Bifurcation of a free boundary equilibrium

1984 ◽  
Vol 31 (1) ◽  
pp. 1-6 ◽  
Author(s):  
A. Turnbull
Keyword(s):  
Simple Model ◽  
Free Boundary ◽  
Unique Solution ◽  
Bifurcation Point ◽  
Numerical Models ◽  
Boundary Equilibrium ◽  
Reversed Current

The problem of bifurcation of a free boundary equilibrium is reconsidered in a simple model. We find which combinations of specified parameters result in a non-unique solution and find the resulting bifurcation point. By including reversed current solutions, the model sheds some light on the equilibrium βp limit obtained in previous analytic and numerical models.

scholarly journals Flow fields associated with in situ mineral leaching

1994 ◽  
Vol 36 (2) ◽  
pp. 133-151 ◽  
Author(s):  
Lawrence K. Forbes ◽  
Anthony M. Watts ◽  
Graeme A. Chandler
Keyword(s):  
Simple Model ◽  
Free Boundary ◽  
Numerical Method ◽  
Flow Rate ◽  
Recovery Rate ◽  
Flow Fields ◽  
Vertical Line ◽  
Rate Limiting ◽  
Mineral Leaching

AbstractA simple model for underground mineral leaching is considered, in which liquor is injected into the rock at one point and retrieved from the rock by being pumped out at another point. In its passage through the rock, the liquor dissolves some of the ore of interest, and this is therefore recovered in solution. When the injection and recovery points lie on a vertical line, the region of wetted rock forms an axi-symmetric plume, the surface of which is a free boundary. We present an accurate numerical method for the solution of the problem, and obtain estimates for the maximum possible recovery rate of the liquor, as a fraction of the injected flow rate. Limiting cases are discussed, and other geometries for fluid recovery are considered.

scholarly journals Baroclinic Instability with a Simple Model for Vertical Mixing

2019 ◽  
Vol 49 (12) ◽  
pp. 3273-3300 ◽  
Author(s):  
Matthew N. Crowe ◽  
John R. Taylor
Keyword(s):  
Growth Rate ◽  
Stability Analysis ◽  
Simple Model ◽  
Numerical Simulations ◽  
Mixed Layer ◽  
Numerical Models ◽  
Baroclinic Instability ◽  
Vertical Mixing ◽  
Small Scale ◽  
Surface Mixed Layer

AbstractHere, we examine baroclinic instability in the presence of vertical mixing in an idealized setting. Specifically, we use a simple model for vertical mixing of momentum and buoyancy and expand the buoyancy and vorticity in a series for small Rossby numbers. A flow in subinertial mixed layer (SML) balance (see the study by Young in 1994) exhibits a normal mode linear instability, which is studied here using linear stability analysis and numerical simulations. The most unstable modes grow by converting potential energy associated with the basic state into kinetic energy of the growing perturbations. However, unlike the inviscid Eady problem, the dominant energy balance is between the buoyancy flux and the energy dissipated by vertical mixing. Vertical mixing reduces the growth rate and changes the orientation of the most unstable modes with respect to the front. By comparing with numerical simulations, we find that the predicted scale of the most unstable mode matches the simulations for small Rossby numbers while the growth rate and orientation agree for a broader range of parameters. A stability analysis of a basic state in SML balance using the inviscid QG equations shows that the angle of the unstable modes is controlled by the orientation of the SML flow, while stratification associated with an advection/diffusion balance controls the size of growing perturbations for small Ekman numbers and/or large Rossby numbers. These results imply that baroclinic instability can be inhibited by small-scale turbulence when the Ekman number is sufficiently large and might explain the lack of submesoscale eddies in observations and numerical models of the ocean surface mixed layer during summer.

scholarly journals VARIOUS LEVELS OF MODELS FOR AEROSOLS

2002 ◽  
Vol 12 (07) ◽  
pp. 903-919 ◽  
Author(s):  
PIERRE-EMMANUEL JABIN
Keyword(s):  
Friction Coefficient ◽  
Simple Model ◽  
Free Boundary ◽  
Boundary Problem ◽  
Free Boundary Problem ◽  
Aerosol Particles ◽  
Distribution Functions ◽  
Macroscopic System ◽  
Dirac Mass ◽  
A Free Boundary Problem

Two limit behaviours of a simple model of aerosol are considered. The only force acting on aerosol particles is a friction due to the flow of gas. It is first proved that in the limit of an infinite friction coefficient, the particles are simply advected by the gas. Then we consider very dilute sprays of aerosol, i.e. with distribution functions which are monokinetic (Dirac mass in velocity). This approach leads to a macroscopic system with a free-boundary problem.

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