Supersonic meteorology of Io: Sublimation-driven flow of SO2

ICARUS 64, 375--390 (1985) Supersonic Meteorology of Io: Sublimation-Driven Flow of S021 ANDREW P. INGERSOLL, MICHAEL E. SUMMERS, 2 AND STEVE G. SCHLIPF Dioision of Geological and Planetary Sciences, California Institute of Technology, Pasadena, California 91125 Received August 19, 1985; revised November 4, 1985 The horizontal flow of SO2 gas from day side to night side of Io is calculated. The surface is assumed to be covered by a frost whose vapor pressure at the subsolar point is orders of magnitude larger than that on the night side. Temperature of the frost is controlled by radiation. The flow is hydrostatic and turbulent, with velocity and entropy per particle independent of height. The vertically integrated conservation equations for mass, momentum, and energy are solved for atmospheric pressure, temperature, and horizontal velocity as functions of solar zenith angle. Formulas from boundary layer theory govern the interaction between atmosphere and surface. The flow becomes supersonic as it expands away from the subsolar point, as in the theory of rocket nozzles and the solar wind. Within 35° of the subsolar point atmospheric pressure is less than the frost vapor pressure, and the frost sublimes. Elsewhere, atmospheric pressure is greater than the frost vapor pressure, and the frost condenses. The two pressures seldom differ by more than a factor of 2. The sublimation rate at the subsolar point is proportional to the frost vapor pressure, which is a sensitive function of temperature. For a subsolar temperature of 130°K, the sublimation rate is l0 ~5 molecules/cm2/sec. Diurnally averaged sublimation rates at the equator are comparable to the 0. l cm/year resurfacing rate required for burial of impact craters. At the poles where both the vapor pressures and atmospheric pressures are low, the condensation rates are 100 times smaller. Surface pressures near the terminator are generally too low to account for the ionosphere discovered by Pioneer 10. The possibility of a noncondensable gas in addition to SO: must be seriously considered. © 1985AcademicPress, Inc. INTRODUCTION This paper describes the flow of condensable gas from day side to night side of a frost-covered planet. The gas is SO2 and the planet is Io, but the model could apply to other planets with thin atmospheres like Triton and Pluto. We are interested in the global resurfacing rate and the distributions of surface pressure, atmospheric temperature, and flow speed. We do not consider atmospheric escape, photochemistry, or the volcanoes, except that some internal process is needed to recycle the SO2 that Contribution Number 3923 of the Division of Geological and Planetary Sciences, California Institute of Technology, Pasadena, Calif. 91125. 2 Current address: Department of Earth and Planetary Sciences, The Johns Hopkins University, Baltimore, Md. 21218. sublimes at low latitudes on the day side and finally condenses at the poles. The observational basis of the model is summarized by Fanale et al. (1982). Perhaps the most important single observation is the deep 4.05-/xm absorption in the reflectance spectrum of Io (Fanale et al., 1979; Howell et al., 1984). The wavelength of the feature is that of SOz, and it fits the frost better than it fits the gas. The depth of the feature is consistent with a surface that is 80% covered by frost. If the feature were due to SO2 gas above a frost-free surface, the gas would have to be supersaturated by a factor of 10 at the prevailing day side temperatures (Pearl et al., 1979). On the other hand, the low albedo at 0.32 to 0.42 /zm, where SO2 frost is highly reflective, suggests that at most 20% of the surface is covered by optically thick frost (Nash et al., 375 0019-1035/85 $3.00 Copyright © 1985by AcademicPress. Inc. All rightsof reproduction in any form reserved. 376 INGERSOLL, SUMMERS, AND SCHL1PF 1980; Nelson et al., 1980). The 4.05- and 0.32- to 0.42-p,m data can probably be reconciled if the SO2 frost and the 0.32- to 0.42-t~m absorber (sulfur?) are in intimate contact over most of the surface (Howell et al., 1984). Direct evidence of SO2 gas comes from the 7.4-t~m (1350 cm -1) absorption seen by the Voyager IRIS in the thermal radiation emitted by a hot spot (Loki) at 250 to 290°K (Pearl et al., 1979). The wavelength matches SO2 gas and not SO2 frost. The depth of the absorption matches a column abundance of 0.2 cm-atm, corresponding to a surface pressure of 10 -v bar if the gas is in hydrostatic equilibrium. At this pressure the SOz would be saturated at 130°K, which is typical of the area around the Loki hot spot at the time of the observation. On the other hand, SO2 absorption features do not appear in the disk-averaged IUE spectrum from 0.20-0.32 ~m (Butterworth et al., 1980). Analysis based on Beer's law implies that the average surface pressure is no more than 25% of that due to saturated SO2 frost with a subsolar temperature of 130°K. However, Beer's law probably underestimates the amount of SO2 present on 1o (Belton, 1982). Also, lowering the frost temperature by 5°K reduces the vapor pressure by a factor of 4, and yields a model that is compatible with the 0.20- to 0.32-t~m data. Thus both SO2 frost and SO2 gas are present on lo, but their abundance and distribution over the surface are uncertain. The Pioneer 10 radio occultation (Kliore et al., 1975) shows that Io has an ionosphere. From the electron density vs altitude one can infer the surface pressure subject to several assumptions, e.g., that the principal constituent is SO2 and the principal ionization sources are solar UV and magnetospheric electrons (Kumar, 1985; Kumar and Hunten, 1982; Summers, 1985). The results are consistent with surface pressures of order 5 x 10~° bar at solar zenith angles of 81.5 and 94.4 °. For comparison, the saturation vapor pressures of SO2 at temperatures of 100, 105, and i 10°K are 4 X l 0 -12, 3 x 10-11, and 2 x 10-10 bar, respectively. These are the highest temperatures one might reasonably expect near the terminator of Io. Thus the inferred pressures are 10 to 100 times the expected vapor pressures of SO2 frost at these locations. The occultation results suggest either that SOz is grossly supersaturated close to the terminator (solar zenith angle = 90°), or else that another gas with a higher vapor pressure is present at these