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Idealized models of reduced complexity are important tools to understand key processes underlying a complex system. In climate science in particular, they are important for helping the community improve our ability to predict the effect of climate change on the earth system. Climate models are large computer codes based on the discretization of the fluid dynamics equations on grids of horizontal resolution in the order of 100 km, whereas unresolved processes are handled by subgrid models. For instance, simple models are routinely used to help understand the interactions between small-scale processes due to atmospheric moist convection and large-scale circulation patterns. Here, a zonally symmetric model for the monsoon circulation is presented and solved numerically. The model is based on the Galerkin projection of the primitive equations of atmospheric synoptic dynamics onto the first modes of vertical structure to represent free tropospheric circulation and is coupled to a bulk atmospheric boundary layer (ABL) model. The model carries bulk equations for water vapor in both the free troposphere and the ABL, while the processes of convection and precipitation are represented through a stochastic model for clouds. The model equations are coupled through advective nonlinearities, and the resulting system is not conservative and not necessarily hyperbolic. This makes the design of a numerical method for the solution of this system particularly difficult. Here, we develop a numerical scheme based on the operator time-splitting strategy, which decomposes the system into three pieces: a conservative part and two purely advective parts, each of which is solved iteratively using an appropriate method. The conservative system is solved via a central scheme, which does not require hyperbolicity since it avoids the Riemann problem by design. One of the advective parts is a hyperbolic diagonal matrix, which is easily handled by classical methods for hyperbolic equations, while the other advective part is a nilpotent matrix, which is solved via the method of lines. Validation tests using a synthetic exact solution are presented, and formal second-order convergence under grid refinement is demonstrated. Moreover, the model is tested under realistic monsoon conditions, and the ability of the model to simulate key features of the monsoon circulation is illustrated in two distinct parameter regimes. [ABSTRACT FROM AUTHOR]

The barotropic instability of horizontal shear flows is investigated by using two numerical algorithms to solve the equatorial β-plane barotropic equations. The first is the Arakawa Jacobian method (Arakawa, in J Comput Phys 1:119-143, ), which is a second-order-centered finite differences scheme that conserves energy and enstrophy, and the second is the fourth-order essentially non-oscillatory scheme for non-linear PDE's of Osher and Shu (SIAM J Numer Anal 28:907-922, ), which is designed to track sharp fronts. We are interested in the performance of these two methods in tracking the long-time behavior of the instability, under the influence of the non-linearity, in the simple case of a Helmholtz shear layer. The associated linear problem is solved analytically, and the linear solution is used as an initial condition for the numerical simulations. We run a series of numerical simulations using both methods with various grid refinements and with two different amplitudes of the initial perturbation. A small viscosity term is added to the vorticity equation to damp the grid-scale waves for Arakawa's method. This is not necessary for the high-order ENO-4 scheme, which has its own grid-scale dissipation. At high resolution, the two methods are in good agreement; they yield qualitatively and quantitatively the same solution in the long run: for small disturbances, the total flow stabilizes into a steady-state meridional shear with a smooth profile near the equator, while strong disturbances merge together to form a single large-scale vortex that propagates westward, along the equator, consistent with the African easterly waves and the monsoons trough circulation. At coarse resolution, however, Arakawa's method seems to be much superior to the fourth-order ENO-4 scheme as it provides solutions that are more consistent with the fine resolution one. [ABSTRACT FROM AUTHOR]

The atmospheric variability in the tropics is primarily driven by convective heating. Observations revealed that convection in the tropics is organized into a hierarchy of multiscale convective systems ranging from the individual cloud cells to planetary scale disturbances that are nested within each other like Russian dolls. Current global climate models simulate very poorly these convectively coupled waves due in part to inadequate treatment of organized convection by the underlying cumulus parameterizations. Here, we present idealized simulations of convectively coupled equatorial waves (CCWs) using the weather research and forecast model in a horizontally limited domain consisting of a 4,500 km-wide square centered at the equator at moderate horizontal resolution of 10 km. We attempted and compared various configuration options, including switching on and off the cumulus parameterization (CP) and nesting a fine resolution 3.33 km domain, a 2,000 km-wide square, in the middle of the domain. It turns out that the results without a CP are much superior than those using a CP. While the cases without a CP resulted in a coherent eastward propagating CCW, which has many common features with observed convectively coupled Kelvin waves, the cumulus parameterization tends to destroy both the coherence of the propagating waves, even in the case with a nested domain, and reduces dramatically the variability. A primary demonstration on how such results could be used to show evidence of energy exchange, through momentum transport, between small-scale circulation due to mesoscale convection and the propagating synoptic scale wave will be reported is also presented. [ABSTRACT FROM AUTHOR]

This paper presents a study of the diurnal cycle of tropical precipitation and its interaction with convectively coupled waves in the context of simple models with crude vertical resolution. One and two baroclinic mode models are tested in both the context of a one-column model and the context of a full spatial dependency that permits waves to propagate and interact with the diurnal cycle. It is found that a one baroclinic mode model is capable of reproducing a realistic diurnal cycle of tropical precipitation both over land and over the ocean provided an adequate switch function is used to mimic the congestus preconditioning mechanism that operates in the multicloud model of Khouider and Majda. However, a full two baroclinic mode multicloud model is needed to capture the interaction of convectively coupled tropical waves with the diurnal cycle. In a more conventional mass flux parameterization framework, both one and two baroclinic mode models fail to capture the diurnal cycle of tropical precipitation. [ABSTRACT FROM AUTHOR]

