http://www.physics.curtin.edu.au/teaching/units/2003/Avp201/?plain Lec02.ppt

Equations of MotionOr: How the atmosphere moves

ObjectivesTo derive an equation which describes the horizontal and vertical motion of the atmosphereExplain the forces involvedShow how these forces produce the equation of motionShow how the simplified equation is produced

Newtons LawsThe fundamental law used to try and determine motion in the atmosphere is Newtons 2nd LawForce = Mass x AccelerationMeteorology 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 ReferenceIf 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 ReferenceFor a non-rotating Earth, these forces are:

Pressure gradient force (Pgf)Gravitational force (ga) andFriction force (F)

Frame of ReferenceHowever, we dont 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 (Ce) and Coriolis Force (Cof)

Equation of MotionWe 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 formIf we now consider the centrifugal force we can combine it with the gravitational force (ga) to produce a single gravitational force (g), since the centrifugal force depends only on position relative to the Earth.Hence, g = ga + Ce

A Useable formWe 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

ConventionsIn Meteorology, the conventions for the components in the horizontal and vertical are;x = E-W flowy = N-S flowZ = Vertical motionAlso, the conventions for velocity areu = velocity E-Wv = velocity N-Sw = Vertical velocity

Pressure Gradient ForceForce acting on air by virtue of spatial variations of pressureThese changes in pressure (or Pressure Gradient) are given by;

Pressure Gradient ForceIf 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 (Pg) on this cube is given by:(Pg) = Force/Volume.

Pressure Gradient ForceWe 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

Pressure Gradient ForceTherefore we have the Pressure Gradient Force (Pgf) given by;

This Pgf acts from High pressure to Low pressure, and so we have a final equation which reads;

Pressure Gradient ForceBecause we are dealing in 3-D, there are components to this Pgf and these are given as follows;

Pressure Gradient ForceCombining the components we get a total Pgf of

The components i and j are the Pgf for horizontal motion and the k component is the Pgf for vertical motion.

Horizontal PgfWe 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.yPgf = x

Vertical PgfIn the synoptic scale (large scale motions such as highs and lows), the Vertical Pgf 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 ForceThis is an apparent force caused by the rotation of the Earth.It causes a change of direction of air parcels in motionIn 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 ForceIt 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 Cof has a maximum at the Poles and is zero at the equator (Sin 0 = 0).

Coriolis Force

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 periodsA rapidly spinning roundabout is betterFrom off the roundabout, a thrown ball travels in a straight line.

Frames of reference- Roundabouts (2)But if youre 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 CofIt can be shown that the horizontal and vertical components of the Cof are as follows;

The complete equationWe can now write the equation of motion which describes the motion of particles on a rotating Earth.Remembering that the equation states that;Acceleration = Pgf + Cof + g + F,We can write the equation as follows;

The complete equation (Ignoring frictional effects)

Scale analysisEven 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

Element

Typical Value

Magnitude

u,v Horizontal velocity

10-20 ms-1

101 m

w Vertical velocity

1 cms-1

10-2 m

L Length (distance)

1000km

106 m

H Depth (Height)

10km

104 m

Horizontal Pressure Change

10 - 20 hPa

103 Pa

Vertical Pressure Change

1000 hPa

105 Pa

L/U (Time)

27 Hours

105 secs

( (Density)

1 kgm-3

100 kgm-3

g (Gravity)

9.8 ms-2

101ms-2

( (Angular velocity)

7.29 x 105

10-4 Radians s-1

Scale Analysis

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 Pgf 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 Pgf and Gravity that it CAN be ignoredWe 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 MotionUsing the assumptions of no friction and negligible vertical motion and using the Coriolis parameter f = 2sin, we can state the Simplified equations of motion as

Simplified Equation of Motion

ReferencesWallace and Hobbs Atmospheric Science pp 365 - 375Thom Meteorology and Navigation pp 6.3 - 6.4Crowder Wonders of the Weatherpp 52 - 53http://www.shodor.org/metweb/session4/session4.html

Parcel of air stationary at A.PGF between A and BAir moves Eward due to Earth rotationAir moves Sward due to PGFMoves to area where Earth spinning lessHowever, air parcel has gained Ely motion from Eq which is greater than Ely of Earth and therefore will move to A and appear to have been deflected to left.Opposite is true for Nwards motion.i.e. Parcel will appear to have been left behind by Earth rotation and again looks as though its been deflected to left.For Eward and Wward moving air the effect of Centirfugal accn needs to be taken into account. If a parcel of air is moving Ewards it will have its own speed added to that of the Earths and will therefore try to fly outwards, much like a stone does when attached to a string and whirled around your head. It is prevented from doing this by being forced to move to an area where the Earths rotation is greater, i.e. towards the equator. The opposite is true for Wward moving air, it moves towards slow moving air i.e. polewards.In both cases it appears to be deflected to the left.