Introduction of the microphysics schemes to the nonhydrostatic σ-coordinate model

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A short history of numerical atmospheric modelling

In 1904, Vilhelm Bjerknes recognised that forecasting is a mathematical physics initial-value problem and that the basic equations that needed to be solved were already known in general form (see Haltiner and Williams, 1980). The rst attempt to forecast weather by solving primitive equations using a numerical analysis approach was made by Lewis Fry Richardson. His forecast, which he produced by hand, was awed because the predicted pressure changes were an order of magnitude larger than those observed (Harper et al., 2007; Holton, 2004). After Richardson’s failure Numerical Weather Prediction (NWP) was not tried again until the 1940s (see Holton, 2004). During World War II computers were developed, observational networks expanded and the number of meteorologists increased (Holton, 2004). These, together with a deeper understanding of the atmospheric equations made forecasting weather using a numerical process a possibibility in the 1950s (Harper et al., 2007). The governing equations that need to be solved are the momentum (equation 1.1.1), continuity (equation 1.1.2) and the thermodynamic (equation 1.1.3) equations.

Development of a Cloud Resolving Model

Clouds are treated di erently by models depending on the models’ horizontal resolution. For models running at resolutions lower than about 4 km, the cloudsenvironment relationship has to be represented statistically because clouds cannot to be captured explicitly since clouds are smaller than the grid resolution (Stensrud, 2007; Jakob, 2010). This option is called cumulus parameterisation which is a way of representing the relationship of clouds and the environment statistically without explicitly simulating the clouds (Weisman et al., 1997). It is thought that this parameterisation of convection is one of the greatest sources of uncertainty in NWP and climate modelling (Randall et al., 2003a; Engelbrecht et al., 2007). For resolutions higher than 4 km, Cloud Resolving Models (CRMs) should be used.
A CRM or a Cumulus Ensemble Model (CEM) is a model that can resolve structures of individual clouds, is integrated over a spatial domain large enough to contain many clouds and for a time long enough to include many cloud life cycles (Randall et al., 1996). Over the past four decades, CRMs have shown much success in replicating the observed structure and evolution of convection, using microphysics parameterisation with cumulus parameterisation neglected. CRMs can also be used to study the atmosphere where observations are lacking, and the ndings be used to improve conventional parameterisation schemes (Emanuel, 1994; Grabowski et al., 2006).

Microphysics

Microphysics processes play a crucial role in the formation, growth, shrinkage, breakup, and fallout of cloud and precipitation particles. Condensation is a microphysics process that converts water vapour to cloud water, while increasing the atmospheric temperature through the release of latent heat. The simplest cloud model is one with only cloud water and water vapour, and should include a microphysics process that allows water vapour to condense when saturation is reached and cloud water to evaporate during subsaturation. In a warm, nonprecipitating cloud, where it is assumed that cloud droplet sizes have the same size, evaporation and condensation are the only processes that need to be considered (Houze, 1993). However, clouds in the atmosphere can be far more complicated, and typically contain liquid and ice particles with di erent sizes, of which some may precipitate. Water in its three phases has to be represented in simulations of such clouds, together with processes that change phases, and those that change the sizes of the hydrometeors. A number of CRMs employ Bulk Microphysics Parameterisations (BMPs), which use a specied functional form for the particle size distributions and predict the particle mixing ratio (Rutledge and Hobbs, 1983; Grabowski, 1998; Stensrud, 2007).

To study the e ect of horizontal resolution, microphysics

and shear on a thunderstorm The eastern part of South Africa frequently experiences the occurrence of isolated thunderstorms during the austral summer season. These thunderstorms are usually triggered by surface heating, and are controlled to a lesser extent by the large scale circulation. In this study, a thunderstorm simulation that is triggered by a warm pertubartion, to represent surface heating is performed. The e ect of the microphysics schemes on the simulated hydrometeors and dynamics of the thunderstorms is studied by comparing simulations made with the two microphysics schemes introduced in the model. The simulations are made with di erent horizontal resolutions and with and without shear in two-dimensions.

