An Analytical Study of Tropical Flows Using an Improvement of the Longwave Model

Alex O. Gonzaleza Academic Aﬃliation, 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 filters 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 filtered 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 filtered. Such procedures have led to the disadvantage that, in the process of filtering the inertia-gravity waves, the Kelvin waves are distorted (e.g., Moura 1976). In the present paper we take a different approach to the filtering 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, ϕ, which satisfies 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 filtering method as mentioned above.

The Signiﬁcant 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 Oﬃce, 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 NiñoSouthern Oscillation (ENSO). The MJO, specifically, 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 β-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 ﬁelds have been nondimensionalized by ¡ ¢1/2 ¡ ¢1/2 taking the units of time and length as [T ] = 1/β(¯c)1/2 , and [L] = (¯c)1/2/β , 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 (Zagarˆ 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 simplified, they are quite good at showing realistic features of the tropical atmosphere (Gonzalez et. al. 2007). By improving our understanding of equatorial waves in simplified 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 different 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 first scientist to present solutions for equatorial waves was Matsuno (1966), deriving and solving the primitive linearized shallow-water equations on the equatorial β-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 filtered out, but Kelvin and long equatorial Rossby waves are retained (Schubert et. al. 2008). However, a deficiency 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 filtered 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 filters 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 filters 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 β-plane using a filtering 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 filtered. Such procedures have led to the disadvantage that, in the process of filtering 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 waves in the equatorial area on the sphere. Curved solid lines represent the equatorial Rossby wave dispersion relation, while the straight solid line represents the Kelvin wave dispersion relation, both using the primitive equations. The dashed lines represent the longwave approximation equatorial Rossby wave dispersion relation. From Stevens et. al. (1990)

Figure 3. On the left are spectral bands of the five filters - Madden-Julian Oscillation (MJO), equatorial Rossby waves (ER), Kelvin waves (Kelvin), Easterly waves (TD-type), and mixed Rossby-gravity waves (MRG) used in the project are outlined with solid curves and plotted over a smoothed tropical OLR spectrum. From Roundy and Frank (2004). On the right is a presentation of the wavenumber vs. frequency spectral peaks of a long record of satellite-observed outgoing longwave radiation for equatorial waves symmetric about the equator. To represent the peaks, the contours show a ratio of the actual power with an estimate of the red-noise background power. A ratio of greater than 1.1 is a statistically significant spectral (95 percent level). Also shown are the dispersion curves for the equatorial waves for equivalent depths of 12, 25, and 50 m. From Wheeler and Kiladis (1999). One can infer from this figure the importance of equatorial Rossby waves (ER) and Kelvin waves (Kelvin) in the MJO, and the rationale for filtering out inertia gravity waves.

In the present paper we take a different approach to the filtering 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, ϕ, which we refer to as the Ripa potential, which satisfies a master equation. We then approximate this master

SOARS°R 2008, Alex O. Gonzalez, 4 equation in such a way that inertia-gravity waves are filtered and equatorial Rossby waves are accurately described. This approach leaves the Kelvin waves undistorted and results in a filtered model that is useful for studying climate phenomena such as the MJO and ENSO. Section 2 outlines the governing primitive equations and the Ripa potential. In section 3 we present the filtered model. In section 4 we present the filtered model with forcing and dissipative terms implemented into the equations. These terms physically represent either diabatic heat sources and sinks, Rayleigh friction, or Newtonian cooling similar to the ones used in Gill (1980) and Chao (1987). The introduction of these heating and dissipative terms in the equations simulates convectively coupled waves, instead of freely oscillating waves, which is representative of how waves in an excited MJO event propagate.

2. Primitive equation model

Principles based on a shallow-water model are used, where the atmosphere is simulated as a incompressible fluid (i.e. water), and the fluid depth is much smaller than the horizontal scale of the flow (Fig. 6). Consider small amplitude motions about a resting basic state on the equatorial β-plane. Vertical accelerations are neglected because of the assumed hydrostatic balance, where horizontal flow is independent of height. Due to these assumptions, the diabatic forcing will excite only one vertical mode. We can write the primitive linearized shallow water equations as ∂u ∂φ − βyv + = 0, (2.1) ∂t ∂x ∂v ∂φ + βyu + = 0, (2.2) ∂t ∂y µ ¶ ∂φ ∂u ∂v +c ¯2 + = 0, (2.3) ∂t ∂x ∂y where u is the eastward component of velocity, v the northward component, φ the perturbation geopotential, c¯ the constant gravity wave speed, and β the equatorial value of the northward gradient of the Coriolis parameter. We seek solutions of (2.1)–(2.3) on a domain that is infinite in y and periodic over −πa ≤ x ≤ πa, where a is the Earth’s radius. The dependent variables u, v, φ are assumed to approach zero as y → ±∞. We shall solve (2.1)–(2.3) by partitioning the solution into the non-Kelvin part and the Kelvin part, i.e.,       u uϕ uK v = vϕ  +  0  . (2.4) φ φϕ φK

To accomplish this partition and to express the non-Kelvin part entirely in terms of the potential function, ϕ, we begin by applying [(∂/∂t) − c¯(∂/∂x)] to (2.1) and [(∂/∂t) +c ¯(∂/∂x)] to (2.3), and then combining the resulting equations to obtain µ ¶ µ ¶ ∂ ∂ ∂ βy +c ¯ (φ +cu ¯ ) = −c¯2 − v, (2.5) ∂t ∂x ∂y c¯ µ ¶ µ ¶ ∂ ∂ ∂ βy − c¯ (φ − cu¯ ) = −c¯2 + v. (2.6) ∂t ∂x ∂y c¯ According to (2.5), the solution for φ +cu ¯ consists of a non-Kelvin part obtained from v by integrating along the characteristic x − ct¯ , plus a Kelvin part obtained from the solution of (2.5) with zero right hand side. Similarly, according to (2.6), the solution for φ − cu¯ consists of a non-Kelvin part obtained from v by integrating along the characteristic x +ct ¯ , plus a Kelvin part obtained from the solution of (2.6) with zero right hand side. The solutions of the homogeneous versions of (2.5) and (2.6) are

