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An Analytical Study of Tropical Flows Using an Improvement of the Longwave Model An Analytical Study of Tropical Flows Using an Improvement of the Longwave Model Alex O. Gonzaleza Academic A±liation, Fall 2008: Senior, The Pennsylvania State University SOARS Summer 2008 Science Research Mentor: Wayne Schubert b Collaborators: Levi G. Silvers b and Matthew T. Masarik b Writing and Communication Mentor: Ingrid Moore c aDepartment of Meteorology, The Pennsylvania State University, University Park, Pennsylvania bDepartment of Atmospheric Science, Colorado State University, Fort Collins, Colorado cNational Center for Atmospheric Research, Boulder, Colorado ABSTRACT: We present a new analytical model of atmospheric waves in the equatorial area that ¯lters out inertia-gravity waves, to gain insight on the Madden-Julian Oscillation (MJO). Since the MJO is a climate phenomenon, one would like to study it using climate models, which are performed exclusively with the primitive equations. Global Climate Models (GCMs) do not simulate the MJO well, so ¯ltered analytical models based on the primitive equations are used. Filtered models are usually derived by partitioning the flow into nondivergent and irrotational parts, which are expressed in terms of the streamfunction and velocity potential. Then certain approximations are introduced into the divergence and potential vorticity equations, with the result that inertia- gravity waves are ¯ltered. Such procedures have led to the disadvantage that, in the process of ¯ltering the inertia-gravity waves, the Kelvin waves are distorted (e.g., Moura 1976). In the present paper we take a di®erent approach to the ¯ltering problem. We partition the flow into Kelvin and non-Kelvin parts (Ripa 1994), expressing the non-Kelvin part in terms of a single potential function, ', which satis¯es a master equation (Schubert et. al. 2008). The methods also yield an improvement of the longwave approximation (Gill 1980), in that they provide a more precise approximation of equatorial Rossby waves of all wavelengths. This is very promising for learning more about the MJO because an accurate representation of the MJO phenomenon requires well represented equatorial Rossby waves propagating west of the heat source, and a Kelvin wave propagating east of the heat source. We also derive an analytical solution for convectively coupled equatorial waves using the same ¯ltering method as mentioned above. The Signi¯cant Opportunities in Atmospheric Research and Science (SOARS) Program is managed by the University Corporation for Atmospheric Research (UCAR) with support from participating universities. SOARS is funded by the National Science Foundation, the National Oceanic and Atmospheric Administration (NOAA) Climate Program O±ce, the NOAA Oceans and Human Health Initiative, the Center for Multi-Scale Modeling of Atmospheric Processes at Colorado State University, and the Coopertative Insititue for Research in Environmental Sciences. SOARS is a partner project with Research Experience in Solid Earth Science for Students (RESESS). 1. Introduction 1.1. Background Many atmospheric phenomena start at the equator as equatorial waves, transporting convection and energy poleward. Equatorial waves are unique in comparison to other latitudinal waves in that they are heat-induced and trapped close to the equator (Matsuno 1966). Many atmospheric models of equatorial wave motions are produced to gain insight into variability of climate phenomena such as the Madden-Julian Oscillation (MJO) and the El Ni~noSouthern Oscillation (ENSO). The MJO, speci¯cally, is the main intraseasonal fluctuation that explains weather variations in the tropics. The more studies performed to better understand this phenomenon, the more long-range weather and climate predictions will improve (Madden and Julian 1994). Figure 1. Horizontal structures of the equatorial wave solutions to the shallow water equations on an equatorial ¯-plane, for nondimensional zonal wavenumber 1, with westward inertia-gravity waves (top left), equatorial Rossby waves (top right), Kelvin waves (bottom left), and mixed Rossby-gravity waves (bottom right). All scales and ¯elds have been nondimensionalized by ¡ ¢1=2 ¡ ¢1=2 taking the units of time and length as [T ] = 1=¯(¹c)1=2 , and [L] = (¹c)1=2=¯ , respectively. The equator runs horizontally through the center of each diagram. Hatching is for convergence and shading for divergence, with a 0.6 unit interval between successive levels. Unshaded contours are for geopotential, with a contour interval of 0.5 units. Negative contours are dashed and the zero contour is omitted. From Wheeler (2002). SOARS°R 2008, Alex O. Gonzalez, 2 The dynamics of tropical weather phenomena, such as the MJO, are not as well understood as the dynamics of mid-latitude weather phenomena