1. Introduction
The propagation of tropical disturbances is strongly affected by the interactions among atmospheric circulation, moisture transport, and convection. These interactions are at the core of many atmospheric issues such as hurricanes, the Madden–Julian oscillation, and planetary circulation (Emanuel 1986; Madden and Julian 1971; Pauluis 2004). The present study focuses on the interaction between deep convection in the intertropical convergence zone (ITCZ) and equatorial waves and, in particular, on how the width of the ITCZ impacts the propagation speed and structure of equatorial Kelvin waves.
In the tropics, energy absorbed at the surface over the tropical oceans is transferred to the lower troposphere through evaporation and then transported to higher altitudes through convection and latent heat release. The ascending branch of the Hadley circulation corresponds to a region of low-level convergence, the ITCZ. The ITCZ is characterized by a region of increased convection, cloudiness, and precipitation; therefore, fluxes of heat, moisture, and momentum vary dramatically inside and outside the ITCZ and impact the tropical circulation.
A number of observational works have been undertaken to clarify the interaction between clouds and atmospheric disturbances. By means of wavenumber–frequency spectral analysis of satellite data, studies such as Wheeler and Kiladis (1999) obtained considerable success in quantifying the relationship between the dynamical structure of equatorial waves and moist convection. These convective propagating disturbances are called convectively coupled equatorial waves (CCEWs) because they present characteristics similar to the equatorially trapped shallow water modes found in Matsuno (1966); however, there are some important differences. Wheeler and Kiladis (1999) found that these modes propagate more slowly than they would in the absence of precipitation and that low-wavenumber Kelvin modes propagate faster than the corresponding high-wavenumber mode. In Straub and Kiladis (2002), evidence is shown of convective activity propagating along the mean axis of the eastern Pacific ITCZ with the spatial structure, propagation speed, and dispersion characteristics of the equatorially trapped Kelvin waves. It is also shown that the coupled Kelvin wave, unlike the traditional Kelvin wave, has a nonnegligible meridional component in regions of deep convection. Moreover, in Wheeler et al. (2000), convectively coupled Kelvin waves are shown to be weakly dispersive.
The theory for equatorial trapping of large-scale wavelike disturbances was discovered by Matsuno (1966), who derived a complete set of linear wave mode solutions of the shallow water equations on the equatorial β plane. Since then, shallow water analytical models have been extensively used to gain basic insight into the response of the tropical atmosphere to heating. For instance, Gill (1980) utilized a linear shallow water model forced by a stationary heat source concentrated in specific areas to study how equatorial propagating modes are affected by diabatic heating. Although these models provide relevant simulations of tropical anomalies, they have the limitation that the active role of tropical convection is not resolved.
The quasi-equilibrium (QE) theory is a convective closure that was originally proposed by Arakawa and Schubert (1974) and assumed that convective motions act to eliminate convective instability over an adjustment time scale of a few hours. Based on the time scale separation between the convective adjustment time and synoptic or planetary circulation, they find that the rate of change of convective instability, quantified in terms of the cloud work function, remains small in convectively active regions. Under similar assumptions, Emanuel et al. (1994) argue that the effect of convection on the large-scale circulation is to reduce the effective static stability of the atmosphere (or the equivalent depth) and therefore gravity waves must slow down. Moreover, based on studies such as Betts (1986) that suggest a convective adjustment time between 2 and 24 h, Emanuel et al. (1994) proposed the strict quasi-equilibrium (SQE) theory, in which the atmosphere is instantaneously relaxed toward a moist adiabatic profile in convective regions. This assumption greatly simplifies theoretical investigations of the interaction of convection and the planetary scale because it eliminates the need for a description of the behavior of individual clouds.
However, the original derivation of the SQE theory has an important limitation: the requirement of an atmosphere in which precipitation is active everywhere. In Frierson et al. (2004, hereafter FMP04), the QE framework is applied to the study of the interface between precipitating and nonprecipitating regions (the precipitation front), assuming SQE and in one space dimension. Utilizing a similar model, Pauluis et al. (2008) found stationary precipitation fronts in an idealized Walker circulation background flow, in both one and two dimensions. In Stechmann and Majda (2006), the precipitation front theory in one space dimension is extended to finite convective adjustment time; they show that for a small convective adjustment time their results are in agreement with the SQE theory.
