Chaotic Convection

By Todd Jones

In the traditional global climate model (GCM) configuration, models simulate atmospheric motions explicitly on spatial grids with spacings on the order of 100 km. Motions on finer scales are not directly simulated. Instead, we use parameterizations, some mathematically simpler, perhaps partially empirical, description of the effects of these motions. To facilitate a harmonious and practical interaction between these scales, they are assumed to be distinct and separate, interacting through time tendencies of temperature and moisture.

Of course, in the real atmosphere, motions occur explicitly in a continuum between and beyond these scales, and it’s unlikely that a specific mean state taken over the area of a typical GCM cell would ever yield exactly the same small-scale motions if that same mean were ever to recur. Instead, it may be similar with some chaotic variability due to the atmosphere’s sensitive dependence on initial conditions. Unfortunately, a deterministic parameterization won’t provide that variability. They’ve historically had no clue about variability on the small scale, the convection that has occurred previously, or how either should influence its representation of convection [Reference 1].

There are clues that these missing pieces are needed for models to produce correct large-scale phenomena, such as the Madden-Julian Oscillation (MJO) or even the stratospheric Quasi-Biennial Oscillation [Refs 2-3], and precipitation statistics, particularly its timing and the occurrence of extreme events [Ref 4]. More recently, there have been efforts to address this deficiency, by adding various representations of small-scale variability and memory to existing convection schemes [Refs 5-7], and one of the remaining questions involves how to represent these ideas correctly in convective parameterizations.

Rather than somewhat arbitrarily stochastically perturbing an existing convective parameterization, some employ the superparameterization (SP) framework [Ref 8; Figure 1]. In SP, the conventional parameterizations are replaced by a cloud permitting model (CPM). A 32-column 2-dimensional curtain on a 4 km grid is placed in each GCM column of the Community Atmosphere Model (CAM). Changes in the state of the GCM column force better-resolved motions within the curtain, while parameterizing the microphysics, radiation, and turbulence on the 4- m grid, that is, at much finer time and space scales that should give more accurate results. Then the CPM reports to the GCM how much precipitation was produced and what temperature and moisture changes resulted from the convective-scale motions.

Figure 1. Schematic representation of the interaction between the global model’s resolved and unresolved scales in CAM, SP-CAM, and the new MP-CAM.

SP-CAM provides individual convective realizations with their sensitive dependence on initial conditions and small-scale structures as well as convective memory, as the convection within the curtain is only initialized at the start of the full simulation, rather than at each GCM time step. We know that it provides an improved solution compared to CAM because of this. What we would like to find out, though, is whether it is possible to create a more deterministic parameterization (like that of CAM) that can retain the benefits of SP-CAM. To this aid in understanding some aspects of this issue, a model was developed for my PhD research at Colorado State University. Shown at the bottom of Figure 1 is the multiple-superparameterization (MP) configuration of the CAM. In MP-CAM we employ 10 CPMs running independently. Each CPM is initialized with different thermal perturbation fields to get things moving, and due to sensitive dependence on initial conditions, they will always be doing something different. In the CPM-domain-mean sense, though, they will remain close together as they each see the same GCM state. Following the CPM computations, their mean column tendencies are averaged in an ensemble sense and passed to the GCM. In this way, the convective effects are more like an “expected mean” that a deterministic parameterization tries to produce and the benefits of simulation at finer scales are retained.

Comparing multi-decadal climate simulations in these frameworks, a number of interesting results emerge [Ref 9]. The models produce slightly different climate features, but of interest to many is the representation of intraseasonal variability (Figure 2). In these wavenumber-frequency power spectra diagrams of outgoing longwave radiation (OLR), we see that MP simulation, with its more smoothed and deterministic representation of the small-scale, shows only slight degradation in the MJO signal (the power peak near eastward wavenumber 1, frequency longer than 30 days). By this estimate, it appears that losing the stochastic nature of the SP tendencies has a negative impact on the result, though one may also reasonably conclude that the bulk of the improvement over the standard CAM is retained, a function of better-resolved motions with convective memory.

