Correspondence

correspondence

Challenges in the science of gradient flow This is not a real force, as Captain Arnold points out, but it is neither a reaction force. It is one of mathematical defini- Owen E. Thompson, University of Missouri, tion only and not of physical reality. To transpose such a Columbia, Mo. term is only incorrect in the use of the word "force" rather We owe a debt of gratitude to Captain Arnold and his dis- than "negative acceleration." One invents terminology for cussion of the "balanced gradient flow" concept, for it brings convenience in other areas such as defining various types of to light an issue concerning the proper teaching of meteo- potential energy by computing the amount of work done by rology which is long overdue if we are to regard meteorology conservative forces (e.g., gravitational potential energy). Also, as an exact science. In this note I would like to reinforce his in the dynamics of atmospheric motions it has proved con- plea for more correct terminology by examining the logical venient to regard the apparent Coriolis acceleration as a framework within which terms such as "centrifugal force" negative force and transpose this term to the "force" side and "balanced flow" might be introduced and, more gen- of the equation. (There may also be a question here as to the erally, the criterion that should be used in deciding when possible confusion introduced by this.) These manipulations it is fruitful to invent such terminology. It is, of course, are absolutely harmless provided everyone knows and obeys clear that the balanced gradient flow idea is an invention the rules and that the manipulation leads to a wore con- of man's imagination but, as we shall see, so also are other venient interpretation of the situation. less criticized ideas in meteorology. The real question then is when and in what situations is Let us first examine two possible interpretations of the it convenient to make such agreements about point of view so-called centrifugal force in the balanced gradient flow and terminology. Our concept of an air parcel in hydrostatic equation. Captain Arnold argues that the centrifugal force equilibrium has proved to be convenient even though it is is a body's reaction to a force applied to it. The reaction no more a truly balanced situation, due to the parcel's con- force to which he refers is probably an imagined analogue tinuous acceleration towards the Earth's axis, than is "bal- to the actual reaction force exerted on the end of a cord anced" gradient flow. The claim of its convenience follows to which is tied a rock undergoing circular motion. This anal- from the fact that one most often observes this phenomenon ogy is quite common for this discussion of circular motion from a frame of reference in which there is no measurable (and many others) mainly because of our rather uniform ex- vertical acceleration. Similarly, geostrophic flow which in an periences in such rock and cord experiments. The forces and inertial coordinate system is unbalanced, appears to be bal- reaction forces in both examples, however, are distinguish- anced in the common meteorological cartesian or natural co- able from the centripetal acceleration. Newton's third law concerning action-reaction pairs addresses itself only to the ordinate system. Such is not the case for gradient flow ob- force in the first and second law and not to the resulting served from these or any other reference frames fixed to a accelerations. The reactions to the pressure gradient force point on the Earth's surface. Certainly, then, Captain Ar- components and the Newtonian gravitational force are, re- nold's point should be well taken. The use of the term spectively, outward pressures exerted on adjacent air parcels "balanced gradient flow" in discussions relative to our com- and a gravitational attraction force exerted on the Earth's mon coordinate systems is improper, it leads to unnecessary center of mass. These reactions do indeed act on bodies other confusion and should consequently be eliminated. than the air parcel in question and for this reason are of no The question brought out by Captain Arnold should not importance in discussing the behavior of the parcel. The be regarded as one merely of semantics. His claim that the centripetal acceleration is, however, a distinct real property concept of centrifugal force and balanced gradient flow leads of the air parcel. Further, one should not even be concerned to confusion is probably a truism in many quarters. This is with the application of Newton's third law to the non-inertial particularly unfortunate since it is no more difficult to teach meteorological coordinate system for even the more impor- the principles of meteorology without such "handwaving." tant second law does not hold in such coordinates. That is, The conveniences of notation and terminology should be in- Newton's third law identifies a reaction force to only those troduced after a careful and complete explanation of the forces which give rise to a change in absolute momentum as fundamentals and then only when it truly leads to a more measured in a non-accelerating, inertial coordinate system. convenient formulation of a concept. Hopefully this will We would be hard pressed to identify real reactions to the prevent a researching scientist of the next generation from apparent forces of which the Coriolis force is an example. "modeling," say, a tornado with both a centrifugal force and This would have to be done with an imagination equal to centripetal acceleration (or at least prevent a published ac- that used in defining these apparent forces themselves. knowledgment of our help and guidance as former teachers). Now if one alternatively defines a specific centrifugal force by transposing an identifiable centripetal acceleration to the Reference "force" side of the equation, then one has merely manufac- Arnold, Charles P., Jr., 1967: Gradient flow—balanced or un- tured a force term using an acceleration as raw material. balanced. Bull. Amer. Meteor. Soc., 48, 715. 24 Vol. 49, No. 1, January 1968

