Axially-symmetric, linear, free modes of global, primitive equation, ocean-atmosphere models are examined to see if they contain decadal (10 to 30 years) oscillation time scale modes. A two-layer ocean model and a two-level atmospheric model are linearized around axially-symmetric basic states containing mean meridional circulations in the ocean and the atmosphere. Uncoupled and coupled, axially-symmetric modes of oscillation of the ocean-atmosphere system are calculated. The main conclusion is that linearized, uncoupled and coupled, ocean-atmosphere systems can contain axially-symmetric, free modes of variability on decadal time scales. These results have important implications for externally-forced decadal climate variability

Mehta, Vikram M.

English

NASA Technical Reports Server

ON THE DECADAL MODES OF 
OSCILLATION OF AN IDEALIZED 
OCEAN-ATMOSPHERE SYSTEM 
Vikram M. Meh ta*  
Department o f  Meteorology 
and 
Supercomputer Computations Research Ins t i tu te  
The Florida State University, Tallahassee, FL 32306 
ABSTRACT 
Axially-symmetric, linear, free modes of global, primitive equation, ocean- 
atmosphere models are examined to see if they contain decadal (10-30 years) 
oscillation time scale modes. A two-layer ocean model and a two-level atmospheric 
model are linearized around axially-symmetric basic states containing mean meridional 
circulations in the ocean and the atmosphere. Uncoupled and coupled, axially-symmetric 
modes of oscillation of the ocean-atmosphere system are calculated. The main conclusion 
of this study is that linearized, uncoupled and coupled, ocean-atmosphere systems can 
contain axially-symmetric, free modes of variability on decadal time scales. These 
results have important implications for externally-forced decadal climate variability. 
INTRODUCTION 
There are numerous observational studies of decadal (10-30 years) time scale 
climate oscillations (see, for example, Pittock (1983), Currie and O'Brien (1988), 
Labitzke and van Loon (1989), and references therein). Most of these studies have tried 
to correlate decadal climate variations with solar and lunar influences. Although the 
existence (or otherwise) of these correlations and the mechanisms by which variability 
of solar and lunar influences might affect terrestrial climate are not yet clear, the 
observational studies show that decadal climate variations do exist. These variations 
may be due to internal variability of the climate system, they may be forced externally 
or a combination of internal variability and external forcing may give rise to these 
modes. Some results from a research effort to study the natural variability of an 
idealized ocean-atmosphere system, especially on decadal time scales, are reported here. 
In addition to the above-mentioned observations, more specific motivation for the work 
reported here was provided by two observational studies dealing with long-period 
(years to decades) fluctuations of the axially-symmetric (symmetric around the 
rotation axis of the earth) components of some climatic variables. Krishnamurti et al. 
(1986)'s analysis of global sea level pressures (SLP) and Semazzi et al. (1988)'s 
analysis of global sea surface temperatures (SST) showed that axially-symmetric 
components of SLP and SST fluctuate with a variety of time scales, including the years to 
decades time scales. In view of the above observations, the emphasis in the modelling 
* C u r r e n t  affi1iation:Code 613,  NASAjGoddard S p a c e  FlighL 
C e n t e r  
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work reported here is on axially-symmetric modes of the ocean-atmosphere system 
oscillating at decadal time scales. 
Anderson and Stevens (1987) (hereafter referred to as AS) found that a multi- 
level model of the tropical atmosphere linearized about a Hadley cell basic state contains 
wavelike, axially-symmetric modes in thermal wind balance. The oscillation time 
scales of these modes range from a few weeks to a few months. Advection of perturbation 
zonal wind and temperature fields by the basic state meridional velocity was suggested as 
the mechanism giving rise to these modes. 
In the ocean, there appears to be a similar but opposite mean meridional 
circulation (MMC) than in the atmosphere. This can be seen clearly in studies of model 
ocean circulations by Bryan and Lewis (1979, fig. 10a) and Meehl et al. (1982, fig. 2). 
This similarity of atmospheric and oceanic MMC raises some interesting questions in 
light of AS'S results. If advection of perturbation fields by the basic state meridional 
velocity is indeed the mechanism giving rise to the axially-symmetric atmospheric 
intra-seasonal modes found by AS, then the same mechanism may give rise to axially- 
symmetric oceanic modes with time scales of several years and longer. This is because 
basic state meridional velocities in the ocean are at least 2 orders of magnitude smaller 
than in the atmosphere. If such long-period, axially-symmetric modes do exist in an 
idealized ocean model, then within the framework of a coupled ocean-atmosphere system, 
such oceanic variability might also result in atmospheric variability on long time 
scales. Moreover, since MMC exist not only in the tropics but also in higher latitudes, 
such axially-symmetric modes might exist in association with higher latitude MMC also. 
Therefore, a modelling study was carried out to test the hypotheses outlined above. 
MODEL DESCRIPTION AND METHOD OF SOLUTION 
