Wow, the silence is deafening! I must really be on to something here!

Mass Flux

The mainstream view of the GRS is that it is rather like a "pancake" (e.g., Marcus 1993, Simon-Miller et al. 2002), that is basically 2-dimensional and much less deep than it is wide. The view taken here, however, is that the GRS is more like a tornado: it is very much a 3-D creature, funnel shaped and penetrating to depths that rival the magnitude of its breadth.

On the theory presented here, the GRS is a plasmavore that feeds on metallic hydrogen at the plasma phase transition zone (PPT). Current models suggest the PPT--if it exists at all--lies between 0.9 and 0.78 R_J, or at about 7,000 to 16,000 km below the cloud deck. So, the GRS would have to reach that far down. But considering that the GRS is a 14,000 km X 24,000 km ellipse at the cloud deck, reaching the PPT is not so far-fetched.

The GRS has a peculiar morphology compared to other Jovian vortices in that it has an organized, rapidly moving "collar" that surrounds an inner turbulent zone. So the GRS can be envisioned as a sort of funnel shaped pipe that pumps material from deep within Jupiter up into the stratosphere. The inner dimensions of the pipe, as mentioned in a previous post, can be described by an ellipse whose semimajor axis is about 5,000 km and whose semiminor axis is about 2,000, with an area of about 3 X 10⁷ km². So it would be of interest to estimate the amount of upwelling through this pipe.

We can obtain an order-of-magnitude estimate of the mass flux through the center of the GRS if we assume an upwelling rate of 50 m s^-1. Since typical upwelling occurs at the rate of tens of meters per second, the 50 m s^-1 figure is not unreasonable. 500 m s^-1 would be the equivalent of over 1,000 mph, which greatly exceeds the maximum winds ever observed on Jupiter, and 5 m s^-1 is much too slow a figure, given the typical wind velocities found all over Jupiter.

Given an area of 3 X 10⁷ km² and a 50 m s^-1 average velocity, that works out to 1.5 X 10¹⁵ m³. Assuming a pressure of 1 bar and a temperature of 170^o K, from the ideal gas law (PV = nRT) we can estimate the molar flux:

10⁵ N 1.5 X 10¹⁵ m³ mole K
-------------------------- = 10¹⁷ moles s^-1
m² s 8.314 N m 170 K

Assuming a helium molar fraction of about 13.6%, a hydrogen molar fraction of 86%, with the rest made up by much heavier molecules, a density of 2.5 gm per mole of atmosphere is not unreasonable. So that yields a mass flux of 2.5 X 10¹⁴ kg s^-1.

Energy Flux

The formula for calculating kinetic energy is (½) mv². This works out to an energy flux of 3 X 10¹⁷ watts or 3 X 10²⁴ erg s^-1. That is a lot of watts. To put this figure in perspective, consider that about 8 X 10²⁴ erg s^-1 is the total radiation emitted by Jupiter of which 5 X 10²⁴ erg s^-1 consists of absorbed solar energy that is re-emitted. In other words, the kinetic energy flux flowing through the GRS is about the same as the total intrinsic power of the whole planet.

Conditions Prevailing at the Bottom of the GRS

Because pressure rapidly increases with depth within the interior of Jupiter, under the plasmavore model, the GRS will take on a pronounced funnel shape, as opposed to the mainstream pancake shape, but with a much more squashed appearance compared to the archetypical terran tornado. In an earlier post, I had estimated the area of the opening at the bottom of the vortex as about 1,000 km²--equivalent to a circle with a 35 km diameter. However, I had assumed that the ideal gas laws prevail in the interior of Jupiter--but the ideal gas laws do not prevail within gas giants. This fact makes the study of gas giants even more problematic and challenging than the study of stars, because at stellar densities, the ideal gas laws once again begin to work. The main problem is the equation of state of hydrogen. Determining the EOS is not an exact science, but the gas giants offer severe constraints that limit what the hydrogen EOS must be like. See Saumon, Chabrier, and Van Horn (1995) for an excellent discussion of EOS theory.