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Comments on "Gradient flow—balanced It is in this system that the pressure gradient force, the or unbalanced?" apparent Coriolis force and the apparent centrifugal force add up to zero. No confusion results when showing this Leon F. Graves, University of Houston, resultant force in a diagram because this resultant force Houston, Tex. is zero. Newtonian equations of motion refer to motion with respect Meteorologists frequently violate an important vector con- to an inertial coordinate system. I believe much of the concept, namely, that the length of a vector arrow represents the fusion referred to by Charles P. Arnold in "Gradient Flow- absolute value of the vector quantity and the direction of the Balanced or Unbalanced" results from failure to specify co- arrow shows the sign of the vector quantity with respect to ordinate systems adequately. The GLOSSARY OF METEOROLOGY the coordinate system selected. states, "For all purposes in meteorology, a system with origin It is naturally possible to introduce sign conventions that on the axis of the Earth and fixed with respect to the stars will enable one to use the same equations in more than one (absolute coordinate system) can be considered an inertial situation. However, no reputable physics book will ever show system. When relative coordinate systems are used, moving a vector arrow with a negative sign on the length, as me- with respect to the inertial system, apparent forces arise in teorology books are prone to do. Such labels seem to result Newton's laws, such as the Coriolis force." from zealous attempts to get more mileage out of the co- In discussing gradient flow, I would prefer to treat cen- ordinate systems. tripetal acceleration with little reference to uniform circular Complicated sign conventions have no place in the ex- motion. Centripetal acceleration is a change in velocity per planation of gradient flow. Furthermore, sign conventions unit time at right angles to the direction of flow. It is an need not be standardized, although it does seem odd for the instantaneous acceleration and should not therefore be tied GLOSSARY OF METEOROLOGY to run the n-axis to the left in specifically to uniform circular motion, which, by its very discussing the geostrophic wind and to the right in discuss- nature, must be uniform, at least for a while. The flow need ing the geostrophic wind scale, both on page 251. Trans- not be. forming a vector diagram into consistent equations may be In the relative coordinate system mentioned above, gra- difficult mathematics or difficult physics, but it is not diffi- dient flow is flow such that the vector sum of the pressure cult meteorology. gradient force (a real force) and the Coriolis force (an appar- I prefer not to confuse readers by discussing the mirror- ent force) becomes a centripetal force which in turn produces image of this flow which occurs in the Southern Hemisphere. If we project Northern Hemisphere flow on the equatorial a centripetal acceleration. In gradient flow, there is no other plane of the Earth, it will coincide with the projection of acceleration. Southern Hemisphere flow. Instructors can easily provide When two or more vector arrows are to be added together the details. in a diagram, the resultant force should be indicated clearly. In diagrams of gradient flow, the curvature and spacing When only two collinear forces are added, it is difficult to of streamlines and isobars represent distribution of pres- show the resultant force arrow. sure in space. Superimposed on this distribution are velocity With cyclonic motion, the pressure gradient force is larger vectors showing variation in position with respect to time. than the Coriolis force and therefore the resultant or net Finally we show forces balanced "isometrically" against each force acts in the same direction as the pressure gradient force other. Frequently we divide the forces by unit mass and but is smaller in magnitude. In anticyclonic motion, the call them specific forces or accelerations. Confusion is to be Coriolis force is larger than the pressure gradient force and expected. therefore the resultant or net force acts in the same direc- The accompanying diagram represents horizontal friction- tion as the Coriolis force but is smaller in magnitude. We less flow at 30° North Latitude, with pressure decreasing uni- are again faced with showing the shorter arrow superimposed formly along the line from right to left. The three pressure on the longer arrow. gradient force (PGF) arrows are therefore equal in length. Some authors attempt to solve this problem by placing the The center diagram shows geostrophic flow, a special case of arrows side by side, but in doing so generally ignore the con- gradient flow in which the isobars and streamlines are projec- cept that they should be placed so that the diagram does not tions of great circles. In geostrophic flow, the Coriolis force introduce a net torque. Collinear forces by themselves cannot (COF) is equal in magnitude to the pressure gradient force. produce a torque. Furthermore these diagrams frequently In cyclonic flow, the Coriolis force is smaller than it is in create the impression that the forces are applied at different geostrophic flow because centrifugal force (CGF) also acts points or in non-parallel directions. against the pressure gradient force. In anticyclonic flow, the Adoption of a rotating coordinate system has several ad- Coriolis force is larger than it is in geostrophic flow be- vantages. For one thing, the centripetal term moves over to cause it acts against both the pressure gradient force and the left side of the equation and becomes an apparent cen- the centrifugal force. Since Coriolis force at a point is pro- trifugal force. Let us take a relative coordinate system rotat- portional to the velocity, the velocity vectors are drawn in ing with respect to the non-rotating system just described. the same ratio of lengths as the Coriolis force vectors. From The coordinate system itself, rotating with angular velocity the diagram it is quite evident that the gradient speed in the a) = v/r, now can represent gradient flow physically without cyclonic flow is less than the gradient speed in anticyclonic any assumptions whatsoever as to the existence of a gradient flow, other conditions being equal. wind. Equations of motion can be written for this system to When an astronaut is sailing along in an Earth-bound describe deviations from gradient flow. If, however, the devia- orbit, is he moving toward the east over the Earth—or is tions are defined to be zero, the gradient flow is uniquely de- the Earth rotating westward under him? It all depends on cribed by this rotating coordinate system itself. (Remember one's point of view! Balanced or unbalanced? McDonald that oj is a function of both v and r.) (1953) says ". . . the non-inertial observer who chooses to ii

