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Winds and the occultation experiment
Planet. Space Sci. 1974, Vol. 22, pp. 789 to 791. Pergamon Press. Printed in Northern Ireland
WINDS A N D THE OCCULTATION EXPERIMENT S. H. GROSS Polytechnic Institute of New York, Long Island Center, Farmingdale, New York 11735, U.S.A.
(Receivedinfinal form 19 November 1973) Abstract The occultation experiment may be used to obtain dynamic meteorologicalinformation when performed from an orbiting spacecraft. It is shown that interpretation of refractivity data in this fashion does not require a composition, as normally used to obtain pressure, density and temperature profiles. 1. INTRODUCTION The occultation experiment has enabled the measurement of pressure, density and temperature profiles of Mars and Venus (Kilore et aL, 1972). These profiles are obtained from measured refractivity data on assuming a chemical composition for the atmosphere. Most of the missions flew past the planets (Mariners 4, 5, 6 and 7), but Mariner 9 orbits Mars. An orbiting spacecraft may be used in this fashion to provide pressure, density and temperature fields over the planet from multiple occultations if the orbit precesses or retrogresses. If the orbital period is short compared with the meteorological time period of interest and the spacing between occultation points is small compared with the meteorological scale length of interest, then successive occultations may be compared to give mean dynamic information, such as wind speeds, over the time and spacing intervals. Aside from inherent inaccuracies in the measurements, computations based on pressure, density or temperature comparisons are somewhat uncertain, however, since the chemical composition needed to deduce such information is, at best, approximately known. Furthermore, uncertainties are introduced as well by the reduction procedure which converts refractivity information to pressure, density or temperature. Such uncertainties are unnecessary, since dynamical information may be extracted directly from refractivity data without assuming a chemical composition. In fact, all the hydrodynamic equations may, in the main, be put in a form where composition is not required, as long as the atmosphere is mixed. Thus, winds may be found by comparing refractivity information rather than pressure or temperature. It is the purpose of this paper to demonstrate that these statements are correct. 2. REFRACTIVITY, COMPOSITION AND WINDS Winds in meteorology are often approximated by geostrophic winds or gradient winds. These winds are related to horizontal pressure gradients through terms like 1/p(Op/Ox) and 1/p(Op/ay), where p is the density, p the pressure, x and y the horizontal coordinates in the east and north directions, respectively. The remaining terms in expressions for these winds depend only on velocity, rotation of the planet, latitude and radius of curvature, quantities not directly related to the composition of the atmosphere. Since p and p both depend on the assumption of a composition to extract them from refractivity data, the intention in this section is to show that 1/p(Op/Ox) and 1/p(~p/ay) are independent of composition and may be obtained from refractivity data alone. Then, if this is true, geostrophic winds or gradient winds may be found without assuming composition. The circulation theorem of Bjerknes and the vorticity theorem (Holton, 1972) also involve similar density-pressure expressions. The solenoidal term ~ (dp/p) is involved in lo 789
7,)0
S. I L G R O S S
tlm circulation equation and the term (l/p'-')V,,pxV,~p, where Vn is the horizontaI gra~.,e:~t operator, is in the vorticity equation. These solenoidal expressions are independe:~i ,,i composition, as may be demonstrated by methods similar to that given below (,)r 1/p(,:??;;).v) and 1/p(Op/Oy). The details for the solenoidal expressions will not be given here, though the method should be apparent. Thus, "~oth the circulation theorem and the ~,::~r i ,~,. theorem may be applied independent of composition, utilizing only refracti'~ity dat> The continuity equation may be similarly treated, since it may be written in tevm o~ expressions like t/p(Op/Ox), I/p(Op/Oy), etc. Identical statements are possible fo,. the temperature-heat or the entropy equations, except for complications arising from net heat accession terms. The implication is that, aside fi'om heat absorptiom latent hear ~md frictional effects, the dynamics cA"the atmosphere may be studied utilizing only refractivity data without assuming a composition.
3.-
lal, O R -l- O -. pay
A mixed atmosphere is assumed of mean mass /fi independent of position and given by ( l / n ) ~ nirni, where the summation is over all types of constituents each of number i
density n~ and mass m~ with assigned value for index i, and n is the total number density --: n~. n~ for each constituent varies with position; however, its mixing ratio to the total i
number density n = ~ n¢ is necessarily fixed by the assumption of a fixed mean mass. i
The refractivity anywhere is N = ~ (nJn~)N, where N~ is the refractivity of constituent i i at STP and n, is the number density at STP. Thus N~ and n, are constants, but N varies with position.
