Natural Convection as a Heat Engine: A Theory for CAPE

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  • 1 Division of Geological and Planetary Sciences, California Institute of Technology, Pasadena, California
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Abstract

On many planets there is a continuous heat supply to the surface and a continuous emission of infrared radiation to space by the atmosphere. Since the heat source is located at higher pressure than the heat sink, the system is capable of doing mechanical work. Atmospheric convection is a natural heat engine that might operate in this system. Based on the heat engine framework, a simple theory is presented for atmospheric convection that predicts the buoyancy, the vertical velocity, and the fractional area covered by either dry or moist convection in a state of statistical equilibrium. During one cycle of the convective heat engine, heat is taken from the surface layer (the hot source) and a portion of it is rejected to the free troposphere (the cold sink) from where it is radiated to space. The balance is transformed into mechanical work. The mechanical work is expended in the maintenance of the convective motions against mechanical dissipation. Ultimately, the energy dissipated by mechanical friction is transformed into heat. Then, a fraction of the dissipated energy is radiated to space while the remaining portion is recycled by the convecting air parcels. Increases in the fraction of energy dissipated at warmer temperature, at the expense of decreases in the fraction of energy dissipated at colder temperatures, lead to increases in the apparent efficiency of the convective heat engine. The volume integral of the work produced by the convective heat engine gives a measure of the statistical equilibrium amount of convective available potential energy (CAPE) that must be present in the planet's atmosphere so that the convective motions can be maintained against viscous dissipation. This integral is a fundamental global number qualifying the state of the planet in statistical equilibrium conditions. For the earth's present climate, the heat engine framework predicts a CAPE value of the order of 1000 J kg−1 for the tropical atmosphere. This value is in agreement with observations. It also follows from our results that the total amount of CAPE present in a convecting atmosphere should increase with increases in the global surface temperature (or the atmosphere's opacity to infrared radiation).

Abstract

On many planets there is a continuous heat supply to the surface and a continuous emission of infrared radiation to space by the atmosphere. Since the heat source is located at higher pressure than the heat sink, the system is capable of doing mechanical work. Atmospheric convection is a natural heat engine that might operate in this system. Based on the heat engine framework, a simple theory is presented for atmospheric convection that predicts the buoyancy, the vertical velocity, and the fractional area covered by either dry or moist convection in a state of statistical equilibrium. During one cycle of the convective heat engine, heat is taken from the surface layer (the hot source) and a portion of it is rejected to the free troposphere (the cold sink) from where it is radiated to space. The balance is transformed into mechanical work. The mechanical work is expended in the maintenance of the convective motions against mechanical dissipation. Ultimately, the energy dissipated by mechanical friction is transformed into heat. Then, a fraction of the dissipated energy is radiated to space while the remaining portion is recycled by the convecting air parcels. Increases in the fraction of energy dissipated at warmer temperature, at the expense of decreases in the fraction of energy dissipated at colder temperatures, lead to increases in the apparent efficiency of the convective heat engine. The volume integral of the work produced by the convective heat engine gives a measure of the statistical equilibrium amount of convective available potential energy (CAPE) that must be present in the planet's atmosphere so that the convective motions can be maintained against viscous dissipation. This integral is a fundamental global number qualifying the state of the planet in statistical equilibrium conditions. For the earth's present climate, the heat engine framework predicts a CAPE value of the order of 1000 J kg−1 for the tropical atmosphere. This value is in agreement with observations. It also follows from our results that the total amount of CAPE present in a convecting atmosphere should increase with increases in the global surface temperature (or the atmosphere's opacity to infrared radiation).

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