The efficiency of escape of material that reaches the exobase is not necessarily the controlling factor determining atmospheric mass loss. For mass to escape from the exobase, it must first be delivered to the exobase, and in many circumstances the rate of transport of mass to the exobase is the limiting factor. When a minor consituent of an atmosphere is escaping, it must first diffuse through the dominant component on its way to the exobase, and even if the escape from the exobase is very effective, mass cannot escape faster than the rate it which it can diffuse up to the exobase. In such cases we can put an upper bound on the rate of escape without knowing much about the precise means of escape from the exobase. This upper bound is the rate of diffusion-limited escape. It has the virtue that it can be computed in a very simple and straightforward fashion.

We consider the diffusion in a gravitational field of a substance A with number density ni(r) through a background gas with density n(z) satisfying dn/dz = —h/H. The equilibrium distribution of A was determined earlier by setting the flux to zero, but now we will determine its distribution assuming a constant nonzero flux. If we let b be the binary diffusion parameter for substance A in the background gas, then Eq. 8.16 for the flux can be re-written i? 1 b b dnA j nA,e d nA ons

Ha n n dr n dr nA,e where nA,e is the equilibrium distribution of substance A, which satisfies dnA,e/dr = —nA,e/HA. As expected, the flux vanishes when nA = nA,e.

For the density distribution to be time-independent, the net flux through a spherical shell, 4nr2Fi(r) must be independent of r. We'll normalize this constant flux to the surface area, writing $ = (-/-s)2Fa as we did for the Jeans flux. For a given constant $, Eq. 8.20 defines a first order differential equation for nA. The upper boundary condition for this equation is applied at the exobase, and states that the flux delivered to the exobase must equal the escape flux from the exobase. The escape flux can be written w*nA(rex), where wt is the escape flux coefficient associated with Jeans escape or some other mechanism. Thus, the upper boundary condition can be written (rex/rs)2w*nA(rex) = $ . This determines nA(rex in terms of $, and we must then solve the equation to get nA(r) and adjust $ so that the lower boundary condition on nA at the homopause is satisfied. Now, when wt becomes very large, molecules are removed essentially instantaneously when they reach the exobase. In this case nA(rex) ^ 0 and we can take a shortcut to determine the limiting flux.

For simplicity we'll assume that the layer is isothermal, so that b is constant. Multiply Eq. 8.20 by n/nA,e and integrate from the homopause to the exobase to yield

which makes use of the assumption nA(rex) « 0. The integral involves only known quantities, so this equation defines the limiting flux in terms of the homopause density of the escaping substance. Let's suppose that the exobase has low altitude in comparison with the radius of the planet. In this case, gravity is nearly constant and n/ni,e varies like exp( —(1/H — 1/Ha)z) where z is the altitude. The ratio decays exponentially if the scale height of the background gas is less than the scale height of minor constituent, i.e. if the minor constituent is lighter than the background gas. If moreover the layer between homopause and exobase is thick enough that the ratio decays to zero at the exobase, then the integral is simply n/ni,e at the homopause divided by (1/H — 1/Ha). Therefore, under these assumptions, which are quite widely applicable, the diffusion-limited escape flux takes on the simple form

where D is the diffusivity at the homopause. The escape of the minor constituent cannot exceed this flux no matter how effective the escape mechanism may be.

Exercise 8.7.2 Derive an expression for the diffusion limited flux in the case when the diffusing constituent has greater molecular weight than the backgound gas. How does the limiting flux depend on the layer depth in this case?

