Abstract

An approach is presented for the construction of linear response functions of a cumulus ensemble to large-scale temperature and moisture perturbations using a cloud system–resolving model (CSRM). A set of time-invariant, horizontally homogeneous, anomalous temperature and moisture tendencies is added, one at a time, to the forcing of the CSRM. By recording the departure of the equilibrium domain-averaged temperature and moisture profiles from those of a control experiment and through a matrix inversion, a sufficiently complete and accurate set of linear response functions is constructed for use as a parameterization of the cumulus ensemble around the reference mean state represented by the control experiment.

This approach is applied to two different mean state conditions in which the CSRM, when coupled with 2D gravity waves, exhibits interestingly different behaviors. With a more strongly convecting mean state forced by the large-scale vertical velocity profile taken from the Tropical Ocean and Global Atmosphere Coupled Ocean–Atmosphere Response Experiment (TOGA COARE), spontaneous development of convectively coupled waves requires moisture variations above the boundary layer, whereas with a mean state of radiative–convective equilibrium (RCE) not forced by large-scale vertical advection, the development of convectively coupled waves is stronger and persists even when moisture variations above the boundary layer are removed. The linear response functions were able to reproduce these behaviors of the full CSRM with some quantitative accuracy. The linear response functions show that both temperature and moisture perturbations at a range of heights can regulate convective heating. The ability for convection to remove temperature anomalies, thus maintaining convective neutrality, decreases considerably from the lower troposphere to the middle and upper troposphere. It is also found that the response of convective heating to a lower tropospheric temperature anomaly is more top-heavy in the RCE case than in the TOGA COARE case. Comparing the linear response functions with the treatment of convection in an earlier simple model by the present author indicates general consistency, lending confidence that the instability mechanisms identified in that model provide the correct explanation to the instability seen in the CSRM simulations and the instability’s dependence on the mean state.

Abstract

Mock Walker cells driven by weak sea surface temperature (SST) forcing are studied using planetary-scale cloud system–resolving simulations and a simplified framework that represents convection with its linear response functions and parameterizes the large-scale flow based on the gravity wave equation. For sinusoidal SST forcings of the same amplitude, as the horizontal domain size increases, the mock Walker cells strengthen substantially and shorter vertical scales in the vertical velocity profile diminish. This is explained by the fact that temperature anomalies required to sustain a vertical velocity profile of given amplitude are stronger in cases of larger horizontal and smaller vertical scales. Such temperature anomalies become significant at planetary scales so that properly accounting for the horizontal momentum balance, including convective momentum transport (CMT), becomes necessary, while a weak temperature gradient approach that neglects horizontal momentum balance is no longer adequate. The downward advection component of the CMT in particular is important for capturing a number of features of the mock Walker cells. The extent of convective organization also affects the mock Walker cell through its effects on the sensitivities of convective heating and moistening to temperature and moisture anomalies. For strongly organized convection with deep inflows, these sensitivities are consistent with a layer mode of convective overturning, instead of the parcel mode as in unorganized convection, resulting in a weaker second baroclinic component in the mock Walker cells.

Abstract

A simple model of two vertical modes is constructed and analyzed to reveal the basic instability mechanisms of convectively coupled waves. The main novelty of this model is a convective parameterization based on the quasi-equilibrium concept and simplified for a model of two vertical modes. It hypothesizes 1) the approximate invariance of the difference between saturation moist static energy in the lower half of the troposphere and moist static energy in the subcloud layer, regardless of free troposphere humidity, and 2) that variations in the depth of convection are determined by moisture-deficit variations in the midtroposphere. Physical arguments for such a treatment are presented. For realistic model parameters chosen based on cloud system resolving model simulations (CSRMs) of an earlier study, the model produces unstable waves at wavelengths and with structures that compare well with the CSRM simulations and observations.