φK +cu ¯ K = Ke(x − ct,¯ y), (2.7)

SOARS°R 2008, Alex O. Gonzalez, 5 φK − cu¯ K = Kw(x +ct, ¯ y), (2.8)

where Ke and Kw are arbitrary functions associated with eastward and westward propagation of information. These solutions need to also satisfy (2.2) with v = 0, which can be written as µ ¶ µ ¶ ∂ βy ∂ βy + K + − K = 0. (2.9) ∂y c¯ e ∂y c¯ w

If Ke = 0, the ﬁrst term on the left hand side of (2.9) vanishes, so that the Kw solution has the unacceptable 1 (β/c¯)y2 behavior e 2 and must be discarded. This behavior is unacceptable because as y → ±∞, the ﬂow → ±∞, which does not make physical sense for waves trapped close to the equator. If Kw = 0, the second − 1 (β/c¯)y2 term on the left hand side of (2.9) vanishes, so that Ke has the acceptable behavior e 2 . Thus, (2.7) and (2.8) become − 1 (β/c¯)y2 φK +cu ¯ K = K(x − ct¯ )e 2 , (2.10)

φK − cu¯ K = 0, (2.11) where the function K(x) is determined from the initial condition. Now consider the non-Kelvin part of the flow. Equations (2.5) and (2.6) motivate the representations (Ripa 1994) µ ¶ µ ¶ ∂ ∂ ∂ βy φ +cu ¯ = − c¯ − ϕ, (2.12) ϕ ϕ ∂t ∂x ∂y c¯ µ ¶ µ ¶ 1 ∂ ∂ ∂ ∂ v = − − c¯ +c ¯ ϕ, (2.13) ϕ c¯2 ∂t ∂x ∂t ∂x µ ¶ µ ¶ ∂ ∂ ∂ βy φ − cu¯ = +c ¯ + ϕ. (2.14) ϕ ϕ ∂t ∂x ∂y c¯ When the Kelvin solution (2.10)–(2.11) and the non-Kelvin representations (2.12)–(2.14) are used in (2.4), we obtain µ 2 ¶ ∂ βy ∂ 1 − 1 (β/c¯)y2 u = − + ϕ + K(x − ct¯ )e 2 , (2.15) ∂x∂y c¯2 ∂t 2¯c µ ¶ ∂2 1 ∂2 v = − ϕ, (2.16) ∂x2 c¯2 ∂t2 µ 2 ¶ ∂ ∂ 1 − 1 (β/c¯)y2 φ = + βy ϕ + K(x − ct¯ )e 2 . (2.17) ∂t∂y ∂x 2 If we substitute (2.15)–(2.17) back into the original shallow water equations, we find that (2.1) and (2.3) are satisfied, and that (2.2) will also be satisfied if ϕ is a solution of the master equation µ ¶ ∂2 ∂2 β2y2 1 ∂2 ∂ϕ ∂ϕ + − − + β = 0. (2.18) ∂x2 ∂y2 c¯2 c¯2 ∂t2 ∂t ∂x

If (2.18) can be solved for ϕ, the u, v, φ fields can be recovered from (2.15)–(2.17) by differentiation of ϕ. Since the ϕ-field yields the non-Kelvin part of the flow, the master equation (2.18) describes the highly divergent flow associated with inertia-gravity waves as well as the quasi-nondivergent, potential vorticity dynamics associated with Rossby waves. In this regard, it is interesting to note that the x-derivative of (2.18) yields the potential vorticity equation, i.e.,

∂q + βv = 0, (2.19) ∂t where µ ¶ ∂v ∂u βy ∂2 ∂2 β2y2 1 ∂2 ∂ϕ β ∂ϕ q = − − φ = + − − + (2.20) ∂x ∂y c¯2 ∂x2 ∂y2 c¯2 c¯2 ∂t2 ∂x c¯2 ∂t

SOARS°R 2008, Alex O. Gonzalez, 6 is the potential vorticity anomaly, while the y-derivative of (2.18) yields the divergence equation, i.e., µ ¶ µ ¶ ∂ ∂u ∂v ∂v ∂u + − βy − + βu + ∇2φ = 0. (2.21) ∂t ∂x ∂y ∂x ∂y In the next section we introduce a filtering approximation that leads to a master equation that is first order in time rather than third order in time. The filtering approximation has no effect on the Kelvin part of the flow, i.e., it filters inertia-gravity waves without distorting Kelvin waves—an extremely useful property for studying the MJO and ENSO.

3. Filtered model The longwave approximation of (2.1)–(2.3) is a filtering approximation obtained by neglecting ∂v/∂t in (2.2). Here we consider a more accurate filtered model obtained by approximating (2.1)–(2.3) by ∂u ∂φ − βyv + = 0, (3.1) ∂t ∂x ∂v˜ ∂φ + βyu + = 0, (3.2) ∂t ∂y µ ¶ ∂φ ∂u ∂v +c ¯2 + = 0, (3.3) ∂t ∂x ∂y wherev ˜ is the approximation of v defined below. Since (3.1) is identical to (2.1), and (3.3) is identical to (2.3), the argument given between (2.4) and (2.17) remains essentially unchanged, but with the inclusion of a representation forv ˜. Thus, the representations of u, v, v,˜ φ are

µ 2 ¶ ∂ βy ∂ 1 − 1 (β/c¯)y2 u = − + ϕ + K(x − ct¯ )e 2 , (3.4) ∂x∂y c¯2 ∂t 2¯c µ ¶ ∂2 1 ∂2 ∂2ϕ v = − ϕ, v˜ = , (3.5) ∂x2 c¯2 ∂t2 ∂x2 µ 2 ¶ ∂ ∂ 1 − 1 (β/c¯)y2 φ = + βy ϕ + K(x − ct¯ )e 2 . (3.6) ∂t∂y ∂x 2 If we substitute (3.4)–(3.6) into the approximate shallow water equations (3.1)–(3.3), we find that (3.1) and (3.3) are satisfied, and that (3.2) will also be satisfied if ϕ is a solution of the (filtered) master equation µ ¶ ∂2 ∂2 β2y2 ∂ϕ ∂ϕ + − + β = 0. (3.7) ∂x2 ∂y2 c¯2 ∂t ∂x

