1. Cloud parameterization

2. Carbon cycle

3. Alto (midlevel) clouds

4. Dry atmospheres in radiative-convective equilibrium

Cloud parameterization . . .

. . . is an important subject because some clouds are too small to be resolved by climate and weather forecast models, even with the most powerful of present-day computers.  Hence models need to account for the small-scale cloud variability that they cannot represent explicitly.


Formulation of CLUBB:

The following papers discuss the formulation of the core of CLUBB.

2017: “CLUBB-SILHS: A parameterization of subgrid variability in the atmosphere” V. E. Larson. arXiv, 1711.03675v3.

This technical document provides a comprehensive overview of CLUBB and its companion Monte-Carlo sampler, SILHS. Consult this document first if you want an introduction to the formulation of CLUBB or SILHS. (Nota bene: the climate models that use CLUBB — such as CAM, E3SM, and NorESM — do not include SILHS in their default versions and do not use CLUBB to parameterize deep clouds. Nevertheless, the description of CLUBB in this tech doc is still relevant to the implementation of CLUBB in those climate models.)

2002: “A PDF-Based Model for Boundary Layer Clouds. Part I: Method and Model Description.” J.-C. Golaz, V. E. Larson, W. R. Cotton. J. Atmos. Sci., 59, 3540-3551.

2002: “A PDF-Based Model for Boundary Layer Clouds. Part II: Model Results J.-C. Golaz, V. E. Larson, W. R. Cotton. J. Atmos. Sci., 59, 3552-3571.

(See also slides 13-35 of the following presentation, and this short conference paper.)

Traditionally, cloud parameterization has been viewed as a multiplicity of tasks. Such tasks include the prediction of heat flux, moisture flux, cloud fraction, and liquid water. In contrast, the papers above adopt the alternative viewpoint that the goal of parameterization consists largely of a single task: the prediction of the joint PDF of vertical velocity, heat, and moisture. Once the PDF is given, the fluxes, cloud fraction, and liquid water can be diagnosed.

The above papers present a parameterization that can model both stratocumulus and cumulus clouds without case-specific adjustments. This avoids the difficulty of having to construct a “trigger function” that determines which cloud type should be modeled under which meteorological conditions.

2002: “Small-Scale and Mesoscale Variability in Cloudy Boundary Layers: Joint Probability Density Functions.” V. E. Larson, J.-C. Golaz, W. R. Cotton. J. Atmos. Sci., 59, 3519-3539.

(See also the following short conference paper.)

Whereas the prior paper discusses one-dimensional PDFs of cloud water and humidity, this paper discusses joint PDFs that include the vertical velocity. Joint PDFs allow us to diagnose the buoyancy flux, which is the means by which convection generates turbulence. Joint PDFs also allow us
to diagnose fluxes of heat and moisture. Therefore, joint PDFs can serve as the foundation of cloud and turbulence parameterizations in numerical models, as proposed and explored in the two following papers.

2005: “Using Probability Density Functions to Derive Consistent Closure Relationships among Higher-Order Moments.” V. E. Larson and J.-C. Golaz. Mon. Wea. Rev., 133, 1023-1042.

(See also slides 26-27 of the following presentation.)

The aforementioned papers show that if we choose an accurate PDF family, then we can solve for many of the unknowns in our one-dimensional cloud parameterization. For some of these unknown terms, the present paper lists simple, analytic approximations. All approximated formulas are based on the same PDF and hence are consistent with each other.

A PDF may be constructed from a set of means, variances, and other moments of velocity, moisture, and temperature. It is possible that a particular set of moments does not correspond to any real PDF
in the family. We call such a set of moments “specifically unrealizable.” For instance, a set that includes asymmetric moments is specifically unrealizable with respect a PDF family of symmetric, bell-shaped curves. This is because the bell shape family is too restrictive to include asymmetric moments. We show that a broad class of moments is specifically realizable with respect to our PDF family. That is, our PDF family is not restrictive.

2007: “Elucidating model inadequacies in a cloud parameterization by use of an ensemble-based calibration framework.” J.-C. Golaz, V. E. Larson, J. A. Hansen, D. P. Schanen, and B. M. Griffin. Mon. Wea. Rev., 135, 4077-4096.

(See also the following oral presentation or slides, and this conference paper.)

It is often easy to see when an atmospheric model disagrees with data. It is usually much harder to locate the ultimate sources of model error.

