Selection of Momentum Variables for a Three-Dimensional Variational Analysis |
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| Authors: | Yuanfu Xie Alexander E MacDonald |
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| Institution: | (1) NOAA Earth System Research Laboratory (ESRL), NOAA/OAR/ESRL/GSD, 325 Broadway, Boulder, CO 80305, USA |
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| Abstract: | Three choices of control variables for meteorological variational analysis (3DVAR or 4DVAR) are associated with horizontal
wind: (1) streamfunction and velocity potential, (2) eastward and northward velocity, and (3) vorticity and divergence. This
study shows theoretical and numerical differences of these variables in practical 3DVAR data assimilation through statistical
analysis and numerical experiments. This paper demonstrates that (a) streamfunction and velocity potential could potentially
introduce analysis errors; (b) A 3DVAR using velocity or vorticity and divergence provides a natural scale dependent influence
radius in addition to the covariance; (c) for a regional analysis, streamfunction and velocity potential are retrieved from
the background velocity field with Neumann boundary condition. Improper boundary conditions could result in further analysis
errors; (d) a variational data assimilation or an inverse problem using derivatives as control variables yields smoother analyses,
for example, a 3DVAR using vorticity and divergence as controls yields smoother wind analyses than those analyses obtained
by a 3DVAR using either velocity or streamfunction/velocity potential as control variables; and (e) statistical errors of
higher order derivatives of variables are more independent, e.g., the statistical correlation between U and V is smaller than the one between streamfunction and velocity potential, and thus the variables in higher derivatives are more
appropriate for a variational system when a cross-correlation between variables is neglected for efficiency or other reasons.
In summary, eastward and northward velocity, or vorticity and divergence are preferable control variables for variational
systems and the former is more attractive because of its numerical efficiency. Numerical experiments are presented using analytic
functions and real atmospheric observations. |
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