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Transient, synoptic-scale vortices produce a large fraction of total rainfall in most monsoon regions and are often associated with extreme precipitation. However, the mechanism of their amplification remains a topic of active research. For monsoon depressions, which are the most prominent synoptic-scale vortex in the Asian–Australian monsoon, recent work has suggested that meridional gradients in zonal wind in the vortex environment may produce growth through barotropic instability, while meridional gradients in environmental humidity have also been proposed to cause amplification through coupling with precipitating convection. Here, a two-dimensional shallow water model on a sphere with parameterized precipitation is used to examine the relative role played by these two environmental gradients. By systematically varying the meridional moisture gradient and meridional wind shear for both weak, quasi-linear waves and finite-amplitude isolated vortices, we show that rotational winds in the initial vortex are amplified most strongly by meridional shear of the environmental zonal wind, while vortex precipitation rates are most sensitive to environmental moisture gradients. The growth rate in the presence of both gradients is less than the sum of growth rates in the presence of isolated gradients, as the phase relation between moisture and vorticity anomalies becomes distorted with increasing shear. These results suggest that background meridional gradients in both zonal wind and environmental humidity can contribute to the amplification of vortices to monsoon depression strength, but with some degree of decoupling of the dry rotational flow and the moist convection.

The composite structure of the Madden-Julian oscillation (MJO) has long been known to feature pronounced Rossby gyres in the subtropical upper troposphere, whose existence can be interpreted as the forced response to convective heating anomalies in the presence of a subtropical westerly jet. The question of interest here is whether these forced gyre circulations have any subsequent effects on divergence patterns in the tropics and the Kelvin-mode component of the MJO. A nonlinear spherical shallow water model is used to investigate how the introduction of different background jet profiles affects the model's steady-state response to an imposed MJO-like stationary thermal forcing. Results show that a stronger jet leads to a stronger Kelvin-mode response in the tropics up to a critical jet speed, along with stronger divergence anomalies in the vicinity of the forcing. To understand this behavior, additional calculations are performed in which a localized vorticity forcing is imposed in the extratropics, without any thermal forcing in the tropics. The response is once again seen to include pronounced equatorial Kelvin waves, provided the jet is of sufficient amplitude. A detailed analysis of the vorticity budget reveals that the zonal-mean zonal wind shear plays a key role in amplifying the Kelvin-mode divergent winds near the equator, with the effects of nonlinearities being of negligible importance. These results help to explain why the MJO tends to be strongest during boreal winter when the Indo-Pacific jet is typically at its strongest. [ABSTRACT FROM AUTHOR]

In this Part II paper we present a fully consistent analytical derivation of the "dry" isentropic 1½-layer shallow-water model described and used in Part I of this study, with no convection and precipitation. The mathematical derivation presented here is based on a combined asymptotic and slaved Hamiltonian analysis, which is used to resolve an apparent inconsistency arising from the application of a rigid-lid approximation to an isentropic two-layer shallow-water model. Real observations based on radiosonde data are used to justify the scaling assumptions used throughout the paper, as well as in Part I. Eventually, a fully consistent isentropic 1½-layer model emerges from imposing fluid at rest (v1 = 0) and zero Montgomery potential (M1 = 0) in the upper layer of an isentropic two-layer model. [ABSTRACT FROM AUTHOR]

An isentropic 1½-layer model based on modified shallow-water equations is presented, including terms mimicking convection and precipitation. This model is an updated version of the isopycnal single-layer modified rotating shallow water (modRSW) model. The clearer link between fluid temperature and model variables together with a double-layer structure make this revised, isentropic model a more suitable tool to achieve our future goal: to conduct idealized experiments for investigating satellite data assimilation. The numerical model implementation is verified against an analytical solution for stationary waves in a rotating fluid, based on Shrira's methodology for the isopycnal case. Recovery of the equivalent isopycnal model is also verified, both analytically and numerically. With convection and precipitation added, we show how complex model dynamics can be achieved exploiting rotation and relaxation to a meridional jet in a periodic domain. This solution represents a useful reference simulation or "truth" in conducting future (satellite) data assimilation experiments, with additional atmospheric conditions and data. A formal analytical derivation of the isentropic 1½-layer model from an isentropic two-layer model without convection and precipitation is shown in a companion paper (Part II). [ABSTRACT FROM AUTHOR]

Shallow convective clouds are important players in Earth's energy budget and hydrological cycle, and are abundant in the tropical and subtropical belts. They greatly contribute to the uncertainty in climate predictions due to their unresolved, complex processes that include coupling between the dynamics and microphysics. Analysis of cloud structure can be simplified by considering cloud motions as a combination of moist adiabatic motions like adiabatic updrafts and turbulent motions leading to deviation from adiabaticity. In this work, we study the sizes and occurrence of adiabatic regions in shallow cumulus clouds during their growth and mature stages, and use the adiabatic fraction (AF) as a continuous metric to describe cloud processes and properties from the core to the edge. To do so, we simulate isolated trade wind cumulus clouds of different sizes using the System of Atmospheric Modeling (SAM) model in high resolution (10 m) with the Hebrew University spectral bin microphysics (SBM). The fine features in the clouds' dynamics and microphysics, including small near-adiabatic volumes and a thin transition zone at the edge of the cloud (∼20–40 m in width), are captured. The AF is shown to be an efficient measure for analyzing cloud properties and key processes determining the droplet-size distribution formation and shape during the cloud evolution. Physical processes governing the properties of droplet size distributions at different cloud regions (e.g., core, edge) are analyzed in relation to AF. Significance Statement: 1) This study investigates the evolution of cumulus clouds (Cu) using a 10-m-resolution LES model with spectral bin microphysics. 2) The study improves the understanding of the mutual effects of adiabatic updrafts and lateral entrainment and mixing. 3) The study demonstrates the existence of an adiabatic core in nonprecipitating Cu. 4) Shapes of the droplet size distributions are closely related to the adiabatic fraction values. 5) Utilization of high resolution reveals the existence of physically significant small features in the cloud structure, such as a narrow cloud interface zone and small adiabatic volumes. [ABSTRACT FROM AUTHOR]

