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Kpanvcue coo6uteuun THE RETROGRESSION OF NON-DIVERGENT WAVES VOJTF.CH VITEK Institute of Physics of the Atmosphere, Czechosl. Acad. Sci., Prague*) Wolff [1] showed that the errors of the equivalent barotropic model are to a certain extent caused by the fact that the model incorrectly forecasts the phase of very long waves corresponding to the first to fourth harmonic of the baric field. Theoretically, this systematic error can be explain- ed by the unreal retrogression of the Rossby waves in the range of very long wave-lengths. The behaviour of very long waves in barotropic and baroclinic models has been dealt with in detail especially by Wiin-Nielsen [2]. In the present paper it is shown that in the middle troposphere (to be quite precise, in the level of non-divergence) there can exist alongside Rossby waves also stable progressive waves the phase velocity of which is controlled by the stability of the atmosphere. The retrogression of such waves is strongly reduced compared with classical Rossby waves. As the initial relation we use the co-equation for adiabatic flow so that in the usual symbols az~ f 60 --fco-~-p2-- ~pp[r0.V($ +f)]-- Rp -I V2(II).VT)= 0. We denoted S : --c~ 60 lg @/60p the measure of stability. In view of the vorticity equation we can write (I) also in the form 602~ (2) f 60p St -- Rp- 1 V2(r0 . VT) + V2(S0)) = 0. Let us further consider wave motion in baroclinic western flow U(p). Let the waves be charac- terized in the standard coordinate system by the quantities vy = V(p)exp [ik(x- ct)], 0)= = P(p) exp [ik(x -- ct)]. Then from the linearized vorticity equation 6020) 600) guy = __f k_ 2( U- c -- ilk-2) -I 60x ap (3) vy = --k-2 f(U -- c -- ilk-2) -1 -~p ; ~ = ax -- " Let us substitute from (3) into the linearized equation (2) for S = S(p) and on the assumption that the term V2(m. VT) can be expressed in the first approximation as (60To/ay) V2vy where 8To/Oy = F(p) is the meridional temperature gradient corresponding to the basic flow U(p). Since we do not consider horizontal wind shear in the basic flow, we obtain 604co d U oa co (4) S(U -- c -- ilk-2) 2 V20) --f2k-2(U - - c - - flk-2)60x60tap--------:~ q - f 2 k - 2 d p 60xat-----~pp+ + Rp-lk-:/60r~ W-- c -- ~-2) V 2 60~ = o. 60y ap We shall assume of the isobaric vertical velocity that it is zero on the atmosphere boundaries 1 and reaches maximum value M in the level of non-divergence p = ~P0. We can thus make the usual choice (5) to = 4Mpp o 1(1 -- PPo 1) exp [ik(x -- ct)]. *) Address: Bo~ni II, Praha 4 - Spo~ilov. 420 Studia geoph, et geod. 9 [1965)

The retrogression of non-divergent waves

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Page 1: The retrogression of non-divergent waves

Kpanvcue coo6uteuun

THE RETROGRESSION OF NON-DIVERGENT WAVES

VOJTF.CH VITEK

Inst i tute o f Physics o f the Atmosphere , Czechosl. Acad. Sci. , Prague*)

Wolff [1] showed that the errors of the equivalent barotropic model are to a certain extent caused by the fact that the model incorrectly forecasts the phase of very long waves corresponding to the first to fourth harmonic of the baric field. Theoretically, this systematic error can be explain- ed by the unreal retrogression of the Rossby waves in the range of very long wave-lengths. The behaviour of very long waves in barotropic and baroclinic models has been dealt with in detail especially by Wiin-Nielsen [2].

In the present paper it is shown that in the middle troposphere (to be quite precise, in the level of non-divergence) there can exist alongside Rossby waves also stable progressive waves the phase velocity of which is controlled by the stability of the atmosphere. The retrogression of such waves is strongly reduced compared with classical Rossby waves. As the initial relation we use the co-equation for adiabatic flow so that in the usual symbols

az~ f 60 --fco-~-p2-- ~pp[r0.V($ +f)]-- Rp -I V2(II).VT) = 0.

