On the Convective-Scale Predictability of the Atmosphere

Lisa Bengtsson Cover image: Lightning in a convective cloud. (c) SMHI, reprinted with permission

°c Lisa Bengtsson, Stockholm 2012

ISBN 978-91-7447-494-7

Printed by Universitetsservice US-AB, Stockholm, 2012 Distributor: Department of Meteorology Stockholm University Stockholm, Sweden Abstract A well-represented description of convection in weather and climate models is essential since convective clouds strongly influence the climate system. Con- vective processes interact with radiation, redistribute sensible and latent heat and momentum, and impact hydrological processes through precipitation. De- pending on the models’ horizontal resolution, the representation of convection may look very different. However, the convective scales not resolved by the model are traditionally parameterized by an ensemble of non-interacting convective plumes within some area of uniform forcing, representing the “large- scale”. A bulk representation of the mass-flux associated with the individual plumes in the defined area provides the statistical impact of moist convection on the atmosphere. Studying the characteristics of the ECMWF ensemble prediction system, it is found that the control forecast of the ensemble system is not variable enough in order to yield a sufficient spread using an initial perturbation technique alone. Such insufficient variability may be addressed in the parameterizations of, for instance, cumulus convection where the sub-grid variability in space and time is traditionally neglected. Furthermore, horizontal transport due to gravity waves can act to organize deep convection into larger-scale structures which can contribute to an upscale energy cascade. However, horizontal advection and numerical diffusion are the only ways through which adjacent model grid-boxes interact in the models. The impact of flow dependent horizontal diffusion on resolved deep convection is studied, and the organization of convective clusters is found to be very sensitive to the method of imposing horizontal diffusion. However, using numerical diffusion in order to represent lateral effects is undesirable. To address the above issues, a scheme using cellular automata to introduce lateral communication, memory and a stochastic representation of the statistical effects of cumulus convection is implemented in two numerical weather models. The behaviour of the scheme is studied in cases of organized convective squall-lines, and initial model runs show promising improvements.

List of Papers

This thesis is based on the following papers, which are referred to in the text by their Roman numerals.

I Bengtsson, L., Magnusson, L., Källén, E. (2008) Independent Estimations of the Asymptotic Variability in an Ensemble Fore- cast System Monthly Weather Review, Volume 136, Issue 11 pp. 4105-4112. (c)American Meteorological Society. Reprinted with permission. II Bengtsson, L., Tijm, S., Vána,ˇ F., Svensson, G. (2012) Impact of ﬂow-dependent horizontal diffusion on resolved convection in AROME. Journal of Applied Meteorology and Climatology, Vol. 51, No.1, pp 54-67. (c)American Meteorological Society. Reprinted with permission. III Bengtsson, L., Körnich, H., Källén, E., Svensson, G. (2011) Large-Scale Dynamical Response to Subgrid-Scale Organization Provided by Cellular Automata. Journal of the Atmospheric Sciences Volume 68, Issue 12 pp. 3132-3144. (c)American Meteorological Society. Reprinted with permission. IV Bengtsson, L., Steinheimer, M., Bechtold, P., Geleyn, J-F. (2012) A stochastic parameterization for deep convection using cellular automata. Submitted to Journal of the Atmospheric Sciences, February 2012.

Reprints were made with permission from the American Meteorological Soci- ety. The idea behind Paper I was initiated by Erland Källén. Originally the paper focused on the relation between ensemble mean and the individual members at inﬁnite forecast length. Linus Magnusson added a contribution on how the two relate to ensemble spread. I did all the analysis and wrote the text. The idea behind Paper II was my own, spawning from of sensitivity studies made together with Sander Tijm. I performed most of the analysis, with contributions from Sander Tijm, and wrote most of the text with invaluable insight from Filip Vána,ˇ and smaller contributions from Gunilla Svensson. The idea behind Paper III was initiated by Erland Källén. I did all of the code developments, analysis and wrote most of the text following invaluable sci- entiﬁc discussion together with Heiner Körnich, Erland Källén and Gunilla Svensson. The general idea behind Paper IV was my own, as a natural exten-

5 sion of Paper III in line with my work at SMHI. The implementation ideas in the ALARO model were developed in close collaboration with Jean-François Geleyn. The model code describing the cellular automata in IFS was introduced by Martin Steinheimer. Modiﬁcations to the code to perform limited area simulations with deterministic rules were done by myself. I did all of the code developments and analysis in regards to the ALARO simulations. The ECMWF implementation ideas, probabilistic rules, and option to run the cellular automata using sub-grid advection are credited to Martin Steinheimer. The ECMWF model runs and analysis in Paper IV was performed by Peter Bechtold.

6 Contents

1 Numerical weather prediction ...... 9 1.1 Representing model uncertainty: ensemble prediction ...... 10 1.2 Asymptotic error growth: results from Paper I ...... 12 2 Weather prediction at differing atmospheric scales ...... 15 2.1 Equatorial wave modes ...... 17 3 Uncertainty due to parameterization of cumulus convection ...... 21 3.1 Parameterization of cumulus convection ...... 23 3.1.1 Cloud model ...... 23 3.1.2 Closure ...... 26 3.2 Lateral communication and convective organization: results from Papers II, III and IV 29 3.3 Stochastic representation of cumulus convection; results from Papers III and IV .... 32 4 Future Outlook ...... 33 5 Acknowledgements ...... 35 Bibliography ...... 37

1. Numerical weather prediction

Numerical Weather Prediction (NWP) models are the main tools used in order to make weather forecasts. In NWP, the state of the atmosphere at some future time is determined using the primitive equations of fluid dynamics and thermodynamics based on an analysis of the current conditions. The very first operational, real-time +72 h NWP in the world was run in Sweden in autumn 1954 for the Swedish Armed Forces by the Department of Meteorology at Stockholm University, and operational production started shortly thereafter (Persson, 2005). The primitive equations consist of conservation of momentum, mass- conservation of dry air, the ideal gas law, the first law of thermodynamics and the thermodynamic equation for conservation of water. Following the notation of Kalnay (2003), the momentum equation and continuity equation used to describe the system are:

dV = α∇p 2Ω V gk + F (1.1) dt − − × − b ∂ρ = ∇ (ρV) (1.2) ∂t − · where V is the three-dimensional velocity vector, F is the frictional force, p is pressure, ρ is density and α is its inverse. The thermodynamic equations for heat and moisture are:

dT dp Q = Cp α (1.3) dt − dt

∂ρq = ∇ (ρVq)+ ρ(e c) (1.4) dt − · − where Q is the diabatic heating or cooling, T is temperature and Cp is the speciﬁc heat at constant pressure. q is the speciﬁc humidity and e and c are the condensation and evaporation rates. Finally the ideal gas-law is:

pα = RT (1.5) where R is the gas constant. Equations 1.1 to 1.4 are non-linear partial differential equations which are impossible to solve exactly through analytical methods. Therefore, the equations are solved using numerical methods by discretization either in spatial or

