David Randall. α p α F. = φ a. is the Lagrangian derivative in the inertial frame, and V a. , and e r. = Dr

Transcription

1 ! Revised Mach 24, :27 PM! 1 Consevation of momentum on a otating sphee David Randall Consevation of momentum on a otating sphee Newton's statement of momentum consevation, as applied in an inetial (i.e., nonacceleating) fame of efeence, can be witten as follows: Hee D a ( ) D a V a = φ a α p α F. is the Lagangian deivative in the inetial fame, and V a is the velocity as seen in the inetial fame. The left-hand-side of (1) epesents the acceleation of the ai as seen in an inetial fame. The gavitational potential is φ a, and F is the stess tenso associated with molecula effects. velocity of The length of a sideeal day is 86,164 s, so the Eath otates about its axis with an angula 2π ( 86,164 s) s 1. This angula velocity can be epesented by a vecto, Ω, pointing towads the celestial Noth Pole, as shown in Fig. 1. In spheical coodinates ( λ,ϕ,), the unit vectos ae e λ, e ϕ, and e, espectively, and the components of Ω ae Ω ( 0,cosϕ,sinϕ ). Let be a position vecto extending fom the cente of the Eath to a paticle of ai whose position is geneally changing with time. The absolute velocity of the ai, V a D a, is elated to its elative velocity, V D, as seen in the otating coodinate system, by D a (1) = D + Ω, (2) o Quick Studies in Atmospheic Science Copyight David Randall, 2011

2 ! Revised Mach 24, :27 PM! 2 V a = V + Ω V + V e. (3) A tansfomation of the fom (3) can be applied to any vecto. In contast, the time ate of change # #!!!!!" " "! "! Fig. 1: Sketch defining vectos used in the text. of a scala quantity, such as tempeatue, is the same in the inetial and otating fames. In the second line of (3), we define V e Ω = ( Ω cosϕ)e λ. Hee V e is the velocity (as seen in the inetial fame) that a paticle at adius and latitude ϕ expeiences due to the Eath's otation (efe to Fig. 1). Note that V e always points towads the east. Accoding to (3), the velocity as seen in the otating fame is diffeent fom the velocity as seen in the inetial fame. Since Ω is a constant, the fist line of Eq. (4) implies that (4) DV e = Ω V. Now we apply the tansfomation used in (2) again, to elate the acceleation as seen in the inetial fame to the acceleation as seen in the otating fame: (5) Quick Studies in Atmospheic Science Copyight David Randall, 2011

3 ! Revised Mach 24, :27 PM! 3 D a V a = DV a + Ω V a. Substituting fo V a in (6), fom (4), and using (5), we find that (6) D a V a = D ( V + V e ) + Ω ( V + V e ) = DV + 2Ω V + Ω V e = DV + 2Ω V + Ω V e. (7) Eq. (7) elates the absolute acceleation, D a V a otating fame, i.e., DV otating fame is, to the appaent acceleation as seen in the. Using (7) in (1), we find that the equation of motion elative to the DV + 2Ω V + Ω V e = φ a α p α F. (8) The tem 2Ω V is the Coiolis acceleation, which points in a diection pependicula to V. The tem Ω V e is the centifugal acceleation, which can be witten as Ω V e = ( Ω 2 cosϕ)e λ e Ω = Ω 2 e, whee e is a vecto that is paallel to the plane of the Equato, as shown in Fig. 1, and e Ω is a unit vecto pointing towad the celestial noth pole. Eq. (9) shows that the centifugal acceleation points outwad, in the diection of e, which is pependicula to the axis of the Eath s otation. With efeence to (9), the centifugal acceleation can be expessed in spheical coodinates as Ω V e = Ω 2 cosϕ ( 0, sinϕ,cosϕ). It has an upwad component that is lagest at the Equato, and an equatowad component that is lagest at the pole, but no zonal component. Quick Studies in Atmospheic Science Copyight David Randall, 2011 (9) (10)

4 ! Revised Mach 24, :27 PM! 4 Kinetic enegy The kinetic enegy equation associated with (1) is D a K a = V a ( φ a α p α F), (11) whee K a 1 ( 2 V V a a ). The ight-hand side of (11) epesents wok done by tue foces. Fom (6), we see that D a K a = DK a This means that the time change of K a is the same in the inetial and otating fames, consistent with ou ealie assetion that the time ate of change of a scala is the same in the inetial and otating fames. Using (12), we can ewite (11) as DK a = V a ( φ a α p α F). The kinetic enegies as seen in the inetial and otating fames ae diffeent. We will deive the kinetic enegy equation fo the otating fame in two diffeent ways. Fist, the had way: Fom (3), we can wite. (12) (13) Then V a V a = ( V + V e ) ( V + V e ). (14) K a = K + V e V V e V e, (15) whee K 1 ( 2 V V ). Fom (15), we see that Quick Studies in Atmospheic Science Copyight David Randall, 2011

