Equatorial Beta Plane Research Articles

Equatorial modons in thermal rotating shallow water model

Exact steady eastward-moving vortex-dipole solutions, the equatorial modons, are constructed in the asymptotic limit of low divergence and small temperature variations in the thermal rotating shallow water (TRSW) model on the equatorial beta-plane. This regime is known to be relevant for the tropical atmosphere. The model itself is a generalization, allowing for horizontal temperature gradients, of the classical rotating shallow water model. The asymptotic modons can carry a temperature anomaly and exist also on the inhomogeneous temperature background. The modon configurations are then used to initialize numerical simulations, in order to check whether such coherent structures can exist in the full TRSW model. The results show that this is, indeed, the case. The parameter regimes and limitations on the structure of the temperature anomaly inside, in order for the modons to persist, are established. It is also shown that the modons keep their coherence even while evolving on the background of meridionally inhomogeneous temperature fields, or while interacting with sharp temperature fronts. A general scenario of disaggregation of the modons, if they are out of the stability domain, is exhibited and analysed.

Dispersion diagrams of linear damped waves on the equatorial beta plane

The linear equations of motion are solved to obtain dispersion diagrams with Rayleigh friction (−γu) and Laplacian friction (ν∂xxu), the latter being solved numerically. Laplacian friction is more efficient at eliminating the short-wavelength Rossby waves, whereas Rayleigh friction is more effective at dissipating long-wavelength Rossby waves. For Rayleigh friction, short-wavelength Rossby waves do not exist at periods longer than the damping time scale (1/γ); for Laplacian friction, they do not exist for wavenumbers less than 3/(2ν3). For both damping forms, the imaginary wavenumber (ki) no longer separates the upper branch (gravity waves) from the lower branch (Rossby waves) of the real wavenumber (kr), and the waves are damped at all frequencies. The two solutions of kr do not overlap, and for Rayleigh damping, they meet at 0.5−γ2, which roughly corresponds to ∼13.7 days for very low friction. For very high Rayleigh damping, which is an unrealistic scenario, only gravity waves exist in the dispersion diagram. The consequence of adding friction, even if negligible, is that the discontinuity evident in the inviscid solution at the critical latitude no longer holds, or the critical latitude ceases to exist.

Minimal Mechanisms Responsible for the Dispersive Behavior of the Madden–Julian Oscillation

An attempt has been made to explore the relative contributions of moisture feedback processes on tropical intraseasonal oscillation or Madden–Julian Oscillation (MJO). We focused on moisture feedback processes, including evaporation wind feedback (EWF) and moisture convergence feedback (MCF), which integrate the mechanisms of convective interactions into the tropical atmosphere. The dynamical framework considered here is a moisture-coupled, single-layer linear shallow-water model on an equatorial beta-plane with zonal momentum damping. With this approach, we aimed to recognize the minimal physical mechanisms responsible for the existence of the essential dispersive characteristics of the MJO, including its eastward propagation (k>0), the planetary-scale (small zonal wavenumbers) instability, and the slow phase speed of about ≈5 m/s. Furthermore, we extended our study to determine each feedback mechanism’s influence on the simulated eastward dispersive mode. Our model emphasized that the MJO-like eastward mode is a possible outcome of the combined effect of moisture feedback processes without requiring additional complex mechanisms such as cloud radiative feedback and boundary layer dynamics. The results substantiate the importance of EWF as a primary energy source for developing an eastward moisture mode with a planter-scale instability. The eastward moisture mode exhibits the highest growth rate at the largest wavelengths and is also sensitive to the strength of the EWF, showing a significant increase in the growth rate with the increasing strength of the EWF; however, the eastward moisture mode remains unstable at planetary-scale wavelengths. Moreover, our model endorses that the MCF alone could not produce instability without surface fluxes, although it has a significant role in developing deep convection. It was found that the MCF exhibits a damping mechanism by regulating the frequency and growth rate of the eastward moisture mode at shorter wavelengths.