The governing equations of the two-level model of the atmosphere formulated for 
this study are primitive equations in spherical co-ordinates with sigma (pressure at 
any level divided by surface pressure) as the vertical co-ordinate. The model 
atmosphere extends around the earth in longitudinal direction and from the South Pole to 
the-North Pole in latitudinal direction. The governing equations of the two-layer model 
of the ocean are primitive equations with height as the vertical co-ordinate. The model 
ocean has the same horizontal domain as the model atmosphere. Vertically, the ocean is 
divided into two homogeneous layers separated by a thermocline. The model ocean has a 
uniform depth of 4000 meters. The model ocean and the model atmosphere interact via 
heat and momentum exchange. Other physical processes included in the models are 
entrainment of mass, momentum, and heat between the oceanic layers, thermal infrared 
radiation, and vertical diffusion within the atmosphere and the ocean. The two models 
and parameterizations of physical processes are described with details in Mehta 
(1 990a, 1990b). 
The two models and parameterizations of physical processes are linearized 
around an axially-symmetric basic climatic state containing mean meridional 
circulations in the atmosphere and the ocean. The resulting linearized perturbation 
climate system is solved as an eigenvalue problem. The eigenvalues are complex-valued, 
where the real parts are the oscillation frequencies and the imaginary parts are the 
growth rates of individual modes. The eigenvectors contain vertical and horizontal 
structures of individual modes. Details of linearization and numerical solution 
procedure are given in Mehta (1990a, 1990b). 
RESULTS AND DISCUSSION 
In the case of a motionless basic state, the oscillation frequencies of axially- 
symmetric modes in a model of the ocean or the atmosphere can be anticipated based on 
analytical calculations (see, for example, Matsuno (1966) and Lindzen (1967). among 
others). The two types of axially-symmetric modes which can exist for motionless basic 
state are fast (oscillation periods of a few hours), inertia-gravity modes and steady 
Rossby and Kelvin modes at zero frequency. Also, in the absence of a reservoir of energy 
in the basic state, these two types of axially-symmetric modes are neutrally stable. The 
two types of modes described above exist (not shown) in uncoupled two-layer ocean 
model and uncoupled two-level atmospheric model in the absence of basic state motion. 
In the next step, basic states containing mean meridional circulations were specified 
based on Meehl et al. (1982) for the ocean and Peixoto and Oort (1984) for the 
atmosphere. Figures 1 a and 1 b show eigenvalue spectra of the uncoupled ocean and the 
uncoupled atmospheric models, respectively. Comparing the oceanic spectrum (fig. 1 a) 
in the presence of mean meridional circulations with the two types of modes described 
above for the motionless basic state case, it is clear that a new family of free, axially- 
symmetric ocean modes with oscillation periods ranging from about two years to several 
centuries appear when the basic oceanic state contains mean meridional circulations. 
These modes have practically neutral growth rates. A similar comparison of uncoupled, 
axially-symmetric atmospheric modes shows that specification of a basic atmospheric 
state containing MMC gives rise to atmospheric modes (fig. Ib )  with oscillation periods 
ranging from about a week to several years. There are a few atmospheric modes with 
decadal and longer oscillation periods also. Unlike the decadal oceanic modes (fig. 1 a), 
however, the low-frequency atmospheric modes have substantially large growth rates. 
To see if these modes of individual systems (ocean and atmosphere) are modified 
due to interaction between the ocean and the atmosphere, the two models were coupled by 
wind stress and heating parameterizations. Coupled, axially-symmetric, ocean- 
atmosphere modes were then calculated for the same oceanic and atmospheric basic states 
as used for the uncoupled calculations. A representative value of the coupling 
coefficients was used. The eigenvalue spectrum resulting from this coupled calculation 
is shown in figure l c .  Comparison with figures l a  ant- 1 b shows that eigenvalues, 
especially growth rates, change dramatically due to ocean-atmosphere coupling. This 
growth rate enhancement is maximum in the decadal time scale range. Ocean modes with 
periods of about 2 years to several hundred years grow much more quickly because of 
ocean-atmosphere coupling. Spatial structures of atmospheric and oceanic components 
of a coupled mode oscillating with 14.6 years period and growing with 3.19 years e- 
folding time are shown in figure 2a and 2b, respectively. 
A number of sensitivity experiments were carried out to test the robustness of 
results by varying parameter values. Figure 3 shows coupled eigenvalue spectra from 
three experiments in which coupling coefficients were varied over a factor of four. As 
the coupling coefficients are increased, the modes in 1-20 years oscillation period range 
are slowed down and their growth rates are increased. This can be seen as general 
displacement of eigenvalues of decadal periods towards higher growth rates and lower 
oscillation frequencies as the coupling coefficients are increased. Modes with oscillation 
periods of weeks to months and periods longer than twenty years are also affected 