We investigate the steady dynamical response of the atmosphere on the equatorial β plane to a steady, localized, midtropospheric heating source. Following Part I, which investigates the case of an equatorial diabatic heating, we explore the sensitivity of the Gill circulation to the latitudinal location of the heating, together with the sensitivity to its horizontal scale. Again, we focus on characteristics of the response that would be particularly important if the circulation interacted with the hydrologic and energy cycles: overturning circulation and low-level wind. In the off-equatorial case, the intensity of the overturning circulation has the same limit as in the equatorial case for small horizontal extent of the diabatic heating, which is also the limit in the f-plane case. The decrease in this intensity with increasing horizontal scale of the diabatic heating is slightly faster in the off-equatorial case than in the equatorial case, which is due to the increase of rotational winds at the expense of divergent winds. The low-level westerly jet is more intense than in the equatorial case, with larger maximum wind and eastward mass transport that tend to infinity for small horizontal extent of the diabatic heating. In terms of spatial characteristics, this jet has a similar latitudinal extent as in the equatorial case but, unlike in the equatorial case, it extends farther equatorward than poleward of the diabatic-heating center. It also extends farther eastward than in the equatorial case. [ABSTRACT FROM AUTHOR]

We investigate the steady dynamical response of the atmosphere on the equatorial β plane to a steady, localized, midtropospheric heating source at the equator. Expanding Gill's seminal work, we vary the latitudinal and longitudinal scales of the diabatic heating pattern while keeping its total amount fixed. We focus on characteristics of the response that would be particularly important if the circulation interacted with the hydrologic and energy cycles: the overturning circulation and the low-level wind. In the limit of very small scale in either the longitudinal or latitudinal direction, the vertical energy transport balances the diabatic heating and this sets the intensity of the overturning circulation. In this limit, a fast low-level westerly jet is located around the center of diabatic heating. With increasing longitudinal or latitudinal scale of the diabatic heating, the intensity of the overturning circulation decreases and the low-level westerly jet decreases in maximum velocity and spatial extent relative to the spatial extent of this heating. The associated low-level eastward mass transport decreases only with increasing longitudinal scale. These results suggest that moisture-convergence feedbacks will favor small-scale equatorial convective disturbances while surface-heat-flux feedbacks would favor small-scale disturbances in mean westerlies and large-scale disturbances in mean easterlies. Part II investigates the case of off-equatorial heating. [ABSTRACT FROM AUTHOR]

A large fraction of tropical cyclones (TCs) are generated near the intertropical convergence zone (ITCZ), and barotropic instability of the related wind shear has been shown to be an important generation mechanism. The latitudinal position of the ITCZ shifts seasonally and may shift poleward in response to global warming. Aquaplanet GCM simulations have shown TC-generation frequency to vary with position of the ITCZ. These results, and that moisture plays an essential role in the dynamics, motivate the present study on the growth rates of barotropic instability in ITCZ-like zonal wind profiles. Base-state zonal wind profiles are generated by applying a prescribed forcing (representing zonally averaged latent heat release in the ITCZ) to a shallow-water model. Shifting the latitudinal position of the forcing alters these profiles, with a poleward shift leading to enhanced barotropic instability. Next, an examination of how latent release impacts the barotropic breakdown of these profiles is considered. To do this, moisture is explicitly represented using a tracer variable. Upon supersaturation, precipitation occurs and the related latent heat release is parameterized as a mass transfer out of the dynamically active layer. Whether moisture serves to enhance or reduce barotropic growth rates is found to depend on how saturation humidity is represented. In particular, taking it to be constant or a function of the layer thickness (related to temperature) leads to a reduction, whereas taking it to be a specified function of latitude leads to an enhancement. Simple arguments are given to support the idea that moisture effects should lead to a reduction in the moist shallow-water model and that a poleward shift of the ITCZ should lead to an enhancement of barotropic instability. [ABSTRACT FROM AUTHOR]

A low-dimensional dynamical system that describes dynamical variability of the stratospheric polar vortex is presented. The derivation is based on a linearized, contour-dynamics representation of quasigeostrophic shallow-water flow on a polar f plane. The model consists of a single linear wave mode propagating on a near-circular patch of constant potential vorticity (PV). The PV jump at the vortex edge serves as an additional degree of freedom. The wave is forced by surface topography, and interacts with the vortex through a simplified parameterization of diabatic wave–mean flow interaction. The approach can be generalized to other geometries. The resulting three-component system depends on four nondimensional parameters, and the structure of the steady-state solutions can be determined analytically in some detail. Despite its extreme simplification, the model exhibits variability that is closely analogous to the Holton–Mass model, a well-known and more complex dynamical model of stratospheric variability. The present model exhibits two stable steady solutions, one consisting of a strong vortex with a small amplitude wave and the second consisting of a weak vortex with a large amplitude wave. Periodic and aperiodic limit cycles are also identified, analogous to similar solutions in the Holton–Mass model. Model trajectories also exhibit a number of behaviors that have been identified in observations. A key insight is that the time-mean state of the vortex is predominantly controlled by the properties of the linear mode, while the strength of the topographic forcing plays a far weaker role away from bifurcations. [ABSTRACT FROM AUTHOR]

