On the bounded derivative initialization method

Abstract

A linearized model in x,z and t coordinates, a primitive equation barotropic model ana baroclinic model based on the NCAR GCM are employed to examine various aspects of the bounded derivative initialization method proposed by Kreiss (1979, 1980). Linearization is performed about a uniform zonal basic current and the method of separation of variables is applied to determine vertical and hori zontal structures of the system. The investigation is confined to the external vertical mode in x,t coordinates. Harmonic dependency is assumed to solve the horizontal structure problem. Solutions associated with the unbalanced inittal state are compared with the case when the bounded derivative initialization method is applied. The results show that the unbalanced -initial state yields solutions dominated by high frequency gravitational oscillations. Application of the first order constraints of the bounded derivative initialization method eliminates the nondivergent wind component of the gravitational motions, but the divergent component still remains. The second order constraints eliminate virtually all the high frequency oscillations in the vorticity. divergence and the geopotential variables. First order constraints arising from the bounded derivative method are similar to geostrophjc balance while second order constraints yield initial specification analogous to quasi-geostrophic balancing of Philiips (1960) and, normal mode initialization after Baer (1977) and Uachenhauer (1977). Concerning the primitive equation barotropic model, we apply the bounded derivative method to design and subsequently carry out successful test calculations of an initialization scheme suitable for equatorial regions in the presence of idealized orography which depends on the zonal and meridional directions. To implement the scheme, we specify a uniform zonal current and then obtain the initial divergent component of the motion and the geopotential through the application of the bounded derivative method. For purposes of comparison we also test unbalanced initial data. In this case again we assume the same· initial uniform zonal current but the geopotential height is obtained by assuming steady-state conditions in absence of orography. The computations indicate a stationary orographic trough also reported by Semazzi (1980a,b), with msximum intensity at the top of the mountain both for easterly and westerly cases. The intensity of the trough is greater in the easterly ease.We observe divergence (of order 10-6 s-1 ) on the windward!ope and convergence on the lee slope irrespective of the direction of the initial uniform zonal current. The vorticity is of order 10-7s-1 thus one order of magnitude weaker than the divergence. It retains the same spatial distribution for easterly and westerly cases. The initialization scheme for the baroclinic model which arise from the application of the bounded derivative method comprises of two constraints. One is the well known classical balance equation for calculating the pressure and the other one is a three dimensional elliptic equation for computing the initial divergence. Suitable boundary conditions for the .divergence equation are also determined as a result of constraining the second order time derivatives of the governine equations to be of order of synoptic .meteorological motion. In order .to implem8nt the proposed scheme, diabatic heating and the nondivergent component ,of the wind field are assumed to be known intially.