The communicating duct between the containers has to have a large bore. With the aid of a V-shaped channel in a stopcock, turnable to a A orientation, hydrostatic stability can be retained under all circumstances. In the experimental section it is shown that accurately constant density gradients are really obtained by using plungers constructed according to the requirements of the theory. It is also shown that, under certain conditions, approximately constant density gradients ...

Penetrative Convective Instabilities

An approximate method for finding the critical conditions for stability with respect to penetrative cellular perturbation is proposed. The method is tested by finding the critical conditions for stability of flow between counter-rotating cylinders and by comparing them with those obtained by more rigorous methods and by experiments. The same method is later applied for Bénard convection arising in a horizontal layer of fluid with constant unstable density gradient and penetrating into a layer of fluid with constant stable density gradient situated above the unstable layer. It is found that the classical rigid-free boundary solution corresponds to the limiting case of infinite stability on top of the unstable layer. For small stability on top the solutions have some of the features of Chandra's columnar perturbations and the build up of cumulus towers in the tropics. The critical value of the Rayleigh number in this case is found to be much smaller than that obtained by not taking penetration into account.

Internal waves produced by a convective layer

Fluid impacts on the base of a stably stratified region of fluid cause internal-wave ripples whose spread is predominantly horizontal if the duration of the impacts is long compared with the natural period of the stratified fluid. The development of a single ripple in a slightly viscous fluid is calculated, first with a constant vertical gradient of potential density and then with a gradient varying linearly with height. The single-ripple results are used to find the intensity of the statistically steady wave motion generated by impacts which are randomly distributed in space and time. Above a critical height, dependent on the viscosity and stability of the fluid and on the time and length scales of the impacts, wave energy falls off as the −5/3 power of the height with a constant density gradient and as the −25/6 power with a linearly varying gradient. The predictions are compared with observations of temperature fluctuations in the stable region of an ice-water convection system and with observations of ‘clear-air turbulence’ over strato-cumulus cloud. Reasonable numerical agreement can be obtained with plausible values for the scales of the convective motion which provides the impacts.

Turbulent Diffusion in a Stratified Fluid

Abstract This paper examines the dynamical effects on a particle drifting in a homogeneous field of turbulence, stationary and uniform except for a constant vertical density gradient. The ordinary “mechanical” impulses due to random normal pressures and shear stresses are assumed to have the same statistical properties as in the absence of density stratification. The buoyancy forces arising from the displacement of the particle are, on the other hand, seen to be more nearly “organized” forces and their presence in the equations of motion modifies Lagrangian velocity autocorrelations significantly. The results show that under unstable stratification the velocity history of a diffusing particle is not stationary to second order, the variance increasing exponentially with time. When the stratification is stable, on the other hand, the standard deviation of vertical displacement tends asymptotically to a constant value. These results are in qualitative agreement with observations of the Prairie Grass project ...

Penetrative convection in stably stratified fluids

The downward convection of discrete masses of heavy fluid, called thermals, in a gravitationally stable environment is studied experimentally. For simplicity the ambient stratification is chosen to have a discontinuous two-layer structure with fluid of uniform density in each layer. When released into the homogeneous upper layer the thermal acquires the circulation of a turbulent ring vortex as reported earlier. For weak stratifications, which are however capable of halting the thermal, this circulation is destroyed in a dramatic fashion during the penetration of the lower layer; restoring buoyancy forces then develop a reversed circulation as the thermal flattens and spreads. For strong stratifications no reversal of the circulation occurs, rather the flattening and spreading of the thermal is associated with a gradual decay of the original circulation. A dimensional argument is proposed to indicate the parameters which determine the extreme penetration of the thermal into the lower layer: this is confirmed by experiment. In order to generalize the results to more realistic geophysical cases a mechanistic description of thermal behaviour is put forward. The treatment is shown to rationalize the data for environments with ( a ) neutral stratification, ( b ) a discontinuous two-layer structure and ( c ) constant density gradient. Thermal behaviour in a stratification consisting of ( a ) surmounted by ( c ) is then predicted. Observations made by the author of the penetration of tall cumulus clouds into the sub-tropical stratosphere permit a test of the applicability of this model in one geophysical area. For every 1 km of penetration it is deduced that convective motions have an average vertical intensity of 10 m/sec: this prediction is shown to be in good agreement with the limited observational data. DOI: 10.1111/j.2153-3490.1962.tb00130.x

Penetrative convection in stably stratified fluids

The downward convection of discrete masses of heavy fluid, called thermals, in a gravitationally stable environment is studied experimentally. For simplicity the ambient stratification is chosen to have a discontinuous two-layer structure with fluid of uniform density in each layer. When released into the homogeneous upper layer the thermal acquires the circulation of a turbulent ring vortex as reported earlier. For weak stratifications, which are however capable of halting the thermal, this circulation is destroyed in a dramatic fashion during the penetration of the lower layer; restoring buoyancy forces then develop a reversed circulation as the thermal flattens and spreads. For strong stratifications no reversal of the circulation occurs, rather the flattening and spreading of the thermal is associated with a gradual decay of the original circulation.
 A dimensional argument is proposed to indicate the parameters which determine the extreme penetration of the thermal into the lower layer: this is confirmed by experiment.
 In order to generalize the results to more realistic geophysical cases a mechanistic description of thermal behaviour is put forward. The treatment is shown to rationalize the data for environments with (a) neutral stratification, (b) a discontinuous two-layer structure and (c) constant density gradient. Thermal behaviour in a stratification consisting of (a) surmounted by (c) is then predicted.
 Observations made by the author of the penetration of tall cumulus clouds into the sub-tropical stratosphere permit a test of the applicability of this model in one geophysical area. For every 1 km of penetration it is deduced that convective motions have an average vertical intensity of 10 m/sec: this prediction is shown to be in good agreement with the limited observational data.

Buoyant vortex rings

It is pointed out that there is a fundamental difference in the behaviour of vortex rings projected upwards, according as they do or do not contain fluid which is lighter than the surroundings. A theory based on the perfect fluid approximation is developed to describe the motion of buoyant rings in a uniform fluid. The essential assumption is that the circulation remains constant with time while the buoyancy force acts to increase the impulse of a ring. This leads to the prediction that increasing the buoyancy will give a greater rate of expansion and a lower velocity of rise. The theory is extended to the case of a ring rising through a stably stratified fluid having a constant density gradient; in this case increasing the buoyancy should lead to a lower final height. The predictions of the theory have been verified by carrying out experiments in the laboratory with small vortex rings formed in water, using methylated spirits and salt to produce the density differences. The observations suggest that substantially the same analysis may be applicable to phenomena on a larger scale in the atmosphere; the velocity and final height of an explosion cloud should be determined by the buoyancy and circulation generated near the ground and the stability conditions of the atmosphere.