A yet unexplained feature of the tropical wavenumber–frequency spectrum is its parity distribution, i.e., the distribution of power between the meridionally symmetric and antisymmetric components of the spectrum. Due to the linearity of the decomposition to symmetric and antisymmetric components and the Fourier analysis, the total spectral power equals the sum of the power contained in each of these two components. However, the spectral power need not be evenly distributed between the two components. Satellite observations and reanalysis data provide ample evidence that the parity distribution of the tropical wavenumber–frequency spectrum is biased toward its symmetric component. Using an intermediate-complexity model of an idealized moist atmosphere, we find that the parity distribution of the tropical spectrum is nearly insensitive to large-scale forcing, including topography, ocean heat fluxes, and land–sea contrast. On the other hand, we find that a small-scale (stochastic) forcing has the capacity to affect the parity distribution at large spatial scales via an upscale (inverse) turbulent energy cascade. These results are qualitatively explained by considering the effects of triad interactions on the parity distribution. According to the proposed mechanism, any bias in the small-scale forcing, symmetric or antisymmetric, leads to symmetric bias in the large-scale spectrum regardless of the source of variability responsible for the onset of the asymmetry. As this process is also associated with the generation of large-scale features in the tropics by small-scale convection, the present study demonstrates that the physical process associated with deep convection leads to a symmetric bias in the tropical spectrum. [ABSTRACT FROM AUTHOR]

In this study, the tangent linear and adjoint (TL/AD) models for the Model for Prediction Across Scales (MPAS) Shallow Water (SW) component are tested and demonstrated. Necessary verification check procedures of TL/AD are included to ensure that the models generate correct results. The TL/AD models are applied to calculate the singular vectors (SVs) with a 48-h optimization time interval (OTI) under both the quasi-uniform-resolution (UR) and smoothly variable-resolution (VR) meshes in the cases of Hurricanes Sandy (2012) and Joaquin (2015). For the global domain, the VR mesh with 30 210 grid cells uses slightly fewer computational resources than the UR mesh with 40 962 cells. It is found that at the points before Hurricanes Sandy and Joaquin made sharp turns, the leading SV from the VR experiment show sensitivities in both areas surrounding the hurricane and those relatively far away, indicating the significant impacts from the environmental flows. The leading SVs from the UR experiments are sensitive to only areas near the storm. Forecasts by the nonlinear SW model demonstrate that in the VR experiment, Hurricane Sandy has a northwest turn similar to the case in the real world while the storm gradually disappeared in the UR experiment. In the case of Hurricane Joaquin, the nonlinear forecast with the VR mesh can generate a track similar to the best track, while the storm became falsely dissipated in the forecast with the UR mesh. These experiments demonstrate, in the context of SW dynamics with a single layer and no physics, the track forecasts in the cases of Hurricanes Sandy and Joaquin with the VR mesh are more realistic than the UR mesh. The SV analyses shed light on the key features that can have significant impacts on the forecast performances. [ABSTRACT FROM AUTHOR]

The Madden–Julian oscillation (MJO) is a large-scale eastward-moving system that dominates tropical subseasonal perturbations with far-reaching impacts on global weather–climate. For nearly a half century since its discovery, there has not been a consensus on the most fundamental dynamics of the MJO, despite intensive studies with a number of theories proposed. In this study, using a simple analytical approach, we found a solution to the linear equatorial shallow-water equations with momentum damping that resembles a harmonic oscillator. This solution exhibits the key characteristics of the observed MJO: its intraseasonal periodicity at the planetary scale and eastward propagation. In contrast to theories that interpret the MJO as a new mode of variability emerging from the evolution in moisture, our solution emphasizes that the core of the MJO resides in the dynamics without explicit fluctuations in moisture. Moisture still plays a role in supplying energy to the core dynamics of the MJO, and determining the value of the equivalent depth required by the theory. The energy source may come from stochastic forcing in the tropics or from the extratropics. The scale selection for the MJO comes from scale-dependent responses to scale-independent Rayleigh damping. We also demonstrate that the MJO solution introduced here reproduces the observed swallowtail structure and the phase relation between zonal wind and geopotential of the MJO, and the continuum nature of the transition between the MJO and Kelvin waves. Roles of feedback mechanisms in the MJO are also discussed using the same simple mathematical framework. [ABSTRACT FROM AUTHOR]

The interactions between the outflow of a tropical cyclone (TC) and its background flow are explored using a hierarchy of models of varying complexity. Previous studies have established that, for a select class of TCs that undergo rapid intensification in moderate values of vertical wind shear, the upper-level outflow of the TC can block and reroute the environmental winds, thus reducing the shear and permitting the TC to align and subsequently to intensify. We identify in satellite imagery and reanalysis datasets the presence of tilt nutations and evidence of upwind blocking by the divergent wind field, which are critical components of atypical rapid intensification. We then demonstrate how an analytical expression and a shallow water model can be used to explain some of the structure of upper-level outflow. The analytical expression shows that the dynamic high inside the outflow front is a superposition of two pressure anomalies caused by the outflow’s deceleration by the environment and by the environment’s deceleration by the outflow. The shallow water model illustrates that the blocking is almost entirely dependent upon the divergent component of the wind. Then, using a divergent kinetic energy budget analysis, we demonstrate that, in a full-physics TC, upper-level divergent flow generation occurs in two phases: pressure driven and then momentum driven. The change happens when the tilt precession reaches left of shear. When this change occurs, the outflow blocking extends upshear. We discuss these results with regard to prior severe weather studies. [ABSTRACT FROM AUTHOR]

