In this talk, we will demonstrate some new diagnostic applications of Fourier analyses to the assessments of vertical velocity profiles in the tropical waves. It has been widely recognized that a significant portion of tropical precipitation is organized in an aggregated convective system over a wide range of spatial and temporal scales. In daily-to-intraseasonal time-scales, precipitation is often organized by waves that propagate eastward or westward along the equator. There exist different kinds of tropical waves (including the Madden-Julian Oscillations, Kelvin waves, equatorial Rossby waves, etc.), each of which is regulated by different mechanisms, and it has been still a challenge to simulate those different kinds of waves in general circulation models (GCMs) with conventional convective parameterizations. Because those waves are a primary source of tropical precipitation, the lack of the ability to simulate them may cause some undesirable biases in the GCMs.

   There have been many theoretical studies which attempt to explain the mechanisms of the tropical waves. In most of them, a vertical velocity profile plays a crucial role, implicitly or explicitly, in setting the phase velocity and instability of the waves. In this talk, we will propose a new diagnostic framework for assessing the vertical velocity profiles in the Fourier space. More specifically, we characterize the vertical velocity profiles with two quantities: top-heaviness ratio, and tilt ratio. And we look at those quantities for different types of waves in the Fourier space. Since most of the past theoretical studies are based on linear wave theories, our diagnoses may be directly compared to those studies, in hoping that we extract some fundamental parameters in the theories for improving the skill for GCMs to simulate tropical wave phenomena.