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Samuel N. Stechmann and Bjorn Stevens

Abstract

Cumulus clouds involve processes on a vast range of scales—including cloud droplets, turbulent mixing, and updrafts and downdrafts—and it is often difficult to determine how processes on different scales interact with each other. In this article, several multiscale asymptotic models are derived for cumulus cloud dynamics in order to (i) provide a systematic scale analysis on each scale and (ii) clarify the nature of interactions between different scales. In terms of scale analysis, it is shown that shallow cumulus updrafts can be described by balanced dynamics with a balance between source terms and ascent/descent; this is a cloud-scale version of so-called weak-temperature-gradient models. In terms of multiscale interactions, a model is derived that connects these balanced updrafts to the fluctuations within the balanced updraft envelope. These fluctuations describe parcels and updraft pulses, and this model encompasses some of the multiscale aspects of entrainment. In addition to this shallow cumulus model, to provide a broad picture of general cumulus dynamics, multiscale models are also derived for other scales; these include models for parcels and subparcel turbulent mixing and models for deep cumulus. Broadly speaking, the differences between the shallow and deep cases convey the notion that shallow cumulus dynamics are parcel dominated, whereas deep cumulus dynamics are updraft dominated; this is largely due to the difference in the apparent magnitude of the background temperature stratification. In addition to their use in guiding theory, the multiscale models also provide a framework for multiscale numerical simulations.

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Samuel N. Stechmann and Scott Hottovy

Abstract

In the tropics, rainfall is coupled with waves in the form of, for example, convectively coupled equatorial waves (CCEWs) and the Madden–Julian oscillation (MJO). In perhaps the simplest viewpoint of CCEWs, the effects of moisture and convective adjustment can predict the basic aspects of their propagation and structure: reduced propagation speeds and reduced meridional length scales. Here, a similar simple viewpoint is investigated for the MJO’s propagation and structure. To do this investigation, budget analyses of a model MJO are first presented to illustrate and motivate the asymptotic scaling assumptions. Asymptotic models are then derived for the MJO. In brief, the structure of the asymptotic MJO is described by a tropical geostrophic balance, and the slow propagation arises from the dynamics of moist static energy. To be specific, if the moist static energy has a background vertical gradient that is asymptotically weak (i.e., a moist stability that is nearly neutral), then it supports a slowly propagating wave. Beyond these main aspects, other processes also have an influence, such as eddy diffusion of moisture. In comparing the simple viewpoints of CCEWs and the MJO, one main difference is in the propagation speeds: relative to a dry wave speed of 50 m s−1, the MJO has a speed of 5 m s−1, resulting from a reduction factor of 0.1 related to moist stability, whereas the basic CCEW speed is 15 m s−1, resulting from a reduction factor of the square root of 0.1, related to the square root of the moist stability.

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Scott Hottovy and Samuel N. Stechmann

Abstract

A linear stochastic model is presented for the dynamics of water vapor and tropical convection. Despite its linear formulation, the model reproduces a wide variety of observational statistics from disparate perspectives, including (i) a cloud cluster area distribution with an approximate power law; (ii) a power spectrum of spatiotemporal red noise, as in the “background spectrum” of tropical convection; and (iii) a suite of statistics that resemble the statistical physics concepts of critical phenomena and phase transitions. The physical processes of the model are precipitation, evaporation, and turbulent advection–diffusion of water vapor, and they are represented in idealized form as eddy diffusion, damping, and stochastic forcing. Consequently, the form of the model is a damped version of the two-dimensional stochastic heat equation. Exact analytical solutions are available for many statistics, and numerical realizations can be generated for minimal computational cost and for any desired time step. Given the simple form of the model, the results suggest that tropical convection may behave in a relatively simple, random way. Finally, relationships are also drawn with the Ising model, the Edwards–Wilkinson model, the Gaussian free field, and the Schramm–Loewner evolution and its possible connection with cloud cluster statistics. Potential applications of the model include several situations where realistic cloud fields must be generated for minimal cost, such as cloud parameterizations for climate models or radiative transfer models.