9 spectral space, to yield an approximate solution. Processes contributing to the evolution of the primitive equations which are not resolved in space or time on the numerical grid need to be represented by a parameterization. Following Bjerknes (1904) weather forecasting should be considered as an initial value problem of mathematical physics, and could be carried out by integrating the Eqns. 1.1 to 1.5 forward in time, starting from the observed, initial state of the atmosphere. In order to construct an initial condition to numerically solve the primitive equations, an analysis of the present state of the atmosphere is constructed using observations of the current conditions combined with the forecast from the previous forecast cycle. Most NWP models today use observations from meteorological observation sites, airports, ships, buoys, airplanes and radiosondes, as well as remote sens- ing such as satellite and radar (Gustafsson et al., 2012). For high-resolution limited area models, radar observations and ground-based GPS observations can be used in order to describe the initial state with increasing accuracy. Due to the spatial and temporal irregularities in observations of the current state of the atmosphere, the information provided by observations are combined with a background field from the previous forecast cycle in order to assimilate the initial state on the uniform NWP grid. The technique to do so varies among different NWP models; examples include variational data assimilation (e.g. Gustafsson et al., 2012) and ensemble Kalman filtering (Kalnay et al., 2007). Since the observations used to describe the atmosphere are irregular in space and time, a continuous complete description of the atmosphere does not exist. Thus, the initial condition given the system of equations will never be perfect. Therefore, there is always an initial error in the model, compared with the true state of the atmosphere, which will grow in time and double in about two to three days (Lorenz, 1969). Even if a perfect initial state could be provided, errors in NWP models are inevitable. Since an analytic solution to the primitive Eqns. 1.1 to 1.5 is only possible when they are linearized and simplified, the solution provided by the numerical methods will only be approximate. Therefore, errors in NWP grow in time due to two factors; 1) the inaccuracy of the initial state of the atmosphere, or 2) model construction error such as discretization of evolution equations, truncation, parameterizations and numerical diffusion. Of course, these two error sources are not independent from one another.

1.1 Representing model uncertainty: ensemble prediction One way of addressing forecast uncertainty due to the errors in the NWP models and the initial conditions is to use ensemble prediction. The idea behind ensemble prediction is to simulate the sensitivity of the forecast to the initial conditions and the model errors discussed above. Although this distinction is useful conceptually, it is stressed in Leutbecher and Palmer (2008) that initial

10 condition error is inseparable from model error for a real physical system like the atmosphere, because model uncertainty contributes to the initial condition uncertainty. Although ensemble prediction yields a good estimate of the uncertainty associated with a certain weather pattern, its ultimate purpose is to predict quantitatively the probability density of the state of the atmosphere at a future time (Leutbecher and Palmer, 2008). The error due to uncertainties in the initial condition can be represented by re-running the model several times starting from slightly different initial conditions. Various methods on how to best construct the initial perturbations have been explored, in particular at the European Centre for Medium-Range Weather Forecasts (ECMWF), and at the National Center for Environmen- tal Prediction (NCEP), where global weather forecast are generated. Methods include singular vectors (Palmer, 1993), Ensemble Transform (ET) perturbations (Wei et al., 2008), breeding vectors (Toth and Kalnay, 1993, 1997), and more recently Ensemble Data Assimilation (EDA, Leutbecher and Palmer, 2008; Buizza et al., 2008), but this list is far from exhaustive. The scientific basis behind ensemble prediction is that in a non-linear dynamical system, the finite-time growth of initial errors is flow dependent, and the various initial perturbation technique is trying to capture this flow de- pendency in one way or another. The singular-vector perturbation approach is based on the idea that it is most important to sample directions in phase space that are characterized by maximal amplification rates (Magnusson et al., 2008). The singular vectors used by ECMWF are computed to maximize total energy growth over a 48-h time interval (Leutbecher and Palmer, 2008). The ET technique, which is an extension of the breeding vector technique, also aims at capturing the fastest growing dynamical instabilities. Breeding vectors are constructed by adding random perturbations to a forecast, and an unper- turbed control forecast is periodically subtracted from the perturbed forecast as they are being integrated in time. The difference is then added back to the control to create the new perturbed initial condition. After some time, this "breeding" process creates vectors dominated by the fastest growing instabilities of the evolving control solution (Kalnay et al., 2002). The ET method transforms the forecast perturbations so that they are orthonormal in the inverse analysis error variance norm (Wei et al., 2008). Ensemble forecasting and data assimilation are closely connected as both require the prediction of uncertainty estimates. The only difference is the time range at which these uncertainty estimates are required (Leutbecher and Palmer, 2008). In the EDA Technique at ECMWF, an ensemble of 20 lower-resolution 4D-Var assimi- lations that differ by perturbing observations, sea-surface temperature fields and model physics are combined with the singular vector perturbations for the complete ensemble prediction system (Buizza et al., 2008). In order to address forecast errors which are due to model imperfections, there are also several methods explored. For instance, in the ECMWF IFS model, the physical tendencies from the different subgrid-scale parameteri-

11 zations are combined and multiplied by a randomly-generated number with uniform probability ranging from 0.5 to 1.5; the random numbers are updated every 6 hours (Buizza et al., 1999).

1.2 Asymptotic error growth: results from Paper I The initial error caused by an inadequate representation of the initial state grows in time throughout the forecast, and if the primitive equations were linear, the error would continue to grow exponentially indefinitely (Lorenz, 1969). However, due to the non-linearity of the primitive equations, the error will saturate after some time, which on average is about two weeks for atmospheric flow (Lorenz, 1969). The characteristics of an ensemble prediction system at infinitely long forecast lengths, when the system has reached its limit of deterministic predictability are studied in Paper I. In this study the Root Mean Square Error (RMSE) of the ensemble at 500 hPa is compared with the ensemble spread at a time ex- trapolated beyond two weeks, using a simple model of error growth suggested by Lorenz (1982). The RMSE for the ensemble mean (EMEAN) and the individual members (EMEMBERS) is related to the ensemble spread (S) as well as that of the control forecast (ECONTROL). Paper I presents the relationships between these statistical measures at infinite forecast time. It is demonstrated that at infinite forecast length:

The asymptotic error limit of ECONTROL should be equal to that of • EMEMBERS.

2 2 EMEMBERS should saturate towards EMEAN + S and also equal √2EMEAN • q

√2EMEAN should be in agreement with the characteristic variability of • the atmosphere, computed from two randomly chosen analyses of the atmosphere.

