Reliability analysis of mass consistent models

Mass Consistent Models (MCM's) are some of the main mesoscale assimilation schemes of wind data by the simplicity of the physics involved and their capacity to accept wind data from different points in a region Ω [1]. The models consist in minimizing the variance between an initial velocity field V⁰ (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)

......... -∂/∂xⁱ 1/α²_i ∂/∂xⁱ q = - Div V⁰ 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/α²_i ∂/∂xⁱ 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/α²_i ∂/∂xⁱ 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 α²=α₁²/ α₃² in the range [10^-4,10¹].

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 α² in the range [10^-4,10¹] 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 V⁰ 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.