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The control of gravity waves
in data assimilation
27 April 1999
By Adrian
Simmons
European Centre for Medium-Range Weather
Forecasts
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3 . Control of gravity waves in the ECMWF variational data assimilation system
The basic variational data assimilation problem is to determine the model state
that minimizes a scalar cost function
. 
comprises three elements:
|
(65) |
where
is the background cost function defined in
terms of the deviation of the analysis from the background state 
and 
is the observation cost function defined in terms of the deviation
of the analysis from the observations in the case of three-dimensional variational
assimilation (3D-Var). In the case of four-dimensional assimilation (4D-Var),
is the observation cost function defined in terms of the deviation
from the observations of a forecast from the state 
. The two elements 
and 
are discussed in the companion lecture notes on Data assimilation
concepts and methods. The primary control of gravity waves in the data
assimilation comes in general through a multivariate formulation of the
term, although in 4D-Var a contribution
can come through the model integration that is implicit in the term 
.The term
represents additional constraints on
the analysis. These could include physical constraints, for example that
the relative humidity be between 0 and 100%, or a constraint on high frequencies
based on digital filtering. In the current implementation at ECMWF, 
is given by
|
(66) |
where
denotes the projection of the tendency 
onto gravity waves and 
denotes a simple energy-based norm defined by a weighted sum of squares
of spectral coefficients. The overall weighting factor 
was adjusted during the development of ECMWF's 3D-Var
system to remove oscillations in surface pressure (typically a fraction
of an hPa in magnitude) that were found to occur in the absence of 
(Courtier et al., 1998).Examples of the evolution of the cost function
and its three components over 70 iterations of a 3D-Var minimization
are presented in Fig. 11 . The upper panel shows how in absolute
terms 
makes only a tiny contribution to the
overall cost function. The background state 
is used as the starting value for the minimization, and 
and 
are thus initially zero. 
is subsequently decreased substantially1, at the expense of some increase in 
and a slight increase in 
. The plots
in the lower panel of Fig. 11 indicate an increase in small-scale
gravity-wave activity when 
is excluded
from the minimization (but computed as a diagnostic). In this case 
grows to about twice the value that develops when it provides
part of the constraint on the analysis. It should be noted that the elements
of the cost function are defined globally; despite its small overall value
may provide
an important local constraint, close to steep orography in particular.The variational data assimilation scheme is implemented in an incremental form in which the minimization is carried out at a lower resolution than that of the background forecast. Until March 1999, initialization was applied twice in the procedure. The low-resolution analysis requires an interpolated low-resolution background field, and this was initialized applying adiabatic non-linear normal-mode initialization for scales with
. Then, on completion of the low-resolution analysis, incremental
adiabatic non-linear normal-mode initialization was applied in forming the
high-resolution analysis 
:
|
(67) |
Here
is the increment of the low-resolution analysis
(the difference between the low-resolution analysis and the low-resolution
background) interpolated to the high resolution. This incremental initialization
was applied operationally only for scales with 
from May 1997 onwards, for the primary purpose of adjusting the low-resolution
analysis to the high-resolution orography. It was, however, applied to all
scales in the original operational implementation of 3D-Var, as discussed
further below.
Figure 11 . Evolution of the cost function
and its components 
, 
and 
during the minimization in a standard cycle of 3D-Var (upper panel)
and plotted on a logarithmic scale (lower panel) both for the standard cycle
(solid) and for a modified cycle (dashed) in which 
was calculated for diagnostic purposes but not activated in
the minimization.Some examples of the time evolution of the surface pressure are presented in Figs. 12 and 13 . Plots are shown for the two points 40oN 90oW and 30oN 90oE used to illustrate results from earlier studies of initialization in Figs. 7 and 8 . The forecasts were carried out after three cycles of 3D-Var using the initialization configuration in question. Fig. 12 shows the impact of excluding the constraint
. Both forecasts shown were from assimilations
in which the incremental initialization was applied on all scales, and this
is evidently sufficient to prevent high-frequency gravity-wave oscillations
whether 
is activated or not. Suppressing 
gives little change at 40oN 90oW over the Great
Plains, but has modified the starting value of surface pressure by about
0.5hPa at the Himalayan point. A pronounced semi-diurnal tidal oscillation
can be seen at the latter point.
