ESCI 343 Atmospheric Dynamics II Lesson 13 Geostrophic/gradient Adjustment

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1 ESCI 343 Amosperic Dynamics II Lesson 13 Geosropic/gradien Adjusmen Reerence: An Inroducion o Dynamic Meeorology (3 rd ediion), J.R. Holon Amospere-Ocean Dynamics, A.E. Gill Reading: Holon, Secion 7.6 GEOSTROPHIC ADJUSTMENT OF A BAROTROPIC FLUID Te amospere is nearly always close o geosropic and ydrosaic balance. I is balance is disurbed roug suc processes as eaing or cooling, e amospere adjuss isel o ge back ino balance. Tis process is called geosropic adjusmen, aloug i may more accuraely be reerred o as gradien adjusmen, since in curved low e amospere ends oward gradien balance. One meod o sudying geosropic adjusmen is o irs sudy adjusmen in a baroropic luid using e sallow-waer equaions. Once we undersand adjusmen in a baroropic luid we can easily eend our resuls o a baroclinic luid by use o e concep o equivalen dep, sudied in a previous lesson. For is we can use e linearized sallow-waer equaions wi zero mean low, u v = g v u = g y u v H = y I we ake / o (1) and add i o /y () we ge (1) () (3) u v y v u = g. (4) y y Rearranging (3) we ge u v 1 =. (5) y H From e deiniion o voriciy we know a v u ζ. (6) y

2 Puing (5) and (6) ino (4) we ge = Hζ y gh. (7) We need one more equaion a relaes and ζ. Tis is e sallow-waer voriciy equaion, ound by aking / o () and subracing /y (1) o ge = y v u ζ, (8) wic, using (5), can be wrien (aer some rearranging) as = H ζ. (9) Inegraing (9) wi respec o ime gives H H ζ ζ =, (1) were ζ and reers o e iniial values o relaive voriciy and eig perurbaion. Using is in (7) resuls in ( ) H y gh = ζ. (11) Since e quaniy gh is e square o e speed o a graviy wave in is luid, we can denoe i by c and wrie is equaion as ( ) H y c = ζ. (1) Equaion (1) governs e geosropic adjusmen process in a baroropic luid. THE STEADY-STATE SOLUTION Les simpliy ings somewa by assuming e iniial sae is a res, and as an abrup sep in e surace eig given by ) ˆsgn( =. 1 (13) We also assume a ere is no dependence in e y-direcion. Equaion (1) en becomes ) ˆsgn( c =, (14) 1 Te sgn() uncion is deined o be 1 or, and 1 or <.

3 a second order, non-omogeneous parial dierenial equaion. Te omogeneous orm o is equaion suppors sallow-waer inerial-graviy waves (see eercises). Aer ese waves ave subsided, ere will remain a seady-sae soluion wic obeys e seady sae equaion d ˆsgn( ) =. (15) d c c Equaion (15) is a second-order, non-omogeneous ordinary dierenial equaion wi consan coeiciens (assuming and c are consan). Te soluion o (15) consiss o a complemenary soluion (e general soluion o e omogeneous equaion) plus a paricular soluion, ( ) = ( ) ( ). (16) c p Te complemenary soluion is ound rom e caracerisic equaion or e omogeneous orm o (15), wic is r ( c) =. (17) Tereore, e complemenary soluion is en α α c ( ) = Ae Be (18) were α c. (19) For e paricular soluion we use e meod o undeermined coeiciens, guessing a e paricular soluion will ave e orm p ( ) = C sgn( ). () Puing () ino (15) sows a C = ˆ, so a e paricular soluion is ( ) = ˆsgn( ). (1) p Tereore, e general soluion o (15) is α ( ) = Ae α Be ˆsgn( ), () or α α Ae Be ˆ ( ) =. α α Ae Be ˆ < (3) All a remains is o apply e boundary condiions, wic require a: () remain bounded as ± : Tis requires a A = or posiive and B = or negaive, so a α Be ˆ ( ) = (4) α Ae ˆ < () be coninuous a = : Tis means a B ˆ = A ˆ (5) Te irs derivaive o () be coninuous a = : Tis means a αb = α A (6) Solving (5) and (6) or A and B yields 3