A Unified Moisture Mode Theory for the Madden–Julian Oscillation and the Boreal Summer Intraseasonal Oscillation

Abstract The Madden–Julian oscillation (MJO) and the boreal summer intraseasonal oscillation (BSISO) are fundamental modes of variability in the tropical atmosphere on the intraseasonal time scale. A linear model, using a moist shallow water equation set on an equatorial beta plane, is developed to provide a unified treatment of the two modes and to understand their growth and propagation over the Indian Ocean. Moisture is assumed to increase linearly with longitude and to decrease quadratically with latitude. Solutions are obtained through linear stability analysis, considering the gravest (n = 1) meridional mode with nonzero meridional velocity. Anomalies in zonal moisture advection and surface fluxes are both proportional to those in zonal wind, but of opposite sign. With observation-based estimates for both effects, the zonal advection dominates, and drives the planetary-scale instability. With a sufficiently small meridional moisture gradient, the horizontal structure exhibits oscillations with latitude and a northwest–southeast horizontal tilt in the Northern Hemisphere, qualitatively resembling the observed BSISO. As the meridional moisture gradient increases, the horizontal tilt decreases and the spatial pattern transforms toward the “swallowtail” structure associated with the MJO, with cyclonic gyres in both hemispheres straddling the equatorial precipitation maximum. These results suggest that the magnitude of the meridional moisture gradient shapes the horizontal structures, leading to the transformation from the BSISO-like tilted horizontal structure to the MJO-like neutral wave structure as the meridional moisture gradient changes with the seasons. The existence and behavior of these intraseasonal modes can be understood as a consequence of phase speed matching between the equatorial mode with zero meridional velocity (analogous to the dry Kelvin wave) and a local off-equatorial component that is characterized by considering an otherwise similar system on an f plane.

The MJO on the Equatorial Beta Plane: An Eastward-Propagating Rossby Wave Induced by Meridional Moisture Advection

AbstractLinearized wave solutions on the equatorial beta plane are examined in the presence of a background meridional moisture gradient. Of interest is a slow, eastward-propagating n = 1 mode that is unstable at planetary scales and only exists for a small range of zonal wavenumbers (). The mode dispersion curve appears as an eastward extension of the westward-propagating equatorial Rossby wave solution. This mode is therefore termed the eastward-propagating equatorial Rossby wave (ERW). The zonal wavenumber-2 ERW horizontal structure consists of a low-level equatorial convergence center flanked by quadrupole off-equatorial gyres, and resembles the horizontal structure of the observed MJO. An analytic, leading-order dispersion relationship for the ERW shows that meridional moisture advection imparts eastward propagation, and that the smallness of a gross moist stability–like parameter contributes to the slow phase speed. The ERW is unstable near planetary scales when low-level easterlies moisten the column. This moistening could come from either zonal moisture advection or surface fluxes or a combination thereof. When westerlies instead moisten the column, the ERW is damped and the westward-propagating long Rossby wave is unstable. The ERW does not exist when the meridional moisture gradient is too weak. A moist static energy budget analysis shows that the ERW scale selection is partly due to finite-time-scale convective adjustment and less effective zonal wind–induced moistening at smaller scales. Similarities in the phase speed, preferred scale, and horizontal structure suggest that the ERW is a beta-plane analog of the MJO.

Theoretical analysis of equatorial near-inertial solitary waves under complete Coriolis parameters

An investigation of equatorial near-inertial wave dynamics under complete Coriolis parameters is performed in this paper. Starting from the basic model equations of oceanic motions, a Korteweg de Vries equation is derived to simulate the evolution of equatorial nonlinear near-inertial waves by using methods of scaling analysis and perturbation expansions under the equatorial beta plane approximation. Theoretical dynamic analysis is finished based on the obtained Korteweg de Vries equation, and the results show that the horizontal component of Coriolis parameters is of great importance to the propagation of equatorial nonlinear near-inertial solitary waves by modifying its dispersion relation and by interacting with the basic background flow.