This study considers a two-layer fluid with constant density in each layer connected by a layer of continuously varying density for flows past topography in which hydraulic jumps with lee vortices are expected based on shallow-water theory. Numerical integrations of the Navier–Stokes equations at a Reynolds number high enough for a direct numerical simulation of turbulent flow allow an examination of the internal mechanics of the turbulent leeside hydraulic jump and how this mechanics is related to lee vortices. Analysis of the statistically steady state shows that the original source of lee-vortex vertical vorticity is through the leeside descent of baroclinically produced spanwise vorticity associated with the hydraulic jump. This spanwise vorticity is tilted to the vertical at the spanwise extremities of the leeside hydraulic jump. Turbulent energy dissipation in flow through the hydraulic jump allows this leeside vertical vorticity to diffuse and extend downstream. The present simulations also suggest a geometrical interpretation of lee-vortex potential-vorticity creation, a concept central to interpretations of lee vortices based on the shallow-water equations. [ABSTRACT FROM AUTHOR]

Strong eastward jets at the equator have been observed in many planetary atmospheres and simulated in numerical models of varying complexity. However, the nature of the transition from a conventional state of the general circulation, with easterlies or weak westerlies in the tropics, to such a superrotating state remains unclear. Is it abrupt or continuous? This question may have far-reaching consequences, as it may provide a mechanism for abrupt climate change in a planetary atmosphere, both through the loss of stability of the conventional circulation and through potential noise-induced transitions in the bistability range. We study two previously suggested feedbacks that may lead to bistability between a conventional and a superrotating state: the Hadley cell feedback and a wave–jet resonance feedback. We delineate the regime of applicability of these two mechanisms in a simple model of zonal acceleration budget at the equator. Then we show using numerical simulations of the axisymmetric primitive equations that the wave–jet resonance feedback indeed leads to robust bistability, while the bistability governed by the Hadley cell feedback, although observed in our numerical simulations, is much more fragile in a multilevel model. [ABSTRACT FROM AUTHOR]

The effects of moisture on the energetics of a statistically stationary, baroclinically unstable jet representing the midlatitude atmosphere are examined using a two-layer, β-plane shallow-water model. Flow is driven by a relaxation of the interface between the two layers to a baroclinically unstable profile. Moisture is input to the lower layer by evaporation. When supersaturation occurs, precipitation is triggered and the related latent heat release drives a mass transfer between the two layers. A comparison between dry and moist reference atmospheres shows that precipitation reduces eddy kinetic energy. This is related to the meridional distribution of precipitation, which occurs on the poleward side of the jet (where the interface field is raised). This latitudinal structure of precipitation is related to a correlation between poleward flow and ascent, which is analyzed using a shallow-water analog to the ω equation. The precipitation effect on the energy budget is predominately due to zonal- and time-averaged terms. Because of this, dry simulations in which the thermal forcing is modified to mimic the effect of zonally averaged precipitation are carried out and compared with their precipitating counterparts. These simulations show a similar reduction of baroclinic eddy kinetic energy; however, the barotropic eddy kinetic energy response shows a larger difference. [ABSTRACT FROM AUTHOR]

This paper investigates the coupling between the rotational and divergent circulations aiming to explain the observations that show that the tropical eddy momentum flux is due to correlations between divergent eddy meridional velocities and rotational eddy zonal velocities. A simple linear model in which the observed eddy divergence field is used to force the vorticity equation can reproduce quite well the observed tropical eddy momentum flux. The eddy momentum flux in the model shows little sensitivity to the basic-state winds and is mainly determined by the characteristics of the divergent forcing. Vortex stretching and divergent advection of planetary vorticity produce eddy momentum flux contributions with the same sign but the former forcing dominates. It is shown that the main factor affecting the direction of the eddy momentum flux response to both forcings is the meridional tilt of the divergence phase lines, albeit with an opposite sign to the classical relation between rotational momentum flux and streamfunction phase tilt. How this divergent structure is determined remains an open question. [ABSTRACT FROM AUTHOR]

The Asian monsoon anticyclone, which develops in the upper troposphere and lower stratosphere during boreal summer, exhibits significant subseasonal variability with a characteristic spatial structure. The dynamics of this variability is investigated using a nonlinear β-plane shallow-water model. The equivalent depth is estimated using reanalysis data to relate the three-dimensional dynamics in isentropic coordinates to the shallow-water model. Composite analysis reveals the resemblance of the horizontal structures between the Montgomery streamfunction and thickness on the 360-K level. However, the coefficients of the linear regressions between those two variables are strongly dependent on latitude. The estimated equivalent depths of the northern region are more than 2 times greater than those of the southern region. This is attributable to the background thermal structure around the tropopause. Based on this, a latitude-dependent mean depth is incorporated into the shallow-water model to numerically investigate responses to a steady localized forcing in the subtropics. With the inclusion of the latitudinal dependence of the mean depth, the vortex shedding state is able to have a longitudinally confined structure, which differs from the conventional case of constant mean depth. The spatial structure of this numerical solution corresponds to the observed structure, in which low-PV air is largely confined to finite longitudes within the Asian monsoon anticyclone. This suggests the possible role of dynamical instability and the interaction with the subtropical jet in determining the characteristic structure of the Asian monsoon anticyclone. [ABSTRACT FROM AUTHOR]

An instability involving the resonant interaction of a Rossby wave and a Kelvin wave has been proposed to drive equatorial superrotation in planetary atmospheres with a substantially smaller radius or a smaller rotation rate than Earth, that is, with a large thermal Rossby number. To pursue this idea, this paper investigates the equilibration mechanism of Kelvin--Rossby instability by simulating the unforced initial-value problem in a shallow-water model and in a multilevel primitive equation model. Although the instability produces equatorward momentum fluxes in both models, only the multilevel model is found to superrotate. It is argued that the shortcoming of the shallow-water model is due to its difficulty in representing Kelvin wave breaking and dissipation, which is crucial for accelerating the flow in the tropics. In the absence of dissipation, the zonal momentum fluxed into the tropics is contained in the eddy contribution to the mass-weighted zonal wind rather than the zonal-mean zonal flow itself. In the shallow-water model, the zonal-mean zonal flow is only changed by the eddy potential vorticity flux, which is very small in our flow in the tropics and can only decelerate the flow in the absence of external vorticity stirring. [ABSTRACT FROM AUTHOR]

