Adding cloud to the Met Office variational analysis system
Sharpe, M. C.
Met Office, FitzRoy Road, Exeter EX1 3PB, United Kingdom; martin.sharpe@metoffice.gov.uk
The strategy for assimilating cloud information in the Met Office 3D and 4D variational data assimilation
systems is to use a total-water control variable in the analysis. Assimilating any moisture information then
requires an operator to partition total-water increments into vapour and cloud components, for use by
observation operators. Conceptually, this partitioning is part of the incrementing operator, which maps
increments from linear model space to non-linear model space and, as such, is separate from the
observation operators but may be non-linear. An operator that considers liquid cloud only and allows
calculation of increments to bulk cloud fraction has been developed, based on the Met Office Unified Model
diagnostic large-scale cloud scheme. Its formulation, linearisation and application within the Met Office
system are described and test results presented.
1. Introduction
2. Incrementing operator formulation
The current Met Office variational analysis system
produces water vapour analyses using a single
moisture control variable (relative humidity).
Motivated by the possibility of assimilating
information from cloud cover analyses and cloud
affected
satellite
observations,
a
revised
incrementing operator for cloud and moisture is
being developed. The new operator is designed to
be used with a total-moisture control variable and
provide increments to water vapour and cloud
water, for use by the observation operators. While
the calculation of these increments may be nonlinear, a linearisation and adjoint are required; the
adjoint is a necessary part of the assimilation
algorithm and the linear forward code may be used
as a computational expedient.
The Met Office variational analysis system is
designed to use non-linear operators and their
linear versions, in the mapping from linear model
space to observation space. The diagnostic liquidcloud scheme used by the forecast model provided
a suitable basis from which to develop an operator
that would derive cloud and moisture quantities
from total-moisture increments produced by the
linear model, but a number of adaptations were
required:
The revised incrementing operator, based on the
Met Office Unified Model diagnostic large-scale
cloud scheme described by Smith 1990, currently
considers
total-moisture
increments
as
representing changes to only water vapour (q) and
liquid cloud (qcl), allowing changes in liquid cloud
fraction (Cl) to be diagnosed. The sections that
follow outline the formulation of this operator and
its linearisation. Some early results and potential
methods for introducing ice cloud into the operator
are also presented along with an outline of the
requirements for further testing and development
of the scheme.
1. remove threshold dependant switches: to
enable straightforward linearisation and
allow cloud observations to influence the
minimisation, even in regions where the
original cloud scheme would report a zero
gradient with respect to its input variables
2. use temperature as an input variable: to
allow the temperature increments output
from the linear model to be applied in a
scheme whose original formulation takes an
input expressed as liquid water temperature;
TL = T – qcl.Lc/Cp
3. output saturated states: to allow adaptation
1 to be used while still reproducing the
features seen in forecast model output and
regularizing the scheme to give sensible
gradients in saturated regions
Adaptations 1 & 2 have been achieved by
changing the function used to describe the gridbox joint-pdf of TL and total specific humidity (qT),
from the triangular distribution used in the forecast
model scheme to a sech2 distribution. The third
adaptation has been achieved by changing the
values of ∂qsat/∂T used; from those at TL, as in the
original formulation, to those at T.
With these adaptations applied, Figure 1 shows
the relationships between cloud fraction and
relative humidity for the simplified scheme
alongside those used in the forecast model, in
which slightly different schemes are used for global
and local area domains.
later calculations to take T increments their as
input. The resulting cloud increments are then
consistent with the changes in total-moisture and
energy, relative to the background states used by
the initial calculation.
Equation 1 shows how increments (denoted by
primes) are calculated for qcl. This method of
adding increments to ‘pseudo-background’ values
derived using TL as an input variable, also applies
to other moisture variables.
Equation 1: q’cl
is
calculated
using
‘pseudobackground’
values
derived
from
background values of TL, p & qT.
3. Linearisation
Although the need for a linear operator was part of
the reason for adapting the forecast model cloud
schemes, the non-linear equations are solved
using a fixed-point iteration. This results in a
technical difficulty with linearisation; the need to
use non-linear iterates in a line-by-line approach
would require either a large amount of extra
storage or repetition of the iterative calculations.
However, taking an alternative approach uses
approximate gradients, moving the analysis
minimisation away from the intended solution.
Figure 1:
The cloud fraction-vs-relative humidity
diagnostic resulting from the global forecast
model cloud scheme (black), the local area
model scheme (blue), and the simplified
version (red)
Early on in the testing of the simplified cloud
scheme, on which the incrementing operator is
based, using TL as its input variable was found to
provide a better energy constraint than using T;
giving cloud liquid water values much more like
those produced in the forecast models.
Unlike the forecast model schemes, the simplified
version is invertible; the original value of T L can be
recovered by applying the scheme to the T values
obtained when using TL as its input. This means
that an initial calculation, using T L as input, allows
Following an approach suggested by Polavarapu &
Tanguay 1998 the analytic equations are linearised
with the intention of using the resulting equations
as an approximation to the line-by-line linearisation
of the non-linear code. This is equivalent to
assuming that the non-linear iteration has
converged to the analytic solution. Linearisation
test results using line-by-line and analytic
approaches showed that, even in regions of large
increments, the two linearisations agreed to 3 s.f.
and globally averaged statistics were also similar.
