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**V**

^{0}(obtained via a suitable interpolation of discrete data) an a field V (called the adjusted field) that satisfies the mass conservation equation

......... Div **V** = 0 .......... (1)

This problem is equivalent to solve the elliptic equation (repeated indices indicate summation)

......... -∂/∂x^{i} 1/α^{2}_{i} ∂/∂x^{i} q = - Div **V**^{0} in Ω ......... (2)

where q is a Lagrange multiplier in terms of which the cpmponents of the adjusted field are V ^{i}= V^{0,i} + 1/α^{2}_{i} ∂/∂x^{i} q.

Two kind of lower boundary conditions are used to compute q: (i) the Neumann boundary condition dq/dn =0 or the the so-called natural boundary condition associated to the elliptic equation (2), which has the form

......... n_{i} 1/α^{2}_{i} ∂/∂x^{i} q = 0, ......... (3)

where n_{i} are the components of the unit vector **n** normal to the topography [denoted by z=h(x,y)]. These boundary conditions coincide when α_{i}=1. Numerical results [1] suggest that both boundary conditions are suitable [1] if α_{i} is not equal to 1. However, in this work we give a careful mathematical deduction which shows that (3) is the correct boundary condition. We report calculations which show that the adjusted field **V** obtained with ∂q/∂n =0 is significantly different from that obtained with (3) for the ratio α^{2}=α_{1}^{2}/ α_{3}^{2} in the range [10^{-4},10^{1}].

All simulations performed with MCM's show that the adjusted field **V** is very sensitive to the values of the ratio α but there is no consensus about its correct value [1]. For instance, Kitada et. al [2] found numerically that the total residual divergence is sensitive to α and has a minimum value for one value of α. We show in this work that the exact Lagrange multiplier q yields an adjusted field that satisfies (1) for any α. This is illustrated by computing the mass flow with an analytic solution of (2). The results yield a mass flow of 10^{-7} for α^{2} in the range [10^{-4},10^{1}] and show that the sensibility of the (numerical) total residual divergence to the value of α is a consequence of the numerical solution of (2).

Some authors [3] compared the adjusted field with an exact field (obtained via map conforming) to study the reliability of MCM's. The analysis is partial because it considers α_{i}=1 and only simple regions Ω were considered, since inherent problems to the map conforming do not permit to consider a region Ω with a complex topography h(x). In other work [4] it is shown that the approximation of h(x) with a natural cubic spline yields analytic solutions of the more general deep continuity equation Div ρ(z) **V**(x,y,z) = 0. We use these analytic solutions to study the reliability of MSM's. Preliminary results suggests that (i) the accuracy of the adjusted field depends critically of the quality of the initial field **V**^{0} and (ii) the density variation ρ(z) is important in a domain Ω with a relatively small height.

[1] C.F. Ratto, et. al., Mass-consistent models for wind fields over complex terrain: The state of the art, Environ. Software 9, 247-268 (1994).

[2] T. Kitada et. al., Atmos. Environ. 17, 2181 (1983)

[3] D. G. Ross et. al., J. Appl. Meteor. 27, 785 (1988).

[4] M. A. Núñez and E. Cruz, work submitted to this conference.