CHAPTER 4. ATMOSPHERIC TRANSPORT - Harvard …

1 40 CHAPTER 4. ATMOSPHERIC TRANSPORTWe saw in CHAPTER 3 that air motions play a key role in determiningthe distributions of chemical species in the are determined by three principal forces: gravity,pressure-gradient, and Coriolis. We previously saw in CHAPTER 2that the vertical distribution of mass in the atmosphere isdetermined by a balance between gravity and the pressure-gradientforce; when these forces are out of balancebuoyant motionsresult,which will be discussed in section In the horizontal direction,where gravity does not operate, the equilibrium of forces usuallyinvolves a balance between the pressure-gradient force and theCoriolis force, and the resulting steady flow is called 1-km altitude, the horizontal flow is modified byfriction with the atmosphere is considerably thinner in its vertical extent (scaleheight 7 km) than in its horizontal extent.

2 The largest scales ofmotion are in the horizontal direction and form the basis for thegeneral circulationof the atmosphere. We will concern ourselvesfirst with these horizontal GEOSTROPHIC FLOWL arge-scale movement of air in the atmosphere is driven byhorizontal pressure gradients originating from differential heatingof the Earth s surface (recall our discussion of the sea breeze effectin section ). As air moves from high to low pressure on thesurface of the rotating Earth, it is deflected by theCoriolis with an explanation of the Coriolis force and then go on toexamine the balance between the pressure-gradient and Coriolis forceConsider an observer fixed in space and watching the Earth the perspective of the observer, an object fixed to the Earth atlatitude is traveling in a circle at a constant translational speed inthe longitudinal direction( )wheret= 1 day (Figure 4-1).

3 For =42o(Boston) we findvE= 1250vE2 R cost-----------------------=41km h-1. We are oblivious to this rapid motion because everythingin our frame of reference (these notes, your ) is traveling atthe same speed. Note thatvEdecreases with increasing latitude; it isthis latitudinal gradient that causes the Coriolis force. Figure 4-1 Spherical geometry of EarthConsider now an observerOfixed to the Earth and throwing a ballat a targetT. To begin with the simplest case, imagine the observerat the North Pole and the target at a lower latitude (Figure 4-2). Ittakes a certain time tfor the ball to reach the target, during whichtime the target will have moved a certain distance xas a result ofthe Earth s rotation, causing the ball to miss the target.

4 Figure 4-2 Coriolis effect for rotating observer at North PoleThe rotating observer at the North Pole does not perceive the targetas having moved, because everything in his/her frame of referenceis moving in the same way. However, the shot missed. From theperspective of this observer, the ball has been deflected to the rightof the target. Such a deflection implies a force (the Coriolis force) Rcos RtimetonnballOTnqnT (old position)T (new position)O timeto+ t xball trajectoryviewed byrotating observer in Oball trajectoryviewed by observerfixed in space42exerted to the right of the direction of motion. An observer fixed inspace over the North Pole notices no such deflection (Figure 4-2).

5 Thus the Coriolis force is fictitious; it applies only in the rotatingframe of reference. However, we must take it into account becauseall our ATMOSPHERIC observations are taken in this rotating frame us now consider the more general case of an observer fixed tothe Earth at latitude 1in the northern hemisphere and throwing aball at a target located at a higher latitude 2(Figure 4-3). Figure 4-3 Coriolis effect for meridional motionAs the ball travels from 1to 2it must conserve its angularmomentummvE( 1)Rcos 1wheremis the mass of the ball,vE( 1)isthe translational velocity of the Earth at 1, andRcos 1is the radiusof rotation at 1. SincevE( 2)<vE( 1), conservation of angularmomentum necessitates that the ball acquire an eastward velocityvrelative to the rotating Earth by the time it gets to latitude , from the perspective of the rotating observer inOthe ballhas been deflected to the the same reasoning, a ballthrown from 2 to 1 would also be deflected to the Coriolis force applies similarly to longitudinal motions(motions at a fixed latitude).

6 To show this, let us first consider aball at rest on the Earth s surface. From the perspective of anobserver fixed to the Earth s surface, the ball experiences acentrifugal force perpendicular to the axis of rotation of the force is balanced exactly by the acceleration of gravity and bythe reaction from the surface (Figure 4-4). Because the Earth is nota perfect sphere, gravity and reaction do not simply oppose eachother. The non-sphericity of the Earth is in fact a consequence of 1 2O(old)O(new)T (old)T(new)ball trajectoryseen byobservernnqq43the centrifugal force applied to the solid Earth; we should not besurprised that the forces of gravity and reaction applied to an objectat rest on the Earth s surface combine to balance exactly thecentrifugal force on the object.

7 Figure 4-4 Equilibrium triangle of forces acting on a ball at rest on the Earth ssurface. The non-sphericity of the Earth is greatly us now throw the ball from west to east in the northernhemisphere. Since the ball is thrown in the direction of the Earth srotation, its angular velocity in the fixed frame of referenceincreases; it experiences an increased centrifugal force. As can beseen from Figure 4-4, the increase in the centrifugal force deflectsthe ball towards the Equator, , to the right of the direction ofmotion of the ball. Conversely, if the ball is thrown from east towest, its angular velocity in the fixed frame of reference decreases.

8 The resulting decrease in the centrifugal force causes the ball to bedeflected towards the pole, again to the right of the direction can generalize the above object movinghorizontally in any direction on the surface of the Earth experiences(from the perspective of an observer fixed to the Earth) a Coriolisforce perpendicular to the direction of motion, to the right in thenorthern hemisphere and to the left in the southern yourself that the Coriolis force in the southernhemisphere indeed acts to deflect moving objects to the left. Onecan derive the Coriolis acceleration capplied to horizontalmotions:( )where is the angular velocity of the Earth andvis the speed ofthe moving object in the rotating frame of reference (not to beconfused withvE, the translational speed of the Earth).

9 TheCoriolis force is zero at the equator and increases with latitude(convince yourself from the above thought experiments that theCoriolis force must indeed be zero at the equator). Note also thatnNorthPoleEquatorgravitycentrifugalf orcereaction c2 v sin=44the Coriolis force is always zero for an object at rest in the rotatingframe of reference (v = 0).The Coriolis force is important only for large-scale motions. Fromequation( )we can calculate the displacement Yincurred whenthrowing an object with speedv at a target at distance X:( )At the latitude of Boston (420N), we find that a snowball traveling10mat20kmh-1incurs a displacement Yof only 1 mm. Bycontrast, for a missile traveling 1000 km at 2000 km h-1, Yis 100km (important!)

10 In the previously discussed case of the sea-breezecirculation (section ), the scale of motion was sufficiently smallthat the Coriolis effect could be Geostrophic balanceWe saw in CHAPTER 2 that a pressure gradient in the atmospheregenerates apressure-gradient forceoriented along the gradient fromhigh to low pressure. In three dimensions the acceleration pfromthe pressure-gradient force is( )where =( / x, / y, / z) is the gradient vector. Consider an airparcel initially at rest in a pressure-gradient field in the northernhemisphere (Figure 4-5). There is no Coriolis force applied to theair parcel since it is at the effect of thepressure-gradient force, the air parcel begins to flow along thegradient from high to low pressure, , perpendicularly to theisobars(lines of constant pressure).