The dynamical evolution of numerically simulated storms is highly dependent on the physical and computational parameters in the model, as well as on the initial and boundary conditions. The deterministic approach to sensitivity analysis, which employs both the tangent linear and adjoint of the original nonlinear model, can provide a wealth of sensitivity information at a very low cost compared with traditional brute force and Monte Carlo methods, which make numerous simulations with the full numerical model and perform various types of statistical analysis on the output.
In the deterministic approach, one develops a set of differential sensitivity equations, which is used to express the gradient of the solution vector with respect to input parameters [21]. Sensitivity coefficients are then computed exactly by solving the differential equations using a nonlinear solution as a basic state (e.g., [8, 17, 5]. In this sense, the sensitivity is defined as the gradient (i.e., the first-order derivative) of the model response with respect to any input parameter [18].
The gradients can be computed efficiently and accurately by using automatic differentiation (AD) tools, which apply the chain rule systematically to elementary operations or functions to generate derivative codes of given nonlinear models [2]. Besides providing basic sensitivity information, AD tools are indispensable in variational data assimilation, whose optimization processes require accurate gradient information.
In meteorology, the adjoint model
(ADJM) has been used substantially in both sensitivity analysis
(e.g., [8]) and variational data assimilation
(e.g., [14]). Although AD tools exist for generating
the ADJM
(e.g., Odyssée [19], AMC [10]),
the ADJMs, especially of 3-D models, are still routinely generated by
hand. Bischof et al. [5] have successfully applied an AD tool to
generate the tangent linear model (TLM) of a 3-D mesoscale model
(the PSU/NCAR MM5). A compilation of currently available AD tools can be
found in [4] and on the World Wide Web at
In this study, we apply the ADIFOR (Automatic DIfferentiation of FORtran) general-purpose AD tool [2, 3] to the 3-D Advanced Regional Prediction System (ARPS) [22] to generate a sensitivity-enhanced (SE-ARPS) code capable of providing derivatives of all model output variables and related diagnostic (derived) parameters as a function of specified control parameters, including initial and boundary conditions as well as physical and computational constants.
In this manner, we obtain exact derivative information, which is used to establish physical/dynamical cause and effect between changes in input and changes in output. Specifically, we compute the sensitivity of model outputs with respect to water vapor, which is a major factor to control storm life and morphology. We also compute sensitivities of the cost function, which measures distance in the Euclidean norm between the observation data and model results, with respect to all forecast aspects. Subsequently, we discuss implications of the sensitivity results on data assimilation.
ARPS is a fully compressible cloud model with full physics. Although an AD tool has been applied to a nonhydrostatic mesoscale model [5], no AD tool has been applied to a compressible model. In a compressible model, meteorologically unimportant acoustic waves are also supported, which severely limit the timestep size of explicit time integration schemes. To improve efficiency, ARPS employs the mode-splitting time integration technique [13]. In this technique, a large integration timestep is divided into a number of small timesteps; the acoustically active terms are updated every small timestep, while all other terms are computed only once every large timestep. This research is the first of its kind to apply an AD tool to a storm-scale model (meteorologically) with a mode-splitting time integration scheme (computationally).