4 A = ˆ B = ˆ so a e seady-sae soluion is α = ˆ e 1 ( ) α 1 e. (7) < ANALYSIS OF THE SOLUTION Te igures below sow e iniial eig ield, e ransien eig ield, and e seady-sae eig and velociy ields aken rom a 1-D sallow-waer numerical model. Te ransien soluion consiss o e sallow-waer inerial graviy waves. Te inal eig soluion is e seady sae soluion rom (7). Te igures are sriking in a, oug e sep a was in e iniial condiions is smooed ou, ere is sill a region near e cener o e domain wi a orizonal pressure gradien, and ereore, wi a geosropic low ou o e page. Te iniial eig ield adjused under e inluence o graviy, and se up a low a is in geosropic balance wi e inal eig ield. Te ecess mass and poenial energy were removed by e inerial-graviy waves wic propagaed away as par o e non-seady sae soluion. Te region in wic ere is a remaining eig gradien is caracerized by an e-olding scale o 1/α. Tis leng scale is o undamenal imporance. I measures e 4

5 scale over wic e inluence o e ear s roaion aecs e low, and is called e Rossby radius o deormaion. I is deined as λ R c. (8) Te radius o deormaion is given by e group velociy o a graviy wave divided by e Coriolis parameer. Te pysical essence o e radius o deormaion can be seen by recalling a e inerial period (e ime scale or wic roaional eecs are imporan) is π 1, so a π λ R can be inerpreed as e disance raveled by a graviy wave during one inerial period. For dispersive graviy wave modes e group velociy, c g, sould be used raer an e pase speed. ROSSBY RADIUS OF DEFORMATION Te Rossby radius o deormaion is a undamenal pysical parameer o a luid on a roaing reerence rame. I gives a leng scale a can be used as a measure o ow large a disurbance as o be in order or roaional eecs o be imporan. Te pysical concep o e radius o deormaion is beer illusraed using e ollowing eample. Imagine a you immerse a umbler ino a lake, urn i upside down, and li i up o e poin jus beore e lip breaks e surace o e waer (a quick calculaion will sow a e radius o e umbler is muc smaller an e radius o deormaion.) Rig as you li e lip o e umbler compleely rom e waer e iniial eig ield would look like a picured below. As soon as you li e umbler rom e waer, e waer surace begins o adjus by generaing graviy waves wic propagae away rom e iniial disurbance. In e seady sae all a remains aer e waer calms is a la surace, as sown. Now, imagine e same eperimen perormed in e ocean, only using an eremely wide umbler a as a radius muc greaer an e radius o deormaion. Perorming e same eperimen will resul in a seady sae soluion picured below, wi a ump o waer remaining, around wic an anicyclonic geosropic (acually, gradien) circulaion as developed! Wa is e dierence beween e wo eperimens? I is ow e orizonal scale o e iniial disurbance compares wi e Rossby radius o deormaion! 5

6 We can illusrae is using a 1-D sallow-waer model or disurbances ranging in size rom very small compared o e Rossby radius o deormaion o ose a are very large. (a) (b) (c) (d) In ese igures, e raio o e orizonal leng o e disurbance, L divided by πλ R is (a) 5; (b) 1.; (c).1; and (d).1 (in e model e pysical size o e disurbances is e same in eac case, bu e Coriolis acceleraion is varied o acieve a variable radius o deormaion). In all cases e inal eig and velociy ields are in geosropic balance. However, or very small disurbances e eig ield adjused o e iniial velociy ield, wile or very large disurbances e velociy ield adjused o e iniial eig ield. Te ollowing rules apply in all cases, and are suggesed or commimen o memory: I e size o e disurbance is muc greaer an π imes e Rossby radius o deormaion (L >> πλ R ), en e velociy ield adjuss o e iniial eig (mass) ield. Te erms eig ield and mass ield are synonymous. 6

