Gravity waves play a fundamental role in driving general circulation of middle atmosphere to transport significant amount of energy and momentum. The process of gravity wave radiation remains an open problem, especially concerning non-orographic waves. Several observational studies suggest gravity wave radiation from strong rotational jet flows.
To investigate what causes gravity wave radiation, we use f-plane shallow water system which is the most simplified system in which both balanced rotational modes and unbalanced gravity wave modes can exist. Here, we adopt a forced-dissipative system and investigate continuous gravity wave radiation from rotational flows. This continuous radiation process does not correspond to the Rossby adjustment process, in which gravity waves are radiated from initial unbalanced states, but corresponds to spontaneous radiation process, in which gravity waves are radiated from balanced states.
F-plane shallow water system is the same as a compressive fluid, if the effect of earth rotation is negligible. Therefore, gravity waves have the same meaning as sound waves in the aero-acoustic theory (the Lighthill theory). It is well known in the aero-acoustic theory that the flux of sound waves is dependent on Mach number. Therefore, we focus on the dependence on Froude number, which have physically the same meaning as Mach number, as well as the effect of earth rotation on the gravity wave radiation in this study.
2.Model description
The basic equation used in this study is shallow water system on an f-plane with forcing and dissipation. The basic state is the zonal jet, which is barotropically unstable. Experimental parameters to sweep in this study are Rossby number (Ro) and Froude number (Fr). Ro determines the effect of earth rotation and Fr determines the depth of fluid, respectively. Gravity wave radiation from unsteady rotational flow is investigated for a wide range of these Ro and Fr. The domain for the numerical experiments is set to be periodic in the longitudinal direction, and infinite in the latitudinal direction. Shallow water equations are computed using a spectral transform methods.
3.The Lighthill equation and estimation of gravity wave flux
Source of gravity wave is introduced on the analogy with the aero-acoustic sound radiation theory (the Lighthill equation). By solving this equation, we can estimate gravity wave amplitude far from jet region by the source in the jet region. We applied scale analyses for this equation and obtained Fr dependence of gravity wave flux in case there is no earth rotation. This Fr dependence corresponds to the aero-acoustic sound radiation theory.
4.Results
Unlike classical Rossby adjustment problem which supposes to the initial unbalanced state, gravity waves are continuously radiated from nearly balanced rotational flow region, where Fr is so small that balance dynamics is thought to be good approximation for the full system. Though time evolution of flow field is qualitatively the same for different parameter values, gravity wave radiation for each experiment have several different features. First, the wave length of gravity wave is proportional to 1/Fr, since the depth of fluid is shallow so that the phase speed of gravity wave is slow for large Fr. While gravity waves having no wavy structure in longitudinal direction can propagate for small Fr only, gravity waves having wavy structure in longitudinal direction can propagate even for large Fr. In addition, for small Ro where the effect of earth rotation is important, radiated gravity waves have strong dispersion. These gravity waves can be calculated from the solution of the Lighthill equation which include the effect of dispersion.
Gravity wave flux is also investigated for Ro - Fr parameter space. For large Ro where the effect of earth rotation is negligible, gravity wave flux is proportional to Fr, which is consistent with the aero-acoustic radiation theory. However, this dependence on Fr is not applied for small Ro, where the effect of earth rotation is important. This is because deformation radius is smaller for large Fr so that the interaction between vortices is inhibited and the unsteadiness of vortex is weakened. In addition, for much smaller Ro, the inertial frequency exceeds the frequency of unsteady rotational flow. Since gravity waves cannot propagate in these parameter ranges, gravity wave flux decreases suddenly. On the other hand, for intermediate Ro, gravity wave flux is larger than that for large Ro. This is because a source term which is related to the inertial frequency increases with Ro. This result suggests that the effect of earth rotation can strengthen gravity wave radiation.
5.Discussion
Balanced dynamics which is introduced by formal scaling analysis for small Ro suppose to no gravity wave radiation from balanced rotational flow. In contrast to the general understanding, smallness of Ro can not guarantee the validity of balanced dynamics. Our next step is to estimate the sudden decrease of gravity wave flux quantitatively for small Ro to discuss the validity of balanced dynamics. It will be also interesting to explore the possibility of similar feature of gravity wave radiation in the real atmosphere.
6.Conclusion
Using forced dissipative system, gravity wave radiation from unsteady rotational flow is investigated numerically in f-plane shallow water system for a wide range of Rossby number and Froude number. Unlike classical Rossby adjustment problem which supposes to initial unbalanced states, gravity waves are continuously radiated from nearly balanced rotational flows. While gravity wave flux is proportional to Fr for large Ro, this Fr dependence does not hold for small Ro because of decreasing deformation radius. In addition, there is a local maximum of the gravity wave flux for middle Ro, where the effect of earth rotation is not negligible and inertial cut-off of gravity wave radiation does not occur. This result suggests that the effect of earth rotation can strengthen gravity wave radiation.
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