### Abstract

When convection is parameterized in an atmospheric circulation model, what types of waves are supported by the parameterization? Several studies have addressed this question by finding the linear waves of simplified tropical climate models with convective parameterizations. In this paper's simplified tropical climate model, convection is parameterized by a nonlinear precipitation term, and the nonlinearity gives rise to precipitation front solutions. Precipitation fronts are solutions where the spatial domain is divided into two regions, and the precipitation (and other model variables) changes abruptly at the boundary of the two regions. In one region the water vapor is below saturation and there is no precipitation, and in the other region the water vapor is above saturation level and precipitation is nonzero. The boundary between the two regions is a free boundary that moves at a constant speed. It is shown that only certain front speeds are allowed. The three types of fronts that exist for this model are drying fronts, slow moistening fronts, and fast moistening fronts. Both types of moistening fronts violate Lax's stability criterion, but they are robustly realizable in numerical experiments that use finite relaxation times. Remarkably, here it is shown that all three types of fronts are robustly realizable analytically for finite relaxation time. All three types of fronts may be physically unreasonable if the front spans an unrealistically large physical distance; this depends on various model parameters, which are investigated below. From the viewpoint of applied mathematics, these model equations exhibit novel phenomena as well as features in common with the established applied mathematical theories of relaxation limits for conservation laws and waves in reacting gas flows.

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

Pages (from-to) | 377-404 |

Number of pages | 28 |

Journal | Theoretical and Computational Fluid Dynamics |

Volume | 20 |

Issue number | 5-6 |

DOIs | |

State | Published - Nov 2006 |

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### Keywords

- Hyperbolic free boundary problems
- Moisture
- Nonlinear relaxation equations
- Tropical atmospheric dynamics
- Tropical convection

### ASJC Scopus subject areas

- Fluid Flow and Transfer Processes
- Oceanography
- Computational Mechanics
- Mechanics of Materials
- Applied Mathematics
- Physics and Astronomy(all)
- Physics and Astronomy (miscellaneous)
- Condensed Matter Physics

### Cite this

*Theoretical and Computational Fluid Dynamics*,

*20*(5-6), 377-404. https://doi.org/10.1007/s00162-006-0014-1

**The structure of precipitation fronts for finite relaxation time.** / Stechmann, Samuel N.; Majda, Andrew J.

Research output: Contribution to journal › Article

*Theoretical and Computational Fluid Dynamics*, vol. 20, no. 5-6, pp. 377-404. https://doi.org/10.1007/s00162-006-0014-1

}

TY - JOUR

T1 - The structure of precipitation fronts for finite relaxation time

AU - Stechmann, Samuel N.

AU - Majda, Andrew J.

PY - 2006/11

Y1 - 2006/11

N2 - When convection is parameterized in an atmospheric circulation model, what types of waves are supported by the parameterization? Several studies have addressed this question by finding the linear waves of simplified tropical climate models with convective parameterizations. In this paper's simplified tropical climate model, convection is parameterized by a nonlinear precipitation term, and the nonlinearity gives rise to precipitation front solutions. Precipitation fronts are solutions where the spatial domain is divided into two regions, and the precipitation (and other model variables) changes abruptly at the boundary of the two regions. In one region the water vapor is below saturation and there is no precipitation, and in the other region the water vapor is above saturation level and precipitation is nonzero. The boundary between the two regions is a free boundary that moves at a constant speed. It is shown that only certain front speeds are allowed. The three types of fronts that exist for this model are drying fronts, slow moistening fronts, and fast moistening fronts. Both types of moistening fronts violate Lax's stability criterion, but they are robustly realizable in numerical experiments that use finite relaxation times. Remarkably, here it is shown that all three types of fronts are robustly realizable analytically for finite relaxation time. All three types of fronts may be physically unreasonable if the front spans an unrealistically large physical distance; this depends on various model parameters, which are investigated below. From the viewpoint of applied mathematics, these model equations exhibit novel phenomena as well as features in common with the established applied mathematical theories of relaxation limits for conservation laws and waves in reacting gas flows.

AB - When convection is parameterized in an atmospheric circulation model, what types of waves are supported by the parameterization? Several studies have addressed this question by finding the linear waves of simplified tropical climate models with convective parameterizations. In this paper's simplified tropical climate model, convection is parameterized by a nonlinear precipitation term, and the nonlinearity gives rise to precipitation front solutions. Precipitation fronts are solutions where the spatial domain is divided into two regions, and the precipitation (and other model variables) changes abruptly at the boundary of the two regions. In one region the water vapor is below saturation and there is no precipitation, and in the other region the water vapor is above saturation level and precipitation is nonzero. The boundary between the two regions is a free boundary that moves at a constant speed. It is shown that only certain front speeds are allowed. The three types of fronts that exist for this model are drying fronts, slow moistening fronts, and fast moistening fronts. Both types of moistening fronts violate Lax's stability criterion, but they are robustly realizable in numerical experiments that use finite relaxation times. Remarkably, here it is shown that all three types of fronts are robustly realizable analytically for finite relaxation time. All three types of fronts may be physically unreasonable if the front spans an unrealistically large physical distance; this depends on various model parameters, which are investigated below. From the viewpoint of applied mathematics, these model equations exhibit novel phenomena as well as features in common with the established applied mathematical theories of relaxation limits for conservation laws and waves in reacting gas flows.

KW - Hyperbolic free boundary problems

KW - Moisture

KW - Nonlinear relaxation equations

KW - Tropical atmospheric dynamics

KW - Tropical convection

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U2 - 10.1007/s00162-006-0014-1

DO - 10.1007/s00162-006-0014-1

M3 - Article

VL - 20

SP - 377

EP - 404

JO - Theoretical and Computational Fluid Dynamics

JF - Theoretical and Computational Fluid Dynamics

SN - 0935-4964

IS - 5-6

ER -