ESCI 34 Atmospheric Dynmics I Lesson 6 Scle Anlysis SCALE ANALYSIS OF THE MOMENTUM EQUATIONS Not ll of the terms in the momentum equtions re sinificnt. If term is much smller thn the others then it is resonble to inore it under certin circumstnces. To ssess which terms cn be nelected, we ssin n order of mnitude to ll the vribles nd prmeters in the equtions. For scle nlysis we often don t ssin exct numbers just orders of mnitude. The orders of mnitude re ssined for specific scles of motion. For instnce, they would be quite different for the study of torndoes thn they would be for the study of hurricnes. The prmeters tht need to be scled re shown in the tble below: Nme Symbol Horizontl velocity U Verticl velocity W Horizontl lenth scle L Verticl lenth scle H Pressure chne P Density Time 1 = L /U Approprite scles for ech re determined s follows: Horizontl velocity, U: For most tmospheric circultions the u nd v components re of similr mnitude, nd so we use sinle scle prmeter, U, to represent both. Verticl velocity, W. Horizontl lenth scle, L: The horizontl lenth scle cn be defined in few wys. For wvelike fetures in the tmosphere it is usully tken to be one-fourth of the totl wvelenth, L 4. This is becuse the scle of the sptil velocity derivtives such s u xre of the order of U L(see fiure below). For vortex, L is tken to be the rdius R (not dimeter), since the sptil derivtive of velocity in vortex will be of the order of U R. 1 The time scle = L /U is clled the dvective time scle. It is the time it would tke for prcel of fluid to trvel the entire horizontl lenth of the flow. In very erly work (Chrney, J.G: On the scle of tmospheric motions, Geof. Publ., 17, 3-17; Burer, A.P., 1957: Scle considertion of plnetry motions of the tmosphere, Tellus, 10, 195-05) horizontl scle of / ws postulted. This ws ltered to /4 in Phillips, N.A., 1963: Geostrophic motion, Rev. Geophys., 1, 13-175
Verticl lenth scle, H: The verticl lenth scle is the heiht of the circultion or disturbnce. Pressure chne, : The pressure chne is needed for terms involvin derivtives of the pressure. In the horizontl, this will be the rne between the mximum nd minimum pressures found movin horizontlly cross the circultion. In the verticl, it will be the mximum nd minimum pressures found movin verticlly throuh the circultion. There my be very lre differences for in the horizontl versus in the verticl. Time, : For the time scle we use the dvective time scle, defined s LU. P This is the time it would tke for prcel of fluid trvelin t speed U to trvel the distnce L. SYNOPTIC SCALE ANALYSIS OF THE HORIZONTAL MOMENTUM EQUATIONS For synoptic scles the followin orders of mnitude re pproprite: p Nme Symbol Order of mnitude Horizontl velocity U 10 m s 1 Verticl velocity W 0.01 m s 1 Horizontl distnce L 1000 km (10 6 m) Verticl distnce H 10 km (10 4 m) Pressure chne P Horizontl: 10 mb (10 3 P) Verticl: 1000 mb (10 5 P) Time = L /U 1 dy (10 5 s)
