Adjoint Equations of Rotunno and Emanuel (1987): Parameter Estimation and Sensitivity Analysis

The adjoint equation effectively transforms the information into the sensitivity of the model variables by backward integration with the aid of the adjoint variables λ(x), where x is the prognostic variable vector. The adjoint equation can be applied to the parameter estimation, though the adjoint equation is commonly known as its application to 4D-Var data assimilation system. Here, we investigate the estimation of air-sea exchange coefficient for high-wind regime of a mature tropical cyclone by using adjoint equations of the axisymmetric cloud-resolving model originally written by Rotunno and Emanuel (1987) coupled to the 1-dimensional ocean mixed layer model(Schade and Emanuel, 1999). The basic idea of adjoint calculations is to define a cost function that quantifies the total misfit between the model results and the observations same as in the data assimilation. The control variables such as drag coefficient (CD), water vapor exchange coefficient (CE) together with initial state are determined to minimize the cost function leading to optimal estimates of the model fields. Using this method, the observational data obtained outside the boundary layer, such as wind velocity, the mixing ratio of water vapor, and ocean mixed layer (ML) momentum are assimilated into the model in order to evaluate the CD and CE values suitable for the actual state through the model physics. This method requires sufficient high-quality observations. We perform an identical twin experiment as a first step toward realistic estimation with the aim of improving both the dynamical representation of TC events as well as the better intensity prediction. In our identical twin experiment, the "True" field is generated by numerical integration of the coupled model, while another run is performed with the same setting except for random errors in the initial state of the assimilation window and the "wrong" coefficients in high-wind regimes. The result is referred to as the "NoAsm" field. Here, the pseudo observations are generated by adding Gaussian noise to variables in the "True" field. The analysis field is generated by digesting the observations with an adjoint data assimilation method. The result of the assimilation is termed "Asm_Coef" field. We confirm the feasibility of the estimation in the light of the model physics for a mature tropical cyclone. As the quantity of campaign observations is sufficient to improve both CD and CE values toward their "True" values, the adjusted CD and CE terms yield substantial improvements in the intensity and in the radius of maximum winds. The estimation of these coefficients in a realistic TC will be shown in the presentaion.