Bifurcation and Stability in a Model of Moist Convection in a Shearing Environment

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  • 1 Department of Meteorology, The Pennsylvania State University, University Park, PA 16802
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Abstract

The six-coefficient spectral model (model I) of two-dimensional shallow moist convection discussed by Shirer and Dutton (1979) is extended to an eleven component system (model II) in order that a height-dependent basic wind V(z) could be added to the problem. In this way, some behavior of atmospheric cloud streets that are usually observed in a shearing environment could be studied qualitatively.

In model II, several different types of solutions exist that correspond to a rich variety of physically relevant possibilities. When a basic wind field is present, stationary, advecting or growing and decaying but propagating cloud bands may occur for different magnitudes of the vertical temperature gradient. These two-dimensional rolls are represented by time-dependent or periodic solutions or by an attractor on a two-dimensional torus. At large values of the lapse rate, the invariant set is probably contained on a three-dimensional torus, but whether or not the limit set is composed of a strange attractor (that may be a model of turbulent behavior) is an open question.

With use of a minimizing principle, the expected cloud band orientations are obtained. The resulting formulas provide predictions that generally agree with the linear studies of previous investigators, but our equations apply to an arbitrary height-dependent wind field. When the wind direction does not vary with altitude, the branching two-dimensional rolls are longitudinal or aligned parallel to the wind shear vector. But when the basic wind direction changes with height, some wind profiles support longitudinal rolls; other wind fields load to transverse rolls that are oriented perpendicular to the shear. Two or three coexisting cloud band alignments can also occur.

Abstract

The six-coefficient spectral model (model I) of two-dimensional shallow moist convection discussed by Shirer and Dutton (1979) is extended to an eleven component system (model II) in order that a height-dependent basic wind V(z) could be added to the problem. In this way, some behavior of atmospheric cloud streets that are usually observed in a shearing environment could be studied qualitatively.

In model II, several different types of solutions exist that correspond to a rich variety of physically relevant possibilities. When a basic wind field is present, stationary, advecting or growing and decaying but propagating cloud bands may occur for different magnitudes of the vertical temperature gradient. These two-dimensional rolls are represented by time-dependent or periodic solutions or by an attractor on a two-dimensional torus. At large values of the lapse rate, the invariant set is probably contained on a three-dimensional torus, but whether or not the limit set is composed of a strange attractor (that may be a model of turbulent behavior) is an open question.

With use of a minimizing principle, the expected cloud band orientations are obtained. The resulting formulas provide predictions that generally agree with the linear studies of previous investigators, but our equations apply to an arbitrary height-dependent wind field. When the wind direction does not vary with altitude, the branching two-dimensional rolls are longitudinal or aligned parallel to the wind shear vector. But when the basic wind direction changes with height, some wind profiles support longitudinal rolls; other wind fields load to transverse rolls that are oriented perpendicular to the shear. Two or three coexisting cloud band alignments can also occur.

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