5 ! Revised Mach 24, :27 PM! 5 DK a = DK + V DV e + V DV e = DK + V DV e + V DV e e = DK + V DV a e. + V e DV e To obtain the second line of (16), we have used (5). To obtain the thid line, we have used (3). Combining (16) with (6) and (1), we find that (16) DK a = DK + V D a V a e Ω V a = DK + ( V V) D a V a a = DK + D K a a V D a V a Ω V a Ω V a = DK + D K a a V ( φ a α p α F) + V ( Ω V a ). (17) Cancelling D a K a obtain in (17), and using V ( Ω V a ) = V ( Ω V e ), we can eaange the esult to DK + V ( Ω V e ) = V ( φ a α p α F). We can also obtain (18) simply by dotting V with (8). That s the easy way. (18) Compaing (18) and (13), we see that D( K a K ) = V ( Ω V e ) + V e ( φ a α p α F). The fist tem on the ight-hand side of (19) appeas to be the wok done by the centifugal acceleation. The second appeas to be the wok done by the tue foces acting on V e. Of couse, neithe of these tems epesents eal wok done. The fist is due to a fictitious foce, and the second is due to a fictitious velocity. (19) Quick Studies in Atmospheic Science Copyight David Randall, 2011

6 ! Revised Mach 24, :27 PM! 6 Effective gavity The centifugal acceleation can be witten as ( Ω V e ) = Ω 2 e = 1 2 Ω 2 = 1 2 V e i.e., it is the gadient of a potential, called the centifugal potential, which is equal to the kinetic enegy associated with V e. Fom (20) and (4), we see that the centifugal potential is given by 1 ( Ω cosϕ 2 )2. 2,, (20) Since both the acceleation due to gavity and the centifugal acceleation can be expessed as gadients of potentials, it is natual and convenient to combine them into an appaent gavity, g, defined by g g a Ω 2 e, whee g a Φ a. Using (23) we see that the potential of g is (21) φ = φ a 1 2 Ω 2, so that g φ. We efe to φ as the geopotential. Fo most puposes g g a = ge, because the centifugal acceleation is small compaed to g a. Hee e is a unit vecto pointing upwad, away fom the cente of the Eath. The acceleation due to the appaent gavity g is pependicula to sufaces of constant φ, which diffe slightly fom spheical sufaces of constant adius. The Equatoial adius of the Eath is about 20 km lage than the pola adius, due to the centifugal acceleation. If the Eath s intenal density distibution was a function of adius only, then φ a, the tue gavitational potential, would also depend only on distance fom the cente of the Eath, and would take the fom (22) Quick Studies in Atmospheic Science Copyight David Randall, 2011

7 ! Revised Mach 24, :27 PM! 7 φ a = GM e, (23) whee G = N m 2 kg -2 is the gavitational constant, and M e is the total mass of the Eath. Eq. (23), which is called Newton s Shell Theoem, would apply at any level at o above the Eath s suface. The geopotential φ is then given by φ = GM e + 1 ( Ω cosϕ 2 )2. Using (22) we can now wite (8), the equation of motion in the otating fame, as (24) DV + 2Ω V = φ α p α F. (25) Similaly, the kinetic enegy equation, (18), becomes DK Component equations in spheical coodinates = V ( φ α p α F). Up to this point, we have not used any coodinate systems, except in connection with a comment on the centifugal acceleation. We now conside spheical coodinates, ( λ,ϕ, ). The unit vectos in the ( λ,ϕ, ) coodinates ae e λ, e ϕ, and e, espectively. You should be able to see that the diection of e λ depends on longitude, and that the diections of e ϕ, and e depend on both longitude and latitude. This means that the diections of e λ, e ϕ, and e ae functions of space, although of couse thei magnitudes ae spatially constant. Simple geometical easoning leads to the following fomulae: De λ = Dλ sinϕe Dλ ϕ cosϕ e = u tanϕ e ϕ u e, (26) (27) Quick Studies in Atmospheic Science Copyight David Randall, 2011

8 ! Revised Mach 24, :27 PM! 8 De ϕ = Dλ sinϕe Dϕ λ e = u tanϕ e λ v e, (28) De = cosϕ Dλ e λ + Dϕ e ϕ = u e λ + v e ϕ = V h. (29) In (29), V h is the hoizontal wind vecto. The vecto opeatos that ae most commonly used in atmospheic science can be expessed in spheical coodinates as follows: 1 A A = cosϕ λ, 1 A ϕ, A, (30) V = 1 V λ cosϕ λ + 1 cosϕ ϕ V cosϕ ϕ ( ) + 1 ( ), 2 V 2 (31) V = 1 V ϕ ( V ϕ ), 1 ( V λ ) cosϕ 1 V cosϕ λ, 1 V ϕ λ ϕ ( V cosϕ) λ (32) 1 1 A 2 A = 2 cosϕ λ cosϕ λ + A cosϕ ϕ ϕ + 2 cosϕ A. Hee A is an abitay scala, and V is an abitay vecto. (33) We can expess the velocity vecto in spheical coodinates as V ue λ + ve ϕ + we, (34) whee Quick Studies in Atmospheic Science Copyight David Randall, 2011