The present work is an extension of FMP04 and Pauluis et al. (2008) for a particular version of the two-dimensional problem in which, assuming SQE, the behavior of large-scale waves propagating along a narrow precipitation band (the ITCZ) is investigated. The main advantage that the SQE framework offers in this case is the simplicity of analysis of the interaction between a precipitating region and propagating disturbances. However, the results have to be carefully interpreted because the applicability of the QE (and SQE) assumption is certainly incorrect at the scale of individual clouds. Nonetheless, the present approach can give insight into dealing with the roll of moisture in the tropics and, because general circulation models are frequently based on QE concepts, it can also be used to explain some of the behavior of these models.
The paper is organized as follows: In section 2, we present the model equations and review the main implications of the SQE assumption. This framework is then applied in section 3 to study the propagation of CCEWs along a precipitating band parallel to the equator. First, we obtain stationary solutions (i.e., the stationary precipitating band, associated with an idealized Hadley circulation). In this model, both the location and width of the ITCZ are controlled by imposing the surface temperature. We then derive the disturbance equations to investigate the behavior of perturbations. In section 4, the analytical and numerical results are presented, including the convectively coupled Kelvin and Rossby waves propagating on precipitating bands of distinct width and location. The effects of the ITCZ on both the meridional flow and the propagation speed of the CCEW, as well as the interaction between the coupled waves and the displacement of the precipitation front, are discussed in section 5. The main results are summarized in section 6, and the analytical and numerical solutions are described in the appendixes.
2. Model description
The model used here is based on a Galerkin truncation of the equations of motion into a finite set of vertical modes, equivalent to the quasi-equilibrium tropical circulation model (QTCM) from Neelin and Zeng (2000). There are a number of versions of models similar to the QTCM and in this paper we utilize the model from FMP04, in which a detailed derivation and discussion of its mathematical aspects is presented.
Note that there is no baroclinic advection term in (1), which is consistent with the model presented in FMP04. We neglect not only the barotropic–baroclinic interaction but also, because of the Galerkin truncation, the baroclinic–baroclinic interaction as well.
The variables are nondimensionalized using the following units: the internal gravity wave speed c = 50 m s−1, the Coriolis parameter β = 2.28 × 10−11 m−1 s−1, the typical equatorial length scale LE =
The spatial domain is a channel around the equator. A dry region is defined as a region where precipitation is inactive (q < q̂) whereas a moist region is defined as a region where precipitation is active (q ≥ q̂). Variables are periodic in the zonal direction and tend to zero at the meridional boundary.
This model allows for the study of the evolution of the interface between dry and moist regions and its interactions with the large-scale circulation. The theory is simpler in one dimension (in a line above the equator) because there is no meridional wind component and the free boundary can be interpreted as a propagating shock along the equator (the precipitation front). The precipitation front theory is developed in FMP04 and Pauluis et al. (2008) where, assuming SQE, the interface displacement and the transfer of signals between the dry and moist regions are investigated. The theory is then extended to a finite convective adjustment time in Stechmann and Majda (2006), again in one space dimension. In two dimensions, the interface between precipitating and nonprecipitating regions is more complex because it is no longer a single location, as it is in the one-dimensional case.
3. Methodology
Figure 1 shows the relation between the parameter σ and the ITCZ width; in Fig. 2, the surface background flow for two distinct choices of σ is displayed. Because of the choice of qs, the precipitating region is located at (−y0, y0) and is surrounded by a region where precipitation is inactive, (−∞, −y0) and (y0, ∞), as can be seen in the bottom right of Fig. 2. The top panels show the Hadley circulation wind pattern at the surface and the bottom left panel shows that the potential temperature is higher nearby the equator.
