__Forces, Accelerations and the Equation of Motion__

The consideration of the forces that influence atmospheric
flow is key in weather forecasting.
Force is the quantitative description of the interaction between two
physical bodies, and was fully defined by Sir Isaac Newton in his Laws of
Motion.

In the case of the example
(illustrating the force of pressure) discussed in class, the net pressure force
across the cube-shaped air parcel was -(Ęp) acting on the faces of the air
parcel. Using simple algebra it
can be shown that the the pressure force acting on each of the faces is:

Newton's Second Law of Motion
states that the acceleration experienced by an object with mass m is
proportional to the sum of the forces acting on the object.

But mass is density X volume or

Put (3) into (2) and the result
into (1) to obtain the three dimensional pressure gradient acceleration.

The pressure gradient acceleration is one of the three major "accelerations" we discussed in Metr 201 in the section on the Equation of Horizontal Motion. These allow us to estimate how air parcels are accelerated horizontally: du/dt and dv/dt. These other two accelrations are Coriolis acceleration and the frictional acceleration. Alternatively, equation (2) can be solved for the net Forces that act to accelerate air horizontally, and those would be the pressure gradient force, the Coriolis "force" and the frictional force.

In this course, we are going to start discussing the Equation of Motion by consideration of its three dimensional version. So, you will find out that there is a vertical equation of motion and it allows us to estimate how air parcels are accelerted vertically: dw/dt. It turns out that although there is a Coriolis acceleration that acts vertically (classroom example discussed today), it's very small and can be neglected on an order of magnitude basis.

The vertical pressure gradient acceleration is very large, two to three orders of magnitude larger than the vertical Coriolis acceleration but also the horizontal pressure gradient accelerations we can estimate on weather maps. Yet we know that observations show that, apart from updrafts in thunderstorms, vertical velocities (and, hence, the vertical accelerations that take air parcels from not moving at all to having a finite vertical speed) are extremely small. If the net vertical acceleration is very small, yet the vertical pressure gradient acceleration is very large, there must be an acceleration that is nearly equal and opposite to the vertical pressure gradient acceleration. That acceleration is gravity. As you will see, the vertical equation of motion, when scaled for the synoptic-scale atmosphere, reduces to the hydrostatic equation.