R2s dr r2rEUv

The heating profile determines the flux (r/rs)2F(r), which is needed for use in the energy equation. The flux integration shows that when the heating is shallow, Qo « 1 FQ/rEUV. The calculations shown in the figure were done with rEUv/rs = 1, yielding a shallow, low-level heating.

Looking at the results, we see that when $/$* is made large, the temperature is monoton-ically decreasing, and at very large escape fluxes the curve looks like the adiabatic escape solution as expected. As $*/$ ^ 1, however, the temperature at the base falls toward zero, and the temperature rises from cool values to a maximum before decaying approximately adiabatically toward the critical point. At lower values of $/$*, solutions fail to exist in the purely radiatively heated case. Although the energy constraint expressed in Eq. 8.37 permits lower escape fluxes when the velocity at infinity is nonzero, the resulting flows cannot be made to satisfy the transonic rule in the configuration under investigation.

The cool-base solutions constitute the desired regime for hydrodynamic escape of hydrogen from Earth or Venus. In these solutions, EUV heating takes the hydrogen entering at the cool temperatures of the lower atmosphere, and heats it as it flows outward, to the point where it is hot enough and far enough out to escape. The EUV heating region is like the boost phase of a rocket, where burning of fuel launches the rocket and accelerates it to escape velocity.

What we have achieved by introducing radiative heating is the ability to sustain hydrody-namic escape with realistic low level temperatures. For the cool-base solutions, the escape flux is on the order of $*, so we can use this quantity to estimate the significance of the escape flux. For Earth conditions, where Fq « .004W/m2, $* = 1.6 ■ 10-nkg/m2s, corresponding to a flux of hydrogen atoms of 1016/m2s. This greatly exceeds the Jeans escape rate we computed earlier, and is


13 800

H 400

Figure 8.5: Temperature structure of an excaping H2 Earth atmosphere subject to radiative heating at low levels. The critical point is held fixed at r/rs = 30. The numbers on curves give the value of the nondimensional escape flux, r/r

Figure 8.5: Temperature structure of an excaping H2 Earth atmosphere subject to radiative heating at low levels. The critical point is held fixed at r/rs = 30. The numbers on curves give the value of the nondimensional escape flux, more than sufficient to keep up with the volcanic outgassing. For the Venus case, where Fq « .008 owing to the closer orbit, we find a mass flux of 3.72 • 10-11 kg/m2s, or an atomic hydrogen escape flux of 2.24 • 1016/m2s. In a billion years, this could get rid of the hydrogen from a 10km deep ocean (of course, one still needs to find something to do with the resulting oxygen). The reader should be cautioned that these numbers should not be taken as definitive; a more sophisticated calculation taking into account heat diffusion and a more realistic model of deposition of radiation could reduce the estimated fluxes. Still we have shown that hydrodynamic escape easily has the potential to get rid of the required amount of hydrogen on Earth or Venus.

In the presence of radiative heating, we can match the desired temperature boundary condition at the base of the escaping atmosphere by varying the escape flux with fixed rc, but the problem now is that we cannot independently specify the density there. For any given base tem-perature,this is fixed by the other parameters in the problem. Examination of the density solutions (not shown) reveals that in the family of curves given in Figure 8.5, the density decreases rapidly as the base temperature increases. In dimensional terms, at 250K the density is 2.5 • 1021 /m3, falling to 3.6 • 1020/m3 at 300K, and to 2.3 • 1019/m3 at 400K. The strong temperature dependence arises mainly because the density scale height is smaller for low temperatures, so that the atmosphere has to start with higher density in order to match with the density needed at the critical point. The additional degree of freedom needed to match the density condition at the base while keeping the temperature fixed at some desired value is provided by varying the critical point position rc. Some numerical experimentation, backed up by a bit of tedious algebra applied to the energy and entropy equations, reveals the following behavior for the density at the base:

• Except for the case of an ideal monatomic gas with y = |, the density at the base can be made arbitrarily large for fixed base temperature by moving rc outward; in fact, for large rc/rs the base density increases linearly with rc, with a slope that increases as the base temperature increases.

