Equations of Motion
The Momentum Equation
For the purposes of this book, the atmosphere will be assumed to be composed of a mixture of gases called dry air (whose proportions are fixed) and water substance which varies in space and time. The combination of these two ingredients will be referred to as moist air. A moist air parcel will always contain water in vapour form but may also contain liquid water (i.e. cloud or rain) or ice. For the most part we will not be concerned with the dynamical effect of either liquid water or ice and on the scales of motion considered here, moist air may be
treated as a fluid continuum whose motion is characterised by a single vector wind field and whose thermodynamic properties (e.g. pressure, temperature and density) are scalar fields representing a state of
(approximate) local thermodynamic equilibrium at a point in the fluid. The molecular nature of air then appears as the fluid properties of viscosity and thermal diffusivity. Neither of these is directly important at the mesoscale though ultimately the former provides the mechanism by which fluid kinetic energy is `dissipated' i.e. converted to internal heat energy.
The vector momentum equation for a system rotating with angular velocity :
where is the material derivative, is wind vector, is the pressure, is the density, is the acceleration due to gravity, is the unit vector pointing vertically upwards, is the kinematic viscosity (equal to the dynamic viscosity ( ) divided by the density ). On a rotating planet represents the combined effect of the gravitation force and the centrifugal force associated with the planet's rotation. At any point, {} is actually normal to the surface of constant geopotential passing through the
point. is the Coriolis force per unit mass and is the pressure gradient force. For a derivation of the momentum equation from first principles the reader is referred to Batchelor (1967) and Gill (1982).
The expanded form of the material derivative is given by:
which represents the rate of change measured {\em following the motion of an air parcel} and should be distinguished from which is the time rate of change measured at a fixed point in space. On some occasions however, the material derivative will be used in circumstances where the above expansion is not appropriate i.e. when the quantity being operated on is not actually a field but an integral over a specified group of fluid particles.
Eq.(1) is too general for our needs and the following simplifications are appropriate. Firstly, the viscous stress term on the right-hand side will be neglected since it is only important for scales of the order of or less. Secondly, the curvature of the Earth may be neglected and the Coriolis force can be replaced by } giving the so called `f-plane approximation'.