It now appears that these climate interactions over land are within reach, as they can be diagnosed in models and in data, and consequently with this understanding the development of an Earth modeling system to represent them is now possible. Are we losing the ability to make essential improvements in model physics because we are more concerned with fine-tuning existing representations? These are different exercises. New physics means the ability to step back and understand what is wrong. This needs a long-range plan and excellent scientific oversight and actually hiring people tasked to do it!
It is easier for understaffed modeling centers to fine-tune. Are we losing the ability and perhaps the will to make critical but often arduous tests of model physics against observations? The real-world link is critical, and too many student projects get lost in virtual reality. Is the emphasis on quantifying model uncertainty diluting efforts to improve model physics?
Yes, quantifying uncertainty, when model interactions between processes are poorly understood, is an illusion. What is the status of and what are the major errors associated with the parameterization of physical processes in atmosphere-land-ocean A-L-O models ranging from local-daily scales to regional-decadal scales? What effects do these errors have on model output compared to other sources of error? Over land it involves the coupling of many processes both at the land surface and in the atmosphere. Energy is transferred by the phase changes of water, both at the surface and in the atmo-.
Precipitation, atmospheric dynamics, and the radiation fields are tightly coupled. At the surface over land, the precipitation, surface hydrometeorology and vegetation, surface and boundary layer fluxes are also tightly coupled. Only with global models can we simulate all the interactions involved, and many processes are parameterized because the range of time and space scales involved is so broad that explicit simulation of them all is not possible. The consequence of this is that the model system must be evaluated carefully to see if the coupling between, say, cloud albedo, boundary layer depth, surface fluxes, and evaporative fraction is properly represented Betts, ; Betts and Viterbo, How can model parameterizations be improved to represent the essential physics in A-L-O models?
How can these parameterizations be tested rigorously, and what supporting infrastructure is needed to do so? Models can be evaluated. What is the appropriate balance between the efforts being directed toward improving physical parameterizations and efforts directed toward other model development and application activities? A good modeling center needs both. The paradigm is straightforward:. Basic research on new representations of physics, in parallel with pragmatic improvements, again tied to data. This appears simple, but very few centers actually complete this cycle e.
Land-atmosphere interactions associated with the carbon cycle can be decomposed into two classes according to the time scales over which they act. Fast ecophysiological processes dominate carbon exchanges on time scales of seconds to years and are strongly coupled to exchanges of water and energy; slow ecological processes are responsible for time-mean sources and sinks of atmospheric CO 2 over scales of decades to centuries.
Gravity waves influence weather and climate -- ScienceDaily
Fast processes are important to model because they help us get exchanges of energy and water right and are relatively well understood. Slow processes dominate uncertainty in future atmospheric CO 2 and are poorly represented in current models. An important strategy in carbon cycle science is to observe variations in carbon compounds in the atmosphere and ocean and use them to understand underlying processes that govern sources and sinks. The trouble is that the fast processes dominate the observations, but the slow processes dominate the future behavior of the Earth system.
How do we model both? They include photosynthesis conversion of atmospheric CO 2 to organic matter , autotrophic respiration, and decomposition of organic matter back into CO 2. They are observed hourly by micrometeorological methods at a network of well over sites around the world. Stomatal physiology enables vegetated plant canopies to modulate the Bowen ratio of surface energy exchange and strongly couples exchanges of carbon at the land surface to exchanges of energy and water.
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Diurnal and seasonal cycles of atmospheric CO 2 are largely explained by these processes and provide strong constraints on their parameterization in climate models. Interannual variability in the fluxes is less well understood and parameterized, especially at regional and larger scales. Important unresolved issues in the parameterization of photosynthesis include 1 representation of physiological stress due to dry soil and dry air; 2 canopy radiative transfer and the effects of direct versus diffuse light on photosynthesis and transpiration; 3 heterogeneity and nonlinear response to drying within grid cell; and 4 management effects such as urban and suburban development, crop fertilization, development, irrigation, and harvest.
Respiration and decomposition are fast processes that provide first-order links to the slow processes. Decomposition is typically parameterized as being directly proportional to the sizes of a set of pools of organic matter e. The initialization problem is compounded by the parameterization of the sensitivity of decomposition to temperature and moisture, which appear not to be universal and are relatively less well constrained by observations relative to influences on photosynthesis.
