Analytic solution of the deep continuity equation over complex terrain

Analytic solution of the deep continuity equation over complex terrain

Analytic solutions of the dynamic governing equations are important (i) to study physical mechanisms involved in specific circulations with flows that are not perturbed by computational errors, (ii) model validation (since they provide the means to check the integrity of the advection pressure and frictional forces), and (iii) as a mean to verify the integrity of all or part of a computer code [1]. In this work we give a scheme to obtain analytic wind velocities

......... U(x,z) = [ u(x,z), w(x,z) ] .........

that satisfy the so-called (two-dimensional) deep continuity equation [1]

......... Div ρ(x,z) U(x,z) = 0 ......... (1)

subject to the slip boundary condition

......... V(x,z) • n = 0 at z=h(x) ......... (2)

where ρ(x,z) and h(x) are arbitrary density and terrain elevation functions, respectively, and n is a normal vector to the topography. There is other motivation to obtain analytic solutions of equations (1,2). They are the base of the main diagnostic models used to compute the velocity U (x,z) with the data obtained from an observational network [2]. The analytic solutions of (1,2) are obtained by Complex Variable Theory (CVT). Several authors [2] have used CVT to obtain analytic solutions of the so-called shallow continuity equation [1]

......... Div V(x,z) = 0 ......... (3)

but they only consider simple terrain elevation functions h(x) because the CVT cannot be applied in the case of an arbitrary h(x). In this work it is shown that the problems posed by the CVT are solved if h(x) is replaced by a natural cubic spline S(x) [3]. If h(x) is known analytically, it can be approximated by S(x) with an arbitrary small error h(x)-S(x). If h(x) is known only on a finite set of points {x_k,h_k} there is a natural spline S(x) determined uniquely by such data. This allows us to obtain analytic fields V(x,z) when we have data from a digital elevation model such as GTOPO30 [4].

The analytic solutions V(x,z) of the shallow continuity equation (3) yield the following solutions of the deep continuity equation (1) subject to condition (2) U = V / ρ. The wind field U is more realistic than V since U has a nonzero vorticity which is suitable to model a two-dimensional shear flow. The continuity of S(x) and its derivatives S'(x), S''(x) guaranty that the field U and its first partial derivatives ∂_x U, ∂_z U are continuous and, by construction, U satisfies the equation (1). This is illustrated by computing U on a region defined by terrain elevation data from the GTOPO30 data base [4]. As an application of the obtained analytic solutions we study the reliability of the main schemes (the kinematic, vorticity and quasi-geostrophic omega-equation methods [5]) to compute the vertical wind velocity. Other application is the study of the sensibility of the field U to the roughness surface, which is estimated with high-order frequencies of the spectral decomposition of h(x).

[1] R.A. Pielke, Mesoscale Meteorological Modeling, 2nd ed. (2002).

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

[3] See, e.g, R.Burden and D. Faires, Numerical Analysis (1985).

[4] GTOPO30, U.S. Geological Survey (1997), http://www.scd.ucar.edu/datasests.

[5] T.N. Krishnamurti, Numerical Weather prediction techniques (1996).