EMEAN should equal that of S at the asymptotic error limit •

Using the above relationships and comparison with the atmosphere’s characteristic variability, three separate arguments are made looking at the ECMWF Ensemble Prediction System, which indicate that the forecast model upon which the ensemble prediction system is based on is under-dispersive, i.e. not variable enough:

12 1. EMEAN does not equal that of S at the asymptotic error limit 2. The atmosphere’s characteristic variability, after accounting for seasonal variability, is in good agreement with √2EMEAN; however the error of the ensemble members EMEMBERS does not saturate to this value. 3. The RMSE of the control forecast, ECONTROL, converges with the error of the ensemble members, EMEMBERS, at an asymptotic error level lower than that of the atmosphere’s characteristic variability. As concluded in Paper I, if the forecast model does not have enough internal variability, it is impossible to obtain sufﬁcient spread by only applying an initial perturbation technique. Model construction errors also need to be accounted for, and as described in the previous section, stochastic physics is one way of approaching this problem. One source of model error stems from physical parameterizations, and in particular that of deep convection. The uncertainty due to the statistical representation of cumulus convection in most NWP models is discussed in Chapter 3, and constitutes the main topic of re- search addressed in Papers III and IV.

2. Weather prediction at differing atmospheric scales

Numerical weather models have to consider a wide range of atmospheric scales, from the molecular scale to the global scale. The energy associated with the different scales is arguably different between, for instance, low pressure systems and small turbulent eddies close to the surface. Figure 2 shows the so-called Nastrom-Gage energy spectrum as a function of horizontal wave-number. Temperature and wind measurements taken from nearly 7000 commercial airline flights during the Global Atmospheric Sampling Program (GASP) from 1975 and 1979 were analysed in Nastrom and Gage (1985). 3 Each spectrum in Figure 2 has two different distinct slopes; k− power- 5/3 law on the synoptic scale, and k− power-law between approximately 600 and 2 km, constituting the “meso-scale”. As explained in Tung and Orlando (2002), there have been many attempts to describe the two slopes theoretically, although a complete understanding of the atmospheric energy spec- 3 trum remains elusive. Kraichnan (1967) predicted a k− power-law for two- dimensional, isotropic and homogeneous turbulence using scaling analysis. This analysis was based on a down-gradient enstrophy cascade from large to small scales due to vorticity advection, and such vortices finally break down through micro-scale turbulence. It was first argued by Charney (1971) that such forward potential enstrophy cascade in two-dimensional turbulence theory carries over to the real atmosphere through Quasigeostrophic (QG) turbulence theory. 5/3 As further summarized in Tung and Orlando (2002), the k− power-law has been derived theoretically by Kolmogorov (1941a,b) to occur for three- dimensional turbulence (isotropic and homogeneous), due to a downscale energy flux, and by Kraichnan (1967) for two-dimensional turbulence (isotropic and homogeneous) due to an upscale energy flux on the large-scale side of energy injection. The transfer of energy from small to large scales according to two-dimensional turbulence arguments arises from the constraints of conservation of enstrophy and energy (Vallis, 2006). In the QG formulation, the two-dimensional picture is augmented by in- cluding the effect of Earth’s rotation and allowing horizontal divergence. Fol- lowing scaling arguments, the rotation term (beta) - and hence the Rossby waves - dominates the large-scale flow, whereas advection and turbulence dominate the small scales (Vallis, 2006). The scaling arguments imply that

15 Figure 2.1: Variance power spectra of wind and potential temperature near the tropopause from GASP aircraft data, from Nastrom and Gage (1985).

an energy source at the short-wave end of the spectra that cascades to larger scales will, at some point, reach the scale where the effect of rotation starts to dominate, i.e. the scale at which Rossby waves dominate the ﬂow (Vallis, 2006). Since two-dimensional turbulence theory and QG turbulence formulations involve an upscale energy cascade, a large source of energy at the short-wave end of the spectrum is required. Lilly (1983) argued that energy generated by convection could participate in an upscale energy cascade through formation 5/3 of convective anvils and gravity waves, consistent with the observed k− 5/3 slope. The important role of gravity waves to generate the k− slope was also suggested by Koshyk and Hamilton (2001). Even though there is some uncertainty on exactly how to theoretically ex- plain the two distinct slopes observed for the different atmospheric scales, results found in Paper III of this thesis support the theory of two-dimensional turbulence and the QG-augmented formulation in that a source at the short- wave end of the spectra can yield increased energy also at the larger scales. Below follows a summary of the equatorial wave modes used to study scale- interaction in Paper III.

16 2.1 Equatorial wave modes In Paper III a shallow water (SW) model is used to study scale interactions. The SW equations which govern the vertically independent motion of a single thin layer of incompressible and homogeneous ﬂuid on a rotating sphere are described by Eqns. 1-3 in Paper III. The SW model is initialized with the transient-free solutions found for the SW equations following Matsuno (1966) and Gill (1982):

u u(y) v = v(y)exp[i(kx ωt)] (2.1) − h h(y)     where k is the zonal wave number and ω is the frequency. Substitution and re- arrangement reduces the set of equations to a single, second-order differential equation in v:

d2v ω2 k β 2y2 + [( k2 β )]v = 0 (2.2) dy2 c2 − − ω − c2

β is the latitudinal gradient of the Coriolis parameter and c = √ghe is the phase speed. he is the average value of the temporally- and spatially-varying depth of the ﬂuid, h, and is assumed to be much smaller than the characteristic horizontal length scale of the motion. Solutions are sought for the meridional distribution of v, subject to the boundary condition y ∞, and are given by: → i(kx ωt) v(x,y,t)= Dn (y)e − (2.3) where the meridional part can be described by parabolic cylinder functions, Dn of order n, where n represents a meridional mode number:

y n/2 y 2 y Dn( )= 2− exp[ ( ) ]Hn( ) (2.4) c/2β − 2 c/2β √2 c/2β p p p Here Hn is a Hermite polynomial of degree n. The expression c/2β is the equatorial radius of deformation, where solutions for h and upare given in terms of linear combinations of the parabolic cylinder functions Dn. Following Matsuno (1966) the solutions to (2.2) exist only when the constant part in the square brackets satisfy the relationship:

ω2 k c/β( k2 β)= 2n + 1 n = 0,1,2,... (2.5) c2 − − ω This equation relates the frequency, ω, to the wave number, k, for each positive integer n, deﬁning the horizontal dispersion relation for the waves. Since the equation is cubic in ω, there are three classes of solutions corresponding to the eastward and westward inertio-gravity waves and equatorial Rossby waves,

17 2 10

1 10

Kelvin EMRG WMRG Rossby n=1 0

ω 10 Rossby n=2 Rossby n=3 WIG n=1 WIG n=2 WIG n=3

−1 10

−2 10 −2 −1 0 1 10 10 10 10 k

Figure 2.2: Dispersion curves for Kelvin, equatorial Rossby, equatorial westward inertia-gravity, eastward and westward moving mixed Rossby gravity waves. Only the lowest three meridional modes (n=1,2,3) for Rossby and inertia-gravity waves are presented. Zonal wave-number, k is normalized by c/2β, and frequency, ω, is normalized by 2cβ p p