Note that (3.7) is ﬁrst order in time, while (2.18) is third order in time. As we shall see, (3.7) ﬁlters inertia-gravity modes while retaining an accurate description of Rossby modes. As in the primitive equation case, it is interesting to note that the x-derivative of (3.7) yields the potential vorticity equation, i.e., ∂q + βv = 0, (3.8) ∂t where µ ¶ ∂v˜ ∂u βy ∂2 ∂2 β2y2 ∂ϕ β ∂ϕ q = − − φ = + − + (3.9) ∂x ∂y c¯2 ∂x2 ∂y2 c¯2 ∂x c¯2 ∂t is the potential vorticity anomaly, while the y-derivative of (3.7) yields the divergence equation µ ¶ µ ¶ ∂ ∂u ∂v˜ ∂v ∂u + − βy − + βu + ∇2φ = 0. (3.10) ∂t ∂x ∂y ∂x ∂y

SOARS°R 2008, Alex O. Gonzalez, 7 Note that (3.9) and (3.10) are identical to (2.20) and (2.21), except that v has been replaced byv ˜ in the vorticity equation and ∂v/∂t has been replaced by ∂v/∂t˜ in the divergence equation. It is interesting to note that (3.7) and (3.9) can be combined to yield µ ¶ ∂2 ∂2 β2y2 ∂q + − v = , (3.11) ∂x2 ∂y2 c¯2 ∂x which can be considered to be a PV invertibility principle, i.e., a relation that can be used to compute the meridional wind v from the potential vorticity q. Note that (3.8) and (3.11) form a closed system in v and q. In the next section, we introduce forcing and dissipative terms to the ﬁltering approximation. We follow the same approach as in the ﬁrst two sections, except we must account for arbitrary diabatic heating and arbitrary frition/cooling.

4. Forced ﬁltered model

We can write the ﬁltered linearized shallow water equations including forcing and dissipative terms as

∂u ∂φ − βyv + = −αu, (4.1) ∂t ∂x

∂v˜ ∂φ + βyu + = −αv,˜ (4.2) ∂t ∂y µ ¶ ∂φ ∂u ∂v +c ¯2 + = −αφ + Q, (4.3) ∂t ∂x ∂y where α the constant coefficient for Rayleigh friction and Newtonian cooling,c ¯ the constant gravity wave speed, and Q the mass source/sink. The mass source/sink is assumed to propagate eastward at the constant speed c, i.e., it is specified as Q(ξ, y), where ξ = x − ct. All the fields u, v, φ, Q are assumed to approach zero as y → ±∞. We shall solve (4.1)–(4.3) by partitioning the solution into the Kelvin wave part and the non-Kelvin wave part, i.e.,       u uϕ uK  v  =  vϕ  +  0  . (4.4) φ φϕ φK To accomplish this partition and to express the non-Kelvin part entirely in terms of the potential function ϕ, we begin by applying [(∂/∂t) − c¯(∂/∂x)] to (4.1) and [(∂/∂t) +c ¯(∂/∂x)] to (4.3), and then combining the resulting equations to obtain µ ¶ µ ¶ ∂ ∂ £ ¤ ∂ βy +c ¯ (φ +cu ¯ − F ) eαt = −c¯2 − veαt, (4.5) ∂t ∂x ∂y c¯ µ ¶ µ ¶ ∂ ∂ £ ¤ ∂ βy − c¯ (φ − cu¯ − G)eαt = −c¯2 + veαt, (4.6) ∂t ∂x ∂y c¯ where we have introduced the new fields F (x, y, t) and G(x, y, t), which are required to satisfy µ ¶ ∂ ∂ +c ¯ F eαt = Qeαt, (4.7) ∂t ∂x µ ¶ ∂ ∂ − c¯ Geαt = Qeαt, (4.8) ∂t ∂x

SOARS°R 2008, Alex O. Gonzalez, 8 − 1 ( β )y2 where F = Fϕ(x, y, t) + FK (x, t)e 2 c¯ . With the initial conditions F (x, y, 0) = G(x, y, 0) = 0, the solutions of (4.7) and (4.8) are

Z t 0 F (x, y, t) = e−α(t−t )Q (x − ct − (¯c − c)(t − t0), y) dt0, (4.9) 0

Z t 0 G(x, y, t) = e−α(t−t )Q (x − ct + (¯c + c)(t − t0), y) dt0. (4.10) 0 According to (4.5), the solution for (φ +cu ¯ − F )eαt consists of a non-Kelvin part obtained from v by integrating along the characteristic x − ct¯ , plus a Kelvin part obtained from the solution of (4.5) with a zero right hand side. Similarly, according to (4.6), the solution for (φ − cu¯ − G)eαt consists of a non-Kelvin part obtained from v by integrating along the characteristic x +ct ¯ , plus a Kelvin part obtained from the solution of (4.6) with a zero right hand side. The solutions of the homogeneous forms of (4.5) and (4.6) are

−αt φK +cu ¯ K = e Ke(x − ct,¯ y) + F, (4.11)

−αt φK − cu¯ K = e Kw(x +ct, ¯ y) + G, (4.12) where Ke and Kw are arbitrary functions associated with eastward and westward propagation of information. These solutions also need to satisfy (4.2) with v = 0, which is ﬁrst written as ∂φ βyu + = 0. (4.13) ∂y

We want to manipulate (4.13) in such a way that the terms φK +cu ¯ K and φK − cu¯ K appear in the equation. This equation can now be written in the following form µ ¶ µ ¶ ∂ βy ∂ βy + (φ +cu ¯ ) + − (φ − cu¯ ) = 0 (4.14) ∂y c¯ K K ∂y c¯ K K

If φK +cu ¯ K = 0, the ﬁrst term on the left hand side of (4.14) vanishes, so that φK − cu¯ K has unacceptable 1 ( β )y2 behavior e 2 c¯ . If φK − cu¯ K = 0, the second term on the left hand side of (4.12) vanishes, so that − 1 ( β )y2 φK +cu ¯ K = 0 has acceptable behavior e 2 c¯ . Thus, (4.11) and (4.12) become

