Structural instability of atmospheric flows under perturbations of the mass balance and effect in transport calculations

Several methods to estimate the velocity field of atmospheric flows, have been proposed to the date for applications such as emergency response systems, transport calculations and for budget studies of all kinds. These applications require a wind field that satisfies the conservation of mass but, in general, estimated wind fields do not satisfy exactly the continuity equation. An approach to reduce the effect of using a divergent wind field as input in the transport-diffusion equations, was proposed in the literature. In this work, a linear local analysis of a wind field, is used to show analytically that the perturbation of a large-scale nondivergent flow can yield a divergent flow with a substantially different structure. The effects of these structural changes in transport calculations are illustrated by means of analytic solutions of the transport equation.


Introduction
Several methods to estimate trajectory models for long-range atmospheric transport calculations, have been proposed [1]. However, conservation of mass is not always exactly satisfied [2,3]. Although several authors have pointed out that the use of a wind field that does not satisfy exactly the continuity equation, can introduce significant errors in transport calculations of chemical species [3,4], the accuracy with which the continuity equation is satisfied, is not always reported in works that propose methods for estimating wind fields for transport calculations. Since large scale atmospheric flows are nearly nondivergent, it is generally accepted that a mass-balanced wind field has to satisfy the continuity equation in its simplest form . Simple methods applied to wind field analysis can yield a divergence of order 10 -5 s -1 . In this work we show that a divergence of this order can change the structure of a large-scale nondivergent flow.
A proposed method to reduce the effect of using a divergent field in transport calculations, consists in subtracting the value of at each grid point where the transport equations are solved with standard spatial-discretization methods [4]. The results of this work show that this procedure can yield results with a significant error because the structure of a nondivergent wind field can be different from that of a divergent one with a divergence of order 10 -5 s -1 .

Types of mass-balanced flows and their perturbations
The structure of an atmospheric flow v=ui+vj in the vicinity of an arbitrary point can be studied by means of the linear terms of its Taylor series. Let us consider a Cartesian system with its origin at , then we have , (1) where the matrix M={M jk } is the velocity gradient at the origin. In terms of the parameters ⁄ ⁄ ⁄ (2) we have , .
(3) Hence the velocity field can be written as v δ = v 0 +u δ where v 0 and u δ = (xi+yj) are the nondivergent and divergent components of v δ . Since large scale atmospheric flows are nearly nondivergent, v 0 will be considered an ideal mass-balanced flow, and u δ will quantify the perturbation of the mass balance of v 0 . Accordingly, the mass-unbalanced flow v δ with and the mass balanced one v 0 will be referred to as a divergent flow (DF) and a nondivergent flow (NDF), respectively.
To analyze the effect of a mass imbalance in transport calculations we consider the transport equation  (8) it takes the form (9) We see that the perturbation term u δ in a DF generates a fictitious reaction term . Since characteristic values of are of order 10 -6 s -1 , a divergence | | of order 10 -5 s -1 yields unrealistic solutions . The solution proposed in the literature [3,4] to reduce the effect of using a DF is to substract the term on the left hand side of equation (8). This is equivalent to solving the equation (10) Transport calculations reported below were made by solving equation (10)  be the corresponding sets of a DF with the same α, β, and . Consider δ, β, such that q<0 and α>0. This yields , , ( ), and the intersection ( ) are values with which an hyperbolic NDF becomes and elliptic DF . Example 1: Consider δ=10 -5 s -1 , β=10 ms -1 , L=10 3 km, which yield q= 10 -10 s -2 , and let α=1 ms -1 , then 0.05 10 -5 s -1 , 5.05 10 -5 s -1 , and any in yields an hyperbolic NDF that becomes an elliptic DF . Figures 1, 2, show , , with 10 -5 s -1 and the corresponding solutions of equation (10) by means of the movement of the iso-line with c=0.75 at t=0. For α<0 and δ, β, such that q<0, we get and the sets ,

Summary and concluding remarks
Standard methods to estimate the wind field from observational data yield, in general, a field with a nonzero divergent component [2,3]. The procedure suggested in the literature to make transport calculations with a DF (subtracting the value of the divergence at each grid point in the transport equation [3,4]), does not consider that a perturbation of a massbalanced flow can yield a divergent flow with a significantly different structure. Thus, the use of a DF in transport calculations can yield trajectories that diverge from those given by a mass-balanced flow . According to Kitada [3], simple data assimilation methods yield a DF with of order 10 -5 s -1 , but the examples reported in [5] show that variational methods conceived to minimize the divergence [6], can yield fields with of order 10 -3 s -1 . In these cases, the use of in transport calculations can yield results with errors larger than those reported above. A more complete study of the effects of perturbing the mass-balance of a wind field in the flow structure and transport calculations will be given in a forthcoming work.