TY - JOUR
T1 - A Model for Staircase Formation in Fingering Convection
AU - Paparella, Francesco
AU - von Hardenberg, Jost
N1 - Publisher Copyright:
© 2014, Springer Science+Business Media Dordrecht.
PY - 2014/8
Y1 - 2014/8
N2 - Fingering convection is a convective instability that occurs in fluids where two buoyancy-changing scalars with different diffusivities have a competing effect on density. The peculiarity of this form of convection is that, although the transport of each individual scalar occurs down-gradient, the net density transport is up-gradient. In a suitable range of non-dimensional parameters, solutions characterized by constant vertical gradients of the horizontally averaged fields may undergo a further instability, which results in the alternation of layers where density is roughly homogeneous with layers where there are steep vertical density gradients, a pattern known as “doubly-diffusive staircases”. This instability has been interpreted in terms of an effective negative diffusivity, but simplistic parameterizations based on this idea, obviously, lead to ill-posed equations. Here we propose a mathematical model that describes the dynamics of the horizontally-averaged scalar fields and the staircase-forming instability. The model allows for unstable constant-gradient solutions, but it is free from the ultraviolet catastrophe that characterizes diffusive processes with a negative diffusivity.
AB - Fingering convection is a convective instability that occurs in fluids where two buoyancy-changing scalars with different diffusivities have a competing effect on density. The peculiarity of this form of convection is that, although the transport of each individual scalar occurs down-gradient, the net density transport is up-gradient. In a suitable range of non-dimensional parameters, solutions characterized by constant vertical gradients of the horizontally averaged fields may undergo a further instability, which results in the alternation of layers where density is roughly homogeneous with layers where there are steep vertical density gradients, a pattern known as “doubly-diffusive staircases”. This instability has been interpreted in terms of an effective negative diffusivity, but simplistic parameterizations based on this idea, obviously, lead to ill-posed equations. Here we propose a mathematical model that describes the dynamics of the horizontally-averaged scalar fields and the staircase-forming instability. The model allows for unstable constant-gradient solutions, but it is free from the ultraviolet catastrophe that characterizes diffusive processes with a negative diffusivity.
KW - Doubly-diffusive convection
KW - Negative diffusion
KW - Perona-Malik equation
KW - Thermohaline staircases
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U2 - 10.1007/s10440-014-9920-1
DO - 10.1007/s10440-014-9920-1
M3 - Article
AN - SCOPUS:84919877526
SN - 0167-8019
VL - 132
SP - 457
EP - 467
JO - Acta Applicandae Mathematicae
JF - Acta Applicandae Mathematicae
IS - 1
ER -