An Analysis of Stochastic Jovian Oscillation Excitation by Moist Convection

The ideal scenario for driving Jovian oscillations occurs when the maximum possible moist convective energy is available and all of it is transferred to the oscillations. In such a scenario, we assume that all the energy of the cloud available for dispersion is kinetic, such that the energy density U = ρv². Since we also use a vertical cloud column entrainment model, we can find the kinetic energy per vertical length dz with

$$\begin{eqnarray}&&{dE}=\pi {r}^{2}U\,{dz}.\end{eqnarray} \tag{ 2 }$$

Integrating over the height of the cloud column then gives the total kinetic energy in the cloud.

The power generated by one of these cloud columns (storms) depends upon the time required for a parcel of air to rise from the cloud base to the maximum cloud height (≈80 km). Utilizing the vertical velocities calculated in Figure 2, the cloud column rise time for Jupiter's moist convection is on the order of 45–135 minutes. Given these times, Table 2 lists the kinetic power outputs of each storm as a function of radius. The α parameter is held constant here at 0.2. In addition, we also calculate the thermal power in the cloud in the same way as the kinetic, except our equation for thermal energy becomes (ignoring latent heat)

$$\begin{eqnarray}&&{dQ}={{mC}}_{p}{\rm{\Delta }}T=\rho \pi {r}^{2}{{dzC}}_{p}({T}_{\mathrm{cloud}}-{T}_{\mathrm{environment}}),\end{eqnarray} \tag{ 3 }$$

where m is the mass, C_p is the specific heat of water, and ΔT is the difference in temperature between the cloud and the environment. These results are also listed in Table 2. Regardless of cloud column radius, the kinetic power comprises ∼1.5% of the total power.

Gierasch et al. (2000) observed a Jovian moist convective storm 1000 km by 1000 km in area with a thermal power output of around 5 × 10²² erg s⁻¹ (5 × 10¹⁵ W) as determined from infrared observations. This value came from measuring the storm's thermal outflow within 1 scale height above 1 bar of pressure. Therefore, it is unclear what proportion of the total outflow energy comes directly from moist convection, and what comes from other sources like simple radiative cooling. The storm is likely comprised of many cloud columns of smaller radii, somewhere around 1–25 km as mentioned previously. Utilizing the thermal powers in Table 2 to match the observed power of the 1000 × 1000 km storm, six 25 km radius cloud columns or ∼39,000 1 km radius cloud columns would be required. However, Gierasch et al. (2000) measured the thermal flux above 1 bar of pressure whereas our thermal powers are integrated over the entirety of the cloud column down to 6 bars. Correcting for this discrepancy and only utilizing the thermal power above 1 bar in our models, 20 25 km or ∼135,000 1 km cloud columns would be required.

Jovian thunderstorms have finite lifetimes and their energy must be dissipated into the near-surface environment. It is worth noting that currently no working theory exists as to exactly how the storm energy and mode excitation are coupled. The precise understanding of the nature of the coupling most likely requires the inclusion of atmospheric microphysics, which is beyond the scope of this paper. However, on a more general scale, we hypothesize that the storm energy is transferred to the modes by the cloud column punching the upper atmosphere as it ascends. By analogy, this would be equivalent to a filled water balloon where someone flicks the balloon skin from the inside, thereby initiating the propagation of waves throughout the balloon. Assuming that the energy can and does transfer to global oscillations, then in order to conclude whether moist convection could drive them, two key points must be understood: (1) how much energy is required to drive the modes (which, to first order, could be found by calculating the energy needed to produce an assumed surface displacement for a given oscillation mode and frequency); and (2) how the energy dissipation is partitioned into thermal transfer, radiation, winds, oscillations, etc. The latter is currently unknown, but the analysis of the former reveals that it may not be necessary to understand in regards to oscillations.

Using a best-case scenario, we are interested in what the maximum possible energy supply can be from moist convection to Jupiter's global oscillations and whether that is sufficient to match the theoretical energies of the modes. Using the information in Table 2, the maximum surface power density from moist convection kinetic energy is ∼9 ×10¹⁶ erg s⁻¹ km⁻². Utilizing 1997 Galileo data of Jovian thunderstorms as a tracer for moist convection, Gierasch et al. (2000) measure moist convective storm surface density on Jupiter to be 0.66 × 10⁻⁹ km⁻². The surface area of Jupiter is 6.423 ×10¹⁰ km² (using an equatorial radius at 1 bar of 71,492 km). Assuming this surface density is temporally typical and a storm surface area of ∼10⁴ km² is typical (Gierasch et al. 2000 specify that these storms are typically a few hundred kilometers in diameter), then at any given moment there is approximately 42.4 moist convective storms occurring on Jupiter, or a total of 4.24 × 10⁵ km² of storms. Using the maximum kinetic surface power density, moist convection could then supply a total of ∼3.8 × 10²² erg s⁻¹ to the global oscillations, assuming all available energy is used in the forcing. Keep in mind this value is a gross overestimate as not all of the observed storm area corresponds directly to a rising cloud column. As noted at the end of Section 3.1, a large storm will consist of several separated cloud columns and thus a large amount of the storm surface area does not contain a rising cloud column directly beneath it, such as ∼20 25 km radius cloud columns for a 1000 km² storm, which would only contribute ∼3.3 × 10²¹ erg s⁻¹ to the modes.