7 I e disurbance is muc less an π imes e Rossby radius o deormaion (L << πλ R ), en e eig ield adjuss o e iniial velociy ield. I e disurbance is o e same order as π imes e Rossby radius o deormaion (L ~ πλ R ), en e eig and velociy ields undergo muual adjusmen. In all cases, e inal eig and velociy ields are in geosropic/gradien balance! ADJUSTMENT IN A VORTEX Our derivaion o e Rossby radius o deormaion was or a 1-D luid, so ere was no curvaure o e low. Te general orm o e Rossby radius o deormaion or a vore is c λ R = ( (9) η v r) were η is e absolue voriciy, v is e angenial velociy, and r is e radius o curvaure o e low (see e noes or Tropical Meeorology, Lesson 9, i you are ineresed in a derivaion or a vore). For lows wose absolue voriciy is primarily due o e ears voriciy (i.e., lows were ζ << ) is becomes λ R = c, wic is e same as wa we derived ere. GEOSTROPHIC ADJUSTMENT IN A MULTI-LAYER FLUID Te principles o geosropic adjusmen in a muli-layer, ydrosaic luid are idenical o a in a baroropic luid, ecep a ere are several modes o inerialgraviy waves generaed: one or e baroropic mode, and one or eac baroclinic mode. Eac mode as is own unique radius o deormaion, given by λ = c λ = c 1 λ = n cn were e subscrip reers o e baroropic mode, and e subscrip n reers o e n baroclinic mode. Te baroropic mode is oen called e eernal mode, and is radius o deormaion e eernal radius o deormaion. In a wo-layer luid e baroclinic mode is oen called e inernal mode, and is radius o deormaion e inernal radius o deormaion. In a coninuously sraiied luid (suc as e ocean or amospere) ere are in eory an ininie number o baroclinic modes possible; owever, mos o e energy is conined o a ew o e lower baroclinic modes, so applicaion o geosropic adjusmen is grealy simpliied, as we can concern ourselves wi a smaller, inie number o modes. In a coninuously sraiied luid e group velociy o e modes o oscillaion can be approimaed as 7 1 (3)

8 cn NH nπ ; n =,1,, (31) were N is e Brun-Vaisala requency and H is e scale eig. In is case e Rossby radii o deormaion are NH λn ; n =,1,,. (3) n π Since e baroclinic modes ave a muc smaller wave speed an e baroropic mode, e baroclinic radius o deormaion is muc smaller an a o e baroropic radius o deormaion (see eercises). SUMMARY AND FURTHER DISCUSSION On e synopic scale, e amospere is close o geosropic and ydrosaic balance. Radiaional eaing and cooling, laen ea release, and oer acors pus e amospere rom geosropic balance. Te amospere adjuss back ino geosropic balance by generaing inerial-graviy waves wic propagae energy away rom e disurbance. Te naure o e adjusmen depends on ow e orizonal scale o e disurbance compares wi e Rossby radius o deormaion. Since e amospere is sraiied, i is usually e baroclinic radii o deormaion a are imporan. For small-scale penomena, suc as individual undersorms, e disurbance is muc smaller an e radius o deormaion. Tereore, e mass ield adjuss o e iniial velociy ield, and no residual synopic scale circulaions are generaed. However, larger-scale penomena can be o e order o e baroclinic radius o deormaion, and can ereore leave a synopic scale circulaion as e velociy ield adjuss o e mass ield. Te radius o deormaion also provides us wi a leng scale by wic o gauge weer penomena are eeced by e ear s roaion. Penomenon wose orizonal leng scales are muc smaller an e radius o deormaion are unlikely o be eeced by e ear s roaion (unless ey persis or a ime scale on e order o e inerial period, π 1. Tereore, undersorms, ornadoes, and oiles are no aeced by e ear s roaion. 8

9 EXERCISES 1. Sow a e omogeneous orm o equaion (1), c =, suppors sallow-waer inerial-graviy waves aving a dispersion relaion o ω = c k.. Te ocean is oen represened as a wo-layer luid. Assume e upper layer as a dep o 7 m and a densiy o 11 kg/m 3, wile e lower layer as a dep o 33 m and a densiy o 13 kg/m 3. a. Find e baroropic (eernal) radius o deormaion a laiude 45 N. b. Find e baroclinic (inernal) radius o deormaion a e same laiude. c. For e disurbances in is ocean sown below, skec e inal posiion o e upper and lower suraces. Assume e disurbance on e le only generaes waves in e baroropic mode, wile e disurbance on e rig only generaes waves in e baroclinic mode. Te op line represens e eernal surace, wile e boom line represens e inernal inerace. 3. Calculae e radius o deormaion or a ypical baub. How large would a disurbance in e ub ave o be in order or roaional eecs o be imporan? 4. Does e radius o deormaion increase or decrease wi laiude? 9