The structure of unstable planetary waves is computed by a quasi-geostrophic model extending from the surface up to 80 km by means of eigenvalue-eigenfunction techniques in spherical coordinates. Three kinds of unstable modes of distinct phase speeds and vertical structures are identified in the winter climate state: (1) the deep Green mode with its maximum amplitude in the stratosphere; (2) the deep Charney mode with its maximum amplitude in the troposphere: and (3) the shallow Charney mode which is largely confined to the troposphere. Both the Green mode and the deep Charney mode are characterized by very slow phase speeds. They are mainly supported by upward wave energy fluxes, but the local baroclinic energy conversion within the stratosphere also contributes in supporting these deep modes. The mesosphere and the troposphere are dynamically independent in the summer season decoupled by the deep stratospheric easterly. The summer mesosphere supports the easterly unstable waves 1-4. Waves 3 and 4 are identified with the observed mesospheric 2-day wave and 1.7-day wave, respectively

Sasamori, T.

Zhang, K. S.

English

NASA Technical Reports Server

1.4A A NUMERICAL ANALYSIS OF TRANSIENT PLANET.MY WAVES AND THE 
VERTICAL STRU m U R E  I N  A MESO-STRATO-TROPOSPHERE MODEL 
Re-Su Zhang and T. Sasamori 
Department of Atmospheric Sciences 
Universi ty of I l l i n o i s  
Urbana, I l l i n o i s  61801 
AB, STRACT 
The s t r u c t u r e  of uns table  p lanetary  waves i s  computed by a quasi-geo- 
s t roph ic  model extending from the  surface up t o  80 km by means of eigenvalue- 
eigenfunction techniques i n  spher ica l  coordinates.  Three kinds of unstable 
modes of d i s t i n c t  phase speeds and v e r t i c a l  s t ruc tu res  a re  i d e n t i f i e d  i n  the  
winter cl imate s t a t e :  (1) the  deep Green 11iode with i t s  maximum amplitude i n  
the  s t ra tosphere ,  ( 2 )  the deep Charney mode with i t s  maximum ampritude i n  the  
troposphere; and (31 the  shallow Charney mode which i s  l a rge ly  confined t o  the  
troposphere. Both the Green mode and the  deep Charney mode a r e  character ized 
by very  slow phase speeds. They a r e  mainly supported by upward wave energy 
f luxes ,  but the loca l  ba roc l in i c  energy conversion wi th in  the  s t r a tosphere  a l s o  
contr ibutes  i n  supporting these deep modes. The mesosphere and the  troposphere 
a r e  dynamically independent i n  the summer season decoupled by the  deep s t r a to -  
spher ic  e a s t e r l y .  The summer mesosphere supports the e a s t e r l y  uns table  waves 
1-4. Waves 3 and 4 a r e  i d e n t i f i e d  with the  observed ruesospheric 2-day wave and 
1.7-day wave, respect ively .  
INTRODUCTION 
Since the  d-iscovery of the  spectacular  na tu ra l  phenomena of s t r a t o s p h e r i c  
sudden warnings, i n t e r e s t  i n  planetary waves has been increasingly enhanced. 
Af ter  s a t e l l i t e  observations became possible.  along with previous observations 
by meteor radar ,  pa r t i a l - r e f l ec t ion  r ada r  and rocket sounding, many t r a n s i e n t  
wave a c t i v i t i e s  i n  the  upper atmosphere have been documented, such as 2-day 
waves i n  the  summer mesosphere (RODGERS and PRATA, 19811, 4-day waves i n  the  
upper s t ra tosphere  (VJZNNE and STANFORD, 1982 1, the slowly moving planetary  waves 
i n  the  winter s t ra tosphere  and mesosphere (HARTMANN, 19761, and a wide wave 
spectrum i n  the  lower s t ra tosphere  (YU e t  a l . ,  1983). Swe  wave a c t i v i t i e s ,  
mainly i n  wave 1 and 2, a r e  in t imate ly  r e l a t e d  t o  the sudden warming phenomena, 