To study the suppressed, transition and deep convection

periods using observed elds of TOGA COARE Most thunderstorms over South Africa occur in well-organised cloud bands, that are caused by a linkage of tropical and temperate systems. To test if the NSM is able to respond well to the large scale forcing, simulations are made using the large scale forcing obtained from the Tropical Oceans Global Atmosphere Coupled Ocean-Atmosphere Response Experiment (TOGA COARE) experiment which took place in November 1992 to February 1993 over the Western Pacic ocean. Three separate periods dominated by suppressed convection which include the end of a previous period of deep convection and start of subsequent period of convection are simulated. The case study was investigated by the Precipitating Cloud Systems Working Group (PCSWG) of the Global Energy and Water Cycle Experiment (GEWEX) Cloud System Study (GCSS). The simulations are made in two-dimensions.

2-D vs 3-D tests

Xu et al. (2000) compared seven CRMs for the simulation of midlatitude continental summer convection. The simulations were made for domain sizes approximately 500 km in the horizontal and 20 km in the vertical for 2-D simulations, and approximately 250 km x 250 km x 20 km for three-dimensional (3-D) simulations. The horizontal grid size used in these simulations was 2 km and the vertical resolution varied from model to model in the range 100mto 1 km, with stretched coordinates applied in the vertical in some of the models. Two of the models were used to perform 3-D simulations. They found 2-D models to produce similar statistical properties of cumulus convection to the 3-D versions, and therefore recommended the application of 2-D models for testing microphysical schemes (because they are computationally economical). They also found the inter-model di erences of temperatures with observations were smaller for CRMs compared to among SCMs. The performance of CRMs was found to be inferior for simulating midlatitude continental convection, relative to that of tropical ocean convection.

Turbulence

Turbulence is a natural response to instabilities in a ow – it acts to reduce the instability (Stull, 2006). Clouds are often turbulent, because of cloud-top radiative cooling and cloud-base radiative warming, as well as latent heat release and absorption. The rate of entrainment is partly determined by turbulent processes (Randall et al., 2003a). Turbulent motions have spatial and temporal variations at scales smaller than those resolved by models and the meteorological observation network. Even if observations would have been taken at very short temporal and spatial separations, a turbulent ow will always have scales that are unresolvable – because there would be eddies with frequencies greater than the observation frequency and spatial scales smaller than the scale separation of the observations (Bryan et al., 2003; Holton, 2004). Bryan and Morrison (2012) made squall line simulations with horizontal resolutions of 4 km, 1 km and 0.25 km resolution, and found that the high resolution simulations rained less because of increased condensation and evaporation rate of cloud water as a result of increased mixing. Bryan et al. (2003) found that simulations made with 1 km grid spacing do not produce equivalent squall-line structure and evolution to higher-resolution simulations of order 100 m. With grid spacing of order 1 km, overturning occurs in a relatively laminar manner.

Richardson based K

Warm bubble simulations were made using vertical di usion coecients that vary with height, buoyancy and shear (i.e. based on the gradient Richardson number (Louis, 1979)), with the same constant horizontal di usion coecients used in Engelbrecht et al. (2007). This scheme allows the eddy di usivity to change with height, so that bigger coecients are found closer to the surface, and increase with increased buoyancy and shear. This scheme is an improvement over the original scheme used in the model, which employed constant values of the coecients throughout the domain and simulation time. The simulation was performed with di usion in the elliptic equation activated. Three updraft cores (Figure 4.2.2c) are visible in the run, which indicate that the simulation is less di used compared to the case of constant K di usion coecients. This is visible also in the potential temperature simulation (Figure 4.2.2a). The values of the coecients (Figure 4.2.2d) di er across the domain with higher values closer to the surface, in areas of strong shear (Figure 4.2.2b) and of high buoyancy

A Thunderstorm

Thunderstorms over South Africa are usually embedded within cloud bands and are strongly controlled by the large-scale circulation (Tyson and PrestornWhyte, 2000). Isolated thunderstorms that are less dependent on mesoscale and synoptic-scale features also form often. In this section a thunderstorm based on the prole of Weisman and Klemp (1982) that represent the latter is discussed. A domain that is 100 km in horizontal extent and that extends to about 25 km above sea level is used. The vertical resolution is stretched so that higher resolution is obtained closer to the surface and lower resolution is used towards the top of the domain with an average resolution of 200 m.

