1. Dynamics I: Basic Equations and approximations Adilson W
Dynamics I: Basic Equations and approximations Adilson W. Gandu
Modelagem Atmosférica com o BRAMS:
Descrição, uso e operacionalização do modelo – Módulo 2
CPTEC, Cachoeira Paulista, 28/07 a 01/08/2008

2. Basic Set of Equations (This is a summary of the book “Mesoscale Meteorological Modeling”, 2nd ed., Academic Press, 2002, by Roger A. Pielke SR. and the book “Clouds”, by Cotton)
The foundation for any model is a set of conservation principles. For mesoscale atmospheric models, these principles are
conservation of mass,
conservation of heat,
conservation of motion,
conservation of water, and
conservation of other gaseous and aerosol

3. Conservation of Mass
In the earth's atmosphere, mass is assumed to have neither sinks nor sources. OR
the mass into and out of an infinitesimal box must be equal to the change of mass in the box.

4. The complete equation for mass flux in the box can be written as
It is also called the continuity equation. In vector notation, it is written as:
For dry air it takes the form
OBS.: We choose not to apply Eq. (2.18) to the total air density, since water vapor
has numerous sources and sinks

5. The Equation of State
The atmosphere on the mesoscale behaves very much like an ideal gas and is considered to be in local thermodynamic equilibrium
(To be in equilibrium, temperature must be controlled by molecular collisions rather than by interaction of the molecules with the radiation field. At levels below 50 km or so in the earth's atmosphere, the density of air is sufficiently great so that over short distances, molecular collisions dominate and a state of local equilibrium occurs)
Where:
R – gas constant (function of the chemical composition of the gas)
R* - universal gas constant (= x 103 J K-1 kmol-1)
μ – molecular weight of the gas

6. In the atmosphere, the apparent molecular weight of air, μatm is determined by the fractional contribution by mass of each component gas (Table 2-1).
Excluding water vapor, μd is 28.98, so that the dry gas constant of the atmosphere, Rd, is
Rd = R* /28.98 = 287 J K-1 kg-1.

7. Inserting, μatm into Eq. (2-11) gives
When water vapor is included, the apparent molecular weight can be written as
where: q – specific humidity (or, approximately, rv,
ratio of mass of vapor to the mass of dry air)
Inserting, μatm into Eq. (2-11) gives or
The total density of the system P is equal to:

8. Conservation of Heat
The first law of ther­modynamics for the atmosphere states that differential changes in heat content, dQ, are equal to the sum of differential work performed by an object, dW, and differential increases in internal energy, dI.
If we define e = I/m, w = W/m and h = Q/m,
and remembering that dw = pdα (α=1/ρ): or

9. For a situation when no heat is gained or lost (adiabatic), this equations can be used to derive the Poisson’s equation
If the change of potential temperature is observed following
a parcel, then the first law of thermodynamics can be written as: or
Where Sθ represents the sources and sinks of heat
as expressed by changes in potential temperature

10. The contributors to Sθ include the sum of the following processes:

11. The Thermodynamic Energy Equation for Moist Convection
Let us consider the thermodynamic energy equation for a moist system containing water vapor, liquid water, and ice particles having mixing ratios rv, rl, and ri, respectively. The liquid particles may be composed of rain drops and small cloud droplets.
the thermodynamic energy equation for an open thermodynamic system is:

12. Following procedures outlined by Tripoli and Cotton (1981):
It is often desirable to obtain a thermodynamic variable that is conservative under adiabatic liquid and ice transformations.
Therefore, one can define the conservative variable ice-liquid-water potential temperature
Therefore, the Thermodynamic Energy Equation
can be written as:
Where p(θil) – represents the influence of precipitation fallout on θil

13. Conservation of Motion
The conservation of motion is expressed by Newton's second law, which states that a force (normalized by mass) exerted on an object causes an acceleration, as given by
where the subscript n refers to a non accelerating coordinate system
Coriolis
acceleration as viewed
from the rotating earth
centripetal

14. forces that cause changes in motion:
pressure gradient force :
modified gravitational force :
dissipation of momentum by molecular motions (viscosity):
In the atmosphere, on the mesoscale, the viscosity is sufficiently small and the velocities are sufficient1y great that the influence of these internal forces is ignored.

15. The conservation-of-motion relation :
Since :
Where:

16. Conservation of Water
where q1, q2, and q3 are defined as the ratio of the mass of the solid, liquid, and vapor forms of water, respectively, to the mass of air in the same volume

17. Conservation of Other Gaseous and AerosoI MateriaIs
Where: Xm refers to any chemical species except water, and is expressed as the mass of the substance to the mass of air in the same volume.
Examples of Xm : CO, CO2, NH4, SO2, O3, etc.
The source-sink terms SXm include changes of state (analogous to that performed for water) as well as chemical transformations, precipitation, and sedimentation.