Synoptic-scale waves that propagate east or west and couple to moist convection frequent the deep Tropics (see review by Kiladis et al., 2009). Even though these convectively coupled equatorial waves (CCEWs) have been studied for decades, their dynamical mechanism remains a matter of debate, and they are poorly represented in many climate models (Lin et al., 2006; Straub et al., 2010). One remarkable feature of CCEWs, noted first by Takayabu (1994) and later by Wheeler and Kiladis (1999), is that they exhibit the dispersive properties of the eigenmodes of a shallow-water system with a linearly varying Coriolis force (e.g. Matsuno, 1966) and fluid depths in the vicinity of 12–60 m. Another characteristic that unifies multiple kinds of CCEWs is that they generate a scale-invariant cycle in convective morphology, producing shallow convection followed by deep convection followed by stratiform precipitation, with longer time-scales for larger waves (e.g. Takayabu et al., 1996; Straub and Kiladis, 2002; Kiladis et al., 2005; Mapes et al., 2006).
The challenges of explaining the dynamics of CCEWs and properly representing them in models have motivated many attempts to simulate them, using simple and complicated models alike. In recent years, a popular approach has been to use a simplified dynamical system with just two vertical normal modes (e.g. Mapes, 2000; Khouider and Madja, 2006; Kuang, 2008), which is based on the observation that much of the temperature and wind structure of tropical convective disturbances projects onto two vertical modes (e.g. Mapes and Houze, 1995; Haertel and Kiladis, 2004). While two-mode models have shed light on possible instability mechanisms for CCEWs, different models suggest different mechanisms, and results are often sensitive to somewhat arbitrarily defined model parameters. For example, the two vertical models of Mapes (2000) suggests stratiform instability as a means of generating waves, while the model of Khouider and Majda (2006) emphasizes the importance of cumulus congestus on the leading edge of waves, and, according to Kuang (2008), Quasi Equilibrium theory (e.g. Arakawa and Schubert, 1974; Emanuel et al., 1994) can be applied to the lower troposphere within the waves. Another approach has been to study waves that spontaneously develop in cloud-system-resolving simulations (e.g. Tulich et al., 2007; Tulich and Mapes, 2008). Such studies have provided strong evidence that convection couples with high-wave-number vertical modes, which could explain the prevailing equivalent depth of CCEWs. However, these studies cannot rule out other mechanisms such as the idea that the equivalent depth of CCEWs results from a reduced gross moist stability for the first baroclinic mode (e.g. Neelin and Held, 1987; Emanuel et al., 1994). To complicate matters, recent observational studies suggest that extratropical forcing is important for generating and possibly maintaining particular CCEWs (e.g. Straub and Kiladis, 2003a; Liebmann et al., 2009). It suffices to say that there is no scientific consensus on explanations for the genesis, equivalent depth, scale-invariant cycle in convective morphology, temperature and wind structure, and propagation mechanism of CCEWs.
In this study we show that additional progress can be made both in simulating CCEWs and understanding their dynamics by employing a Lagrangian approach to fluid modelling. Over the past decade, the lead author and collaborators have developed a fully Lagrangian ocean model (Haertel and Randall, 2002, hereafter HR02; Haertel et al., 2004, hereafter H04; Haertel et al., 2009, hereafter H09; Van Roekel et al., 2009). We show here that it is easy to adapt this model to the atmosphere, and that when a simple parametrization for moist convection is used in an aquaplanet version of the resulting Lagrangian atmospheric model, robust convectively coupled Kelvin waves spontaneously develop. This study is a first attempt to decipher what these results say about the potential usefulness of Lagrangian models for simulating moist convective disturbances, and the mechanism of convectively coupled Kelvin waves.
This paper is organized as follows. Section 2 describes the Lagrangian atmospheric model. Section 3 introduces the concept of Lagrangian Overturning (LO) as a means of parametrizing moist convection. Results of single-column experiments that combine LO with simple physics parametrizations are presented in section 4. Section 5 discusses how a three-dimensional model employing LO spontaneously generates convectively coupled Kelvin waves. Section 6 is a summary and discussion.
2. The Lagrangian atmospheric model
In this section we show that with just a few modifications, a recently developed Lagrangian method for simulating ocean circulations (HR02; H04; H09; Van Roekel et al., 2009) can be adapted to the atmosphere. We describe the Lagrangian modelling approach, detail the equations of motion for the atmosphere, and present a test simulation conducted with the new Lagrangian atmospheric model (LAM).