For obvious reasons, the Hadley circulation in our highly idealized models differs in many ways from observation. For instance, the Galerkin formulation used here does not enforce angular momentum conservation, which is a key aspect of the Hadley circulation (Held and Hou 1980). However, for the purpose of this study, the only feature of the basic flow that impacts the CCEWs is the location and width of the precipitating region. Thus, the first baroclinic structure of the basic flow seems adequate.
The solution for a fixed y0 ∈ (0,∞) is derived in the following manner: First, the two linear independent solutions in each region (|y| < y0 and |y| ≥ y0) are obtained. Second, they must satisfy two constraints: solutions must vanish at infinity and must obey the matching condition at the interface that requires variables (u, υ, T) to be continuous. Utilizing these two constrains, the complete solution is obtained; that is, the dispersion relation ω = ω(k; y0) and the meridional amplitude of the waves are determined. For a detailed derivation and discussion of the analytical solutions in SQE, see the appendixes.
Figure 3 shows the analytical dispersion relation for antisymmetric solutions of (13), where for simplicity only two coupled modes are displayed. On the right side of the plot, the dispersion curve corresponds to convectively coupled Kelvin waves for two distinct ITCZ widths. When the ITCZ is wider, the curve is closer to the moist theoretical curve (ω = cmk), whereas for a narrower ITCZ, it lies closer to the dry theoretical curve (ω = cdk). Analogous results are found for Rossby M = 1 waves, displayed on the left side of the plot. These results are further discussed in the next section.
To validate these solutions, we compare them to numerical solutions that are obtained utilizing the nonoscillatory balanced scheme, introduced by Khouider and Majda (2005a,b). Their scheme solves the full equations in (1); thus, in the appendix we describe the method to obtain the CCEW along the ITCZ that corresponds to the analytical solution. Furthermore, to understand how the location of the ITCZ affects these modes, we utilize the numerical scheme to extend the theory of CCEW to the case in which the ITCZ is shifted to the Northern Hemisphere—that is, qs = σ exp[−σ2(y − a)2], where a is the location of the ITCZ. We then obtain propagating modes along the off-equatorial ITCZ that are similar to the traditional equatorial modes.
4. Results
In this section the dynamical structure of some CCEWs are illustrated for various ITCZ widths and locations. First, the ITCZ is centered at the equator and a Kelvin wave disturbance propagating along a narrow ITCZ (y0 ∼ 250 km) is described. This case is then compared to a Kelvin wave along a wider ITCZ (y0 ∼ 1000 km). Wavenumbers 1 and 6 are tested and numerical and analytical solutions are discussed. Next, the Kelvin wave disturbance propagating along an ITCZ off the equator is compared to the symmetric ITCZ case. The section ends with a comparison between a Rossby wave M = 1 disturbance along a narrow and wide symmetric ITCZ.
a. Kelvin waves along a symmetric ITCZ
Many interesting features can be seen in the dispersion diagram in Fig. 3. The right-hand side (i.e., for positive wavenumbers) shows that the convectively coupled Kelvin waves propagate eastward and its phase speed (cp = ω/k) lies in the region cm < cp < cd. The dispersion curve when y0 = 250 km (thin line) is closer to the dry curve than with y0 = 1000 km (thick line), as previously predicted. Remarkably, the convectively coupled Kelvin waves are only weakly dispersive; the dispersion curve is very close to a straight line in the frequency–wavenumber diagram.
Figure 4 shows a convectively coupled Kelvin wave for wavenumber 6 and y0 = 250 km. The left (analytical) and right (numerical) panels indicate that the solutions are in close agreement. The zonal wind and potential temperature are both (top) in phase and then (bottom) out of phase with respect to the disturbance of the precipitation rate. The precipitation rate maximum occurs in the same region as the maximum wind convergence and the precipitation bandwidth and location remain unchanged. The structure of the potential temperature and zonal wind disturbance is similar to the free dry and moist modes (i.e., with opposite signs and a single maximum amplitude at the equator); however, they are not exactly proportional to each other, as they are for the free mode. Moreover, the meridional circulation is noticeable in this case, whereas for free modes, it is absent; in the next section we argue that these features are necessary to distribute the latent heat associated with the ITCZ.