EIG, WIG, and ER, respectively. There is also the special case of n = 0, corresponding to mixed Rossby gravity (MRG) waves, and the solution in which v(y)= 0 and n = 0 corresponding to the Kelvin wave, often referred to as the n = 1 solution (Kiladis et al., 2009). The− choice of eigenmodes included in 2.3, and the spectral variance distribution follows closely that of Zagar et al. (2004). The selected eigenmodes are based on a Fourier truncation, elliptic truncation, and frequency cut-off based on a choice of a critical frequency ωc, derived from the dispersion relations for the equatorial waves. The elliptic truncation criterion is:

k n ( )2 + ( )2 1 (2.6) Nm Nn ≤ and maximal values for k and n are given by Nm and Nn and correspond to the values for truncation in the zonal and meridional directions, respectively. The dispersion curves for the equatorially trapped Kelvin, ER, MRG, and eastward and westward IG waves corresponding to the lowest meridional modes are plotted in Figure 2.2. Wheeler and Kiladis (1999), demonstrate using a wave number-frequency spectral analysis of satellite-observed outgoing longwave radiation (OLR), as a proxy for cloudiness, where spectral peaks of convective clouds can be found along these dispersion curves. Following Zagar et al. (2004), a critical fre-

18 quency ωc appropriate for mid latitudes based on Figure 2.2 would be:

βc ω = (2.7) c r 2 However, this frequency rejects all the EIG and EMRG waves and most of the WMRG and Kelvin waves, important for the tropical dynamics. Therefore, instead of applying (2.7) strictly, all Kelvin and/or WMRG modes which are allowed by (2.6) are included. The spectral variance distribution, or the amplitude of the waves, is decided by: Γ2 γ2 (Km)= (2.8) 1 + L2(Km)2 2 { } and is applied to the eigenmodes which satisfy 2.6 and 2.7. Km is a one dimen- 2 2 sional wave number and deﬁned as Km = (c1k) + (c2n) . The total amplitude is decided by Γ, and L is the horizontalp length scale. The total variance is split between the different waves such that the ER waves contain about 50 percent, Kelvin modes and WMRG modes about 15 percent each, while EMRG and WEIG modes contain about 10 percent of the total variance as chosen in Zagar et al. (2004) based on OLR observations by Wheeler and Kiladis (1999), demonstrating a similar variance distribution of the mass ﬁeld. The interaction between the convective-scales and the larger-scale equatorial wave-modes as studied in Paper III is summarized in Chapter 3.2.

3. Uncertainty due to parameterization of cumulus convection

As discussed in previous chapters, cumulus convection interacts with the large-scale flow of the atmosphere. Thus, a large contribution to model uncertainty stems from the statistical representation of cumulus convection, as highlighted in Chapter 1.1. This final chapter will, in some detail, describe the cumulus parameterization challenge, and the main results from Papers II, III and IV are summarized. In Papers II, III, and IV of this thesis, three different NWP models are used. There are thus three separate methods in which cumulus convection is represented. Two essential uncertainties in the representation of cumulus convection, common for all parameterizations considered in this work, are addressed in this thesis: 1) In all three models, the sub-grid vertical eddy transport of heat, moisture and momentum due to convection is parameterized using a so called bulk mass-flux concept. That is, all active cloud elements in a grid-box are represented in one steady-state updraft representing the whole cloud ensemble. Thus the sub-grid variability in time and space of convective updrafts is not accounted for in the parameterizations. 2) In all three parameterizations, horizontal advection of mean grid-box properties, and numerical diffusion are the only processes which are able to produce exchanges between adjacent grid-boxes. There is no parameterization of horizontal transport of heat, moisture or momentum due to cumulus convection. In reality, mass transport due to gravity waves that propagate in the horizontal can trigger new convection, important for the organization of deep convection (Huang, 1988). In fact, cumulus parameterizations used in most operational weather and climate models today are based on the mass-flux concept which took form in the early 1970’s, first formulated by Ooyama (1971), and developed further in the fundamental work of Yanai et al. (1973) and Arakawa and Schubert (1974). In the mass-flux approach, the amount of air rising through each model layer in the clouds is determined by a one-dimensional cloud model. Environ- mental subsidence is assumed to compensate for this vertical motion as a consequence of mass continuity. A schematic figure of how the one-dimensional cloud model influences the environment is depicted in Figure 3.1. Parameterization schemes of cumulus convection may look very different depending on model resolution and the representation of physical processes interacting with the cumulus scheme. The configuration of the ECMWF IFS cycle 37r2, (IFS-documentation, 2011) model used in Paper IV has a hori-

21 Figure 3.1: Schematic view of how the cumulus convection parameterization inﬂu- ences the large-scale atmosphere. In the mass-ﬂux approach the amount of air rising through each model layer in the clouds is determined by a one-dimensional cloud model. Environmental subsidence is assumed to compensate for this vertical motion, as a consequence of mass continuity, yielding a warming and drying of the atmosphere. Detrainment of cloudy air into the environment yields a cooling and moistening. The evaporation of cloud condensate and precipitation in the downdraft yields a cooling and stabilization of the boundary layer.

zontal grid-spacing of roughly 16 km. At these scales, vertical transport of heat and moisture due to cumulus convection is certainly a sub-grid process that needs to be parameterized. The conﬁguration of the ALARO model cycle 36h1, (Bénard et al., 2010; Vánaˇ et al., 2008; Gerard et al., 2009) used in the same paper has a horizontal grid-spacing of 5.5 km; this scale is com- monly referred to as the “grey-zone” of cumulus convection, as deep convection is partly resolved and partly sub-grid. Therefore, the model uses a cumulus convection parameterization developed to be satisfying on the scales of 4-7 km grid-spacing by using prognostic closure equations (Gerard et al., 2009). Finally, the conﬁguration of the AROME model (Seity et al., 2011; Bénard et al., 2010) used in Paper II has a horizontal grid-spacing of 2 km, and it is assumed that deep convection at this scale is explicitly resolved by the model. However the AROME model is far from a cloud resolving model; the evolution equations of heat, moisture and momentum are mean grid-box variables, and vertical transport of heat, moisture and momentum from turbulence, dry convective plumes and non-precipitating shallow convection still need to be parameterized. Below follows a summary of the building blocks used in deep convection parameterizations, with focus on the aspects addressed in this thesis. The dis-

22 tinguishing features of the deep convection schemes in the three different models studied in Papers II and IV are discussed.

3.1 Parameterization of cumulus convection In a review paper of cumulus parameterizations, Arakawa (2004, p 2496), deﬁnes the cumulus parameterization problem as “formulating the statistical effects of moist convection to obtain a closed system for predicting weather and climate.” It is a delicate problem, or challenge for the more optimistic reader, and a fair representation of the statistical effects of moist convection on the large-scale atmosphere is crucial for weather forecasts and climate simulations. Although parameterization schemes of cumulus convection may look very different depending on NWP model, convective parameterization consists of three general building blocks (Emanuel, 1994): 1) cloud model: effect of convection on the environment. 2) closure: regulation of the amount of convection by the larger-scale processes or grid-scale variables, and 3) triggering: when to activate the convection scheme.