The followin prmeters re lso used: density 1 k m 3 kinemtic viscosity 1.4610 5 m s 1 ome 7.910 5 rd s 1 ltitude 45 rdius of Erth 6.37810 6 m Usin these scles nd prmeters, the terms in the u-momentum eqution hve the followin orders of mnitude u t u u u v x y w u z uv tn uw 1 p x v sin w cos U /L U /L WU/H U / UW/ P/( L) U sin 45 W cos 45 10 4 10 4 10 5 10 5 10 8 10 3 10 3 10 6 u u x y u z u (tn 1) v tn x w x U/L U/H U/ U/L W/L 10 16 10 1 10 18 10 18 10 0 A similr nlysis for the v-momentum eqution is v t v v u v x y w v z u tn vw 1 p y u sin U /L U /L WU/H U / UW/ P/( L) U sin 45 10 4 10 4 10 5 10 5 10 8 10 3 10 3 v v x y v z v tn 1 u tn x w tn w y U/L U/H U/ U/L W/ W/L 10 16 10 1 10 18 10 18 10 1 10 0 Mny of the terms re very smll compred to others, nd cn be inored without sinificnt loss of ccurcy. We cn therefore inore the curvture terms, the viscous terms, nd the Coriolis term tht involves the verticl velocity. Inorin these terms yields much simpler version of the horizontl equtions of motions: u u u u 1 p u v w v sin (1) t x y z x v v v v 1 p u v w u sin () t x y z y 3
Note: We could hve lso inored the verticl dvection terms, but it is not too much of n inconvenience to keep them. By definin the Coriolis prmeter s f sin (3) the horizontl momentum equtions ssume the form Du 1 p fv (4) Dt x Dv 1 p fu (5) Dt y In vector form the horizontl momentum eqution is DV 1 p kˆ f V. (6) Dt In (6), ll derivtives nd vectors re horizontl, V u iˆv ˆj ˆ i ˆ j. x y The totl derivtive terms in (4), (5), nd (6) re known s the inertil terms. The terms on the riht-hnd-side re the pressure rdient nd Coriolis terms respectively. THE ROSSBY NUMBER Dividin the horizontl momentum eqution, (6), throuh by fv we et 1 DV 1 ˆ f V p k. f V Dt f V f V Usin the representtive scles the order of mnitude of these terms re U P 1 f L f U. The dimensionless combintion U/f L is defined s the Rossby number (nmed for Gustv Rossby), Ro U f L (7) GEOSTROPHIC BALANCE (VERY SMALL ROSSBY NUMBER) When the Rossby number is much less thn unity (Ro << 1), then the ccelertion (inertil) term cn be inored nd the only two terms left re the pressure rdient term nd the Coriolis term, which must be nerly in blnce. This is known s eostrophic blnce, nd the velocity in this cse is known s the eostrophic wind. 4
The momentum eqution in this cse reduces to kˆ 1 f V p which is solved for the eostrophic wind to yield 1 V ˆ k p, (8) f with wind speed (mnitude) equl to V p f 5. (9) CYCLOSTROPHIC BALANCE (VERY LARGE ROSSBY NUMBER) When the Rossby number is much reter thn unity (Ro >> 1) then the Coriolis term cn be inored. In this instnce the only terms tht re left re the ccelertion nd the pressure rdient terms, nd so the ccelertion is direct result of the pressure rdient force DV 1 p. (10) Dt This type of blnce is clled cyclostrophic. In cyclostrophic blnce the pressure rdient ccelertion is exctly tht required for the centripetl ccelertion, nd so we hve V c p r or r p Vc (11) where r is the rdius of curvture of the flow. GRADIENT BALANCE (ROSSBY NUMBER NEAR UNITY) If the Rossby number is of the order of unity (Ro ~ 1), then ll three terms must be retined. This is known s rdient blnce, nd the wind in this cse is known s the rdient wind. Detils of rdient wind will be discussed in future lesson. The followin tble summrized these results Ro Terms Blnce << 1 pressure rdient nd Coriolis eostrophic ~1 ccelertion, pressure rdient, nd Coriolis rdient >> 1 ccelertion nd pressure rdient cyclostrophic For lre-scle (synoptic scle) motion, the Rossby number is of the order (10m / s) Ro ~ 0.1 4 1 6 (10 s )(10 m), which shows tht on these scles the tmosphere is close to bein in eostrophic blnce. Hence, the ctul wind should be close to the eostrophic wind.