£ ¤ β 2 −αt − 1 ( )y φK +cu ¯ K = e K(x − ct¯ ) + FK (x, t) e 2 c¯ , (4.15)

φK − cu¯ K = 0, (4.16) where the function K(x) is an arbitrary function determined from the initial condition and the function FK (x, t) is determined from

Z ∞ − 1 ( β )y2 FK (x, t) = F (x, y, t)e 2 c¯ dy. (4.17) −∞ Now consider the non-Kelvin part of the ﬂow. Equations (4.5) and (4.6) motivate the representations (Ripa 1994) µ ¶ µ ¶ ∂ ∂ ∂ βy (φ +cu ¯ − F )eαt = − − c¯ − ϕeαt, (4.18) ϕ ϕ ∂t ∂x ∂y c¯ µ ¶ ∂2 1 ∂2 ∂2 (ϕeαt) v eαt = − ϕeαt, v˜ eαt = , (4.19) ϕ ∂x2 c2 ∂t2 ϕ ∂x2 µ ¶ µ ¶ ∂ ∂ ∂ βy (φ − cu¯ − G)eαt = +c ¯ + ϕeαt. (4.20) ϕ ϕ ∂t ∂x ∂y c¯

SOARS°R 2008, Alex O. Gonzalez, 9 When the Kelvin solution (4.15)-(4.16) and the non-Kelvin representations (4.18)-(4.20) are used in (4.4), we obtain

" # µ 2 ¶ µ ¶ −αt ∂ βy ∂ αt 1 −αt − 1 β y2 u = −e + ϕe + (F − G ) + e K(x − ct¯ ) + F (x, y) e 2 ( c¯ ) , (4.21) ∂x∂y c¯2 ∂t 2¯c ϕ ϕ K µ ¶ ∂2 1 ∂2 ∂2 (ϕeαt) v = e−αt − ϕeαt, v˜ = e−αt , (4.22) ∂x2 c2 ∂t2 ∂x2

" # µ 2 ¶ µ ¶ −αt ∂ ∂ αt 1 −αt − 1 β y2 φ = e + βy ϕe + (F + G ) + e K(x − ct¯ ) + F (x, y) e 2 ( c¯ ) . (4.23) ∂t∂y ∂x 2 ϕ ϕ K

If we substitute these representations back into the original shallow water equations, we find (making use of (4.9) and (4.10)) that (4.1) and (4.3) are satisfied, and that (4.2) will also be satisfied if ϕ is a solution of the master equation µ ¶ "µ ¶ µ ¶ # ∂2 ∂2 β2y2 ∂(ϕeαt) ∂ϕ 1 ∂ βy ∂ βy e−αt + − + β = − + F + − G . (4.24) ∂x2 ∂y2 c¯2 ∂t ∂x 2 ∂y c¯ ϕ ∂y c¯ ϕ

If (4.22) can be solved for ϕ, the uϕ, vϕ, φϕ fields can be recovered from (4.21)-(4.23) by differentiation of ϕ. Note that the ϕ-field yields the non-Kelvin part of the flow. It is interesting to note that the x-derivative of (4.24) yields the potential vorticity equation, i.e., µ ¶ ∂ βy + α q + βv˜ = − Q, (4.25) ∂t c¯2 where µ ¶ ∂v˜ ∂u βy ∂2 ∂2 β2y2 ∂ϕ ∂ϕ q = − − φ = + − − β (4.26) ∂x ∂y c¯2 ∂x2 ∂y2 c¯2 ∂x ∂t is the potential vorticity anomaly. In addition, the y-derivative of (4.22) yields the divergence equation, i.e., µ ¶ µ ¶ µ ¶ ∂ ∂u ∂v˜ ∂v ∂u + α + − βy − + βu + ∇2φ = 0. (4.27) ∂t ∂x ∂y ∂x ∂y

In the next section we solve the master equation for each model using Fourier and Hermite transforms, in order to remove dependence of x and y. Then we will have an ordinary diﬀerential equation in time, which is much easier to solve.

5. Solution of the master equation for the PE model We can solve the partial differential equation (2.18) by transforming it into an ordinary differential equation in time, using a Fourier transform in x and a Hermite transform in y. We first take the Fourier transform of (2.18), defining the transform pair

Z πa 1 −imx/a ϕm(y, t) = ϕ(x, y, t)e dx, (5.1) 2πa −πa

X∞ imx/a ϕ(x, y, t) = ϕm(y, t)e , (5.2) m=−∞

SOARS°R 2008, Alex O. Gonzalez, 10 where the integer m denotes the zonal wavenumber. In this way, (2.18) reduces to ( µ ¶ ) ∂2 ² ∂2 ∂ϕ m2 − ²1/2 − yˆ2 + m = 2Ωimϕ , (5.3) ∂yˆ2 (2Ω)2 ∂t2 ∂t m wherey ˆ = (β/c)1/2y = ²1/4(y/a), with ² = 4Ω2a2/c¯2 denoting Lamb’s parameter. Lamb’s parameter is a dimensionless parameter that is useful for describing changes in ﬂuid depth (h¯), becausec ¯ = gh¯ is the only nonconstant variable in the equation. It also introduces periodicity into the solution, because wavenumbers can only be of integer values, which is useful for describing waves in the real atmosphere. We now convert (5.3) into an ordinary diﬀerential equation by transforming iny ˆ. We use the transform pair

Z ∞ ϕmn(t) = ϕm(ˆy, t)Hn(ˆy)dy,ˆ (5.4) −∞ X∞ ϕm(ˆy, t) = ϕmn(t)Hn(ˆy), (5.5) n=0 where the Hermite functions Hn(ˆy)(n = 0, 1, 2,...) are related to the Hermite polynomials Hn(ˆy)(n = 1 n − 1 − 1 yˆ2 0, 1, 2,...) by Hn(ˆy) = (π 2 2 n!) 2 Hn(ˆy)e 2 . The Hermite functions Hn(ˆy) satisfy the recurrence relation

µ ¶ 1 2 ³ ´ 1 n + 1 n 2 yˆH (ˆy) = H (ˆy) + H (ˆy), (5.6) n 2 n+1 2 n−1 and the derivative relation

µ ¶ 1 2 ³ ´ 1 dH (ˆy) n + 1 n 2 n = − H (ˆy) + H (ˆy). (5.7) dyˆ 2 n+1 2 n−1

Figure 4. The Hermite functions Hn(ˆy) for n = 0, 1, 2, 3, 4. These satisfy the orthonormality condition (5.8) and serve as the basis functions for the transform pair (5.4) and (5.5).

