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Models for Stratiform Instability and Convectively Coupled Waves

Andrew J. MajdaCourant Institute of Mathematical Sciences, and Center for Atmosphere–Ocean Science, New York University, New York, New York

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Michael G. ShefterCourant Institute of Mathematical Sciences, New York University, New York, New York

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Abstract

A simplified intermediate model for analyzing and parameterizing convectively coupled tropical waves is introduced here. This model has two baroclinic modes of vertical structure: a direct heating mode and a stratiform mode. The key essential parameter in these models is the area fraction occupied by deep convection, σc. The unstable convectively coupled waves that emerge from perturbation of a radiative convective equilibrium are discussed in detail through linearized stability analysis. Without any mean flow, for an overall cooling rate of 1 K day−1 as the area fraction parameter increases from σc = 0.0010 to σc = 0.0014 the waves pass from a regime with stable moist convective damping to a regime of “stratiform” instability with convectively coupled waves propagating at speeds of roughly 15 m s−1; instabilities for a band of wavelengths in the supercluster regime, O(1000)–O(2000) km; and a vertical structure with a “wave tilt” where the temperature structure in the upper troposphere lags behind that in the lower troposphere. Thus, these convectively coupled waves in the model reproduce several key features of convectively coupled waves in the troposphere processed from recent observational data by Wheeler and Kiladis. As the parameter σc is increased further to values such as σc = 0.01, the band of unstable waves increases and spreads toward a mesoscale wavelength of O(100) km while the same wave structure and quantitative features mentioned above are retained for O(1000) km.

A detailed analysis of the temporal development of instability of these convectively coupled waves is presented here. In the first stage of instability, a high convective available potential energy (CAPE) region generates deep convection and a front-to-rear ascending flow with enhanced vertical shear in a stratiform wake region. Thus, these intermediate models may be useful prototypes for studying the parameterization of upscale convective momentum transport due to organized convection. In the second stage of instability, detailed analysis of the CAPE budget establishes that the effects of the second baroclinic mode in the stratiform wake produce new CAPE, which regenerates the first half of the wave cycle. Finally, since these convectively coupled stratiform waves do not require a barotropic mean flow, a barotropic mean flow, which alters the surface fluxes, is added to study its effect on their stability. These effects of a barotropic mean flow are secondary; an easterly mean flow enhances instability of the eastward-propagating convectively coupled waves and diminishes the instability of the westward-propagating waves through a wind-induced surface heat exchange mechanism.

Corresponding author address: Prof. Andrew J. Majda, Courant Institute of Mathematical Sciences, New York University, 251 Mercer Street, New York, NY 10012. Email: jonjon@cims.nyu.edu

Abstract

A simplified intermediate model for analyzing and parameterizing convectively coupled tropical waves is introduced here. This model has two baroclinic modes of vertical structure: a direct heating mode and a stratiform mode. The key essential parameter in these models is the area fraction occupied by deep convection, σc. The unstable convectively coupled waves that emerge from perturbation of a radiative convective equilibrium are discussed in detail through linearized stability analysis. Without any mean flow, for an overall cooling rate of 1 K day−1 as the area fraction parameter increases from σc = 0.0010 to σc = 0.0014 the waves pass from a regime with stable moist convective damping to a regime of “stratiform” instability with convectively coupled waves propagating at speeds of roughly 15 m s−1; instabilities for a band of wavelengths in the supercluster regime, O(1000)–O(2000) km; and a vertical structure with a “wave tilt” where the temperature structure in the upper troposphere lags behind that in the lower troposphere. Thus, these convectively coupled waves in the model reproduce several key features of convectively coupled waves in the troposphere processed from recent observational data by Wheeler and Kiladis. As the parameter σc is increased further to values such as σc = 0.01, the band of unstable waves increases and spreads toward a mesoscale wavelength of O(100) km while the same wave structure and quantitative features mentioned above are retained for O(1000) km.

A detailed analysis of the temporal development of instability of these convectively coupled waves is presented here. In the first stage of instability, a high convective available potential energy (CAPE) region generates deep convection and a front-to-rear ascending flow with enhanced vertical shear in a stratiform wake region. Thus, these intermediate models may be useful prototypes for studying the parameterization of upscale convective momentum transport due to organized convection. In the second stage of instability, detailed analysis of the CAPE budget establishes that the effects of the second baroclinic mode in the stratiform wake produce new CAPE, which regenerates the first half of the wave cycle. Finally, since these convectively coupled stratiform waves do not require a barotropic mean flow, a barotropic mean flow, which alters the surface fluxes, is added to study its effect on their stability. These effects of a barotropic mean flow are secondary; an easterly mean flow enhances instability of the eastward-propagating convectively coupled waves and diminishes the instability of the westward-propagating waves through a wind-induced surface heat exchange mechanism.

Corresponding author address: Prof. Andrew J. Majda, Courant Institute of Mathematical Sciences, New York University, 251 Mercer Street, New York, NY 10012. Email: jonjon@cims.nyu.edu

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