### Abstract

Analysis and computations are presented for singularities in the solution of the steady Boussinesq equations for two-dimensional, stratified flow. The results show that for codimension 1 singularities, there are two generic singularity types for general solutions, and only one generic singularity type if there is a certain symmetry present. The analysis depends on a special choice of coordinates, which greatly simplifies the equations, showing that the type is exactly that of one dimensional Legendrian singularities, generalized so that the velocity can be infinite at the singularity. The solution is viewed as a surface in an appropriate compactified jet space. Smoothness of the solution surface is proved using the Cauchy-Kowalewski Theorem, which also shows that these singularity types are realizable. Numerical results from a special, highly accurate numerical method demonstrate the validity of this geometric analysis. A new analysis of general Legendrian singularities with blowup, i.e., at which the derivative may be infinite, is also presented, using projective coordinates.

Original language | English (US) |
---|---|

Pages (from-to) | 369-414 |

Number of pages | 46 |

Journal | Selecta Mathematica, New Series |

Volume | 2 |

Issue number | 3 |

State | Published - 1996 |

Externally published | Yes |

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### ASJC Scopus subject areas

- Mathematics(all)
- Physics and Astronomy(all)

### Cite this

*Selecta Mathematica, New Series*,

*2*(3), 369-414.

**Geometry of singularities for the steady boussinesq equations.** / Caflisch, Russel E.; Ercolani, Nicholas M; Steele, Gregory.

Research output: Contribution to journal › Article

*Selecta Mathematica, New Series*, vol. 2, no. 3, pp. 369-414.

}

TY - JOUR

T1 - Geometry of singularities for the steady boussinesq equations

AU - Caflisch, Russel E.

AU - Ercolani, Nicholas M

AU - Steele, Gregory

PY - 1996

Y1 - 1996

N2 - Analysis and computations are presented for singularities in the solution of the steady Boussinesq equations for two-dimensional, stratified flow. The results show that for codimension 1 singularities, there are two generic singularity types for general solutions, and only one generic singularity type if there is a certain symmetry present. The analysis depends on a special choice of coordinates, which greatly simplifies the equations, showing that the type is exactly that of one dimensional Legendrian singularities, generalized so that the velocity can be infinite at the singularity. The solution is viewed as a surface in an appropriate compactified jet space. Smoothness of the solution surface is proved using the Cauchy-Kowalewski Theorem, which also shows that these singularity types are realizable. Numerical results from a special, highly accurate numerical method demonstrate the validity of this geometric analysis. A new analysis of general Legendrian singularities with blowup, i.e., at which the derivative may be infinite, is also presented, using projective coordinates.

AB - Analysis and computations are presented for singularities in the solution of the steady Boussinesq equations for two-dimensional, stratified flow. The results show that for codimension 1 singularities, there are two generic singularity types for general solutions, and only one generic singularity type if there is a certain symmetry present. The analysis depends on a special choice of coordinates, which greatly simplifies the equations, showing that the type is exactly that of one dimensional Legendrian singularities, generalized so that the velocity can be infinite at the singularity. The solution is viewed as a surface in an appropriate compactified jet space. Smoothness of the solution surface is proved using the Cauchy-Kowalewski Theorem, which also shows that these singularity types are realizable. Numerical results from a special, highly accurate numerical method demonstrate the validity of this geometric analysis. A new analysis of general Legendrian singularities with blowup, i.e., at which the derivative may be infinite, is also presented, using projective coordinates.

UR - http://www.scopus.com/inward/record.url?scp=53349128417&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=53349128417&partnerID=8YFLogxK

M3 - Article

VL - 2

SP - 369

EP - 414

JO - Selecta Mathematica, New Series

JF - Selecta Mathematica, New Series

SN - 1022-1824

IS - 3

ER -