2.1. Lagrangian framework
Our Lagrangian models represent fluids as piles of conforming parcels. A single function defines the vertical thickness distribution for every parcel, which is shaped like a bell and is easy to discern when a parcel is on a level surface (Figure 1(a)). Parcel surfaces conform to variable bottom topography and to the surfaces of parcels above and beneath them, yielding a variety of parcel shapes in typical applications (Figure 1(b)). Deformations of parcels occur because although vertically aligned cores of a given parcel have the same horizontal velocity, they typically have different vertical velocities (see also Figs. 2–3 of HR02 for additional illustrations of parcel deformations). Horizontal accelerations of parcels are predicted using Newton's 2nd Law of Motion, and effects of turbulent mixing are included by allowing parcels to exchange mass and momentum with their nearest neighbours. There are several advantages of this Lagrangian framework for fluid modelling: (1) advection does not alter tracer properties; (2) trajectories for every fluid parcel are provided with no extra computational cost; (3) effects of eddies and convection are represented in unique ways; and (4) including variable bottom topography adds no complexity. Although the atmospheric and oceanic model equations are similar, there are some differences owing to the fact that the water parcels are weakly compressible, whereas the density of air parcels varies greatly.
2.2. Atmospheric equations of motion
Horizontal motions of air parcels are calculated using classical physics:
where x denotes horizontal position, v is horizontal velocity, t is time, f is the Coriolis parameter, k is the unit vector in the vertical, Ap is the acceleration resulting from pressure, and Am is the acceleration resulting from turbulent mixing of momentum.
Equations (1)–(2) are ordinary differential equations that are easy to approximate using standard numerical methods once Ap and Am are determined. To calculate Ap for every parcel in a computationally efficient manner, we assume that each parcel has the following characteristics: (1) a uniform potential temperature; (2) hydrostatic pressure; and (3) a time-invariant pressure thickness distribution δ p. We construct δ p using a bell shape b defined with a polynomial:
where δ pmax is the maximum pressure thickness, rx and ry are the parcel radii in the and directions, the prime (′) notation denotes a coordinate system centred on the parcel, and for d < 1 and b(d) = 0 for d ≥ 1 (Figure 1(a)). Now consider the Montgomery potential (or dry static energy) M = CpT+ gz where Cp is the specific heat of dry air, T is temperature, g is gravity, and z is height. In a hydrostatic atmosphere, the horizontal acceleration due to pressure at a particular location equals the horizontal gradient of the Montgomery potential on a constant potential temperature surface (e.g. Holton, 2004, pp 55, 109). We define the pressure acceleration of a parcel as the mass-weighted average of the gradient in the Montgomery potential across the parcel:
where the second equality provides the form of the equation that we discretize, and the parcel weight W is calculated by integrating the pressure thickness over its horizontal projection H:
The integrals in (4)–(5) are approximated using a Riemann sum, and the Montgomery potential is calculated following the method of Haertel et al. (2001). At the surface, M is set to gz (i.e. T is defined to be zero), and the change in the Montgomery potential (δ M) rising vertically across a parcel interface is given by:
where the pressure at the interface equals the sum of pressure thickness functions above it, and temperature is calculated from potential temperature using Poisson's equation. Note that there are no approximations resulting from this vertical discretization, which is a consequence of the fact that the Montgomery potential is independent of height within an isentropic layer.
We include a vertical viscosity (0.01 m2 s−1) to parametrize the turbulent mixing of momentum, and its implementation is identical to that in our Lagrangian ocean model (H04). Note that, for simplicity, we introduce the equations for the LAM in Cartesian coordinates, but they are adapted to spherical coordinates following the method described by H04 for applications on the aquaplanet (see below). Finally, we mention that the number of computations required to solve (1)–(6) scales linearly with the number of parcels. This is also the case for our Lagrangian ocean model, which has been found to be computationally competitive with Eulerian ocean models when run at a similar resolution (H04).
2.3. Test simulation
Tests conducted during the development of the Lagrangian ocean model have shown that internal and external gravity waves and Rossby waves propagate through piles of conforming parcels much like they do through fluids (HR02), and that our Lagrangian method can be used to simulate lake upwelling fronts (H04), western boundary currents (H09), and oceanic thermocline structure and overturning (H09; Van Roekel et al., 2009). In general, we have succeeded in reproducing flow features predicted by analytic solutions or generated by other numerical models in all applications we have attempted, including unpublished simulations of the equatorial undercurrent and tropical instability waves. Moreover, since the focus of this paper is on the effects of the Lagrangian convective parametrization and its implications for CCEW dynamics, we do not present a thorough suite of test simulations for the LAM here. However, we do quantitatively compare the behaviour of nonlinear internal gravity waves simulated with the LAM to those generated by another atmospheric model (Haertel et al., 2001). There are two reasons for this test: (1) it would likely reveal errors in the formulation and/or coding of the atmospheric equations of motion; and (2) it tests the model's representation of some of the most relevant dynamics to convectively coupled Kelvin waves. Even though the dynamics CCEWs are predominantly linear (e.g. Haertel and Kiladis, 2004); we use the LAM to simulate nonlinear waves with large displacements of potential temperature surfaces, because an improper representation of compressibility is more likely to become manifest in such a wave. For careful tests of our Lagrangian method's ability to simulate linear internal gravity waves, including a discussion of numerical convergence, the reader is referred to HR02.