Figure 2 shows an example of the linearisation test
output, for globally averaged correlations between
qcl increments calculated using the non-linear and
linear simplified schemes.
The conclusion from the linearisation tests is that
although the analytic linearisation may not be
perfect, with respect to the non-linear iteration
code, it is acceptable for use in the analysis
system given that the simplified cloud scheme is
already an approximation to that used in the
forecast model.
largely similar. More impact on verification scores
may be obtained when the scheme is used
alongside a newly implemented latent heating
scheme for the linear model and applied in a
system with finer grid spacing.
Figure 2:
Area-weighted correlations between nonlinear and linearised qcl increments for lineby-line linearisation (left hand figure) and for
the analytic linearisation (right hand figure)
4. Testing
Initial testing of the simplified cloud scheme
confirmed that the adaptations do result in an
adequate reproduction of the cloud and water
vapour values diagnosed in the forecast model and
the linearisation tests confirmed that using the
analytic method is a reasonable approach.
Initial results of using the new incrementing
operator within a global 4D-Var system (but not
using
cloud
observations)
showed
small
differences in the evolution of the penalty function
during minimisation; apparently fitting the
observations less closely.
The operator was then trialed in a reduced
resolution (N48) 4D-Var version of the Met Office
global NWP system, over a month-long period
during the summer of 2004. In this trial, a change
to the initialization of the forecast model was also
introduced; to derive increments in water vapour
and liquid cloud, given total-water analysis
increments, by applying the same method as is
used in the incrementing operator. Although the
moisture observations used in the trial related only
to water vapour, the results are of value in
assessing the effectiveness of initialising cloud
fields from the total-water analysis. (When the trial
was carried out, the linear model used in the Met
Office 4D-Var system did not include any latent
heating terms.)
The trial results showed a neutral overall impact on
rms scores in forecast verification against both
analyses and observations. However, the
background fit to observations was improved for
nearly all observations that are sensitive to water
vapour, as shown in Figure 3.
Additionally, as would be expected from the
improved background fit to observations, the size
of the analysis increments was reduced for all
variables, even though the overall synoptic
evolution of both the test and control runs was
Figure 3:
The improvement in initial fit to observations
when applying the incrementing operator for
vapour and liquid cloud over a 31 day global
NWP system (points lower down on the plot
show those observations that have the
largest effect on the analysis)
5. Further developments to include frozen
cloud in the incrementing operator
In order to take full advantage of current and future
observations, especially those from satellites and
cloud analysis products, the incrementing operator
must be extended to diagnose increments to cloud
ice content and grid-box ice-cloud fraction. It is
hoped that a diagnostic approach will simulate the
prognostic ice scheme used in the forecast model
sufficiently well that the liquid cloud scheme
already developed will not need to be changed
radically in order to incorporate a scheme for ice.
Given cloud ice increments, calculating the gridbox ice-cloud fraction for use with cloud analysis
data is straightforward; following the method
applied in the forecast model.
Work in this area has focused, so far, on assuming
an equilibrium between sublimation & deposition,
as used (but not run to convergence) in the
existing Met Office forecast system. An alternative
approach, intended to make better use of
background information, is also being investigated.
It is based on diagnosing the cloud ice content by
using the scheme for liquid cloud in combination
with a parameterization of increments to the
partitioning of cloud water between liquid and ice.
Although this work is still in its early stages, it is
clear that the form of the equilibrium assumption
will need to be modified in order to obtain a smooth
transition between cold and warm regions (where it
does not apply). Additionally, to obtain ice cloud
increments that are not unreasonably large, it may
be necessary to constrain the calculated
increments to the equilibrium state by requiring
them to reproduce approximate conservation
properties of cloud ice evolution in the forecast
model. If using the parameterized partitioning
approach, care is required in constraining the
incremented partitioning to remain within physical
limits.
 Developing the incrementing operator to
include ice cloud, in terms of cloud water
and grid-box cloud fraction, is under way
and must include work to derive increments
to bulk cloud fraction; combining increments
in liquid and frozen cloud fractions
 Testing the completed operator alongside
the planned Met Office prognostic cloud
scheme will be necessary, as reformulation
of the cloud incrementing operator may be
required; when the new cloud scheme is
used in the forecast model, the incrementing
operator must adequately approximate
increments that would be obtained by
applying the forecast model scheme.
6. Future work
Acknowledgements:
 Testing the incrementing operator for vapour
and liquid cloud alongside a newly
implemented latent heating scheme, in the
linear model used with the Met Office 4DVar system, and in a system with finer grid
spacing is under way
 Testing the performance of the incrementing
operator for vapour and liquid cloud in
assimilating cloud information is planned to
consider 23 & 31 GHz AMSU-A channels
and the 19GHz SSMI channel, when the
required work on satellite data assimilation
code is completed
Sue Ballard & Damian Wilson: Guidance on cloud
scheme formulation
Rick Rawlins: Analysis of results from the global 4D-Var
trial
References:
Polavarapu, S., Tanguay, M., Linearizing iterative
processes
for
four-dimensional
data-assimilation
schemes. QJRMS, 124, 1715-1742, 1998
Smith, R. N. B., A scheme for predicting layer clouds
and their water content in a general circulation model.
QJRMS, 116, 435-460, 1990