• For any gas, the base density can be made small by moving rc toward the surface, though for ideal monatomic gases the decrease in density only becomes effective as rc moves into the heating region. However, if the base density one is trying to match gets too small, then the required rc approaches the surface, meaning that the atmosphere escapes only because it is supersonic already at low levels - i.e. that it escapes by virtue of having been given high kinetic energy at low levels. These solutions are of little physical interest, as they do not patch to an atmosphere in hydrostatic balance at low altitudes.

• For ideal monatomic gases, solutions satisfying the transonic rule cease to exist once rc is moved moderately outside the heating region. This feature is connected with the insensitivity of Tb to the critical point position, expressed in Eq. 8.30 (applied to the base of the adiabatic region rather than all the way to the base of the heating region). As a result, monatomic gases appear to be subject to a maximum allowable base density. The implications of this peculiarity for the important case of monatomic hydrogen escape seem not to have been remarked on or explored in the literature. We speculate that when the base density is "too large," the flow tries to become subsonic out to infinity, whereafter upstream influence in some way acts to alter conditions at the base. Because of the way the EUV heating affects the transonic condition, the behavior of density in the monatomic case is likely to be highly sensitive to the radial distribution of the heating, and in particular to the extent to which some heating persists out to great distances from the planet's surface.

• For low base temperatures, the escape flux remains near the limiting flux and does not increase significantly as the base density increases. Increasing the base density in this regime moves the critical point outward, but does not increase the flux. This behavior contrasts with both the high-temperature regime and with Jeans escape, and comes about because the escape flux is constrained by the energy supplied by EUV heating.

Our understanding of hydrodynamic escape so far can be summarized in the following recipe for estimating the escape flux due to this mechanism: First compute the characteristic temperature T*, defined by cpT* = 2gsrs. Is the temperature at the base comparable to or greater than this value? If so, you can use the adiabatic escape formulation as an estimate, which yields the proportionality constant between the escape flux and the atmospheric density at the base, which can take on whatever value is dictated by the lower boundary condition. The proportionality constant was computed in Exercise 8.7.6. On the other hand, if the temperature at the base is much less than T* your atmosphere is in the low temperature limit, and EUV absorption plays a critical role in sustaining escape. In this case there is a threshold base density which must be exceeded in order to permit hydrodynamic escape from a level in the atmosphere which is nearly in hydrostatic balance. The threshold density must be determined by integrating the entropy equation inward from some chosen critical point, and moving the critical point inward until the velocity at the base becomes unacceptably large. This is the only hard part of the calculation. However, once you know that the base density exceeds the threshold, the escape flux in the low temperature regime can be estimated by simply evaluating the limiting flux, « | Fo,/gsrs, where Fq is the EUV flux impinging on the planet.

A rather startling aspect of the radiatively heated escape calculation is that the limiting escape flux is independent of the molecular weight of the substance making up the escaping atmosphere. From this, it would appear that a given amount of absorbed EUV could make a kilogram of oxygen escape just as easily as a kilogram of hydrogen. Continuum hydrodynamics appears to know nothing about the fact that the fluid is made up of discrete particles, so it is not immediately obvious how the molecular weight should enter the problem. In fact, the microscopic nature of the fluid enters through cp, which, through equipartion of energy, does know about the existence of particles. A kilogram of H2 has more degrees of freedom than a kilogram of O2, and this is reflectied in a higher value of cp for H2. The specific heat enters into the determination of the density at the base, and it is this density boundary condition which tells us that it is essentially impossible for heavy species to escape hydrodynamically, unless the temperature becomes very large. Using dimensional analysis, we can use our H2 escape calculations to make some inferences about what happens if we try to make a heavier species escape. The value of cp for atomic oxygen is nearly ten times smaller than the value for H2, so that the characteristic temperature T* for oxygen is more than ten times that of H2. That means that to achieve the same particle density for oxygen as was achieved at 250K in the H2 case, the base temperature has to be raised to over 2500K. An escape calculation with atomic oxygen shows that one can indeed achieve a temperature of 250K at the base by making $/$* sufficiently close to the limiting value, but when one does so the base particle density is 1032/m3 and the pressure there is over three million bars!. There is not likely to be any planet that can meet such a high density threshold.