Consider an atmosphere comprising two isentropic layers with potential temperatures 297 and 300 K respectively. Suppose that some physical process (e.g. a thunderstorm) has created a bulge in the lower layer (Figure 2(a)). Figure 2(b)–(c) show the evolution of the layer interface height and lower layer velocities predicted by the LAM, compared with those calculated with another atmospheric model (Haertel et al., 2001). In both cases nonlinear internal gravity waves propagate away from the disturbed area, with zonal velocity perturbations either in phase or π radians out of phase with height perturbations (Figure 2(b)–(c)). Both models produce the same phase speed for the waves, the same amplitudes for the layer interface height perturbations, and the same velocity perturbations (with small differences attributable to numerical truncation errors). This experiment provides evidence that the LAM properly models nonlinear internal gravity waves, and that it can handle the relevant dry dynamics of CCEWs (e.g. Kiladis et al., 2009).
3. Lagrangian overturning
In this section we introduce a parametrization for moist convection designed for use in the LAM. It is a natural extension of the convective parametrization we use in our Lagrangian ocean model, and it is arguably the simplest possible convective parametrization to implement in the Lagrangian framework. However, as will be shown below, it supports convectively coupled disturbances with complex and realistic vertical structures.
Consider a column containing a number of parcel centres in a LAM simulation (Figure 3(a)). Let A denote a particular parcel, and B the parcel immediately above it. Let θ denote potential temperature. Lagrangian Overturning (LO) amounts to swapping the vertical positions of A and B in the event that doing so yields θ (A) > θ (B). It is assumed that excess water vapour is condensed out of parcel A as it rises to a lower pressure. In this process A can be thought of as representing a collection of cumulus updraughts and B the air that subsides around them (Figure 3(b)). Note that there is not an arbitrary limit on the distance parcels can ascend or descend through this convective process (through repeated vertical position swaps); rather, parcels' buoyancies relative to those of their environment determine the depth of convective updraughts and downdraughts as they do in nature. For simplicity, we only allow a given parcel to be involved in one vertical position swap per time step, but one could, for example, allow a parcel to ascend or descend the entire column in a single time step. Here we describe how LO is implemented in a Lagrangian atmospheric model; its implementation in a Lagrangian ocean model can be described in the same words with θ denoting density. In the oceanic case there are only very small changes in θ through the swapping process, if these are considered at all.
3.2. Potential advantages
There are a number of potential advantages of using LO as a convective parametrization. First, it simply and elegantly captures the essence of convection–LO generates vertical transports of air that occur in response to convective instability, and which remove that instability. Second, it can represent both dry and moist convection. Third, ascending and descending parcels (i.e. updraughts and downdraughts) can have different properties within a given column of the model. For example, LO-generated updraughts are typically more moist and warmer than the air that subsides around them. A fourth advantage is that the existence and/or depth of convective plumes respond to local perturbations in thermodynamic profiles. For example, a low-level inversion (i.e. a cap) can prevent convection altogether, and a mid-level inversion can cause shallow convection to develop instead of deep convection. A fifth advantage is that LO supports persisting temperature and moisture perturbations in air descending around updraughts, which are strongly damped in more traditional parametrizations, and which can have feedbacks on convection in large-scale disturbances. A sixth advantage of LO is that parcel trajectories can be traced through the convective process, and these are provided at no additional computational cost. Finally, implementing LO requires few (if any) tunable parameters, which means its effects can freely respond to changing environments (e.g. in climate change studies). Many of these potential advantages of LO are illustrated in the following sections.
Our current implementation of LO also has a few limitations as well, including an extremely simple treatment of cloud microphysics (see below), and the fact that the velocities of convective updraughts are determined by the modellers' choices of time step and vertical thickness for parcels. However, these are not necessarily limitations of the LO concept. For example, one could incorporate more sophisticated microphysics within the LO framework, and even constrain the frequency of vertical position swaps by the amplitude of buoyancy perturbations.