Incorrect parameterization of initial carbon pools and environmental dependence of respiration rates can then hide errors in models of the slower carbon flux processes that control future atmospheric CO 2. Eddy covariance measurements of surface fluxes of heat, moisture, and CO 2 are tremendously valuable for elucidating the dependence of land-atmosphere exchanges on radiation, temperature, and precipitation soil moisture.
The network of tower sites has unfortunately been oversold as a way to directly observe the time-mean source or sink of carbon, which is probably the weakest aspect of their measurements. The very small footprints observed by this method severely limit their utility for quantifying slow processes that control the carbon balance. These processes are more difficult to observe, or rather to extrapolate to large spatial scales based on limited observations than their fast counterparts.
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To test mechanistic models of changes in slow carbon cycle processes, parameterizations in climate models must be evaluated against observations at spatially-aggregated scales. It is necessary but not sufficient that these parameterizations be evaluated against local data. For example, forward modeling of carbon storage and biomass over time scales of decades following forest fire or harvest must be compared to biometric inventory measurements such as are available for over , plots within the United States.
It is quite likely, however, that a parameterization of ecosystem biogeochemistry, plant competition, and succession could reproduce the broad statistics of these observations and still fail to capture variations at larger spatial scales due to poorly constrained extrapolation. There is also a need, therefore, for predictions made by forward models of these slow processes to be compared quantitatively to integral properties of the coupled system such as changes in atmospheric carbon gases. This is analogous to comparison of predictions made by cloud models to observable quantities at larger than climate model grid scales, or the comparison of spatially-integrated predictions of runoff to discharge from large river basins.
This is a generic requirement for subgrid-scale physical parameterizations in climate models, not a particular requirement for carbon cycle science. There is an emerging consensus in the carbon science community that diagnostic modeling and data assimilation provide a framework for leveraging. This requires treatment of unresolved time and space variations in the fluxes, especially to the extent they are coupled to or covary with transport processes e. Excellent parameterization of fast processes and specification of highly-resolved spatial variability must be included in such calculations, or errors in the space-time patterns are unavoidably aliased into errors in the time-mean, regionally-integrated fluxes through aggregation error.
Other aspects of the physical problem that are highly relevant for this exercise include the parameterization of clouds and planetary boundary layer processes in transport models used for inversions of atmospheric CO 2 as they also show up in other parameterization problems. Cultural issues that hamper progress in carbon cycle parameterization include the traditional divides between observationalists and modelers, as well as perhaps more unique divides between modelers of fast versus slow processes.
Thirty years after the initial publication of a cumulus parameterization including a cloud submodel Arakawa and Schubert, , the representation of convection in atmospheric general circulation models AGCMs at major climate research institutions is problematic. In general, these centers are at best employing the method of Arakawa and Schubert without further advances, and in many cases the methods used are not even at the level of Arakawa and Schubert.
This is despite significant new observational knowledge of convection during that period and substantial success in modeling convective systems using high-resolution models, which can resolve the largest individual deep convective elements and many aspects of organized convective systems.
Cumulus parameterizations in major-center AGCMs generally continue to use as cumulus submodels only convective mass fluxes. Momentum transport is often treated crudely, if at all. There has been over the past several decades only limited research on closure for cumulus parameterizations, despite its central importance and evidence from observations of problems with current approaches, especially at subdiurnal time scales Donner and Phillips, Treatments are absent or extremely limited of interactions between deep convective towers and mesoscale circulations, interactions between convection and boundary layers, and interactions between deep and shallow convection.
In many cases this is despite compelling observational evidence of the importance of these interactions; for example, observational evidence of the importance of mesoscale circulations associated with deep convection has existed for at least 20 years. The sociological reasons for this situation probably can be found by examining the role of convection in AGCM development at major modeling centers.
This development is strongly driven by goals of reducing biases in climate simulations and reducing uncertainty in climate sensitivity to anthropogenic changes in atmospheric composition. Convection and most other physical processes tend to be viewed as tools toward reducing these biases and uncertainties. Since it is often unclear which physical processes are responsible for particular biases and sensitivity uncertainty, there has been a serious lack of focus on improving the fundamental physical soundness of convective parameterizations and an emphasis on tuning physical parameterizations to produce realistic climate simulations.
