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The Weak Temperature Gradient Approximation and Balanced Tropical Moisture Waves

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  • 1 Department of Applied Physics and Applied Mathematics, and Department of Earth and Environmental Sciences, Columbia University, New York, New York
  • | 2 Department of Meteorology, Stockholm University, Stockholm, Sweden
  • | 3 Department of Applied Physics and Applied Mathematics, and Department of Earth and Environmental Sciences, Columbia University, New York, New York
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Abstract

Horizontal temperature gradients are small in the tropical atmosphere, as a consequence of the smallness of the Coriolis parameter near the equator. This provides a strong constraint on both large-scale fluid dynamics and diabatic processes. This work is a step toward the construction of a balanced dynamical theory for the tropical circulation that is based on this constraint, and in which the diabatic processes are explicit and interactive.

The authors first derive the basic fluid-dynamical scaling under the weak temperature gradient (WTG) approximation in a shallow water system with a fixed mass source representing an externally imposed heating. This derivation follows an earlier similar one by Held and Hoskins, but extends the analysis to the nonlinear case (though on an f plane), examines the resulting system in more detail, and presents a solution for an axisymmetric “top-hat” forcing. The system is truly balanced, having no gravity waves, but is different from other balance models in that the heating is included a priori in the scaling.

The WTG scaling is then applied to a linear moist model in which the convective heating is controlled by a moisture variable that is advected by the flow. This moist model is derived from the Quasi-equilibrium Tropical Circulation Model (QTCM) equations of Neelin and Zeng but can be viewed as somewhat more general. A number of additional approximations are made in order to consider balanced dynamical modes, apparently not studied previously, which owe their existence to interactions of the moisture and flow fields. A particularly interesting mode arises on an f plane with a constant background moisture gradient. In the limit of low frequency and zero meridional wavenumber this mode has a dispersion relation mathematically identical to that of a barotropic Rossby wave, though the phase speed is eastward (for moisture decreasing poleward in the background state) and the propagation mechanism is quite different. This mode also has significant positive growth rate for low wavenumbers. The addition of the β effect complicates matters. For typical parameters, when β is included the direction of phase propagation is ambiguous, and the growth rate reduced, as the effects of the background gradients in moisture and planetary vorticity appear to cancel to a large degree. Possible relevance to intraseasonal variability and easterly wave dynamics is briefly discussed.

Corresponding author address: Dr. Adam H. Sobel, Department of Applied Physics and Applied Mathematics, Columbia University, 500 West 120th St., Rm. 217, New York, NY 10027. Email: sobel@appmath.columbia.edu

Abstract

Horizontal temperature gradients are small in the tropical atmosphere, as a consequence of the smallness of the Coriolis parameter near the equator. This provides a strong constraint on both large-scale fluid dynamics and diabatic processes. This work is a step toward the construction of a balanced dynamical theory for the tropical circulation that is based on this constraint, and in which the diabatic processes are explicit and interactive.

The authors first derive the basic fluid-dynamical scaling under the weak temperature gradient (WTG) approximation in a shallow water system with a fixed mass source representing an externally imposed heating. This derivation follows an earlier similar one by Held and Hoskins, but extends the analysis to the nonlinear case (though on an f plane), examines the resulting system in more detail, and presents a solution for an axisymmetric “top-hat” forcing. The system is truly balanced, having no gravity waves, but is different from other balance models in that the heating is included a priori in the scaling.

The WTG scaling is then applied to a linear moist model in which the convective heating is controlled by a moisture variable that is advected by the flow. This moist model is derived from the Quasi-equilibrium Tropical Circulation Model (QTCM) equations of Neelin and Zeng but can be viewed as somewhat more general. A number of additional approximations are made in order to consider balanced dynamical modes, apparently not studied previously, which owe their existence to interactions of the moisture and flow fields. A particularly interesting mode arises on an f plane with a constant background moisture gradient. In the limit of low frequency and zero meridional wavenumber this mode has a dispersion relation mathematically identical to that of a barotropic Rossby wave, though the phase speed is eastward (for moisture decreasing poleward in the background state) and the propagation mechanism is quite different. This mode also has significant positive growth rate for low wavenumbers. The addition of the β effect complicates matters. For typical parameters, when β is included the direction of phase propagation is ambiguous, and the growth rate reduced, as the effects of the background gradients in moisture and planetary vorticity appear to cancel to a large degree. Possible relevance to intraseasonal variability and easterly wave dynamics is briefly discussed.

Corresponding author address: Dr. Adam H. Sobel, Department of Applied Physics and Applied Mathematics, Columbia University, 500 West 120th St., Rm. 217, New York, NY 10027. Email: sobel@appmath.columbia.edu

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