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Leslie M. Smith and Samuel N. Stechmann

Abstract

Precipitating versions of the quasigeostrophic (QG) equations are derived systematically, starting from the equations of a cloud-resolving model. The presence of phase changes of water from vapor to liquid and vice versa leads to important differences from the dry QG case. The precipitating QG (PQG) equations, in their simplest form, have two variables to describe the full system: a potential vorticity (PV) variable and a variable M including moisture effects. A PV-and-M inversion allows the determination of all other variables, and it involves an elliptic partial differential equation (PDE) that is nonlinear because of phase changes between saturated and unsaturated regions. An example PV-and-M inversion is provided for an idealized cold-core cyclone with two vertical levels. A key point illustrated by this example is that the phase interface location is unknown a priori from PV and M, and it is discovered as part of the inversion process. Several choices of a moist PV variable are discussed, including subtleties that arise because of phase changes. Boussinesq and anelastic versions of the PQG equations are described, as well as moderate and asymptotically large rainfall speeds. An energy conservation principle suggests that the model has firm physical and mathematical underpinnings. Finally, an asymptotic analysis provides a systematic derivation of the PQG equations, which arise as the limiting dynamics of a moist atmosphere with phase changes, in the limit of rapid rotation and strong stratification in terms of both potential temperature and equivalent potential temperature.

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Andrew J. Majda and Samuel N. Stechmann

Abstract

A minimal, nonlinear oscillator model is analyzed for the Madden–Julian oscillation (MJO) “skeleton” (i.e., its fundamental features on intraseasonal/planetary scales), which includes the following: (i) a slow eastward phase speed of roughly 5 m s−1, (ii) a peculiar dispersion relation with /dk ≈ 0, and (iii) a horizontal quadrupole vortex structure. Originally proposed in recent work by the authors, the fundamental mechanism involves neutrally stable interactions between (i) planetary-scale, lower-tropospheric moisture anomalies and (ii) the envelope of subplanetary-scale, convection/wave activity. Here, the model’s nonlinear dynamics are analyzed in a series of numerical experiments, using either a uniform sea surface temperature (SST) or a warm-pool SST. With a uniform SST, the results show significant variations in the number, strength, and/or locations of MJO events, including, for example, cases of a strong MJO event followed by a weaker MJO event, similar to the Tropical Ocean and Global Atmosphere Coupled Ocean–Atmosphere Response Experiment (TOGA COARE). With a warm-pool SST, MJO events often begin as standing oscillations and then propagate slowly eastward across the warm pool, a behavior imitating MJOs in nature. While displaying the fundamental features of the MJO skeleton, these MJO events had significant variations in their lifetimes and regional extents, and they displayed intense, irregular fluctuations in their amplitudes. The model reproduces all of these features of the MJO skeleton without including mechanisms for the MJO’s “muscle,” such as refined vertical structure and upscale convective momentum transport from subplanetary-scale convection/waves. Besides these numerical experiments, it is also shown that the nonlinear model conserves a total energy that includes a contribution from the convective activity.

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Samuel N. Stechmann and Andrew J. Majda

Abstract

It is known that gravity waves in the troposphere, which are often excited by preexisting convection, can favor or suppress the formation of new convection. Here it is shown that in the presence of wind shear or barotropic wind, the gravity waves can create a more favorable environment on one side of preexisting convection than the other side.

Both the nonlinear and linear analytic models developed here show that the greatest difference in favorability between the two sides is created by jet shears, and little or no difference in favorability is created by wind profiles with shear at low levels and no shear in the upper troposphere. A nonzero barotropic wind (or, equivalently, a propagating heat source) is shown to also affect the favorability on each side of the preexisting convection. It is shown that these main features are captured by linear theory, and advection by the background wind is the main physical mechanism at work. These processes should play an important role in the organization of wave trains of convective systems (i.e., convectively coupled waves); if one side of preexisting convection is repeatedly more favorable in a particular background wind shear, then this should determine the preferred propagation direction of convectively coupled waves in this wind shear.

In addition, these processes are also relevant to individual convective systems: it is shown that a barotropic wind can lead to near-resonant forcing that amplifies the strength of upstream gravity waves, which are known to trigger new convective cells within a single convective system. The barotropic wind is also important in confining the upstream waves to the vicinity of the source, which can help ensure that any new convective cells triggered by the upstream waves are able to merge with the convective system.

All of these effects are captured in a two-dimensional model that is further simplified by including only the first two vertical baroclinic modes. Numerical results are shown with a nonlinear model, and linear theory results are in good agreement with the nonlinear model for most cases.