Figure 2. Ratios of symmetric spectral power to a smoothed background power for OLR for NOAA observations, CAM, SP-CAM (Control), and MP-CAM (Ensemble). Dispersion curves of the linear shallow water equations are shown in solid black for equivalent depths of 12, 25, and 50 metres. Wave types are Equatorial Rossby (ER), inertio-gravity (IG), and Kelvin.

The nature of the MP approach also allows for study of the range of potential solutions under the same large-scale state. For instance, each CPM produces a different value for grid-cell precipitation, and analysis of that spread can provide insight into the geographic locations and large-scale atmospheric structures that are associated with unpredictable convective precipitation (Figure 3). I encourage those interested in seeing how difficult-to-predict precipitation related to measures of CAPE, atmospheric stability, and critical column water vapour to check out my dissertation [Ref 9] and keep an eye out for two papers currently in preparation for submission to J. Adv. Model. Earth Syst. (JAMES).

Figure 3. Average values across 5 Aprils of CPM-ensemble mean (left) and standard deviation (right). The MP-CAM framework allows for identification of regions of difficult-to-predict precipitation.


[1] Jones, T. R., and D. A. Randall, 2011: Quantifying the limits of convective parameterizations. J. Geophys. Res. Atmos., 116 (D8), doi:10.1029/2010JD014913.

[2] Ricciardulli, L., and R. R. Garcia, 2000: The excitation of equatorial waves by deep convection in the NCAR Community Climate Model (CCM3). J. Atmos. Sci., 57 (21), 3461–3487, doi: 10.1175/1520-0469(2000)057⟨3461:TEOEWB⟩2.0.CO;2.

[3] Neelin, J. D., O. Peters, J. W. B. Lin, K. Hales, and C. E. Holloway, 2008: Rethinking convective quasi-equilibrium: Observational constraints for stochastic convective schemes in climate models. Philos. Trans. R. Soc. A, 366 (1875), 2581–2604, doi:10.1098/rsta.2008.0056.

[4] Li, F., D. Rosa, W. D. Collins, and M. F. Wehner, 2012: “Super-parameterization”: A better way to simulate regional extreme precipitation? J. Adv. Model. Earth Syst., 4 (2), doi:10.1029/ 2011MS000106.

[5] Buizza, R., M. Miller, and T. N. Palmer, 1999: Stochastic representation of model uncertainties in the ecmwf ensemble prediction system. Q.J.R. Meteorol. Soc., 125 (560), 2887–2908, doi: 10.1002/qj.49712556006.

[6] Plant, R. S., and G. C. Craig, 2008: A stochastic parameterization for deep convection based on equilibrium statistics. J. Atmos. Sci., 65 (1), 87–105, doi:10.1175/2007JAS2263.1.

[7] Berner, J., U. Achatz, L. Batté, L. Bengtsson, A.d. Cámara, H.M. Christensen, M. Colangeli, D.R. Coleman, D. Crommelin, S.I. Dolaptchiev, C.L. Franzke, P. Friederichs, P. Imkeller, H. Järvinen, S. Juricke, V. Kitsios, F. Lott, V. Lucarini, S. Mahajan, T.N. Palmer, C. Penland, M. Sakradzija, J. von Storch, A. Weisheimer, M. Weniger, P.D. Williams, and J. Yano, 2017: Stochastic Parameterization: Toward a New View of Weather and Climate ModelsBull. Amer. Meteor. Soc., 98, 565–588,

[8] Khairoutdinov, M., D. Randall, and C. DeMott, 2005: Simulations of the atmospheric general circulation using a cloud-resolving model as a superparameterization of physical processes. J. Atmos. Sci., 62 (7), 2136–2154, doi:10.1175/JAS3453.1.

[9] Jones, T. R. (2017), Examining chaotic convection with super-parameterization ensembles, PhD Dissertation, Colorado State University, Fort Collins, CO.



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