In Fig. 5 we plot the phase speed of the Kelvin mode with respect to the ITCZ width for wavenumbers 1 and 6. In agreement with Fig. 3, the disturbance propagates faster for longer waves. Note that observed Kelvin waves (e.g., Wheeler and Kiladis 1999) show a similar faster propagation speed at smaller wavenumbers and also develop a meridional circulation (Wheeler et al. 2000, their Fig. 5). In the absence of the ITCZ, the mode propagates at the gravity wave speed (50 m s−1). As the ITCZ becomes wider, the speed of the Kelvin coupled mode decreases; for an ITCZ width of about 1000 km, it propagates at the free moist speed (15 m s−1). To further investigate the sensitivity of the dispersion relation to the ITCZ width, Fig. 6 shows the relation between the ratio between the meridional flow and the zonal flow and the location of the interface, once more for wavenumbers 1 and 6. The ratio shown was computed using the analytical solution and we found a good agreement with the numerical result (not shown). For both wavelengths, the ratio is zero if there is no ITCZ (corresponding to the free dry Kelvin mode) and increases until the ITCZ width is about 400 km, when it starts to decrease with increasing ITCZ width. For narrower ITCZ widths, it is noticeable that the shorter wave (k = 6) develops a more intense meridional circulation. In the next section, we argue that the meridional wind increases with wavenumber.
b. Kelvin waves along an ITCZ off the equator
The analytical solutions used so far rely on the symmetry of the background state. Without this assumption, the disturbance response to the background state cannot be split into linear independent symmetric and antisymmetric components, as shown in the appendixes. We do not compute this analytical solution; instead, we show numerical results for the Kelvin wave disturbance propagating along a precipitating band off the equator (for details of this computation, see the appendixes). We test the same widths and wavenumbers from the last section, but now the ITCZ is centered at 10°N.
Figure 7 shows the potential temperature contour with the wind fields overlaid for a Kelvin wave k = 1 and ITCZ width y0 = 250 km. The overall structure is similar to the case in which the ITCZ is centered at the equator; that is, the zonal wind and potential temperature are in phase (top plot), both are out of phase with respect to the disturbance of the precipitation rate (bottom plot), and its peak coincides with the maximum wind convergence zone. In agreement with Straub and Kiladis (2002, e.g., their Fig. 16), the inflow toward the precipitating region in regions of wind convergence is noticeable and, because the amplitude of the waves is very small at the northern interface, we observe a considerable stronger inflow at the southern interface. In Fig. 8 the amplitude of the disturbance is displayed for the Kelvin wave propagating along the symmetric ITCZ (dashed line) and along the off-equatorial ITCZ (solid line). Although no longer symmetric, the general structure of zonal wind and potential temperature meridional amplitude is fairly close to the “free” dry Kelvin wave and to the equatorial coupled mode. Importantly, these results suggest that the maximum precipitation rate (Fig. 8, bottom right) is smaller when the ITCZ is away from the equator; in the next section we argue that this is a consequence of the fact that the off-equatorial coupled mode propagates faster than the symmetric coupled mode for the same ITCZ width, as shown in Fig. 5. For instance, if the ITCZ width is y0 = 250 km, the Kelvin mode when the ITCZ is centered at the equator propagates at about 28 m s−1; in contrast, when the ITCZ is centered at 10°N, it propagates at about 35 m s−1.
It is also worth noting that the meridional flow in Fig. 8 (right and left) has a sharp peak at the interface that indicates that the vertical velocity is discontinuous at the precipitation front, in agreement with the results from FMP04.
c. Rossby wave M = 1 along a symmetric ITCZ
Here we investigate the propagation of Rossby waves along a symmetric ITCZ. Analogous to the Kelvin wave experiment, the solution is known when y0 = 0 and y0 = ∞ because in both cases the equivalent depth is uniform in the whole domain and linear theory is valid. We examine the Rossby M = 1 mode, which propagates westward; it is dispersive and the zonal wind and potential temperature are symmetric across the equator. For small wavenumbers this mode propagates at about one third of the gravity wave speed; therefore, for long waves it determines the region in the dispersion diagram where we look for solutions: −cmk/3 < ω < −k/3.