A moisture–stratiform instability and a direct–stratiform instability are identified as the main instability mechanisms in the model. The former relies on the effect of midtroposphere humidity on the depth of convection. The latter relies on the climatological mean convective heating profile being top heavy, and it is identified to be the same as the stratiform instability mechanism proposed by B. E. Mapes. The moisture–stratiform instability appears to be the main instability mechanism for the convectively coupled wave development in the CSRM simulations. The finite response time of convection has a damping effect on the waves that is stronger at high wavenumbers. The net moistening effect of the second-mode convective heating also damps the waves, but more strongly at low wavenumbers. These effects help to shape the growth rate curve so that the most unstable waves are of a few thousand kilometers in scale.

Abstract

A limited-domain cloud system–resolving model (CSRM) is used to simulate the interaction between cumulus convection and two-dimensional linear gravity waves, a single horizontal wavenumber at a time. With a single horizontal wavenumber, soundings obtained from horizontal averages of the CSRM domain allow the large-scale wave equation to be evolved, and thereby its interaction with cumulus convection is modeled. It is shown that convectively coupled waves with phase speeds of 8–13 m s⁻¹ can develop spontaneously in such simulations. The wave development is weaker at long wavelengths (>∼10 000 km). Waves at short wavelengths (∼2000 km) also appear weaker, but the evidence is less clear because of stronger influences from random perturbations. The simulated wave structures are found to change systematically with horizontal wavelength, and at horizontal wavelengths of 2000–3000 km they exhibit many of the basic features of the observed 2-day waves. The simulated convectively coupled waves develop without feedback from radiative processes, surface fluxes, or wave radiation into the stratosphere, but vanish when moisture advection by the large-scale waves is disabled. A similar degree of vertical tilt is found in the simulated convective heating at all wavelengths considered, consistent with observational results. Implications of these results to conceptual models of convectively coupled waves are discussed. In addition to being a useful tool for studying wave–convection interaction, the present approach also represents a useful framework for testing the ability of coarse-resolution CSRMs and single-column models in simulating convectively coupled waves.

Abstract

Gross moist stability (GMS), a measure of how efficiently divergent flow exports column-integrated moist static energy (MSE), is a widely used quantity in current simplified models of the tropical mean circulation and intraseasonal variabilities such as the Madden–Julian oscillation (MJO), where it is often assumed to be constant. In this paper, it is shown, with cloud-system-resolving model experiments that incorporate feedback from the large-scale flow, that the GMS is smaller at longer wavelengths. The reason for this wavelength dependence is that temperature anomalies required to maintain a given divergent flow increase with wavelength. At long wavelengths, the required temperature anomalies become sufficiently strong to affect the shape of convective heating. As a consequence, the divergent flow is forced to be less top heavy in order to maintain the balance of momentum, heat, and moisture, as well as consistency with the behavior of cumulus convection. A simple model is constructed to illustrate this behavior. Given the ongoing theoretical efforts that view the MJO as resulting from instability in column-integrated MSE, the results presented here provide a planetary-scale selection for such instability, which is absent in current theoretical models that assume a constant GMS.

Abstract

For a linearized system such as ∂ ψ /∂t = 𝗠 ψ , singular vector analysis can be used to find patterns that give the largest or smallest ratios between the sizes of 𝗠 ψ and ψ . Such analyses have applications to a wide range of atmosphere–ocean problems. The resulting singular vectors, however, depend on the norm used to measure the sizes of 𝗠 ψ and ψ , as noted in various applications. This causes complications because the choices of norm are generally nonunique. Based on perturbation theory, a derivation of how singular vectors change with norms typically used in the atmosphere–ocean literature is provided, and it is shown that the norm dependences observed in previous studies can be understood as general properties of singular vectors. This will hopefully clarify the interpretation of these observed norm dependencies, and provide guidance to new studies on how singular vectors would vary for different norms. It is further argued, based on these results, that there may not be as much norm-related ambiguity in problems, such as designing targeted observations or ensemble forecasts, as is often assigned to them.

