Computer programming was completed for digital acquisition of temperature and velocity data generated by the Geophysical Fluid Flow Cell (GFFC) during the upcoming Spacelab 3 mission. A set of scenarios was developed which covers basic electro-hydrodynamic instability, highly supercritical convection with isothermal boundaries, convection with imposed thermal forcing, and some stably stratified runs to look at large-scale thermohaline ocean circulations. The extent to which the GFFC experimental results apply to more complicated circumstances within the Sun or giant planets was assessed

Gilman, P.

Hart, J.

Toomre, J.

English

NASA Technical Reports Server

THEORETICAL STUDIES IN SUPPORT OF T 
GEOPHPSICAL FLUID FLOW EXPERI 
April  13, 1984 
John H a r t ,  J u r i  T o o m r e ,  and P e t e r  G i l m a n  
Department of Astrophysical,  P lane tary  and Atmospheric Sciences 
University of Colorado 
Boulder, Colorado 80309 
(303)  492-8568 
(303)  492-7854 
Meteorologis ts  and as t rophys ic i s t  i n t e r e s t e d  i n  g loba l  p lane tary  and s o l a r  
c i r cu la t ions  have long recognized t h e  importance of r o t a t i o n  and s t r a t i f i c a t i o n  i n  
constraining the  cha rac t e r  of these  flows. In p a r t i c u l a r ,  t he  e f f e c t  of 
lati tude-dependent Coriol is  force ,  t he  so-cal led be ta -ef fec t ,  is thought t o  play 
a c r u c i a l  r o l e  i n  such phenomena as d i f f e r e n t i a l  r o t a t i o n  on t h e  Sun, cloud band 
or ien ta t ion  on Jupi te r ,  and t h e  generat ion of magnetic f i e l d s  i n  thermal ly  driven 
dynamos. Most t h e o r e t i c a l  works and a l l  l abora tory  s tud ie s  on these  problems t o  
d a t e  have t r e a t e d  t h e  curva ture  e f f e c t s  on ly  l o c a l l y ,  and t h e  l abora to ry  e f f o r t s  
have only been a b l e  t o  s tudy be ta  e f f e c t s  i n  layered  quasi-geostrophic models. 
The continuous low-g environment of t h e  orb i t ing  space s h u t t l e  o f f e r s  a unique 
s e t t i n g  f o r  conducting geophysical f l u id  model experiments with a completely 
cons is ten t  representa t ion  of spher ic i ty  and the  r e s u l t a n t  r a d i a l  g rav i ty  found on 
astro-geophysical objects .  This is possible  because i n  zero gravi ty  one can 
cons t ruc t  an experiment t h a t  has its own r a d i a l  buoyancy forces .  The d i e l e c t r i c  
forces  i n  a l iquid,  which are l i n e a r l y  dependent on f l u i d  temperature,  give rise 
t o  an e f f e c t i v e l y  r ad ia l  buoyancy f o r c e  when a r a d i a l  e l e c t r o s t a t i c  f i e l d  is 
applied. The Geophysical Fluid Flow C e l l  (GFFC) is an implementation of t h i s  idea 
i n  which f l u i d  is contained between two r o t a t i n g  hemispheres t h a t  are 
d i f f e r e n t i a l l y  heated and s t r e s s e d  with a l a r g e  a-c voltage.  Our group has 
developed a f a c i l i t y  t o  analyze the  temeprature  and ve loc i ty  da t a  t h a t  w i l l  come 
out of the  GFFC experiment on spacelab 111. In  addi t ion w e  have been working on 
many theo re t i ca l  and l abora to ry  problems which serve  as models of expected GFFC 
motions, o r  bridges between t h e  ideal ized s i t u a t i o n  of t h e  GFFC and terrestrial, 
planetary,  and s o l a r  c i rcu la t ions .  
R e s e a r c h  Accompl ishments  i n  the  t i m e  p e r f o d  May 1983 - A p r i l  1984. 
1) GFFC Data Analysis and Programming 
Programming f o r  d i g i t a l  acquis i t ion of thermal  maps and f l u i d  ve loc i t i e s  from 
