1 a Coriolis tutorial, Part 1: 2 the Coriolis force, inertial and geostrophic motion 3 4 James F. Price 5 Woods Hole Oceanographic Institution 6 Woods Hole, Massachusetts, 02543 7 www.whoi.edu/science/PO/people/jprice [email protected] 8 Version8 October15,2018 Figure 1: The annual mean sea surface height (SSH) of the North Atlantic for 2007. Colorbar at right is in meters. The principal features are a high over the subtropics and a low over the subpolar region. The inferred geostrophic current is sketched at a few locations. Geostrophic currents are parallel to lines of SSH, with higher SSH to the right of the current in the northern hemisphere. A central goal of this essay is to understand how Earth’s rotation leads to this key relationship between SSH and currents. 1 9 Abstract: This essay is the first of a four-part introduction to the Coriolis force and its consequences 10 for the atmosphere and ocean. It is intended for students who are beginning a quantitative study of 11 geophysical fluid dynamics and who have some background in classical mechanics and applied 12 mathematics. 13 The equation of motion appropriate to a steadily rotating reference frame includes two terms that 14 account for accelerations that arise from the rotation of the reference frame, a centrifugal force and a 15 Coriolis force. In the special case of an Earth-attached reference frame of interest here, the centrifugal 16 force is effectively subsumed into the gravity field. The Coriolis force has a very simple mathematical 0 0 17 form, −2Ω×V M, where Ω is Earth’s rotation vector, V is the velocity observed from the rotating 18 frame and M is the parcel mass. The Coriolis force is perpendicular to the velocity and so tends to 19 change velocity direction, but not velocity amplitude. Hence the Coriolis force does no work. 20 Nevertheless the Coriolis force has a profound importance for the circulation of the atmosphere and 21 oceans. 22 Two direct consequences of the Coriolis force are considered in this introduction: If the Coriolis 23 force is the only force acting on a moving parcel, then the velocity vector of the parcel will be turned 24 anti-cyclonically (clockwise in the northern hemisphere) at the rate − f , where f = 2Ωsin(latitude) is 25 the Coriolis parameter. These free motions, often termed inertial oscillations, are a first approximation 26 of the upper ocean currents generated by a transient wind event. If the Coriolis force is balanced by a 27 steady force, say a horizontal component of gravity as in Fig.1, then the associated geostrophic wind or 28 current will be in a direction that is perpendicular to the gradient of the SSH and thus parallel to isolines 29 of SSH. In the northern hemisphere, higher SSH is to the right of the current. This geostrophic balance 30 is the defining characteristic of the large scale, low frequency, extra-tropical circulation of the 31 atmosphere and oceans. 32 A little more on Figure 1: The 2007 annual mean of sea surface height (SSH) observed by satellite 33 altimetry and compiled by the Aviso project, http://www.aviso.oceanobs.com/duacs/ SSH is a constant 34 pressure surface that is displaced slightly but significantly from level and hence there is a horizontal 35 component of gravity along this surface that is proportional to the gradient of SSH. What keeps the SSH 36 displaced away from level? We can be confident that the horizontal gravitational force associated with 37 this tilted SSH is balanced locally (at a given point) by the Coriolis force acting upon currents that flow 38 parallel to isolines of SSH. This geostrophic relationship is a central topic of this essay. Notice that by 39 far the largest gradients of SSH and so the largest geostrophic currents are found on the western 40 boundary of the gyres. This east-west asymmetry is a nonlocal consequence of Earth’s rotation that will 41 be taken up in Part 3 of this three-part series. 2 42 Contents 43 1 Large-scale flows of the atmosphere and ocean 5 44 1.1 Modelsandreferenceframes . ........ 7 45 1.1.1 Classical mechanics observed from an inertial referenceframe.......... 7 46 1.1.2 Classical mechanics observed from a rotating, noninertialreferenceframe . 8 47 1.2 Thegoalsandtheplanofthisessay . ......... 9 48 1.3 Abouttheseessays ................................ ..... 11 49 2 Noninertial reference frames 12 50 2.1 Kinematics of a linearly accelerating reference frame . .................. 13 51 2.2 Kinematicsofarotatingreferenceframe . ............. 15 52 2.2.1 Transforming the position, velocity and accelerationvectors ........... 15 53 2.2.2 Stationary ⇒ Inertial; Rotating ⇒ Earth-Attached. 21 54 2.2.3 Remarksonthetransformedequationofmotion. ........... 24 55 2.3 Problems ........................................ .. 25 56 3 Inertial and noninertial descriptions of elementary motions 25 57 3.1 Switchingsides .................................. ..... 26 58 3.2 TogetafeelfortheCoriolisforce . .......... 32 59 3.3 Anelementaryprojectileproblem . .......... 35 60 3.4 AppendixtoSection3;SphericalCoordinates . .............. 