3.1.1 Cloud model As mentioned above, the most common way to describe the effect of convection on its environment is to apply a so called mass-flux scheme. In such schemes, the mass-flux in convective “plumes” dominates the vertical transport, and are based on simple one-dimensional models of entraining plumes. Following Ooyama (1971), Yanai et al. (1973) and Arakawa and Schubert (1974), a horizontal area is assumed to contain enough plumes such that they can be treated in a statistical manner. The horizontal area should be “large enough to contain the ensemble of clouds, but small enough to be regarded as a fraction of the large-scale system” (Yanai et al., 1973 p. 612) Since it is the thermodynamic forcing by cumulus convection that is of interest, the equations of heat and moisture averaged over the hypothesized area are used as a starting point (more recent cumulus convection schemes based on the mass-flux concept also consider momentum e.g. Bechtold et al. (2008)):

∂s ∂sω ∂ + ∇ sV + = R + L(c e) s ω (3.1) ∂t · ∂ p − − ∂ p ′ ′ ∂q ∂qω ∂ + ∇ qV + = e c q ω (3.2) ∂t · ∂ p − − ∂ p ′ ′ where s is the dry static energy: s CpT + gz, which is conserved for dry ≡ adiabatic ascent (Cp is the speciﬁc heat capacity of air at constant pressure, T is absolute temperature, g is gravitational acceleration, and z is height). The equations are written on pressure-coordinates and take a slightly differ-

23 ent form from Eqns. 1.4 and 1.5. q is the specific humidity, ω is the vertical velocity in pressure coordinates, R is the heating/cooling rate due to radiation, c is the rate of condensation per unit mass of air, and e is the rate of evaporation of cloud droplets. L is the latent heating/cooling of condensation and evaporation. The horizontal averages are denoted by (), deviation from these are denoted by primes. The mixing of the mean variables by the mean wind is done by horizontal and vertical advection of the mean quantities. The last terms on the RHS of Eqns. 3.1 and 3.2 are the vertical eddy transport by heat and moisture, respectively. The processes contributing to this vertical sub-grid mixing is arguably different depending on if the model grid-spacing is 16 km, 5.5 km, or 2 km. It may contain sub-grid mixing from cumulus convection and turbulent motions in the sub-cloud layer, which may cause significant vertical eddy transport of heat and moisture (Yanai et al., 1973). From here on, only the transport due to moist convection is considered; the transport due to, for instance, turbulence is assumed to be parameterized separately. In the AROME model, a combined parameterization of dry convection and turbulence using the Eddy Diffusivity Mass Flux (EDMF) scheme (Pergaud et al., 2009; Soares et al., 2004) is used. However, dry convection and turbulence will not be discussed here. In all of the models used in this thesis, it is assumed that small-scale eddies in the horizontal due to convection are small compared to the horizontal transport of the resolved flow. Therefore, horizontal communication between adjacent grid-boxes is done through advection of mean variables and numerical diffusion only. There is no parameterization of horizontal transport due to deep convection, i.e. the deep convection parameterizations are applied in each vertical column separately. In Papers II, III and IV, studies of the impact of horizontal effects caused by deep convection and organization of convection are presented. The main findings are summarized in the next section. The mass-flux concept considers the effect of a whole ensemble of clouds over the hypothesized area (one grid-box in practice) rather than a single cloud element. With the exception of the work of Arakawa and Schubert (1974), where a spectral method is used depending on cloud type, most mass-flux parametrizations employ a so-called bulk approach, first suggested by Yanai et al. (1973), in which all active cloud elements are represented in one steady-state updraft representing the whole cloud ensemble. The vertical eddy transport of dry static energy and moisture for the bulk updraft is expressed using the mass-flux as:

s ω σωc(sc s)= Mc(sc s) (3.3) − ′ ′ ≈ − − −

q ω σωc(qc q)= Mc(qc q) (3.4) − ′ ′ ≈ − − − where σ is the cloudy fraction of the grid-box, and the subscript c is used to indicate the cloudy fraction of the grid-box. It should be noted that the notation is for the bulk representation of the updraft, and a summation over

24 individual cloud types is assumed. Furthermore, the equations are based on the assumption that a small fraction is used (σ 1) so that environmental values can be replaced by the horizontal averages.≪ Following Yanai et al. (1973), it is further assumed that the activity of the cumulus clouds is statistically in equilibrium with the imposed “large-scale” motion, with the consequence that the cloud ensemble maintains the heat and moisture balance with the environment. The cloud budget equations, i.e. the equations of balance for mass, heat, water-vapour, and liquid water between the cloud and its environment are:

∂M c = E D (3.5) ∂ p −

∂(Mcsc) = Es Dsc + Lcc (3.6) ∂ p −

∂(Mcqc) = Eq Dqc cc (3.7) ∂ p − −

∂(Mcl) = Dl + cc r (3.8) ∂ p − − where E and D are the rates of mass entrainment and detrainment per unit length, l is the cloud liquid water content, cc is the net condensation in the cloudy updraft, r is the rate of precipitation, and L is the latent heating/cooling of condensation and evaporation.. Using Eqns. 3.5-3.8 together with 3.3 and 3.4, Eqns. 3.1 and 3.2 can be expressed as:

∂s ∂sω ∂s + ∇ sV + R = Mc + D(sc s) L(e) (3.9) ∂t · ∂ p − ∂ p − −

∂q ∂qω ∂q + ∇ qV + = Mc + D(qc q)+ e (3.10) ∂t · ∂ p ∂ p − where the RHS of Eq. 3.9 and 3.10 are the contribution to the evolution of heat and moisture due to moist convection, respectively. It states that convection changes the large-scale atmosphere giving a i) warming and drying through compensating subsidence, ii) cooling and moistening by detrainment of cloudy air into the environment, and iii) cooling and stabilization of the boundary layer through evaporation of cloud condensate and precipitation in the downdraft. The assumption of a stationary budget has been widely used to develop convective parameterizations, and many schemes have adapted the bulk mass-ﬂux approach; some popular schemes are e.g Bougeault (1985); Tiedtke (1989); Gregory and Rowntree (1990); Kain and Fritsch (1993); Bechtold et al. (2001). However, to date, it is not clear whether a description

25 of a single cloud type of the cloud ensemble would be preferable or if a bulk representation is sufficient. The concept of one single entraining plume is clearly a great over-simplification of the thermodynamics of an individual cloud (Plant, 2010). Arakawa (2004, p 2495) states “because of the statistical nature of the parameterization problem, it is rather obvious that we should introduce stochastic effects at some point in the future development of parameterizations.” Papers I, III and IV discuss the need for stochastic parameterizations, and in Paper IV a stochastic parameterization of cumulus convection using cellular automata is proposed. The main findings will be discussed in Chapter 3.3.