because of a lack of numerical data collected (Zagar^ 2004). Observational data is sparse in the tropics due to the fact that the tropical atmosphere is surrounded by water. Therefore, it is harder to get information using data analysis techniques. The Global Climate Models (GCMs) often do a poor job at simulating the dynamics and structure of convectively coupled waves, especially the MJO (Frierson 2008, Majda and Khouider 2008); much of what we know about climate phenomena in the equatorial region is due to theoretical studies. Even though these theoretical studies are simpli¯ed, they are quite good at showing realistic features of the tropical atmosphere (Gonzalez et. al. 2007). By improving our understanding of equatorial waves in simpli¯ed theoretical studies, it will improve our understanding of climate phenomena that initialize near the equator in the real atmosphere. Waves in the tropics are excited by localized atmospheric forcings. A physical interpretation of these forcings is diabatic heating produced by latent heat release. Latent heat release occurs as a result of water vapor condensing into liquid in deep convective clouds. Waves in the atmosphere are anisotropic, i.e., their response is not the same in all directions, producing di®erent types of wave structures (Gonzalez et. al. 2007). There are four types of equatorially trapped waves in the tropics: Kelvin waves, inertia-gravity waves, equatorial Rossby waves, and mixed Rossby-gravity waves (Fig. 1). The ¯rst scientist to present solutions for equatorial waves was Matsuno (1966), deriving and solving the primitive linearized shallow-water equations on the equatorial ¯-plane (2.1{2.3). He found that Kelvin waves and mixed Rossby-gravity waves actually do exist in theory, and thus could exist in the real atmosphere. Many studies of equatorial wave theory and related atmospheric phenomena cite this primitive equation model. Numerous models attempting to represent atmospheric waves in the tropics have also referred to the longwave approximation of Gill (1980). Gill (1980) initially evaluated the steady-state response of equatorial waves to prescribed heat sources by using an analytical model based on the linearized shallow-water equations. An advantage to this approximation of tropical circulations is that inertia-gravity waves are ¯ltered out, but Kelvin and long equatorial Rossby waves are retained (Schubert et. al. 2008). However, a de¯ciency of this approximation is its distortion of the dispersion relation for short equatorial Rossby waves (Stevens et. al. 1990). An improvement is needed when studying flows that include features that require higher resolution, such as diabatic heating, which is the main forcing in the tropics. A graphical comparison of the dispersion relation for equatorial Rossby waves between the Gill (1980) model and the primitive equation model is shown in Figure 2. Many studies have utilized the longwave approximation for simulating the MJO, but an accurate representation of the MJO requires requires taking account of accurate Rossby wave dynamics to the west of the region of interest, and Kelvin wave dynamics to the east of the region of interest (Fig. 3). Inertia-gravity waves can be ¯ltered when studying the MJO because they do not contain as much outgoing longwave radiation compared to Kelvin and equatorial Rossby waves (Fig. 3). A model that ¯lters out inertia-gravity waves while retaining an accurate representation of equatorial Rossby waves for all wavelengths, and just as accurate Kelvin waves may lead to a better understanding of, not just these variations of climate, but the underlying dynamics in the tropics. 1.2. New analytical model A new analytic theoretical model is presented in this study that ¯lters out intertia-gravity waves, retains Kelvin waves, and provides a more accurate approximation of equatorial Rossby waves of all wavelengths (Schubert et. al. 2008). The primitive linearized shallow-water equations are solved on the equatorial ¯-plane using a ¯ltering method similar to that used by Ripa (1994). Filtered models are usually derived by partitioning the flow into rotational and divergent parts, which are expressed in terms of the streamfunction and velocity potential, respectively. Then certain approximations are introduced into the divergence and potential vorticity equations, with the result that inertia-gravity waves are ¯ltered. Such procedures have led to the disadvantage that, in the process of ¯ltering the inertia-gravity waves, the Kelvin waves are distorted (e.g., Moura 1976). This distortion of the Kelvin waves makes such models of limited use in studying the MJO and ENSO, which as mentioned before require accurate Kelvin and equatorial Rossby wave dynamics. SOARS°R 2008, Alex O. Gonzalez, 3 Figure 2. Dispersion diagram for atmospheric
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