Effects of heat diffusion

EUV heating is not the only diabatic effect of importance in the outer atmosphere. The other main heating and cooling effect that needs to be taken into account is the redistribution of heat by diffusion. Heat diffusion is much like diffusion of mass, and becomes strong when the mean free path is large. When the temperature gradient is dT/dr, the flux of heat is KdT/dr, where k is the same kind of thermal conductivity we encountered in earlier chapters, only this time applied to a gas rather than a solid. One of the many important effects of heat diffusion is that it allows heat from deposition of EUV at high levels to be diffused down to the lower atmosphere, which is often necessary to sustain hydrodynamic escape. Indeed, the radiative escape calculations we presented above in some sense implicitly assume some diffusion, since it is unlikely that enough EUV would penetrate to rb to allow the fairly strong heating we assumed there in our calculation.

In contrast with radiative heating, heat diffusion fundamentally changes the mathematical character of the problem. With radiative heating only, the energy equation 8.36 is an algebraic equation that is solved for the Mach number. It doesn't involve any derivatives. However, with heat diffusion, the heat flux F appearing in the equation is no longer just a function of r, but instead involves a term KdT/dr. Thus, diffusion changes the energy equation from an algebraic equation for the Mach number to a differential equation for temperature. This is true no matter how small the thermal conductivity, though the equation will become very ill posed when k is small. One can also see the singular effect of diffusion by examining the entropy equation. With only radiative heating, the highest order derivative is on the left hand side, and constitutes a first order differential equation for 0. With diffusion, the heat heating term on the right hand side has a contribution from diffusion, which introduces the gradient of KdT/dr. The entropy equation thus becomes instead a second order differential equation for T. The extra derivative can provide some additional scope for meeting both a temperature and density boundary condition at the base of the atmosphere, though this advantage is somewhat constrained by the fact that the heat diffusion is weak near the base, where the density is relatively large.

The second order diffusion equation based on the entropy equation can in principle be used as the basis of a solution method in the diffusive case. A more common approach to incorporating heat diffusion proceeds from a variation on the reasoning we used in deriving the transonic rule. Instead of relating the pressure gradient to the potential temperature, as we did in our derivation of 8.26, we can instead write p = pRT and use the same manipulations as previously to eliminate dp/dr using conservation of mass. In that case, we obtain the following alternate form of Eq. 8.26

dr dr dr r2

where cT = RT. The left hand side looks similar to the previous form, except that the isothermal sound speed cT (Problem ??) appears in place of the adiabatic sound speed. From this equation one would be tempted to draw the paradoxical conclusion that the critical point should be defined with regard to the isothermal sound speed rather than the adiabatic sound speed. This would be a fallacy. It's true that the right hand side must vanish where w equals the isothermal sound speed, but this does not give us a useful transonic rule since we do not a priori know the value of dT/dr. This contrasts with the original form, Eq. 8.26, where we know dO/dr from the entropy equation, and in particular that it vanishes if there is no heating near the critical point. But that's not the end of the story. If the diffusion is nonzero everywhere, then the energy equation, Eq. 8.36 can be solved for dT/dr in terms of w, T, r and the radiative flux; the resulting expression has no derivatives, and so when substituted into Eq. 8.41 it does not change the location of the singular point of the equation where the coefficient of the derivative vanishes. It would appear that the introduction of even infinitesimal heat diffusion discontinuously changes the mathematical character of the problem, so that we are left with a transonic rule that is applied at the critical point defined by isothermal rather than adiabatic sound speed. In the diffusive case, the most common approach to obtaining a steady solution is to simultaneously integrate Eq. 8.41 and Eq. 8.36, which jointly define a coupled set of differential equations for the pair (w,T). The integration is supplemented by a transonic condition applied at the point defined by the isothermal sound speed. This is a technically correct approach to the problem, but as a numerical scheme it is sure to become badly behaved in one way or another when the heat diffusivity becomes small. This may account for the difficulty some researchers have reported in finding consistent solutions to the escape equations in the presence of both heating and diffusion.

