Significant accomplishments during 1987 to 1988 are summarized with regard to each of the major project components. Model 1 requires satisfaction of two nonlinear horizontal momentum equations, the integrated continuity equation, and the hydrostatic equation. Model 2 requires satisfaction of model 1 plus the thermodynamic equation for a dry atmosphere. Model 3 requires satisfaction of model 2 plus the radiative transfer equation. Model 4 requires satisfaction of model 3 plus a moisture conservation equation and a parameterization for moist processes

Achtemeier, Gary L.

English

NASA Technical Reports Server

Ngo-' . 1 o644
TITLE: Variational Objective Analysis for Cyclone Studies
INVESTIGATOR:
Gary L. Achtemeier
Climate and Meteorology Section
Illinois State Water Survey
2204 Griffith Drive
Champaign, IL 61820
(217) 333-0689
SIGNIFICANT ACCOMPLISBMENTS DURING THE PAST YEAR:
Five major components of this project are:
i. MODEL I. Requires satisfaction of the two nonlinear horizontal momentum
equations, the integrated continuity equation, and the hydrostatic
equation.
2. MODEL II. Requires satisfaction of MODEL I plus the thermodynamic equation
for a dry atmosphere.
3. MODEL llI. Requires satisfaction of MODEL II plus the radiative transfer
equation. Brightness temperatures and skin temperature are introduced as
dependent variables.
4. MODEL IV. Requires satisfaction of MODEL III plus a moisture conservation
equation and a parameterization for moist processes. Moisture is
introduced as a dependent variable.
5. INITIAL A_YSIS. Makes the variational models more responsive to the
observations. Thorough analysis of all phases of the data
representation, grldding and initial analysis/model interface.
Significant accomplishments during 1987-88 are summarized with regard to each
of the major project components.
I. MODEL I
A. The completion of MODEL II allowed for the testing of the variational
method for the complete set of dynamic equations. The tests were determined
whether the MODEL I vertical velocity formalism was posed correctly as it
wasn't as rigorous as are the other model formalisms. The results of the test
revealed that there was little or no coupling of the "observed" vertical
velocity with the other adjusted variables. In other words, the adjustment
simply didn't "see" the divergent part of the observed wind. It was found
that the adjusted vertical velocities converged toward a solution of the
vorticity theorem that gave vertical velocity patterns similar to those of one
of the NMC numerical prediction models that was operational during 1979.
This information was used to develop a new MODEL I that includes as
constraints the two nonlinear horizontal momentum equations, the hydrostatic
equation, and a particular solution of the vorticlty theorem that satisfies
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the constraints that the vertical integrals of the vorticity theorem and of
the divergence vanish at the top of the model domain. In theory, this
formulation strongly couples the divergence of the observed wind into the
adjustment for th8 winds and there are no "variational models within
variational models" as was required for the first version of MODEL I.
The variational equations for the new version, MODEL 1.2, are more
complex than for the old version, MODEL I.l. After the higher order terms are
moved into forcing functions, the equation set reduces to three diagnostic
equations; geopotential height, divergence, and vorticlty. The latter two are
solved for the velocity potential and stream function in order to get the
adjusted winds. Boundary condition problems inherent in these Polsson
equations are well known. We tested four methods for retrieving the velocity
potential and steam function (BiJlsma, 1986, MWR 114, 1547-1551; Shukla and
Saha, 1974, MWR 102, 419-425_ Endlich, 1967, J. METEOR, 6, 837-844; Schaefer
and Doswell, 1979, MWR 107, 458-476.) The method of Schaefer and Doswell
seems to work best for the staggered grid of MODEL 1.2. This problem of
boundary conditions is a subject of continuing investigation including
personal communications with R. Endlich.
Theoretical development and programming of MODEL 1.2 (extensive
programming was required to implement MODEL 1.2) have been completed and tests
will begin soon.
%
B. Precision modull sensitivity testing was done with MODEL I.l to determine
which variables are crucial to the convergence of the method. Fifteen runs of
MODEL I.l, beginning with "standard" precision moduli calculated from
observations or formulas linking with observations, were done with standard
precision modull selectively multiplied by 0.1 or 10.0. RMS residuals, a
measure of how well the constraints are satisfied, were plotted for each cycle
out to four cycles. The MODEL I.l solution method was found to be sensitive
to the horizontal velocity (if too accurate) and the developmental component
of the horizontal velocity tendencies (if too inaccurate); the solution for
the horizontal momentum equations diverged. A similar sensitivity test will
be done for MODEL 1.2.
II. MODEL If.
MODEL II was completed near the end of the first year of the current 3-
year effort. As reported in the NASA MSFC FY87 Global Scale Atmospheric
Processes Research Program Review, slow divergence was found in the solution
for the middle layers of the model atmosphere. SubsequeSt analysis of the
behavior of MODEL II has revealed the following:
a) The adjusted vertical velocities in MODEL I were weakly coupled with the
initial vertical velocities. The solution wandered toward the vortlcity
equation rather than the continuity equation. The solution order that
readjusted the winds to satisfy the continuity equation did not force the
solution toward the initial vertical velocity. A solution for the problems
with vertical velocity coupling has been briefly described in connection with
