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  • covariance  (1)
  • linear stability analysis  (1)
  • 1
    Electronic Resource
    Electronic Resource
    Advection-diffusion in spatially random flows: Formulation of concentration covariance (1997)
    Springer
    Stochastic environmental research and risk assessment 11 (1997), S. 397-422 
    Details
    ISSN: 1436-3259
    Keywords: Random flow field ; diffusion ; concentration ; fluctuations ; covariance ; variance
    Source: Springer Online Journal Archives 1860-2000
    Topics: Architecture, Civil Engineering, Surveying , Energy, Environment Protection, Nuclear Power Engineering , Geography , Geosciences
    Notes: Abstract The concentration c(x,t) of a nonreactive solute undergoing advection and diffusion in a spatially random divergence-free flow field is analyzed. A leading order formulation for the spatial covariance of the concentration field, $$\overline {c'\left( {x,t} \right)c'\left( {x,t} \right)} $$ , is made. That formulation includes the velocity variability induced macrodispersive flux of the covariance field, and the smoothing effects of diffusion. Previous formulations of the concentration covariance had dropped at least one of these effects. It is shown that both these effects need to be included to obtain a qualitatively correct description of the concentration fluctuations.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF02427926
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  • 2
    Electronic Resource
    Electronic Resource
    Criterion for instability of steady-state unsaturated flows (1996)
    Springer
    Transport in porous media 25 (1996), S. 313-334 
    Details
    ISSN: 1573-1634
    Keywords: unsaturated flow ; instabilities ; fingering ; linear stability analysis ; cutoff wavelength ; instability criterion ; nonlinearity ; spatial-temporal complexity ; exponential growth ; moisture profiles
    Source: Springer Online Journal Archives 1860-2000
    Topics: Geosciences , Technology
    Notes: Abstract The stability of steady-state solutions to the unsaturated flow equation is examined. Conditions under which infinitesimal disturbances are amplified are determined by linear stability analysis. Uniform suction head profiles are shown to be linearly stable to three-dimensional disturbances. The stability of nonuniform suction head profiles to planar (gc 1 − χ 2) disturbances is examined. When the steady-state suction head solution (Ψ) increases with depth, χ3, (dΨ/dχ3 〉 0), a condition for the amplification of infinitesimal planar disturbances is identified as $$\frac{{d^2 K(\Psi )}}{{d\Psi ^2 }} 〉 \frac{{\left( {\frac{{dK(\Psi )}}{{d\Psi }}} \right)^2 }}{{K(\Psi )}},$$ , where K(Ψ) is the hydraulic conductivity versus suction head characteristic of the porous medium. The same condition applies when dΨ/dχ3 〈 -1. Therefore when the rate of change of the slope of the K - Ψ characteristic curves is larger than the squared slope divided by K, even small disturbances can be amplified exponentially. The smallest wavelength of unstable planar perturbations is shown to be inversely related to the coarseness of the soil. Conditions under which the instability criterion is met are delineated for some commonly employed K - Ψ curves.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF00140986
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