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Samuel N. Stechmann and Andrew J. Majda

Abstract

The Madden–Julian oscillation (MJO) skeleton model offers a theoretical prediction of the MJO’s structure. Here, a method is described for identifying this structure in observational data. The method utilizes projections onto equatorial wave structures, and a main question is: Can this method isolate the MJO without using temporal filtering or empirical orthogonal functions? For the data projection, a wide range of data is incorporated: multiple variables (wind, geopotential height, water vapor, and, as a proxy for convective activity, outgoing longwave radiation), multiple pressure levels (850 and 200 hPa), and multiple latitudes (both equatorial and off-equatorial). Such a data variety is combined using a systematic method, and it allows for a distinction between the Kelvin and Rossby components of the MJO’s structure. Results are illustrated for some well-known cases, and statistical measures are presented to quantify the variability of the MJO skeleton signal, MJOS(x, t), and its amplitude, MJOSA(t). The robustness of the methods is demonstrated through a suite of sensitivity studies, including tests with two projection methods. When the projection is based on the skeleton model’s energy, as opposed to the standard L 2 energy, water vapor is seen to be of primary importance. Finally, a simple interpretation is given for the MJO skeleton structure: it is related to the wave response to a moving heat source. From either perspective, the methods here identify signals that project onto coupled convection–circulation patterns, and the results suggest that a large portion of the MJO’s structure is consistent with such a coupled pattern.

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Andrew J. Majda and Samuel N. Stechmann

Abstract

It is well known that the envelope of the Madden–Julian oscillation (MJO) consists of smaller-scale convective systems, including mesoscale convective systems (MCS), tropical cyclones, and synoptic-scale waves called “convectively coupled equatorial waves” (CCW). In fact, recent results suggest that the fundamental mechanisms of the MJO involve interactions between the synoptic-scale CCW and their larger-scale environment (Majda and Stechmann). In light of this, this chapter reviews recent and past work on two-way interactions between convective systems—both MCSs and CCW—and their larger-scale environment, with a particular focus given to recent work on MJO–CCW interactions.

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Andrew J. Majda and Samuel N. Stechmann

Abstract

Convective momentum transport (CMT) plays a central role in interactions across multiple space and time scales. However, because of the multiscale nature of CMT, quantifying and parameterizing its effects is often a challenge. Here a simple dynamic model with features of CMT is systematically derived and studied. The model includes interactions between a large-scale zonal mean flow and convectively coupled gravity waves, and convection is parameterized using a multicloud model.

The moist convective wave–mean flow interactions shown here have several interesting features that distinguish them from other classical wave–mean flow settings. First an intraseasonal oscillation of the mean flow and convectively coupled waves (CCWs) is described. The mean flow oscillates due to both upscale and downscale CMT, and the CCWs weaken, change their propagation direction, and strengthen as the mean flow oscillates. The basic mechanisms of this oscillation are corroborated by linear stability theory with different mean flow background states.

Another case is set up to imitate the westerly wind burst phase of the Madden–Julian oscillation (MJO) in the simplified dynamic model. In this case, CMT first accelerates the zonal jet with the strongest westerly wind aloft, and then there is deceleration of the winds due to CMT; this occurs on an intraseasonal time scale and is in qualitative agreement with actual observations of the MJO. Also, in this case, a multiscale envelope of convection propagates westward with smaller-scale convection propagating eastward within the envelope. The simplified dynamic model is able to produce this variety of behavior even though it has only a single horizontal direction and no Coriolis effect.

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Samuel N. Stechmann and J. David Neelin

Abstract

Prototype models are presented for time series statistics of precipitation and column water vapor. In these models, precipitation events begin when the water vapor reaches a threshold value and end when it reaches a slightly lower threshold value, as motivated by recent observational and modeling studies. Using a stochastic forcing to parameterize moisture sources and sinks, this dynamics of reaching a threshold is a first-passage-time problem that can be solved analytically. Exact statistics are presented for precipitation event sizes and durations, for which the model predicts a probability density function (pdf) with a power law with exponent −. The range of power-law scaling extends from a characteristic small-event size to a characteristic large-event size, both of which are given explicitly in terms of the precipitation rate and water vapor variability. Outside this range, exponential scaling of event-size probability is shown. Furthermore, other statistics can be computed analytically, including cloud fraction, the pdf of water vapor, and the conditional mean and variance of precipitation (conditioned on the water vapor value). These statistics are compared with observational data for the transition to strong convection; the stochastic prototype captures a set of properties originally analyzed by analogy to critical phenomena. In a second prototype model, precipitation is further partitioned into deep convective and stratiform episodes. Additional exact statistics are presented, including stratiform rain fraction and cloud fractions, that suggest that even very simple temporal transition rules (for stratiform rain continuing after convective rain) can capture aspects of the role of stratiform precipitation in observed precipitation statistics.

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