Instability of Frontal Motions in the Atmosphere

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  • 1 National Center for Atmospheric Research, Boulder, Colo., and Institute of Meteorology, University of Stockholm, Sweden
  • | 2 Dept. of Energetics and the Center for Great Lakes Studies, University of Wisconsin-Milwaukee
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Abstract

Stability properties of a two-layer model of homogeneous and incompressible fluid subject to gravity and rotation are investigated by the small-perturbation method. The upper and lower fluids correspond, respectively, to warm and cold air. The interface between the warm and cold layers intersects the ground and forms the surface front. The model has been used to investigate the development of frontal cyclones. Linearized equations of warm and cold layers are solved as an eigenvalue problem to find solutions growing exponentially with time using a finite-difference technique.

For prescribed values of the density ratio ε of warm and cold layers, the north-south extent D of the frontal interface, the Coriolis parameter f, the external gravity wave speed C0, and the basic state cold air velocity ū1, we vary the values of the wavenumber k of perturbations and the basic state warm air velocity ū2, through the use of the Rossby number [Ro≡½(ū2&minusū1)k/f] and the Richardson number [Ri≡C0(1−∈)/(ū2&minusū1)].

Orlanski has investigated the stability of a similar model in the parameter domain of Ri≲5 and Ro≲3. Eliasen has studied the stability of a frontal model in the domain of 3 Ri 6 and Ro≲0.5. In the present work, we cover the domain of 1.25≲Ri<12 and Ro≲2 which includes the region that is not investigated by either Eliasen or Orlanski.

The stability characteristics for Ri≲2 are fairly complex. For Ri≳3, there are two modes of instability, One appears in the region of Ro≲0.4 which corresponds to the type of instability found by Eliasen and it is a quasi-geostrophic baroclinic instability. The other appears for Ro≲0.9 which has an unbounded growth rate as the wavelength decreases. Orlanski pointed out the presence of instability in this parameter domain, but he has not investigated in detail the characteristics of the instability.

The kinematics for Ri=5.0 and Ro=1.2 reveals that the unstable motion is highly nongeostrophic and has a small latitudinal width. Since this instability has an unbounded growth rate as the wavelength decreases, the determination of the preferred scale of this unstable motion is very much dependent on the mechanism of momentum dissipation which is not considered in this study.

Abstract

Stability properties of a two-layer model of homogeneous and incompressible fluid subject to gravity and rotation are investigated by the small-perturbation method. The upper and lower fluids correspond, respectively, to warm and cold air. The interface between the warm and cold layers intersects the ground and forms the surface front. The model has been used to investigate the development of frontal cyclones. Linearized equations of warm and cold layers are solved as an eigenvalue problem to find solutions growing exponentially with time using a finite-difference technique.

For prescribed values of the density ratio ε of warm and cold layers, the north-south extent D of the frontal interface, the Coriolis parameter f, the external gravity wave speed C0, and the basic state cold air velocity ū1, we vary the values of the wavenumber k of perturbations and the basic state warm air velocity ū2, through the use of the Rossby number [Ro≡½(ū2&minusū1)k/f] and the Richardson number [Ri≡C0(1−∈)/(ū2&minusū1)].

Orlanski has investigated the stability of a similar model in the parameter domain of Ri≲5 and Ro≲3. Eliasen has studied the stability of a frontal model in the domain of 3 Ri 6 and Ro≲0.5. In the present work, we cover the domain of 1.25≲Ri<12 and Ro≲2 which includes the region that is not investigated by either Eliasen or Orlanski.

The stability characteristics for Ri≲2 are fairly complex. For Ri≳3, there are two modes of instability, One appears in the region of Ro≲0.4 which corresponds to the type of instability found by Eliasen and it is a quasi-geostrophic baroclinic instability. The other appears for Ro≲0.9 which has an unbounded growth rate as the wavelength decreases. Orlanski pointed out the presence of instability in this parameter domain, but he has not investigated in detail the characteristics of the instability.

The kinematics for Ri=5.0 and Ro=1.2 reveals that the unstable motion is highly nongeostrophic and has a small latitudinal width. Since this instability has an unbounded growth rate as the wavelength decreases, the determination of the preferred scale of this unstable motion is very much dependent on the mechanism of momentum dissipation which is not considered in this study.

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