[1] Arakawa, A. 2004 The cumulus parameterization problem: Past, present and
future. J. Climate, 17, 2493–2525.
[2] Arakawa, A. & Schubert, W. H. 1974 Interaction of a cumulus cloud ensemble
with the large-scale environment. Part I. J. Atmos. Sci., 31, 674–701.
[3] Ball, M. A. & Plant, R. S. 2008 Comparison of stochastic parameterization
approaches in a single-column model. Phil. Trans. Roy. Soc. A, 366, 2605–
2623.
[4] Bennett, L. J., Browning, K. A., Blyth, A. M., Parker, D. J. & Clark, P. A. 2006
A review of the initiation of precipitating convection in the United Kingdom.
Quart. J. Roy. Meteorol. Soc., 132, 1001–1020.
[5] Byun, Y.-H. & Hong, S.-Y. 2007 Improvements in the subgrid-scale representation
of moist convection in a cumulus parameterisation scheme: The singlecolumn
test and its impact on seasonal prediction. Mon. Weather. Rev., 135,
2135–2154.
[6] Cohen, B. G. & Craig, G. C. 2006 Fluctuations in an equilibrium convective
ensemble. Part II: Numerical experiments. J. Atmos. Sci., 63, 2005–2015.
[7] Craig, G. C. & Cohen, B. G. 2006 Fluctuations in an equilibrium convective
ensemble. Part I: Theoretical formulation. J. Atmos. Sci., 63, 1996–2004.
[8] Davies, L. 2008 Self organisation of convection as a mechanism for memory.
Ph.D. thesis, Department of Meteorology, University of Reading.
[9] Davoudi, J., McFarlane, N. A. & Birner, T. 2010 Fluctuation of mass flux in
a cloud resolving simulation with interactive radiation. J. Atmos. Sci., 67,
400–418.
[10] Emanuel, K. A. 1993 A cumulus representation based on the episodic mixing
model: The importance of mixing and microphysics in predicting humidity.
In The representation of cumulus convection in numerical models (eds K. A.
Emanuel & D. J. Raymond), vol. 24 of Meteorological Monographs, chap. 19,
pp. 185–192. American Meteorological Society.
[11] Emanuel, K. A. & Bister, M. 1996 Moist convective velocity and buoyancy
scales. J. Atmos. Sci., 53, 3276–3285.
[12] Frenkel, Y., Majda, A. J. & Khouider, B. 2011 Using the stochastic multicloud
model to improve tropical convective parameterization: A paradigm example.
Submitted to: J. Atmos. Sci.
[13] Gerard, L., Piriou, J.-M., Bro˘zkov´a, R. & Geleyn, J.-F. 2009 Cloud and precipitation
parameterization in a meso-gamma-scale operational weather prediction
model. Mon. Wea. Rev., 137, 3960–3977.
[14] Gregory, D. & Rowntree, P. R. 1990 A mass flux convection scheme with representation
of cloud ensemble characteristics and stability-dependent closure.
Mon. Weather Rev., 118, 1483–1506.
[15] Guichard, F., Petch, J. C., Redelsperger, J.-L., Bechtold, P., Chaboureau, J.-P.,
Cheinet, S., Grabowski,W., Grenier, H., Jones, C. G. et al. 2004 Modelling the
diurnal cycle of deep precipitating convection over land with cloud resolving
models and single-column models. Quart. J. Roy. Meteorol. Soc., 130, 3139–
3171.
[16] Kain, J. S. & Fritsch, J. M. 1990 A one-dimensional entraining/detraining
plume model and its application in convective parameterization. J. Atmos.
Sci., 47, 2784–2802.
[17] Khouider, B., Biello, J. & Majda, A. J. 2010 A stochastic multicloud model
for tropical convection. Comm. Math. Sci., 8, 187–216.
[18] Khouider, B., Majda, A. J. & Katsoulakis, A. 2003 Coarse grained stochastic
models for tropical convection and climate. Proc. Nat. Acad. Sci., 100, 11 941–
11 946.
[19] Landau, L. D. & Lifshitz, E. M. 1980 Statistical physics part 1. Elsevier, 3rd
edn. 544pp.
[20] Lin, J.-L., Kiladis, G. N., Mapes, B. E., Weickmann, K. M., Sperber, K. R.,
Lin, W., Wheeler, M. C., Schubert, S. D., Genio, A. D. et al. 2006 Tropical
intraseasonal variability in 14 IPCC AR4 climate models. Part I: Convective
signals. J. Climate, 19, 2665–2690.
[21] Lin, J.-L., Lee, M.-I., Kim, D., Kang, I.-S. & Frierson, D. M. W. 2008 The
impacts of convective parameterization and moisture triggering on AGCMsimulated
convectively coupled equatorial waves. J. Climate, 21, 883–909.
[22] Liu, Y., Guo, L., Wu, G. & Wang, Z. 2010 Sensitivity of ITCZ configuration
to cumulus convective parameterizations on an aqua planet. Clim. Dyn., 34,
223–240.
[23] Lugo, C. A. & McKane, A. J. 2008 Quasicycles in a spatial predator-prey
model. Phys. Rev. E, 78, 051 911.
[24] Majda, A. J., Franzke, C. & Khouider, B. 2008 An applied mathematics perspective
on stochastic modelling for climate. Phil. Trans. Roy. Soc. A, 366,
2429–2455.