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FIG. 1 think in terms of 'Coriolis forces' and 'centrifugal forces' 715) proposes "that we change our description of gradient can give just as internally consistent an account of dynami- wind flow to make it perfectly clear, particularly to our stu- cal problems as can an inertial observer who analyzes these dents, that this condition is an unbalanced one with the net same problems." force or centripetal force being the algebraic sum of the I agree wholeheartedly with Capt. Arnold when he says pressure gradient and Coriolis forces." This must be done. that physics has been neglected in some well-known meteo- We in education, and especially those writing and revising rology books. I wish to point out, however, that physics has books dealing with the dynamics of air motion, should go also been neglected in some well-known physics books. When farther in clarification than Arnold proposes. It is wrong to we combine physics and meteorology, we need to make a have been leaving the impression that geostrophic flow, gra- firm alloy, not a spot weld. dient flow, and their modification by friction in the bound- ary layer sufficiently characterize the horizontal dynamics References and motion. Air flow is generally accelerated, and, in viola- Huschke, R. E., 1959: Glossary of Meteorology. Boston, Amer. tion of assumptions in the familiar gradient wind equation, Meteor. Soc., 646 pp. the resultant acceleration is very seldom both normal to the McDonald, James E., 1953: Remarks on the development of velocity and also equal in magnitude to the resultant of the Coriolis principle. Bull. Amer. Meteor. Soc., 48, p. 192. pressure gradient and Coriolis accelerations. Unless we cover the full spectrum of resultant horizontal accelerations occur- ring in the real settings, in which the restrictive gradient Comments on "Gradient-flow—balanced or wind condition is only one of many special cases, then good unbalanced?" surpasses harm in analysis of atmospheric motions by com- pletely omitting discussion of the gradient wind equation, W. J. Saucier, University of Oklahoma, whose cuteness as an exercise in algebra has long covered Norman, Okla. up its deficiency in fulfilling the intended purpose of mod- In concluding his correspondence note on the same subject, elling accelerated motion. And consequently the non-existent Charles P. Arnold (Bull. Amer. Meteor. Soc., 48, No. 9, p. centripetal (centrifugal) force will be left to antiquity.

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