We denoted S : --c~ 60 lg @/60p the measure of stability. In view of the vorticity equation we can write (I) also in the form

602~ (2) f 60p St - - R p - 1 V2(r0 . VT) + V2(S0)) = 0.

Let us further consider wave motion in baroclinic western flow U(p). Let the waves be charac- terized in the standard coordinate system by the quantities vy = V ( p ) e x p [ i k ( x - ct)], 0 ) = = P(p) exp [ik(x -- ct)]. Then from the linearized vorticity equation

6020) 600) guy = __f k _ 2( U - c -- i l k -2 ) - I 60x ap (3) vy = - - k - 2 f ( U -- c - - i l k - 2 ) - 1 -~p ; ~ = ax - - "

Let us substitute from (3) into the linearized equation (2) for S = S(p) and on the assumption that the term V2(m. VT) can be expressed in the first approximation as (60To/ay) V2vy where 8To /Oy = F(p) is the meridional temperature gradient corresponding to the basic flow U(p). Since we do not consider horizontal wind shear in the basic flow, we obtain

604co d U oa co (4) S ( U -- c -- i lk -2) 2 V20) - - f 2 k - 2 ( U -- c -- flk-2)60x60tap--------:~ q - f 2 k - 2 dp 60xat-----~pp +

+ R p - l k - : / 6 0 r ~ W - - c - - ~ - 2 ) V 2 60~ = o . 60y ap

We shall assume of the isobaric vertical velocity that it is zero on the atmosphere boundaries 1 and reaches maximum value M in the level of non-divergence p = ~P0. We can thus make the

usual choice

(5) to = 4 M p p o 1(1 -- PPo 1) exp [ik(x -- c t ) ] .

*) Address: Bo~ni II, Praha 4 - Spo~ilov.

420 Studia geoph, et geod. 9 [1965)

Page 2: The retrogression of non-divergent waves

Shor ter Contributions

Due to (5) and in view of the geostrophic approximation we then obtain from (4) the frequency equation

(6) k 2 S p ( p o -- p) ( U - c -- i lk -2) 2 -- 2f2 c ( U - - c -- f lk - 2 ) - -

d U d U -- 2f 2 ~-p-p (Po -- 2p) c + f 2 ~ p (p 0 -- 2 p ) ( U - - f l k - 2 ) : O.

1 Let us now deal only with the conditions in the level of non-divergence p = ~Po, where the simplifying assumptions used are probably satisfied the best. Then

(7) [ k 2 S p 2 ( U - - c - - f lk - 2 ) -- 8/2e1 ( U -- c - - flk - 2 ) : O.

One solution of Eq. (7) are the Rossby waves

(8) c = U - - f l k - 2 = c ( 8 ) .

The second solution gives stable waves, their phase velocity being

8s (9) c : 1 -[- k2p20S J ( U -- f lk - 2 ) = e(9 ) .

These waves are thus controlled by the stability of the atmosphere. For S ~ 0, c(9 ) ~ 0, if S grows in the theoretical abstraction beyond all limits, c(9 ) ~ c(8 ). The waves described by relation (9) thus always move more slowly in the stable atmosphere than the Rossby waves. The stationary wave-length in both types of waves is the same. For sufficiently long waves ( U - - f lk - 2 < O)

retrogression occurs for both types. With the Rossby waves the greatest retrogression is for the longest waves. On the other hand, the retrogression of very long waves of type (9) is obviously greatly damped since

p~Sf l (10) lira c(9 ) - - 8 f z c t .

k~O

For illustration let us consider a very long wave L = 4=. 106 m. For f = 10-'* sec -1 , S : = 3.2. 10 - z m 2 mb -2 sec - z , Po = 103 mb, we have A = 1 + 8fZ/k2p2o$ = 11. This means that the retrogression of waves of type (9) compared with Rossby waves for this wave-length is reduced in the ratio 1 : 11. In the extreme case (10), we find that for fl45" we have c i - - --6"5 msec -1 . This value is numerically close to an analogous limiting value obtained from the baroclinic model proposed by Wiin-Nielsen (see [2], Eq. (6.8)). It is also seen from (10) that the damping of the retrogression may be effective especially in high geographical latitudes.