locations. STATEMENT OF THE PROBLEM Fanale et al. (1982) and Trafton (1984) have discussed these planetary-scale, frostdriven flows in qualitative terms. At the subsolar point where the temperature is near 130°K, the vapor pressure of SO2 frost exceeds 10 -7 bar. At the terminator and on the dark side, the vapor pressure is orders of magnitude smaller. The horizontal pressure gradient drives a flow from day side to night side. The flow transports mass, and tends to lower the atmospheric pressure on the day side and raise it on the night side. Therefore at the subpolar point the atmospheric pressure is less than the vapor pressure of the frost, and frost sublimes. On the night side and near the terminator the atmospheric pressure is greater than the vapor pressure, and the gas condenses. As long as there is frost on the surface and solar energy to keep it warm, a steady pattern of flow follows the Sun as the planet turns. The hydrodynamic model developed below calculates this flow pattern subject to a few simplifying assumptions. Fortunately the solution is insensitive to the least wellknown parameters, e.g., those involving turbulent transfer between atmosphere and surface. A major simplification occurs if the temperature of the frost can be calculated in advance from radiative balance. Trafton and Stern (1983) and Trafton (1984) discuss the conditions under which this is a good approximation by assuming that the flow speeds do not exceed the speed of sound by too wide a margin. For SO2 on Io the ap- SUPERSONIC METEOROLOGY OF IO proximation is a good one. The atmosphere has only a small effect on surface temperatures, which are controlled mainly by radiation. The shortwave and longwave radiative fluxes at the surface are much larger than the latent heat of sublimation for these low atmospheric densities. For example, with an evaporation rate of 1015 molecules/ cm2/sec (see below) and a latent heat of 5 × I0 -z° J/molecule, the latent heat flux is 0.5 W/m 2. This flux is only a few percent of the incident sunlight at Jupiter's orbit. A second simplification occurs if the molecules behave as a continuous fluid that is hydrostatically bound to the planet. There are several conditions that must be met: The mean free path, which is about 20 cm at a surface pressure of 10-7 bar, must be much less than the pressure scale height H. Also H must be much less than the planetary radius r, which for Io is 1820 km. Finally, the kinetic energy of the flow must be much less than the gravitational binding energy o f l o . In fact H is about 9 km for SO2 at a temperature of 130°K in Io's gravitational field, so the first two conditions are met. The last condition is met if the flow speed does not exceed the speed of sound by too wide a margin. A third major simplification occurs if the flow is turbulent. This allows us to treat both the horizontal velocity and the entropy per unit mass as constants with respect to the vertical coordinate z. Modifications to this adiabatic, constant-velocity profile are confined to the thin atmosphere at high altitudes and to the thin boundary layer which controls exchange of energy, mass, and horizontal momentum between atmosphere and surface. Justification comes partly from the fact that the Reynolds number VHp/rI is large (greater than 105) over most of the flow. Here V is the horizontal velocity in a direction away from the subsolar point, p is the density, and "0 is the dynamic viscosity of the gas. Justification also comes from the fact that the turbulent diffusion t i m e - - t h e time for growth of the turbulent profile--is short compared to 377 the travel time across the disk. In other words H/V, is less than r/V, where the friction velocity V, is typically about 0.05 to 0.10 times V (Schlichting, 1979). This condition is satisfied at the 10 to 20% level, which is adequate for a first treatment of the problem given our ignorance about Io's atmosphere. A final simplification concerns symmetry about the subsolar/antisolar axis and neglect of Io's rotation. This simplification requires that the planetary rotation rate fl = 2zr/(1.77 days) be much less than V/r. For V -- 300 m/sec or twice the speed of sound the condition is satisfied at the 30% level, which is probably adequate for a first treatment. The model is described in the next four sections. It contains a large number of feat u r e s - d e t a i l s of the turbulent boundary layer, saturated adiabats, and interaction with the surface--that do not significantly affect the results. We felt it was easier to model these processes explicitly than to try to argue them away. However, the casual reader is advised to turn immediately to the results section, which is reasonably selfcontained. A more serious reader might read the two sections on basic equations and hydraulic jumps but skip those on surface fluxes and saturated adiabats. Anyone may, of course, read the paper straight through. BASIC EQUATIONS The problem reduces to finding the three variables P0, V0, To as functions of 0, the angular distance (solar zenith angle) from the subsolar point. Here P and T are pressure and temperature, respectively. Both P and T depend on the vertical coordinate z. The subscript zero denotes the value at the top of the boundary layer, which is assumed to be thin compared to the scale height H. Thus Po/g is approximately equal to the atmospheric mass per unit area and VoPo/g is approximately equal to atmospheric momentum per unit area, where g = 1.8 m/sec 2 is Io's gravitational accelera- 378 INGERSOLL, SUMMERS, AND SCHLIPF tion. The three equations expressing conservation of mass, m o m e n t u m , and energy are 1 rsin0d0 ' "(' rsin 0d0 d(I g ) VoPo sin O = mE, (1) ) g VgPo sin O _ _1 d f~; Pdz + r. r dO where S is e n t r o p y per unit mass. The fact that dS = 0 for vertical ascent in an adiabatic a t m o s p h e r e implies that W + • is independent of height. At the top of the b o u n d a r y layer, qb is close to zero compared with its value elsewhere, e.g., one scale height a b o v e the surface, so W + qb is equal to W0. T h e r e f o r e the integral (4) becomes (2) (6) Wo f~ pVdz = WoVoPo/g, r sin 0 dO \ 2 + CpTo)VoPo sin 0] = Q, (3) where mE, r, and Q are the rates of mass, m o m e n t u m , and energy transfer from the surface per unit area. The mass per molecule is m, and the specific heat at constant pressure is Cp. Both