ESCI 343 Atmopheric Dynami II Leon 9 Internal Gravity Wave Reference: An Introduction to Dynamic Meteoroloy (3 rd edition), J.R. olton Atmophere-Ocean Dynami, A.E. Gill Wave in Fluid, J. Lihthill Readin: olton, 7.4.1 TE BRUT-VÄISÄLÄ FREQUECY Before prorein with an analyi of internal wave we hould review the important concept of the Brunt-Väiälä frequency. The vertical acceleration on an air parcel i Dw 1 p Dt z where quantitie with a ~ character are propertie of the air parcel, while thoe with an overbar are for the urroundin environment. Aumin that the atmophere i in hydrotatic balance we can write p () z and o Dw 1. (3) Dt Definin the perturbation denity a the difference in denity between the parcel and it urroundin air at the ame level,, (4) then (3) can be written a (1) Dw. (5) Dt If the parcel tart out at level z 0 and ha the ame denity a it environment, ( z ) ( z ) (6) 0 0 and i diplaced adiabatically a mall vertical ditance z, then it new denity will can be expreed a a Taylor erie expanion p ( z0 z) ( z0) z ( z0) z. (7) z p z A parcel rie and expand their preure intantaneouly adjut to be equal to that of the urroundin environment, o that
and therefore Furthermore, we know that (8) p p. (9) z z p p 1. (10) p c Uin (9) and (10), (7) become ( z z) ( z ) z. (11) 0 0 The denity of the environment can alo be expanded uin Taylor erie, ( z0 z) ( z0) z. (1) z From (11) and (1) the perturbation denity, (4), become and o (5) become Dw Dt z z (13) z z. (14) For mall diplacement the denominator can be approximated a (15) and o (14) i now Equation (16) ha the form of where Dw Dt z c Dz z 0 Dt z. (16) (17). (18) z Equation (17) i a homoeneou nd -order differential equation. Althouh i a function of z, in thoe cae were i a contant then (17) ha olution iven by t t z() t Ae Be. (19)
Thee olution are fundamentally different dependin on whether i poitive or neative. If i poitive, itelf if real, and olution to (19) are it it z() t Ae Be (0) which are ocillation havin an anular frequency of. i therefore a fundamental frequency of the ocillation, and i referred to a the Brunt-Väiälä frequency (or buoyancy frequency). If i neative, then itelf i imainary, and olution to (19) are t t z() t Ae Be. (1) Thee olution are exponential with time, and are not ocillatory. The Brunt-Väiälä frequency frequency i directly related to the tatic tability of the atmophere. Solution for z(t) Static Stability poitive real ocillation table neative imainary exponential rowth untable For the atmophere z c and o we can et away with definin a z. (3) Alo, for an ideal a, the Brunt-Väiälä frequency can be written in term of potential temperature a () z. (4) Equation (4) i valid for an ideal a only, wherea (18) i true for any fluid. For an ideal a, (18) and (4) are equivalent (ee exercie). An additional reult that will be of ue in the next ection i that from (13) and (18) we can derive a direct relationhip between the Brunt-Väiälä frequency and the perturbation denity, z. (5) DISPERSIO RELATIO FOR PURE ITERAL WAVES For the preent dicuion we will inore chane in denity due to local compreion or expanion, which i a valid aumption a lon a the wave are hort compared to the cale at which the denity chane with heiht (lare value of wave 3
number). We will therefore ue the linearized, anelatic continuity equation, o that the overnin equation are u 1 p (6) t x v 1 p t y w 1 p t z d u v w w dz x y z (9) which when written in flux form are p u t x (30) p v t y (31) x (7) (8) p w (3) t z u v w. (33) y z 0 Equation (30) thru (33) are four equation in five unknown (u, v, w, The fifth equation i found by takin t t of (5) to et w t w t p, and ). Thouh we could write inuoidal olution for the five dependent variable and olve a 5 5 determinant to et the diperion relation, thi would be tediou. We can eliminate u, v, and w from the equation and reduce the number of equation to two a follow: Take of (33), and combinin it with of (30), of (31), and t of (3) to et 1 z Eliminate w between (3) and (34) to et t x y (34) z p. (35) p. (36) z Equation (35) and (36) are two equation in two unknown. Subtitutin the inuoidal olution 4
p Ae Be i kxlymz i kxlymz into equation et (35) and (36) yield the followin diperion relation for internal ravity wave of where t t (37) k l (38) k l m k l i the horizontal wave number, and wave number. k l m i the total AALYSIS OF ITERAL WAVE DISPERSIO The diperion relation for internal wave (38) how that for purely horizontal wave (K = K) the frequency i equal to the Brunt-Väiälä frequency. For nonhorizontally travelin wave the frequency i le than the Brunt-Väiälä frequency. Therefore, the Brunt-Väiälä frequency i an upper-limitin frequency for internal wave. In other word, for internal wave. The phae peed for internal wave i iven by c. (39) The phae velocity i iven by c kiˆ l ˆj mkˆ. (40) 3 The roup velocity i m c iˆ ˆ j kˆ kiˆ l ˆ j kˆ 3 k l m. (41) m Inpection of (40) and (41) how a curiou fact that for internal wave, if there i a downward component to the phae velocity, then there i an upward component to the roup velocity, and vice-vera. In fact, by takin the dot product of (40) and (41) we find that c c 0. (4) which how that the roup velocity and phae velocity are actually oriented at 90 to each other in the vertical plane! Thi i illutrated in the diaram below. 5
The link below contain an animated GIF loop howin internal wave diperion for a wave number pointin toward the upper riht. ote that individual cret propaate toward the upper-riht corner, while the roup of wave propaate toward the lowerriht corner. http://www.atmo.millerville.edu/~adecaria/esci343/internal-waveloop.if. EXERCISES 1. Show that for an ideal a d d. dz dz c. Subtitute inuoidal olution into equation (35) and (36) to derive the diperion relation for internal ravity wave,. 3. a. Show that the roup velocity for internal wave i c m 3 kiˆ l ˆ j m kˆ. b. What i the manitude of the roup velocity for purely vertically propaatin wave? c. What i the manitude of the roup velocity for purely horizontally propaatin wave? 4. Ue equation (40) and (41) to how that for internal wave, c c 0. 6
5. For an ideal, incompreible a the linearized overnin equation in the x-z plane can be written a u 1 p (a) t x a. Subtitute (d) into (b) and then take w 1 p t z u w 0 x z (b) (c) z z (d) t of the reultin equation to et w 1 p t tz b. Subtitute inuoidal olution into (a), (c), and (e) to find the diperion relation and phae peed. c. What kind of wave are thee? w (e) 6. Show that for the atmophere, d. dz c 7