Zonally long tropical waves in the presence of a large-scale meridional and vertical overturning circulation are studied in an idealized model based on the intraseasonal multiscale moist dynamics (IMMD) theory. The model consists of a system of shallow-water equations describing barotropic and first baroclinic vertical modes coupled to one another by the zonally symmetric, time-independent background circulation. To isolate the effects of the meridional circulation alone, an idealized background flow is chosen to mimic the meridional and vertical components of the flow of the Hadley cell; the background flow meridionally converges and rises at the equator. The resulting linear eigenvalue problem is a generalization of the long-wave-scaled version of Matsuno’s equatorial wave problem with the addition of meridional and vertical advection. The results demonstrate that the meridional circulation couples equatorially trapped baroclinic Rossby waves to planetary, barotropic free Rossby waves. The meridional circulation also causes the Kelvin wave to develop an equatorially trapped barotropic component, imparting a westward-tilted vertical structure to the wave. The total energy of the linear system is positive definite, so all waves are shown to be neutrally stable. A critical layer exists at latitudes where the meridional background flow vanishes, resulting in a minimum frequency cutoff for physically feasible waves. Therefore, linear Matsuno waves with periods longer than the vertical transport time of the meridional circulation do not exist in the equatorial waveguide. This implies a low-frequency cutoff for long equatorial waves. [ABSTRACT FROM AUTHOR]

Analytic results and numerical experimentation reveal that a “backward” integration of the adjoint of the shallow water system linearized about a basic state at rest on an f plane is characterized by a radiation of gravity wave–like structures and the emergence of a steady adjoint state. The earlier adjoint states are linked to the prescribed adjoint state (i.e., the adjoint forcing) through the locally conserved dynamical adjoint variables of the shallow water system: the sensitivity to “balanced height” ( ) and the sensitivity to potential vorticity (PV) . The sensitivity to balanced height is determined by the prescribed adjoint sensitivity forcings for the flow, and , and height (fluid depth), : − (g/f)( ). The sensitivity to PV is diagnosed from the inversion of an elliptic operator relating the sensitivity to PV to the distribution of . The sensitivity to PV determines the long-time (t → ∞), steady behavior of the adjoint sensitivity to height, = −(f/H2). In the vicinity of the initial adjoint forcing, the long-time, steady-state behavior of the adjoint system (linearized about a state at rest) is characterized by nondivergent sensitivities to the flow that resemble geostrophic balance: = (1/H) /∂y and = (1/H) /∂x. The process by which this long-time, nondivergent, adjoint state emerges is termed adjoint adjustment. For the system considered, sensitivities to the ageostrophic and irrotational components of the flow vanish for the adjusted state near the prescribed adjoint forcing. [ABSTRACT FROM AUTHOR]

In the present study a simplified multiscale atmosphere-ocean coupled model for the tropical interactions among synoptic, intraseasonal, and interannual scales is developed. Two nonlinear equatorial β-plane shallow-water equations are considered: one for the ocean and the other for the atmosphere. The nonlinear terms are the intrinsic advective nonlinearity and the air-sea coupling fluxes. To mimic the main differences between the fast atmosphere and the slow ocean, suitable anisotropic multispace/multitime scalings are applied, yielding a balanced synoptic-intraseasonal-interannual-El Niño (SInEN) regime. In this distinguished balanced regime, the synoptic scale is the fastest atmospheric time scale, the intraseasonal scale is the intermediate air-sea coupling time scale (common to both fluid flows), and El Niño refers to the slowest interannual ocean time scale. The asymptotic SInEN equations reveal that the slow wave amplitude evolution depends on both types of nonlinearities. Analytic solutions of the reduced SInEN equations for a single atmosphere-ocean resonant triad illustrate the potential of the model to understand slow-frequency variability in the tropics. The resonant nonlinear wind stress allows a mechanism for the synoptic-scale atmospheric waves to force intraseasonal variability in the ocean. The intraseasonal ocean temperature anomaly coupled with the atmosphere through evaporation forces synoptic and intraseasonal atmospheric variability. The wave-convection coupling provides another source for higher-order atmospheric variability. Nonlinear interactions of intraseasonal ocean perturbations may also force interannual oceanic variability. The constrains that determine the establishment of the atmosphere-ocean resonant coupling can be viewed as selection rules for the excitation of intraseasonal variability (MJO) or even slower interannual variability (El Niño). [ABSTRACT FROM AUTHOR]

The Martian polar atmosphere is known to have a persistent local minimum in potential vorticity (PV) near the winter pole, with a region of high PV encircling it. This finding is surprising, since an isolated band of PV is barotropically unstable, a result going back to Rayleigh. Here the stability of a Mars-like annular vortex is investigated using numerical integrations of the rotating shallow-water equations. The mode of instability and its growth rate is shown to depend upon the latitude and width of the annulus. By introducing thermal relaxation toward an annular equilibrium profile with a time scale similar to that of the instability, a persistent annular vortex with similar characteristics as that observed in the Martian atmosphere can be simulated. This time scale, typically 0.5-2 sols, is similar to radiative relaxation time scales for Mars's polar atmosphere. The persistence of an annular vortex is also shown to be robust to topographic forcing, as long as it is below a certain amplitude. It is therefore proposed that the persistence of this barotropically unstable annular vortex is permitted owing to the combination of short radiative relaxation time scales and relatively weak topographic forcing in the Martian polar atmosphere. [ABSTRACT FROM AUTHOR]