N--~indY'---n~'t~N, ,_ 11s
11s "
rl
from which n
N
rls
~ - - N,
~ , tli
(0
n
The total pressure
P __ gj'~'thgn dz,
(2)
(the hydrostatic assumption), the weight of atmosphere above, g is the gravitational acceleration and the vertical is taken as the z direction. Substituting for n from Equation (1) co
p = l~g (
nsN dz
Jz
.... lfign~ ~ ° N dz,
ni_N~ •
R
WINDS A N D T H E O C C U L T A T I O N E X P E R I M E N T
791
all quantities outside the integral being constant. The density ndhN p = mfi -- - -
(4)
i tl
using Equation (1). With O/(Ox, y) meaning the partial derivative with respect to x or y, on using Equations (4) and (3) one obtains 1 Op p Ox, 3,
n
l~gn8
_ _
0
Ndz--
nsrfiN ~ n~ N~ Ox, y
g
0
Ndz,
(5)
N Ox, y
i n
a quantity depending only on the refractivity N and its derivatives and integral. Thus, (1/p)ap/(ax, y) has been shown to be independent of composition. If N falls off nearly exponentially with altitude with some effective scale height H~ then fz ° N dz --= NH~, where
_l 0N1 1 H~ = I
N Oz J
"
Then Equation (5) may be rewritten p Ox, y
N Ox, y
"~ • Oz
(6)
Equation (6), in contrast with Equation (5), does not involve integration. However, one must determine ON/Oz to utilize the expression, which may be less accurate than Equation (5). In practice finite differences would be used to obtain derivatives. 4. CONCLUSIONS
It should be noted that the present form of the occultation experiment, as performed from a precessing, orbiting spacecraft, provides a two-dimensional rather than threedimensional refractivity field. Assumptions or approximations may be necessary to make up for the missing one dimensional variation. Aside from this limitation which is peculiar to the occultation experiment, it is possible to derive dynamical structure without resorting to any particular composition, an important simplification. The only mission for possible demonstration is Mariner 9, one that may be poor because of its long orbital period, 12 hr. The computation of winds from occultation measurements is a new use for the experiment which may prove of greater value in future missions. Acknowledgement--This work was supported by NASA under Grant NGR33-006-047.
REFERENCES HOLTON, J. R. (1972). An Introduction to Dynamic Meteorology. Academic Press, New York and London. KLIORE, A. J., CAIN, D. L., FJELDBO, G., SEmEL, B. L., SYKES, M. J. and RASOOL, S. I. (1972). The atmosphere of Mars from Mariner 9 radio occultation measurements. Icarus 17, 484-516.
WINDS A N D THE OCCULTATION EXPERIMENT S. H. GROSS Polytechnic Institute of New York, Long Island Center, Farmingdale, New York 11735, U.S.A.
(Receivedinfinal form 19 November 1973) Abstract The occultation experiment may be used to obtain dynamic meteorologicalinformation when performed from an orbiting spacecraft. It is shown that interpretation of refractivity data in this fashion does not require a composition, as normally used to obtain pressure, density and temperature profiles. 1. INTRODUCTION The occultation experiment has enabled the measurement of pressure, density and temperature profiles of Mars and Venus (Kilore et aL, 1972). These profiles are obtained from measured refractivity data on assuming a chemical composition for the atmosphere. Most of the missions flew past the planets (Mariners 4, 5, 6 and 7), but Mariner 9 orbits Mars. An orbiting spacecraft may be used in this fashion to provide pressure, density and temperature fields over the planet from multiple occultations if the orbit precesses or retrogresses. If the orbital period is short compared with the meteorological time period of interest and the spacing between occultation points is small compared with the meteorological scale length of interest, then successive occultations may be compared to give mean dynamic information, such as wind speeds, over the time and spacing intervals. Aside from inherent inaccuracies in the measurements, computations based on pressure, density or temperature comparisons are somewhat uncertain, however, since the chemical composition needed to deduce such information is, at best, approximately known. Furthermore, uncertainties are introduced as well by the reduction procedure which converts refractivity information to pressure, density or temperature. Such uncertainties are unnecessary, since dynamical information may be extracted directly from refractivity data without assuming a chemical composition. In fact, all the hydrodynamic equations may, in the main, be put in a form where composition is not required, as long as the atmosphere is mixed. Thus, winds may be found by comparing refractivity information rather than pressure or temperature. It is the purpose of this paper to demonstrate that these statements are correct. 