Formal proof of the existence of an atmospheric base–state and its estimation

In mesoscale meteorology it is customary the use of the governing equations to calculate the deviations from a reference state that is horizontally homogeneous [1], namely,

......... ξ(x,y,z,t)= ξ⁽⁰⁾(z) + ξ⁽¹⁾(x,y,z,t) ......... (1)

where ξ⁽⁰⁾(z) is a reference value that depends of the height z with respect to a tangent-plane xy. As is pointed out by several authors, a base state that is closer to the total fields will give better accuracy. In this work a rigorous proof of a decomposition like (1) is given and its reliability region is estimated. Results obtained with analytic solutions of the governing equations on a complex topography show that (1) has a small validity region that is smaller than some regions used in mesoscale modeling.

The key of our argument consists in rewrite the momentum equation

......... dV/dt = - 1/ρ grad p + g – 2 Ω x V + f ......... (2)

in terms of the curvilinear coordinates xs = a (λ-λc) cos(φ), ys = a cos(φ) and zs=r-a (a=earth radius) to obtain the Ordinary Differential Equation (ODE) for the isobars defined by the intersection between a pressure-constant surface p(xs,ys,zs,t)=cte. and a θ-constant surface where θ together with s are cuasi-polar coordinates defined by xs = s cos(θ) and ys = s sin(θ). These equations together with zs = f(s,θ,t) (where θ and t are taken as constant parameters) constitute the parametric equations of the desired isobar. The ODE in question is

........df(s,θ)/ds = -cos(θ) ∂p/∂xs /∂p/∂zs - sin(θ) ∂p/∂ys / ∂p/∂zs ......... (3)

where the ratios ∂p/∂xs / ∂p/∂zs and ∂p/∂ys / ∂p/∂zs are obtained from (2). The eq. (3) has the feature of being independent of the density ρ since it disappears when the ratios are calculated. Let Ls, H, t0, U, W be the characteristic values of s, zs, t, and the horizontal and vertical components of the velocity, respectively. In terms of dimensionless variables s'=s/Ls, zs'=zs/H, etc., the eq. (3) has the form

......... df'(s',θ,t')/ds' = (μ/e) F(s',θ,t',μ) ......... (4)

where e=H/L, μ=U²/gLs. The characteristic values for midlatitude synoptic systems yield e=10^-2 and μ=10^-5, so that the right side of (4) is small. The analytic perturbation theory for ordinary differential equations [2] guarantees that the solution of eq. (4) and, therefore, (3) has a convergent power series of μ,

......... f(s,θ,t)= f⁽⁰⁾(zs,t) + ∑_k=1 μ^k f^(k)(xs,ys,zs,t) .........

where the zero-order term is simply a constant value f⁽⁰⁾=zs0. From this it follows that the pressure field has the series

......... p(xs,ys,zs,t)= p⁽⁰⁾(zs,t) + ∑_k=1 μ^k p^(k)(xs,ys,zs,t) ......... (5)

This rigorous result is worthy because is independent of the density ρ and temperature T fields. The use of (5) into the equation of state yields a functional equation for T and ρ,

......... p(xs,ys,zs,t,μ)= R T(xs,ys,zs,t,μ) ρ(xs,ys,zs,t,μ),

which has the formal solution

......... ρ(xs,ys,zs,t)= ρ⁽⁰⁾(zs,t) + ∑_k=1 μ^k ρ^(k)(xs,ys,zs,t)

......... T(xs,ys,zs,t) = T⁽⁰⁾(zs,t) + ∑_k=1 μ^k T^(k)(xs,ys,zs,t)

where the zero-order terms only depend of zs and t; that is, there exists a thermodynamic reference state that only depends of zs and t. If we put these series into the equation (2), a similar series is obtained for the velocity field. Thus we can conclude that each dependent variable has the decomposition

......... ψ(xs,ys,zs,t)= ψ⁽⁰⁾(zs,t) +∑_k=1 μ^k ψ^(k)(xs,ys,zs,t) ......... (6)

The standard mesoscale decomposition (1) is an approximation of (6) because the relevant vertical coordinate is the height zs with respect to the earth, rather than the height z on a tangent-plane xy. Analytic solutions of the two-dimensional deep continuity equation on a complex topography [3] are used to estimate the base-state terms fs⁽⁰⁾(zs,t) and the domain of validity of the decomposition (1). In the case ρ(zs)=cte. the pressure is obtained from the Bernoulli equation. The results show that the magnitude of relative error

......... [<p(xs,zs)>/ps(xs,zs)-1] x 100

of the (spherical) horizontal average

......... <p(xs,zs)> = 1/Ls ∫ ^Ls/2 _-Ls/2 p(xs,zs) dxs

with respect to the total field p(xs,ys,zs,t) is lower than 1% for Ls in the range 100 to 800 km and zs between 2 and 10 km. In contrast, if the zero-order term in (1) is estimated by means of the (cartesian) horizontal average

......... <p(x,z)> = 1/L ∫ ^L/2 _-L/2 p(x,z) dx ......... (7)

its relative error is bounded by 10% only for L < 150 km and z between 2 and 10 km. This means that the decomposition (1) is reliable for an horizontal L scale lower than 150 km. Similar results are obtained for isothermal and adiabatic atmospheres. The estimation L=150 km is a theoretical bound obtained from an analytic velocity field via (7) whereas in practice <p(x,z)> is estimated by data assimilation schemes and, therefore, its relative error with respect to the total field is larger, so that the validity region of the standard mesoscale decomposition (1) may have a scale L significanlty smaller than 150 km. This validity region is small with respect to the region used or recommended in the mesoacale literature (see, e.g.,[1,4]). [1] R.A. Pielke, Mesoscale Meteorological Modeling, 2nd ed. (2002).

[2] See., e.g.,H. Hochstadt, Differential equations, Dover (1964).

[3] M. A. Núñez and E. Cruz, Analytic solutions of the deep continuity equation on a complex topography, work submitted to this conference.

[4] See, e.g., M. Xue et. al., ARPS version 4.0 User's Guide (1995).