Chapter 11: Relation between vorticity, divergence and the vertical velocity

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1 Cater 11: Relation between ticity, diergence and te ertical elocity Te diergence equation In cater 3 we used a simle ersion of te continuity equation. Here we deelo it furter, artly because it will gie useful relationsi between izontal diergence and ertical elocities and artly because it is one of te fundamental equations wic we will need wen we attemt to sole te equations of motion to deduce te roerties of arious meteological enomena. Te fundamental idea is tat te mass of a marked lum of fluid stays constant, so tat f a small lum of olume δτ, say, and density ρ, te roduct of olume and density remains constant, i.e. ( ρδτ = 0. If te mass is constant, so is its olume, so tat we also ae ln ( ρδτ = 0. Tis can be re-written 1 ρ 1 δτ = 0. ρ δτ Now we remarked in cater 11 tat 1 δτ δτ = di. (It is te full 3- diergence wic aears in tis exression. We did not roe tis result, but you were inited to roe te 2- analogue of it as one of te roblems on tat cater, and te generalisation to 3- is straigtfward. Putting tat result into Eq 1 and exanding te deriatie gies Eq 1 Oter ways of writing tis are and 1 u w ρ t z ρ ρdi = 0 di = 0 Eq 2 Eq 3 di t ( ρ = 0 Eq 3 and Eq 4 are te continuity equation. Being written in ect fm tose equations make no assumtion about te co-dinate system. Tere is an aroximation to tese equations wic we can make. To demonstrate tis, we will Eq 4

2 write te equation in our tangent lane rectangular co-dinate system. Anoter way of grouing te terms is u w t u t z ρdi w ρ = 0 z ( wρ ρdi = 0. z Te term in curly brackets in te last equation can be sown to be small comared wit te oter terms, so to good aroximation te continuity equation is ρdi ( wρ z = 0 Eq 5 We see tat tere is a relationsi between te ertical elocity and te izontal diergence. Vertical elocities usually ae teir largest magnitude in te middle of te troosere. Tis is not surrising, as te must mean tat ertical elocities are zero actually at te. In addition te ig static stability of te stratosere suresses ertical elocities in te icinity of te. Tus te largest ertical elocities occur in te middle of te troosere. If tere is uward ( wρ motion in te middle troosere we sall ae > 0 in te lower troosere z ( wρ and < 0 in te uer troosere. It follows tat in tis case di < 0 in te z lower troosere and di > 0 in te uer troosere. Tat is to say tat middle leel ascending motion as (izontal conergence below it and diergence aboe it, wile descending motion in middle leels as diergence below it and conergence aboe it. diergence conergence conergence diergence diergence conergence conergence diergence

3 Te sketces aboe sow some ossible configurations of ertical and izontal elocities and ence izontal conergences and diergences. Magnitude of te ertical elocity We can use te aroximate continuity equation (Eq 5 to estimate tyical alues of te ertical elocity. Exanding tat equation we obtain w w = di z ρ z Now if we use U, W,, H f tyical ders of magnitude f, resectiely, te izontal, and ertical elocities and izontal and ertical distances, ten te terms on te left-and side of te equation are of der H W We ae seen in te reious cater tat te rigt and side is of der U R o. Hence we must ae tat H W ~ RoU. H 10km 1 Te ratio f mid-latitude systems is ~ =, so we migt ae exected 1000km 100 tat ertical elocities were about one undredt of izontal elocities, but our analysis sows tat we must multily by an additional fact of te Rossby number (namely 1/10 in f te mid-latitude synotic scale. Tus te ertical elocities are tyically 1/1000 time te izontal ones. Tis gies a tyical large-scale mid-latitude ertical elocity of 1cm/sec 1km er day. W H ~ R A consequence of tese magnitudes is tat te ertical adection terms in te material deriatie are small. Te material deriatie is u w. Te second t z and tird terms are of der of magnitude U / and we ae reiously sown tat te timescales are suc tat te first term is of similar magnitude. Howeer te final W Ro UH U term is of magnitude = = Ro wic is an der of magnitude smaller and H H may be igned to der of te Rossby number, in symbols:- = u t O y ( R o o U Te ticity equation An imtant relation between te ticity and diergence (and ence ertical elocities emerges wen we derie te rate of cange of ticity. To obtain tis we

4 start from te equations of motion, in wic we ae neglected te ertical adection term. 1 u f = t ρ 1 u fu = t ρ Take of te first equation from of te second, giing u t x y = ρ ρ f f f We ae igned te terms in te izontal gradients of density, as being small comared to te retained terms. Te terms on te rigt cancel and te remainder of te equation can be re-arranged to gie ( ζ f = ( ζ f rel ( ζ = ( ζ di rel di Eq 6 Tis is known as te ticity equation. We ae added a subscrit to te material deriatie to remind ourseles tat only te izontal terms matter. Sometimes tis is described as te rate of cange following te izontal motion f obious reasons. Absolute ticity is usually ositie in te ntern emisere (and negatie in te soutern. Accding to tis equation, te magnitude of te olute ticity is decreased by izontal diergence and increased by izontal conergence. Tis is a consequence of te conseration of angular momentum. Consider a lum of air. Conergence decreases te izontally rojected area of te lum. Hence it decreases its moment of inertia. To maintain a constant angular momentum te lum as to sin faster. Te oosite occurs f te case of izontal diergence. Using te continuity equation in te fm of Eq 5 to eliminate te diergence from Eq 6 gies ζ ( ( ρw ζ = ρ z Te term on te rigt is ositie if w ρ is increasing uwards. Te dominant effect is weter w increases uwards not. If it does it means tat te air columns are stretcing. Stretcing air columns lead to an increase in te magnitude of te olute

5 ticity, wile srinking air columns lead to a decrease in te magnitude of te olute ticity. Vticity and iergence equations in ressure co-dinates We sall find it useful to use ressure co-dinates in later deeloments. It can be sown, by analysis wic will need to be taken on trust in tis course, tat f ydrostatic atmoseres, te continuity equation in ressure co-dinates becomes ϖ = 0 di ϖ = 0 Eq 7 Te tree-dimensional diergence is zero in ressure co-dinates. Note tat tere are no time deriaties in tis equation.. Since te ydrostatic equation imlies z δ = ρgδz, we migt exect ϖ = ~ ρg = ρgw. Te similarity between Eq 7 and te aroximate fm of te continuity equation in te reious cater is tus aarent. Howeer, it turns out tat, wile tat equation is only aroximate, Eq 7 is exact to te extent tat te ydrostatic equation olds. Te ticity equation in ressure co-dinates becomes ( ζ = ζ ϖ Eq 8 a a ( ζ = ζ di Eq 9 b