Recent experience at major modeling centers suggests that the tuning approach with current convective parameterizations may be reaching limits as to its usefulness in attacking model biases. The unphysical nature of the tuning process is also increasingly apparent and unsatisfactory to its practitioners. It is also clear that tuning to past or present climate conditions may not capture future climate change and is thus of limited use as a means of reducing uncertainty in climate sensitivity. There are possibilities for advancing current parameterization capabilities using observations and high-resolution model results, but this will require enhanced efforts.
A particularly promising avenue is emerging with current computational advances. This is a major conceptual advance on past methods, which have required that central controls on the problem be based on closure assumptions, whose very existence has never been established. The embedded convection-resolving models draw on well-established dynamics of convection and rationalize many of the choices on how to treat issues related to cloud submodels.
They have an enormous potential to indicate outstanding research issues that must be addressed, for example, the roles of microphysics and smaller-scale circulations. My remarks are limited to consideration of parameterizations of fluxes at the air-sea interface.
The fluxes of interest include the traditional meteorological forcing momentum, sensible heat, and latent heat , precipitation, and net solar and infrared radiative fluxes. For climate purposes we must expand our considerations to include fluxes of trace gases e. Oceanographic and meteorological energy and buoyancy forcing are made up with different weightings of the basic fluxes.
Precipitation is a critical variable, but in the global climate modeling context is not parameterized in terms of surface variables. Radiative fluxes at the surface do involve surface variables, but the principal source of variability and uncertainty is clouds. Although surface radiative flux parameterizations do exist, again from a climate model point of view the surface radiative flux is viewed as part of the entire atmospheric column problem. For turbulent fluxes meteorological, gas, and particle , the bulk flux model, where fluxes are computed as the product of wind speed, sea-air contrast, and a semi-empirical transfer coefficient, is essentially universally used in GCMs and in higher-resolution models.
In the last decade advances in ship-based measurement technologies and in physically-based formulations of bulk models have resulted in major progress. Meteorological transfer coefficients are now known, on average, to about 5 percent for wind speeds from 0 to 20 m s In the last five years, direct covariance measurements of CO 2 flux from ships have reduced the uncertainty in CO 2 transfer significantly but have also illuminated the importance of presumably wave-breaking processes.
Computations of global mean oceanic CO 2 flux show large sensitivity factor of 2 to the choice of simple wind-speed-based transfer formulations. Direct measurement of particle fluxes is still exploratory, and interpretation of such measurements is uncertain and particle size dependent. A number of parameterization issues for surface fluxes are listed below:. The central theme is that the parameterizations are in good shape in regimes where we have good observations. Note that wave processes on the oceanic and atmospheric sides of the interface dominate this list.
Existing research programs in the United States provide a good venue for attacking many of these issues, but there are major gaps. Research on oceanic gas transfer suffers from fragmented sources of support, but the large, highly organized U. Carbon Cycle program has in my opinion too much emphasis on observations to constrain oceanic PCO 2. Unfortunately, the U. The withdrawal of the Office of Naval Research as a major player in funding ocean wave and particle flux research has also hurt.
Because the gas transfer, particle, and wave problems are all tied together, it makes sense to address these problems in a holistic fashion, which requires an initiative such as SOLAS. Overall, the U. There are clear concerns about the aging workforce and uncertainties about the training of the next generation of flux scientists this is part of a general pattern of a declining population of students interested in getting their hands dirty.
There is another issue of concern to me—that is, the recent trend of changing the emphasis of the national government laboratories toward performance measures, deliverables, and products and away from strategic technology development. The problem is that funds tend to move down agency stove-. Much of the influence of the ocean in climate involves the oceanic uptake and transport of scalar properties such as heat, fresh water, carbon, fluoro-carbons CFCs , and the like.
On daily to decadal time scales, this uptake and transport is controlled by upper ocean processes occurring in the surface mixed-layer and the wind-driven gyres. The turbulent mixing in the surface mixed-layer sets the rate at which properties are exchanged between the ocean and the atmosphere. The wind-driven gyres transport meridionally these properties, once the surface waters are subducted in the interior. In this session we will discuss what the key subgrid-scale processes are, which need to be parameterized to properly simulate the uptake and transport of properties in ocean models used for climate studies.