3.1.2 Closure In order to close the parameterization, the large-scale model variables need to be related to the vertical distribution of the mass-flux, and to the mass- flux at cloud base. The choice of appropriate closure assumptions is crucial for the performance of a cumulus convection parameterization scheme (Arakawa, 2004). One of the earliest convection schemes used a closure based on moist convective adjustment (Smagorinsky, 1956; Manabe et al., 1965). In adjustment schemes, a target atmospheric state is produced through the ac- tion of convection and then the model state is adjusted towards that target (Emanuel, 1994). The most well-known standard adjustment scheme is that of Betts and Miller (1986). In their scheme, layers unstable to convection are adjusted to reference profiles of temperature and moisture. In such standard adjustment schemes, no cloud model needs to be specified. Many closures in cumulus convection parameterizations based on the more process oriented mass-flux concept have evolved through the adjustment paradigm. In Arakawa and Schubert (1974), the concept of adjustment is important in the underlying theory, but it is not explicit in the actual computation (Arakawa, 2004). The scheme uses a so-called quasi-equilibrium closure, where it is assumed that a quasi-equilibrium exists between convection and the “large-scale” forcing on CAPE. For models that resolve the meso-scale it has been recognized that convection depends not only on the availability of Convective Available Potential Energy (CAPE), as in the quasi-equilibrium closure, but also on the amount of CAPE and Convective Inhibition, CIN (Fritsch and Chappell, 1980). Many convection schemes developed for the meso-scale include a so-called “CAPE closure” e.g. Bechtold et al. (2001); Kain and Fritsch (1993); Fritsch and Chappell (1980); Zhang and McFarlane (1995). In these schemes, equilibrium states are also defined, but adjustments toward those states are either relaxed (partially each time-step) and/or performed only when certain conditions for triggering are met (Arakawa, 2004). The quasi-equilibrium closure was constructed under the assumption that there is a separation in scales between the slow dynamic forcing and

26 the fast convective response. At higher resolution, such a separation in scales becomes less obvious (Gerard et al., 2009). Convection schemes using prognostic closures have been explored in order to avoid the strict enforcement of quasi-equilibrium. Pan and Randall (1998) introduced a prognostic convective kinetic energy equation in the Arakawa-Schubert, 1974 scheme. Gerard and Geleyn (2005) and Gerard et al. (2009) introduced a prognostic updraft vertical velocity and a prognostic updraft mesh-fraction in order to depart from the quasi-equilibrium hypothesis.

Diagnostic closure in ECMWF The cumulus convection scheme in the ECMWF model is explained in Tiedtke (1989), Gregory et al. (2000) and Bechtold et al. (2004, 2008). It is a bulk mass-flux scheme with a trigger procedure that determines the occurrence of convection. The total cloud mass-flux considers the updraft (UD) and downdraft (DD) change of mass-flux with height, η, scaled with the updraft mass- flux at the cloud base (subscript B), and the downdraft mass-flux at the cloud top (subscript T). The cloud mass-flux in Eqns. 3.9 and 3.10 is thus written as (Gregory et al., 2000):

UDηUD DDηDD Mc = MB + MT (3.11) The change of the mass-flux with height, η is determined by specifying the entrainment (ε) and detrainment (δ) rates. These are split up between dynamical entrainment due to larger-scale organized inflow (εdyn), and turbulent entrainment caused by mixing at the cloud edge (εturb); similarly, the detrainment is split up between a dynamic δdyn and turbulent δturb component (Bechtold et al., 2008): ∂ 1 M η = M− = εdyn + εturb δdyn δturb (3.12) ∂z − − εturb depends on the saturation specific humidity, and εdyn is based on relative humidity. The scheme uses a CAPE closure which in practice means that η is scaled with the updraft mass-flux at the cloud base, and the downdraft mass-flux at the cloud top, and these are related to the consumption of CAPE over a convective time-scale, τ. In the current version of the ECMWF model, τ is set proportional to the convective turnover time scale as: τ = f (N) H , where ωu f (N) is a linear function of the horizontal resolution, H is the cloud depth, and ωu is the updraft velocity in the cloud (Bechtold, 2008)

Prognostic closure in ALARO The cumulus convection scheme in ALARO is explained in Gerard et al. (2009) and references therein. In ALARO, where deep convection is in the “grey-zone”, the mass-ﬂux is described using a prognostic closure. Using prognostic variables addresses both space and time-scale inconsistencies,

27 allowing precipitation to fall in another grid-box and at another time-step than where and when the convective conditions initially appear (Gerard et al., 2009). Furthermore, contrary to the procedure ﬁrst proposed by Yanai et al. (1973), the evolution equations of heat and moisture due to cumulus convection are expressed directly in the ALARO model, without introducing the cloud budget equations, and assume steady-state. Approximations are then made, leading to a system of equations from which detrainment no longer requires speciﬁcation (Piriou et al., 2007). Eqns. 3.9 and 3.10 may instead be written as (Gerard et al., 2009):

∂s ∂sω ∂[Mc(sc s)] + ∇ sV + R = − + L(c e) (3.13) ∂t · ∂ p − ∂ p −

∂q ∂qω ∂[Mc(qc q)] + ∇ qV + = − + c e (3.14) ∂t · ∂ p ∂ p − Note, that compared with Eqns. 3.9 and 3.10, condensation is now included in the equations, and there is no longer a term representing detrainment. The cloudy mass-ﬂux is given by: ω ω u∗ d Mc = σu + σd (3.15) − g g ω σ where u∗ is the updraft vertical velocity relative to its environment, and u is the updraft mesh-fraction. ωd is the absolute downdraft vertical velocity, and σd is the downdraft mesh-fraction. Both σ and ω for the updraft and moist downdraft are given by prognostic equations. For the updraft we have:

∂ω ∂ω 2 u∗ 2 u∗ = B + Eω∗ A (3.16) ∂t u − ∂ p where B is the buoyancy force, E includes entrainment and the drag coefﬁ- cient, and the term A represents the auto-advection. The updraft mesh-fraction is:

∂σu dp δqc dp hu h = L σuω∗ + L CVGQ (3.17) ∂t Z − g Z u g Z g ¡ ¢ The left hand side of Eq. 3.17 represents the storage of moist static energy, h, through the increase of the updraft fraction. The source to the updraft mesh fraction is moisture convergence, whereas the sink is condensation in the updraft air. qc is the cloud condensation along the ascent, and CVGQ stands for resolved moisture convergence. The equation constitutes the closure of the updraft parameterization. The downdraft calculation is done similarly as the updraft. The downdraft mesh-fraction σd considers the work of the buoyancy force, where negative buoyancy of the parcel is required to generate a downdraft.

