Background

1
Background In deriving basic understanding of
atmospheric phenomena, the analysis often
revolves around discovering and exploiting
relationships between fields, and between
locations e.g. for extratropical cyclones,
geostrophic balance relates wind and pressure,
and the tropopause has important relations to
surface development. Up until now, these
relationships have been determined by sampling
methods involving long periods of time, such as
multiple case studies, composites, and
time-series analysis. The advent of probabilistic
analyses offers the opportunity to establish
these relationships at an instant in time. We
call this new analysis technique ensemble
synoptic analysis because probabilistic samples
are derived using an ensemble technique.

• Potential Vorticity Inversion
• Typically, PV inversion involves the
  specification of balance constraints between wind
  and mass and mass and temperature, boundary
  conditions, and numerical approximations i.e.
  the inversion operator is specified. Here we
  determine the operator statistically using the
  analysis ensemble. Let
• P L X where P is the ensemble Ertel PV
  matrix and L is a matrix operator (N x N).
• X L-1 P defines the PV inversion recovering
  the state from the PV.
• XT LT PT has form A X B. Undetermined
  since A XT M x N M ltlt N.
• Solve for L using SVD for
  the M non-zero singular values of X.
• Also, since L cov(X, X)-1 cov(X, P), observe
  that L depends on the covariance between the
  state and PV. This suggests a new definition for
  balance the subspace that covaries with PV.

The Idea Probabilistic analyses provide the
best-estimate of the state of the atmosphere
(expected value), state uncertainty (e.g., the
variance), and the desired relationships between
all locations and all variables. An ensemble
Kalman filter (EnKF) is used here to generate the
probabilistic analyses assuming Gaussian
statistics. The ensemble covariance matrix
contains the information needed to perform
ensemble synoptic analysis. We illustrate the
covariance relations for an extratropical
cyclone, and show the usefulness of statistically
determined Ertel potential vorticity (PV)
operators for piecewise PV inversion. We note
that in traditional data assimilation (e.g.
3D-VAR), knowledge of established dynamical
relationships (e.g. geostrophic balance) is
imposed on the covariances and the state is
discovered. Here we turn that around and use
the covariances determined by the EnKF to
discover the dynamical relationships.
Results at 500hPa for a blob of PV near Oklahoma
show low geopotential height (dashed lines every
20 m), cyclonic circulation (left panel) and
subgeostrophic wind around the low (middle
panel). A zonal cross section shows shows warm
air above and cold air below the PV anomaly, and
maximum winds normal to the section (gold lines
every 5 m/s) at the level of the PV anomaly
(right panel).
advantages
disadvantages

• invert unapproximated Ertel PV.
• no balance assumptions.
• no boundary conditions, map factors, SOR.
• irregular boundaries grids are easy.
• linear model straightforward superposition.
• trivial coding (10 lines of Matlab code)

• results depend on ensemble size (M).
• M may need to be large.
• Spurious cov for large distance.
• L not unique.
• matrices get very large.

The Method Probabilistic analyses are generated
using a limited-area EnKF for the WRF model (in
collaboration with C. Snyder, NCAR). The model is
run for 100 ensemble members with 100 km
horizontal resolution, 28 vertical levels, and
warm-rain microphysics. Further details are given
in poster 1.10 in the WAF/NWP conference.
Observations consist of 250 randomly spaced
surface pressure observations sampled from a
truth run using GFS boundary conditions. Here we
focus on analyses at 06 UTC 29 March 2003.
Summary of Ensemble Synoptic Analysis Uses
probabilistic analyses to discover kinematic and
dynamic relationships. Allows a powerful array
of statistical tools to be used on instantaneous
fields. Simplifies piecewise Ertel PV
inversion. May be useful in mesoscale
applications, where conventional balances are
intermittent.
This poster may be downloaded from
http//www.atmos.washington.edu/hakim/hakim_torn_
ams03.ppt