4. Single-column experiments
In this section we present simulations of radiative convective equilibria in a single-column version of the LAM. Simple parametrizations for radiation, surface fluxes and the evaporation of rain are included in the LAM in addition to LO. Simulated moisture and temperature profiles are compared with observed profiles from the Tropical Ocean Global Atmosphere Coupled Ocean–Atmosphere Response Experiment (TOGA COARE: Webster and Lukas, 1992; Ciesielski et al., 2003). The experiments reveal that LO is capable of producing realistic temperature and moisture profiles for the Tropics, and also some interesting oscillations that depend on feedbacks between descending temperature anomalies and variations in the depth of convection.
4.1. Single-column model configuration
Unlike traditional single-column models, our single-column version of the LAM is actually a three-dimensional model with meridional and zonal dimensions equal to a parcel width, and with periodic boundaries in both horizontal directions. This allows us to use the same dynamical core and physics packages for both the single-column and the aquaplanet versions of the LAM, and it ensures that the behaviour of the single-column model is like that of an actual column in the aquaplanet model (e.g. in which parcel centres need not be perfectly aligned in the vertical). Note that using this model configuration still yields typical advantages of single-column modelling such as a greatly reduced computational cost, and a simplified modelling environment for introducing and studying new model features.
4.2. Column physics
We use minimal parametrizations for radiation, surface fluxes and cloud microphysics. The net effects of short-wave and long-wave radiation are included by cooling the troposphere by ∼1 K day−1, but fixing the stratospheric temperature near observed values. Surface fluxes of heat and moisture are included by restoring the temperature of the lowest air parcel to the sea-surface temperature (SST), and by restoring the specific humidity of the lowest parcel to the saturation specific humidity for the SST. In both cases we use the decay equation with the same time-scale for moisture and temperature. Condensed water (rain) falls one parcel down per time step, and a fixed percentage evaporates during each time step (until the parcel containing the rain is saturated). The temperature change (ΔT) of a parcel in which water condenses or evaporates is given by CpΔT = LΔq where L is the latent heat of vaporization, Cp is the specific heat of dry air, and Δq is the mass of condensed (or evaporated) water. Ice processes are neglected.
We present four single-column simulations: a control run that includes only radiation and surface fluxes (in addition to LO), and three sensitivity tests that also include mixing and/or the evaporation of rain. In all cases we set the initial tropospheric temperature and SST to 302.5 K, which is the observed value of SST for the COARE Intensive Flux Array (IFA). The time step is 20 minutes for both model physics and dynamics, and the restoring time for surface fluxes is in the range of 12–15 hours.
4.3.1. Control run
For the first experiment there is absolutely no mixing in the free troposphere; a parcel's potential temperature and moisture are modified only by surface fluxes and/or radiation. Figure 4 shows average temperature and moisture profiles for days 100–200 (solid lines), compared with the mean conditions for the COARE IFA (dotted lines). The simulated temperature profile has the gross shape of a moist adiabat, and it is a few degrees warmer that the COARE IFA profile, with greater differences in the upper troposphere. The simulated moisture profile is much drier than the corresponding COARE profile, except in the boundary layer, where moisture differences are small.
Perhaps the most interesting and unanticipated feature of the control run is its non-steady behaviour. On a 40–50-day time-scale, warm anomalies descend through the troposphere, and when they reach mid-levels they enhance the frequency of relatively shallow convection, which has the effect of moistening the lower troposphere (Figure 5(a))1.
When the lower troposphere is relatively moist, parcels descending into the boundary layer start out more moist, and reach a greater specific humidity before they rise out of the boundary layer (Figure 5(a)). Consequently, they have a greater temperature when they reach the upper troposphere in convective updraughts (e.g. around days 50 and 100). These positive temperature anomalies then descend, repeating the cycle.
Sample parcel trajectories are shown in Figure 5(b). Parcels rapidly ascend in convective updraughts, and then slowly descend at a rate dictated by the radiative cooling and the stability profile. In most cases convective plumes are deep, but one example of a relatively shallow convective ascent occurs around day 35, when the middle troposphere is relatively warm.
The first sensitivity test explores the effects of including mixing in LO. In other words, when two parcels exchange vertical positions, they exchange a small percentage of their mass, momentum and tracers as well. In this case we set the mixing rate to μ = 5 × 10−6 Pa−1 (the ratio of mixed air is calculated by multiplying μ by Δp, the pressure distance travelled by each parcel when they exchange vertical positions). The introduction of mixing causes one important change: it reduces the tropospheric temperature, especially at upper levels, making it more like the observed profile (compare Figures 6(a) and 4(a)). The mixing also slightly increases the low- to mid-tropospheric humidity (compare Figures 6(b) and 4(b)); however, the changes are small when compared with the differences between the observed and simulated humidities. Even with the mixing the non-steady behaviour remains strong, and exhibits the same pattern, although the period of the cycle is reduced slightly (compare Figures 7(a) and 5(a)). The parcel trajectories are also similar (Figure 7(b)), with most parcels ascending through the entire troposphere in convective updraughts, and descending at a rate determined by radiative cooling.
