### Abstract

The dynamics of two compositionally buoyant columns of fluid rising in an infinite less buoyant fluid is studied. The fluid within and outside the columns is thermally stably stratified and has a kinematic viscosity, ν, and a thermal diffusivity, K. The mean vertical flow and the associated temperature profiles depending on the horizontal coordinate normal to the surfaces of the columns are obtained and the associated buoyancy, heat and material fluxes are discussed. The stability of the mean state to infinitesimal disturbances is governed by the five dimensionless parameters: the number R (=UL/ν, where U and L are characteristic velocity and length, respectively) which measures the strength of the compositional buoyancy, x_{0}, x_{1}, d the dimensionless measures of the thickness of the two plumes and the distance between them, respectively, and G{cyrillic} representing the ratio of the strengths of the two plumes. The stability is examined for small values of R. The preferred mode of instability is studied in the space (x_{0}, x_{1}, d, G{cyrillic}). When a single plume is present (and x_{1}, d, G{cyrillic} = 0), it is already known that the perturbations are neutral at O(R^{0}) and linearly unstable at O(R^{1}) for all values of x_{0} and Prandtl number, σ (=ν/κ). It is shown here that at O(R^{0}), the presence of a second interacting plume has the significant effect that the system of two plumes becomes unstable for a certain range of the parameters while it remains neutrally stable for the others. In contrast with the instability of the single plume, the instability of the two interacting plumes does not depend, at the O(R^{0}) level, on the Prandtl number of the fluid. However, the neutrally stable two plumes might suffer instability at O(R^{0}) and that instability may depend on the Prandtl number. The instability is found to be driven by the gradients of all of the concentration of light material, the flow and temperature of the basic state. The presence of the second plume destroys the definite parity of the perturbations' solution and this can lead to non-zero total helicity.

Original language | English |
---|---|

Pages (from-to) | 29-63 |

Number of pages | 35 |

Journal | Geophysical and Astrophysical Fluid Dynamics |

Volume | 104 |

Issue number | 1 |

DOIs | |

Publication status | Published - Feb 2010 |

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### Keywords

- Earth's core
- Helicity
- Instability
- Plumes

### ASJC Scopus subject areas

- Geochemistry and Petrology
- Geophysics
- Mechanics of Materials
- Computational Mechanics
- Astronomy and Astrophysics

### Cite this

**The dynamics of two interacting compositional plumes.** / Elbashir, T. B A; Al Lawati, M. A.; Eltayeb, I. A.

Research output: Contribution to journal › Article

}

TY - JOUR

T1 - The dynamics of two interacting compositional plumes

AU - Elbashir, T. B A

AU - Al Lawati, M. A.

AU - Eltayeb, I. A.

PY - 2010/2

Y1 - 2010/2

N2 - The dynamics of two compositionally buoyant columns of fluid rising in an infinite less buoyant fluid is studied. The fluid within and outside the columns is thermally stably stratified and has a kinematic viscosity, ν, and a thermal diffusivity, K. The mean vertical flow and the associated temperature profiles depending on the horizontal coordinate normal to the surfaces of the columns are obtained and the associated buoyancy, heat and material fluxes are discussed. The stability of the mean state to infinitesimal disturbances is governed by the five dimensionless parameters: the number R (=UL/ν, where U and L are characteristic velocity and length, respectively) which measures the strength of the compositional buoyancy, x0, x1, d the dimensionless measures of the thickness of the two plumes and the distance between them, respectively, and G{cyrillic} representing the ratio of the strengths of the two plumes. The stability is examined for small values of R. The preferred mode of instability is studied in the space (x0, x1, d, G{cyrillic}). When a single plume is present (and x1, d, G{cyrillic} = 0), it is already known that the perturbations are neutral at O(R0) and linearly unstable at O(R1) for all values of x0 and Prandtl number, σ (=ν/κ). It is shown here that at O(R0), the presence of a second interacting plume has the significant effect that the system of two plumes becomes unstable for a certain range of the parameters while it remains neutrally stable for the others. In contrast with the instability of the single plume, the instability of the two interacting plumes does not depend, at the O(R0) level, on the Prandtl number of the fluid. However, the neutrally stable two plumes might suffer instability at O(R0) and that instability may depend on the Prandtl number. The instability is found to be driven by the gradients of all of the concentration of light material, the flow and temperature of the basic state. The presence of the second plume destroys the definite parity of the perturbations' solution and this can lead to non-zero total helicity.

AB - The dynamics of two compositionally buoyant columns of fluid rising in an infinite less buoyant fluid is studied. The fluid within and outside the columns is thermally stably stratified and has a kinematic viscosity, ν, and a thermal diffusivity, K. The mean vertical flow and the associated temperature profiles depending on the horizontal coordinate normal to the surfaces of the columns are obtained and the associated buoyancy, heat and material fluxes are discussed. The stability of the mean state to infinitesimal disturbances is governed by the five dimensionless parameters: the number R (=UL/ν, where U and L are characteristic velocity and length, respectively) which measures the strength of the compositional buoyancy, x0, x1, d the dimensionless measures of the thickness of the two plumes and the distance between them, respectively, and G{cyrillic} representing the ratio of the strengths of the two plumes. The stability is examined for small values of R. The preferred mode of instability is studied in the space (x0, x1, d, G{cyrillic}). When a single plume is present (and x1, d, G{cyrillic} = 0), it is already known that the perturbations are neutral at O(R0) and linearly unstable at O(R1) for all values of x0 and Prandtl number, σ (=ν/κ). It is shown here that at O(R0), the presence of a second interacting plume has the significant effect that the system of two plumes becomes unstable for a certain range of the parameters while it remains neutrally stable for the others. In contrast with the instability of the single plume, the instability of the two interacting plumes does not depend, at the O(R0) level, on the Prandtl number of the fluid. However, the neutrally stable two plumes might suffer instability at O(R0) and that instability may depend on the Prandtl number. The instability is found to be driven by the gradients of all of the concentration of light material, the flow and temperature of the basic state. The presence of the second plume destroys the definite parity of the perturbations' solution and this can lead to non-zero total helicity.

KW - Earth's core

KW - Helicity

KW - Instability

KW - Plumes

UR - http://www.scopus.com/inward/record.url?scp=77951012292&partnerID=8YFLogxK

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U2 - 10.1080/03091920903255142

DO - 10.1080/03091920903255142

M3 - Article

VL - 104

SP - 29

EP - 63

JO - Geophysical and Astrophysical Fluid Dynamics

JF - Geophysical and Astrophysical Fluid Dynamics

SN - 0309-1929

IS - 1

ER -