In present climate models, the ocean horizontal grid resolution is O km or larger, and the vertical grid resolution is tens to a hundred meters. At this resolution the subgrid ocean processes that need to be parameterized can be divided into two categories:. More powerful computers may decrease these scales to a marginal mesoscale eddy resolution of O 25 km in the next 10 years, but horizontal grids of better than O 10 km are needed to adequately resolve the fluxes produced by mesoscale motions. This has elicited a large literature in the last 10 years on parameterization schemes for mesoscale eddies in the oceanic interior and microscale turbulence in the surface mixed layer.
This has not always been the case. One of the most significant problems with early ocean models was the high level of microscale turbulent mixing inherent in the numerics. These high levels of mixing affected the heat transport, stability, and variability of simulated climate in coupled models.
Thus the emphasis in model development was on hydrodynamic codes. New numerical methods now allow scientists to construct models, which can operate with far smaller levels of spurious mixing. Although research continues on improving the hydrodynamic codes, it is clear that modern ocean models suffer more from errors in the parameterization of subgrid-scale motions than from errors in the simulation of resolved hydrodynamic processes. Parameterizations of mesoscale processes in oceanic general circulation models represent the adiabatic release of potential energy by baroclinic instability as well as the stirring and mixing of material tracers along isopycnal surfaces Gent and McWilliams, These quasi-adiabatic conservation properties have led to a series of dramatic improvements in oceanic models.
However, close to the boundaries, eddy fluxes develop a diabatic component, both because of the vigorous microscale turbulence in boundary layers and because eddy motions are constrained to follow the topography or the upper surface, while density surfaces can and often do intersect the boundaries. The dynamics of these diabatic near-boundary fluxes are not well understood, and there is as yet no standard parameterization.
Recently a CPT has been funded to develop new approaches to mesoscale eddy parameterizations at the ocean boundaries, based on better dynamical understanding and analysis of available observations. The physics of microscale turbulence in the oceanic boundary layers is the subject of a vast literature and parameterizations exist. Less is known about microscale turbulence in the ocean interior, and parameterizations are very rudimentary. The difference in development between the two fields has historical and practical reasons. The boundary layer problem benefited from the similarities with the well-developed corresponding atmospheric problem.
Furthermore, surface boundary layers are fairly accessible and observations are available to test the proposed parameterization schemes. The situation is opposite for microscale turbulence in the ocean interior. The physics is very different from the atmospheric case, mostly because of the lack of radiative processes. Observations are very sparse and do not allow careful testing of numerical schemes. Ray Schmitt gives a comprehensive review of the progress being made on these issues. Although it is recognized that the diapycnal mixing coefficient for heat, salt, and tracers is much less than the isopycnal mixing rate, fluxes may actually be larger because vertical gradients are so much larger than horizontal gradients.
Kim, S. Eckermann, and H.
An overview of the past, present and future of gravity-wave drag parameterization for numerical climate and weather prediction models. Atmosphere-Ocean , 41 1 —98, Koren, J. Schaake, K. Mitchell, Q. Duan, F. Chen, and J. A parameterization of snowpack and frozen ground intended for ncep weather and climate models. Koster, Y. Chang, and S. A mechanism for land-atmosphere feedback involving planetary wave structures. Krueger, Q. Fu, K.
Liou, and H-N. Improvement of an ice-phase microphysics parameterization for use in numerical simulations of tropical convection. Journal of Applied Meteorology , —, January Near sea surface temperatures nsst analysis in ncep gfs. Investigation of aerosol indirect effects on simulated moist convections. Lin, R. Farley, and H. Bulk parameterization of the snow field in a cloud model. Climate Appl. Lin, W. Chao, Y. Sud, and G. A class of the van leer-type transport schemes and its application to the moisture transport in a general circulation model.
Turbulence and stress due to gravity wave and tidal breakdown. A numerical experiment on chandrasekhar's discrete-ordinate method for radiative transfer: Applications to cloudy and hazy atmospheres. A new boundary layer mixing scheme. Part I: Scheme description and single-column model tests. An general unified similarity theory for the calculation of turbulent fluxes in the numerical weather prediction models for unstable condition. Office Note , U. An economical and compatible scheme for parameterizing the stable surface layer in the medium-range forecast model.