r and Q include a part that is associated with the mass flux m E as well as an eddy part that is independent of mE. In addition the m o m e n t u m equation (2) contains the gradient of the vertically integrated pressure on the right side. The left sides of Eqs. (1), (2), and (3) are the vertically integrated divergences of the horizontal fluxes of mass, m o m e n t u m , and energy, respectively. In (3) the term involving V2o/2 is due to kinetic energy. The term involving CpTo is the sum of internal and gravitational energies and work due to pressure forces. This particular form of the energy flux holds for any adiabatic fluid, provided CpTo is interpreted as the enthalpy per unit m a s s W evaluated at the top of the b o u n d a r y layer. T o show this, consider the vertically integrated energy flux (e.g., Haltiner and Williams, 1980) f ( p U •+ P + pdp) Vdz. (4) where p is density, U is internal energy per unit mass, and qb is gravitational energy per unit mass. The sum ( p U + P) is pW. F r o m the first law of t h e r m o d y n a m i c s and the hydrostatic equation dP = - p d ~ , we have 0 = TdS = d W - dP/p = d ( W + ap), (5) which agrees with the left side of (3) with Wo = CpTo. To integrate Eq. (2) it is useful to write the pressure gradient as the sum of a divergence t e r m and an undifferentiated term, so that the equation b e c o m e s r sin 0 dO _ 1 (~ Pdz + "r. (7) r tan 0 30 The quantity on the left is the m o m e n t u m flux divergence including the effect of pressure. The first t e r m on the right is the downstream (upstream) pressure force due to expansion (contraction) of the coordinate s y s t e m for 0 < 90 ° (0 > 90°). The final step is to relate the vertically integrated pressure to P0 and T0. For an ideal gas with an adiabatic lapse rate the vertically integrated pressure is proportional to CpToPo/g. ~ pdz = f:° ~ l g = f~°(Wo- w) l g = _1 Co (.t'. (Tb - T) d P = -! CpToPo/3. g Jo g (8) The last step m a y be taken as a definition of /3. F o r an adiabatic profile we have T = To(P/Po) mcp, where R = k/m is the SO2 gas constant and k is B o l t z m a n n ' s constant. Then/3 is given by R/(R + Cp), and Eq. (7) becomes SUPERSONIC METEOROLOGY OF IO I d [I (vg + flCpTo)Po sin 0] r sin 0 dO 1 - gr tan 0 flCpToPo + r. (9) Equations (1), (3), and (9) are the basic equations of the system. HYDRAULICJUMPS The equations may be integrated with respect to 0 provided E, Q, and r are expressible in terms of the dependent variables P0, V0, and To. These functions are discussed in the next section. Here we discuss the solution in general terms. To integrate from an angle 0 to a nearby angle 0 + A0, one starts with values of P0, V0, and To at the old location 0. One first calculates the right sides of Eqs. (1), (3), and (9). Next one calculates the fluxes, the bracketed terms on the left sides of Eqs. (I), (3), and (9) at the new location 0 + A0. Finally one solves for P0, V0, and To at the new location by means of the algebraic equations VoPo = f l , (10) (V g + t~CpTo)Po = f2, (11) (V2/2 + CpTo)VoPo = j%, (12) where f l , J~, J% are proportional to the known fluxes. Eliminating P0 and To yields a quadratic equation for Vo whose solution is V0 = [J~ -+ (f2 _ 2/3(2 -- [ 3 ) f l f 3 ) l / 2 ] / f l ( 2 -- [3). (13) The situation is reminiscent of rocket nozzles and the solar wind (e.g., Parker, 1958, 1965; Landau and Lifshitz, 1959). There is a positive (supercritical) branch and a negative (subcritical) branch, corresponding to the + and - in Eq. (13), respectively. The flow can always drop from the positive to the negative branch, leading to a stable, steady discontinuity. However, the only way to go from the negative to the positive branch is on the special solution that goes through the critical point, where 379 the radical of Eq. (13) vanishes. Then the two solutions coincide; the fluid properties behind the discontinuity are the same as those ahead of it. This is a weak, steady disturbance whose upstream propagation speed exactly matches the downstream flow speed. Solving for V0 when the radical vanishes therefore yields the signal velocity of the fluid: f2 V0--fl(2 - /~) - V~ + flCpTo (14) V0(2 - /3) ' Vg= (1_~flfl)CpT0 = RTo=gH. (15) We call the discontinuity an atmospheric hydraulic jump rather than a shock wave, because of the dominant influence of gravity in this hydrostatically bound system. To obtain the complete solution one starts at the subsolar point 0 = 0, setting V0 equal to zero and To equal to the temperature of the frost. One chooses a trial value for the atmospheric pressure P0 and integrates forward in 0 on the negative branch. If the trial value is too high the atmosphere will condense onto the surface. Conservation of mass then demands a reverse flow toward the subsolar point (V0 < 0), which violates the downstream boundary condition V0 = 0 at 0 = 180°. If the trial value is too low the frost will sublime at too high a rate. Conservation of momentum then demands an impossibly large pressure drop, which leads to negative values of the radical in Eq. (13). The correct value of P0 at 0 = 0 is the one that avoids both pitfalls. It is found by iterat i o n - b y repeatedly splitting the difference between the lowest " h i g h " solution and the highest " l o w " solution. By iterating to 16digit accuracy and restarting at intermediate values of 0, one can approach the critical point to arbitrary accuracy. After a small extrapolation across the critical point, one continues the solution on the positive branch. The downstream boundary condition V0 = 0 at 0 = 180° is usually satisfied automatically: Near the terminator (0 380 INGERSOLL, SUMMERS, AND SCHLIPF = 90 °) the surface pressure is usually so low that the gas is no longer a fluid. Molecules take a few hops of order one scale height, and then stick to the cold surface. Otherwise one could bring the flow to zero at the antisolar point (0 = 180°) by placing a hydraulic jump at the appropriate 0, the value of which depends on upstream flow conditions. Figure 1 shows the family of solutions, only one of which satisfies the downstream boundary condition. The ordinate is the Mach number, defined from (15) as M = V0[(I - fl)/(flCpTo)] w2. (16) The abscissa is the angle 0. All solutions start at the origin O with V0 = M = 0. The solutions OA have the radical less than zero and cannot be continued beyond point A. The solutions