We compare the performances of two different numerical methods to solve the equatorial shallow water equations in a background meridional shear: A coarse-resolution Galerkin-truncation-based method and a finite-volume method, for the case of an equatorial Rossby wave. In the presence of the barotropic shear, the Rossby wave quickly loses its energy through shear interaction and excites other equatorially trapped waves despite the fact that the PDE system is linear. The two methods handle this sudden energy exchange across scales very differently. While the finite volume converges statistically to a coherent large-scale solution, the Galerkin truncation method results in large-scale and slow oscillations where energy is bumped back and forth within the resolved-scale spectrum, involving higher-order equatorial wave modes heavily depending on numerical resolution, that is, the number of Galerkin basis functions. The addition of an artificial viscosity for the coarse-resolution Galerkin method heavily damps the energy oscillations without significantly changing its transient dynamics. This provides an example of interaction across scales where the resolution of scales that are apparently not representative of the statistical solution are important for the transient dynamics. Therefore, non-linear interactions of equatorially trapped waves may provide an interesting test bed for the validation of climate model dynamical cores. [ABSTRACT FROM AUTHOR]

Stratified hydrostatic fluids have linear internal gravity waves with different phase speeds and vertical profiles. Here a simplified set of partial differential equations (PDE) is derived to represent the nonlinear dynamics of waves with different vertical profiles. The equations are derived by projecting the full nonlinear equations onto the vertical modes of two gravity waves, and the resulting equations are thus referred to here as the two-mode shallow water equations (2MSWE). A key aspect of the nonlinearities of the 2MSWE is that they allow for interactions between a background wind shear and propagating waves. This is important in the tropical atmosphere where horizontally propagating gravity waves interact together with wind shear and have source terms due to convection. It is shown here that the 2MSWE have nonlinear internal bore solutions, and the behavior of the nonlinear waves is investigated for different background wind shears. When a background shear is included, there is an asymmetry between the east- and westward propagating waves. This could be an important effect for the large-scale organization of tropical convection, since the convection is often not isotropic but organized on large scales by waves. An idealized illustration of this asymmetry is given for a background shear from the westerly wind burst phase of the Madden–Julian oscillation; the potential for organized convection is increased to the west of the existing convection by the propagating nonlinear gravity waves, which agrees qualitatively with actual observations. The ideas here should be useful for other physical applications as well. Moreover, the 2MSWE have several interesting mathematical properties: they are a system of nonconservative PDE with a conserved energy, they are conditionally hyperbolic, and they are neither genuinely nonlinear nor linearly degenerate over all of state space. Theory and numerics are developed to illustrate these features, and these features are important in designing the numerical scheme. A numerical method is designed with simplicity and minimal computational cost as the main design principles. Numerical tests demonstrate that no catastrophic effects are introduced when hyperbolicity is lost, and the scheme can represent propagating discontinuities without introducing spurious oscillations. [ABSTRACT FROM AUTHOR]

Recent observational analysis reveals the central role of three multi-cloud types, congestus, stratiform, and deep convective cumulus clouds, in the dynamics of large scale convectively coupled Kelvin waves, westward propagating two-day waves, and the Madden–Julian oscillation. The authors have recently developed a systematic model convective parametrization highlighting the dynamic role of the three cloud types through two baroclinic modes of vertical structure: a deep convective heating mode and a second mode with low level heating and cooling corresponding respectively to congestus and stratiform clouds. The model includes a systematic moisture equation where the lower troposphere moisture increases through detrainment of shallow cumulus clouds, evaporation of stratiform rain, and moisture convergence and decreases through deep convective precipitation and a nonlinear switch which favors either deep or congestus convection depending on whether the troposphere is moist or dry. Here several new facets of these multi-cloud models are discussed including all the relevant time scales in the models and the links with simpler parametrizations involving only a single baroclinic mode in various limiting regimes. One of the new phenomena in the multi-cloud models is the existence of suitable unstable radiative convective equilibria (RCE) involving a larger fraction of congestus clouds and a smaller fraction of deep convective clouds. Novel aspects of the linear and nonlinear stability of such unstable RCE’s are studied here. They include new modes of linear instability including mesoscale second baroclinic moist gravity waves, slow moving mesoscale modes resembling squall lines, and large scale standing modes. The nonlinear instability of unstable RCE’s to homogeneous perturbations is studied with three different types of nonlinear dynamics occurring which involve adjustment to a steady deep convective RCE, periodic oscillation, and even heteroclinic chaos in suitable parameter regimes. [ABSTRACT FROM AUTHOR]

We propose a non-oscillatory balanced numerical scheme for a simplified tropical climate model with a crude vertical resolution, reduced to the barotropic and the first baroclinic modes. The two modes exchange energy through highly nonlinear interaction terms. We consider a periodic channel domain, oriented zonally and centered around the equator and adopt a fractional stepping–splitting strategy, for the governing system of equations, dividing it into three natural pieces which independently preserve energy. We obtain a scheme which preserves geostrophic steady states with minimal ad hoc dissipation by using state of the art numerical methods for each piece: The f-wave algorithm for conservation laws with varying flux functions and source terms of Bale et al. (2002) for the advected baroclinic waves and the Riemann solver-free non-oscillatory central scheme of Levy and Tadmor (1997) for the barotropic-dispersive waves. Unlike the traditional use of a time splitting procedure for conservation laws with source terms (here, the Coriolis forces), the class of balanced schemes to which the f-wave algorithm belongs are able to preserve exactly, to the machine precision, hydrostatic (geostrophic) numerical-steady states. The interaction terms are gathered into a single second order accurate predictor-corrector scheme to minimize energy leakage. Validation tests utilizing known exact solutions consisting of baroclinic Kelvin, Yanai, and equatorial Rossby waves and barotropic Rossby wave packets are given. [ABSTRACT FROM AUTHOR]