SOARS°R 2008, Alex O. Gonzalez, 11 − 1 − 1 yˆ2 1 − 1 − 1 yˆ2 The ﬁrst two Hermite functions are H0(ˆy) = π 4 e 2 and H1(ˆy) = 2 2 π 4 yeˆ 2 , from which all succeeding structure functions can be computed using the recurrence relation (5.6). Plots of Hn(ˆy) for n = 0, 1, 2, 3, 4 are shown in Fig. 4. Note that (5.4) can be obtained through multiplication of (5.5) by Hn0 (ˆy), followed by integration overy ˆ and use of the orthonormality relation Z ( ∞ 1 n0 = n, 0 Hn(ˆy)Hn (ˆy)dyˆ = 0 (5.8) −∞ 0 n 6= n.

Multiplying (5.3) by Hn(ˆy) and integrating overy ˆ (i.e., taking the Hermite transform of (5.3)) we obtain the third order ordinary diﬀerential equation ½ ¾ ² d2 dϕ m2 + ²1/2(2n + 1) + mn = 2Ωimϕ . (5.9) (2Ω)2 dt2 dt mn

In the derivation of (5.9) we have used two integrations by parts (with vanishing boundary terms) 2 2 2 2 2 2 and the fact that Hn(ˆy) is an eigenfunction of the operator (d /dyˆ − yˆ ), i.e., (d /dyˆ − yˆ )Hn(ˆy) = −(2n + 1)Hn(ˆy). The solution of (5.9) is

X2 −iνmnr t ϕmn(t) = ϕmnr(0)e , (5.10) r=0

Figure 5. The dimensionless frequenciesν ˆmnr, determined from the primitive equation dispersion relation (5.11) with ² = 500 and n = 0, 1, 2, ··· , 9. For completeness, the dimensionless Kelvin wave frequencyν ˆ = ²−1/2m is also shown.

SOARS°R 2008, Alex O. Gonzalez, 12 where the dimensionless frequenciesν ˆmnr = νmnr/(2Ω) are solutions of

2 2 m 1/2 ²νˆmnr − m − = ² (2n + 1) (5.11) νˆmnr

for n = 0, 1, 2, ··· , with r = 0, 1, 2 serving as an index for the three roots of the dispersion relation (5.11). The resulting dispersion diagram is shown in Fig. 5. Thus, using (5.2), (5.5), and (5.10), we conclude that the solution of the master equation (2.18) is

X∞ X∞ X2 i(mx/a−νmnr t) ϕ(x, y, t) = ϕmnr(0)Hn(ˆy)e . (5.12) m=−∞ n=0 r=0

Using the solution (5.12) in the right hand sides of (2.15)–(2.18), and then making use of (5.6) and (5.7), we obtain the ﬁnal solution       −1 u(x, y, t) X∞ X∞ X2 umnr c¯ i(mx/a−ν t) 1 − 1 (β/c)y2 v(x, y, t) = ϕ (0) v  e mnr +  0  K(x − ct)e 2 , (5.13) mnr mnr 2 φ(x, y, t) m=−∞ n=0 r=0 φmnr 1

where "µ ¶ # i²1/4 n + 1 1/2 ³n´1/2 u (y) = (²1/2νˆ + m)H (ˆy) + (²1/2νˆ − m)H (ˆy) (5.14) mnr a2 2 mnr n+1 2 mnr n−1

1 v (y) = (²νˆ2 − m2)H (ˆy) (5.15) mnr a2 mnr n "µ ¶ # ic²¯ 1/4 n + 1 1/2 ³n´1/2 φ (y) = (²1/2νˆ + m)H (ˆy) − (²1/2νˆ − m)H (ˆy) (5.16) mnr a2 2 mnr n+1 2 mnr n−1

are the eigenfunctions for the Rossby modes (r = 0) and the inertia-gravity modes (r = 1, 2).

6. Solution of the master equation for the ﬁltered model

The solution of (3.7) proceeds in a manner analogous to the solution of (2.18). After transforming in x and y, we obtain the ﬁrst order ordinary diﬀerential equation

dϕ 2Ωim mn = ϕ . (6.1) dt m2 + ²1/2(2n + 1) mn

The solution of (6.1) is −iνmnt ϕmn(t) = ϕmn(0)e , (6.2) where the dimensionless frequencyν ˆmn = νmn/(2Ω) is given by m νˆ = − (6.3) mn m2 + ²1/2(2n + 1) for n = 0, 1, 2, ··· , which is an approximation of the low frequency solutions of the cubic equation (4.11). The resulting dispersion diagram for the ﬁltered model is shown in Fig. 6. Thus, we conclude that the solution of (3.7) is X∞ X∞ i(mx/a−νmnt) ϕ(x, y, t) = ϕmn(0)Hn(ˆy)e . (6.4) m=−∞ n=0

SOARS°R 2008, Alex O. Gonzalez, 13 Figure 6. The dimensionless frequenciesν ˆmn, determined from the ﬁltered model dispersion relation (6.3) with ² = 500 and n = 0, 1, 2, ··· , 9. For completeness, the dimensionless Kelvin wave frequencyν ˆ = ²−1/2m is also shown.

