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p201/?plain p201/?plain Lec02.ppt Equations of Motion Or: How the atmosphere moves
Objectives To derive an equation which describes the horizontal and vertical motion of the atmosphere Explain the forces involved Show how these forces produce the equation of motion Show how the simplified equation is produced
Newtons Laws The fundamental law used to try and determine motion in the atmosphere is Newtons 2nd Law Force = Mass x Acceleration Meteorology is a science that likes the KISS principle, and to further simplify matters we shall consider only a unit mass. i.e. Force = Acceleration
Frame of Reference If we were to consider absolute acceleration relative to “fixed” stars (i.e. a non-rotating Earth ) Then our equation would read something like this; “The rate of change of velocity with time is equal to the sum of the forces acting on the parcel”
Frame of Reference For a non-rotating Earth, these forces are: Pressure gradient force (P gf ) Gravitational force (g a ) and Friction force (F)
Frame of Reference However, we don’t live on a non-rotating Earth, and we have to consider the additional forces which arise due to this rotation, and these are: Centrifugal Force (C e ) and Coriolis Force (C of )
Equation of Motion We now have a new equation which states that: i.e. The relative acceleration relative to the Earth is equal to the real forces (Pressure gradient, Gravity and Friction) plus the “apparent forces” Centrifugal force and Coriolis force
A Useable form If we now consider the centrifugal force we can combine it with the gravitational force (g a ) to produce a single gravitational force (g), since the centrifugal force depends only on position relative to the Earth. Hence, g = g a + C e
A Useable form We can now write our equation as: We now look at the equation in its component forms, since we are considering the atmosphere as a 3- Dimensional entity
Conventions In Meteorology, the conventions for the components in the horizontal and vertical are; x = E-W flow y = N-S flow Z = Vertical motion Also, the conventions for velocity are u = velocity E-W v = velocity N-S w = Vertical velocity
Pressure Gradient Force “Force acting on air by virtue of spatial variations of pressure” These changes in pressure (or Pressure Gradient) are given by;
Pressure Gradient Force If we now consider this pressure gradient acting on a unit cubic mass of air with volume given by x y z we can say that the Pressure gradient (P g ) on this cube is given by: (P g ) = Force/Volume.
Pressure Gradient Force We can also say that the volume of this unit mass is the specific volume which is given by 1/ , where is density. This has the dimensions of Volume/Mass. The dimensions of the Pressure gradient are Force/Volume
Pressure Gradient Force
Therefore we have the Pressure Gradient Force (P gf ) given by; This P gf acts from High pressure to Low pressure, and so we have a final equation which reads;
Pressure Gradient Force Because we are dealing in 3-D, there are components to this P gf and these are given as follows;
Pressure Gradient Force Combining the components we get a total P gf of The components i and j are the P gf for horizontal motion and the k component is the P gf for vertical motion.
Horizontal P gf We can simplify matters still further if we take the x axis or y axis normal to the isobars, i.e. in the direction of the gradient. We then only have to consider one of the components as the other one will be zero. y P gf = x
Vertical P gf In the synoptic scale (large scale motions such as highs and lows), the Vertical P gf is almost exactly balanced by gravity. So we can say that This is known as the Hydrostatic equation, and basically states that for synoptic scale motion there is no vertical acceleration
Coriolis Force This is an “apparent” force caused by the rotation of the Earth. It causes a change of direction of air parcels in motion In the Southern Hemisphere this deflection is to the LEFT. Is proportional towhere is the local latitude Its magnitude is proportional to the wind strength
Coriolis Force It can be shown that the Coriolis force is given by 2 sin V The term 2 sin is known as the Coriolis parameter and is often written in texts as f. Because of the relationship with the sine of the latitude C of has a maximum at the Poles and is zero at the equator (Sin 0 ° = 0).
Frames of reference- Roundabouts (1) Our earth is spinning rather slowly (i.e. once per day) and so any effects are hard to observe over short time periods A rapidly spinning roundabout is better From off the roundabout, a thrown ball travels in a straight line.
Frames of reference- Roundabouts (2) But if you’re on the roundabout, the ball appears to take a curved path. And if the roundabout is spinning clockwise, the ball is deflected to the left
Components of C of It can be shown that the horizontal and vertical components of the C of are as follows;
The complete equation We can now write the equation of motion which describes the motion of particles on a rotating Earth. Remembering that the equation states that; Acceleration = P gf + C of + g + F, We can write the equation as follows;
The complete equation (Ignoring frictional effects)
Scale analysis Even though the equation has been simplified by excluding Frictional effects and combining the Centrifugal force with the Gravitational force, it is still a complicated equation. To further simplify, a process known as Scale analysis is employed. We simply assign typical scale values to each element and then eliminate those values which are SIGNIFICANTLY smaller than the rest
Scale Analysis (Horizontal Motion) From the previous slide we can see that for the horizontal equations of motion du/dt and dv/dt, the largest terms are the P gf and the Coriolis term involving u and v. The acceleration is an order of magnitude smaller but it cannot be ignored.
Scale Analysis (Vertical Motion) For the vertical equation we can see that there are two terms which are far greater than the other two. The acceleration is of an order of magnitude so much smaller than the P gf and Gravity that it CAN be ignored We can say therefore that for SYNOPTIC scale motion, vertical acceleration can be ignored and that a state of balance called the Hydrostatic Equation exists
Simplified Equation of Motion Using the assumptions of no friction and negligible vertical motion and using the Coriolis parameter f = 2 sin , we can state the Simplified equations of motion as
Simplified Equation of Motion
References Wallace and Hobbs Atmospheric Science pp Thom Meteorology and Navigation pp Crowder Wonders of the Weather pp ession4.htmlhttp://www.shodor.org/metweb/session4/s ession4.html