It is particularly difficult to diagnose errors in a model’s structure, that is, errors in the functional form of the model equations. One technique that may help is parameter estimation, that is, the optimization of model parameter values. Typically, parameter estimation is used solely to improve the fit between a model and observational data. In the process, however, parameter estimation may cover up structural model errors.

In a quite opposite application, parameter estimation may be used to uncover the ways in which a model is wrong. The basic idea is to separately optimize model parameters to two different data sets, and then identify parameter values that differ between the two optimizations. When no single value of a particular parameter fits both datasets, then there must exist a related structural error.


CAM-CLUBB papers:

In the following 7 papers, the implementation of CLUBB in CAM is tested using a variety of configurations and observational datasets.


Bogenschutz, P.A., A. Gettelman, H. Morrison, Vincent E. Larson, C. Craig, and D. P. Schanen (2013). “High-order turbulence closure and its impact on climate simulations in the Community Atmosphere Model.” J Climate. 26, 9655-9676.

In this paper, CLUBB is implemented in CAM and tested in global simulations. CLUBB is used in these simulations to parameterize all shallow (stratocumulus and cumulus) clouds.

Bogenschutz, P. A., Gettelman, A., Morrison, H., Larson, V. E., Schanen, D. P., Meyer, N., et al. (2012). “Unified parameterization of the planetary boundary layer and shallow convection with a higher-order turbulence closure in the Community Atmosphere Model: single-column experiments.” Geoscientific Model Development, 5, 1407-1423. 

This paper simulates a variety of boundary-layer cloud types using the single-column version of CAM-CLUBB. The solutions are fairly robust to changes in time step and vertical grid spacing.

Wang, M., V. E. Larson, S. Ghan, M. Ovchinnikov, D. P. Schanen, H. Xiao, X. Liu, P. Rasch, and Z. Guo (2015). “A Multi-scale Modelling Framework model (Super-parameterized CAM5) with a higher-order turbulence closure: model description and low cloud simulations” Journal of Advances in Modeling Earth Systems

Here CLUBB is implemented in a cloud-resolving model with 4-km horizontal grid spacing, which in turn is implemented in CAM5. The model behavior is similar to CAM-CLUBB at 100-km horizontal grid spacing. This indicates that CLUBB behaves similarly over a range of horizontal grid spacings. 

Kubar, T., Stephens, G. L., Lebsock, M., Larson, V. E., and Bogenschutz, P. A., (2015). Regional Assessments of Low Clouds Against Large-Scale Stability in CAM5 and CAM-CLUBB Using MODIS and ECMWF-Interim Reanalysis Data, J. Climate.

Here, CAM-CLUBB’s depiction of low clouds is evaluated using satellite data. CAM-CLUBB simulates an improved transition between marine stratocumulus and shallow cumulus clouds.

Guo, Z., Wang, M., Qian, Y., Larson, V. E., Ghan, S., Ovchinnikov, M., et al. (2014). A sensitivity analysis of cloud properties to CLUBB parameters in the single‐column Community Atmosphere Model (SCAM5). Journal of Advances in Modeling Earth Systems, 6, 829-858.

Guo, Z., Wang, M., Qian, Y., Larson, V. E., Ghan, S., Ovchinnikov, M., et al. (2015). Parametric Behaviors of CLUBB in Simulations of Low Clouds in the Community Atmosphere Model (CAM). Journal of Advances in Modeling Earth Systems.

In these two papers, the sensitivity of CAM-CLUBB to changes in parameter values is tested using single-column and global simulations. These papers provide valuable guidance not only on the practical issue of tuning CAM-CLUBB, but also on the issue of understanding how changes in the strength of various small-scale processes affects the emergent cloud behavior.

Implementation of CLUBB in cloud-resolving and regional models:

The fact that CLUBB works in host models with a wide range of grid spacings (4 to 100 km) suggests that CLUBB is relatively insensitive to horizontal grid spacing.

2012: “PDF Parameterization of boundary layer clouds in models with horizontal grid spacings from 2 to 16 km.” V. E. Larson, D. P. Schanen, M. Wang, M. Ovchinnikov, and Ghan, S. Mon. Wea. Rev., 140, 285-306.