significantly by increasing coupling coefficients. Results from other sensitivity 
experiments are described and discussed in Mehta (1990b). The most important result 
of the sensitivity studies is that decadal modes exist in the coupled models over a wide 
range of parameter values and even in the presence of dissipative processes. 
Like any model, the ocean-atmosphere models formulated for this study also have 
their limitations. The most important ones are absence of continents, absence of fully 
interactive hydrological cycle in the atmosphere, and absence of salinity, ice-snow, and 
high-latitude convection in the ocean. In spite of these limitations, it is felt that the 
results are reliable enough, computationally and physically, to proceed to 
experimentation with models of higher complexity. There are two main implications of 
the results presented here for the Sun-climate relationship. One implication is that i f  
there is external forcing on decadal time scales, there is a possibility of resonant 
response of the climate system since !he system appears to have natural variability on 
decadal time scales. The other implication is that it will be necessary to separate 
internal climate variability from externally-forced climate variability, i f  any. It can 
be speculated that transient correlations between solar activity and climate, such as 
solar cycle-level of Lake Victoria and solar magnetic field sector boundary-atmospheric 
vorticity area index, may be due to internal variability. It is suggested that climate data 
analysis studies be carried out with an aim of separating internal variability from 
externally-forced variability. 
ACKNOWLEDGEMENTS 
This work was supported partially by a Supercomputer Computations Research 
Institute Fellowship and partially by NSF grant ATM-8714674 to the Florida State 
University. The Florida State University Supercomputer Computations Research 
Institute is partially funded by the U.S. Department of Energy through contract no. DE- 
FC05-85ER250000. 
REFERENCES 
Anderson, J.R., and D.E. Stevens, 1987: The presence of linear wavelike modes in a 
zonally symmetric model of the tropical atmosphere. 3. Atmos. Sci., 44, 21 15-21 27. 
Bryan, K., and L.J. Lewis, 1979: A water mass model of the world ocean. J. Geophys. 
Res., 84, 2503-251 7. 
Currie, R.G., and D.P. O'Brien, 1988: Periodic 18.6-year and cyclic 10 to 11 year 
signals in northeastern United States precipitation data. J. Climatol.. 8, 255-281. 
Krishnamurti, T.N., S. Chu, and W. Iglesias, 1986: On the sea level pressure of the 
Southern Oscillation. J. Met. Atmos. Phys., 34, 385-425. 
Labitzke, K., and H. van Loon, 1989: Association between the 11-year solar cycle, the 
QBO, and the atmosphere. Part Ill: Aspects of the association. J. Climate, 2, 554-565. 
Lindzen, R.D., 1967: Planetary waves on beta planes. Mon. Wea. Rev., 95, 441-451. 
Matsuno, T., 1966: Quasi-geostrophic motions in the equatorial area. J. Met. Soc. 
Japan, 44, 24-42. 
Meehl, G.A., W.M. Washington, and A.J. Semtner, Jr., 1982: Experiments with a global 
ocean model driven by observed atmospheric forcing. J. Phys. Oceanogr., 12, 301- 
31 2. 
Mehta, V.M, 1990a: On the axially-symmetric modes of oscillation of an idealized 
ocean-atmosphere system. Submitted to Climate Dynamics. 
----, 1990b: On the decadal modes of oscillation of an idealized ocean-atmosphere 
system. Ph. D. dissertation, Dept. of Meteorology, The Florida State University, 179pp. 
Peixoto, J.P., and A.H. Oort, 1984: Physics of climate. Rev. Mod. Phys., 56, 365-429. 
Pittock, A.B., 1983: Solar variability, weather, and climate: An update. Quart. J. Roy. 
Met. Soc., 109, 23-55. 
Semazzi, F.H.M., V. Mehta, and Y.C. Sud, 1988: An investigation of the relationship 
between sub-Saharan rainfall and global sea surface temperatures. Atmosphere-Ocean, 
2 6 ,  118-138. 
O S C l U A l l O N  PERIOD (YEARS) 
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O s C l U A l l O N  FREQUENCY (RADIANS PER SECOND) 
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symmetric modes in the presence of basic state with motion. Only positive values are 
plotted. M is the zonal wavenumber and N is the number of north-south grid points 
including the poles. 2.5 degree north-south resolution. (a) Uncoupled ocean (b) 
Uncoupled atmosphere (c) Coupled ocean-atmosphere with nominal values of coupling 
coefficients. 
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Figure 2. Latitudinal structure of an axially-symmetric coupled ocean-atmosphere mode 
oscillating at 14.6 years period with amplitude e-folding time of 3.19 years. All 
variables are non-dimensional. Upper (lower) level variables are plotted with heavy 
(light) solid lines. u and v are zonal and meridional velocities, respectively. T is 
temperature. (a) Coupled atmospheric component. p is surface pressure. (b) Coupled 
oceanic component. h is layer thickness. 
Figure 3. Oscillation frequency (radtsec) . versus growth rate (Jsec) of axially- 
symmetric coupled ocean-atmosphere modes in the presence of basic states with motion. 
Only positive values are plotted. W s  the zonal wavenumber and N is the number of 
north-south grid points including the poles. 2.5 degree north-south resolution. H, N, 
and D denote eigenvalues for half, nominal, and double values of coupling coefficients. 
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On the Decadal Modes of Oscillation of an Idealized Ocean-atmosphere System

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