28 Explicit deep convection in AROME In the AROME model, deep convection is explicitly resolved by the model, so there is no parameterization for deep convection present. However, the mass- flux and detrainment of dry convective plumes in the boundary layer and non- precipitating shallow convection need to be specified. This is done following Pergaud et al. (2009). It is a bulk mass-flux scheme based on the eddy diffusivity mass-flux (EDMF) scheme (Soares et al., 2004), that parameterizes dry thermals and shallow convection. It is thus no longer assumed that the vertical eddy transport of heat and moisture in Eqns. 3.1 and 3.2 (and momentum) are only due to cumulus convection; the term also represent the eddy transport due to turbulent motions. As described in Seity et al. (2011), in the boundary layer, entrainment and detrainment depend on the buoyancy and on the vertical speed of the updraft, whereas in clouds, they are computed using a buoyancy sorting model as in Kain and Fritsch (1993).

3.2 Lateral communication and convective organization: results from Papers II, III and IV The effects of horizontal communication are discussed in various ways in Pa- pers II, III and IV. In Paper II, the impact on deep convection by parameterized horizontal diffusion, in this cased the SLHD (Semi-Lagrangian Horizon- tal Diffusion) is studied. The SLHD is implemented in the AROME model, with horizontal grid- spacing of 2 km, and deep convection is explicitly resolved by the model. There is no parameterization of the horizontal transport due to cumulus convection, thus the only way adjacent grid-boxes can communicate is through horizontal advection of mean grid-box variables and horizontal diffusion. The spectral horizontal diffusion in the AROME model is a fourth-order linear diffusion, with the same strength for each spectral prognostic variable (the dynamical fields of temperature, vertical and horizontal winds). The tun- ing of the numerical diffusion has been chosen to ensure as low as possible damping (Seity et al., 2011). Some prognostic variables, such as the hydrometeors, are never converted into spectral space during the model integration, therefore, they cannot be diffused using the linear spectral horizontal diffusion (Seity et al., 2011). In order to improve the forecast skill of precipitation, water condensates are diffused using SLHD. The implementation is inside the Semi-Lagrangian advection scheme and uses information from the dynamical deformation field in order to diffuse the variables in a flow-dependent manner. In Paper II, we examine how deep convective precipitation and vertical velocities respond to the use of SLHD on the dynamical fields of temperature and wind (horizontal and vertical) instead of applying it on the water con-

29 Wavelength [km]

2400 40 4 10 REF k−5/3 1 10

0 10

K −1

∆ 10 ]* −2 s 2 −2 10

−3 10 KE(K) [kgm

−4 10

−5 10

−6 10 0 1 2 10 10 10 2−dimensional total wave−number, K

Figure 3.2: Two-dimensional kinetic energy spectra of AROME model using 2 km 5/3 horizontal resolution at level 50 (close to 850 hPa) (black), k− power-law (red). densates. We study the organization of convection by examining the size of the convective structures. Furthermore, the intensity of precipitation and vertical velocity is studied. The main findings are that the intensity of convective precipitation and vertical velocities are very sensitive to the strength of the damping imposed by horizontal diffusion, and that the spatial structure and horizontal clustering of convective cells showed sensitivity to the method of imposing horizontal diffusion (flow-dependent or linear). The large sensitivity to horizontal diffusion may be explained by looking at the spatial scales of the smallest resolved waves in the model. Figure 3.2 shows the kinetic energy spectra of the reference AROME configuration (Jan- 5/3 uary 2011), used in Paper II, together with k− slope discussed in Chap- ter 2. It can be seen that, at this resolution, the model is able to reproduce the expected slope for the meso-scale, but around wavelengths of 20 km, the 5/3 modelled kinetic energy spectra is starting to fall below this k− slope. The point at which this drop in kinetic energy starts is sometimes referred to as the “effective horizontal resolution”, implying that even though the model has a grid-spacing of 2 km, the shortest wavelengths resolved by the model in its spectral representation, is about 10 times as large. Thus, even a grid-spacing of 2 km lies within the grey-zone-resolution for deep convective parameterization. Since the smallest atmospheric scales are the ones targeted with horizontal diffusion, used in order to remove unpredictable noise, the “effective hori-

30 zontal resolution” is undoubtedly sensitive to the amount (and method) of damping imposed by horizontal diffusion, as discussed in Paper II. However, as concluded in Paper II, a more pragmatic way of addressing horizontal communication between adjacent grid-boxes, rather than changing the horizontal diffusion, would be to introduce a parameterization of the horizontal effects. Papers III and IV propose such a parameterization which aims to mimic the effect of horizontal mass transport due to gravity waves triggered by convection, using self-organizational properties of a cellular automata (CA). The CA is a lattice acting in two dimensions, sub-grid of the “host” model. Each CA cell consists of a state which is either 0 or 1. The state in a CA cell can change or remain the same at the next time-step depending on the state of its surrounding eight neighbours, and the rule given to evolve the CA in time. In Paper III, sub-grid scale organization is investigated in an idealized set- ting using a shallow water (SW) model. It was examined whether a CA can be used in order to enhance sub-grid scale organization and form clusters rep- resentative of convective scales. The main ﬁnding from Paper III is that cells that organize into larger clusters with a time-scale on the order of hours, have the ability to interact with the large-scale ﬂow. Such interaction is not present using a purely random equivalent. Simulations using a single Kelvin wave show that clusters generated by the CA yield an interaction between the convective-sale and the large-scale which reduces the phase speed of the Kelvin wave - an observed behaviour of convectively coupled equatorial waves (Straub and Kiladis, 2002). Similarly to Paper II, the kinetic energy spectra is also examined in Paper III when using a wide range of equatorial wave-modes, explained in detail in Chapter 2.1. However, in Paper II energy is removed at the smallest scales by introducing more damping via horizontal diffusion. In Paper III energy is instead inserted at the smallest scales through the parameterization, which leads to an increase of energy also at the larger scales. The results support the theory of two-dimensional turbulence and the quasi-geostrophy turbulence arguments discussed in Chapter 2, in that an inverse energy cascade can be found through non-linear interactions due to a source of energy at the short- wave end of the spectra. On one hand, numerical diffusion is necessary in NWP models in order to remove unpredictable noise at the smallest atmospheric scales. On the other hand, as seen in Paper II, excessive damping by horizontal diffusion leads to dissipation of energy which causes the kinetic energy spectra generated 5/3 from the model to deviate from the expected k− slope. The parameterization proposed in Paper III, in which the effect of horizontal mass transport due to gravity waves was mimicked, instead leads to an up-scale energy cascade by organization of deep convection beyond the sub-grid. This is another argu- ment for introducing a parameterization of the horizontal effects, rather than changing the horizontal diffusion in order to enhance horizontal communication.

31 Finally, in Paper IV the CA scheme is implemented in two state-of-the-art NWP models: the global ECMWF IFS and the limited area ALARO model. In case studies using the ALARO model, the CA-scheme has potential to organize convective cells along the leading edge of the system. When implemented in the ECMWF model, the scheme has potential to modulate wintertime convective activity so that convection forming over sea could penetrate inland.