Contents :

• 1 Introduction
  • 1.1 Background of the research
  • 1.1.1 Introduction
  • 1.1.2 A short history of numerical atmospheric modelling
  • 1.1.3 Vertical coordinates
  • 1.1.4 Nonhydrostatic models in p and σ-coordinates
  • 1.1.5 Development of a nonhydrostatic σ-coordinate model (NSM) in South Africa
  • 1.2 Motivation of the research
  • 1.2.1 Convective processes over South Africa
  • 1.2.2 Development of a Cloud Resolving Model
  • 1.2.2.1 Microphysics
  • 1.2.2.2 Turbulence
  • 1.2.3 Model development in South Africa
  • 1.3 Objectives of the research
• 2 Convection in atmospheric models
  • 2.1 Introduction
  • 2.2 Cloud Resolving Models
  • 2.2.1 Cloud Resolving Models in Large Scale Models
  • 2.3 Microphysics Parameterisations
  • 2.3.1 The Single Moment Scheme
  • 2.3.1.1 Warm clouds
  • 2.3.1.2 Cold clouds
  • 2.3.2 Multimoment schemes
  • 2.3.3 2-D vs 3-D tests
  • 2.4 Turbulence
  • 2.4.1 K-Theory – Richardson based
  • 2.5 Summary
• 3 Introduction of the microphysics schemes to the nonhydrostatic σ-coordinate model
  • 3.1 Introduction
  • 3.2 Basic Equations with moisture and radiation
  • 3.3 The microphysics parameterisations
  • 3.3.1 The PURDUE-LIN microphysics schemes
  • 3.3.2 The SBU-YLIN scheme
  • 3.4 Turbulence
  • 3.4.1 The basic equations
  • 3.4.1.1 Momentum equations
  • 3.4.1.2 Thermodynamic energy equation
  • 3.4.1.3 The water continuity equations
  • 3.4.1.4 The elliptic equation
  • 3.4.2 K-theory and conversion from geometic height to the σ-coordinate
  • 3.4.2.1 K-Theory
  • 3.4.2.2 Conversion from height to σ
  • 3.4.3 Surface Fluxes
  • 3.5 The numerical technique used in the NSM
  • 3.6 Summary
• 4 Numerical Experiments: Convective bubbles and an Isolated Thunderstorm
  • 4.1 Introduction
  • 4.2 Warm Dry Bubbles
  • 4.2.1 Constant K
  • 4.2.2 Richardson based K
  • 4.3 A Thunderstorm
  • 4.3.1 The e ect of the microphysics scheme
  • 4.3.2 The e ect of resolution on individual thunderstorms
  • 4.3.2.1 1 km resolution simulations
  • 4.3.2.2 2 km resolution simulations
  • 4.3.3 The e ect of shear on thunderstorms
  • 4.3.3.1 500 m resolution simulations with shear
  • 4.3.3.2 1 km resolution simulations with shear
  • 4.3.3.3 2 km resolution simulations with shear
  • 4.4 Discussion
  • 4.5 Summary and Conclusions
• 5 Numerical Experiments: Thunderstorms controlled by large-scale conditions – TOGACOARE suppressed and active periods
  • 5.1 Introduction
  • 5.2 TOGA COARE
  • 5.3 Comparing Simulations with Observations
  • 5.3.1 Temperature
  • 5.3.1.1 A0 experiment
  • 5.3.1.2 B0 experiment
  • 5.3.1.3 C0 experiment
  • 5.3.2 Specic Humidity
  • 5.3.2.1 A0 experiment
  • 5.3.2.2 B0 experiment
  • 5.3.2.3 C0 experiment
  • 5.3.3 Horizontal Winds
  • 5.3.3.1 A0 experiment
  • 5.3.3.2 B0 experiment
  • 5.3.3.3 C0 experiment
  • 5.4 Microphysics E ects
  • 5.4.1 The spin-up period
  • 5.4.1.1 A0 experiment
  • 5.4.1.2 B0 experiment
  • 5.4.1.3 C0 experiment
  • 5.4.2 The suppressed period
  • 5.4.2.1 A0 experiment
  • 5.4.2.2 B0 experiment
  • 5.4.2.3 C0 experiment
  • 5.4.3 The transition period
  • 5.4.3.1 A0 experiment
  • 5.4.3.2 B0 experiment
  • 5.4.3.3 C0 experiment
  • 5.4.4 The active period
  • 5.4.4.1 A0 experiment
  • 5.4.4.2 B0 experiment
  • 5.4.4.3 C0 experiment
  • 5.5 Summary and Conclusions
• 6 Conclusions
  • 6.1 Di usion Scheme Improvement
  • 6.2 Bulk Microphysics Parameterisation schemes
  • 6.3 Thunderstorm simulations
  • 6.4 TOGA COARE simulations
  • 6.5 Recommendations

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Simulations of moist convection using the quasi-elastic equations