MODEL 1.2.
b) The initial vertical velocity must not have large error or else the
adjusted temperature tendencies will have large error. Kinematic vertical
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velocity methods make no allowance for the discontinuity in stability between
the troposphere and stratosphere. Large stratospheric kinematic vertical
velocities produce large erroneous temperature tendencies. In an attempt to
correct this problem, Barb Chance coupled adiabatic vertical velocities with
the kinematic divergence of the horizonal wind through a variational algorithm
that weighted the velocities through static stability, relative humidity and
cloudiness. Her method was modified to blend adiabatic and kinematic vertical
velocities through static stability alone.
III. MODEL III.
Three variational models that incorporate the four radiative transfer
equations for the four TOVS microwave channels as constraints and the four
brightness temperatures and the skin temperature as new dependent variables
are in various stages of development. These models are:
a) MODEL 3a consists of
the radiative transfer
equations with rawinsonde
temperature and
brightness temperatures
as adjustable variables.
It was apparent that this
formalism would be Just
another retrieval
algorithm. Therefore the
main reason for
derivation of MODEL 3a
was to gain understanding
of the behavior of four
integral equations as
dynamic constraints.
Solutions were by matrix
inversion and an
iterative technique. An
unexpected result from
MODEL 3a was that the
retrieval is sensitive to
the vertical distribution
of the temperature
weights. Figure I gives
an example of the
response of the
adjustment to the
vertical distribution of
temperature weights. The
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700
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standard atmosphere was used as the true temperature and the true brightness
temperatures were calculated from it. Then a 3C temperature anomaly (dashed
line in Fig. I) was introduced into the first guess temperature and MODEL 3a
was asked to retrieve the true temperature. The best retrieval more closely
matches the anomaly curve. The curve connected by the squares in Fig. I shows
the results of this effort when the temperature weights were uniform (as is
done with most retrieval methods). If the temperature uncertainty is doubled
at the location of the anomaly, then the retrieval follows the line given by
19
the pluses; the correct temperature profile has been mostly restored. This
finding may be useful for retrievals across the tropopause.
b) MODEL 3b consists of MODEL 3a plus the u and v geostrophic equations and
the hydrostatic equation. This model couples the satellite temperatures to
the three dimensional wind and height fields. It was tested with the 00 GMT
ii April 1979 SESAME data with the result that the SAT analysis filled a deep
trough over the high plateau areas of the southwest U.S. by 30 m. This
anomaly may be traceable to high skin temperatures. (Skin temperature was
treated as true in MODEL 3b.)
c) MODEL 3c consists of MODEL 3b but with skin temperature as an adjustable
variable. Varying the skin temperature increases the complexity of the
variational equations. MODEL 3c is still under development.
IV. MODKL IV.
No significant progress has been made to date.
V. INITIAL ANALYSIS
a) A three-pass version of the Barnes objective analysis method has been
implemented. This method has been found to reduce short wavelength noise in
derivatives of gridded fields by up to 70% (Laplacians of height fields.)
b) A vertical velocity method that blends adiabatic vertical velocities with
the divergence of the observed wind has been implemented. The blending is
done as a function of the static stability.
RESEARCH PIANS I_OR'I'HEI_J_"TY']_.R
I. Complete MODEL 1.2, run, and verify (top priority)
2. Combine the new MODEL 1.2 with the thermodynamic equation to produce a new
MODEL II.
3. Complete the studies with the radiative transfer models (3a-3c) and develop
theory to combine with model II into model III.
4. Develop a variational methodology for moisture.
5. Investigate a new objective interpolation method developed by F. Caracena
(JAS 44, 3753-3768) that allows for the direct interpolation of derivatives
from the observations and modify the vertical velocity algorithm to include
vorticlty advection.
LIST OF PUBLICATIONS PREPARED IN 1987-1988
Achtemeier, G. L., 1987: On the Concept of Varying Influence Radii for a
Successive Corrections Objective Analysis. Mon. Wea. Rev._ I151 1760-1771.
2O
Achtemeier, G. L., 1987: A 3-pass near optimum Barnes method for the
univariate objective analysis of mesoscale phenomena. 3rd Conf. Mesoscale
Proc., Vancouver, B.C., Canada, 21-26 Aug. 1987, 126-127. (Poster paper)
Achtemeier, G. L., 1988: Varlatlonal Blending of Space Observed Radiance with
Conventional Weather Data Through Coupling Radiative Transfer with the
Truncated Navler-Stokes Equations. European Geophysical Society XIII General
Assembly, Bologna, Italy, 21-25 March 1988, Annales Geophysicae_ 6_ 97.
Achtemeier, G. L., and H. T. Ochs III, 1988: A Variational Objective Analysis
- Assimilation Method, Part I: Development of the Basic Model. (Submitted to
Tellus 20 April 1988)
Achtemeier, G. L., S. Q. Kidder, and R. W. Scott, 1988: A Multivariate
Variational Objective Analysis Assimilation Method, Part II: Case Study
Results with and without Satellite Data. (Submitted to Tellus 20 April 1988)
Achtemeier, G. L., 1988: Modification of a Successive Corrections Objective
Analysis for Improved Higher Order Calculations (Submitted to Mon. Wea. Rev.
15 April 1988)
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Variational objective analysis for cyclone studies

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