## Abstract

The authors show that the linear approximation errors in the presence of a discontinuous convective parameterization operator are large for a small number of grid points where the noise produced by the convective parameterization is largest. These errors are much smaller for “smooth convective” points in the integration domain and for the nonconvective regions. Decreasing of the amplitude of initial perturbations does not reduce the errors in noisy points. This result indicates that the tangent linear model solution is erroneous in these points due to the linearization that does not include the linear variations of regime changes (i.e., due to use of *standard method*).

The authors then show that the quality of local four-dimensional variational (4DVAR) data assimilation results is correlated with the linearization errors: Slower convergence is associated with large errors. Consequently, the 4DVAR assimilation results are different for different convective points in the integration domain. The negative effect of linearization errors is not, however, significant for the cases that are studied. Erroneous points slightly degrade 4DVAR results in the remaining points. This degradation is reflected in decreased monotonicity of the cost function gradient reduction with iterations.

These results suggest that there is a probability for locally *bad* 4DVAR assimilation results when using *standard adjoints* of discontinuous parameterizations. In practice, when using for example observations, this is unlikely to cause errors that are larger than errors associated with other approximations and uncertainties in the data assimilation integrations such as the linear approximation errors and the uncertainties associated with the background and model errors statistics. This conclusion is similar to the conclusions of prior 4DVAR assimilation studies that use the *standard adjoints* but unlike in these studies the results in the current study show that 1) the linearization errors are nonnegligible for small-amplitude initial perturbations and 2) the assimilation results are locally and even globally affected by these errors.

*Corresponding author address:* Dr. Tomislava Vukicevic, NCAR, P.O. Box 3000, Boulder, CO 80307-3000.

Email: tomi@milena.cgd.ucar.edu