the  16mm GFFC da ta  f i lms  was completed. One remaining task  is t o  a t t a c h  our 
camera t o  the  VAX-11 computer so t h a t  f a s t e r  precessing of each d a t a  frame can 
be e f fec ted .  
On the  bas is  of computer modeling and l i n e a r  s t a b i l i t y  theory a set of 
scenarios  (on-board programs t h a t  run the  GFFC during the  Spacelab I11 mission) 
were generated.  The runs cover s tud ie s  of basic  electro-hydrodynamic 
i n s t a b i l i t y ,  highly s u p e r c r i t i c a l  convection with i so thermal  boundaries, 
convection with imposed thermal  forcing t h a t  has both r a d i a l  - and l a t i t u d i n a l  
s t ruc tu re ,  and some s t a b l y  s t r a t i f i e d  runs t o  look a t  some very simple quest ions 
concerning la rge-sca le  thermohaline ocean c i rcu la t ions .  
13 
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2) Studies of Compressible Convection. 
To apply t h e  Boussinesq r e s u l t s  from GFFC t o  p lane tary  and s o l a r  f lows, 
some ideas  about t he  e f f e c t s  of compressibi l i ty  are needed. Hurlburt  and Toomre 
have s tudied the  e f f e c t s  of compressibi l i ty  on thermal  convection tha t  is allowed 
t o  pene t r a t e  i n t o  s t a b l e  surroundings, as is o f t e n  typ ica l  of deep p lane tary  o r  
s o l a r  convection, F u l l y  non-linear simulations of two-dimensional f lows have 
been car r ied  out  on t h e  Cyber 205 and on the VAX. These s tud ie s  a l low us t o  
assess t o  what ex ten t  t he  GFFC experimental  r e s u l t s  without penetrat ion may 
apply t o  more complicated circumstances within the  Sun o r  t h e  giant  p lane ts .  
For motions spanning seve ra l  scale heights  t he  c e l l u l a r  s t r u c t u r e  obtained i n  t h e  
compressible models are q u a l i t a t i v e l y  similar t o  those predicted by our earlier 
incompressible s tud ie s  However, compressibi l i ty  l eads  t o  d i s t inc t ive  
asymmetries between t h e  s t r eng ths  of t he  upflow and downflow: pressure  
f luc tua t ions  cause s t ronge r  downward d i rec ted  plumes. O n  encountering a s t a b l e  
l aye r  these  plumes can excite i n t e r n a l  grav i ty  waves and these  i n  tu rn  can 
couple back i n t o  t h e  convecting l a y e r  and modulate its eff ic iency.  The 
penetrat ion e f f e c t  is s t ronger  i n  compressible models than Boussinesq ones 
because of the  f a s t e r  pressure-augmented plumes. The f u l l  numerical s imulat ions 
of pene t ra t ive  convection have been complemented by a n e l a s t i c  modal so lu t ions  by 
Toomre and co-workers 
3 )  Laboratory experiments on mul t ip le  equi l ibr ia  and chaot ic  o s c i l l a t o r s .  
In 1979 Charney proposed t h a t  the  atmosphere may contain mul t ip l e  quasi- 
s t a b l e  states. Since then Hart, Ghil and many o the r s  have looked f o r  multi- 
equi l ibr ia  i n  models of t he  Earth 's  atmosphere, of low Prand t l  number Jovian 
convection, and of GFFC type convection. Hart and s tuden t s  have r e c e n t l y  been 
studying a s i m p l e  l abora tory  experiment t h a t  contains  a convection f r o n t .  This 
very s i m p l e  experiment exhibi ts  extremely complex dynamics including pe r s i s t en t  
s ta tes  (hys te res i s ) ,  f r o n t a l  o sc i l l a t ions ,  in te rmi t tency  i n  f r o n t a l  formation, 
quasi-periodic o s c i l l a t i o n s  and chaos. Some of t he  motions may be i n t e r p r e t e d  by 
