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. Convective processes interact with radiation, redistribute sensible and latent heat
and momentum, and impact hydrological processes through precipitation. Depending 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 “largescale”. 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.
3
List of Papers
This thesis is based on the following papers, which are referred to in the text
by their Roman numerals.
I
II
III
IV
Bengtsson, L., Magnusson, L., Källén, E. (2008) Independent
Estimations of the Asymptotic Variability in an Ensemble Forecast System Monthly Weather Review, Volume 136, Issue 11 pp.
4105-4112. (c)American Meteorological Society. Reprinted with
permission.
Bengtsson, L., Tijm, S., Váňa, 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.
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.
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 Society. 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 Váňa, 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 scientific discussion together with Heiner Körnich, Erland Källén and Gunilla
Svensson. The general idea behind Paper IV was my own, as a natural exten5
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 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.1
Representing model uncertainty: ensemble prediction . . . . . . . . . . . . . . . . . . . . .
1.2
Asymptotic error growth: results from Paper I . . . . . . . . . . . . . . . . . . . . . . . . . .
2 Weather prediction at differing atmospheric scales . . . . . . . . . . . . . . . . . . .
2.1
Equatorial wave modes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3 Uncertainty due to parameterization of cumulus convection . . . . . . . . . . . .
3.1
Parameterization of cumulus convection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.1.1
Cloud model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.1.2
Closure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2
Lateral communication and convective organization: results from Papers II, III and IV
3.3
Stochastic representation of cumulus convection; results from Papers III and IV . . . .
4 Future Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5 Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9
10
12
15
17
21
23
23
26
29
32
33
35
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, massconservation 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
k+F
= −α ∇p − 2Ω × V − gb
dt
(1.1)
∂ρ
(1.2)
= −∇ · (ρ V)
∂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:
Q = Cp
dT
dp
−α
dt
dt
(1.3)
∂ ρq
(1.4)
= −∇ · (ρ Vq) + ρ (e − c)
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 sensing 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 Environmental 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 dependency 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 unperturbed 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 assimilations 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 parameteri11
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 extrapolated 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 (ECONT ROL ). 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 ECONT ROL should be equal to that of
EMEMBERS .
q
√
2
+ S2 and also equal 2EMEAN
• EMEMBERS should saturate towards EMEAN
•
√
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, ECONT ROL , 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 research addressed in Papers III and IV.
13
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).
Each spectrum in Figure 2 has two different distinct slopes; k−3 powerlaw on the synoptic scale, and k−5/3 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 spectrum remains elusive. Kraichnan (1967) predicted a k−3 power-law for twodimensional, 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.
As further summarized in Tung and Orlando (2002), the k−5/3 power-law
has been derived theoretically by Kolmogorov (1941a,b) to occur for threedimensional 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 including the effect of Earth’s rotation and allowing horizontal divergence. Following 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
of convective anvils and gravity waves, consistent with the observed k−5/3
slope. The important role of gravity waves to generate the k−5/3 slope was
also suggested by Koshyk and Hamilton (2001).
Even though there is some uncertainty on exactly how to theoretically explain 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 shortwave 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 scaleinteraction 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 − ω t)]
h
h(y)
(2.1)
where k is the zonal wave number and ω is the frequency. Substitution and rearrangement reduces the set of equations to a single, second-order differential
equation in v:
ω2
β 2 y2
k
d2v
2
+
[(
−
k
−
)]v = 0
β
−
dy2
c2
ω
c2
(2.2)
√
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:
v(x, y,t) = Dn (y) ei(kx−ω t)
(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
y
y
) = 2−n/2 exp[−( p
)2 ]Hn ( √ p
)
(2.4)
Dn ( p
c/2β
2 c/2β
2 c/2β
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 u are 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:
c/β (
ω2
k
− k2 − β ) = 2n + 1 n = 0, 1, 2, ...