Figure 9 is similar to Fig. 4, but for Rossby M = 1 and wavenumber k = 1. The analytical and numerical solutions are again in good agreement and the overall structure of this mode is similar to the free Rossby mode. Once more, potential temperature is out of phase with precipitation rate and convergence in the ITCZ is located at the peak of precipitation rate. However, wind also converges at higher latitudes, whereas for Kelvin waves it converges only at the ITCZ.
As can be seen on the left side of Fig. 3, the dispersion curve for the convectively coupled Rossby wave indeed lies between the dry and moist curves and exhibits a similar behavior to the free modes; that is, the dispersion relation is approximately linear for small wavenumbers and it decreases for large wavenumbers. However, the dispersion curves for both a narrow and a wide ITCZ are closer to the dry curve than for Kelvin waves, which indicates that Rossby waves are less affected by the ITCZ. Further evidence of this fact can be seen in Fig. 5, which shows a smoother and slower transition between the free dry and moist Rossby modes than the corresponding Kelvin mode (solid lines). That is, it is necessary to have a wider ITCZ to achieve the moist speed than for Kelvin waves. One explanation for this behavior is that because Rossby modes are strongly affected by the earth’s rotation, the maximum convergence–divergence is located off the equator, in contrast to Kelvin waves.
5. Meridional circulation and modulations of the precipitation on the ITCZ
The propagation of convectively coupled waves along the ITCZ provides a useful framework to study the interactions between moist convection and the atmospheric dynamics. Here, we focus on some of the key aspects of the propagation of a Kelvin wave along the ITCZ. First, in contrast to the classic equatorial Kelvin wave on an equatorial beta plane, the Kelvin wave exhibits a meridional circulation. We argue that this circulation acts to redistribute the latent heat released in the ITCZ to the dry regions. This circulation modulates both the amount of precipitation and the width of the ITCZ itself, playing an important role in setting the propagation speed of the Kelvin wave.
a. The meridional circulation
b. The displacement of the interface
The ITCZ expands during the ascent phase of the Kelvin wave. Conversely, the ITCZ becomes narrower during the subsidence phase. It ensues that the variations of the ITCZ width by the Kelvin waves are out of phase by a quarter of a period with the variations of precipitation.
c. The speed of propagation
In section 4, we obtained a relation between the width of the ITCZ (y0) and the speed of propagation of Kelvin and Rossby waves. Moreover, we found that this curve is in agreement with the numerical results. Because Kelvin waves have a simple structure, we focus our analysis on this mode. To better understand how the ITCZ affects this wave, it would be useful to relate its phase speed to the meridional circulation and the total precipitation rate in the ITCZ. To carry on this analysis, we average the equations in the meridional direction.
6. Conclusions
We presented an extension to the SQE theory for convectively coupled gravity waves to cases in which the precipitating region only occupies a fraction of the domain. In particular, the conceptual simplification inherent in the SQE assumption leads us to build a simplified theory for CCEWs propagating over a precipitating band.
The key implication of the SQE within our model is that moisture is instantaneously adjusted to the saturation values in regions where precipitation is active; consequently, the problem was reduced to matching solutions of two shallow water systems, with distinct equivalent depths. Fundamentally, the matching comes from the requirement of the tendency in potential temperature to be continuous at the interface, and the discontinuity in the equivalent depth translates to a discontinuity in the vertical velocity.
This study is focused on convectively coupled Kelvin and Rossby waves propagating along the ITCZ, and we found that the propagation speed of the coupled waves is between the dry and moist speed, depending on the width and location of the precipitating band. Kelvin waves have speeds comparable to the moist case (15 m s−1), except for narrow (less than 1000 km) precipitating regions; they propagate faster as the ITCZ moves away from the equator, they are weakly dispersive, and shorter waves propagate slower than long waves. The results also indicate that the ITCZ must be wider to have an impact on Rossby waves.
