The Impact of Rough Forcing on Systems with Multiple Time Scales

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  • 1 CIRA, Colorado State University—Foothills Campus, Fort Collins, Colorado, and NOAA/ERL Forecast Systems Laboratory, Boulder, Colorado
  • | 2 Department of Mathematics, University of California, Los Angeles, Los Angeles, California
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Abstract

In a series of numerical experiments, Williamson and Temperton demonstrated that the interaction of the high-frequency gravity waves with the low-frequency Rossby waves in a three-dimensional adiabatic model is very weak. However, they stated that this “might not be the case when the model includes realistic physical processes, such as release of latent heat, which are strongly influenced by the vertical motion.” The bounded derivative theory is valid for inhomogeneous hyperbolic systems with multiple time scales, but the magnitude of any forcing term must be less than or equal to that of the horizontal advection terms in the same equation. When diabatic effects are added to the basic dynamical equations for the atmosphere, in the smaller scales of motion forcing terms can appear in both the entropy and pressure equations that do not satisfy this restriction. Assuming that the heating terms are only functions of the independent variables, the forcing term in the entropy equation can be eliminated so that only a large forcing term in the pressure equation remains. It is proved that a large forcing term in the pressure equation does not by itself preclude a smooth (in the bounded derivative sense) solution. However, the proof shows that the smoothness of the derivatives of the forcing determines the smoothness of the solution. If the spatial variation of the forcing in the pressure equation is much larger than that of the advective component of the solution of the homogeneous system, then no mathematical estimates of smoothness can be obtained and examples show a smooth solution does not exist. On the other hand, if the spatial derivatives of the forcing are smooth, but the temporal derivatives are not, a smooth solution exists and the effect of the large variation of the forcing in time on that smooth solution is small. When both spatial and temporal derivatives of the forcing are smooth, a smooth solution also exists, and it is proved that it is extremely accurately described by the corresponding reduced system; that is, the effect of the interaction of any gravity waves generated by the prescribed forcing with the smooth solution is minimal. The implications of these results for atmospheric prediction models are discussed.

Abstract

In a series of numerical experiments, Williamson and Temperton demonstrated that the interaction of the high-frequency gravity waves with the low-frequency Rossby waves in a three-dimensional adiabatic model is very weak. However, they stated that this “might not be the case when the model includes realistic physical processes, such as release of latent heat, which are strongly influenced by the vertical motion.” The bounded derivative theory is valid for inhomogeneous hyperbolic systems with multiple time scales, but the magnitude of any forcing term must be less than or equal to that of the horizontal advection terms in the same equation. When diabatic effects are added to the basic dynamical equations for the atmosphere, in the smaller scales of motion forcing terms can appear in both the entropy and pressure equations that do not satisfy this restriction. Assuming that the heating terms are only functions of the independent variables, the forcing term in the entropy equation can be eliminated so that only a large forcing term in the pressure equation remains. It is proved that a large forcing term in the pressure equation does not by itself preclude a smooth (in the bounded derivative sense) solution. However, the proof shows that the smoothness of the derivatives of the forcing determines the smoothness of the solution. If the spatial variation of the forcing in the pressure equation is much larger than that of the advective component of the solution of the homogeneous system, then no mathematical estimates of smoothness can be obtained and examples show a smooth solution does not exist. On the other hand, if the spatial derivatives of the forcing are smooth, but the temporal derivatives are not, a smooth solution exists and the effect of the large variation of the forcing in time on that smooth solution is small. When both spatial and temporal derivatives of the forcing are smooth, a smooth solution also exists, and it is proved that it is extremely accurately described by the corresponding reduced system; that is, the effect of the interaction of any gravity waves generated by the prescribed forcing with the smooth solution is minimal. The implications of these results for atmospheric prediction models are discussed.

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