There is one special case, however, where one can take a short-cut to avoid the complexities and the problematic features of the integration sketched out above. When the heat diffusion is so strong that it keeps the outer atmosphere isothermal, then dT/dr vanishes, and Eq. 8.41 provides a usable constraint on the conditions at the critical point defined with reference to the isothermal sound speed. The isothermal case is worth pursuing, as it provides the opposite extreme to the no-diffusion case we considered previously, and thus serves to highlight the effects of diffusion. Numerical solutions to the full problem for Earth and Venus have temperature variations that somewhat resemble the cool-base nondiffusive cases shown in Fig. 8.5, so the isothermal limit cannot be relied on for an accurate estimate of the actual escape flux, however.

As for the adiabatic case, the isothermal case admits a very simple explicit solution for the escape flux. When the atmosphere is isothermal, we don't need to invoke the First Law of Thermodynamics to derive a conservation law from the momentum equation. Since T is constant, substituting p = pRT immediately yields the result

Note that this energy expression is a constant even though no explicit heating term appears in the equation. Moreover, cT is a constant as well, because the atmosphere is isothermal. By equating the values of this expression at the base and at the critical point, we find

, p(rc) 1 2gsrs ,rs rs. 1 ,rc 1 , ln =---^(— - —) — = —2( — - 1) — (8.43)

As usual, we have assumed that the kinetic energy is small at the base. The second equality arises from application of the isothermal form of the transonic rule. Eq. 8.43 links the density at the base to the density at the critical point once the critical point position is specified. The result implies that p(rc)/p(rb) gets exponentially small as the critical point is removed to infinity. Further, the escape flux is $ = p(rc)cT(rc)(rc/rs)2, so $ is known once the position of the critical point and the density there are known. So far, the nature of the isothermal solution is rather similar to the adiabatic case, in the sense that the base temperature and base density can be fixed by adjusting rc and p(rc), whereafter the escape flux is also determined. An important difference with the adiabatic case is that the base temperature can be made quite cool, given that the temperature is uniform and its value is set by the (low) speed of sound at the distant critical point. The quantitative behavior of the escape flux in a few illustrative cases is explored in Problem ??.

Because diffusion only redistributes heat and is not in itself a source of energy for escape, it would be bizarre if introduction of strong diffusion were to allow an atmosphere to escape at low temperature without any further conditions being met. Indeed, we are not quite done with this problem. The solution must be completed by making use of the energy constraint linking escape flux to EUV heating. For the isothermal case, this comes in via the entropy equation. For an isothermal medium ln 0 = —(R/cp)ln p + const. Further, since the temperature is constant, when Eq. 8.38 is integrated over an interval of r, the heating term integrates out to a difference in the fluxes at the endpoints. Putting the two results together, integrating the entropy equation from the base to the critical point, and dividing through by R yields the expression

p(r'b) 4 r"2 cTp r2 dr c r2 dr b where we have assumed that the net radiative flux vanishes at the base and becomes constant above rabs. rabs is also assumed to be below rc. Although the system is nearly isothermal, the diffusive heat fluxes out of the boundaries are not necessarily small, since the small (but nonzero) gradients are multiplied by a large diffusivity. If, however, we assume that the diffusion only redistributes heat provided by radiation and doesn't import heat from the lower atmosphere or export it through the critical point, then the last two terms on the right hand side of Eq. 8.44 can be dropped. Further, the isothermal transonic rule says that 2cT = gsr'2/rc, while Eq. 8.43 allows us to rewrite the log of the density ratio in terms of rc as well. With these substitutions, the flux becomes simply $ = 4FQ/gsrs, assuming rb/rs « rabs/rs « 1 and rc/rb ^ 1. The latter assumption is in fact necessary to assure that the kinetic energy at the base is indeed negligible, as was assumed in the derivation of Eq. 8.43. This flux is precisely the same as the low-temperature limiting flux derived earlier for the nondiffusive case. This satisfactory result provides a consistency check on our reasoning; that the two results should be the same was a foregone conclusion, given the requirements of energy conservation and the assumptions made regarding the energy available to drive the escape.

