Imperial College London
Professor John Gibbon

ProfessorJohnGibbon

Faculty of Natural Sciences, Department of Mathematics

Senior Research Investigator
 
 
 
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Contact

 

j.d.gibbon Website

 
 
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Location

 

6M41Huxley BuildingSouth Kensington Campus

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Summary

 

Publications

Citation

BibTex format

@article{Gibbon:2017:1361-6544/aa6946,
author = {Gibbon, JD and Holm, DD},
doi = {1361-6544/aa6946},
journal = {Nonlinearity},
pages = {R1--R24},
title = {Bounds on solutions of the rotating, stratified, incompressible, non-hydrostatic, three-dimensional Boussinesq equations},
url = {http://dx.doi.org/10.1088/1361-6544/aa6946},
volume = {30},
year = {2017}
}
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RIS format (EndNote, RefMan)

TY  - JOUR
AB  - We study the three-dimensional, incompressible, non-hydrostatic Boussinesq fluid equations, which are applicable to the dynamics of the oceans and atmosphere. These equations describe the interplay between velocity and buoyancy in a rotating frame. A hierarchy of dynamical variables is introduced whose members ${{ \Omega }_{m}}(t)$  ($1\leqslant m<\infty $ ) are made up from the respective sum of the L 2m -norms of vorticity and the density gradient. Each ${{ \Omega }_{m}}(t)$  has a lower bound in terms of the inverse Rossby number, Ro −1, that turns out to be crucial to the argument. For convenience, the ${{ \Omega }_{m}}$  are also scaled into a new set of variables D m (t). By assuming the existence and uniqueness of solutions, conditional upper bounds are found on the D m (t) in terms of Ro −1 and the Reynolds number Re. These upper bounds vary across bands in the $\left\{{{D}_{1}},\,{{D}_{m}}\right\}$  phase plane. The boundaries of these bands depend subtly upon Ro −1, Re, and the inverse Froude number Fr −1. For example, solutions in the lower band conditionally live in an absorbing ball in which the maximum value of ${{ \Omega }_{1}}$  deviates from Re 3/4 as a function of $R{{o}^{-1}},\,Re$ and Fr −1.
AU  - Gibbon,JD
AU  - Holm,DD
DO  - 1361-6544/aa6946
EP  - 24
PY  - 2017///
SN  - 0951-7715
SP  - 1
TI  - Bounds on solutions of the rotating, stratified, incompressible, non-hydrostatic, three-dimensional Boussinesq equations
T2  - Nonlinearity
UR  - http://dx.doi.org/10.1088/1361-6544/aa6946
UR  - http://hdl.handle.net/10044/1/57670
VL  - 30
ER  -
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