3.3 Stochastic representation of cumulus convection; results from Papers III and IV Describing the statistical effect that deep convection has on the large-scale ﬂow in a stochastic manner means that the resolved scale variables, such as the vertical and horizontal wind, or temperature would respond slightly different to the convection scheme each time the model was run. Thus, the stochastisity of the proposed parameterizations in Papers III and IV can be studied by examining the ensemble spread in resolved model variables generated from the sub-grid scheme. Such a spread is not achieved by conventional deterministic parameterizations. In Paper III, the stochasticity in the sub-grid scheme comes from a random initial condition driving the CA. It is found that the non-linearity of the CA scheme gives rise to an exponential growth of the spread of the model wind speed, although the amplitude of the spread remains modest. It is hypothesised that the spread in a full NWP model would be larger, mainly since due to the scaling of the equivalent depth in the SW model, the wind speeds are substantially lower than in reality. Therefore, the same analysis is repeated in Paper IV, where the proposed parameterization is implemented in a full NWP model. In this implementation the stochasticity in the sub-grid scheme stems from a random initial condition driving the CA, and a random seeding of new cells each time-step in regions where CAPE exceeds a threshold. It is found that a spread in the wind-ﬁeld can be found in regions where deep convection is present, and the amplitude of the spread of the wind-speed at 850 hPa now seem more reasonable than in the SW experiments in Paper III. The maximum amplitude of the spread was comparable to the maximum amplitude found in the operational ECMWF Ensemble Prediction System for the same case. As concluded in Paper IV the ensemble spread generated by the CA scheme can be viewed as the uncertainty due to sub-grid variability of deep convection. It could be an interesting complement to an ensemble prediction system with initial perturbations aimed for the synoptic scale, and serve as “stochastic physics”, which are part of the parameterization in the deterministic model.

32 4. Future Outlook

Representing cumulus convection in weather and climate models presents an arduous challenge. First of all, as highlighted in this thesis, convection in the atmosphere represents a wide range of scales, which need to be considered in the model’s parameterization. Furthermore, the vertical transport generated by convection is strongly inter-linked to the vertical transport generated by turbulent eddies in the boundary layer, however, these processes are often represented by two separate parameterization schemes. The convective clouds are also strongly inter-linked with the model’s cloud and microphysical represen- tations, as clouds generated in the convection scheme contribute to the total cloud fraction, and evaporation of precipitation below convective clouds stabi- lizes the sub-cloud layer. Consequently, a desirable direction for the representation of sub-grid scale physical processes in weather and climate models as a whole would be to parameterize the effects of turbulence, convection, clouds and cloud microphysics in a more general and unified manner. In connection, another desirable path would be to represent the uncertainty associated with sub-grid scale processes in a unified manner. The general con- sensus today is to do so by targeting the physical parameterization schemes with respect to their inherent uncertainties. Such parameterizations that explicitly introduce random elements to represent the uncertain response to a given forcing were explored in Papers III and IV of this thesis with focus on deep convection. Can similar methods be applicable on other sub-grid processes? For the future, the challenge at hand would be to assess the amplitude of uncertainty that each physical process represents, which is further complicated by the fact that there are complex interactions between physical processes in the models. For instance, radiation uncertainty generally derives from cloud uncertainty. When it comes to parameterization of deep convection, most weather and climate models today use the bulk-mass-flux approach with underlying con- ceptual framework originating in the 1970’s. With increasing horizontal resolution of the numerical grid, some assumptions made in the early work becomes questionable. In particular the assumption that the fraction of the grid- box which is occupied by clouds is much smaller than one. Furthermore, the assumption of steady state in the cloud-environment balance equations is a crude approximation, as deep convection clearly is not a steady-state process. Lastly, as presented in the work of this thesis, horizontal effects due to deep convection are important for organization into larger-scale structures which

33 can interact with the large-scale flow. In Paper III and IV of this thesis the effect of horizontal transport was represented using cellular automata. As a future outlook, it would be interesting to go back to the evolution equations of heat, moisture and momentum and introduce a horizontal eddy transport term due to deep convection. For instance, this term could be introduced on the momentum equation such that there is an increase of divergence where there is a decrease of mass flux in the vertical, and an increase of convergence where there is an increase of mass flux in the vertical. The impact of this additional transport term could be explored at different horizontal resolutions.

34 5. Acknowledgements

I am grateful to SMHI and MISU for providing me with the opportunity to pursue my PhD part-time. In particular, I would like to thank Per Undén, Nils Gustafsson and Erland Källén for orchestrating the initial arrangements of my PhD studies. I would like to express my deepest gratitude to my supervisor, Gunilla Svensson, for accepting me as a PhD student well after the project had been defined. Your broad scientific understanding has been an invaluable resource in order for me to finish my thesis. I would in particular like to thank you for bringing me to NCAR in Boulder, Colorado. I recall summarizing my PhD project during a few hours in the Mesa Laboratory with a growing suspicion that you were regretting having accepted me as your student. However, after a few hours of listening you asked a few questions that were right to the core of my work, and I found myself impressed by the width of your expertise in the field of atmospheric science, and immensely relieved knowing I was in good hands. I would also like to thank my co-supervisor Klaus Wyser for discussions, even though we seldom met, I hope that we will collaborate more in the future. Thank you to my former supervisor Erland Källén for scientific guidance and inspiring discussions throughout my PhD work. Even though you didn’t see the end of my studies at MISU, you always found time to discuss the science, and I am truly grateful for your timely openness. Splitting my time between MISU and SMHI, and working within a large consortia of numerical weather prediction development, I have been given the opportunity to collaborate with a number of people. I would like to express a special thank you to my co-authors: Linus Magnusson, Heiner Körnich, Sander Tijm, Filip Vána,ˇ Martin Steinheimer, Peter Bechtold and Jean-François Geleyn. Hopefully our collaborations do not stop here. I am profoundly thankful to a number of people for inspiring scientific discussions, technical assistance, administrative help, or simply laugh-out-loud coffee break moments: Ulf Andræ, Magnus Lindskog, Stefan Gollvik, Per Kållberg, Karl-Ivar Ivarsson, Anneli Arkler and Eva Tiberg. A special thank you to Paul Roebber, Anastasios Tsonis and Jonathan Kahl of University of Wisconsin, Milwaukee for encouraging me to pursue my studies in Atmospheric Science in the first place. On a personal level, I would like to thank my parents for your unceasing encouragement throughout my entire life, and for supporting my pursuits no

35 matter where they have taken me. Lastly, I would like to thank my husband Joseph. Your deep understanding of science is inspiring, and your creative ideas have lead to interesting discussions which have progressed this work forward. More importantly, I would like to thank you for your never-ending encouragement and patience during this time, and for being a wonderful father to our son.

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