### Abstract

The initial stage of the adjustment of a gravity current to the effects of rotation with angular velocity f/2 is analysed using a short time analysis where Coriolis forces are initiated in an inviscid von Kármán-Benjamin gravity current front at t_{F} = O. It is shown how, on a time-scale of order 1/f, as a result of ageostrophic dynamics, the slope and front speed U_{F} are much reduced from their initial values, while the transverse anticyclonic velocity parallel to the front increases from zero to O(NH_{0}), where N = √g′/H_{0} is the buoyancy frequency, and g′= gΔρ/ρ_{0} is the reduced acceleration due to gravity. Here ρ_{0} is the density and Δρ and H_{0} are the density difference and initial height of the current. Extending the steady-state theory to account for the effect of the slope σ on the bottom boundary shows that, without rotation, U_{F} has a maximum value for σ = π/6, while with rotation, U_{F} tends to zero on any slope. For the asymptotic stage when f^{t}
_{F} ≫ 1, the theory of unsteady waves on the current is reviewed using nonlinear shallow-water equations and the van der Pol averaging method. Their motions naturally split into a 'balanced' component satisfying the Margules geostrophic relation and an equally large 'unbalanced' component, in which there is horizontal divergence and ageostrophic vorticity. The latter is responsible for nonlinear oscillations in the current on a time scale f^{-1}, which have been observed in the atmosphere and field experiments. Their magnitude is mainly determined by the initial potential energy in relation to that of the current and is proportional to the ratio √Bu = L_{R}/R_{0}, where L_{R} = N H_{0}/f is the Rossby deformation radius and R_{0} is the initial radius. The effect of slope friction also prevents the formation of a steady front. From the analysis it is concluded that a weak mean radial flow must be driven by the ageostrophic oscillations, preventing the mean front speed U_{F} from halting sharply at ft_{F} ∼ 1. Depending on the initial value of L_{R}/R_{0}, physical arguments show that U_{F} decreases slowly in proportion to (ft_{F})^{-1/2}, i.e. U_{F}/U F_{0} = F (ft_{F}, Bu). Thus the front only tends to the geostrophic asymptotic state of zero radial velocity very slowly (i.e. as ft_{F} → ∞) for finite values of L_{R}/R_{0}. However, as L_{R}/R_{0} → O, it reaches this state when ft_{F} ∼ 1. This analysis of the overall nonlinear behaviour of the gravity current is consistent with two two-dimensional non-hydrostatic (Navier-Stokes) and axisymmetric hydrostatic (shallow-water) Eulerian numerical simulations of the varying form of the rotating gravity current. When the effect of surface friction is considered, it is found that the mean movement of the front is significantly slowed. Furthermore, the oscillations with angular frequency f and the slow growth of the radius, when ft_{F} ≥ 1, are consistent with recent experiments.

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

Pages (from-to) | 285-315 |

Number of pages | 31 |

Journal | Journal of Fluid Mechanics |

Volume | 537 |

DOIs | |

State | Published - Aug 25 2005 |

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### ASJC Scopus subject areas

- Mechanics of Materials
- Computational Mechanics
- Physics and Astronomy(all)
- Condensed Matter Physics

### Cite this

*Journal of Fluid Mechanics*,

*537*, 285-315. https://doi.org/10.1017/S0022112005005239

**Effects of rotation and sloping terrain on the fronts of density currents.** / Hunt, J. C R; Pacheco, J. R.; Mahalov, Alex; Fernando, H. J S.

Research output: Contribution to journal › Article

*Journal of Fluid Mechanics*, vol. 537, pp. 285-315. https://doi.org/10.1017/S0022112005005239

}

TY - JOUR

T1 - Effects of rotation and sloping terrain on the fronts of density currents

AU - Hunt, J. C R

AU - Pacheco, J. R.

