Fall 2016 Semester METR 3113 Atmospheric Dynamics I: Introduction to Atmospheric Kinematics and Dynamics

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1 Fall 06 Semeste METR 33 Atmospheic Dynamics I: Intoduction to Atmospheic Kinematics Dynamics Lectue 7 Octobe 3 06 Topics: Scale analysis of the equations of hoizontal motion Geostophic appoximation eostophic wind Scalin the thid equation of motion (equation of vetical motion) Geostophic appoximation eostophic wind Appoximate foms of hoizontal ponostic equations Rossby numbe ydostatic appoximation Pessue density deviations fom hydostatic values Readin: Section 6 Chapte in olton akim Scale analysis of the equations of hoizontal motion Geneal equations of motion (also called the momentum balance equations) deived in Class 7 descibe atmospheic motions on a vey lae ane of scales The impotance of paticula scales of motion may be estimated thouh the analyses of manitudes of tems in the scaled equations of motion Motions of some scales can be unimpotant fo a iven poblem thus can be excluded fom consideation (filteed out dopped off) by elimination of the coespondin tems in the equations of motion The notions of scale analysis scalin in atmospheic dynamics wee biefly discussed in Class Let us conside chaacteistic scales of atmospheic motion elated to a mid-latitude synoptic system: L~ 3 km= 6 m is the lenth scale; ~ km = m is the depth scale; - U~ m s is the hoizontal velocity scale; W~ cm s m s is the vetical velocity scale; L/U~ 5 s is the time scale; 3 - p / m s is the (nomalized) hoizontal pessue fluctuation scale Now we can estimate the manitude of each tem in the fist two equations of motion usin the intoduced scales Fo 5 (exactly the mid-latitude): f0 sin cos s - The esultin scale estimates fo the tems (all in of motion (see Class 7): - m s ) of the hoizontal (in X Y diections) equations Total time deivatives of hoizontal velocity components: du u u u u dv U u v w dt t x y z dt L Coiolis tems: 3 u sin vsin f0u Pessue adient tems: p p p x y L 3 Cuvatue tems: uv tan u tan U 5 a a a uw vw UW a a a 8

2 Fiction tems: F x F y vu Scalin the thid equation of motion (equation of vetical motion) The thid equation of motion (it pesents the balance of foces alon Z axis see Class 7) eads: w w w w u v p u v w u cos Fz t x y z a z We may now evaluate individual manitudes of tems of this equation fo a synoptic-scale motion in midlatitudes usin a scalin appoach simila to the one applied to analysis of tems in the equations of the hoizontal motion (see p ) To do this we aain conside chaacteistic scales of such atmospheic motion: L~ 3 km= 6 m is the lenth scale; ~ km = m is the depth scale; - U~ m s is the hoizontal velocity scale; W~ cm s m s is the vetical velocity scale; P0 hpa N m is the scale of vetical pessue diffeence acoss the atmosphee; f0 sin cos s - (takin 5 use these scales fo the estimation of the tems of the above equation The followin estimates of individual tems (all in Total time deivative of w: ) - m s ) may be obtained dw w w w w UW u v w dt t x y z L 7 Coiolis tem: cos 3 u f0u p P0 Pessue adient tem: z Cuvatue tem: Gavity tem: u v U a a 5 ( is the density scale in the laye of depth ) Fiction tem: F z vw 5 3 Geostophic appoximation eostophic wind Scalin consideations pesented in p eadin the fist two equations of motion u u u u uv tan uw p u v w v sin Fx t x y z a a x

3 v v v v u tan vw p u v w u sin Fy t x y z a a y demonstated that the main tems in these equations when applied to descibe the synoptic-scale motion in mid-latitudes ae the pessue adient tems the Coiolis tems p x p y vsin usin If only these majo tems ae kept in the equations of the hoizontal motion they educe to p fv x p fu y whee f= sin is the aleady familia Coiolis paamete The above appoximate fom of the equations of hoizontal motion coesponds to the so-called eostophic appoximation The atmospheic hoizontal motion (wind) unde this assumption is called the eostophic wind whose vecto whee V (note that it has only hoizontal components!) is iven by ae the x y components of the eostophic wind u V iu j v p f y p v f x Usin popeties of the vecto poduct the eostophic wind vecto can be witten in the fom: V k p f whee del opeato is applied on the hoizontal (X-Y) plane theefoe the pessue adient is iven by p p p i j Please be able to deive the above fomula fo V usin popeties of the vecto coss poduct x x Geostophic appoximation woks athe well fo lae-scale hoizontal motions away fom the equato sufficiently hih above the ound Appoximate foms of hoizontal ponostic equations Geostophic appoximation allows to ewite the hoizontal pessue adient foce components p y as fv fu espectively p x

4 If we take the equations of the hoizontal motion with etained acceleation tems (these tems ae the next to the pessue adient Coiolis tems with espect to the manitude see the scale analysis in p ) we may wite them down as the followin ponostic equations fo the hoizontal wind components: u u u u v f ( v v ) t x y v v v u v f ( u u ) t x y Thus the hoizontal acceleation on the synoptic scales of motion is popotional to the diffeence between the actual eostophic wind (the so-called aeostophic wind) In vecto fom these ponostic equations fo hoizontal motion may be witten (please be able to show it youself) as one equation: dv dv fk ( V V) fk V a 0 dt dt whee V iuj v is the hoizontal velocity vecto V a is the aeostophic wind vecto with components u u (in x diection) v v (in y diection) 5 Rossby numbe The Rossby numbe Ro is intoduced in atmospheic dynamics as convenient measue of the manitude of the hoizontal acceleation compaed to the action of the Coiolis foce Calculatin the atio of scales intoduced in p fo the hoizontal acceleation U L fo the Coiolis foce pe unit mass fu 0 we have: U Ro fl The small values of Ro (Ro<<) indicate that the manitude of acceleation is small compaed to the manitude of the Coiolis foce pe unit mass thus the eostophic appoximation is valid 0 6 ydostatic appoximation Scale analysis of the thid equation of motion (see p ) indicates that to a hih deee of accuacy the pessue field in the atmosphee on synoptic scales of motion is in the hydostatic equilibium that coesponds to the state when the vetical component of the pessue adient foce is balanced by the avity foce The pessue p density in the idealized hydostatic atmosphee (they ae also called the stad pessue the stad density) ae thus elated by the hydostatic balance equation: p z which povides the hydostatic appoximation of the thid equation of motion

5 7 Deviations fom the hydostatic equilibium It is convenient to conside actual pessue density fields in the atmosphee p( x y z t ) ( x y z t) in tems of small deviations p'( x y z t ) '( x y z t) fom thei stad (elated to each othe thouh the hydostatic balance dependin on z only) values p( z ) ( z) p( x y z t) p ( z) p'( x y z t) ( x y z t) ( z) '( x y z t) ie in the fom: In this case takin into account that may be witten as ' p z the main tems of the thid equation of motion p = p p ' ' ' ' ' ' p p p z z z z z z Fo synoptic-scale motions takin 3 p' p z m s - - ' - m s - so p ' ' z p ' ' z appeas to be a easonable appoximation fo the elation between pessue density petubation fields in this case