Using the solution (6.4) in the right hand sides of (3.4)–(3.6) we obtain       −1 u(x, y, t) X∞ X∞ umn(y) c¯ i(mx/a−ν t) 1 − 1 (β/c)y2 v(x, y, t) = ϕ (0) v (y) e mn +  0  K(x − ct)e 2 , (6.5) mn mn 2 φ(x, y, t) m=−∞ n=0 φmn(y) 1 where "µ ¶ # i²1/4 n + 1 1/2 ³n´1/2 u (y) = (²1/2νˆ + m)H (ˆy) + (²1/2νˆ − m)H (ˆy) (6.6) mn a2 2 mn n+1 2 mn n−1

1 v (y) = (²νˆ2 − m2)H (ˆy) (6.7) mn a2 mnr n "µ ¶ # ic²¯ 1/4 n + 1 1/2 ³n´1/2 φ (y) = (²1/2νˆ + m)H (ˆy) − (²1/2νˆ − m)H (ˆy) (6.8) mn a2 2 mn n+1 2 mn n−1 are the eigenfunctions for the Rossby modes. In comparing the balanced model results (6.6)–(6.8) with the primitive equation model results (5.14)–(5.16), we note that the subscripts r and the sum over r are missing in (6.6)–(6.8) since inertia-gravity waves have been ﬁltered. The eigenfunctions in (6.6)–(6.8) are accurate approximations to the Rossby wave eigenfunctions in (5.14)–(5.16) since the only diﬀerence is that νmn is computed from (6.3) for use in (6.6)–(6.8), while νmn0 is the low frequency solution of (5.11) for use in (5.14)–(5.16).

SOARS°R 2008, Alex O. Gonzalez, 14 7. Solution of the master equation for the forced ﬁltered model

The solution of (4.24) proceeds in a manner analogous to the solution of (3.7). After a Fourier transform pair in x, with similar Fourier transform pairs existing for F and G, (4.24) reduces to ( µ ¶) "µ ¶ µ ¶ # ∂2 ∂ (ϕ eαt) ²1/4a ∂ ∂ e−αt −m2 + ²1/2 − yˆ2 m + 2Ωimϕ = − +y ˆ F ϕ (ˆy, t) + − yˆ Gϕ (ˆy, t) . ∂yˆ2 ∂t m 2 ∂yˆ m ∂yˆ m (7.1) We now convert (7.1) into an ordinary differential equation by transforming in y as done in the previous sections. After multiplying (2.14) by Hn(ˆy) and taking the Hermite transform of (7.1) we obtain the first order ordinary differential equation

( ) "µ ¶ 1 # αt 1/4 2 ³ ´ 1 d (ϕ ne ) ² a n + 1 n 2 e−αt −m2 − ²1/2 (2n + 1) m + 2Ωimϕ = Gϕ (t) − F ϕ (t) dt m 2 2 m,n+1 2 m,n−1 (7.2) In the derivation of (7.2) we have used two integrations by parts (with vanishing boundary terms) and the fact 2 2 2 2 2 2 that Hn(ˆy) is an eigenfunction of the operator (d /dyˆ − yˆ ), i.e., (d /dyˆ − yˆ )Hn(ˆy) = −(2n + 1)Hn(ˆy). The solution of (7.2) is

Z "µ ¶ 1 # t 2 ³ ´ 1 1 0 0 n + 1 ϕ n 2 ϕ 4 −α(t−t ) −iνmn(t−t ) 0 0 0 ϕmn(t) = ² a e e Gm,n+1(t ) − Fm,n−1(t ) dt (7.3) 0 2 2

where the frequencies νmn are solutions of m νˆ = − (7.4) mn m2 + ²1/2(2n + 1) for n = 0, 1, 2, ··· , which is analogous to (6.3), which means the dispersion diagram is the same as Fig. 6. Thus, using (5.2), (5.5), and (7.3), we conclude that the solution of (4.24) is " # ∞ ∞ Z µ ¶ 1 X X t 2 ³ ´ 1 1 imx/a −α(t−t0) −iν (t−t0) n + 1 ϕ 0 n 2 ϕ 0 0 ϕ(x, y, t) = ² 4 a H (ˆy)e e e mn G (t ) − F (t ) dt . n 2 m,n+1 2 m,n−1 m=−∞ n=0 0 (7.5) Using the solution (7.5) in the right hand sides of (4.21)–(4.23), and then making use of (5.6) and (5.7), we obtain   u(x, y, t) v(x, y, t) = φ(x, y, t)   " # ∞ ∞ Z µ ¶ 1 X X umn(y) t 2 ³ ´ 1 1 0 0 n + 1 ϕ n 2 ϕ 4   imx/a −α(t−t ) −iνmn(t−t ) 0 0 0 ² a vmn(y) e e e Gm,n+1(t ) − Fm,n−1(t ) dt 0 2 2 m=−∞ n=0 φmn(y)  " µ ¶ ³ ´ # 1/2 − 1 ² y2  1 (F − G ) + e−αtK (x − ct¯ ) + F (x, y) e 2 a2   c¯ ϕ ϕ K  1   +  0  , 2  " µ ¶ ³ ´ #   1 ²1/2 2   −αt − 2 2 y  (Fϕ + Gϕ) + e K (x − ct¯ ) + FK (x, y) e a

(7.6)

SOARS°R 2008, Alex O. Gonzalez, 15 where "µ ¶ # i²1/4 n + 1 1/2 ³n´1/2 u (y) = (²1/2νˆ + m)H (ˆy) + (²1/2νˆ − m)H (ˆy) (7.7) mn a2 2 mn n+1 2 mn n−1

1 v (y) = (²νˆ2 − m2)H (ˆy) (7.8) mn a2 mn n "µ ¶ # ic²¯ 1/4 n + 1 1/2 ³n´1/2 φ (y) = (²1/2νˆ + m)H (ˆy) − (²1/2νˆ − m)H (ˆy) (7.9) mn a2 2 mn n+1 2 mn n−1 are the eigenfunctions for the Rossby modes. In summary, we have introduced a new equatorial β-plane filtered model that retains Rossby and Kelvin modes, and that acts as an effective filter of inertia-gravity modes. The new filtered model leads to the Rossby wave dispersion relation (6.3), which is more accurate than the one obtained from the longwave approximation. In fact, the Rossby wave dispersion relation obtained from the longwave approximation is similar to (6.3), but does not contain the m2 factor in the denominator of (6.3). Thus, the longwave approximation leads to a catastrophe for high wavenumber equatorial Rossby waves.