Quasi-Equilibrium and Weak Temperature Gradient Balances in an Equatorial Beta-Plane Model

AbstractConvective quasi-equilibrium (QE) and weak temperature gradient (WTG) balances are frequently employed to study the tropical atmosphere. This study uses linearized equatorial beta-plane solutions to examine the relevant regimes for these balances. Wave solutions are characterized by moisture–temperature ratio (q–T ratio) and dominant thermodynamic balances. An empirically constrained precipitation closure assigns different sensitivities of convection to temperature (εt) and moisture (εq). Longwave equatorial Kelvin and Rossby waves tend toward the QE balance with q–T ratios of εt:εq that can be ~1–3. Departures from strict QE, essential to both precipitation and wave dynamics, grow with wavenumber. The propagating QE modes have reduced phase speeds because of the effective gross moist stability meff, with a further reduction when εt > 0. Moisture modes obeying the WTG balance and with large q–T ratios (>10) emerge in the shortwave regime; these modes exist with both Kelvin and Rossby wave meridional structures. In the υ = 0 case, long propagating gravity waves are absent and only emerge beyond a cutoff wavenumber. Two bifurcations in the wave solutions are identified and used to locate the spatial scales for QE–WTG transition and gravity wave emergence. These scales are governed by the competition between the convection and gravity wave adjustment times and are modulated by meff. Near-zero values of meff shift the QE–WTG transition wavenumber toward zero. Continuous transitions replace the bifurcations when meff < 0 or moisture advection/WISHE mechanisms are included, but the wavenumber-dependent QE and WTG balances remain qualitatively unaltered. Rapidly decaying convective/gravity wave modes adjust to the slowly evolving QE/WTG state in the longwave/shortwave regimes, respectively.

Eastward-moving equatorial modons in moist-convective shallow-water models

It is shown that steady large-scale slowly eastward-moving twin-cyclone coherent structures, the equatorial modons, exist in both one- and two layer versions of the rotating shallow water model on the equatorial beta plane. They arise via the process of “ageostrophic adjustment” from the analytic asymptotic modon solutions of the vorticity equation obtained in the limit of small pressure perturbations. Evolution of these structures in adiabatic and moist-convective environments, and also in the presence of topography is analysed, showing their robustness in the one-layer model. It is demonstrated that moist convection enhances and helps maintain the modons. In the two-layer model the barotropic and quasi-barotropic modons display similar to one-layer modon features, while increasing baroclinicity leads to eventual loss of coherence and arrest of the eastward propagation. Some features of equatorial modons resemble those observed in the Madden-Julian Oscillation events in tropical atmosphere, which hints at their possible relevance to the dynamics of this phenomenon.

Geostrophic adjustment on the equatorial beta-plane revisited

The process of geostrophic adjustment of localized large-scale pressure anomalies in the standard adiabatic shallow-water model on the equatorial beta-plane is revisited, and it is shown that the standard scenario of generation of westward-moving Rossby and eastward-moving Kelvin waves, which underlies the classical Gill theory of tropical circulation due to a localized heating, is not unique. Depending on the strength and aspect ratio of the initial perturbation, the response to the initial perturbation in the western sector can be dominated by inertia-gravity waves. The adjustment in the diabatic moist-convective shallow water model can be totally different and produces, depending on parameters, either Gill-like response or eastward-moving coherent dipolar structures of the type of equatorial modons, which do not appear in the “dry” adjustment, or vortices traveling, respectively, northwest in the Northern and southwest in the Southern hemispheres.

Eastward-moving convection-enhanced modons in shallow water in the equatorial tangent plane

We report a discovery of steady long-living slowly eastward moving large-scale coherent twin cyclones, the equatorial modons, in the shallow water model in the equatorial beta-plane, the archetype model of the ocean and atmosphere dynamics in tropics. We start by constructing analytical asymptotic modon solutions in the non-divergent velocity approximation and then show by simulations with a high-resolution numerical scheme that such configurations evolve into steady dipolar solutions of the full model. In the atmospheric context, the modons persist in the presence of moist convection, being accompanied and enhanced by specific patterns of water-vapour condensation.