Lord, H. Willoughby, and J. Role of a parameterized ice-phase microphysics in an axisymmetric, nonhydrostatic tropical cyclone model. Lott and M. A new subgrid-scale orographic drag parametrization: Its formulation and testing. A parametric model of vertical eddy fluxes in the atmosphere. Boundary-Layer Meteorology , —, MacVean and P. Cloud-top entrainment instability through small-scale mixing and its parameterization in numerical models. Journal of the Atmospheric Sciences , 47 8 —, McCormack, S.
Eckermann, D. Siskind, and T. Chem2d-opp: A new linearized gas-phase ozone photochemistry parameterization for high-altitude nwp and climate models. McCormack, K. Hoppel, and D. Parameterization of middle atmospheric water vapor photochemistry for high-altitude nwp and data assimilation.
Milovac, K. Warrach-Sagi, A. Behrendt, F. Spath, J. Ingwersen, and V. Investigation of pbl schemes combining the wrf model simulations with scanning waver vapor differential absorption lidar measurements. Miyakoda and J. Manual of the E-physics. Princeton University Press, Taubman, P. Brown, M. Iacono, and S. Radiative transfer for inhomogenerous atmospheres: Rrtm, a validated correlated-k model for the longwave.
PaiMazumder and J. Potential predictability sources of the u. Palmer, G. Shutts, and R. Alleviation of a systematic westerly bias in circulation and numerical weather prediction model through an orographic gravity wave drag parameterization. Pan and W. Implementing a mass flux convection parameterization package for the nmc medium-range forecast model. Paulson and J. The temperature difference across the cool skin of the ocean. Peters-Lidard, M. Zion, and E. A soil-vegetation-atmosphere transfer sheme for modeling spatially variable water and energy balance processes.
Peters-Lidard, E. Blackburn, X. Liang, and E. The effect of soil thermal conductivity parameterization on surface energy fluxes and temperatures. An essay on the parameterization of orographic wave drag. An empirical technique for diagnosing precipitation type from model output. Fifth Int. On four-dimensional variational assimilation of ozone data in weather-prediction models. Rutledge and P. The mesoscale and microscale structure and organization of clouds and precipitation in mid-latitude cyclones. The mesoscale and microscale structure and organization of clouds and precipitation in midlatitude cyclones.
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Stratospheric aerosol optical depth, Schuur, H. Park, A. Ryzhkov, and H. Classification of precipitation types during transitional winter weather using the ruc model and polarimetric radar retrievals. Journal of Applied Meteorology and Climatology , —, Pier Siebesma, Pedro M. Soares, and Joao Teixeira. A combined eddy-diffusivity mass-flux approach for the convective boundary layer. Soares, P.
Miranda, A. Siebesma, and J. Soloviev and N. The vertical structure of the thin surface layer of the ocean under conditions of low wind speed. Deep Sea Research Part A. Oceanographic Research Papers , 29 12 —, Sundqvist, E. Berge, and J. Condensation and cloud studies with a mesoscale numerical weather prediction model. Physically-based modeling and simulation of climate and climatic changes, Part I , chapter Parameterization of condensation and associated clouds in models for weather prediction and general circulation simulation, pages — Schlesinger, Ed.
A simple model of the atmospheric boundary layer; sensitivity to surface evaporation. Boundary-Layer Meteorology , 37 —, Untch, A. Simmons, M. Hortal, and C. Increased stratospheric resolution in the ecmwf forecasting system. Netherlands, The Warner-McIntyre scheme uses an empirically-based saturation condition that limits the energy in the tail of the spectrum.
Role of gravity waves in vertical coupling during sudden stratospheric warmings
The momentum flux spectra imposed near the tropopause level was prescribed to be identical in the two schemes. There were systematic differences between the performance of the two schemes, with much more of the wave spectrum being removed lower down in the atmosphere by the WMP than the DSP.
The profiles of flux and flux divergence computed by the two schemes becomes similar when the saturation fluxes for the WMP are raised by a factor of 25 over their standard values. An important development described in several talks at the workshop has been the systematic application of known constraints on the mean flow forcing to adjust aspects of parameterisations. Alexander presented calculations in which the input spectrum of momentum flux versus horizontal phase speed for the Alexander-Dunkerton parameterisation ADP is adjusted to account for the needed gravity wave mean-flow forcing in the middle atmosphere determined through analysis of large-scale observations.