4.3.3. Evaporation of rain
Including the evaporation of rain causes only small changes to the temperature profile (compare Figures 8(a) and 4(a)), but dramatic changes to the moisture profile (compare Figures 8(b) and 4(b)). We experimented with a few different values of the evaporation rate, and found that using a value of 7 per cent per time step yields a good agreement with observations (Figure 8(b)). Of course, our simple model lacks several processes that make important contributions to the temperature and moisture profiles over the western Pacific warm pool, such as the horizontal convergence of moisture into the atmospheric column and the horizontal divergence of dry static energy out of the column, and the fact that we can reproduce the observed profiles by tuning model parameters does not necessary mean that the same physical processes are determining the profiles in the model as they are in nature. Nevertheless, even when the potential effects of these other processes are included (e.g. by adding a moisture source or enhancing the column cooling), the model consistently produces a drier free troposphere than is observed when the evaporation of rain is excluded.
The evaporation or rain also damps the 40–50 day variability (compare Figures 9(a) and 5(a)). Apparently, because the evaporation of rain keeps the lower troposphere consistently moist, there is less of a variation in the equivalent potential temperature of boundary layer parcels, which leads to small variations in upper-tropospheric temperature. There are also a couple of important differences in the parcel trajectories (compare Figures 9(b) and 5(b)): (1) parcels descend through the troposphere more rapidly, because they are cooled by both radiation and the evaporation of rain; and (2) there is more evidence of shallow convective plumes. When a smaller evaporation rate is used (e.g. a few per cent per time step), there is still a signal of intraseasonal variability, albeit much weaker, and simulated moisture profiles are not drastically different from those observed (not shown). Moreover, in nature, regions with intense rainfall and evaporation interact with drier regions, so it is possible that the descending warm anomaly/convective depth interaction that the model generates could play a role in actual intraseasonal oscillations.
4.3.4. Mixing and the evaporation of rain
The most realistic temperature and moisture profiles are obtained by including both mixing and the evaporation of rain (Figure 10); both temperature and moisture profiles generally follow those for the COARE IFA. Once again, the 40–50-day variability is weak in this simulation (Figure 11(a)), apparently because the evaporation of rain keeps the lower troposphere consistently moist. Including mixing enhances the variability in the depth of convective plumes (Figure 11(b)) compared with that for the run with only the evaporation of rain (Figure 9(b)). The variety of convective cell depths in this particular run seems the most consistent with observations (Johnson et al., 1999).
Overall, the single-column experiments are encouraging; when combined with simple parametrizations for radiation, surface fluxes and the evaporation of rain, LO is capable of producing realistic temperature and moisture profiles for the Tropics, a variety of depths of convective plumes, and interesting feedbacks between descending temperature anomalies and the depth of convection in certain circumstances.
5. Aquaplanet simulations
We now discuss the results of applying LO and the column physics described in the previous section in a tropical aquaplanet version of the LAM. In keeping with the focus of the paper, we primarily consider the organization of moist convection. We first describe the model configuration and then discuss simulations that include convectively coupled Kelvin waves.
5.1. The tropical aquaplanet version of the LAM
We have implemented spherical geometry in the LAM following the approach we used for our Lagrangian ocean model (H04; H09). The model domain extends from about 35°S to 35°N and spans 360° in longitude. The SST is prescribed using a zonally symmetric analytic function that approximates the observed SST over the western Pacific (Figure 12). The parametrizations for surface fluxes, radiation, rain evaporation and convective mixing are identical to those used in the single-column simulations presented in section 4, except we also include a form of surface friction in this case to prevent easterly trade winds from becoming too strong. The model atmosphere has a free surface for an upper boundary (above 30 km), which is consistent with the assumption that potential temperature is constant within the uppermost air parcel (Haertel et al., 2001). The meridional boundaries are slanted walls (a requirement of the Lagrangian method) that span the vertical extent of the atmosphere and are a parcel radius wide. There are no fluxes of heat, moisture, momentum or mass across them. The atmosphere is initialized to be motionless and dry, and to have a constant potential temperature of 302.5 K. Simulations are run out to 100 days, but we are primarily interested in convective disturbances that become organized in a matter of weeks. For the simulation we present in detail, parcels have a 10° radius in the meridional direction, a 20° radius in the zonal direction, and a maximum vertical thickness of 28 hPa. While we also conducted a similar run using a 10° zonal resolution with very similar results, we decided to present the lower resolution run to demonstrate that the LAM can generate realistic convectively coupled Kelvin waves at an impressively low resolution, a feature that will make the LAM an attractive tool for future studies of convectively coupled waves using low-power computers. Convecting columns are one parcel radius across in both zonal and meridional directions. We use net radiative coolings of 0.5–1 K day−1, which are in the range of observed and theoretically derived values for the equatorial western Pacific (Ciesielski et al., 2003, and references therein).
