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with the advent of helioseismology and specially over last few years remarkable progress has been made in characterising the differential rotation of the sun .
analysis of the first 4 month s global oscillation network group ( gong ) data @xcite have qualitatively confirmed the results of earlier analyses that the surface differential rotation changes to rigid rotation below the outer 30% of the solar radius .
in addition , data from gong network and medium-@xmath0 program of michelson doppler imager ( mdi ) on board solar and heliospheric observatroy ( soho ) have allowed more detailed study of the rotational dynamics .
this is achieved due to the splitting of the sun s global oscillation frequencies by large scale flows .
details of this can be found in a recent review by @xcite .
@xcite studied the time variation of equatorial rotation rate for different @xmath0 ranges and suggested a small variation from year to year .
inversions for the rotation rate in solar interior have confirmed that the temporal variations in the rotation rate penetrate to some what deeper layers @xcite further , the changing pattern of solar rotation rate in the interior also agrees with the torsonal oscillations observed at the solar surface @xcite .
these zonal variations of the sun s differential rotation were first observed by @xcite using mdi _
f_-mode data .
@xcite have confirmed these zonal flows using both _ p_- and _ f_-mode splittings derived from mdi data . in this paper , we analyse the odd order splitting coefficients for detecting any periodic or systematic variation with solar cycle . extending the analytical formulation of @xcite to include coefficients up to 9th order
, we infer the solar ratation rate as a function of depth and latitude .
these rotation profiles are compared with those obtained from a 1.5d inversion technique .
we also estimate the variation in the rotation rate from the solar minimum to maximum period of the current cycle .
it is known that the solar differential rotation and other symmetry breaking factors like magnetic field can lift the degeneracy of the solar acoustic modes and split the eigen frequencies .
individual mode splittings can be represented by polynomial expansion @xmath1 where @xmath2 is the mean multiplet frequency , @xmath3 s are orthogonal polynomials of degree @xmath0 and the expansion coefficients @xmath4 are known as the splitting coefficients .
the odd - order coefficients measure the solar rotation while the even - order coefficients probe the symmetry about the equator .
the nonzero values of these even coefficients reflect the pole 
equator asymmetries in the solar structure @xcite .
we use data from both gong @xcite and mdi @xcite to determine the rotation rate in the outer convection zone .
each of the 45 gong data sets covers a period of 108 days and is centered on dates 36 days apart .
the month 1 starts from may 7 , 1995 and month 46 ends on december 23 , 1999 .
the mdi data consists of twenty 72 days time series starting from may 1 , 1996 and ending on august 31 , 2000 and includes the breakdown period of mid 1998 . in case of the gong data sets , splitting coefficients
are fitted up to @xmath5 while in mdi data , the fits obtain coefficients up to @xmath6 . in the present work
, we further restrict the frequency range to 1.5 @xmath7 3 mhz and @xmath8 70 .
@xcite studied the variation in gong odd - order @xmath9 coefficients but could not find significant correlation with solar activity measures .
later the same authors @xcite reported small trends up to @xmath10 coefficients . we have carried out a detailed analysis of these coefficients and observe that first few odd - order coefficients ( up to @xmath11 ) from mdi show significant correlation with activity measures ( see table-1 ) . however , gong odd - order coefficients reveal weaker correlation with activity .
the temporal variation in odd - order coefficients is shown in figure  1 .
we notice that there is no clear trend in higher order coefficients while lower order coefficients vary in a systematic way .
further , the @xmath12 coefficients from mdi reveal strong anti - correlation with activity while these from gong do not show any systematic trend .
.correlation statistics for odd order coefficients @xmath13 . [ cols="<,^,^,^,^,^,^ " , ]     [ tb1 ]      the solar rotation rate using the helioseismic data are generally obtained through two different methods : in _ forward approach _ the frequency splittings are computed for a chosen solar rotation model and then compared with the observed splittings . in the _ inverse method _ the measured frequency splittings are used directly to produce a single function for the angular velocity . in this study
, we use the analytical method of @xcite by including higher order splitting coefficients up to @xmath5 where the appropriate combination of odd order splitting coefficients reflects the depth variation of angular velocity at chosen latitude  @xmath14 ( see tripathy , jain , & bhatnagar , 2000 for details ) .
epsf    = 3.2 in = 3.2 in    = 3.2 in    it is conventional to express the observed rotation rate in terms of even powers of cos@xmath14 : @xmath15 where @xmath14 is the latitude and @xmath16 is restricted to 2 . with the availability of higher order @xmath9-coeffcients ,
we extend and retain terms up to @xmath16 = 4 . in the asymptotic limit
, we can express surface constants @xmath17 in terms of splitting coefficients as : @xmath18 , \\
a_2 & = & 21 a_5 + 99 a_7 + 286 a_9 , \\
a_3 & = & -\ [ { \frac{429}{5 } } a_7 + 572 a_9 ] , \\   a_4 & = &   { \frac{2431}{7 } } a_9 . \end{aligned}\ ] ]    various authors have calculated these surface coefficients using different data sets and considering only three terms in the expression ( 2 ) .