The seasonality of the Madden-Julian oscillation (MJO) is documented in observational data, and a nonlinear shallow-water model is used to help interpret some of the contrasts in MJO structure between the boreal winter season [November-March (NDJFM)] and the Asian summer monsoon period [June-September (JJAS)]. At upper-tropospheric levels, the flanking Rossby waves remain centered around 28°N/S year-round, but they tend to be stronger in the winter hemisphere, where the climatological-mean jet stream is stronger, rendering the subtropical circulation more sensitive to forcing by a near-equatorial heat source. Amplitudes of the MJO-related deep convection and lower-tropospheric zonal wind are stronger in the summer hemisphere, where the column-integrated water vapor is larger. During NDJFM, the equatorial asymmetry is subtle: as in the annual mean, moisture convergence into swallowtail-shaped regions of enhanced deep convection is an integral part of the equatorial Rossby wave signature, and the eastward propagation is due to moistening of the air to the east of the enhanced convection by poleward moisture advection. During the Asian summer monsoon in JJAS, the convection assumes the form of northward-propagating, west-northwest-east-southeast-oriented rainbands embedded within cyclonic shear lines. These features are maintained by frictional convergence of moisture, and their northward propagation is mainly due to the presence of features in the climatological-mean fields: that is, the west-east moisture gradient over India and the Arabian Sea and the southwesterly low-level monsoon flow over the northwest Pacific. [ABSTRACT FROM AUTHOR]

A shallow-water model is used to study the role of critical layers in tropical cyclogenesis. Forced and unforced problems of disturbances on a parabolic jet associated with weak basic-state meridional potential vorticity (PV) gradients, leading to Kelvin cat's-eye formation around the jet axis, are first investigated. Numerical simulations with various initial disturbance magnitudes and structures suggest that the results of previous studies can be extended to the next level of complexity toward the more realistic atmosphere. The model is therefore initialized using an observed jet profile obtained from the reanalysis data presented in Part I of this study. For this asymmetric marginally stable basic-state profile, unforced (free) and forced linear integrations show spatial contraction of the perturbation structures in the meridional direction, similar to what occurred in experiments on the parabolic jet. Nonlinear free simulations highlight the role of nonlinear processes in redistributing PV within the critical-layer region. However, they do not yield a realistic time scale for the formation of the cat's-eye. By including diabatic heating as a mass sink term to represent convective PV generation, the nonlinear forced simulation is found to produce a realistic time scale for cat's-eye formation, and confirms the analytical solution of τe τQ ~ O( ε−1) obtained in Part I. These results highlight the synergic role of the dynamical mechanisms, including wave breaking and PV redistribution within the nonlinear critical layer characterized by weak PV gradients and the thermodynamical mechanisms such as convectively generated PV anomalies in the cat's-eye formation in tropical cyclogenesis. [ABSTRACT FROM AUTHOR]

Giant planet tropospheres lack a solid, frictional bottom boundary. The troposphere instead smoothly transitions to a denser fluid interior below. However, Saturn exhibits a hot, symmetric cyclone centered directly on each pole, bearing many similarities to terrestrial hurricanes. Transient cyclonic features are observed at Neptune's South Pole as well. The wind-induced surface heat exchange mechanism for tropical cyclones on Earth requires energy flux from a surface, so another mechanism must be responsible for the polar accumulation of cyclonic vorticity on giant planets. Here it is argued that the vortical hot tower mechanism, claimed by Montgomery et al. and others to be essential for tropical cyclone formation, is the key ingredient responsible for Saturn's polar vortices. A 2.5-layer polar shallow-water model, introduced by O'Neill et al., is employed and described in detail. The authors first explore freely evolving behavior and then forced-dissipative behavior. It is demonstrated that local, intense vertical mass fluxes, representing baroclinic moist convective thunderstorms, can become vertically aligned and accumulate cyclonic vorticity at the pole. A scaling is found for the energy density of the model as a function of control parameters. Here it is shown that, for a fixed planetary radius and deformation radius, total energy density is the primary predictor of whether a strong polar vortex forms. Further, multiple very weak jets are formed in simulations that are not conducive to polar cyclones. [ABSTRACT FROM AUTHOR]

Instabilities of hurricane-like vortices are studied with the help of a rotating shallow-water model, including the effects of moist convection. Linear stability analysis demonstrates that dominant unstable modes are mixed Rossby-inertia-gravity waves. It is shown that, depending on fine details of the vorticity profile, a wavenumber selection of the instability may operate or not, leading in some cases to an unstable mode with a distinctively maximal growth rate and in other cases to an ensemble of unstable modes with close growth rates. Numerical simulations are performed in order to investigate nonlinear saturation of the instability and to understand the dynamical role of moisture. In agreement with previous studies, the authors confirm axisymmetrization of vorticity in the course of the development of the instability, which induces changes of intensity of the hurricane. In 'dry' simulations, winds are intensified only inside the radius of maximum wind, while the maximum value of the wind decreases. 'Moist precipitating' simulations (with and without evaporation) exhibit a net increase of winds, also at the radius of maximum wind, as compared to the dry simulations. Dynamical effects of moisture on the reorganization of the vortex and on the efficiency of inertia-gravity wave emission are quantified and shown to be considerable. Periodic bursts in the emission of waves related to the development of the unstable modes inside the vortex are evidenced, as well as the appearance of convectively coupled waves in the moist precipitating simulations with evaporation. [ABSTRACT FROM AUTHOR]