5.2. Simulations of convectively coupled Kelvin waves
When the aquaplanet model is run with surface fluxes, radiation and the evaporation of rain, intense eastward-moving precipitation anomalies spontaneously develop. For example, Figure 13 shows a time series of rainfall within 15 degrees of the Equator for a case with a net radiative cooling of 0.7 K day−1. Convective rain develops within a couple of days, and it becomes organized into eastward-propagating disturbances within several weeks. Linear approximations of paths of long-lived disturbances are depicted with dotted white lines (Figure 13). We construct a composite disturbance by remapping model variables to a disturbance-relative coordinate system (centred on a given dotted white line), and averaging over the lifetimes of all of the disturbances. Figure 14 shows the horizontal structure of surface pressure and velocity for the composite disturbance. Regions of high and low pressure are centred on the Equator, and are accompanied by westerlies and easterlies respectively. These features are generally consistent with the theoretical structure of an Equatorial Kelvin wave (Matsuno. 1966), but there are also a couple of deviations from classical structure: maximum surface westerlies are shifted eastward of the highest surface pressure, and regions of heaviest precipitation have a smaller meridional extent and are shifted westward of the upward motion in the classical wave, which is centred between the high and the low. Since observed convectively coupled Kelvin waves also exhibit these deviations from classical structure (Straub and Kiladis, 2003b), hereafter we refer to the composite disturbance as a convectively coupled Kelvin wave.
Panels (a) and (b) of Figures 15–17 compare the vertical structures of temperature, zonal wind and humidity for the composite LAM-simulated wave to those in a composite observed convectively-coupled Kelvin wave constructed by Straub and Kiladis (2003b). Note that although the observed wave uses time for a horizontal coordinate, and the LAM composite uses longitude, it is reasonable to make a direct comparison because the simulated disturbances are in a steady state, and travel approximately 16 degrees longitude per day. Considering the simplicity of the LAM, the LO convective parametrization, and the other column physics, the model does a remarkable job of reproducing key features of observed Kelvin waves. In both cases just ahead of the wave the lower troposphere is warm, and this warm anomaly extends into the upper troposphere in the precipitating region, with weakest perturbations around 500 hPa (Figure 15(a)–(b)). Just to the left of longitude/time 0 the lower troposphere is relatively cool with a local minimum around 700 hPa. The simulated and observed zonal wind fields also have many common features (Figure 16(a)–(b)). On the leading edge of the wave there are easterlies near the surface that progress upward across the precipitating region with local maxima at mid levels and just above 200 hPa. Surface westerlies occur just to the left of time/longitude 0, and these also rise towards the left. Ahead of the wave the boundary layer is relatively moist beneath a dry middle troposphere (Figure 17(a)–(b)), with a transition to a dry boundary layer and a moist middle troposphere occurring across the precipitating region. Apparently, the combination of LO for a convective parametrization, and our simple parametrization for the evaporation of rain, just happens to capture the most fundamental physics of Kelvin waves.
The fact that the LAM generates realistic Kelvin waves is particularly surprising in light of the fact that most climate models do not. Straub et al. (2010) analysed convectively coupled Kelvin waves in 20 climate models that were run for the Fourth Assessment Report of the Intergovernmental Panel on Climate Change (Solomon et al., 2007). They found that less than half of the models generated convectively coupled Kelvin waves, and only a few of these produced waves with realistic vertical structures. Panels (c)–(d) of Figures 15–17 show vertical structures of temperature, zonal wind and humidity for the two models with the most realistic Kelvin waves. The quality of the LAM-generated waves (Figures 15(b), 16(b), 17(b)) is competitive with the climate models with the best representations of convectively coupled equatorial waves (Figures 15(c)–(d), 16(c)–(d), 17(c)–(d)).