2. REFRACTIVITY, COMPOSITION AND WINDS Winds in meteorology are often approximated by geostrophic winds or gradient winds. These winds are related to horizontal pressure gradients through terms like 1/p(Op/Ox) and 1/p(Op/ay), where p is the density, p the pressure, x and y the horizontal coordinates in the east and north directions, respectively. The remaining terms in expressions for these winds depend only on velocity, rotation of the planet, latitude and radius of curvature, quantities not directly related to the composition of the atmosphere. Since p and p both depend on the assumption of a composition to extract them from refractivity data, the intention in this section is to show that 1/p(Op/Ox) and 1/p(~p/ay) are independent of composition and may be obtained from refractivity data alone. Then, if this is true, geostrophic winds or gradient winds may be found without assuming composition. The circulation theorem of Bjerknes and the vorticity theorem (Holton, 1972) also involve similar density-pressure expressions. The solenoidal term ~ (dp/p) is involved in lo 789
7,)0
S. I L G R O S S
tlm circulation equation and the term (l/p'-')V,,pxV,~p, where Vn is the horizontaI gra~.,e:~t operator, is in the vorticity equation. These solenoidal expressions are independe:~i ,,i composition, as may be demonstrated by methods similar to that given below (,)r 1/p(,:??;;).v) and 1/p(Op/Oy). The details for the solenoidal expressions will not be given here, though the method should be apparent. Thus, "~oth the circulation theorem and the ~,::~r i ,~,. theorem may be applied independent of composition, utilizing only refracti'~ity dat> The continuity equation may be similarly treated, since it may be written in tevm o~ expressions like t/p(Op/Ox), I/p(Op/Oy), etc. Identical statements are possible fo,. the temperature-heat or the entropy equations, except for complications arising from net heat accession terms. The implication is that, aside fi'om heat absorptiom latent hear ~md frictional effects, the dynamics cA"the atmosphere may be studied utilizing only refractivity data without assuming a composition.
3.-
lal, O R -l- O -. pay
A mixed atmosphere is assumed of mean mass /fi independent of position and given by ( l / n ) ~ nirni, where the summation is over all types of constituents each of number i
density n~ and mass m~ with assigned value for index i, and n is the total number density --: n~. n~ for each constituent varies with position; however, its mixing ratio to the total i
number density n = ~ n¢ is necessarily fixed by the assumption of a fixed mean mass. i
The refractivity anywhere is N = ~ (nJn~)N, where N~ is the refractivity of constituent i i at STP and n, is the number density at STP. Thus N~ and n, are constants, but N varies with position.
N--~indY'---n~'t~N, ,_ 11s
11s "
rl
from which n
N
rls
~ - - N,
~ , tli
(0
n
The total pressure
P __ gj'~'thgn dz,
(2)
(the hydrostatic assumption), the weight of atmosphere above, g is the gravitational acceleration and the vertical is taken as the z direction. Substituting for n from Equation (1) co
p = l~g (
nsN dz
Jz
.... lfign~ ~ ° N dz,
ni_N~ •
R
WINDS A N D T H E O C C U L T A T I O N E X P E R I M E N T
791
all quantities outside the integral being constant. The density ndhN p = mfi -- - -
(4)
i tl
using Equation (1). With O/(Ox, y) meaning the partial derivative with respect to x or y, on using Equations (4) and (3) one obtains 1 Op p Ox, 3,
n
l~gn8
_ _
0
Ndz--
nsrfiN ~ n~ N~ Ox, y
g
0
Ndz,
(5)
N Ox, y
i n
a quantity depending only on the refractivity N and its derivatives and integral. Thus, (1/p)ap/(ax, y) has been shown to be independent of composition. If N falls off nearly exponentially with altitude with some effective scale height H~ then fz ° N dz --= NH~, where
_l 0N1 1 H~ = I
N Oz J
"
Then Equation (5) may be rewritten p Ox, y
N Ox, y
"~ • Oz
(6)
Equation (6), in contrast with Equation (5), does not involve integration. However, one must determine ON/Oz to utilize the expression, which may be less accurate than Equation (5). In practice finite differences would be used to obtain derivatives. 4. CONCLUSIONS
It should be noted that the present form of the occultation experiment, as performed from a precessing, orbiting spacecraft, provides a two-dimensional rather than threedimensional refractivity field. Assumptions or approximations may be necessary to make up for the missing one dimensional variation. Aside from this limitation which is peculiar to the occultation experiment, it is possible to derive dynamical structure without resorting to any particular composition, an important simplification. The only mission for possible demonstration is Mariner 9, one that may be poor because of its long orbital period, 12 hr. The computation of winds from occultation measurements is a new use for the experiment which may prove of greater value in future missions. Acknowledgement--This work was supported by NASA under Grant NGR33-006-047.
REFERENCES HOLTON, J. R. (1972). An Introduction to Dynamic Meteorology. Academic Press, New York and London. KLIORE, A. J., CAIN, D. L., FJELDBO, G., SEmEL, B. L., SYKES, M. J. and RASOOL, S. I. (1972). The atmosphere of Mars from Mariner 9 radio occultation measurements. Icarus 17, 484-516.