Equatorial long-wave theory applies where a small horizontal aspect ratio between meridional and zonal length scales is assumed. In an idealized setting, the theory suggests that (i) meridional wind is small, (ii) geostrophic balance holds in the meridional direction, and (iii) inertio-gravity waves are small in amplitude or 'filtered out.' In this paper a spectral data analysis method is used to quantitatively assess the spatial and temporal scales on which each of these aspects of long-wave dynamics is observed in reanalysis data. Three different perspectives are used in this assessment: primitive variables, characteristic variables, and wave variables. To define each wave variable, the eigenvectors and theoretical wave structures of the equatorial shallow-water equations are used. Evidence is presented that the range of spatial and temporal scales on which long-wave dynamics holds depends on which aspect of the dynamics is considered. For example, while meridional winds are an order of magnitude smaller than zonal winds over only a very narrow range of spatial scales (planetary wavenumber ), an examination of meridional geostrophic balance and inertio-gravity waves indicates long-wave dynamics for a broader range of scales (). A simple prediction is also presented for this range of scales based on physical and mathematical reasoning. [ABSTRACT FROM AUTHOR]

A new formulation of the weak pressure gradient approximation (WPG) is introduced for parameterizing large-scale dynamics in limited-domain atmospheric models. This new WPG is developed in the context of the one-dimensional, linearized, damped, shallow-water equations and then extended to Boussinesq and compressible fluids. Unlike previous supradomain-scale parameterizations, this formulation of WPG correctly reproduces both steady-state solutions and first baroclinic gravity waves. In so doing, this scheme eliminates the undesirable gravity wave resonance in previous versions of WPG. In addition, this scheme can be extended to accurately model the emission of gravity waves with arbitrary vertical wavenumber. [ABSTRACT FROM AUTHOR]

Applicability of the reduced-gravity shallow-water (RGSW) theory to a shallow atmospheric layer capped by an inversion underneath a deep stratified atmosphere over a two-dimensional ridge has been investigated using linear analysis and nonlinear numerical simulations. Two key nondimensional parameters are identified: namely, and , where g′ is the reduced-gravity acceleration; H0 is the RGSW layer depth; and N and U are the buoyancy frequency and wind speed, respectively, in the layer above the inversion. If J and γ are around unity or larger, the response of the RGSW flow over the ridge can be significantly modified by pressure perturbations aloft. Any jumplike perturbations in the RGSW layer rapidly decay while propagating away from the ridge as the perturbation energy radiates into the upper layer. With J and γ much less than unity, RGSW theory is more adequate for describing RGSW flows. In addition, inversion splitting occurs downstream of a jump when , where N i is the buoyancy frequency in the inversion and h m stands for the ridge height. A less stratified upper layer with slower winds in general has less influence on the RGSW flow below and favors the application of the RGSW theory. For a thick inversion ( d), the equivalent RGSW flow depth is approximately given by H + d/2, where H is the depth of the neutral layer below the inversion. [ABSTRACT FROM AUTHOR]

Using an asymptotic expansion, a balance model is derived for the shallow-water equations (SWE) on the equatorial β plane that is valid for planetary-scale equatorial dynamics and includes Kelvin waves. In contrast to many theories of tropical dynamics, neither a strict balance between diabatic heating and vertical motion nor a small Froude number is required. Instead, the expansion is based on the smallness of the ratio of meridional to zonal length scales, which can also be interpreted as a separation in time scale. The leading-order model is characterized by a semigeostrophic balance between the zonal wind and meridional pressure gradient, while the meridional wind υ vanishes; the model is thus asymptotically nondivergent, and the nonzero correction to υ can be found at the next order. Importantly for applications, the diagnostic balance relations are linear for winds when inferring the wind field from mass observations and the winds can be diagnosed without direct observations of diabatic heating. The accuracy of the model is investigated through a set of numerical examples. These examples show that the diagnostic balance relations can remain valid even when the dynamics do not, and the balance dynamics can capture the slow behavior of a rapidly varying solution. [ABSTRACT FROM AUTHOR]

The dynamics of convectively coupled equatorial waves (CCEWs) is analyzed in an idealized model of the large-scale atmospheric circulation. The model is composed of a linear rotating shallow-water system with a variable equivalent height, or equivalent gravity wave speed, which varies in space. This model is based on the hypothesis that moist convection acts to remove convective instability, therefore modulating the equivalent height of a shallow-water system. Asymptotic solutions are derived in the case of a small perturbation around a constant coefficient, which is assumed to be a mean moist equivalent height derived from satellite observations. The first-order solutions correspond to the free normal modes of the linear shallow-water system and the second-order flow is derived solving a perturbation eigenvalue problem. The asymptotic solutions are documented in the case of a zonally varying equivalent height and for wavenumbers and frequencies that are consistent with observations of CCEWs. This analysis shows that the dynamics of the secondary divergence and its impact on the full divergence varies mode by mode. For instance, for a negative equivalent height anomaly, which is interpreted as a moister background, the secondary divergence is nearly in phase with the primary divergence in the case of Kelvin waves-in contrast to mixed Rossby-gravity waves where the secondary divergence acts to attenuate the primary divergence. While highly idealized, the modeled waves share some features with observations, providing a mechanism for the relationship between CCEWs phase speed, amplitude, and horizontal structure. [ABSTRACT FROM AUTHOR]

The interaction of midlatitude eddies and the thermally driven Hadley circulation is studied using an idealized shallow-water model on the rotating sphere. The contributions of the annually averaged differential heating, vertical advection of momentum from a stationary boundary layer, and the gross effect of eddies, parameterized by Rayleigh damping, including a hemispherically asymmetric damping, are examined at steady state. The study finds that the relative dominance of eddies, as quantified by the local Rossby number, is predicted by an effective macroturbulent Hadley circulation Prandtl number Pr. In addition, viscous solutions of the Hadley circulation width and strength, subtropical jet amplitude, and equator-to-pole temperature difference scale as deviations from the respective inviscid solutions. Semianalytic solutions for the steady circulation are derived in the limit of weak eddy dominance (small Pr) as deviations from the respective inviscid solutions. These solutions follow a three-region paradigm: weak temperature gradient at the ascending branch of the Hadley circulation, monotonically decreasing angular momentum at the descending branch, and modified radiative-convective equilibrium at the extratropics. Using the three-region solutions, scaling relations found in the full solutions are reproduced analytically. The weak eddy-dominance solutions diverge from the full solutions as Pr increases and may become invalid for Pr > 1 due to the breakdown of the three-region global circulation structure. The qualitative predictions of the response of the Hadley circulation to heating based on the weak eddy-dominance solutions and Pr are in agreement with the findings of more complex models and the observed atmosphere. [ABSTRACT FROM AUTHOR]