From the considerations so far we can deduce that the reduction factor A - 1 could be used for barotropic numerical forecasts as a means of semi-empirical correction in eliminating the unreal retrogression of very long waves. In regions of shorter waves moving to the east ( U - - f lk - 2 > 0) the use of this correction is in our opinion doubtful and a decision on it cannot be reached without practical applications.

Received 26. 3. 1965 Reviewer: S. Brandeis

References

[1] P. M. W o l f f : The Error in Numerical Forecasts due to Retrogression of Ultra-long Waves. Monthly Weather Rev., 86 (1958), 245.

[2] A. W i i n - N i e l s e n : On Barotropic and Baroclinic Models, with Special Emphasis on Ultra- long Waves. Monthly Weather Rev., 87 (1959), 171.

Studia geoph, et geod. 9 (1965] 421

Page 3: The retrogression of non-divergent waves

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X P O H H K A - N E W S

Z U M 65. G E B U R T S T A G D E S P R O F . D R . M I K U L A S K O N C E K

Anl/isslich des 60. Geburtstages von Prof. Kon~ek (12.4. 1960) wurde in dieser Zeitschrift*) die verdienstvolle Arbeit bewertet, die der Jubilar seinerzeit dadurch vollbracht hat , dass er das klassische Lehrbuch der synoptischen Meteorologie yon S. P. Chromov dutch seine tschechische und deutsche IJbersetzung sowohl ffir die tschechoslovakische als auch ftir die Weltrneteorologie zug/i.nglich gemacht, sowie auch ffir die Meteorologie und Klimatologie in der Slovakei die Forschungsarbeitsst i i t ten an der Universit/it und an der SAV (Slovakische Akademie der Wissen- schaften) gegrtindet hatte.

Im Zeitraum von den ftinf vergangenen Jahren konzentrierte sich die T~tigkeit des korrespon- dierenden Mitgliedes der SAV Kon~ek vor allem auf die Zusammenarbei t mit den Fachinst i tut io- nen, insbesondere in der UdSSR, in Polen, der DDR, Ungarn und Osterreich. Die Forschung fiber das Klima und die meteorologischen Verhiiltnisse der Hohen Tatra geht als eine polnisch- tschechoslovakische Kollektivarbeit zu Ende und die fiber die Hohe Tatra einhei t l ich unter Kon~ek's Redakt ion verarbeitete Monographie wird ihr Ergebnis darstellen. Es wird als ein interessanter Beitrag zur Meteorologie von Europa gelten, der zu einem Beispiel filr die/ ihnl iche Zusammenarbei t auch in einer Reihe yon anderen Gebieten werden sollte.

Die Ehrenmitgliedschaft der polnischen und ungarischen meteorologischen Gesellschaft sowie auch die Wahl zum Mitglied des Priisidiums der Internationalen Assoziation ffir Meteorologie und Physik der Atmosphere in den Jahren 1960 und 1963 bringen die Bewertung der erfolgreichen wissenschaftlichen und organisatorischen T/itigkeit Prof. Kon~ek's genau so zum Ausdruck, wie seine Betrauung von Seiten der mitteleurop/iischen limnologischen Kommission, die klimato- logischen Verh/iltnisse des Donaugebietes in allen Donaustaaten zu verarbeiten und zu koordinieren.

Auf dem slovakischen Boden, in der SAV hat Prof. Kon~ek unter Mitarbeit seiner Schiller die Forschungsbasis im Labora tor ium ftir Meteorologie weiter ausgestaltet. Prof. Kon~.ek ist, t rotz aller Belastung durch organisatorische Funktionen, auch weiterhin publizist isch tfitig, was weitere 14 Arbei ten bezeugen, die der Jubilar in den vergangenen ffinf Jahren publ izier t hat.

Wir wfinschen dem Jubilar, er m6ge filr das weitere Aufblt ihen der tschechoslovakischen Meteorologie und Klimatologie noch lange Zeit aktiv arbeiten. Alois Gregor

*) A. G r e g o r : lllecTri~ecrrrmaerrte npoqb. ~-pa Mnzyzlama Kormera. Studia geoph, et geod., 4 (1960), 416.

422 Studia geoph, et geod. 9 (1965)