A Simple Model of Convectively Coupled Equatorial Rossby Waves

AbstractIntraseasonal oscillations affect the weather not just in the tropics but all around the globe. The convectively coupled equatorial Rossby wave is observed as the westward‐moving intraseasonal oscillation. The fundamental physics of its coupling is still unknown; thus, many questions remain unanswered. How is its phase speed altered by convection? What makes it unstable? Why is it an intraseasonal oscillation? Using the Fuchs and Raymond model with linearized governing equations on an equatorial beta plane, first baroclinic mode vertical structure, and moisture and wind‐induced surface heat exchange (WISHE) convective parametrizations, this paper seeks a fundamental analytical theory that can explain the basic features of the convectively coupled equatorial Rossby wave. The WISHE‐moisture theory leads to a large‐scale, unstable westward propagating mode in the n = 1 case, which we call the westward propagating WISHE‐moisture mode. We find that the westward propagating WISHE‐moisture mode is indeed the free equatorial Rossby wave in the absence of moisture closure and WISHE. It is propagating westward due to the beta effect, and it slows down when it is convectively coupled. Its phase speed decreases mainly due to WISHE and cloud‐radiation interactions. The x‐y structure of the pressure and horizontal winds is similar to the free and observed Rossby wave, with convergent net flow. The strongest easterlies are to the west of the precipitation maximum increasing the moisture in that area. The mode is unstable due to the interplay of surface fluxes and moisture, which increases as a function of zonal wavelength.

Intraseasonal Variability in a Cloud-Permitting Near-Global Equatorial Aquaplanet Model

Abstract Recent studies have suggested that the Madden–Julian oscillation is a result of an instability driven mainly by cloud–radiation feedbacks, similar in character to self-aggregation of convection in nonrotating, cloud-permitting simulations of radiative–convective equilibrium (RCE). Here we bolster that inference by simulating radiative–convective equilibrium states on a rotating sphere with constant sea surface temperature, using the cloud-permitting System for Atmospheric Modeling (SAM) with 20-km grid spacing and extending to walls at 46° latitude in each hemisphere. Mechanism-denial experiments reveal that cloud–radiation interaction is the quintessential driving mechanism of the simulated MJO-like disturbances, but wind-induced surface heat exchange (WISHE) feedbacks are the primary driver of its eastward propagation. WISHE may also explain the faster Kelvin-like modes in the simulations. These conclusions are supported by a linear stability analysis of RCE states on an equatorial beta plane.

Solitary waves in shallow water magnetohydrodynamics

We consider a perfect electrically conducting rotating fluid in the presence of an ambient toroidal magnetic field, governed by the shallow water magnetohydrodynamic (MHD) equations in a modified equatorial beta plane approximation. Using a multiple scale asymptotic technique, previously developed by Boyd (1980) for equatorial solitary hydrodynamic waves, we look for solitary MHD waves. In the case of a weak ambient magnetic field, the leading order equations governing the poleward velocities can be solved using associated Legendre functions. These functions are multiplied by an amplitude function of slow length and time variables. They are solved for at second order via a compatibility condition and have the form of equatorial Rossby solitary waves. When the ambient magnetic field is moderately strong, the equations governing the poleward velocities cannot be solved using special functions. Instead, we are able to apply a WKB type approximation to solve this problem using a large parameter, , which arises naturally in the governing equations. The solution has the form of an equatorial magneto-Rossby solitary wave. When the ambient magnetic field is very strong, the solutions are bounded away from the equator in the form of magnetostrophic solitary waves. The possible relationship between all of these waves and solar phenomena such as the solar cycle, sunspots and active longitudes is discussed.

Eddy Influences on the Strength of the Hadley Circulation: Dynamic and Thermodynamic Perspectives

Abstract The strength of the equinoctial Hadley circulation (HC) is investigated in idealized simulations conducted on an equatorial beta plane in which the zonal width of the domain is varied to either permit or suppress large-scale eddies. The presence of such eddies is found to amplify the HC by a factor of 2–3 in simulations with slab-ocean boundary conditions or with a simple representation of ocean heat transport. Additional simulations in which the eddy forcing is prescribed externally indicate that this amplification is primarily associated with large-scale eddy momentum fluxes rather than large-scale eddy heat fluxes. These results contrast with results from simulations with fixed distributions of sea surface temperature (SST), in which the HC strength has been found to be relatively insensitive to large-scale eddy momentum fluxes. In both the interactive- and fixed-SST cases, the influence of nonlinear momentum advection by the mean flow complicates efforts to use the angular-momentum budget to constrain the HC strength. However, a strong relationship is found between the HC strength and a measure of the meridional gradient of boundary layer entropy, indicating a possible thermodynamic control on the HC strength. In simulations with interactive SSTs, meridional eddy momentum fluxes affect the boundary layer entropy by inducing a low-level frictional flow that reduces the ability of the HC to transport heat poleward. This allows for the maintenance of a large meridional entropy gradient in the presence of a strong HC. The results highlight the potential utility of a thermodynamic perspective for understanding the HC in flow regimes for which dynamical constraints may be difficult to apply.