Space-time spectral analysis has been used frequently in studying observational evidence of convectively coupled equatorial waves. Here 23 yr of brightness temperature Tb data and dynamical reanalysis data are analyzed by an appropriate projection onto the meridional basis functions of the β-plane linear shallow-water equations. Evidence of peaks in power along linear equatorial mode dispersion curves in Tb, zonal and meridional wind, divergence, and geopotential spectra are presented. Another feature of all space-time spectra considered is the redness in frequency, zonal wavenumber, and meridional mode number. It is found that spectral peaks in the dynamical variable spectra are largely consistent with linear shallow-water waves, but peaks related to barotropic waves and extratropical storm track activity are also apparent. The convectively coupled wave signals are seen to be confined to the first few meridional basis functions, suggesting a filtering method to reduce noise for these signals. This result also has implications for future modeling and theoretical work. A comparison of the results herein for two different reanalysis products shows only minor differences, adding confidence in the robustness of the results presented. This work implies that any comprehensive theory of tropical convection should explain the ubiquity of moist linear waves as well as the spectral redness with respect to all temporal and horizontal scales. [ABSTRACT FROM AUTHOR]

The authors undertake a detailed analysis of the influence of water vapor condensation and latent heat release upon the evolution of the baroclinic instability. The framework consists in a two-layer rotating shallow-water model with moisture coupled to dynamics through mass exchange between the layers due to condensation/precipitation. The model gives all known in literature models of this kind as specific limits. It is fully nonlinear and ageostrophic. The reference state is a baroclinic Bickley jet. The authors first study its 'dry' linear instability and then use the most unstable mode to initialize high-resolution numerical simulations of the life cycle of the instability in nonprecipitating (moisture being a passive tracer) and precipitating cases. A new-generation well-balanced finite-volume scheme is used in these simulations. The evolution in the nonprecipitating case follows the standard cyclonic wave-breaking life cycle of the baroclinic instability, which is reproduced with a high fidelity. In the precipitating case, the onset of condensation significantly increases the growth rate of the baroclinic instability at the initial stages due to production of available potential energy by the latent heat release. Condensation occurs in frontal regions and wraps up around the cyclone, which is consistent with the moist cyclogenesis theory and observations. Condensation induces a clear-cut cyclone-anticyclone asymmetry. The authors explain the underlying mechanism and show how it modifies the equilibration of the flow at the late stages of the saturation of the instability. In spite of significant differences in the evolution, only weak differences in various norms of the perturbations remain between precipitating and nonprecipitating cases at the end of the saturation process. [ABSTRACT FROM AUTHOR]

Persistent superrotation is seen in the atmospheres of other terrestrial bodies (Venus, Titan) but not in that of present Earth, which is distinguished by equatorial easterlies. Nevertheless, superrotation has appeared in numerical simulations of Earth's atmosphere, from two-layer models to multilevel comprehensive GCMs. Simulations of warm climates that generate enhanced tropical convective variability seem particularly prone to superrotation, which has led to hypotheses that the warmer atmospheres of the early Pliocene and Eocene may have been superrotating, and that the phenomenon may be relevant to future climate projections. This paper considers a positive feedback leading to superrotation based on an equatorial wave resonance that occurs in a westerly background flow. The authors present simulations with an idealized multilevel GCM forced with a zonally varying equatorial heating, which show abrupt transitions to strongly superrotating states. Linear shallow water theory is used to show that these transitions occur as the superrotating jet velocity approaches the phase speed of free equatorial Rossby wave modes, leading to a resonant amplification of the response to eddy heating and its associated equatorward momentum flux. The resonance and transition are most prominent in simulations where the meridional temperature gradient has been reduced, and hysteresis behavior is seen when the gradient is eliminated completely. No evidence is found in these simulations for the midlatitude wave feedback believed to drive abrupt transitions in two-layer models, and there is only a minor role for the axisymmetric feedback based on vertical advection by the Hadley circulation. [ABSTRACT FROM AUTHOR]

The fundamental dynamics of 'vortex splitting' stratospheric sudden warmings (SSWs), which are known to be predominantly barotropic in nature, are reexamined using an idealized single-layer f-plane model of the polar vortex. The aim is to elucidate the conditions under which a stationary topographic forcing causes the model vortex to split, and to express the splitting condition as a function of the model parameters determining the topography and circulation. For a specified topographic forcing profile the model behavior is governed by two nondimensional parameters: the topographic forcing height M and a surf-zone potential vorticity parameter Ω. For relatively low M, vortex splits similar to observed SSWs occur only for a narrow range of Ω values. Further, a bifurcation in parameter space is observed: a small change in Ω (or M) beyond a critical value can lead to an abrupt transition between a state with low-amplitude vortex Rossby waves and a sudden vortex split. The model behavior can be fully understood using two nonlinear analytical reductions: the Kida model of elliptical vortex motion in a uniform strain flow and a forced nonlinear oscillator equation. The abrupt transition in behavior is a feature of both reductions and corresponds to the onset of a nonlinear (self-tuning) resonance. The results add an important new aspect to the 'resonant excitation' theory of SSWs. Under this paradigm, it is not necessary to invoke an anomalous tropospheric planetary wave source, or unusually favorable conditions for upward wave propagation, in order to explain the occurrence of SSWs. [ABSTRACT FROM AUTHOR]