As in the low-temperature nondiffusive case, then, $ is fixed by the gravity and EUV heating, whence evaluating $ at the critical point implies that p(rc) decreases algebraically (specifi-cally,like r^/2) as the critical point is moved outwards. Eq. 8.43 then implies that the density at the base can be made arbitrarily large by moving the critical point outward, since the exponential growth of p(rb) trumps the algebraic decay of p(rc). Conversely,there are limits to how small the base density can be made by moving the critical point inward, since one ultimately violates the condition that the flow be nearly at rest near the base. The qualitative behavior is the same as for the nondiffusive case, in that the EUV flux fixes the escape rate, but that one must have enough density at the base if the escaping solution is to exist at all. Unlike the nondiffusive case, though, nothing special happens for monatomic gases.

Wrap-up and summary of hydrodynamic escape

An important feature of the blowoff state is that the escaping flow of a light gas like hydrogen can carry heavier minor constituents along with it, if the outward velocity of the hydrogen is greater than the characteristic fall speed of the heavy constituent. This works only if the concentration of heavy constituents is small enough that it doesn't increase the mean molecular weight of the gas sufficiently to choke off escape. To determine which species can escape, we note that the fall speed (relative to the background current) of a species with molecular weight M is wf = (R*T/Mg)(b/n), where b is the binary diffusion parameter for the heavy species diffusing through hydrogen and n is the number density of the hydrogen. For a heavy gas diffusing through a much lighter gas, the binary parameter is nearly independent of M, so that the fall speed is inversely proportional to M. Species heavier than a threshold value do not escape at all, whereas lighter ones escape at a rates which depend linearly on 1 /M - in contrast to Jeans escape, which depends exponentially on molecular weight. This differential escape implies a characteristic pattern of enrichment, which is most readily detected in noble gases like Xenon, which are not complicated by chemical reactions. The effect may not be very important for climate evolution, but it provides the main means of determining whether an atmosphere ever experienced a blowoff state in its past.

The one-dimensional treatment of hydrodynamic escape we have given may be elegant, but it suffers from a glaring physical inadequacy: The escape is energized by EUV absorption from the planet's star, which illuminates only the dayside. Yet, despite the fact that the thin outer atmosphere has little thermal inertia to even out the dayside/nightside constrast, the escaping flow has been modelled as spherically symmetric. In reality, the escape is likely to take the form of a complex three dimensional flow, with an outward directed jet centered on the subsolar point while some of the outward directed circulation on the dayside will close in the cold nightside exosphere instead of escaping to space. While more comprehensive treatments of hydrodynamic escape have added much sophistication in terms of atmospheric chemistry and radiative transfer, none at the time of writing have taken on the grand challenge of modelling the three-dimensional structure of an escaping atmosphere. Another challenge is that the traditional hydrodynamic escape formulation assumes local thermodynamic equilibrium and treats the fluid as a continuum, whereas the actual dynamics become nearly collisionless when one goes sufficiently far out in the atmosphere. If the transition to nearly collisionless dynamics occurs past the transonic point, this may matter little, given that information cannot propagate upstream in supersonic flow. However, if the transition occurs below the transonic point, the very notion of "speed of sound" breaks down and the controlling role of the transonic rule is likely to change substantially. Dealing with the transition from continuum to collisionless flow in this case is a considerable challenge. Given the nonlinearities and thresholds in the escape problem, it is likely that major revisions in our conception of escape rate are in store once somebody rises to these challenges.

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