As mentioned above, one characteristic common to multiple kinds of convectively coupled waves is a cycle in convective morphology with shallow convection followed by deep convection followed by stratiform precipitation (e.g. Takayabu et al., 1996). We now examine the vertical structure of heating for the composite Kelvin wave (Figure 18) to look for a signal of such a life cycle. In the total heating field there is indeed evidence of shallow heating in the extreme lower troposphere on the leading edge of the wave (around 40° in Figure 18(a)). We further break down the heating into contributions from condensation (Figure 18(b)), evaporation (Figure 18(c)), their sum (Figure 18(d)) and the residual (Figure 18(e)), which includes vertical transports of heat by convective eddies (i.e. the local heating caused by swapping vertical positions of parcels). The superposition of condensational heating and evaporative cooling (Figure 18(d)) has a strong tilt in the positive heating perturbation in the lower troposphere, and the residual heating (Figure 18(e)) contributes the shallow heating in the extreme lower troposphere ahead of the wave. Note that the model generates a fairly realistic tilted heating profile without strong evidence of cumulus congestus clouds (see the distinction between shallow convection and cumulus congestus in Johnson et al. (1999)), which are also absent from Takayabu's schematic of the life cycle in 2-day waves (Takayabu et al., 1996). We also mention that because evaporative cooling is largely contained in the lower troposphere (Figure 18(c)), it projects strongly onto the second vertical mode (e.g. Mapes and Houze, 1995; Haertel et al., 2008), and it is likely responsible for this mode's strong contribution to the composite wave's vertical structure (Figures 15–16).
5.3. Sensitivity tests
We performed a number of sensitivity tests to see how the existence and structure of Kelvin waves depends on our model's numerical and physical parameters. First, we doubled the vertical resolution, adjusting the time step and boundary layer restoring time-scale to minimize changes in the rate of parcel ascents and average surface fluxes. The model produced Kelvin waves that were quite similar in horizontal and vertical structures and propagation speeds (not shown). Second, we doubled the horizontal resolution of the model, and again the model produced very similar Kelvin waves. These results suggest that our simulations are not overly sensitive to numerical parameters. We also conducted runs that included convective mixing, a different prescribed height for the stratosphere, and different boundary layer restoring time-scales. In all cases the model produced Kelvin waves much like those illustrated in Figures 13–18.
We did find one physical parameter that has an important impact on the frequency and zonal wave number of Kelvin waves, however. When we increased (reduced) the net radiative cooling to 1 (0.5) K day−1, we found that frequencies of Kelvin waves increased (decreased). Figures 19 and 20 show time–longitude series of precipitation and vertical structures of temperature perturbations for the lower and higher frequency Kelvin waves. It is readily apparent that the run with the higher radiative cooling produces waves with a higher frequency and lower zonal wavelength. However, the vertical structure of the waves is quite similar apart from the change in zonal wavelength (e.g. compare Figure 20(a) and (b)).
Note that increasing radiative forcing also increases the magnitude of the average precipitation and all components of heating perturbations, because on a global basis radiative cooling is largely balanced by convective heating. Curiously, we found in all cases the predominant frequency of Kelvin waves to be roughly proportional to the descent time of evaporatively and radiatively cooled parcels from 700 hPa to the surface. For example, there is roughly a 20 K difference between the mean 700 and 1000 hPa potential temperatures near the Equator. In the runs depicted in Figures 19(b) and 20(b), there is a radiative cooling of 1 K day−1 and an average evaporative cooling of about 4 K day−1 below 700 hPa. This implies a descent time of about 4 days, which is the predominant period of the Kelvin waves (e.g. by examining the right side of Figure 19(b) one sees that each Kelvin wave comes about 4 days after its predecessor). It is possible that this time-scale is favoured because mid-tropospheric moisture anomalies in the wake of Kelvin waves have some remnant that descends to the surface, and enhances convection as the next wave arrives. Or, perhaps the parcel descent time indicates the time required for a complete recovery of the lower troposphere after the passage of a wave.
5.4. Implications for theories of convectively coupled equatorial waves
While we are leaving a detailed analysis of the evaporative-descent time-scale question and of the mechanism of LO-generated Kelvin waves to a subsequent paper, we do mention a few conclusions that we have reached based on our initial simulations. First, because it is excluded by our simple surface flux parametrization, we can say that wind-induced surface heat exchange is not important for generating or maintaining the simulated waves. Second, while including convective mixing alters waves' vertical structures slightly, and makes them slightly slower, it does not appear to be part of the most fundamental dynamics of the waves. Third, the evaporation of rain does appear to be important for the development and mechanism of the waves; when this process is excluded, Kelvin waves are much less prevalent and shorter lived. Finally, while shallow convection (following the definition of Johnson et al. (1999)) is active in the Kelvin waves simulated with the LAM, there is not a strong signal of cumulus congestus in the waves, even though the LAM is capable of simulating congestus and their low-level moistening effects (e.g. Figures 5 and 7). In an observational study Haertel et al. (2008) also found much stronger perturbations in congestus cloud populations in the Madden–Julian Oscillation (Madden and Julian, 1972; Zhang, 2005) than in convectively coupled 2-day waves.