OB have V0 < 0 beyond point B, and do not satisfy the downstream boundary condition. The correct solution goes through the critical point C and joins to a supercritical solution D. There is only one such solution; it corresponds to a unique value of the pressure at 0 = 0. The topology of the solutions resembles that for the solar wind (Fig. 1 of Parker, 1965) except for the flow reversal at points B. SURFACE FLUXES Details o f the solution depend on interaction with the surface. The integration requires that we specify the vertical fluxes E, r, and Q as functions of the dependent variables P0, V0, and To. Specifying these functions is a messy business, but fortunately the most important quantities are the most straightforward ones, such as the relation between the pressure difference and the sublimation rate E. The solution is relatively insensitive to T and Q. The flux of molecules E is derivable from kinetic theory (Hirschwald and Stranski, 1964), and is proportional to the difference between the vapor pressure Pv of the frost and the atmospheric pressure P0 of the gas. The sticking coefficient a, analogous to the emissivity in radiative transfer theory, en- Number M O=© = O B Angle From Subsolar Point (e) ~ BB FIG. I. Family of solutions, similar to Fig. 1 of Parker (1965). The solutions OA do not exist beyond 0 = 0A. The solutions OB have inward velocities beyond 0 = 0B. Only the critical solution OCD can satisfy the downstream boundary condition. This solution is supersonic beyond 0 = 0c. The flow drops to zero at the antisolar point either by having all the molecules stick to the surface or else by means of a hydraulic jump. ters in the constant of proportionality. Disregarding temperature differences in the lowest mean free path of the atmosphere, we adopt the formula m E = ctp~vs(2~r)1/2(1 - - PolPv), (17) where ps is density and v~ is molecular speed ( k T / m ) 1i2 evaluated for a vapor in equilibrium with the surface frost. The heat and momentum fluxes Q and ~are obtained as functions of P0, V0, and T0 from measurements of boundary layers with suction and injection (Schlichting, 1979). We found no completely general formulas, and have adapted several to our own use. The fluxes are assumed to depend linearly on the quantities being transported, with the same positive transfer coefficients w~ and ws for heat and momentum: Q/ps = Wsqs - Waq,, (18) r/ps = W s U s - (19) WaUa. Here qs is W~, the enthalpy at the surface per unit mass of vapor; qa is (V2/2 + Wo), the sum of kinetic energy per unit mass and enthalpy per unit mass at the top of the atmospheric boundary layer. Similarly us is zero, the m o m e n t u m per unit mass at the surface; Ua is V0, the momentum per unit SUPERSONIC METEOROLOGY OF IO 381 mass at the top of the atmospheric bound- discontinuous. F o r ws we use the analogous expressions ary layer. Transfer o f heat and m o m e n t u m between atmosphere and surface takes place in two ws = - V e + 2Vd' Ve --< 0, (23a) w a y s - - a d v e c t i o n by the mean flow normal + + to the surface and turbulent exchange by eddies. Accordingly, we parameterize the W s ~--V e + 2Vd ' behavior o f the transfer coefficients Wa and Ve > O. (23b) ws in terms of two velocities, V~ and Va. When Vo[Vd ~ V2,, the heat flux Q beThe first velocity V~ = mE/ps represents the comes effect of the mean flow; it may be either positive or negative. The second velocity Q _ v2. v~ Vd = V2,/Vo represents the effect of the edPs V0 (qs - qa) + "~- (qs + qa). (24) dies, and is always positive. Here E is computed from P0, V0, and To using (17). The The first term on the right agrees with Eq. friction velocity V, is computed from P0, (23.15) of Schlichting (1979). The second V0, and To using formulas for turbulent term is at least reasonable; it says that in a boundary layers without suction or injec- boundary layer with weak suction or inject i o n - t h e case E = 0. These formulas are tion, the energy per unit mass advected across the layer is the average of the value given below. We have constructed simple functions of at the wall and the value at infinity. T o find Vd = vE/Vo, we use formula Ve and Vd that model the behavior of wa and ws in known limits. F o r V~ ~> Vd, advection (20.14) of Schlichting (1979), which can be from surface to atmosphere is the dominant written process, so ws ~ lie and Wa ~ 0. F o r - V~ >> V(z) = 2.5V. log(9.0zV.p/q). (25) Vd, advection from atmosphere to surface is dominant, so w~ --* - Ve and w~ ~ 0. F o r The numbers 2.5 and 9.0 are determined Ivol ~ v~, eddy transfer is dominant, so we from observations of turbulent boundary expand formula (21.26) of Schlichting layers in smooth pipes and o v e r smooth flat plates. We find V. by setting V(H/2) = Vo in (1979) to yield Eq. (25). F o r either an isothermal or an adi- r i p s = V 2 - V~Vo/2. (20) abatic atmosphere, the altitude z = H/2 is With us = 0 and Ua = Vo, Eqs. (19) and (20) approximately the point at which the logarithmic velocity is equal to its densityimply weighted average with respect to z. Using W a = V d - - VJ2, Ivol ~ Va. (21) this average is consistent with our use of V0 Two expressions that approach the right as the average m o m e n t u m per unit mass in limits for IV~[ large and agree with (21) for Eqs. (1) to (3). Equation (25) is solved iteratively by writing IV~I small are Vn+~ = V0/[2.5 log(9.0VnHp/(2"O))], V 2 - 2VdVe + 2V~ Wa = - V e + 2Vd Ve < 0, wa 2V 2 V~ + 2Va' V~ >-- O. (22a) (22b) These expressions have continuous first and second derivatives at V~ = 0, although the third derivative with respect to Ve is (26) where V, and V,+I are successive approximations to V,. Equation (25) is valid when the flow is turbulent. F o r laminar viscous flow we use the expression -~lps = (V,) 2 = ~qVol(pH/2). (27) In both (26) and (27), 7/is the dynamic vis- 382 INGERSOLL, SUMMERS, AND SCHLIPF cosity of S O 2 at the surface (Weast, 1975): • / = 1.3 x 10 6(T/To) 1/2 kg/m/sec, (28) where To = 273.16°K. As density falls, vl/p increases, and eventually V, calculated from (27) exceeds that calculated from (26). In the calculations we use the largest of the two values. The final results are insensitive to the factor H/2 appearing in (26) and (27); it can be doubled or halved with less than 1% change in the derived values of V0, P0, and To. Equations (25)-(27) give V, for smooth surfaces. We include the effects of