This paper implements CLUBB in a convection-permitting model, SAM. The use of CLUBB in SAM is tested for various boundary-layer cloud cases. We introduce a simple method for damping CLUBB’s effects at high resolution, thereby reducing undesirable sensitivities to horizontal grid spacing. We find that the use of CLUBB can improve the simulations for grid spacings of 4 km or greater.

2013: “SILHS: A Monte Carlo interface between clouds and microphysics.” V. E. Larson, C. Harlass, and J. Höft. Preprints, Fourteenth Annual WRF Users’ Workshop, Boulder, CO, Natl. Cent. for Atmos. Res.

2012: “Implementation and early tests of a PDF parameterization in WRF.”
V. E. Larson, C. Harlass, and J. Höft. Preprints, Thirteenth Annual WRF Users’ Workshop, Boulder, CO, Natl. Cent. for Atmos. Res.

These conference papers show simulations of a marine stratocumulus case using CLUBB implemented in a weather-forecast model, WRF, at moderate resolution.

Coupling CLUBB to microphysical variability:

Larson, V. E., B. J. Nielsen, J. Fan, and M. Ovchinnikov (2011). “Parameterizing correlations between hydrometeor species in mixed-phase Arctic clouds.” J. Geophys. Res., 116, D00T02, doi:10.1029/2010JD015570.

In order to drive microphysics using subgrid variability, we need to know the correlations between hydrometeor species. For instance, the correlation between cloud water and rain water influences the rate of accretion of cloud droplets by rain drops. If cloud and rain are correlated, then cloud and rain co-exist, and accretion occurs rapidly. This paper proposes a method to diagnose correlations based on information that is typically available in cloud models.

Larson, V. E., and B. M. Griffin (2013). “Analytic upscaling of a local microphysics scheme. Part I: Derivation.” Quart. J. Roy. Meteor. Soc., 139, 46-57.

Griffin, B. M., and V. E. Larson (2013). “Analytic upscaling of a local microphysics scheme. Part II: Simulations.” Quart. J. Roy. Meteor. Soc., 139, 58-69.

One reason to predict the subgrid PDF is to drive microphysical parameterizations more accurately. For instance, once we know the subgrid PDF, then we know what percentage of a grid box is precipitating strongly, and so forth. In these papers, we integrate a microphysics scheme analytically over CLUBB’s PDF. We are able to do this exactly for the drizzle parameterization of Khairoutdinov and Kogan, which is relatively simple in formulation. We find that, for a marine stratocumulus case, accounting for subgrid variability leads to significantly more simulated drizzle at the ocean surface.

Larson, V.E., J.-C. Golaz, H. Jiang, and W. R. Cotton (2005). “Supplying local microphysics parameterizations with information about subgrid variability: Latin hypercube sampling.” J. Atmos. Sci., 62, 4010-4026.

(See also slides 36-60 of the following presentation.)

Larson, V. E. , and D. P. Schanen (2013). “The Subgrid Importance Latin Hypercube Sampler (SILHS): a multivariate subcolumn generator.” Geosci. Model Dev., 6, 1813–1829.

The most accurate way to drive microphysics using a PDF is to integrate the relevant microphysical formulas analytically over the PDF. However, this may be intractable for some microphysics schemes or may require rewriting the microphysics code. To avoid this, one may draw sample points from the PDF and input them into the microphysics code one at a time. This allows the use of existing microphysics codes, but it also introduces statistical noise due to imperfect sampling. To reduce the noise, sample points may be spread out in a quasi-random fashion using “Latin hypercube sampling,” and the sample points may be clustered in important regions, such as cloud.

Chowdhary, K., Salloum, M., Debusschere, B., and Larson, V. E. (2015). Quadrature Methods for the Calculation of Subgrid Microphysics Moments. Mon. Wea. Rev.

Analytic integration over microphysics is restricted in applicability, and Monte Carlo sampling introduces sampling noise. Here, the integration is performed using a third alternative: deterministic quadrature. This method is more general than analytic integration and more accurate than Monte Carlo integration.

Storer, R. L., B. M. Griffin, J. Höft, J. K. Weber, E. Raut, V. E. Larson, M. Wang, and P. J. Rasch (2014). “Parameterizing deep convection using the assumed probability density function method.” Geosci. Model Dev. Discuss., 7, 3803–3849.

In this paper, variability in ice is included in CLUBB’s subgrid PDF, and a fully unified cloud parameterization is created. CLUBB’s single equation set is used to do single-column simulations of stratocumulus, shallow cumulus, and deep cumulus layers.