It has been shown that a meridional circulation, present even for Kelvin waves (Figs. 4, 6 and 8), plays an important role in redistributing energy between the dry and moist regions. At the surface and where the zonal wind converges, the meridional wind flows toward the ITCZ at the interface, ascends inside the precipitating region, and then flows toward the poles (Fig. 11). We proposed some interpretation of the mechanisms involved in this process by deriving a direct relationship among zonal wind convergence inside the ITCZ, total precipitation, and the speed of the coupled waves [(33) and (34)], which indicates that enhanced wind convergence is correlated with an increase in the precipitation rate and with the slowdown of waves.
In keeping with the secondary meridional circulation due to precipitation and in agreement with the theory of precipitation fronts presented in FMP04, we found that for temperature to match at the interface there must be a discontinuity in the vertical velocity that is proportional to the prescribed gross moisture stratification. Moreover, upward motions at the interface are related to the displacement of the ITCZ toward the poles; conversely, the interface moves toward the equator during the subsidence phase, as shown in Fig. 11.
Some key features of observed convectively coupled waves were well captured by the model: Kelvin waves propagate more slowly when convection is active; they are weakly dispersive and develop a meridional circulation. Because relatively few vertical modes seem to participate in many tropical phenomena involving the ensemble mean effects of deep convection, we believe that the extension of the method used here to other configurations (e.g., Rossby waves propagating over an off-equatorial ITCZ or a double ITCZ) can provide further insight into the coupling between convection and large-scale phenomena.
Acknowledgments
We thank Samuel Stechmann and Boualem Khouider for their help and advice regarding the numerical results. We also thank Paul Roundy, Dargan Frierson, Adam Sobel, and Zhiming Kuang for their careful and thoughtful suggestions to improve this paper.
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APPENDIX A
Analytical Solutions for a Symmetric ITCZ
Stationary Hadley circulation
We still have to show that in fact there will be a unique (β1, β2, y0) that satisfy Eqs. (A5), (A6), and (A7). Because the dependence in β1 and β2 is linear, we can find a single equation only depending on y0 that has the form F(y0, σ) = 0 (recall that σ is fixed and comes from the forcing term), and then the width of the precipitating band is well defined as long as we can define y0 = f (σ) such that F[σ, f (σ)] = 0. The particular choice of qs indicates that this is indeed the case. However, for simplicity, we obtain this solution numerically (it will be described later) and we verify that it satisfies the properties shown in this section.
Convectively coupled waves
APPENDIX B
Numerical Solutions
At this point the speed of propagation of the strongest signal of the disturbance needs to be estimated; next, to filter out any signal that propagates at a different speed, the data are averaged along the path: (x − c(σ)t = 0). This process needs to be repeated a few times until the disturbance corresponds to a single propagating mode with a speed given by c(σ). In addition, at each iteration the disturbance is amplified by eκt because the modes decay at the reciprocal of this rate due to linear damping.
Because the solutions are continuous as long as the increment of the ITCZ width is small compared to the size of the domain, the transition between the disturbance in one equilibrium state to the next one is smooth. Thus, we successively increase σ and repeat the method described above until the ITCZ width is close to zero. In the limit y0 → 0, the disturbance must converge to the free dry mode; that is, c(σ) → cd. As a result, for each σ, a solution of the form (10) is obtained.
The transition between the ITCZ centered at the equator to one centered at a higher latitude is done in small steps and σ is fixed; hence, the propagating mode obtained for the new ITCZ location is denoted by the original mode corresponding to the ITCZ centered at the equator. That is, if one begins with a convectively coupled Kelvin wave disturbance along an ITCZ centered at the equator of width y0(σ), the mode obtained after shifting the ITCZ to a higher latitude is also denoted a convectively coupled Kelvin wave. In fact, it was numerically verified that these modes converge to the corresponding free dry mode when y0 → 0.