surface roughness by increasing V~/Vo by an amount Crd that represents the contribution of roughness to the drag coefficient. Thus the final value of Vd is given by Vd = V~(s)/Vo + Cr~Vo, (29) where V,(s) is the friction velocity for smooth surfaces calculated from (25)-(27). The constant Crd may be positive or zero. In the Earth's atmosphere a value Cr~ = 0.01 is considered large (e.g., Priestley, 1959; Sellers, 1965). Schlichting discusses the influence of the Mach number on heat and momentum transfer. F r o m M = 0 to M = 4 the transfer coefficients vary by 50%. Our decision to neglect this influence is not that serious, however, since Q and ~- do not affect the solution in any major way. The main effect of the high velocity is the contribution of kinetic energy to the heat flux. This effect is included in our treatment, since q~ in (18) includes both enthalpy and kinetic energy. SATURATED ADIABAT In going from (7) to (9), we assumed that the atmosphere follows a dry adiabat. Such an assumption implies that the atmosphere is supersaturated above a certain level. Most of our examples are for this case. An alternate assumption developed in this section is that T(P) follows a saturated adiabat. Besides affecting atmospheric temperatures and energy budgets, the change affects the effective thickness of the atmo- s p h e r e - - t h e relation between the vertically integrated pressure and the surface pressure. This relation is the main point of feedback from the energy equation to the momentum e q u a t i o n - - t h e point at which temperature affects the dynamics. A mist of solid and vapor in thermal equilibrium can be regarded as a pure substance with 2 df, P and W, where W is enthalpy per unit mass of the mixture. For a saturated adiabatic profile, the three dependent variables are V0, P0, and W0. The three governing equations are (1), (3), and (9), but now CpTois replaced by W0 and/3 = R / ( R + Cp) is replaced by a function of W0 and P0. The remainder of this section describes the method used to calculate this function /3( Wo, Po). The third and fifth terms of Eq. (8) are a definition of/3. Thus we need (W0 - W) as a function of P under the condition S = So = constant, where So is entropy per unit mass of the mixture at the top of the boundary layer. The latter condition follows if the atmospheric vertical structure is adiabatic. We neglect particle fallout, so the profile is a true adiabat rather than a pseudoadiabat. The vapor c o m p o n e n t of the mixture behaves as an ideal gas: Wv = CpT, Wv0 = CpT0, [ (PoU0|]' S v - Sv0= Cplog T0\-P-! (30) (31) where Wv and S~ are the enthalpy and entropy per unit mass of the vapor, and Wv0 and Sv0 are their values at the top of the boundary layer. The temperature dependence may be eliminated by means of the saturation condition P = A e ,iT, which can be written T = -B/log(P/A). (32) Here A = 1.516 x 1013 Pa and B = 4510°K for SO: in this temperature range (Wagman, 1979). F r o m the relation TdS = d W - dP/p, the enthalpy and entropy of the mixture are related to that of the vapor by the relation SUPERSONIC METEOROLOGY OF IO S - Sv = ( W - Wv)/T, (33) since condensation takes place at constant T and P. At the top of the boundary layer, (33) becomes S o - Sv0 = (Wo - Wvo)/To. (34) We eliminate S from the above two equations using S = So = constant, and then solve for W. Subtracting W0 from the result we obtain, with the aid of (30), W 0 - W = W0(1 - T/To) + T(Sv - Sv0). (35) Equation (35) in conjunction with (8), (31), and (32) determines fl(Wo, Po). Thus in going from (7) to (9) the vertically integrated pressure is replaced by the integral of (W0 - W) with respect to pressure: fl-- WoPo = f~ P d z g = fe°(Wo - w) ldp. g (36) To evaluate the integral, the program uses a four-term asymptotic expansion in the small parameter 6 = - l / l o g ( P o / A ) , with T/To as the variable of integration. Since/3 is not constant, Eqs. (10)-(12) are solved by iteration: Calculate V0, P0, and W0 from (13) using the old value o f / 3 ; calculate a new /3 from (36) and repeat, holding the fluxes jq, f2, and f~ constant. 383 base 10 (log P) are plotted on linear scales at right. Negative values of E represent frost deposition. A single division along the right ordinate is equal to a factor of 10 change in the pressure. The units of P are dynes per square centimeter, so log P = 0 corresponds to a pressure of 10 -6 bar. Table I gives the parameter values. For all but one of the cases (Fig. 5) the temperature of the frost T~ follows an idealized radiative equilibrium model TF = (Tss - TAS) COS1/40 + TAS, (37) where Tss is the subsolar temperature and TAS is the dark side (antisolar) temperature. We use (37) in the range 0 ° -< 0 -< 80 °. Temperature falls linearly from its value at 0 = 80 ° to the value TAS at 0 = 100°. We set T~ = TAS for 0 > 100 °. The vapor pressure Pv of the frost is computed from its temperature [Eq. (37)] using the saturation condition (32). Figure 2 is a representative case with Tss = 130°K and sticking coefficient a = 1. This model describes a frost-covered planet whose temperature at the subsolar point is equal to that observed by Voyager IRIS (Pearl et al., 1979). The solution becomes supercritical (M > I) at 0 = 34 °, which is near the point where the mass flux changes TABLEI SUMMARY OF PARAMETER VALUES RESULTS OF THE CALCULATIONS Figures 2 through 9 show the results for some representative cases. The variables plotted are V0, To, P0, M, and E, but the subscript zero is omitted. Henceforth, V, T, and P are the velocity, temperature, and pressure at the top of the boundary layer. The velocity V, Mach number M, and temperature T are plotted on linear scales at left. The unit of each quantity, equal to a single division along the ordinate, is given by dividing the value shown in the figure by the n u m b e r o f divisions. The sublimation rate E and logarithm of the pressure to the Figure Tss TAS 2 3 4 5 6 7 8 9 130 130 130 130 120 120 120 120 50 50 50 90 50 50 50 50 Tss = TAS = a = S = Cra = a 1.0 1.0 1.0 1.0 1.0 0.1 0.01 0.01 and 1.0 S/D Crd D D S D D D D D 0 0.01 0 0 0 0 0 0 Surface temperature at subsolar point (°K). Surface temperature at antisolar point (°K). Molecular sticking coefficient. Saturated, D = dry adiabatic profile. Drag coefficient due to surface roughness. 