As a first simple example of diffusion limited escape, let's take a look at the escape of hydrogen from an anoxic early Earth. Suppose that H2 diffuses through pure N2 above the homopause. At 300K, the binary parameter for this pair of species is 2 • 1021/ms. Then, noting that nH2/n is the molar concentration nH2 of hydrogen at the homopause and that the scale height for H2 is much greater than the scale height for N2, Eq. 8.22 implies that $ « (2 • 1021 /HN2 )nH2 = 2.2 • 1017nH2/m2s. We'll assume that H2 has nearly uniform concentration below the homopause,as is reasonable in the absence of oxygen. With this assumption, we can directly determine how high the concentration has to go in order to lose the volcanic hydrogen source, assuming the loss to be diffusion limited. Thus, nH2 = 1015/2.2 • 1017 = .0045. Thus, while Jeans escape would allow H2 to build up to very high concentrations in the tropospahere, diffusion-limited escape of H2 could hold the concentration to well under a percent. Of course, for the diffusion limit to be reached, we would need a much more efficient means than Jeans escape of removing H2 once it reaches the outer atmosphere. If the H2 were to dissociate into atomic hydrogen near the homopause,the diffusion limited escape flux would be much greater and the equilibrium diffusion limited concentration would be much lower, since the binary coefficient for atomic hydrogen diffusing through N2 is 1024 at 300K.

Exercise 8.7.3 Suppose that an N2/H2 atmosphere has some initial hydrogen concentration which is escaping at the diffusion limited rate, but which is not being replenished by volcanic outgassing or any other source. Compute the exponential decay time of the hydrogen in the atmosphere assuming (a) H2 diffuses through N2 above the homopause, or (b) H2 dissociates and diffuses as H through N2 above the homopause.

As a second example, let's consider diffusion limited water escape for a "dry runaway" state on a planet like Venus. For the dry runaway, we assume that the entire ocean is evaporated into the atmosphere as water vapor, and that in the absence of liquid water a dense CO2 atmosphere accumulates because of weak or absent silicate weathering. In this case, water may escape in the form of H2O diffusing through CO2. The binary parameter for this pair is 8.4 • 1020/ms at 300K based on the hard sphere approximation with 140 picom radii. Plugging in the appropriate scale heights for Venus Eq. 8.22 implies that $ « 7.8 • 1016nH2o/m2s, where nH2o is the water vapor concentration at the homopause. For the present application, we are interested in how long it takes to lose the hydrogen in an ocean, assuming it diffuses upwards as water vapor which then dissociates at high altitudes. We have seen earlier that nH2O is determined by the cold trap concentration in a dry runaway, which can range from miniscule values of a few parts per million to values approaching unity as wet-runaway conditions are approached. As a point of reference, let's assume that the cold trap concentration is 10%, which is typical of hot, moist conditions. Then, in one billion years, 2.4 • 1032 molecules of water could be lost per square meter of the planet's surface. This amounts to 7.2 • 106kg/m2, or just over a 7km depth of ocean. It would thus appear that diffusion limitation is not a major impediment to loss of an ocean if the cold trap concentration is 10% or more. When the cold trap concentration falls below 1%, though, the diffusion limitation becomes a serious bottleneck. On the other hand, if water dissociates near the homopause and hydrogen escapes by diffusing as H or H2, the diffusion limited escape rate becomes much greater, and lower cold trap concentrations can be tolerated without making it difficult to lose an ocean, as is illustrated in the following exercise

Exercise 8.7.4 For the conditions given in the preceding paragraph, compute the diffusion limited escape flux assuming water dissociates into H2 at the homopause, so that nH2 = Vh2o, the latter being the cold trap concentration. Compute the lowest value of nH2O that permits the hydrogen in a 7 km deep ocean to escape. Do the same for the case in which water dissociates into H (in which case nh = 2-qH2O). In both cases you may use the hard-sphere formula in computing the binary parameter b, assuming a collision radius of 140 picom for the H2 case and a collision radius of 7 picom for both colliding species for the atomic hydrogen case.

Further examples of diffusion limited escape are developed in the Workbook for this chapter.

The concept of diffusion-limited escape applies in a straightforward fashion only to the escape of a minor-constituent, which makes up a small proportion of the diffusing layer. The removal of substantial quantities of a major constituent from the top of the diffusing layer has the potential to create large, unbalanced pressure gradients, which would drive an upward mass flux far in excess of the diffusive flux. There is no magic concentration threshold defining a "major" constituent, but certainly one should begin to worry about the induced flow when the concentration exceeds 10% or so. This regime is largely unexplored territory, but is somewhat related to the hydrodynamic escape mechanism which we shall discuss a bit later in this section.

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