37 61 3.5 Problems ........................................ .. 40 62 4 A reference frame attached to the rotating Earth 41 63 4.1 Cancellation of the centrifugal force by Earth’s (slightlychubby)figure . 41 64 4.2 The equation of motion for an Earth-attached reference frame............... 44 65 4.3 Coriolisforceonmotionsina thin,spherical shell . ................. 44 66 4.4 Onelastlookattheinertialframeequations . .............. 46 67 4.5 Problems ........................................ .. 49 68 5 A dense parcel released onto a rotating slope with friction 51 69 5.1 Thenondimensionalequations;Ekmannumber . ............ 53 70 5.2 (Near-)Inertialmotion . ........ 55 71 5.3 (Quasi-)Geostrophicmotion . ......... 59 72 5.4 Energybalance ................................... .... 61 3 73 5.5 Problems ........................................ .. 63 74 6 Summary and Closing Remarks 64 75 6.1 WhatistheCoriolisforce? . ........ 64 76 6.2 What arethe consequences of the Coriolisforcefor thecirculation of the atmosphere and 77 ocean? ........................................... 65 78 6.3 What’snext? ..................................... ... 66 79 6.4 Supplementarymaterial. ........ 66 80 Index .............................................. 68 4 1 LARGE-SCALE FLOWS OF THE ATMOSPHERE AND OCEAN 5 81 1 Large-scale flows of the atmosphere and ocean 82 The large-scale flows of Earth’s atmosphere and ocean take the form of circulations around centers of 83 high or low gravitational potential (the height of a constant pressure surface relative to a known level, 84 the sea surface height, SSH, of Fig. 1, or the 500 mb height of Fig. 2). Ocean circulation features of this 85 sort include gyres that fill entire basins, and in the atmosphere, a broad belt of westerly wind that 86 encircles the mid-latitudes in both hemispheres). Smaller scale circulations often dominate the weather. 87 Hurricanes and mid-latitude storms have a more or less circular flow around a low, and many regions of 88 the ocean are filled with slowly revolving eddies having a diameter of several hundred kilometers. The 89 height anomaly that is associated with these circulation features is the direct result of a mass excess or 90 deficit (high or low height anomaly). 91 What is at first surprising and deserving of an explanation is that large scale mass anomalies 92 implicit in the SSH and height fields of Figs. (1) and (2) persist for many days or weeks even in the 93 absence of an external momentum or energy source. The winds and currents that would be expected to 94 accelerate down the height gradient (in effect, downhill) and disperse the associated mass anomaly are 95 evidently strongly inhibited. Large-scale, low frequency winds and currents are observed to flow in a 96 direction almost parallel to lines of constant height; the sense of the flow is clockwise around highs 97 (northern hemisphere) and anti-clockwise around lows. The flow direction is reversed in the southern 98 hemisphere, anti-clockwise around highs and clockwise around lows. From this we can infer that the 99 horizontal gravitational force along a pressure surface must be balanced approximately by a second 100 force that acts to deflect horizontal winds and currents to the right of the velocity vector in the northern 101 hemisphere and to the left of the velocity vector in the southern hemisphere (you should stop here and 1,2 102 make a sketch of this). This deflecting force is the Coriolis force and is the theme of this essay. A 103 quasi-steady balance between the horizontal gravitational force (or equivalently, pressure gradient) and 104 the Coriolis force is called a geostrophic balance, and an approximate or quasi- geostrophic balance is 3 105 the defining characteristic of large scale atmospheric and oceanic flows. 106 We attribute profound physical consequences to the Coriolis force, and yet cannot point to a 107 physical interaction as the cause of the Coriolis force in the straightforward way that height anomalies 1The main text is supplemented liberally by footnotes that provide references and background knowledge. Many of these footnotes are important, but they may nevertheless be skipped to facilitate a first reading. 2After the French physicist and engineer, Gaspard-Gustave de Coriolis, 1792-1843, whose seminal contributions include the systematic derivation of the rotating frame equation of motion and the development of the gyroscope. An informative history of the Coriolis force is by A. Persson, ’How do we understand the Coriolis force?’, Bull. Am. Met. Soc., 79(7), 1373-1385 (1998). 3To be sure, it’s not quite this simple. This ’large scale’ is a shorthand for (1) large spatial scale, (2) low frequency, (3) extra-tropical, and (4) outside of frictional boundary layers. It is important to have a quantitative sense what is meant by each of these (which turn out to be linked in interesting ways) and we will come to this in Parts 2 and 3. For now, suffice it to say that this present use of ’large scale’ encompasses everythingthat you can readily see inFigs.