384 INGERSOLL, SUMMERS, AND SCHLIPF V=Z520 m/s , T = 8 0 cosW4(O) + 5 0 K T , , , log p log P =0 E = 10t5 mol.,ec /cn~/s ! r=iOOK units of P: dyne /cm 2 M=4 E:O V,T~= ~5 30 45 60 75 Angle From Subsoler Point (deg) 90I°gP=-8 Fro. 2. Solution with subsolar temperature of 130°K. For other p a r a m e t e r s , see Table I. The velocity, temperature, and pressure at the top of the boundary layer are V, T, and P, respectively. The Mach number and evaporation rate are M and E. The scales are indicated along the ordinate. Thus, the units of V, T, M, and E are 40 m/sec, 20°K, 1 and 0.2 x 10~5molecules/cm:/ sec, respectively. The logarithm of P is to the base 10, so that log P = -8 is a pressure of 10 s d y n / c m 2. from subliming (E > 0) to condensing (E < 0). Atmospheric temperature falls somewhat faster than the surface temperature (not shown), because atmospheric thermal and gravitational energies are converted into kinetic energy during the expansion. Atmospheric pressure P is 3% less than the frost vapor pressure Pv (not shown) at 0 = 0 °, but P falls more slowly than Pv as 0 increases. The two pressures are equal when E = 0 a t 0 = 37 ° . In the supercritical regime, velocity and Mach number rise and pressure falls. The flow starts to run out of mass at 0 = 50 ° as a result of decreasing pressure; the deposition rate - E decays to zero beyond this point. At 0 = 70 ° the atmospheric pressure is 1.5 times the vapor pressure, and the mass flux velocity mE/p is 0.13 times the speed of sound. H o w e v e r , density is so low that the deposition rate - E is close to zero. B e y o n d 0 = 70 °, a drag crisis occurs. The pressure and density drop to such low vai- ues that the stress r becomes important. This change can be traced to the dependence o f t on the viscosity ~/p. Beyond 0 = 70 ° the velocity falls; the gas heats up as kinetic energy is turned into heat. By 0 = 82 ° the pressure is 10 -~j bar, at which point the mean free path is comparable to a scale height. B e y o n d 82 ° the flow ceases; the molecules stick after traveling one scale height, which is a fraction of 1°. At 0 = 82 °, the frost temperature for this model is 96.5°K and the frost vapor pressure is 10 ~2 bar. Figure 3 shows the effect of adding roughness drag (Cra = 0.01). In the Earth's atmosphere in neutral (adiabatic) conditions, such values are associated with forests, cities, and other aerodynamically rough surfaces (Priestley, 1959; Sellers, 1965). The velocity and Mach number are clearly smaller in Fig. 3 than in Fig. 2, especially in the supercritical regime. Temperatures do not drop as much in Fig. 3, because less thermal energy is turned into kinetic energy. Rates of sublimation and deposition are lower, although the associated difference in pressure between Fig. 2 and Fig. 3 is hardly noticeable on the logarithmic graph. Other runs (not shown) in which the constant H/2 in Eqs. (26) and (27) V=320/ m/s ROUGH SURFACE (CDRAG= 0.01) , r ~ _ ~ togP=O molec /cm2/s T:/OOK units of P: dyne /cm 2 M : 4- E:O V,T,M =0 0 15 30 45 60 75 Angle From Subsolor Point (dog) l o g P =-8 90 FIG. 3. As in Fig. 2 but with additional drag due to surface r o u g h n e s s (Cra = 0.01). SUPERSONIC METEOROLOGY OF IO LATENTHEATRELEASE V =320 - ~ , m/s logP =0 \ T=IOOK ~P E=IO 15 T /cmZ/s molec E units of P: dyne M=4 /cm z E=O V,T,M= 0 IogP=-8 I I 15 90 15 30 415 60 AnGle From Subsolor Point (deg) FIG. 4. As in Fig. 2 but the atmospheric profile follows a saturated adiabat. The flow receives additional energy from latent heat release. was varied by factors of 2 and 4 produced changes less than the line thickness. Figure 4 shows results for the saturated adiabatic model [Eqs. (30)-(36)]. The curve labeled T is Wo/Cp, where W0 is enthalpy. The saturated atmosphere (Fig. 4) releases energy of condensation while the dry adiabatic atmosphere (Fig. 2) does not. ThereStick I.O, T = 110+20xcos(O) K V=520 m'Is i i r I r IogP=O J 385 fore, more kinetic energy is generated, and velocities are higher in Fig. 4 than in Fig. 2. Again the pressure follows the frost vapor pressure to within a factor of 2. Figure 5 shows the effect of raising temperatures at the terminator. The subsolar temperature is the same as in Fig. 2, but the temperature at 0 = 90 ° is 110°K instead of 75.8°K. The main change is in the pressure P and evaporation rate E at 0 > 70 °. Since P follows Pv within a factor of 2, the higher frost temperature near the terminator causes P and E to increase dramatically. At 0 = 85 and 95 °, the values of P are 5 × 10 -1° and 2 × 10 -l° bar, respectively. These values are within the range of those inferred from the Pioneer 10 ionospheric profiles. The onset of free molecular flow (P -< 10 1~ bar) moves downstream from 0 = 82 ° in Fig. 2 to 0 = 110° in Fig. 5. Both P and E are essentially zero b e y o n d this point. The difference between Fig. 2 and Fig. 5 is due to the high value (Ts = 110°K) of the surface temperature at the terminator in the latter case. Such a high value seems improbable, although direct observations do not rule it out (Pearl et at., 1979). Figure 6 shows the effect of lowering the subsolar temperature Tss to 120°K. ComSTICK=I.O, T(8=O)=I20K ~ , , , , V=520m t E = Io '5 E jO14 ! mol_ec /COm~ffS T= M I OOF ~ /cm~/s / of P: 4 /cm dyne v''Xol Ioo Angle From Subsolor Point (deg) IoOgP=-8 FIG. 5. As in Fig. 2 but surface temperature falls gradually to a relatively warm 90°K at the antisolar point. log P=O l M=4 I V'T'oM. 0 / units I of P: I /cm dyne~ ~ . . . . . 