Participation by CLUBB in single-column model intercomparisons:

In single-column intercomparisons, CLUBB has been tested in a wide variety of cloud regimes.

2011: “Evaluation of the diurnal cycle in the atmospheric boundary layer over land as represented by a variety of single column models — the second GABLS experiment.” G. Svensson, A.A.M. Holtslag, V. Kumar, T. Mauritsen, G. J. Steeneveld, W. M. Angevine, E. Bazile, A. Beljaars, E.I.F. de Bruijn, A. Cheng, L. Conangla, J. Cuxart, M. Ek, M. J. Falk, F. Freedman, H. Kitagawa, V. E. Larson, A. Lock, J. Mailhot, V. Masson, S. Park, J. Pleim, S. Soderberg, M. Zampieri, and W. Weng, Bound. Layer Met., 140, 177–206.

2014: “The third GABLS intercomparison case for evaluation studies of boundary-layer models: Part B: results and process understanding.” F. C. Bosveld et al. (including V. E. Larson), Bound. Layer Met., 152, 157–187.

These two intercomparisons demonstrate that CLUBB can simulate stable boundary layers, including those that form at night after the occurrence of daytime boundary-layer turbulence.

2009: “Intercomparison of model simulations of mixed-phase clouds observed during the ARM Mixed-Phase Arctic Cloud Experiment. Part I: Single layer cloud.” S. A. Klein and Co-authors (including V. E. Larson). Quart. J. Royal Met. Soc., 135, 979-1002.

2009: “Intercomparison of model simulations of mixed-phase clouds observed during the ARM Mixed-Phase Arctic Cloud Experiment. Part II: Multilayer cloud.” H. Morrison and Co-authors (including V. E. Larson). Quart. J. Royal Met. Soc., 135, 1003-1019.

Clouds in the Arctic are often mixed-phase: that is, they often contain both liquid and ice. Long-lived mixed-phase clouds are difficult to simulate because ice naturally tends to grow at the expense of liquid. Models may overdeplete liquid unless the ice particles are limited in number and sediment out of cloud base rapidly enough. Our cloud parameterization, CLUBB, was used to simulate mixed-phase clouds during the M-PACE experiment. CLUBB was able to maintain liquid water in these clouds, as was observed.

2013: “A single-column model ensemble approach applied to the TWP-ICE experiment.”
L. A. Davies and Co-authors (including V. E. Larson). J. Geophys. Res., 118, 6544-6563.

This paper compares several internationally recognized parameterizations of deep convection. The simulated observations were obtained during the Tropical Warm Pool International Cloud Experiment (TWP-ICE) near Darwin, Australia. CLUBB simulated this deep convective case using the same configuration that is used to simulate boundary-layer clouds. CLUBB’s results for TWP-ICE are competitive with those of the other participating parameterizations. The results suggest that CLUBB contains enough physics to serve as a unified parameterization of both shallow and deep clouds.

2013: “CGILS: Results from the first phase of an international project to understand the physical mechanisms of low cloud feedbacks in single column models.”
M. Zhang et al. (including V. E. Larson), J. Adv. Model. Earth Syst., 5, 826–842.

This intercomparison demonstrates that CLUBB can simulate marine shallow clouds that are driven to equilibrium in month-long simulations.

2007: “A single-column model intercomparison of a heavily drizzling stratocumulus-topped boundary layer.” M. C. Wyant and Co-Authors. J. Geophys. Res., 112, D24204, doi:10.1029/2007JD008536.

This paper compared the output from numerous single-column model that were set up identically to simulate a cloud layer observed during the DYCOMS-II field experiment. Part of the challenge was simulating drizzle. In order to couple drizzle to the cloud fields, instead of drawing sample points from the PDF using the Latin hypercube method discussed above, we analytically integrated over the PDF.


Carbon cycle . . .

. . . is important for climate studies because not all the carbon dioxide that is emitted by humans remains in the atmosphere.  Rather, some CO2 is taken up by vegetation or dissolved in the oceans.

2008: “An idealized model of the one-dimensional carbon dioxide rectifier effect.” V. E. Larson and H. Volkmer.  Tellus B, 60B, 525-536.

(See also this shorter conference paper.)