which extend from the  s t ra tosphere  we l l  i n t o  the  mesosphere. The planetary  
sca le  waves possess deep v e r t i c a l  s t r u c t u r e  i n  winter ,  but v e r t i c a l l y  decoupled 
i n  summer (LABITZKE. 1981a,b). 
The purpose of the present study i s  t o  inves t iga te  those t r ans ien t  plane- 
t a ry  waves based on i n s t a b i l i t y  theory. The growth r a t e ,  phase speed, merid- 
ional  and v e r t i c a l  s t r u c t u r e  of unstable modes a r e  examined f o r  the winter and 
summer s o l s t i c e  c l imate  s t a t e s .  A comparison study. of s t a b i l i t i e s  f o r  the 
bas i c  s t a t e s  before and a f t e r  the 197611977 sudden warming event i s  a l s o  dis-  
cussed. 
. &%"ETHOD OF ANALYSIS 
( a )  Governing Equation 
. Assuming the  large-scale wave motions a r e  ad iaba t i c  and nondiss ipat ive ,  we 
use the  l inea r i zed  quasi-geostrophic, spher ica l  model formulated by I?ATSUNO 
(1971 : 
https://ntrs.nasa.gov/search.jsp?R=19850024160 2020-03-20T18:08:50+00:00Z
where 
Equations 1 and 2 a r e  v o r t i c i t y  and thermodynamic equations, respect ively ,  where 
1 denotes the  longitude, 0 the l a t i t u d e .  c = cos 0, s = s i n  8, i2 the angular 
ve loc i ty  of the  earth.  a the radius  of the ea r th ,  4 the per turbat ion geopo- 
t e n t i a l ,  U the  bas ic  zonal flow, and S the s t a t i c  s t a b i l i t y .  
(b) Grid Point Arrangenent and Boundary Conditions 
The model i s  d i sc re t i zed  v e r t i c a l l y  by ten computation l eve l s  covering 
from the surface up t o  80 km and it i s  re fe r red  t o  as  an MST model (11 f o r  meso- 
sphere, S f o r  s t ra tosphere  and T f o r  troposphere). The computation l e v e l s  a r e  
arranged a t  p 0.01, 0.04, 0.2, 1 mb i n  the mesosphere; a t  p a 3 ,  10, 30, 100 
mb i n  the s t ra tosphere  and a t  p = 250, 750 mb i n  the troposphere. The v e r t i c a l  
boundary condition w = 0 i s  assumed a t  p = 0 and p = 1000 mb. I n  the  meridion- 
a1  d i rec t ion  the  model i s  bounded by the  f ixed  boundary conditions 4' u' V '  
= 0 a t  the pole and the equator with 10' meridional mesh s izes .  The g r i d  point 
arrangement i s  shown i n  Figure 1. 
Figure 1. Ver t ica l  and meridional g r i d  arrangement 
and boundary conditions used ir, the spher ical  model. 
( c )  Basic S t a t e  Parameters 
The winter and summer s o l s t i c e  cl imate bas ic  flows a r e  shown i n  Figure 2. 
The zonal-mean temperature f i e l d  and the  wind f i e l d  cons i s t  of the  parameter 
space (S, U), where the s t a t i c  s t a b i l i t y  i s  ca lcula ted  from the  temperature 
f i e l d .  
( d )  Normal Hode Solution 
Figure 2. The s o l s t i c e  c l imate  s t a t e s .  The upper 
panel i s  temperature (K) and the lower panel i s  
the  zonal mean wind i n  m/s. 
We assume a normal mode solut ion:  
0 = @(e,  p)eim(X-ctl 
and ca lcu la te  the phase speed C r ,  growth r a t e  oi = mCi, and the eigen- 
funct ion @(e ,p ) ,  where C Cr + i C i  and 0 = @r + ia i  a re  complex 
numbers . 
RESULTS 
(a) Winter 
Figure 3 shows the spectrum of the growth r a t e s  and phase speeds of the 
unstable waves computed f o r  the winter  bas ic  flow. The values of a and as 
a r e .  representa t ive  f o r  "dry" and "wet" tropospheric conditions. Both reveal 
the presence of th ree  kinds of unstable modes: 
1. Green mode. denoted by MST i n  Figure 3 ,  fo r  waves 1 and 2,  
2. Deep Charney mode, denoted by ST i n  Figure 3 ,  f o r  wave 3 ,  and 
3. Shallow Charney mode. denoted by T i n  Figure 3 ,  fo r  waves 5-9. 
m 
