Dynamics I: Aproximate Equations Adilson W. Gandu (adwgandu@model.iagp.br)

1. Dynamics I: Aproximate EquationsAdilson W. Gandu(adwgandu@model.iag.usp.br) Modelagem Atmosférica com o BRAMS: Descrição, uso e operacionalização do modelo – Módulo 2 CPTEC, Cachoeira Paulista, 28/07 a 01/08/2008

2. Introduction • The basic equations can be simplified for specific mesoscale meteorological simulations, to permit their solution in an easier, more economical fashion. • By mathematical operations, some of these relations can also be changed in form. • The method of scale analysisis often used to determine the relative importance of the individual terms in the conservation relations. • This technique involves estimation of their order of magnitude

3. First, we define a reference (synoptic) state • an arbitrary • horizontally uniform • dry (rv0 = rT0 = 0) • hydrostatic (α0 ∂p0 / ∂z = -g) • denoted by the subscript “O” • Which obeys the equation of state:

4. We can now express the thermodynamic variables as: where : the prime quantities represent a deviation from the reference state

5. Equation of State for a moist atmosphere (linearized) • Using those definitions, can be written as

6. Equations of Motion • Multiplying Eq. (2.17) by αm • now, considering the second and the third terms and • expanding about the reference state gives • This equation expresses the essence of the Boussinesq approximation; • that is, variations in density or specific volume are ignored • except when multiplied by gravity. • The second term on the RHS is the so-called buoyancy term • which is affected by perturbations of θ, rv, p • and by the presence of condensate rw

7. The equations of motion expressed in terms of the Exner function where : Poois an arbitrary reference pressure (1000 mbar) and k = Ra / cpa • considering: gives : OBS.; the main advantage of the π - system is that pressure anomalies do not appear in the buoyancy term.

8. The Continuity Equation for Dry Air • considering the equation of continuity, Eq. (2.18), rewritten in the form • Letting αa= αo+ αa’ • assuming that the basic state • is horizontally homogeneous: • and steady: • and: • The analysis of scale shows that an appropriate • continuity equation for deep convectionis (anelastic): • and appropriate for shallow convection is (incompressible):

9. time-splitting scheme • The approach is simply to rewrite Eq. (2.87) or (2.99) in the form LHS RHS • an appropriate approximate continuity equation in which sound waves are not filtered out • Linearization of the equation of state • In this system the LHS of Eq. (2.150) can be integrated using a small time step in which acoustic waves are resolved. • Equation (2.152) is then used to diagnose p'. • The RHS of Eq. (2.150), along with the thermodynamic • energy equation and other scalar equations can be integrated on longer time steps that are controlled by advective time scales.

10. Averaging the Conservation Relations • The basic equations are defined in terms of the differential operators (∂/∂t, ∂/∂xi), and thus in terms of mathematical formalism are valid only in the limit when δt, δx, δy, and δz approach zero. • In terms of practical application, however, they are valid only when the spatial increments δx, δy, and δz are much larger than the spacing between molecules but are small enough so that the differential terms over these distances and over the time interval δt can be represented accurately by a constant (δt ~ 1 sec and δx ~ 1 cm) • If these terms vary significantly within the intervals, however, then the Equations must be integrated over the distance and time intervals over which they are being applied, whose sizes are determined by the available computer capacity, including its speed of operation • For a specific mesoscale system, the smaller these scales, the better the resolution of the circulation

11. In performing the integration, it is convenient to perform the following decomposition: where Φrepresents any one of the dependent variables and • represents the average of Φ over the finite time increment Δt and space intervals Δx, Δy, and Δz. • The variable Φ" is the deviation of Φ from this average and is often called the subgrid-scale perturbation. • In a numerical model, Δt is called the time stepand Δx, Δy, and Δz represent the model grid intervals

12. Another used decomposition: where Φrepresents any one of the dependent variables and is called the layer domain-averaged variable • Dxand Dyrepresent distances that are large • compared to the mesoscale system of interest • {the horizontal size (domain) of the mesoscale model representation} • Φ0is assumed to represent the (horizontal – function of z) synoptic-scale atmospheric conditions, as referred to previously. • Φ ' represents the mesoscale deviations from this larger scale.

13. An Example of the Reynolds Averaging Procedure(to the anelastic equations of motion) • begin with the equations of motion in the form of • Multiplying by ρ0, expanding the substantial derivative into • local and advective changes and using the anelastic approximation gives

14. We now decompose each variable into a mean and a turbulent fluctuating component: • Substitution of Eq. (3.17) into Eq. (3.16) gives

15. which can be written after averaging as : • The usual procedure is to assume that for any variable Φ :

16. Resulting: OBS.: Equations are not closed. Some means of defining the correlations among velocity fluctuations and among velocity and scalar fluctuations must be found.