15 5o 45 60 75 Angle From Subsolor Point (deg) ogP=-8 FIG. 6. As in Fig. 2 but the subsolar temperature is 120°K. The sticking coefficient is 1.0. Note change of scale for E. 386 INGERSOLL. SUMMERS, AND SCHLIPF pared to Fig. 2 the v a p o r pressure of the frost is lower b y a factor of 20; both P and E are lower by a factor of 20 as well. Neither the velocity V, M a c h n u m b e r M, nor atmospheric t e m p e r a t u r e T are strongly affected by the change, the main effect being to m o v e the onset point for free molecular flow u p s t r e a m toward the subsolar point. The curves V(O) and M(O) start o f f t h e same in Figs. 2 and 6. T h e y diverge only because the pressures in Fig. 6 are 20 times lower; the flow runs out of m a s s sooner in Fig. 6. The model was also run with Tss = I I0°K (not shown). C o m p a r e d to Fig. 6, the v a p o r pressure Pv as well as P and E are smaller by a factor of 20. Yet V and M are almost unchanged. When Tss = II0°K, the maxim u m value of M is 1.8. The supersonic nature of the flow is a general feature of the problem. Figures 7 - 9 show the effects of varying the sticking coefficient c~, with Fig. 6 as a reference case (Tss = 120°K in all four cases). N o t e the change of scale in the figures. In going f r o m a = 1.0 (Fig. 6) to a = 0.1 (Fig. 7), the e v a p o r a t i o n rate E only decreases by 25%. Referring to Eq. (17), the tenfold decrease of a is largely compensated by an increase of (I - P/Pv). Instead of P being 3% less than Pv at the subsolar STICK =0.1, T(O=O)= 120K IogP=O V :320 m/s E =4xld 3 molec /cmZ/s T=IOOK units of P: dyne,, /cm" M=2 E=O V,T,M =0 o L 20 4'0 8o 8'0 ,;o ,2JolOgP =-8 Angle From Subsolor Point (deg) FIG. 7. As in Fig. 6 but the sticking coefficient is 0. I. Note change of scale for E and M. V=I60 STICK= 0.01, T(8 =0) = 120K ~ ~ ~ ~ ~ rn/s t E =2 x I013 molec v T=IOOK M= 1 tlogP=O J//cm2//s [ units | of P: 4 dyne /cm t V,T,M =0 o ogP=-8 3o 60 90 12o 15o i Angle From Subsoler Point (deg) FIG. 8. As in Fig. 6 but the sticking coefficient is 0.01. Note change of scale for E and M. point as in Fig. 6, the difference (Pv - P)/P is closer to 25%. This 25% reduction in P causes a 25% reduction in dP/dO, V, M, and E. Reducing a by another factor of 10 to 0.01 (Fig. 8) causes a twofold reduction of P, dP/dO, V, M, and E c o m p a r e d to the a = 0.1 case (Fig. 7). In Eq. (17), a twofold decrease of P/Pv is required to c o m p e n s a t e for this hundredfold decrease of a. The corresponding twofold reduction in dP/dO causes slower velocities, smaller mass flux divergences, and smaller sublimation rates. The reductions of P, V, M, and E are much less than the hundredfold reduction of c~ from Fig. 6 to Fig. 8, however. Figure 8 is interesting because the velocity n e v e r quite goes supersonic. Also the pressure remains high (P -> 3 × 10 m bar) well onto the night side. Despite the large supersaturation, the a t m o s p h e r e cannot easily condense on the night side because a is so low. The result would be consistent with the a t m o s p h e r i c pressures inferred from the Pioneer 10 radio occultation experiment except that values of a tend to be large (a > 0.5) for polyatomic molecules impinging on cold surfaces (Bryson et al., 1974). Figure 9 describes a case where the frost SUPERSONIC METEOROLOGY OF IO STICK= 0.01 and 1.0, T(O=O)=I20K V=320 , m/s , , IogP=O , E:2xld 3 molec /cm2/s T =lOOK units of P: dyne /cm 2 M=2 E=O p a r a m e t e r i z a t i o n and a t m o s p h e r e - s u r f a c e interaction. G r e a t e s t sensitivity is associated with the t e m p e r a t u r e of the frost at the subsolar point. Although velocity and temperature are relatively insensitive, the quantities that involve the mass of the atm o s p h e r e - d e n s i t y , pressure, mass transport, sublimation rate, condensation r a t e - all vary as the v a p o r pressure of the frost, which is a sensitive function of frost temperature. DISCUSSION V,T,k' =0 387 0 15 30 45 60 7[5 IogP=-8 9o Angle From Subsolor Point (deg) FIG. 9. As in Fig. 6 but the sticking coefficient is 0.01 where the frost is subliming (E -> 0) and 1.0 where the frost is condensing (E < 0). This choice of parameters describes a mixture of frost and bare ground, with the frost occupying 1% of the area at a subsolar temperature of 120°K. is in small patches that c o v e r 1% of the area. N e a r the subsolar point where the frost is subliming (E -> 0), we set a = 0.01 to represent the p a t c h y coverage. When the frost is condensing (E < 0), we set a --- 1.0 to represent the fact that frost can form a n y w h e r e on the surface. The sublimation rate E near 0 = 0 ° is the same as in Fig. 8, but the flow runs out of m a s s much sooner b e c a u s e the condensation rate is so m u c h greater. A positive value of E is an average b e t w e e n patches of frost and bare ground. F o r a = 0.01, the e v a p o r a t i o n rate for the frost patches alone is I00 times greater than the value shown. Thus, a frost patch at the subsolar point sublimes at a rate 1.5 × 1015 molecules/cm2/sec according to this model. We s u m m a r i z e the results as follows: The flow speed is supersonic for all realistic values of p a r a m e t e r s . T h e surface pressure follows the frost v a p o r pressure within a factor of 2 despite d a y - n i g h t pressure ratios of 10 4 or more. A t m o s p h e r i c temperature is generally below the surface t e m p e r a ture as a result of d e c o m p r e s s i o n in the expanding flow. The solution is relatively insensitive to details of the b o u n d a r y layer T w o p r o b l e m s arise when the results of the model are c o m p a r e d with observation. The first concerns the mass budget of SO2. W h e r e does it c o m e f r o m and where does it go? W h y does Io not h a v e polar caps? H o w does frost deposition c o m p a r e with other m a s s exchange p r o c e s s e s ? The second p r o b l e m c o n c e r n s the night side ionosphere. The electron density profile seems to imply a surface pressure near the terminator that is orders of magnitude higher than the frost v a p o r pressure there. H a v e the profiles b e e n correctly interpreted? Is the night side t e m p e r a t u r e w a r m e r than currently believed? Is there another gas with a higher v a p o r pressure than SO2 on the night side of Io? The mass budget may be discussed by comparing the diurnally averaged evaporation rate to the global resurfacing rate, the e s c a p e rate, and other mass exchange processes. We c o m p u t e the diurnally averaged evaporation rate E as follows: E(x) = E [ c o s -1 (cos h cos 4))] d~b, (38) where k = latitude, 4) -- longitude, and E(O) is the e v a p o r a t i o n rate c o m p u t e d as a function of solar zenith angle as in the last section. Diurnally a v e r a g e d e v a p o r a t i o n rates are given in Table II. F o r a frost density of I g/ cm 3, an e v a p o r a t i o n rate of 1014 molecules/ cm2/sec c o r r e s p o n d s to 0.3 c m / y e a r of frost 388 INGERSOLL, SUMMERS, AND SCHLIPF nificant resurfacing process, at least at low and mid-latitudes. Yet other processes are DIURNALLY AVERAGED SUBLIMATION RATES also at work. The surface is dominated by t"igure [ ,atit ude local features centered around the volcaNo -nos, whereas atmospheric transport is a global process. Features that look the most 2 I.