Current theory suggests that the density of the molecular envelope at the upper boundary of the PPT is maybe 0.7 gm cm^-3. Now, since energy and mass are conserved, and if the flow through the pipe that consists the GRS is approximately adiabatic and isobaric with respect to the surroundings, then the vertical velocity component at the mouth of the GRS must be the same as the vertical velocity at the top. We can calculate the volume flow from the given density (0.7 gm cm^-3) and given mass (2.5 * 10¹⁴ kg): that works out to 3.6 X 10¹¹ m³ s^-1. And from the volume flow and the known vertical velocity component, the area of the bottom must be: 3.6 X 10¹¹ m³ / 50 m s^-1 = 7.2 X 10⁹ m².

If the opening was round, that would work out to a 100 km pipe diameter! That's about 1% of the diameter of the opening at the visible top of the GRS. The shape would be rather like a one meter long straight trumpet-style bugle with a bell the size of a tuba.

Caloric Requirements of a Plasmavore

Clearly, there is much more kinetic energy wrapped up in the outer collar of the GRS than in the central upwelling zone. However, given the large sizes and low viscosities involved, there may not be very much friction for the rotating wall to overcome. And although there is a vast amount of kinetic energy stored within the whirling wall, since the material mostly goes in circles, little actual work gets done. So it's hard to calculate the ongoing energy requirements the whirling wall.

On the other hand, if the turbulent interior is in fact the interior of a pipe that's continously pumping fluid from the bottom of the molecular envelope up into the stratosphere, that represents real work requiring constant energy input, and the energy required to power that work must be accounted for. At a minimum, there must be enough power to accelerate 2.5 X 10¹⁴ kg s^-1 up to a velocity of 50 km s^-1 on the particular model being described here. In other words, the power source must be equivalent to the kinetic power going through the top of the GRS--3 X 10¹⁷ J s^-1.

On the theory that the GRS is a plasmavore, the required energy comes from the conversion of metallic hydrogen to molecular hydrogen. Saumon et al. (1995)'s Table 1. lists the ∆S across the PPT for various pressures and temperatures.

Let's assume that the temperature is 6,000 K (log[6000]=3.78), so that's the second line down, where it says the difference in entropy between the metallic and molecular states (∆S) is 0.590k_B. Multiplying by the gas constant R (which is equal to the product of Boltzmann's and Avogadro's constants) converts this figure into SI units, which equals 4.91 J mole^-1 K^-1. If we further assume that the Gibbs free energy of the phase change is zero (which is reasonable, since we aren't compressing any gasses), according to the standard formula for calculating the enthalpy (latent heat) of a phase change is:

∆H = T∆S

that yeilds:

6000^o K * 4.91 J mole^-1 K^-1 ~ 30 kJ mole^-1

Since 1 mole of protons is equivalent to 1 gm of hydrogen, the latent heat liberated by converting metallic hydrogen to molecular hydrogen is roughly 3 X 10⁷ J kg^-1 K^-1. Now we can calculate the mass flux of metallic hydrogen required for a power source:

3 X 10¹⁷ J kg
------------- = 10¹⁰ kg s^-1
s 3 X 10⁷ J

If we assume a helium mass fraction of ~0.25, then the total flow of material extracted from the metallic region is about 1.3 X 10¹⁰. This is only about 0.005% of the total mass flux through the interior of the GRS. Earlier I had suggested that perhaps all the material sucked out by the GRS comes from beyond the PPT. The above calculations show what a crazy idea that was. If all the hydrogen that was pumped started out as metallic hydrogen, that would generate ~ 5 X 10²¹ watts or about 10,000 times the Jupiter's intrinsic power.

How it Works

Very little material is being scooped up from the PPT. What happens is that like a waterspout, a small "cloud" of metallic hydrogen "droplets" will be kicked up a the base of the GRS. Ordinarily, within terran tornados, such debris tends to weaken the tornado, but with the GRS the effect is the opposite. As the metallic hydrogen droplets get sucked up into the GRS, they rise and the pressure decreases to the point where they will undergo the phase transition to molecular hydrogen, thus releasing their latent heat and reinforcing the convection.



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