Table of Contents
Dynamics
Second Law of Motion: The acceleration of an object is directly proportional to the net force acting on it and is inversely proportional to its mass. The direction of the acceleration is in the direction of the applied net force.
Law of Universal Gravitation: Every particle in the universe attracts every other particle with a force that is proportional to the product of their masses and inversely to the square of the distance between them. This force acts along the line joining them.
A force is a vector, having both magnitude and direction, expressed as a symbol with an arrow above it.
Equations of Motion
Writing this in the U, V components of the wind gives: Since we can write, then substituting into the above equations gives,
So, if we know the initial wind speed components and the forces acting on air parcels, we should be able to determine the wind speed at some time in the future, (t + Dt).
Forces acting on air parcels
Advection
Advection terms are written as: where, are gradient terms.
Pressure Gradient Force
Centripital Acceleration
From the point of view of an observer in fixed space, the speed of the ball is constant, but its direction of travel is continuously changing, so its velocity is not constant; i.e., it has an acceleration.
If we divide by Dt and note that in the limit as , Dv is directed toward the axis of rotation, (it is a -Dv), then: Since, tangential velocity Then, In our convention we can write:
Therefore, when viewed from fixed coordinates, the motion is one of uniform acceleration directed toward the axis of rotation and equal to the square of the angular velocity times the distance from the axis of rotation. This acceleration is the Centripital Acceleration.
Now, suppose the motion is observed in a coordinate system rotating with the ball.
Thus, the Centrifugal Force is equivalent to the inertial reaction of the ball on the string and is equal but opposite to the centripital acceleration.
Since, for wind at rest on Earthís surface, M = wr, then and To get the components in the x and y direction, remember that and to get the sign right, (Centrifugal force going in the proper direction), consider the following low pressure center.
By writing the equations as: and using the sign below, the direction of the force is proper.
Effective Gravity
Thus, the weight of the particle of mass, m, at rest on the Earthís surface will generally be less than the gravitational force, mg*, because the centrifugal force partly balances the gravitational force.
Coriolis Force
Expanding gives:
The discrepancy between the magnitude of the two is:
This outward directed Coriolis Force (acceleration) can be divided into components in the vertical and parallel to the Earthís surface along the meridional directions (North - South).
That change in direction for our object moving eastward with velocity U, is toward the south, i.e., to the right of the direction of motion in the northern hemisphere. (In other words, it is causing a change in the north-south velocity of the parcel).
Objects moving north or south.
Its initial angular momentum, L, is: , where w is initially equal to zero. For other than unit mass it is:
We can write the following showing conservation of angular momentum. Where, is the angular momentum of the unit mass object at rest on the surface of the earth at latitude F.
If we expand the right side we get: If we neglect second-order, and higher differentials, we get: and,
We can see that
PPT Slide
Turbulent Drag Force
Stress: The amount of frictional force per contact area where the force is parallel to the area rather than vertical, as is pressure.
(2) Turbulence
PPT Slide
The Turbulent Drag Force per mass (acceleration) for horizontal motion is: where, wT = the Turbulent Transport Velocity, zi = height above ground.
For windy conditions of near statically neutral conditions, turbulence is generated primarily by shear, then the Turbulent Transport Velocity is given by: where, CD = Drag Coefficient M = wind speed.
For statically unstable conditions of light winds and strong surface heating, the Turbulent Transport Velocity is given by: where, wB = buoyancy velocity scale (the effectiveness of thermals in producing vertical transport) bD = 1.83 x 10-3.
Equations of Motion
Centrifugal force is not included because it is an imbalance of the forces in the full equations of motion.
Geostropic Winds
Gradient Wind
If we let G represent the Geostrophic Wind and Mr be the Gradient wind, then where the negative sign is used around low pressure centers and positive around high pressure centers.
Solving the quadratic equation for Mr gives for low pressure centers: and for high pressure centers: R = the radius of curvature of the airflow.
Boundary-Layer Wind
These can be written in terms of the Geostrophic Wind as
Boundary layer winds around curved isobars require the Centrifugal force term.
Cyclostrophic Winds
Problems: N1 (b, h), N2 (b, e), N3, N4 (a, d, e) (assume z = 1000m), N5 (a, d, k) N8, (a, d, g) (assume zi = 1000 m), N9 (b, d), N10 (a, c), N11 (b, e), N13, N16 (a, g)
End
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Author: Alcorn
Email: alcorn@ariel.met.tamu.edu
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