9 ! Revised Mach 24, :27 PM! 9 u cosϕ Dλ, v Dϕ D, and w. (35) Using (35), the Lagangian time deivative can then be expanded as D t + Dλ = t + u cosϕ λ + Dϕ λ + v ϕ + D ϕ + w. (36) Eq. (34) shows that the diections in which the u, v, and w components actually point depend on whee you ae. Taking this into account, fo an abitay vecto Q ae λ + be ϕ + ce, we can wite DQ D ( ae λ + be ϕ + ce ) Da = e λ + a De λ Db + e ϕ + b De ϕ Dc + e + c De Da = e λ + a u tanϕ e ϕ u e + e Db ϕ + b u tanϕ e λ v e + e Dc + c u e + v λ e ϕ Da = e λ + bu tanϕ + uc + e ϕ Db + au tanϕ + vc Dc + e ua + vb. (37) As a special case of (37), the acceleation in the otating fame can be expanded as DV D ( ue λ + ve ϕ + we ) Du = e λ + u De λ Dv + e ϕ + v De ϕ Dw + e + w De Du = e λ + u u tanϕ e ϕ u e + e Dv ϕ + v u tanϕ e λ v e + e Dw + w u e + v λ e ϕ Du = e λ + uv tanϕ + uw + e ϕ Using (38), we can sepaate (8) into component fom: Dv + u2 tanϕ + vw + e Dw u2 + v 2. (38) Quick Studies in Atmospheic Science Copyight David Randall, 2011

10 ! Revised Mach 24, :27 PM! 10 Du 2Ω + u vsinϕ wcosϕ cosϕ ( ) = α Dv + 2Ω + u cosϕ usinϕ + vw = α Dw 2Ω + u v2 u cosϕ cosϕ p cosϕ λ α F p ϕ α( F), ϕ p + g = α α( F). ( ) λ, These ae the components of the equation of motion in spheical coodinates. (39) We can modify (39) to show tue gavity and the centifugal acceleation explicitly: Du 2Ω + u cosϕ vsinϕ wcosϕ ( ) = α Dv + 2Ω + u cosϕ usinϕ + vw + Ω 2 sinϕ cosϕ = α Dw 2Ω + u cosϕ p cosϕ λ α F p ϕ α ( F), ϕ v2 u cosϕ Ω 2 cos 2 ϕ + g a = α p α ( F). ( ) λ, By using the continuity equation in spheical coodinates, we can ewite (39) in flux fom: t t ( ρu) + ρvu t ( ) ρ 2Ω + ( ρv) + ρvv ( ρw) + ρvw ( ) + ρ 2Ω + ( ) ρ 2Ω + u cosϕ vsinϕ wcosϕ u cosϕ ( ) = 1 u cosϕ usinϕ + ρ vw = 1 p cosϕ λ F p ϕ ( F), ϕ v2 u cosϕ ρ + ρg = p ( F). These equations ae faily exact. Vaious appoximations ae intoduced below. Solid-body otation ( ) λ, It is instuctive to conside the special case of zonal solid-body otation, in which u = ω cosϕ, v = 0, w = 0, and p = p( ϕ,), whee ω = constant. This type of motion is called solid-body otation, because the fluid otates as if it wee a solid, i.e., neighboing paticles emain neighbos fo all time. The flow is puely zonal. Eqs. (41) educe to Quick Studies in Atmospheic Science Copyight David Randall, 2011 (40) (41) (42)

11 ! Revised Mach 24, :27 PM! 11 t ( ρu) = 0, ρu 2 tanϕ = ρ fu 1 ρ u2 p ϕ, = ρ fu p ρg. (43) Hee we have assumed that the stess tenso vanishes (it does). The zonal wind equation is in tivial balance. The meidional momentum equation is in gadient-wind balance, which is a genealization of geostophic balance. The vetical momentum equation is in a modified hydostatic balance, in which the centifugal and Coiolis acceleations ente. Appoximations Up to hee the discussion has been faily exact. We now intoduce some vey useful appoximations: Replace by a eveywhee, whee a is the adius of the Eath. An appoximation of this fom can be justified fo an atmosphee which is thin compaed to the adius of the planet, and so it is called the thin atmosphee appoximation. It is a good appoximation fo Eath, but would not apply, e.g., to Jupite. Replace (33) by ( ) 1 V = acosϕ λ V + λ ϕ V cosϕ ϕ + V. (44) This is also justified by the thinness of the Eath s atmosphee. Dop the tems involving the hoizontal component of Ω, i.e., Ω cosϕ. This is often called the taditional appoximation. Thee is an ongoing discussion as to whethe o not this is always justified. Neglect uw and vw, the cuvatue tems involving w, in the equations fo u and v, espectively, neglect u2 + v 2 in the equation of vetical motion. Simplify the vetical component of (40) to p z = ρg. (45) Quick Studies in Atmospheic Science Copyight David Randall, 2011