Relationship between the ITCZ width and the variance (σ) of the forcing due to evaporation (qs).
Citation: Journal of the Atmospheric Sciences 66, 8; 10.1175/2009JAS3020.1
Low-level stationary flow for σ = 14 (solid) and σ = 6 (dashed–dotted), normalized by their maximum value of the zonal wind; U is the zonal wind, V is the meridional wind, T is the potential temperature, and Pr is the precipitation rate.
Citation: Journal of the Atmospheric Sciences 66, 8; 10.1175/2009JAS3020.1
Analytical dispersion relation for convectively coupled Kelvin and Rossby M = 1 waves, for y0 = 250 km (thin lines) and y0 = 1000 km (thick lines). The dashed lines correspond to the free dry and moist modes. The nondimensional zonal wavenumber is scaled by 1/LE and nondimensional frequency is scaled by 1/TE.
Citation: Journal of the Atmospheric Sciences 66, 8; 10.1175/2009JAS3020.1
Comparison between (left) analytical and (right) numerical solutions for a Kelvin wave (k = 6) with y0 = 250 km (ITCZ centered at the equator). Overlaid are (top) a snapshot of contours of potential temperature and lower-level velocity profiles and (bottom) contours of the precipitation rate and lower-level velocity profiles. Dark contours correspond to negative values; the thick solid line corresponds to the ITCZ width. All variables are nondimensional.
Citation: Journal of the Atmospheric Sciences 66, 8; 10.1175/2009JAS3020.1
Relationship between phase speed and ITCZ width for some of the coupled modes. The solid lines correspond to Kelvin and Rossby M = 1 modes along an ITCZ centered at the equator and wavenumber k = 1. The dotted–dashed line corresponds to a Kelvin mode along an ITCZ at the equator and wavenumber k = 6; the diamonds correspond to a Kelvin mode, k = 1, along an ITCZ centered at 10°N.
Citation: Journal of the Atmospheric Sciences 66, 8; 10.1175/2009JAS3020.1
Ratio between meridional and zonal wind at the interface [|υ(y0)/u(y0)|] for Kelvin wave k = 1 (crosses) and k = 6 (diamonds).
Citation: Journal of the Atmospheric Sciences 66, 8; 10.1175/2009JAS3020.1
Convectively coupled Kelvin wave (k = 1) along an ITCZ off the equator. Overlaid are (top) contours of potential temperature and velocity profiles and (bottom) contours of precipitation rate and velocity profiles. The ITCZ is centered at 10°N and y0 = 250 km. Darker contours correspond to negative values. All variables are nondimensional.
Citation: Journal of the Atmospheric Sciences 66, 8; 10.1175/2009JAS3020.1
Meridional structure for the convectively coupled Kelvin wave (k = 1) along the ITCZ centered at the equator (dashed line) and at 10°N (solid line), both with y0 = 250 km. All variables are normalized by the maximum of the meridional velocity.
Citation: Journal of the Atmospheric Sciences 66, 8; 10.1175/2009JAS3020.1
As in Fig. 7, but for convectively coupled Rossby M = 1 wave (k = 1) along the ITCZ centered at the equator. The thick solid lines correspond to the ITCZ width.
Citation: Journal of the Atmospheric Sciences 66, 8; 10.1175/2009JAS3020.1
Relationship between phase speed of the coupled Kelvin wave (k = 1) and the ITCZ width; comparison between the analytical phase speed (solid line) and the two approximations. The crosses correspond to the estimate in (26) and the diamonds to the estimate in (33).
Citation: Journal of the Atmospheric Sciences 66, 8; 10.1175/2009JAS3020.1
Sketch of a cross section of the secondary circulation due to the ITCZ. The thick dotted–dashed vertical lines mark the two interfaces separating the dry regions and the moist (shaded) region; the thin dashed line represents the displacement of the interface. The arrows indicate the circulation pattern in a region where the zonal wind converges toward the equator.
Citation: Journal of the Atmospheric Sciences 66, 8; 10.1175/2009JAS3020.1