The net flux of carbon dioxide (CO2) from the land surface into the atmospheric boundary layer has a diurnal cycle.  Drawdown of CO2 occurs during daytime photosynthesis, and return of CO2 to the atmosphere occurs during night. Even when the net diurnal-average surface flux vanishes, the diurnal-average profile of atmospheric CO2 mixing ratio is usually not vertically uniform. This is because of the diurnal rectifier effect, by which atmospheric vertical transport and the surface flux conspire to produce a surplus of CO2 near the ground and a deficit aloft. 

This paper constructs an idealized, 1-D, eddy-diffusivity model of the rectifier effect and provides an analytic series solution. When non-dimensionalized, the intensity of the rectifier effect is related solely to a single ‘rectifier parameter’.


Alto clouds . . .

. . . could be called the “forgotten clouds” of meteorology because they are less studied than other cloud types.  But we think they are well worth remembering!

2002: “Observed microphysical structure of mid-level, mixed-phase clouds.” R. P. Fleishauer, V. E. Larson, and T. H. Vonder Haar. J. Atmos. Sci., 59, 1779–1804. 

(See also this related presentation.)

Altostratocumulus (ASc) clouds are not merely very high stratocumulus clouds.  ASc are distinctive because they are often mixed-phase and also because they are often decoupled from surface fluxes of heat, moisture, and momentum.  This paper presents some observations from the CLEX-5 field experiment.  In most cases we examined, there were weak temperature inversions and wind shears at cloud top.  This contrasts with many observations of low-level stratocumulus clouds.  We conjecture that the differences are related to the fact that ASc clouds are usually not sustained by surface moisture fluxes, and they are usually not frictionally coupled to the ground by turbulent updrafts and downdrafts.

CLEX-5 frequently encountered alto clouds containing both liquid and ice.  In the thin, single-layer clouds that we observed, we found that near cloud top, where the cloud is coldest, liquid predominates over ice.  Near cloud bottom, where the cloud is warmest, ice predominates.  Prior authors have found the same vertical structure.  Presumably it is due to gravitational settling of the ice crystals. 

2001: “The death of an altocumulus cloud.” V. E. Larson, R. P. Fleishauer, J. A. Kankiewicz, D. L. Reinke, and T. H. Vonder Haar. Geophys. Res. Lett., 28, 2609–2612.

This is a case study of an altostratocumulus cloud that “died,” or dissipated, as an aircraft observed it.  There are four mechanisms that can cause an ASc to die: solar heating, incorporation into the cloud of dry air from outside, heating induced by large-scale subsidence of air, and precipitation.  In this particular case, subsidence seemed to be the major culprit.  Solar radiative heating was weak because the cloud formed over Montana in November.

2006: “What determines altocumulus dissipation time?” V. E. Larson, A. J. Smith, M. J. Falk, K. E. Kotenberg, and J.-C. Golaz. J. Geophys. Res., 111, D19207, doi:10.1029/2005JD007002.

(See also the following two animations, courtesy of David Schanen. The first shows dissipation of liquid water, with redder colors representing higher amounts of liquid; notice the strong turbulence.  The second movie shows the evolution of cloud top and cloud base surfaces; notice that although the cloud base rises, the cloud remains overcast (100% cloud cover) until near the end of the simulation.)

This paper further investigates the causes of altostratocumulus death using numerical simulations.  A particular subject of study is feedbacks or interactions between the 4 aforementioned processes: solar heating, incorporation into the cloud of dry air from outside, heating induced by large-scale subsidence of air, and precipitation of ice.  To quantify these, we construct a “budget term feedback matrix.”  It shows that precipitation of ice is a negative feedback on the other processes.  For instance, if solar heating dissipates the cloud, precipitation of ice dissipates the cloud less than it would have otherwise, thereby diminishing the effectiveness of solar heating on cloud dissipation rate.

2009: “Processes that generate and deplete liquid water and snow in midlevel, mixed-phase cloudsA. J. Smith, V. E. Larson, J. Niu, J. A. Kankiewicz, L. D. Carey. J. Geophys. Res., 114, D12203, doi:10.1029/2008JD013131.