2
c
ω
(2.5)
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
ω
10
Kelvin
EMRG
WMRG
Rossby n=1
Rossby n=2
Rossby n=3
WIG n=1
WIG n=2
WIG n=3
0
10
−1
10
−2
10
−2
−1
10
0
10
10
1
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
p and inertia-gravity waves are
c/2β , and frequency, ω , is norpresented. Zonal
wave-number,
k
is
normalized
by
p
malized by 2cβ
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:
(
n
k 2
) + ( )2 ≤ 1
Nm
Nn
(2.6)
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 fre18
quency ωc appropriate for mid latitudes based on Figure 2.2 would be:
r
βc
(2.7)
ωc =
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
γ 2 (K m ) =
(2.8)
{1 + L2 (K m )2 }2
and is applied to the eigenmodes which satisfy
p 2.6 and 2.7. Km is a one dimensional wave number and defined as Km = (c1 k)2 + (c2 n)2 . The total amplitude is decided by Γ, and L is the horizontal 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.
19
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. Environmental 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 hori21
Figure 3.1: Schematic view of how the cumulus convection parameterization influences 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; Váňa et al., 2008; Gerard et al., 2009) used in
the same paper has a horizontal grid-spacing of 5.5 km; this scale is commonly 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 dis22
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) −
sω
∂t
∂p
∂p
(3.1)
∂q
∂ qω
∂ ′ ′
qω
+ ∇ · qV +
= e−c−
∂t
∂p
∂p
(3.2)
where s is the dry static energy: s ≡ Cp T + 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 differ23
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 ′ ≈ −σ ωc (sc − s) = Mc (sc − s)
(3.3)
− q′ ω ′ ≈ −σ ωc (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:
∂ Mc
= E −D
∂p
(3.5)
∂ (Mc sc )
= Es − Dsc + Lcc
∂p
(3.6)
∂ (Mc qc )
= Eq − Dqc − cc
∂p
(3.7)
∂ (Mc l)
= −Dl + cc − r
∂p
(3.8)
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)
∂t
∂p
∂p
(3.9)
∂q
∂ qω
∂q
+ ∇ · qV +
= Mc
+ D(qc − q) + e
∂t
∂p
∂p
(3.10)
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 massflux 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 action 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 massflux 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):
Mc = MBUD η UD + MTDD η DD
(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):
∂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) ωHu , where
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)
η = M −1
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)
∂t
∂p
∂p
(3.13)
∂q
∂ qω
∂ [Mc (qc − q)]
+ ∇ · qV +
=
+c−e
∂t
∂p
∂p
(3.14)
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:
Mc = −σu
ωu∗
ωd
+ σd
g
g
(3.15)
where ωu∗ is the updraft vertical velocity relative to its environment, and σu 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:
∂ ωu∗
∂ ω ∗2
= B + E ωu∗2 − A u
∂t
∂p
(3.16)
where B is the buoyancy force, E includes entrainment and the drag coefficient, and the term A represents the auto-advection. The updraft mesh-fraction
is:
∂ σu
∂t
Z ¡
Z
Z
¢ dp
dp
∗ δ qc
hu − h
= L σu ω u
+ L CV GQ
g
g
g
(3.17)
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 CV GQ 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 massflux and detrainment of dry convective plumes in the boundary layer and nonprecipitating 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 Papers II, III and IV. In Paper II, the impact on deep convection by parameterized horizontal diffusion, in this cased the SLHD (Semi-Lagrangian Horizontal Diffusion) is studied.
The SLHD is implemented in the AROME model, with horizontal gridspacing 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 tuning 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
10
40
4
REF
k−5/3
1
10
0
KE(K) [kgm2s−2]*∆K
10
−1
10
−2
10
−3
10
−4
10
−5
10
−6
10
0
10
1
10
2
10
2−dimensional total wave−number, K
Figure 3.2: Two-dimensional kinetic energy spectra of AROME model using 2 km
horizontal resolution at level 50 (close to 850 hPa) (black), k−5/3 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 (January 2011), used in Paper II, together with k−5/3 slope discussed in Chapter 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
modelled kinetic energy spectra is starting to fall below this k−5/3 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 hori30
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 setting 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 representative 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 shortwave 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
from the model to deviate from the expected k−5/3 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 argument 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 representations, as clouds generated in the convection scheme contribute to the total
cloud fraction, and evaporation of precipitation below convective clouds stabilizes 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 consensus 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 conceptual 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 gridbox 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 Váňa, 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.
36
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