simple convection loop theory. Others can be described by simple one-dimensional 
mappings (which unfor tuna te ly  don't  get  a t  the  underlying i n t e r n a l  physics). A s  
seve ra l  of the  GFFC experiments are designed t o  look f o r  hys t e re s i s  and 
t r ans i t i on ,  these  terrestrial labora tory  s tud ie s  are use fu l  i n  building our 
in tu i t i on  about how convective f lows with competing i n s t a b i l i t i e s  work, and how 
t o  best  ana lyse  complex t i m e  series. 
4 )  Transit ion t o  chaos i n  numerical models of double d i f fus ive  convection. 
Toomre, Knobloch, Moore, and Weiss have performed some numerical 
ca l cu la t ions  on 2-dimensional but many degree-of -f reedom thermohaline 
convection. Remarkably the  r e s u l t s  show period doubling sequences (a l a  
Feigenbaum) t o  chaos. The r e s u l t s  are of i n t e r e s t  t o  GFFC and t o  atmospheric 
dynamicists because they i l l u s t r a t e  how period doubling can arise i n  r e l a t i v e l y  
unconstrained f l u i d  systems. Period doubling has  been observed i n  labora tory  
models of baroc l in ic  i n s t a b i l i t y ,  and may occur i n  GFFC. 
1) Provide science support f o r  t he  Spacelab 111 f l i g h t  of GFFC. 
2) Complete high-speed fi lm-to-digital  t r a n s l a t o r ,  and a l l  da t a  ana lys i s  
programs- 
3)  G e t  l i n e a r  convective i n s t a b i l i t y  and non-linear GFFC hydrodynamic codes up 
and running on the  Cyber 205. 
d 
4) Analyse the GFFC da ta  f i l m  and compare r e s u l t s  with theories  of hydrodynamic 
s t ab i l i t y ,  non-linear numerical models, and terrestrial  laboratory experiments. 
5) Continue our laboratory study of o sc i l l a t ions  i n  convecting systems, as w e l l  
as theo re t i ca l  and numerical s tudies  of diffusive and compressible convection. 
L i s t  of publications and submissions o l l y  or partial ly  
by NASA contract 
Hart, J.E,, "Low Prandtl  number convection between d i f f e ren t i a l ly  heated end 
w a l l s " ,  In t .  J. Heat and Mass Transport, 26, 1069-1074, 1983. 
Hart, J.E., "A new analysis of the  closed l o o p  thermosyphon", Int .  J. Heat 
and mass transfer", 27, 125-136, 1984. 
Hart, J.E., "Alternative experiments using the Geophysical Fluid Flow C e l l " ,  
NASA repor t  CR-3766, 44pp, Jan. 1984. 
Hurlburt, N.E., "Nonlinear compressible convection with penetration", Phd. 
thesis ,  Univ. Colorado, August, 1983. 
Moore, D.R., J. Toomre, E. Knobloch and N.O. Weiss, "Period doubling and chaos 
i n  p a r t i a l  d i f f e ren t i a l  equations f o r  thermosolutal convection", Nature, 
103, 663-667, June 1983. 
M a s G u e r ,  J., J. Latour, J, Toomre, and J-P Zahn, "Penetrative convection i n  
a s t r a t i f i e d  atmosphere", Astronomy and Astrophysics, i n  press,  1984, 
Hurlburt, N.E., J. Toomre, and J . M .  Massaguer, "Two-dimensional compressible 
convection extending over multiple scale heights", Astrophys. J., i n  press, 
1984. 
Toomre, J., N.E. Hurlburt, and J.P. Massaguer, "Strong downward plumes 
resu l t ing  from compressibility i n  nonlinear convection and the i r  
coupling t o  gravity waves", Proc . Small-scale Dynamical Processes i n  
Quiet S te l la r  Atmospheres (ed. S. K e i l ) ,  i n  press, 1984. 
i n  two-dimensional double-dif fusive convection", J. Fluid Mech., t o  be 
submitted, 1984. 
Knobloch E., D.R. Moore, J. Toomre, and N.0. Weiss, "Transitions t o  chaos 
15 


Theoretical and experimental studies in support of the geophysical fluid flow experiment

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