@xcite found @xmath19 = 462.8 nhz , @xmath20 = @xmath21  56.7 nhz and @xmath22  =  @xmath21  75.9 nhz for @xmath230.723  @xmath24 using caii k intensity data taken from south pole . based on 100-day observations made at bbso for @xmath0 between 10 to 60
, @xcite found the best fit with @xmath19  =  461 nhz , @xmath20 = @xmath21 60.5 nhz and @xmath22  =  @xmath21  75.4 nhz .
we have derived these constants from the gong data for the 70 @xmath25 @xmath0 @xmath25 150 and obtained the best fit with @xmath19= 460.55 nhz , @xmath20 = @xmath21  55.76 nhz and @xmath22 = @xmath21  154 nhz , @xmath26 = 220.36 nhz and @xmath27 =  @xmath21  164.43 nhz .
similarly , mdi data yields @xmath19= 458.43 nhz , @xmath20 = @xmath21  40.58 nhz and @xmath22 = @xmath21  182.81 nhz , @xmath26 = 223.893 nhz and @xmath27 = @xmath21  136.09 nhz .
the values obtained for @xmath19 are in close agreement with the earlier values .    in figure  2 , we plot the average surface rotation rate as a function of latitude using the derived coefficients . in the same plot ,
we have also shown the results from 1.5d rls inversion and doppler surface measurements @xcite .
it is clear that the rotation rate changes significantly in mid latitude while the change in rotation rate near pole and equator is small .
we find that the rotation rate derived from the analytical approach agrees well with other results .
however , the inverted rotation profile departs from other rotation rates beyond the latitude of 70 degree , probably due to the resolution limitation in inversion techniques .
figure  3 shows the residual surface rotation rate at two different latitudes obtained after subtracting the average angular velocity .
it is evident that at higher latitudes , the residual rotation rate , commonly known as zonal flows , is time dependent .
this has a magnitude of approximately 3 nhz at equator .
the residual rotation rate at equator for two different depths is calculated using 1.5d rls inversion @xcite and is illustrated in figure  4 .
it is clear that in the outer layers , the residual flows has a systematic variation while at deeper layers , the variation does not appear to be systematic .
recently @xcite , @xcite using gong data , and @xcite using mdi data , have also reported a small but significant time variation in the rotation rate .
the first few odd - order coefficients are found to vary systematically with the solar cycle .
we also detect a small but significant variation in the rotation rate derived from the linear combination of odd order coefficients over a period of four and half years .
the residual rotation rate in the outer layers seem to be correlated with the solar activity while no such signature is found at the base of the convection zone .
= 3.2 in    = 3.2 in
we thank h. m. antia for allowing us to use the inversion code .
this work utilises data obtained by the global oscillation network group project , managed by the national solar observatory , a division of the national optical astronomy observatories , which is operated by aura , inc . under cooperative agreement with the national science foundation .
the data were acquired by instruments operated by big bear solar observatory , high altitude observatory , learmonth solar obsrvatory , udaipur solar observatory , instituto de astrophsico de canaris , and cerro tololo interamerican observatory .
this work also utilises data from the solar oscillations investigation / michelson doppler imager on the solar and heliospheric observatory and we thank j. schou for providing us the data sets .
soho is a mission of international cooperation between esa and nasa .
nso / kitt peak magnetic , and helium measurements used here are produced cooperatively by nsf / noao ; nasa / gsfc and noaa / sel .
this work is partially supported under the csir emeritus scientist scheme and indo  us collaborative programme
 nsf grant int9710279 .
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we present results on variation in rotation rate in the upper convection zone using data from gong and mdi / soho covering a period of more than four years . 
 we find that the first few odd - order splitting coefficients vary systematically with the solar cycle . 
 the rotation rate near the solar surface calculated from analytical methods agrees well with that of inversion techniques . 
 the residual rotation rate in the outer layers seem to be correlated with the solar activity .    
 [ 1999/12/02 1.01 ( pwd ) ]