AU - Mahalov, Alex

AU - Fernando, H. J S

PY - 2005/8/25

Y1 - 2005/8/25

N2 - The initial stage of the adjustment of a gravity current to the effects of rotation with angular velocity f/2 is analysed using a short time analysis where Coriolis forces are initiated in an inviscid von Kármán-Benjamin gravity current front at tF = O. It is shown how, on a time-scale of order 1/f, as a result of ageostrophic dynamics, the slope and front speed UF are much reduced from their initial values, while the transverse anticyclonic velocity parallel to the front increases from zero to O(NH0), where N = √g′/H0 is the buoyancy frequency, and g′= gΔρ/ρ0 is the reduced acceleration due to gravity. Here ρ0 is the density and Δρ and H0 are the density difference and initial height of the current. Extending the steady-state theory to account for the effect of the slope σ on the bottom boundary shows that, without rotation, UF has a maximum value for σ = π/6, while with rotation, UF tends to zero on any slope. For the asymptotic stage when ft F ≫ 1, the theory of unsteady waves on the current is reviewed using nonlinear shallow-water equations and the van der Pol averaging method. Their motions naturally split into a 'balanced' component satisfying the Margules geostrophic relation and an equally large 'unbalanced' component, in which there is horizontal divergence and ageostrophic vorticity. The latter is responsible for nonlinear oscillations in the current on a time scale f-1, which have been observed in the atmosphere and field experiments. Their magnitude is mainly determined by the initial potential energy in relation to that of the current and is proportional to the ratio √Bu = LR/R0, where LR = N H0/f is the Rossby deformation radius and R0 is the initial radius. The effect of slope friction also prevents the formation of a steady front. From the analysis it is concluded that a weak mean radial flow must be driven by the ageostrophic oscillations, preventing the mean front speed UF from halting sharply at ftF ∼ 1. Depending on the initial value of LR/R0, physical arguments show that UF decreases slowly in proportion to (ftF)-1/2, i.e. UF/U F0 = F (ftF, Bu). Thus the front only tends to the geostrophic asymptotic state of zero radial velocity very slowly (i.e. as ftF → ∞) for finite values of LR/R0. However, as LR/R0 → O, it reaches this state when ftF ∼ 1. This analysis of the overall nonlinear behaviour of the gravity current is consistent with two two-dimensional non-hydrostatic (Navier-Stokes) and axisymmetric hydrostatic (shallow-water) Eulerian numerical simulations of the varying form of the rotating gravity current. When the effect of surface friction is considered, it is found that the mean movement of the front is significantly slowed. Furthermore, the oscillations with angular frequency f and the slow growth of the radius, when ftF ≥ 1, are consistent with recent experiments.

AB - The initial stage of the adjustment of a gravity current to the effects of rotation with angular velocity f/2 is analysed using a short time analysis where Coriolis forces are initiated in an inviscid von Kármán-Benjamin gravity current front at tF = O. It is shown how, on a time-scale of order 1/f, as a result of ageostrophic dynamics, the slope and front speed UF are much reduced from their initial values, while the transverse anticyclonic velocity parallel to the front increases from zero to O(NH0), where N = √g′/H0 is the buoyancy frequency, and g′= gΔρ/ρ0 is the reduced acceleration due to gravity. Here ρ0 is the density and Δρ and H0 are the density difference and initial height of the current. Extending the steady-state theory to account for the effect of the slope σ on the bottom boundary shows that, without rotation, UF has a maximum value for σ = π/6, while with rotation, UF tends to zero on any slope. For the asymptotic stage when ft F ≫ 1, the theory of unsteady waves on the current is reviewed using nonlinear shallow-water equations and the van der Pol averaging method. Their motions naturally split into a 'balanced' component satisfying the Margules geostrophic relation and an equally large 'unbalanced' component, in which there is horizontal divergence and ageostrophic vorticity. The latter is responsible for nonlinear oscillations in the current on a time scale f-1, which have been observed in the atmosphere and field experiments. Their magnitude is mainly determined by the initial potential energy in relation to that of the current and is proportional to the ratio √Bu = LR/R0, where LR = N H0/f is the Rossby deformation radius and R0 is the initial radius. The effect of slope friction also prevents the formation of a steady front. From the analysis it is concluded that a weak mean radial flow must be driven by the ageostrophic oscillations, preventing the mean front speed UF from halting sharply at ftF ∼ 1. Depending on the initial value of LR/R0, physical arguments show that UF decreases slowly in proportion to (ftF)-1/2, i.e. UF/U F0 = F (ftF, Bu). Thus the front only tends to the geostrophic asymptotic state of zero radial velocity very slowly (i.e. as ftF → ∞) for finite values of LR/R0. However, as LR/R0 → O, it reaches this state when ftF ∼ 1. This analysis of the overall nonlinear behaviour of the gravity current is consistent with two two-dimensional non-hydrostatic (Navier-Stokes) and axisymmetric hydrostatic (shallow-water) Eulerian numerical simulations of the varying form of the rotating gravity current. When the effect of surface friction is considered, it is found that the mean movement of the front is significantly slowed. Furthermore, the oscillations with angular frequency f and the slow growth of the radius, when ftF ≥ 1, are consistent with recent experiments.

UR - http://www.scopus.com/inward/record.url?scp=23944468277&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=23944468277&partnerID=8YFLogxK

U2 - 10.1017/S0022112005005239

DO - 10.1017/S0022112005005239

M3 - Article

AN - SCOPUS:23944468277

VL - 537

SP - 285

EP - 315

JO - Journal of Fluid Mechanics

JF - Journal of Fluid Mechanics

SN - 0022-1120

ER -