Jan Erik Weber

THE BOUNDARY-LAYER REGIME FOR CONVECTION IN A VERTICAL POROUS LAYER

Abstract Buoyancy-driven convection in a differentially heated vertical porous layer is studied theoretically by the method developed by Gill [;5];. The model is of finite extent, and the temperature difference between the vertical walls is assumed to be large. Satisfactory agreement with experiment has been obtained for the interior temperature distribution and the Nusselt number. The applied method is also extended to include some effects of a variable viscosity. This is shown to introduce asymmetry into the solutions.

On steady convection in a porous medium

For convection in a porous medium the dependence of the Nusselt number on the Rayleigh number is examined to sixth order using an expansion for the Rayleigh number proposed by Kuo (1961). The results show very good agreement with experiment. Additionally, the abrupt change which is observed in the heat transport at a supercritical Rayleigh number may be explained by a breakdown of Darcys law.

Steady wind- and wave-induced currents in the open ocean

Abstract Steady wind-drift currents in a deep viscous rotating ocean are studied theoretically. The analysis is based on the Lagrangian description of motion. A mean wind-stress at the surface yields the traditional Ekman current. In addition, the wind-stress is assumed to contain a fluctuating part which transfers energy to the surface waves and compensates for loss due to viscous dissipation. The induced drift due to such waves is investigated. The wave-drift depends on the eddy viscosity as well as the earths rotation. We assume a fully developed sea, and take the eddy viscosity to be proportional to the friction velocity times a characteristic depth. Hence the total current (Ekman current plus wave-induced current) can be expressed as functions of the wind speed. The results show that the magnitude of the total surface current lies between 3.1 and 3.4% of the wind speed at 10 m height for winds between 5 and 30 m s−1. The deflection angle away from the wind direction varies from 23 to 30° in this ran...

Convection in a porous medium with horizontal and vertical temperature gradients

Abstract The stability of convection in a horizontal porous layer subjected to horizontal as well as vertical temperature gradients is investigated. The boundaries are taken to be perfectly conducting and the horizontal temperature gradient is assumed to be small. The analysis shows that the critical Rayleigh number is always larger than for the ordinary Benard problem in a porous medium. The preferred mode of disturbance is stationary, being longitudinal rolls, i.e. rolls having axes aligned in the direction of the basic flow. This particular mode minimizes the potential energy. Assuming that the initially preferred mode also dominates at supercritical Rayleigh numbers, a finite amplitude solution is obtained. The vertical heat flux is computed to second order. Compared with Benard convection in a porous medium, the perturbation heat flux is diminished. The flux due to the basic flow is increased, however, so the total vertical heat flux is increased.

Wave Attenuation and Wave Drift in the Marginal Ice Zone

Abstract Surface gravity waves in a viscous rotating ocean are studied theoretically when they penetrate an area covered by highly concentrated brashlike ice. The motion is described by a Lagrangian formulation, and the brash is modeled by a viscous Newtonian fluid. Results for wave attenuation and wave drift are obtained in the asymptotic limit of a thin, very viscous upper layer. The derived damping rate compares favorably with field data from the marginal ice zone (MIZ). The drift velocity in the ocean exhibits a marked maximum in the viscous boundary layer near the ice-ocean interface. At the outer edge of the boundary layer it exceeds the inviscid Stokes drift by a factor of 7/4. Computed values for the mean viscous drag on the ice induced by the wave motion show that this effect may compete with the frictional effected of the wind in packing the ice. Finally it is demonstrated that the integrated horizontal mass transports in the open ocean and under the ice do not match, which leads to upwelling in...

On thermal convection between non-uniformly heated planes

Abstract The stability of natural convection in a thin, horizontal layer subjected to horizontal as well as vertical temperature gradients is investigated on the basis of linear theory. The boundaries are taken to be stress-free and perfectly conducting and the horizontal temperature gradient is assumed to be small. The analysis shows that the critical Rayleigh number is always larger than that for the ordinary Benard problem. The preferred mode of disturbance is stationary, and will be a transverse roll (having axes normal to the basic flow) or a longitudinal roll (having axes aligned in the direction of the basic flow) depending on whether the Prandtl number is less or larger than 5.1. Finally, some calculations are made of the converted energy associated with the unstable perturbations, indicating that the mechanism of instability is of thermal (convective) origin.

Thermal convection in a tilted porous layer

It seems, therefore. that for departure diameters less than about 1.6 mm or for Jakob numbers less than about 16, the bubble departure is controlled by surface tension force while for D,, > 1Omm (approx.) or N,, > 100 (approx.) the inertia forces control bubble departure. For departure diameters between 1.6 and 10 mm (approx.) or N,, between 16 and 100 (approx.) the surface tension and inertia forces are of nearly equal importance. Figure 3 shows Dd as function of N,,. Curve 1 is a plot of experimental departure diameters while curves 3 and 3 show respectively the values of Dd obtained from the following equations:

Attenuated wave-induced drift in a viscous rotating ocean

Mean drift currents due to spatially periodic surface waves in a viscous rotating fluid are investigated theoretically. The analysis is based on the Lagrangian description of motion. The fluid is homogeneous, the depth is infinite, and there is no continuous energy input at the surface. Owing to viscosity the wave field and the associated mass transport will attenuate in time. For the non-rotating case the present approach yields the time-decaying Stokes drift in a slightly viscous ocean. The analysis shows that the drift velocities are finite everywhere. In a rotating fluid it is found that the effect of viscosity implies a non-zero net mass transport associated with the waves, as opposed to the result of no net transport obtained from inviscid theory (Ursell 1950).

Eulerian versus Lagrangian Approaches to the Wave-Induced Transport in the Upper Ocean

Abstract It is demonstrated that the Eulerian and the Lagrangian descriptions of fluid motion yield the same form for the mean wave-induced volume fluxes in the surface layer of a viscous rotating ocean. In the Eulerian case, the volume fluxes are obtained in the familiar way by integrating the horizontal components of the Navier–Stokes equation in the vertical direction, as seen, for example, in the book by Phillips. In the direct Lagrangian approach, the perturbation equations for the second-order mean drift are integrated in the vertical direction. This yields the advantage that the form drag, which is a source term for the wave-induced transports, can be related to the virtual wave stress that acts to transfer dissipated mean wave momentum into mean currents. In particular, for waves that are periodic in space and time, comparisons between empirical and theoretical relations for the form drag yield an estimate for the wave-induced bulk turbulent eddy viscosity in the surface layer. A simplistic approa...

Ekman Currents and Mixing due to Surface Gravity Waves

Abstract The steady Ekman boundary-layer current is studied theoretically for the case when the eddy viscosity is proportional to the shear of the wave orbital velocity in a turbulent wave, times the square of a mixing length (Kitaigorodsky, 1961). Assuming a fully developed sea, the wave characteristics, and hence the eddy viscosity distribution with depth, are determined by the wind. The momentum equation is solved numerically to yield the Ekman current as a function of the wind speed. The results show that the magnitude of the Ekman surface current lies between 2.1 and 3% of the wind speed at 10 m height. The deflection angle away from the wind direction is a monotonic decreasing function of wind speed. It varies from 36 to 25° for winds between 5 and 30 m s−1.