This paper extends the study of Larson et al. (2006) by adding simulations of two new observed mixed-phase altostratocumulus cases and by constructing budgets of snow.  As before, the new clouds, in both observations and simulations, consist of a mixed-phase layer with a quasi-adiabatic profile of liquid, and a virga layer below that consists of snow.  The snow budgets show that snow grows by deposition in and below the liquid (mixed-phase) layer, and sublimates in the remainder of the virga region below. The deposition and sublimation are balanced primarily by sedimentation, which transports the snow from the growth region to the sublimation region below.

2009: “An analytic scaling law for glaciation rate in mixed-phase layer clouds.” V. E. Larson and A. J. SmithJ. Atmos. Sci., 66, 2620-2639.

In various practical problems, such as assessing the threat of aircraft icing or calculating radiative transfer, it is important to know whether or not mixed-phase clouds contain significant liquid water content. Some mixed-phase clouds remain predominantly liquid for an extended time, and others glaciate, or become converted to ice, quickly.

The glaciation rate of mixed-phase layer clouds is thought to depend on various factors. This paper attempts to quantify some of these factors by deriving scaling laws (i.e.~power laws) for the amount of snow at cloud base. The scaling laws are derived from the governing equation for snow concentration.

The scaling laws agree adequately with high-resolution simulations over one order of magnitude for snow flux and over two orders of magnitude for snow mixing ratio. They indicate, for instance, that cloud base snow amount increases faster than linearly with increasing cloud thickness and supersaturation with respect to ice.

By varying the exponents and pre-factors of the scaling laws, one may explore the sensitivity of glaciation rate to ice particle shape. The relationship is complex, but for our cloud cases, dendrites tend to glaciate cloud more rapidly than plates.

2007: “What causes partial cloudiness to form in multilayered midlevel clouds?  A simulated case study.M. J. Falk and V. E. Larson. J. Geophys. Res., 112, D12206, doi:10.1029/2006JD007666.

(See also the following short conference paper.)

At first glance, one might expect that a lower cloud layer would be little affected by a separated upper cloud layer that does not deposit snow or other quantities into it.  However, we find that the cloud fraction of such a lower layer can increase from 15% to 100% if the upper layer is removed.  The reason is that the removal of the upper layer allows cloud-top radiative cooling in the lower layer, thereby stabilizing it. 

2007: “An analytic longwave radiation formula for liquid layer clouds.” V. E. Larson, K. E. Kotenberg, and N. B. Wood. M. Wea. Rev., 135, 689–699.

(See also the following short conference paper.)

This paper discusses an idealized longwave radiative transfer parameterization that is used in two papers above, Falk and Larson (2007) and Larson et al. (2006).  This radiation parameterization is easy to implement in a numerical model, rendering it especially useful for numerical model intercomparisons.


Dry atmospheres in radiative-convective equilibrium:

The goal of the two papers below is to move theory one step away from Rayleigh-Benard convection, which has proved so fruitful for understanding of buoyant fluids, and one step closer to atmospheric convection.  The problem considered here adds infrared radiation to the classical problem of fluid flow between two plates, the lower being heated and the upper being cooled.  When radiation is added, the stability properties do not change qualitatively as long as one substitutes a radiative Rayleigh number for the classical Rayleigh number.  However, when fluid motion occurs, the turbulent heat flux does change because the heat flux is strongly constrained by radiation.

(The following article has been made available by the permission of Dynamics of Atmospheres and Oceans.  Single copies of the following article can be downloaded and printed for the reader’s personal research and study.)

2001: “The effects of thermal radiation on dry convective instability.” V. E. Larson. Dynamics of Atmospheres and Oceans, 34, 45–71. 

2000: “Stability properties of and scaling laws for a dry radiative-convective atmosphere.” V. E. Larson. Q. J. R. Meteorol. Soc., 126, 145-171.

Theory of fluid mechanical mixing:

1999:  “The relationship between the transilient matrix and the Green’s function for the advection-diffusion equation.” V. E. Larson.  J. Atmos. Sci., 56, 2447-2453.

(See also slide 7 of the following presentation.)

The point of this paper is that if a single-column model contains information only about horizontal averages, it discards crucial information about  horizontal structure.  For instance, a single-column model may predict the average concentration of a pollutant at some altitude perfectly.  But a model that only predicts averages doesn’t know whether the pollutant resides in an updraft or downdraft.  Hence the model doesn’t know whether to transport the pollutant up or down at the next time step.  Therefore, the transport prediction degrades rapidly.  This problem was termed “convective structure memory” by Roland Stull.  It can be quantified using Green’s function theory.