8. Discussion and Conclusion

Being able to understand the role of equatorial Rossby and Kelvin waves in their fullest, most accurate form is vital to understanding the 30-60-day tropical intraseasonal oscillation (MJO). The MJO is the dominant component of intraseasonal variability in the tropical atmosphere, and its effects are communicated throughout the global atmosphere. Because convection of all scales drives the atmospheric response, we would also like to learn more about convective coupling in these equatorial waves. Many previous models (ie. Chao (1987), Wu et. al. (2001), Biello and Majda (2004), Schubert and Masarik (2006)) utilize a Gill-type (longwave) model, but we still do not fully understand all aspects of the MJO. Hopefully we can learn more about the role of atmospheric waves, forced and unforced, in the equatorial region with future utilization of the filtered model produced here. Also, understanding the role of potential vorticity, divergence, and the invertibility principle in the horizontal structure of these waves is quite important, and future studies would benefit from more discussion of these topics. The Ripa (1994) method of solving the shallow-water equations emphasizes the ever-growing importance of cross discipline research in the fields of applied mathematics and physical oceanography. Ripa’s study was focused on improving knowledge on ray theory, not the MJO, yet it yields a convenient way of filtering equatorial waves. The concept of only needing one potential function is suitable for deriving the analytical solution, but can it be generalized to the sphere? Also, what atmospheric variable is described when taking ∂/∂t of the master equations ((2.18),(3.7),(4.24))? We still have much to learn about the analytical solution of this type of filtered model. But with more studies done in this area, hopefully we will learn more about variations of climate that start at the equator and about the underlying dynamics in the tropical atmosphere.

9. References

Blandford, R. 1966: Mixed gravity-Rossby waves in the oceans. Deep-Sea Res., 13, 941–961. Chao, W. C. 1987: On the origin of the tropical intraseasonal oscillation. J. Atmos. Sci., 44, 2324-2340. Frierson, D. M. W. 2008: Midlatitude static stability in simple and comprehensive general circulation models. J. Atmos. Sci., 65, 1049-1062. Gill, A. E. 1980: Some simple solutions for heat induced tropical circulation. Quart. J. Roy. Meteor. Soc., 106, 447–462.

SOARS°R 2008, Alex O. Gonzalez, 16 Gonzalez, A. O., Zagar, N., Schubert, W. H., and Masarik, M. T. 2007: Large-scale tropical circulations induced by heat sources using two simple models. SOARS Program 2007. Madden, R. A., and Julian P. R. 1994: Observations of the 40–50-day tropical oscillation—A review. Mon. Wea. Rev., 122, 814–837. Masunaga, H. 2007: Seasonality and regionality of the Madden-Julian oscillation, Kelvin wave, and equatorial Rossby wave. J. Atmos. Sci., 12, 4400-4416. Matsuno, T. 1966: Quasi-geostrophic motions in the equatorial area. J. Meteor. Soc. Japan, 44, 25–43. Moura, A. D. 1976: The Eigensolutions of the Linearized Balance Equations over a Sphere. J. Atmos. Sci., 33, 877-907. Ripa, P. 1994: Horizontal wave propagation in the equatorial waveguide. J. Fluid Mech., 271, 267–284. Rui, H. and Wang, B. 1990: Development characteristics and dynamic structure of tropical intraseasonal convection anomalies, J. Atmos. Sci., 47, 357-379. Schubert, W. H., M. T. Masarik, L. G. Silvers, and A. O. Gonzalez. 2008: A Filtered Model of Tropical Wave Motions. J. Adv. Model. Earth Syst., 1, 1-12 In Progress. Stevens, P. L., Kuo H.-C., Schubert, W. H., and Ciesielski, P. 1990: Quasi-balanced dynamics in the tropics. J. Atmos. Sci., 47, 2262–2273. Wu, Z., Sarachik E. S., and Battisti D. S. 2001: Thermally driven tropical circulations under Rayleigh friction and Newtonian cooling: Analytic solutions. J. Atmos. Sci., 58, 724–741. Zagar,ˆ N. 2004: Dynamical aspects of atmospheric data assimilation in the tropics. Diss. Stockholm Univ. Stockholm: Universitetsservice, 2004.

10. Glossary circulation - a precise measure of the average large-scale flow of fluid along a given closed curve. There are two types of atmospheric circulations, cyclonic (referred to as a low-pressure system) and anticyclonic (referred to as a high-pressure system). In the Northern Hemisphere cyclonic circulation is counterclockwise rotation and anticyclonic circulation is clockwise rotation. At the equator the atmospheric circulation changes direction so that in the Southern Hemisphere cyclonic circulation is clockwise rotation and anticyclonic circulation is counterclockwise rotation Coriolis parameter - frequency of the Coriolis force. As one gets closer to the equator in the earth’s atmosphere, the magnitude of the Coriolis parameter decreases, with opposite signs on either side of the equator (its value is zero at the equator). This strength in magnitude of this parameter is one main factor that allows for equatorial waves to be distinctly different in physical structure when compared to other latitudinal waves. Coriolis force - an apparent deflection of moving objects when they are viewed from a rotating frame of reference. Refer to Fig. 7. diabatic heating - temperature changes in the atmosphere that do not necessarily depend on whether a parcel (small volume) of air is rising or sinking. The prime contributor to diabatic heating is the sun. dispersion relation - a functional relation between wavenumber and wave frequency. For a wave propagating in the x direction, Acos(kx − νt), (10.1) where k is wavenumber in units of m−1 and ν is wave frequency in units of s−1, k(ν) is a dispersion relation. divergence - the expansion or spreading out of a vector field. See potential vorticity as well. El NiñoSouthern Oscillation (ENSO) - an interannual oscillation in weather that begins near the equator that results in a cyclic warming and cooling of the surface ocean in the eastern Atlantic and central/eastern Pacific. It is intialized by a downwelling ocean/atmospheric Kelvin wave in the the western Pacific and Atlantic, leading to the upwelling of cold deep water off the eastern coast. equatorial waves - can occur in either the atmosphere(or ocean). Atmospheric waves, in general, are the mechanism by which a localized forcing sends information to the rest of the atmosphere (Gonzalez et. al.