Shear-driven instabilities and shocks in the atmospheres of hot Jupiters

General circulation models of the atmosphere of hot Jupiters have shown the existence of a supersonic eastward equatorial jet. In this paper, we investigate the effects of compressibility on the atmospheric dynamics by solving the standard Euler equations. This is done by means of a series of simulations performed in the framework of the equatorial beta-plane approximation using the finite volume shock-capturing code RAMSES. At low resolution, we recover the classical results described in the literature: we find a strong and steady supersonic equatorial jet of a few km/s that displays no signature of shocks. We next show that the jet zonal velocity depends significantly on the grid meridional resolution. When that resolution is fine enough to properly resolve the jet, the latter is subject to a Kelvin-Helmholtz instability. The jet zonal mean velocity displays regular oscillations with a typical timescale of few days and a significant amplitude of about 15% of the jet velocity. We also find compelling evidence for the development of a vertical shear instability at pressure levels of a few bars. It seems to be responsible for an increased downward kinetic energy flux, significantly affecting the temperature of the deep atmosphere, and appears to act as a form of drag on the equatorial jet. This instability also creates velocity fluctuations that propagate upward and steepen into weak shocks at pressure levels of a few mbars. We conclude that hot Jupiter equatorial jets are potentially unstable to both a barotropic Kelvin-Helmholtz instability and a vertical shear instability. Upon confirmation using more realistic models, both instabilities could result in significant time variability of the atmospheric winds, may provide a small scale dissipation mechanism in the flow, and might have consequences for the internal evolution of hot Jupiters.

Exploring the Role of Eddy Momentum Fluxes in Determining the Characteristics of the Equinoctial Hadley Circulation: Fixed-SST Simulations

Abstract The influence of eddy momentum fluxes on the equinoctial Hadley circulation is explored using idealized simulations on an equatorial beta plane in which the sea surface temperature (SST) distribution is fixed. By comparing simulations run in a wide-domain configuration, in which large-scale eddies are present, to simulations in which the model domain is too narrow to permit baroclinic instability, the role of large-scale eddies in determining the characteristics of the Hadley circulation is elucidated. The simulations also include an explicit representation of deep convection, allowing for an evaluation of the influence of convective momentum transport on the zonal-mean circulation. The simulated eddy momentum fluxes are much weaker in the narrow-domain configuration than in the wide-domain case, and convective momentum transport is found to be of secondary importance. As a result, many characteristics of the narrow-domain Hadley circulation are well described by axisymmetric theory and differ from those of the wide-domain case. Nevertheless, the strength of the Hadley circulation is similar irrespective of the domain width. The sensitivity of this result to the strength of the eddy forcing is investigated using narrow-domain simulations forced by artificial sinks of zonal momentum. As the magnitude of the momentum sink increases, the Hadley circulation strengthens, but the increase is relatively modest except at very strong forcing magnitudes. The results suggest that the fixed-SST boundary condition places a strong thermodynamic constraint on the Hadley circulation strength and that one should consider the energy budget as well as the angular momentum budget in order to fully understand the influence of large-scale eddies on the zonal-mean circulation in the tropics.