The results were somewhat different in midlatitudes and the tropics, with a broader phase speed spectrum indicated for low latitudes. Ortland discussed a systematic approach to the inverse problem of finding input spectra in the ADP that can reproduce the required wave drag as determined from simulations of the middle atmospheric circulation obtained with a two-dimensional zonally-averaged model. Vincent and P.
Love discussed the use of mesospheric radar measurements of the winds near the equator to constrain the tropospheric input spectra employed in a ray-tracing model with sources associated with regions of convection as seen in satellite imagery. It appears that, with appropriate assumptions about the source spectra, much of the observed mean-flow forcing inferred from wind observations in the equatorial mesosphere can be explained by tropospheric convective sources.
One issue that is involved in such adjustment of parameterisations is the determination of how much of the mean flow forcing is attributable to gravity waves versus the motions that should be resolved in current climate models or current global observational analyses. This determination has typically been based on monthly-mean data used to infer the Coriolis and advective effects of the residual mean meridional circulation along with some other observational estimate for the contributions of planetary waves. In principle, a more satisfying approach might be based on the analysis increments obtained in data assimilation procedures within a forecast-analysis cycle.
Tan discussed this issue but noted that the inadequacy of current data sources and assimilations may severely limit the utility of this approach, at least at present. Beres discussed linear theory results for the gravity wave field forced by localised transient heating. She then used these results as the basis for a practical scheme to determine the input spectrum appropriate for convective forcing. Her approach basically takes the grid-scale latent forcing computed in the convective parameterisation and assumes some subgrid-scale structure for the heating.
Then the linear theory is used to obtain a source spectrum for the gravity wave parameterisation that depends on the grid-scale heating and the resolved horizontal winds. This is a rational way to begin to consider the effects of variability of convection in the gravity wave parameterisation problem. Chun discussed another parameterisation for convective gravity waves that also related the source spectrum to grid-scale winds and convective heating. Song discussed results with this source spectrum determination incorporated into Lindzen and Warner-McIntyre parameterisations.
Shaw discussed the issue of how gravity wave drag parameterisations are affected by the effective truncation of the model domain at some finite altitude that is inherent in the numerical discretisation. She showed that the residual meridional circulation differs dramatically if the parameterised gravity wave fluxes that reach the model top are assumed to be absorbed at the top level or are simply neglected. It seems that, in terms of simulating the residual circulation, assuming that the flux is absorbed at the top will lead to a result much closer to what would be obtained by explicitly including a very high model domain.
In particular, she investigated the issue of interhemispheric asymmetry in summer mesopause temperatures. Giorgetta discussed results with a version of the ECHAM model showing that a combination of resolved equatorial waves and an appropriately tuned gravity wave parameterisation could allow the model to produce a quite realistic quasi-biennial oscillation QBO of the tropical stratosphere.
He then used the model to investigate the effect of changed carbon dioxide concentrations on the QBO. A number of papers dealt with detailed simulations of gravity wave generation and propagation in high-resolution limited-area models. Horinouchi discussed 3D cloud-resolving simulations of gravity waves forced by convection in a tropical squall line.
He showed that the model can convincingly simulate the entire life cycle of convectively-forced waves: generation, propagation through the middle atmosphere, and nonlinear breakdown near the mesopause. He showed that his simulated meteorological fields could be used as the basis for a calculation of airglow emission, thus allowing a direct comparison of his model results near the mesopause with airglow imager observations.
Two papers dealt with model studies of the convectively-generated gravity waves during this experiment. Alexander discussed the wave field computed for one day in DAWEX using a dry model forced with time-dependent, 3D heating fields based on detailed meteorological radar observations of precipitation. The radar can be expected to give a good estimate of the overall space-time evolution of the pattern of precipitation, but J.
Alexander notes some uncertainty in overall amplitudes. Stenchikov described a simulation of the circulation and convection for one day during DAWEX using a 3D cloud-resolving mesoscale model. Lane simulated isolated convection and resultant stratospheric gravity waves in a 2D version of a cloud-resolving model. The restriction to 2D allowed him to examine results obtained over a range of model grid resolutions. He finds that the momentum flux spectrum of the waves emerging into the stratosphere above the convection depends significantly on model resolution even down to rather fine grid spacings.
Notably, convergence of results for the momentum flux occurs only when the horizontal grid spacing is reduced substantially below 1 km which is typical of the horizontal resolution of most 3D models that have been applied to this problem.