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Determination of Eddy Fluxes of Heat and Eddy Temperature Variances Using Weakly Nonlinear Theory

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  • 1 Geophysical Fluid Dynamics Institute and Metrology Department, Florida State University, Tallahassee, 32306
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Abstract

Drazin's (1972) weakly nonlinear theory of the self interaction of a single, slightly unstable, normal mode in a viscous, baroclinic fluid with a continuous density distribution is used to determine eddy fluxes of heat and eddy temperature variances as a function of rotation and internal thermal gradients. The calculations are applied to annulus experiments in that portion of the regular wave regime in which the observed flow is dominated by a single mode. V. Barcilon's (1964) linear theory for a viscous fluid is used for the purpose of mode selection at each point in dimensionless-parameter space. The geostrophic eddy heat flux and temperature variance are then determined from Drazin's lowest order nonlinear theory (which is a direct extension of Barcilon's theory).

It is found that the eddy beat flux and temperature variance depend upon internal thermal gradients at marginal stability and upon the distance of the point in dimensionless-parameter space from marginal stability for each mode. This result is different from those obtained from linear theory (e.g., by Green, 1970; Saltzman and Vernekar, 1971; Stone, 1972) but agrees in essence with that obtained by Hart (1974) from a nonlinear analysis of a two-layer model.

Numerical calculations from the theoretically derived formulas show that the eddy beat flux and temperature variance corresponding to each mode increase with increasing rotation rate when the imposed temperature contrast is held constant, but that there is an abrupt drop in the magnitude of both quantities when the wavenumber changes to the next higher integral value. Experimental evidence obtained by Pfeffer et al. (1978) verifies that real fluids behave in this way. Another feature observed in laboratory experiments and predicted by the theory is that at fixed rotation rate (or Taylor number) the eddy heat flux and the eddy temperature variance may be either larger or smaller at larger values of the imposed temperature contrast, depending on the location of the experiments in dimensionless-parameter space. If we think of seasons being brought about by changes in imposed temperature contrast at constant rotation rate, this result implies that only in certain regions of dimensionless-parameter space is winter (defined as the season with the highest imposed temperature contrast) the season with the largest eddy heat flux or eddy temperature variance.

Abstract

Drazin's (1972) weakly nonlinear theory of the self interaction of a single, slightly unstable, normal mode in a viscous, baroclinic fluid with a continuous density distribution is used to determine eddy fluxes of heat and eddy temperature variances as a function of rotation and internal thermal gradients. The calculations are applied to annulus experiments in that portion of the regular wave regime in which the observed flow is dominated by a single mode. V. Barcilon's (1964) linear theory for a viscous fluid is used for the purpose of mode selection at each point in dimensionless-parameter space. The geostrophic eddy heat flux and temperature variance are then determined from Drazin's lowest order nonlinear theory (which is a direct extension of Barcilon's theory).

It is found that the eddy beat flux and temperature variance depend upon internal thermal gradients at marginal stability and upon the distance of the point in dimensionless-parameter space from marginal stability for each mode. This result is different from those obtained from linear theory (e.g., by Green, 1970; Saltzman and Vernekar, 1971; Stone, 1972) but agrees in essence with that obtained by Hart (1974) from a nonlinear analysis of a two-layer model.

Numerical calculations from the theoretically derived formulas show that the eddy beat flux and temperature variance corresponding to each mode increase with increasing rotation rate when the imposed temperature contrast is held constant, but that there is an abrupt drop in the magnitude of both quantities when the wavenumber changes to the next higher integral value. Experimental evidence obtained by Pfeffer et al. (1978) verifies that real fluids behave in this way. Another feature observed in laboratory experiments and predicted by the theory is that at fixed rotation rate (or Taylor number) the eddy heat flux and the eddy temperature variance may be either larger or smaller at larger values of the imposed temperature contrast, depending on the location of the experiments in dimensionless-parameter space. If we think of seasons being brought about by changes in imposed temperature contrast at constant rotation rate, this result implies that only in certain regions of dimensionless-parameter space is winter (defined as the season with the highest imposed temperature contrast) the season with the largest eddy heat flux or eddy temperature variance.

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