SOARS°R 2008, Alex O. Gonzalez, 17 2007). Without atmospheric waves, clouds and atmospheric systems would have no mechanism, other than advection by the mean flow, by which to propagate. Waves in the atmosphere are anisotropic, i.e., their response is not the same in all directions, producing different types of wave structures. There are four types of equatorially trapped waves in the tropics: Kelvin waves, inertia-gravity waves, Rossby waves, and mixed Rossby-gravity waves. In Figure 8 are the horizontal structures of the waves at some given vertical mode. equatorial β-plane - assumes that the Coriolis parameter varies linearly with latitude. Global Climate Models - are a class of numerical computer models used mainly for weather forecasting, understanding climate, and projecting climate change. The atmospheric response observed by these models derives from governing dynamical, chemical, and sometimes biological equations, including the primitive equations. There are approximations made to the equations and the contributing variables, therefore their result is never error-free. heat source and dissipation - can simulate the addition of heat added to a model atmosphere, as would happen in the real atmosphere due to latent heat release in convection. Rayleigh friction simulates the thermodynamic heat loss, i.e. radiation from cloud tops. Newtonian cooling simulates the mechanical energy loss. The reason to include both heating and cooling is to have a mechanism to add energy to the system, and a counteracting mechanism if the model is to reach a balanced state. Lamb’s parameter - a dimensionless parameter describing the effect on fluid flow changing in fluid depth. See also nondimensionalized equation. latent heat release - when water, in any of the three phases (vapor, liquid, or solid), moves from one state to another, and the air surrounding the water has energy added to it. Madden-Jullian Oscillation (MJO) - a 30-60-day oscillation explaining weather variations in the tropics. The MJO involves variations in wind, sea surface temperature (SST), cloudiness, and rainfall. The MJO is most obvious in the variation of outgoing longwave radiation (OLR) because most tropical rainfall is produced by deep convective clouds, and convective clouds absorb the majority of outgoing radiation. A vertical structure of this phenomenon is shown in Fig. 9 nondimensionalized equation - an equation containing terms with values that are independent of the units of measurement; usually with a physical interpretation. These terms arise naturally in the scale analysis of equations. outgoing longwave radiation (OLR) - energy leaving the earth as infrared radiation at low energy. Earth’s radiation balance is very closely achieved since the OLR very nearly equals the radiation received at high energy from the sun. OLR is dependent on the temperature of the radiating body. potential vorticity - rotational flow that, following a parcel of air or water, can only be changed by diabatic or frictional processes. Vorticity is the rotation in a flow (not as large-scale as the term circulation). By Stokes’s theorem, the total vorticity of the fluid enclosed by a curve is equal to the circulation about a plane curve. A spinning ice skater with her arms spread out laterally can accelerate her rate of spin by contracting her arms. Similarly, when a vortex of air is broad, it is in turn, slow. When the air converges, to maintain potential vorticity, the air speed increases, resulting in a stretched vortex. Divergence causes the vortex to spread, slowing down the rate of spin. primitive equations - Eulerian equations of motion of a fluid in which the primary dependent variables are the fluid’s velocity components. They govern a wide variety of fluid motions in the atmosphere and ocean and are derived from the equations of motion/Navier Stokes equations. scale analysis - a method usually using the nondimensional equations to determine which terms are dominant for a particular phenomenon or situation so that the smaller terms can be neglected, resulting in a simplified set of equations. shallow-water equations - come from the Navier-stokes equations, in the case where the horizontal length scale is much greater than the vertical length scale. Under this condition, conservation of mass implies that the vertical velocity of the fluid is small. It can be shown from the momentum equation that vertical pressure gradients are nearly hydrostatic, and that horizontal pressure gradients are due to the displacement of the pressure surface, implying that the velocity field is nearly constant throughout the depth of the fluid. Taking the vertical velocity and variations throughout the depth of the fluid to be exactly zero in the Navier-Stokes equations, the shallow water equations are derived. Shallow water equation models have only one vertical level, so they cannot directly encompass any factor that varies with height. However, in cases where the mean state is sufficiently simple, the vertical variations can be separated from the horizontal and several sets

SOARS°R 2008, Alex O. Gonzalez, 18 [b]

Figure 7. Coordinate system at latitude ϕ with x-axis east, y-axis north and z-axis upward (that is, radially outward from center of sphere). From en.wikipedia.org. of shallow water equations can describe the state. A physical representation of these equations is shown in Figure 10. steady-state - a fluid motion in which the velocities, and many times other physical variables such as pressure, density, etc., at every point of the field are independent of time. streamfunction - a parameter of two-dimensional, nondivergent flow, with a value that is constant along each streamline. A streamline has its tangent at any point in a fluid parallel to the instantaneous velocity of the fluid at that point. velocity potential - a scalar function with its gradient equal to the velocity vector of an irrotational flow. If a velocity potential exists, it is simpler to describe the motion by means of the potential rather than the vector velocity, since the former is a single scalar function whereas the latter is a set of three scalar functions.

SOARS°R 2008, Alex O. Gonzalez, 19 Figure 8. Horizontal structures of the equatorial wave solutions to the shallow water equations on an equatorial β-plane, for nondimensional zonal wavenumber 1, with westward inertia-gravity waves (top left), eastward inertia-gravity waves (top right), equatorial Rossby waves (middle left), Kelvin waves (middle right), and mixed Rossby-gravity waves (bottom). All scales and ¡ ¢1/2 ¡ ¢1/2 ﬁelds have been nondimensionalized by taking the units of time and length as [T ] = 1/β(¯c)1/2 , and [L] = (¯c)1/2/β , 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, 20 [t]

Figure 9. Schematic of the vertical three-dimensional structure of an established MJO. Figure adapted from Rui and Wang (1990). Blue (red) ovals indicate anticyclonic (cyclonic) circulations. Black arrows indicate wind direction and rising (sinking) motion.

[b]

Figure 10. The shallow-water representation. u(x,y,t) is the zonal acceleration of the fluid flow, v(x,y,t) is the meridional acceleration of the fluid flow, phi(x,y,t) (φ) is the geopotential/depth of the fluid flow, and Omega (Ω) is the earth’s rotation rate.

SOARS°R 2008, Alex O. Gonzalez, 21