Geostrophic balance. John Marshall, Alan Plumb and Lodovica Illari. March 4, 2003

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1 Geostophic balance John Mashall, Alan Plumb and Lodovica Illai Mach 4, 2003 Abstact We descibe the theoy of Geostophic Balance, deive key equations and discuss associated physical balances. 1 1 Geostophic balance If the flow is such that the Rossby numbe is small R o 1 whee R o = U fl (1) (hee U is a typical hoizontal cuent speed, f is the Coiolis paamete and L is a typical hoizontal scale ove which U vaies), then the Coiolis foce is balanced by the pessue gadient foce in the hoizontal component of the momentum equation, which educes to: fbz u+ 1 p =0 (2) ρ whee bz is a unit vecto in the vetical diection. In situations whee (as is always the case) Eq.(2) is only an appoximate balance, the velocity u defined by (2) involving only the hoizontal components of u isknownasthe geostophic wind (o cuent). Rewiting Eq.(2), we can define the geostophic wind (since bz bz u = u) as u g = 1 bz p, (3) ρf o, witing out its Catesian components, u g = 1 p ρf y ; (4) v g = 1 p ρf. 1

2 1 GEOSTROPHIC BALANCE 2 Figue 1: Schematic of two isobas on a hoizontal suface. The magnitude of u inceases as the siobas become close togethe. We see fom Eq.(2) that the geostophic flow is nomal to the pessue gadient: i.e. along isobas (lines of constant pessue) and its speed is popotional to the pessue gadient. Conside Fig.1, the cuved lines show two isobas on which pessue has the constant values p and p + δp. Thei sepaation is δs. Fom Eq.(3), the flow speed is u g = 1 ρf p = 1 δp ρf δs. Since δp is constant along the flow, u g (δs) 1 : the flow is stongest whee the isobas ae closest togethe. Since the geostophic flow cannot coss the isobas, the latte act like banks of a ive, causing the flow to speed up whee the ive is naow and slow down whee it is wide. AsshowninFig.2, theflow is (in the nothen hemisphee) anticlockwise (cyclonic) aound a low pessue cente, and clockwise (anticyclonic) aound a cente of high pessue. ( Cyclonic means in the same sense as the vetical component of the Eath s otation, and anticyclonic the opposite. So, in the southen hemisphee whee f < 0, theflow is clockwise, but still cyclonic, aound a low pessue cente.) This ule is summaized in 1 Notes to accompany : Weathe and Climate Laboatoy. Fo a moe detailed desciption see notes on web page hee:

3 1 GEOSTROPHIC BALANCE 3 Figue 2: Geostophic flow aound (left) a low pessue cente and (ight) a high pessue cente. (Nothen hemisphee case, f>0.) The effect of Coiolis delecting flow to the ight is balanced by the hoizontal component of the pessue gadient foce, 1 p, diected fom high to low pessue. ρ Figue 3: Buys-Ballot s law: If you stand with you back to the wind in the nothen hemisphee, low pessue is on you left. We now conside pessue coodinate vesions of the geostophic and continuity equations which allow some simplifications (with egad to density vaiations) to be made. 1.1 The geostophic wind in pessue coodinates In ode to apply the geostophic equations to atmospheic obsevations and paticulaly uppe ai analyses (see below), we need to expess them in tems of height gadients on a pessue suface, athe than, as in Eq.(4), of pessue gadients at constant height. Conside Fig.3. The figue depicts a suface of constant height z 0, and one of constant height p 0,which intesect at A, whee of couse pessue is p A = p 0 and height is z A = z 0. Atconstantheight,

