Lecture 12: Angular Momentum and the Hadley Circulation
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1 Lecture 12: Angular Momentum and the Hadley Circulation September 30, 2003 We learnt last time that there is a planetary radiative drive net warming in the tropics, cooling over the pole which induces an equator pole temperature gradient = (due to expansion of warm columns, contraction of cold columns) a equator pole pressure gradient force at upper levels, as sketched in Fig.1. Today we discuss the consequence of this poleward pressure gradient force. Since air tends to move down the pressure gradient and we must get a circulation. Onemightexpectringsofaircirclingtheglobetobedrivenpolewards by pressure gradient forces. Indeed, in 1753 Hadley suggested one giant meridional cell with rising motion in the tropics and descending motion at the pole, as sketched schematically in Fig.2. As the rings contract conserving the angular momentum imparted to them by the spinning Earth, eastward winds will be induced. At the poles Hadley imagined that the rings would sink and then expand outward as they flow equatorward below, generating eastward winds. Let s look at the circulation postulated by Hadley in a little more detail. 1 Angular Momentum The earth is a rather rapidly rotating planet and its spin imparts angular momentum to the atmosphere above. This angular momentum, and its conservation, imposes rather rigid constraints on the possible motions. Consider a ring of air circling the globe as shown in Fig.3. Under the influence of the poleward pressure gradient force it will move poleward, its 1
2 COLD Pole p Pressure gradient force from high to low pressure WARM p Equator Figure 1: Figure 2: The circulation envisaged by Hadley comprising one giant meridional cell stretching from equator to pole. 2
3 distance, r, from the axis of rotation will decrease and hence, conserving its angular momentum, its eastward wind, u, relative to the Earth, will increase. The angular momentum of the ring, per unit mass, is Figure 3: A = Ωr 2 + ur, the first term being the contribution from planetary rotation, and the second from the eastward wind u relative to the Earth, where r is the distance from the rotation axis. Here Ω is the rotation rate of the earth (in radians per second). Since r = a cos ϕ, A = Ωa 2 cos 2 ϕ + ua cos ϕ. (1) If we assume (for simplicity, but motivated by observation) the circulation is axisymmetric (no longitudinal variations) then angular momentum is conserved by the ring as it moves around. Now, consider the upper branch of the Hadley circulation, and suppose that, at the equator, u =0. Then the angular momentum at the equator is simply A 0 = Ωa 2. Now, if we follow this ring of air as it moves poleward and, very reasonably, assume it retains its absolute angular momentum A 0, it will acquire an eastward velocity u (ϕ) = (A 0 Ωa 2 cos 2 ϕ) a cos ϕ 3 = Ωa sin2 ϕ cos ϕ. (2)
4 This can produce a very large wind: at ϕ =10 o, u(ϕ) =14ms 1 ;at20 o, u(ϕ) =58ms 1 ;at30 o, u(ϕ) =130ms 1, and of course u(ϕ) as ϕ 90 o. By the time middle latitudes are reached, these values are completely unrealistic and are not significantly changed by invoking frictional effects at the ground. We must conclude that rings do not conserve their angular momentum as they are driven poleward in the manner postulated. Clearly, the circulation cannot go all the way to the pole as envisaged by Hadley. How far this axisymmetric circulation will extend depends (according to theory) on many factors; observations see Fig.4 tell us that it extends to about 30 o latitude 1. We conclude that something must balance the pressure gradient force. What? It turns out to be centrifugal forces (Coriolis forces) directed outward associated with the zonal motion of the ring of air relative to the earth. 1.1 Visualising the Coriolis force Consider again our ring of air. There is a centrifugal acceleration directed outwards perpendicular to the earth s axis of rotation, the vector A sketched in Fig.5: V 2 (u + Ωr)2 = = Ω 2 r +2Ωu + u2 (3) r r r Here V is the absolute velocity the fluid has viewed from an observer fixed in space looking back at the earth. Let s now consider the terms in turn: Ω 2 r - this is the centrifugal acceleration acting on a particle fixed to the earth. It is included in the gravity which is usually measured and is the reason that the earth is not a perfect sphere a fluid planet (and the earth is a fluid on geological timescales) becomes an oblate spheroid. Indeed, the equatorial radius of the earth is 24 km larger than the polar radius see Fig.6 2Ωu+ u2 - the additional centrifugal acceleration due to motion relative r u to the earth. Note that if << 1, we may neglect the term in Ωr u2. For the earth u 0.02 and so the 2Ωu term dominates. It is directed Ωr 1 If the Earth were rotating less (or more) rapidly, and other things being equal, the Hadley circulation would extend further (or less far) poleward. 4
5 Figure 4: The meridional overturning streamfunction of the atmosphere in annual mean, DJF and JJA conditons. Units are in kg/sec. Flow circulates around positive (negative) centers in a clockwise (anti-clockwise) sense. Thus, in the annual mean, air rises just north of the equator and sinks around ±30 o. 5
6 Figure 5: outward perpendicular to the axis of rotation and can be resolved: perpendicular to the earth s surface - vector B in the diagram and parallel to the earth s surface - vector C in the diagram. Component B changes the weight of the ring slightly - it is very small compared to g, the acceleration due to gravity, and so unimportant. Component C, parallel to the earth s surface, is the Coriolis acceleration: 2Ω sin ϕ u So there is a centrifugal force directed toward the equator because of the motionoftheringofairrelativetotheearth. Itisthisforcethatbalancesthe pressure gradient force associated with the sloping isobaric surfaces induced by the pole-equator temperature gradient. Let s postulate a balance between the Coriolis force and the pressuregradient force ρadϕdz {z } 2Ω sin ϕu {z } mass acceleration = p ϕ dϕdz {z } p_grad 6
7 Figure 6: Introducing a coordinate y which points northwards on the earth s surface, dy = adϕ, the above reduces to: where fu+ 1 p ρ y =0 (4) f =2Ω sin ϕ (5) is known as the Coriolis parameter, the component of the Earth s rotation resolved in the direction of gravity. Eq.(4) is a special case of the geostrophic balance which we will be discussing in much more detail in later lectures a balance between pressure gradient forces and Coriolis forces - it is one of the most important concepts in the dynamics of rotating fluids. Whatdoesitimplyaboutthemagnitude of the wind? Using the hydrostatic relation to write 1 z as g where z is the geopotential height of, for example, the 500mb surface, geostrophic balance implies ρ y awindofstrength: u = g f p y z L = =10ms 1 7
8 Figure 7: wherewehaveassumedthatthe500mbsurfaceslantsdownby z =500m in L = 5000km, roughly consistent with the observations we looked at last time. Thus centrifugal forces acting on a ring of air moving at a speed of 10ms 1 relative to the ground, are of sufficient magnitude to balance the poleward pressure gradient force associated with the pole-equator temperature gradient. The above considerations, then, suggest we should observe zonal winds of a magnitude of some 10ms 1. This, indeed, is just what we observe see Fig.8. 8
9 Figure 8: Meridional cross-section of zonal wind (ms 1 ) under annual mean conditions (top), DJF (December, January, February ) (middle) and JJA (June, July, August) (bottom) conditions. 9
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