{R) 56 .30 .N) .4[1 .04 l} like frost deposits are small and bright, and .94 .47 .37 .65 29 .03 0 are located near volcanic vents (Strom and 4 125 .69 .tl .85 .4{~ .1)5 (I s 1.03 .g9 .2~ .74 .58 .28 .16 Schneider, 1982; Schaber, 1982). The most 6~ 5~ 20 .18 41 .18 0 l) obvious global-scale white areas are on the 7* 4t .2s .09 .~1 2~ .12 .()9 8" 15 1[) .01 I0 I~ 13 12 side of Io facing away from Jupiter (Ma9~ .15 I() ,01 II 14 0 [) sursky et al., 1979). The distribution of white material seems to reflect surface proN,m,. U n i t s are 10 t4 m o l e c u l e s / c m 2 / s e c t o t figure N o s 2 ~,, and arc I() I~ m o l e c u l e s / c m 2 / s e c for figure Nos. 6 9 I m a r k e d by a s l e r l s k ! cesses more than atmospheric processes. In other words, the Voyager images prothickness. The rates shown in Table II are vide little direct evidence that atmospheric therefore comparable to the resurfacing transport is shaping the surface. rate of 0.1 cm/year required for removal of It is difficult to place SO2 sublimation in impact craters (Johnson et al., 1979). Both the hierarchy of resurfacing processes, the SO2 sublimation rate and the required since the sublimation rate is so ill defned. resurfacing rate are uncertain by an order A key unknown in Table II is the frost temof magnitude, however. perature near the subsolar point. Knowing The diurnally averaged sublimation rate the frost temperature or the surface presis positive only within 25 ° of the equator. sure, one could calculate mass transports Both sublimation and condensation occur and sublimation rates, since the flow speed during the course of the day in this region, is almost the same for all models. Other key the peak values at noon being an order of unknowns are the fraction of surface covmagnitude larger than the diurnal averages. ered by frost and the frost albedo. All these Condensation rates are greatest at latitudes unknowns are measurable, and should have from 30 to 60 °, according to Table II. At high priority in future missions to Jupiter. still higher latitudes the condensation rate The night side ionosphere is difficult to is essentially zero (less than 10 ~ molecules/ explain with SO2 alone. If the interpretation cm2/sec in most cases). The absence of po- (Kumar, 1980, 1985; K u m a r and Hunten, lar caps on Io may be a reflection of the fact 1982; Summers, 1985) of the Pioneer data that the atmosphere does not go that far: are correct, the value of log P at 0 = 90 ° in The SO2 condenses out at mid-latitudes and Figs. 2-9 must be about - 3 (P = 10-9 bar). the flow ceases. Any SOz that does reach Several models come close, but they are the poles is either buried by volcanic sur- the most unrealistic ones. Figure 5 is one face flows, covered by volcanic plume de- such model, but it has TF = 130°K at the posits, or sputtered away by magneto- subsolar point, 110°K at the terminator, and spheric bombardment. For comparison, the 90°K at the antisolar point. The terminator escape flux o f SO2 required to populate the and night side temperatures are much magnetosphere is in the range 10 l° to 1012 warmer than what one would expect for a molecules/cm2/sec (Johnson et al.. 1979; surface like Io's. Figure 8 also has log P of K u m a r and Hunten, 1982). Sputtering of at- order - 3 at 0 = 90 °, but the assumed stickoms from Io by magnetospheric particles is ing coefficient is 0.01, which is much too likely to be in the same range. low for a surface below 120°K (Bryson et The present study indicates that atmo- al., 1974). One possibility is that another gas such spheric transport of SO: is a potentially sigT A B L E 11 SUPERSONIC METEOROLOGY OF IO as 02 is present on the night side. The gas would have to have a vapor pressure greater than 10-9 bar at Io dark side temperatures. The surface pressure would have to match the value required to explain the electron densities measured by Pioneer. 02 is a candidate because it is a photochemical by-product of SOz and has a high vapor pressure. If a background gas were present, the supersonic flow of SO2 would hold it on the night side behind a hydraulic jump. The location of the jump would depend on the pressure of the background gas. Figure 10 shows how this works for the flow pictured in Fig. 2. The solid curves show V, T, and log P in the upstream region ahead of the jump, and are the same as those of Fig. 2. The dashed curves exist only where the upstream flow is supersonic. For each angle O they show V, T, and log P in the downstream region immediately behind the jump, for a jump located at that same angle 0. The difference between the dashed curve and the solid curve is the size of the jump. V=520 m/s , T = 80 cosI/4 (0) + 50 K , , ~ log P - ~ IogP=O \ 389 Note that P and T increase and V decreases in going from the upstream to the downstream side. The increase in P is less than an order of magnitude in all cases. Although details have not been worked out, the pressure of the background gas is likely to be nearly constant with respect to 0. SO2 will condense out in the narrow zone behind the jump. Beyond this zone the flow velocity is zero, and therefore the pressure gradient OP/O0 is zero. Thus, the dashed pressure curve of Fig. 10 is a relation between log P of the background gas on the dark hemisphere of Io and the angle 0 at which the jump is located. For this and most other cases, the jump is located well in from the terminator on the day side of the planet. A key unknown is the absolute altitude of the occultation profiles (Kliore et al., 1975). The technology of radio occultation experiments has improved since the Pioneer days, and the measurements should be repeated. Ionospheric models should be repeated for gases besides SO2--particularly 02, N2, Ar, and CH4--those with high vapor pressures. Finally, a chemical model of the dark side background gas should be developed. In the case of 02, the supply of O and 02 crossing the jump must be balanced by the destruction of O and 02 by oxidation of S and loss to the magnetosphere. T=IOOK ACKNOWLEDGMENT This research was supported by the Planetary Atmospheres Program and the Planetary Astronomy Program of NASA. REFERENCES V,T=O 15 I 10 415 /0 715 9010gP=-8 Angle From Subsolor Poinl (dog) F1G. 10. C h a n g e o f V, T, and log P across a h y d r a u - lic jump. 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