Figure 3. The growth r a t e  (ordinate)  and the phase speed a t  
the equator (labeled ic m/s) fo r  winter s o l s t i c e  bas ic  
s t a t e :  a )  the dry troposphere; b) the moist troposphere. 
QRfGr;i:;$iL ,s 
OF POOR QUALITY. 
The phase speeds given i n  Figure 3 reveal an in te res t ing  i n s t a b i l i t y  ex- 
change between the slow modes (waves 1-4, including the Green mode and the deep 
Charney modes) and the f a s t  modes (waves 5-9, a l l  of which a r e  confined t o  the 
troposphere). I t  implies some i n t r i n s i c  r e l a t i o n  between the phase speed and 
the v e r t i c a l  s t ructure .  The v e r t i c a l  wave energy fluxes (Figure 4 )  depic t  t h a t  
the deep waves a re  v e r t i c a l l y  coupled a s  an e n t i r e t y .  They may be i d e n t i f i e d  
with the slowly moving waves 1, 2 and 3 i n  the r e a l  atmosphere. 
(b) Summer 
.The spectra  of the growth r a t e s ,  phase speeds fo r  the summer c i r c u l a t i o n  
a r e  given i n  Figure 5. The mesospheric modes, with large  negative phase speeds 
Cr < 0, and the tropospheric modes. with the pos i t ive  phase speeds Cr > 0, 
exchange t h e i r  s t a b i l i t i e s  a t  wave number 4. The computed most unstable mode 
i s  wave 3 with a period of 2 days which compares favorably with the observed 
2-day wave. Figure 6 shows s t ruc tu res  of the computed mesospheric waves with 
m 1-4 and the tropospheric wave with m = 6. They a r e  s p a t i a l l y  separated by 
the e a s t e r l y  wind i n  the stratosphere.  The planetary scale  waves dominate i n  
the  mesosphere and the synoptic scale  waves dominate i n  the  troposphere. 
VEXTICAL COUPLING AND DECOUPLING 
I n  order  t o  understand the  s t a b i l i t y  due to  v e r t i c a l  coupling between the 
mesosphere, the s t ra tosphere  and the troposphere i n  the winter season, a case 
study i s  performed t o  compare the eigenmodes i n  the bas ic  s t a t e s  before and 
a f t e r  the 1976/1977 sudden warming event. Figure 7 shows the  observed var ia-  
t i o n  of geopotential  amplitude fo r  waves 1 and 2 during the warming phase. 
0 ,  degree! 
Figure 4. Geopotential eigenfunctions (upper and v e r t i c a l  wave energy 
f l u x  (lower) i n  the NH winter bas ic  flow with the "dry" troposphere. 
Units a r e  a rb i t r a ry .  
Figure 5. The growth ra te s  (ordinate). and the p h a ~ e  speeds 
(labeled i n  m/s) fo r  summer s o l s t i c e  bas ic  s t a t e  with 
a )  the  "dry" troposphere and b) the  ' ho i s t "  troposphere. 
m indicates  the wave number, M the  mesospheric mode and 
T ' the tropospheric node. 
Both waves 1 and 2 were growing during December 1976 and decreasing i n  January 
1977. The bas ic  flow i n  December 1976 i s  characterized by a broad, weak 
throughout the  troposphere and the s t ra tosphere ,  and i n  January 1977 i s  
characterized by e a s t e r l i e s  a t  high l a t i tudes .  Figure 8 s?mmarizes the r e s u l t s  
of frequency computations f o r  December and January. 
Three s ign i f i can t  changes occur a f t e r  the sudden warming : 
1. The maximum growth r a t e  s h i f t s  from planetary- scale  (m = 2 )  t o  synop- 
t i c  sca le  (m = 7) and the planetary sca le  waves a re  subs tan t i a l ly  
s tabi l ized.  
2. Transi t ion occurs f ron  the slow modes t o  the f a s t  modes a t  wave number 
4 before the warning, but a l l  waves change t o  the f a s t  modes a f t e r  it. 
3. A 1 1  the  deep modes (BET modes and ST modes) a r e  suppressed t o  the 
shallow, tropospheric modes a f t e r  the sudden warming a s  shown i n  
Figures 9 and 10. 