This study discusses the representation of moist convective processes in a new Lagrangian atmospheric model (LAM). The LAM is developed by modifying a Lagrangian ocean model to accommodate compressible fluid parcels. Its ability to model the dynamics of internal gravity waves is tested by comparing a simulation of internal gravity waves conducted with the LAM to a similar run carried out with another atmospheric model.
The LAM employs a unique convective parametrization referred to as ‘Lagrangian Overturning (LO)’. The vertical positions of overlapping air parcels are swapped when doing so yields a greater potential temperature for the rising parcel than for the subsiding parcel. There are a number of potential advantages of this parametrization including: (1) it simply and elegantly captures the essence of convection–LO generates vertical transports of air that occur in response to convective instability and which remove that instability; (2) it can model the effects of both moist and dry convection; (3) convective plumes and the air descending around them can have different properties; (4) the existence and depth of convection responds to local perturbations in thermodynamics profiles; (5) parcel trajectories can be traced through the convective process; (6) temperature and moisture perturbations in subsiding air can be maintained for long periods of time; and (7) LO's implementation requires few (if any) tunable parameters.
We first implement LO in a single-column version of the LAM along with simple parametrizations for radiation, surface fluxes and the evaporation of rain. Radiative convective equilibrium experiments reveal that LO can generate realistic profiles for temperature and moisture for the Tropics, and also support some interesting intraseasonal variability involving descending temperature anomalies that modulate the depth of convective plumes.
When LO is tested in a tropical aquaplanet model, convectively coupled Kelvin waves spontaneously develop. They have realistic vertical structures including out-of-phase upper- and lower-tropospheric temperature anomalies and tilted wind perturbations with local maxima near the surface, at mid levels, and near the tropopause. The simulated waves also propagate at about the same speed as Kelvin waves do in nature. The Kelvin waves generated by the LAM appear to be as realistic, if not more so, than those generated by the few climate models that have good representations of Kelvin waves. Sensitivity tests suggest that while the waves' frequencies are not sensitive to numerical parameters, there is a preferred period equal to the time it takes an evaporatively cooled parcel to descend from 700 hPa to the surface.
Overall, the results of this study suggest that a Lagrangian modelling framework might provide certain advantages for simulating moist convective processes. Convectively coupled waves with realistic vertical structures can be modelled with a simple convective parametrization, and simple representations of radiation, surface fluxes and cloud microphysics. We are optimistic that we can pin down the dynamical mechanism of the simulated waves in future work.
Finally, we will mention that the usefulness of Lagrangian Overturning for simulating Kelvin waves is not the end of the story. In other configurations of our aquaplanet model we have also simulated westward-propagating equatorial waves, and slow, eastward-moving envelopes of convection. We show an example of the latter in Figure 21. An unfiltered time series of precipitation (Figure 21(a)) shows eastward-propagating Kelvin waves much like those discussed in section 5. However, in this case the waves are organized into slower-moving (∼5 m s−1) envelopes of convection that are easily discerned in a low-pass filtered time series of precipitation (Figure 21(b)). Within these envelopes, active deep convection is preceded by congestus heating and moistening (Figure 21(c)–(d)), as in some of our single-column experiments with LO (Figures 5 and 7). Could these envelopes of convection be Madden–Julian oscillations? We plan to address this question in a subsequent paper.
We thank Ming Cai for suggesting we explore the behaviour of moist convective disturbances in a Lagrangian modelling framework, Kerry Emanuel for helping to formulate LO, the organizers of the Banff International Research Station workshop on multi-scale tropical processes for inviting the talk that was the precursor to this paper, and Brian Mapes for suggesting the idea that the lower-tropospheric descent time of evaporatively and radiatively cooled parcels might determine the frequency of simulated Kelvin waves. We also thank Richard Johnson, George Kiladis and an anonymous reviewer for their comments and suggestions. This research was supported by NSF Grants ATM-0500061, ATM-0754088, and ATM-0849323, and by the DOE Office of Science (DE-FG02-08ER64590) and the Packard Foundation.
Parcels that ascend to the upper troposphere lose almost all of their water vapour, and because of a lack of mixing and of the evaporation of rain water they remain very dry as they descend through the troposphere. In contrast, parcels that stop their ascents at low- or mid-levels retain signiﬁcant amounts of water vapour (i.e. have a speciﬁc humidity equal to the saturation value for the low- or mid-tropospheric temperature).