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Hydrostatic Adjustment in Nonisothermal Atmospheres

Dean G. DuffyNASA Goddard Space Flight Center, Greenbelt, Maryland

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Abstract

The author examines hydrostatic adjustment due to heating in two nonisothermal atmospheres. In the first case both the temperature and lapse rate decrease with height; in the second case the atmosphere consists of a troposphere with constant lapse rate and a colder, isothermal, semi-infinitely deep stratosphere. In both cases hydrostatic adjustment, to a good approximation, follows the pattern found in the Lamb problem: initially the Eulerian available potential energy remains essentially constant so that an increase (a decrease) in the kinetic energy occurs with a corresponding decrease (increase) of Eulerian available elastic energy. After this initial period the acoustic–gravity waves disperse and all three forms of energy—Eulerian kinetic, Eulerian available potential, and available elastic—interact with each other. Relaxation to hydrostatic balance occurs rapidly, within the first acoustic cutoff period 4πHS/c, where c is the speed of sound and HS is the scale height. In the Lagrangian description, the available potential energy remains constant with time. The kinetic energy is coupled to the Lagrangian available elastic energy so that an increase (a decrease) in the kinetic energy always occurs with a corresponding decrease (increase) of Lagrangian available elastic energy.

The primary effect of a positive lapse rate is a decrease in the percentage of the total perturbation energy available for wave motions. In the two-layer atmosphere, the discontinuity in the static stability at the tropopause results in imperfect ducting of the acoustic–gravity waves within the troposphere.

Corresponding author address: Dr. Dean G. Duffy, NASA Goddard Space Flight Center, Mail Code 912, Greenbelt, MD 20771-0001. Email: duffy@agnes.gsfc.nasa.gov

Abstract

The author examines hydrostatic adjustment due to heating in two nonisothermal atmospheres. In the first case both the temperature and lapse rate decrease with height; in the second case the atmosphere consists of a troposphere with constant lapse rate and a colder, isothermal, semi-infinitely deep stratosphere. In both cases hydrostatic adjustment, to a good approximation, follows the pattern found in the Lamb problem: initially the Eulerian available potential energy remains essentially constant so that an increase (a decrease) in the kinetic energy occurs with a corresponding decrease (increase) of Eulerian available elastic energy. After this initial period the acoustic–gravity waves disperse and all three forms of energy—Eulerian kinetic, Eulerian available potential, and available elastic—interact with each other. Relaxation to hydrostatic balance occurs rapidly, within the first acoustic cutoff period 4πHS/c, where c is the speed of sound and HS is the scale height. In the Lagrangian description, the available potential energy remains constant with time. The kinetic energy is coupled to the Lagrangian available elastic energy so that an increase (a decrease) in the kinetic energy always occurs with a corresponding decrease (increase) of Lagrangian available elastic energy.

The primary effect of a positive lapse rate is a decrease in the percentage of the total perturbation energy available for wave motions. In the two-layer atmosphere, the discontinuity in the static stability at the tropopause results in imperfect ducting of the acoustic–gravity waves within the troposphere.

Corresponding author address: Dr. Dean G. Duffy, NASA Goddard Space Flight Center, Mail Code 912, Greenbelt, MD 20771-0001. Email: duffy@agnes.gsfc.nasa.gov

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