Abstract

A procedure is presented to systematically construct simple models for the linear stability of moist convecting atmospheres. First, linear response functions of a cumulus ensemble constructed from cloud-system-resolving models are coupled with matrices expressing two-dimensional large-scale linear wave dynamics. For a radiative–convective equilibrium reference state, this model gives two branches of unstable modes: a propagating convectively coupled wave branch and a stationary branch related to storage of column-integrated moist static energy (MSE). The stationary branch is unstable only when radiative feedback is included, while the convectively coupled wave branch is less affected by radiative feedback. With a modular order-reduction procedure from control theory, the linear-response-function-based model is reduced to a system of six ordinary differential equations while still capturing the essential features of the unstable modes (eigenvalues and structures). The six-dimensional system is then split into a slow and a fast manifold. The slow manifold (again, reflecting column MSE storage) is essential for the stationary mode but not for the convectively coupled waves. The fast manifold is then transformed into a form similar to that of prior simple models of convectively coupled waves, thus placing those models and the insights derived from them on a firmer footing. The procedure also better quantifies the parameters of such simple models and allows the stability difference between different reference states to be better understood.

Abstract

Methods in system identification are used to obtain linear time-invariant state-space models that describe how horizontal averages of temperature and humidity of a large cumulus ensemble evolve with time under small forcing. The cumulus ensemble studied here is simulated with cloud-system-resolving models in radiative–convective equilibrium. The identified models extend steady-state linear response functions used in past studies and provide accurate descriptions of the transfer function, the noise model, and the behavior of cumulus convection when coupled with two-dimensional gravity waves. A novel procedure is developed to convert the state-space models into an interpretable form, which is used to elucidate and quantify memory in cumulus convection. The linear problem studied here serves as a useful reference point for more general efforts to obtain data-driven and interpretable parameterizations of cumulus convection.

Abstract

Responses of shallow cumuli to large-scale temperature/moisture perturbations are examined through diagnostics of large-eddy simulations (LESs) of the undisturbed Barbados Oceanographic and Meteorological Experiment (BOMEX) case and a stochastic parcel model. The perturbations are added instantaneously and allowed to evolve freely afterward. The parcel model reproduces most of the changes in the LES-simulated cloudy updraft statistics in response to the perturbations. Analyses of parcel histories show that a positive temperature perturbation forms a buoyancy barrier, which preferentially eliminates parcels that start with lower equivalent potential temperature or have experienced heavy entrainment. Besides the amount of entrainment, the height at which parcels entrain is also important in determining their fate. Parcels entraining at higher altitudes are more likely to overcome the buoyancy barrier than those entraining at lower altitudes. Stochastic entrainment is key for the parcel model to reproduce the LES results. Responses to environmental moisture perturbations are quite small compared to those to temperature perturbations because changing environmental moisture is ineffective in modifying buoyancy in the BOMEX shallow cumulus case.

The second part of the paper further explores the feasibility of a stochastic parcel–based cumulus parameterization. Air parcels are released from the surface layer and temperature/moisture fluxes effected by the parcels are used to calculate heating/moistening tendencies due to both cumulus convection and boundary layer turbulence. Initial results show that this conceptually simple parameterization produces realistic convective tendencies and also reproduces the LES-simulated mean and variance of cloudy updraft properties, as well as the response of convection to temperature/moisture perturbations.

Abstract

An idealized Walker cell with prescribed sea surface temperature (SST) and prescribed radiative cooling is studied using both a two-dimensional cloud-resolving model (CRM) and a simple conceptual model. In the CRM, for the same SST distribution, the width of the warm pool (area of strong precipitation) varies systematically with the magnitude of the radiative cooling, narrowing as radiative cooling is increased. The simple model is constructed to interpret these behaviors. Key aspects of the simple model include a surface wind determined from the boundary layer momentum budget, which in turn sets evaporation assuming a spatially uniform surface relative humidity, prescribed gross moist and dry stratification as a function of column water vapor and precipitation, and a gustiness enhancement on evaporation in areas of precipitation. It is found that the gustiness enhancement, likely due to mesoscale systems, creates a feedback that narrows the warm pool. This process has not been included in previous formulations of the simple model and its role is emphasized here.