INERTIAL FLOW (ROSSBY NUMBER EXACTLY EQUAL TO ONE) If the pressure rdient is exctly zero, then the inertil terms must exctly blnce the Coriolis term. 3 The blnce in this cse is clled inertil blnce, with the speed iven by V f r. (1) in In inertil blnce the flow is circulr, with rdius of R Vin f. Since by definition V is lwys positive, then R must be netive nd so inertil flow is nticyclonic. The period of the inertil flow is found by dividin the circumference of the inertil circle by the speed, r. (13) V f in The inertil period is shorter t hiher ltitudes, nd is infinity t the Equtor. MORE ON THE GEOSTROPHIC WIND The eostrophic wind is definition! On the synoptic scle the ctul wind should be close to the eostrophic wind (becuse Ro << 1), but will rrely be exctly equl to the eostrophic wind. The components of the eostrophic wind re 1 p u (14) f y 1 p v (15) f x The eostrophic wind is prllel to the isobrs with lower pressure to the left (in the Northern Hemisphere). The eostrophic wind speed is directly proportionl to the pressure rdient. In pressure coordintes, the eostrophic wind nd components re 1 ˆ 0 V ˆ k k Z f f (16) 1 0 Z u (17) f y f y 1 0 Z v (18) f x f x Therefore, on constnt pressure surfce The eostrophic wind is prllel to the isohypses with lower heihts to the left (in the Northern Hemisphere). 3 In this instnce the Rossby number would be exctly equl to one. However, Rossby number of one does not utomticlly imply inertil blnce, becuse when clcultin Rossby number we use chrcteristic orders of mnitude for the flow prmeters, not exct vlues. Only if the pressure rdient is exctly zero do we hve inertil blnce. Pure inertil flow rrely if ever occurs. However, there often is n inertil component to the flow. Inertil blnce plys role in some tmospheric phenomen such s nocturnl low-level jets. 6
The eostrophic wind speed is directly proportionl to the eopotentil heiht rdient. Another importnt feture of the eostrophic wind is tht it is non-diverent ( V 0 ) if f is constnt. The eostrophic wind is sometimes written in terms of the stremfunction,, defined s p f (in pressure coordintes ). where f nd re constnt. In this cse, the eostrophic wind is V kˆ (19) THE AGEOSTROPHIC WIND u v y x The difference between the ctul wind nd the eostrophic wind is clled the eostrophic wind. V V V () f (0) (1) Since the tmosphere is usully close to eostrophic blnce, the eostrophic wind is typiclly smll in comprison to the eostrophic wind. Horizontl diverence is very importnt mechnism for risin nd sinkin motions in the tmosphere. Since the eostrophic wind is non-diverent, ny diverence must be due to the eostrophic wind. Therefore, even thouht the eostrophic wind is smll, it is very importnt! 7
SCALE ANALYSIS OF THE VERTICAL MOMENTUM EQUATION Scle nlysis of the verticl momentum eqution proceeds s follows (note tht in this cse P is the verticl vrition in pressure, which is ~ 1000 mb or 10 5 P). w t w w u v x y w w z u v 1 p z u cos UW/L UW/L W /H U / P/H U cos 45 10 7 10 7 10 8 10 5 10 10 3 10 w w x y w z w v tn u v x y W/L W/H W/ U/ U/L 10 19 10 15 10 0 10 18 10 17 This nlysis shows tht the pressure rdient nd rvity terms re dominnt. Therefore, on the synoptic scle, the tmosphere cn be ssumed to be in hydrosttic blnce, nd the verticl momentum eqution simplifies to p. z A more riorous nlysis (see Mesoscle Meteoroloicl Modelin by Pielke) shows tht the hydrosttic reltion is pproprite if the verticl lenth scle is much smller thn the horizontl lenth scle, H << L. This condition certinly pplies on the synoptic scle. 8
EXERCISES 1. Perform scle nlysis of the horizontl momentum equtions (in component form) for the whirlpool formed s your bthroom sink drins. Which terms re importnt in this cse? Wter hs density of 1000 k/m 3 nd kinemtic viscosity of = 1.810 6 m s 1. The horizontl pressure difference cross the whirlpool is ~ 10 P. (Use resonble estimte for the horizontl velocity bsed on your own experiences.). Wht is the Rossby number for torndo? Does the Coriolis force effect torndo? 3. Expnd the horizontl momentum eqution DV 1 p kˆ f V Dt to show tht in pure Crtesin-component form it is Du 1 p ˆ Dv 1 p fvi fu ˆj 0 Dt x Dt y, nd therefore yields the two component equtions Du 1 p fv Dt x 4. Show tht kˆ ( kˆ V ) V. Dv 1 p Dt y fu 9
5. At the four points shown in the picture below, estimte the mnitude of the eostrophic wind. Assume density of 1.3 k/m 3 nd ltitude of 45. The isobrs re lbeled in mb. 6. Perform scle nlysis of the verticl momentum eqution for midltitude thunderstorm to find out wht terms cn be inored. 10