Extended Abstract (844K)P1M.14
Reliability analysis of mass consistent models
Marco A. Nuñez, Universidad Autonoma Metropolitana Iztapalapa, Mexico City, Mexico; and H. Juarez and C. Flores
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/α2i ∂/∂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= V0,i + 1/α2i ∂/∂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
......... ni 1/α2i ∂/∂xi q = 0, ......... (3)
where ni are the components of the unit vector n normal to the topography [denoted by z=h(x,y)]. These boundary conditions coincide when αi=1. Numerical results [1] suggest that both boundary conditions are suitable [1] if αi 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 αi=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.
Poster Session 1M, Mesoscale Model Development & Data Assimilation
Monday, 24 October 2005, 1:15 PM-3:00 PM, Alvarado F and Atria
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