Figure 12 . Surface pressure (hPa) as a function of time for a control forecast (solid) and a forecast following three cycles of 3D-Var in which
was not activated (dashed), at 40oN 90oW (upper)
and 30oN 90oE (lower).Fig. 13 shows the effect of removing the incremental initialization of the larger scales (
), and of removing initialization (and the
constraint) completely. Removing the larger-scale
initialization allows some high- frequency oscillations to develop, although
even at the Himalayan point the amplitude is barely over 0.1hPa. Completely
removing initialization has a larger effect, especially in the first few
steps at the Himalayan point. Gravity-wave oscillations are nevertheless
much smaller than in the forecast from the uninitialized analysis shown
in Fig. 7 , presumably because
the present forecasts come from a consistent and much more modern data assimilation
system.Further examination of these issues was carried out as part of the development of the 50-level version of the model which became operational in March 1999. It was found that
continues to play a small but useful role, but that the initialization
steps could be eliminated without significant deterioration of analysis
and forecast quality. As the amplitudes of the internal modes vary approximately
as 
for small 
, elimination of the initialization steps avoided
a problem of large initialization increments close to the top of the 50-level
model.
Figure 13 . Surface pressure (hPa) as a function of time for a control forecast (solid) and for forecasts following three cycles of 3D-Var with no large-scale initialization (dashed) and no initialization at all (dotted), at 40oN 90oW (upper) and 30oN 90oE (lower).
The removal of the incremental initialization of the scales with
was implemented operationally at ECMWF in May 1997
at the same time as a change to the background term 
, moving from the formulation described by Courtier et al.(1998) (referred
to as "old" 
) to that reported by Bouttier et al.(1997) (the "new"
). The new 
was used for the forecasts shown in Figs. 12 and 13 . Some idealized tests carried out prior
to the change provide examples of the working of the initialization and
of the background constraint.Fig. 14 shows increments in 850hPa height due to several idealized isolated observations of this field, specified such that the observed deviation from the background field was the same at each point. Results are shown for a single cycle of 3D-Var. The old
reduces to a univariate formulation in the tropics, and produces
localized increments of similar magnitude at all locations (upper-left panel).
However, the incremental initialization (applied to all scales) removes
most of the increment in the tropics (lower-left panel). The new 
imposes a semi-empirical (close to linear) balance. It produces more
of a large-scale increment, and smaller local increments in the tropics
(upper-right panel). More of each local increment survives initialization.
Incremental initialization thus plays a smaller role in imposing balance
on the analysis in the case of the new 
. This was an important factor in the decision to
remove initialization for scales 
in the operational system.
Figure 14 . Analysis increments in 850hPa height for a set of idealized height observations at 850hPa, for the
operational prior to May 1997 (left)
and that operational after May 1997 (right), with no initialization (upper)
and after incremental non-linear normal-mode initialization (left).Close-ups of the height increments at one location and the associated wind increments are presented in Fig. 15 . The multivariate formulations of the old and new
both produce increments which are close to being in geostrophic balance,
and initialization changes are much smaller than in the tropics. They are
slightly smaller with the new than the old 
. The height increments are reduced by initialization and the
wind increments are increased.The new
produces a divergent component to the
wind increment at the ground, as can be seen in the plots for 1000hPa shown
in Fig. 16 . The increment in divergence is
shallow, and survives initialization because the latter is applied only
to the first five, relatively deep, vertical modes.
Figure 15 . Analysis increments in 850hPa height and wind for a set of idealized height observations at 850hPa, for the
operational prior to May 1997 (left)
and that operational after May 1997 (right), with no initialization (upper)
and after incremental non-linear normal-mode initialization (left).
Figure 16 . Analysis increments in 1000hPa height and wind for a set of idealized height observations at 850hPa, for the
operational after May 1997, with no
initialization (left) and after incremental non-linear normal-mode initialization
(right).The thermal tide provides a final example. The variational analysis is able to "draw" to the tidal signal present in the surface pressure observations, but the signal is not fully retained in the ensuing forecast. The analysis thus produces increments which improve the description of the tides. A fraction of the improvement is lost, however, if incremental initialization is applied to large scales. Fig. 17 illustrates how the analysis generally fits better the surface-pressure observations from a frequently reporting tropical island station when the initialization is restricted to scales
.
Figure 17 . Surface pressure (hPa) from 00UTC 8 February to 18UTC 14 February 1997, as observed at Seychelles International Airport (5oS, 56oE; dashed line) and as analysed at this location (solid) with (upper) and without (lower) large-scale initialization.
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1 The sharp fall near iteration number 30 is due to the initiation of variational quality control.
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