[25] Majda, A. J. & Khouider, B. 2002 Stochastic and mesoscopic models for tropical
convection. Proc. Nat. Acad. Sci., 99, 1123–1128.
[26] McKane, A. J., Nagy, J. D., Newman, T. J. & Stefanini, M. O. 2007 Amplified
biochemical oscillations in cellular systems. J. Stat. Phys., 128, 165–191.
[27] McKane, A. J. & Newman, T. J. 2004 Stochastic models in population biology
and their deterministic analogs. Phys. Rev. E, 70, 041 902.
[28] Neelin, J. D., Peters, O., Lin, J. W.-B., Hales, K. & Holloway, C. E. 2008 Rethinking
convective quasi-equilibrium: Observational constraints for stochastic
convective schemes in climate models. Phil. Trans. R. Soc. A, 366, 2579–2602.
[29] Pan, D.-M. & Randall, D. A. 1998 A cumulus parameterization with prognostic
closure. Quart. J. Roy. Meteorol. Soc., 124, 949–981.
[30] Patti, F. D. & Fanelli, D. 2009 Can a microscopic stochastic model explain the
emergence of pain cycles in patients? J. Stat. Mech. P01004.
[31] Peters, O. & Christensen, K. 2002 Rain: Relaxations in the sky. Phys. Rev.
Lett., 66, 1–9.
[32] Plant, R. S. 2010 A review of the theoretical basis for bulk mass flux convective
parameterization. Atmos. Chem. Phys., 10, 3529–3544.
[33] Plant, R. S. & Craig, G. C. 2008 A stochastic parameterization for deep convection
based on equilibrium statistics. J. Atmos. Sci., 65, 87–105.
[34] Plant, R. S. & Yano, J.-I. 2010 Comment on ”An ensemble cumulus convection
parameterization with explicit cloud treatment” by T. M. Wagner and H.-F.
Graf. Submitted to: J. Atmos. Sci.
[35] Randall, D. A. & Huffman, G. J. 1980 A stochastic model of cumulus clumping.
J. Atmos. Sci., 37, 2068–2078.
[36] Randall, D. A. & Pan, D.-M. 1993 Implementation of the Arakawa-Schubert
cumulus parameterization with a prognostic closure. In The representation of
cumulus convection in numerical models (eds K. A. Emanuel & D. J. Raymond),
vol. 24 of Meteorological Monographs, chap. 11, pp. 137–144. American
Meteorological Society.
[37] Raymond, D. J. & Blyth, A. M. 1986 A stochastic mixing model for nonprecipitating
cumulus clouds. J. Atmos. Sci, 43, 2708–2718.
[38] Shutts, G. J. & Gray, M. E. B. 1999 Numerical simulations of convective equilibrium
under prescribed forcing. Quart. J. Roy. Meteorol. Soc., 125, 2767–
2787.
Article submitted to Royal Society
A new modelling framework for statistical cumulus dynamics 19
[39] Shutts, G. J. & Palmer, T. N. 2007 Convective forcing fluctuations in a cloudresolving
model: Relevance to the stochastic parameterization problem. J.
Climate, 20, 187–202.
[40] Stensrud, D. J. 2007 Parameterization schemes: Keys to understanding nu-
merical weather prediction models. Cambridge University Press. 478pp.
[41] Takeuchi, Y. 1996 Global dynamical properties of Lokta-Volterra systems.
World Scientific. 302pp.
[42] Teixeira, J. & Reynolds, C. A. 2008 The stochastic nature of physical parameterisations
in ensemble prediction: A stochastic convection approach. Mon.
Weather. Rev., 136, 483–496.
[43] Tiedtke, M. 1989 A comprehensive mass flux scheme for cumulus parameterization
in large-scale models. Mon. Weather Rev., 117, 1779–1800.
[44] Tompkins, A. 2001 Organisation of tropical convection in low vertical wind
shears: The role of cold pools. J. Atmos. Sci., 58, 1650–1672.
[45] van Kampen, N. G. 2007 Stochastic processes in physics and chenistry. Elsevier,
3rd edn. 463pp.
[46] Wagner, T. M. & Graf, H.-F. 2010 An ensemble cumulus convection parameterization
with explicit cloud treatment. J. Atmos. Sci., 67, 3854–3869.
[47] Xu, K.-M., Arakawa, A. & Krueger, S. K. 1992 The macroscopic behavior of
cumulus ensembles simulated by a cumulus ensemble model. J. Atmos. Sci.,
49, 2402–2420.
[48] Yanai, M., Esbensen, S. & Chu, J.-H. 1973 Determination of bulk properties of
tropical cloud clusters from large-scale heat and moisture budgets. J. Atmos.
Sci., 30, 611–627.
[49] Yano, J.-I., Blender, R., Zhang, C. & Fraedrich, K. 2003 1/f noise and pulselike
events in the tropical atmospheric surface variabilities. Quart. J. Roy.
Meteorol. Soc., 130, 1697–1721.
[50] Yano, J.-I., Guichard, F., Lafore, J.-P., Redelsperger, J.-L. & Bechtold, P. 2004
Estimations of mass fluxes for cumulus parameterizations from high-resolution
spatial data. J. Atmos. Sci., 61, 829–842.
[51] Yano, J.-I. & Plant, R. S. 2011 Finite departure from convective quasiequilibrium:
Periodic cycle and discharge-recharge mechanism. Submitted to:
Quart. J. R. Meteorol. Soc.