4 1 GEOSTROPHIC BALANCE 4 the gadient of pessue in the x-diection is µ p = p C p 0 z δx whee δx is the (small) distance between C and A. Now, the gadient of height along the constant pessue suface is µ z = z B z 0. p δx Since z C = z 0,andp B = p 0, we can use the hydostatic balance equation Eq.(??) towite p C p 0 z B z 0 = p C p B z B z C = p z =+gρ Theefoe (and invoking a simila esult in the y-diection), it follows that µ µ p z = gρ ; z p µ µ p z = gρ. y y Theefoe Eq.(3) becomes, in pessue coodinates, z p u g = g f bz p p z, (5) whee bz p is the upwad unit vecto in pessue coodinates and p denotes the gadient opeato in pessue coodinates. In component fom, (u g,v g )= µ g z f y, g z f. (6) Note that, like p contous on sufaces of constant z, z contous on constant p ae steamlines of the geostophic flow. Note that, in pessue coodinates, the geostophic wind is, as befoe, nondivegent if f is taken as constant: u g = u + v y =0. Let s now look at some synoptic chats to see some of these ideas in action.

5 1 GEOSTROPHIC BALANCE 5 Figue 4: 500mb wind and geopotential height field on Octobe 9th The wind blows away fom the quive: one full quive denotes a speed of 5ms 1, one half-quive a speed of 2.5ms 1. The geopotential height is in metes. 1.2 Highs and Lows; synoptic chats Fig.4 shows the height of the 500mb suface (in geopotential metes, contoued) plotted with the obseved wind vecto (one full quive epesents a wind speed of 5 ms 1 ). Note how the wind blows along the isobas and is stongest the close the isobas ae togethe - see the schematic diagam in Fig.2. At this level, away fom fictional effects at the gound, the wind is close to geostophic.

6 1 GEOSTROPHIC BALANCE 6 Figue 5: 1.3 Visualizing geostophic balance on the sphee To get a feel fo geostophic balance on the sphee it is vey useful to conside a ing of ai moving eastwad at speed u elative to the undelying otating eath. Thee is a centifugal acceleation diected outwads pependicula to the eath s axis of otation, the vecto A sketched in fig(5): V 2 (u + Ω)2 = = Ω 2 +2Ωu + u2 Hee V is the absolute velocity the fluid has viewed fom an obseve fixedinspacelooking back at the eath. Let s now conside the tems in tun: Ω 2 - this is the centifugal acceleation acting on a paticle fixed to the eath. As discussed above, this acceleation is included in the gavity which is usually measued and is the eason that the eath is not a pefect sphee. 2Ωu + u2 - the additional centifugal acceleation due to motion elative to the eath. u Note that if << 1, we may neglect the tem in Ω u2. Fo the eath R o = u 0.02 Ω and so the 2Ωu tem dominates. It is diected outwad pependicula to the axis of otation and can be esolved: pependicula to the eath s suface - vecto B in the diagam - and paallel to the eath s suface - vecto C in the diagam. Component B changes the weight of the ing slightly - it is vey small compaed to g, the acceleation due to gavity, and so unimpotant. (7)

7 1 GEOSTROPHIC BALANCE 7 Figue 6: Component C, paallel to the eath s suface, is the Coiolis acceleation: 2Ω sin ϕ u So thee is a centifugal foce diected towad the equato because of the motion of the ing of ai elative to the eath. It is this foce that balances the pessue gadient foce associated with the sloping isobaic sufaces induced by the pole-equato tempeatue gadient. Let s postulate a balance between the Coiolis foce and the pessue-gadient foce diected fom equato to pole associated with the tilted isobaic sufaces - see Fig.6. ρadϕdz {z } 2Ω sin ϕu {z } mass acceleation = p ϕ dϕdz {z } p_gad Intoducing a coodinate y whichpointsnothwadsontheeath ssuface,dy = adϕ, the above educes to: fu+ 1 p =0 (8) ρ y whee f =2Ωsin ϕ is the Coiolis paamete. This is just one component of the geostophic elation, Eq.(4), between pessue gadient foces and Coiolis foces.

2 r2 θ = r2 t. (3.59) The equal area law is the statement that the term in parentheses,

2 r2 θ = r2 t. (3.59) The equal area law is the statement that the term in parentheses, 3.4. KEPLER S LAWS 145 3.4 Keple s laws You ae familia with the idea that one can solve some mechanics poblems using only consevation of enegy and (linea) momentum. Thus, some of what we see as objects