12 ! Revised Mach 24, :27 PM! 12 Eq. (45) is called the hydostatic equation. With an appopiate bounday condition, (45) allows us to compute p z appoximation to p( z), simply because Dw ( ) fom ρ( z). Even when the ai is moving, (45) gives a good and the vetical component of the fiction foce ae small compaed to g. Eq. (45) as applied to moving ai is called the hydostatic appoximation, and it is applicable to vitually all meteoological phenomena, including violent thundestoms. Fo lage-scale ciculations, the appoximate p( z) detemined though the use of (45) can be used to compute the pessue gadient foce in the equation of hoizontal motion. To do so is to use the quasi-static appoximation. The quasi-static appoximation applies vey well fo lage-scale motions, but it is not applicable to many small-scale motions, such as thundestoms. When the quasi-static appoximation is made, the effective kinetic enegy is due entiely to the hoizontal motion; the contibution of the vetical component, w, is neglected. Fo lage-scale motions, w << u,v ( ), so that this quasi-static kinetic enegy is vey close to the tue kinetic enegy. Futhe discussion is given in the QuickStudy on the quasi-static appoximation. With the appoximations discussed above, (39) is eplaced by Du uv tanϕ a = fv Dv + u2 tanϕ = fu α a a α p acosϕ λ α( F), λ p ϕ α( F), ϕ 0 = g α p z. (46) Component equations in a local Catesian coodinate system Now conside a locally defined ( x, y) o Catesian coodinate system, as shown in Fig. 2. Hee locally defined means that the oigin of the coodinate system is attached to a specific point on the otating Eath, e.g., Fot Collins. With this coodinate system, Ω = Ω ( 0, cosϕ, sinϕ ), whee ϕ is the latitude of the oigin of the coodinate system, and (47) Quick Studies in Atmospheic Science Copyight David Randall, 2011

13 ! Revised Mach 24, :27 PM! 13 Ω V = 2Ω i j k 0 cosϕ sinϕ u v w The diections of the unit vectos i, j, and k ae independent of position, although they do of couse depend on the feely chosen position of the oigin of the coodinate system. The components of the equation of motion in this local Catesian coodinate system ae. (48) Hee we define Du p = fw + fv α x α ( F), x Dv = fu α p y α ( F), y Dw = fu α p α ( F ) z z g. (49) V ui + vj + wk, (50) u Dx, v Dy, w Dz, (51) f 2Ω sinϕ, and f 2Ω cosϕ, (52) D = t + u x + v y + w z. (53) At the oigin, and at all points with the same longitude as the oigin, the x -coodinate points east; but because this Catesian coodinate system is defined with espect to a fixed location on the Eath s suface, the x -coodinate does not point east at othe longitudes. Fo example, at points 90! to the east of the oigin, the x-coodinate points up, and on the opposite side of the Quick Studies in Atmospheic Science Copyight David Randall, 2011

14 ! Revised Mach 24, :27 PM! 14 Eath fom the oigin, the x -coodinate points west. Such a coodinate system could be used to Fig. 2: Sketch depicting a local Catesian coodinate system. study the geneal ciculation, but clealy it is not vey well suited to such an application. Compaing (49) with (39), we see that (49) does not contain the tems uw, uv tanϕ, etc. These so-called metic tems aise in (39) because in spheical coodinates the diections of the unit vectos e λ, e ϕ, and e vay with λ and ϕ. This does not happen in the local Catesian coodinate system. Summay We have pesented the momentum equation that descibes motion on a otating sphee, as seen in the otating fame of efeence. We have also intoduced seveal commonly used appoximations. Refeences and Bibliogaphy Phillips, N. A., 1966: The equations of motion fo a shallow otating atmosphee and the taditional appoximation. J. Atmos. Sci., 23, White, A. A., and R. A. Bomley, 1995: Dynamically consistent, quasi-hydostatic equations fo global models with a complete epesentation of the Coiolis foce. Quat. J. Roy. Meteo. Soc., 121, Quick Studies in Atmospheic Science Copyight David Randall, 2011