The Relationship between Equatorial Mixed Rossby–Gravity and Eastward Inertio-Gravity Waves. Part I

Abstract The relationship between n = 0 mixed Rossby–gravity waves (MRGs) and eastward inertio-gravity waves (EIGs) from Matsuno’s shallow-water theory on an equatorial beta plane is studied using statistics of satellite brightness temperature Tb and dynamical fields from ERA-Interim data. Unlike other observed convectively coupled equatorial waves, which have spectral signals well separated into eastward and westward modes, there is a continuum of MRG–EIG power standing above the background that peaks near wavenumber 0. This continuum is also present in the signals of dry stratospheric MRGs. While hundreds of papers have been written on MRGs, very little work on EIGs has appeared in the literature to date. The authors attribute this to the fact that EIG circulations are much weaker than those of MRGs for a given amount of divergence, making them more difficult to observe even though they strongly modulate convection. Empirical orthogonal function (EOF) and cross-spectral analysis of 2–6-day-filtered Tb isolate zonally standing modes of synoptic-scale convection originally identified by Wallace in 1971. These display antisymmetric Tb signals about the equator that propagate poleward with a period of around 4 days, along with westward-propagating MRG-like circulations that move through the Tb patterns. Further analysis here and in Part II shows that these signatures are not artifacts of the EOF approach but result from a mixture of MRG or EIG modes occurring either in isolation or at the same time.

A Formulation of Three-Dimensional Residual Mean Flow and Wave Activity Flux Applicable to Equatorial Waves

Abstract The large-scale waves that are known to be trapped around the equator are called equatorial waves. The equatorial waves cause mean zonal wind acceleration related to quasi-biennial and semiannual oscillations. The interaction between equatorial waves and the mean wind has been studied by using the transformed Eulerian mean (TEM) equations in the meridional cross section. However, to examine the three-dimensional (3D) structure of the interaction, the 3D residual mean flow and wave activity flux for the equatorial waves are needed. The 3D residual mean flow is expressed as the sum of the Eulerian mean flow and Stokes drift. The present study derives a formula that is approximately equal to the 3D Stokes drift for equatorial waves on the equatorial beta plane (EQSD). The 3D wave activity flux for equatorial waves whose divergence corresponds to the wave forcing is also derived using the EQSD. It is shown that the meridionally integrated 3D wave activity flux for equatorial waves is proportional to the group velocity of equatorial waves.

Effects of Varying the Shape of the Convective Heating Profile on Convectively Coupled Gravity Waves and Moisture Modes

Abstract The analytical model of convectively coupled gravity waves and moisture modes of Raymond and Fuchs is extended to the case of top-heavy and bottom-heavy convective heating profiles. Top-heavy heating profiles favor gravity waves, while bottom-heavy profiles support moisture modes. The latter behavior results from the sensitivity of moisture modes to the gross moist stability, which is more negative with bottom-heavy heating. A numerical implementation of the analytical model allows calculations in the two-dimensional nonrotating case as well as on a three-dimensional equatorial beta plane. In the two-dimensional case the analytical and numerical models are mostly in agreement, although minor discrepancies occur. In three dimensions the gravity modes become equatorial Kelvin waves whereas the moisture modes are more complex and require further investigation.

Observed Structure of Convectively Coupled Waves as a Function of Equivalent Depth: Kelvin Waves and the Madden–Julian Oscillation

Abstract The view that convectively coupled Kelvin waves and the Madden–Julian oscillation (MJO) are distinct modes is tested by regressing data from the Climate Forecast System Reanalysis against satellite outgoing longwave radiation data filtered for particular zonal wavenumbers and frequencies by wavelet analysis. Results confirm that nearly dry Kelvin waves have horizontal structures consistent with their equatorial beta-plane shallow-water-theory counterparts, with westerly winds collocated with the lower-tropospheric ridge, while the MJO and signals along Kelvin wave dispersion curves at low shallow-water-model equivalent depths are characterized by geopotential troughs extending westward from the region of lower-tropospheric easterly wind anomalies through the region of lower-tropospheric westerly winds collocated with deep convection. Results show that as equivalent depth decreases from that of the dry waves (concomitant with intensification of the associated convection), the ridge in the westerlies and the trough in the easterlies shift westward. The analysis therefore demonstrates a continuous field of intermediate structures between the two extremes, suggesting that Kelvin waves and the MJO are not dynamically distinct modes. Instead, signals consistent with Kelvin waves become more consistent with the MJO as the associated convection intensifies. This result depends little on zonal scale. Further analysis also shows how activity in synoptic-scale Kelvin waves characterized by particular phase speeds evolves with the planetary-scale MJO.