Since the  deep Green mode and the deep Charney mode have been calcula ted 
i n  the winter cl imate bas ic  flow, we suspect t h a t  the deep modes e x i s t  i n  the 
whole season except i n  periods a f t e r  sudden warnings, i n  which a l l  deep modes 
a r e  suppressed p r j m r i l y  by the  reversed zonal flow a t  high l a t i tudes .  
r c -  QR;p:<:L :r. 
OF, FOOZ Q G A i i T f  
0 uu 
15 30 45 60 75 15 30 45 60 75 
8 (degree) 
F i g u r e  6. The g e o p o t e n t i a l  e i g e n f u n c t i o n s  f o r  waves m = 1 - 4 
i n  t h e  summer mesosphere, f o r  wave m = 6 i n  t h e  summer t ropo-  
sphere.  The u n i t s  of ampl i tude  ( s o l i d  l i n e )  a r e  a r b i t r a r y .  
The phases (broken l i n e )  a r e  i n  n r a d i a n s .  
0 
f "  100 Wave 2. 20 mb 
F i g u r e  7. Lat i tude-t ime c r o s s  s e c t i o n s  of g e o p o t e n t i a l  h e i g h t  wave 
ampl i tude  ( m e t e r s )  f o r  waves 1 and 2 d u r i n g  t h e  1976/1977 sudden 
warming e v e n t  ( a f t e r  O'NEILL and TAYLOR, 1978). 
Figure 8 .  The growth rate (ordinate) and the phase speed 
a t  the equator ( labeled i n  m/s) as a function of wave- 
number m. 
8. degree 
Figure 9 .  Geopotential (top) and wave energy f lux  (bottom) 
i n  December 1976. Units are arbitrary. 
ORIGINAL PALE ,S 
OF P O O ~  GUAL~TP;' 
s.aorn . 
Figure 10. A s  i n  Figure 9 except i n  January 1977. 
CONCLUSION 
I n  conclusion, we propose a ba-clinic i n s t a b i l i t y  for the generation of 
some planetary t r ans ien t  waves i n  the  upper atmosphere. The usually observed 
t r ans ien t  waves 1 and 2 i n  the winter may have an o r ig in  of ba roc l in ic  ins tabi-  
l i t y  i n  addi t ion t o  the response t o  the external  forcing. The mesosphere, t h e  . 
s t ra tosphere  and the  troposphere a r e  in t imately  coupled by the  deep planetary 
waves i n  the  winter c i r cu la t ion ,  while the  summer c i rcu la t ion  is  v e r t i c a l l y  de- 
coupled by the s t r a tospher ic  e a s t e r l y  zonal wind. The computed unstable  wave 
- 
3 ,  confined t o  the mesosphere and the. upper stratosphere,  compares favorably 
with the observed 2 d a y  wave. 
Hartmann, D. L. (19761, The s t ruc tu re  of the  s t ra tosphere  i n  the southern hemi- 
sphere during l a t e  winter 1973 by s a t e l l i t e ,  J. Atmos. Sci., a, 1141- 
1154. 
Labitzke,  K. (1981a), The ampl i f ica t ion of the height wave 1 i n  January 1979: A 
c h a r a c t e r i s t i c  precondition f o r  the major warning, Mon. Wea. Rev,, lq% 
983-989. 
Labitzke, K. (1981b), Stratospheric-mesospheric midwinter disturbances : A 
summary of observed c h a r a c t e r i s t i c s ,  J. Geo~hys. RevL, a, 9665-9678. 
Fatsuno, T. (1971); A dynamical model of the  s t r a tospher ic  sudden wazming, +ZL 
Atmos. Sci., 28, 1479-1494. 
O'Neill ,  A. and B. F. Taylor (19791, A study of the  major s t r a tospher ic  warming 
of 1976-1977, Ouart. J. Rov. Meteorol. Soc., 1QS, 71-92. 
Rodgers, C. D. and A. J. P ra ta  (19811, Evidence f o r  a t r ave l ing  Z d a y  wave i n  
the middle atmosphere, J, Geoohvs, Res, , 86, 9661-9664. 
Venne, D. E. and J. L. Stanford (19821, An observational study of high-lati tude 
s t r a tospher ic  planetary waves i n  winter,  J. Atmos. Sci., a, 1026-1034. 
Yu, W.-B., R. L. Martin and J. L. Stanford (19831, Long and medium-scale waves 
i n  the 'lower s t ra tosphere  from sa te1  li te-derived microwave measurepents , 
J. Geophys. Res., 8505-8511. 


A numerical analysis of transient planetary waves and the vertical structure in a meso-strato-troposphere model, part 1.4A

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