Gudkova & Zharkov (1999) calculate theoretical estimates of the energies for these Jovian global oscillations. For the fundamental n = 0, ℓ = 2 mode, they find that the energy in this mode per period is 6.4 × 10²⁶ erg when a 1 m surface displacement is assumed. For the various models utilized to arrive at this value, the period ranges from 75 to 1600 minutes. The ideal scenario for driving oscillations would be a minimum mode energy, thus the longest period results in a power of 6.67 × 10²¹ erg s⁻¹ for this particular mode.

At first glance, it appears that there is enough kinetic power available to drive the modes. However, while the n = 0, ℓ = 2 mode, for example, requires an average power of 6.67 × 10²¹ erg s⁻¹ to survive, the storms do not produce the aforementioned kinetic powers, on average, as the kinetic power is only produced over the rise time of the cloud column, or about 45 minutes for a 25 km radius cloud. If we assume that a 1000 km × 1000 km storm erupts every day on Jupiter, then when we examine the total power required to drive just the n = 0, ℓ = 2 mode, the kinetic energy from the storms falls two orders of magnitude short of the required driving energy.

In addition, the actual Jovian surface displacements for global oscillations are unknown. Thus, this is a free parameter in the calculation of mode energies. Gudkova & Zharkov (1999) compute these energies assuming a surface displacement of 1 m. Using the shortest mode period, this translates to surface velocities of the order of 0.02 cm s⁻¹. However, Vorontsov et al. (1976) and Bercovici & Schubert (1987) predict global oscillations to have surface velocities of 10–100 cm s⁻¹. For the 75 minute mode period, this results in surface displacements on the order of 50–100 m. Observational efforts place surface amplitudes at 49₋₁₀⁺⁸ cm s⁻¹ (Gaulme et al. 2011). Therefore, the 1 m estimate is a conservative estimate, adding to the disparity between moist convection kinetic energy and mode energies.

Given this analysis, if all the kinetic energy from moist convection does excite modes, the lowest fundamental mode possible to be excited would be of angular degree ℓ ≈ few tens (according to the predictions of Vorontsov et al. 1976), which would require all the energy to drive this singular mode, a mode with a surface amplitude much too small to observe. Higher-order radial modes could also be possible (n > 10), again requiring the entirety of the moist convective kinetic energy. A far more likely scenario would be a partitioning of energy to excite several high order modes, yet it is likely that none of these modes would be even close to observable limits. To reiterate, this scenario assumes maximum storm kinetic energy, minimum mode energy, 100% energy conversion between the two, every square kilometer of a moist convective storm carries kinetic energy readily available to the oscillations, and a highly conservative surface displacement of 1 m. Therefore, we can conclude that moist convective kinetic energy alone is insufficient to immediately excite any fundamental, low-order, observable global oscillations.

As a final note, Bercovici & Schubert (1987) estimate the power of turbulence from kinetic motions radiated into acoustic waves to be described by

$$\begin{eqnarray}&&W=\displaystyle \frac{\rho {u}^{3}}{\lambda }({M}^{3}+{M}^{5}),\end{eqnarray} \tag{ 4 }$$

where u is the eddy velocity, λ is the eddy wavelength, and M is the Mach number. If we use Equation (4) with our cloud models having velocities of 10–50 m s⁻¹ and the sound speed within Jupiter being ∼1 km s⁻¹, we find that only 10⁻⁴%–10⁻²% of the kinetic cloud energy actually gets converted to acoustic modes due to the very small Mach number (10⁻⁶ to 10⁻⁴). If this is truly the case, then moist convective kinetic energy most certainly cannot excite Jovian modes.

The modes we utilize in this simulation are a subset of eigenfunctions, both the radial and horizontal components, and their corresponding eigenfrequencies calculated from state-of-the-art Jupiter models (Jackiewicz et al. 2012). We only consider those with angular degrees ℓ = 0–15 and radial orders n = 0–10. The temporal eigenfrequencies span from ∼0.1 mHz to ∼2.5 mHz, commensurate with the expectations of low-degree oscillations of Jupiter.

To establish a proxy for the initial energy of a given mode that takes into account its inertia we consider

$$\begin{eqnarray}&&E=\pi {\omega }^{2}{\int }_{0}^{R}[{| {\xi }_{r}(r)| }^{2}+l(l+1){| {\xi }_{h}(r)| }^{2}]\rho {r}^{2}{dr}.\end{eqnarray} \tag{ 5 }$$

Here, ξ_r and ξ_h are the radial and horizontal eigenfunctions, respectively, of the mode in question and ρ is the radial density profile of the Jupiter model. As is common, the eigenfunctions are normalized at the surface according to the boundary conditions. At each timestep, once the energy imparted to each mode is computed, its radial (surface) amplitude is determined according to

$$\begin{eqnarray}&&{\xi }_{r}(R)=A\tilde{{\xi }_{r}}(R)=A,\end{eqnarray} \tag{ 6 }$$

where $\tilde{{\xi }_{r}}$ is the eigenfunction that is arbitrarily normalized to one at the surface. The ratio between the radial and horizontal eigenfunction amplitudes is fixed by the solution of the oscillation equations. In what follows though, we only consider A, the surface displacement in real units of the mode in question.

The eigenfunctions we use are not necessarily meant to be the most representative of the planet, as we have ignored higher-order effects due to rotation in the computation. As has been shown in similar efforts, rapid rotation causes subtle, but important changes to the eigenmode solutions of giant planets (Fuller 2014). However, these functions provide a good approximation of how the oscillation inertia is distributed throughout the interior of the planet.

Having computed the available kinetic energy from the storms as discussed in Section 3.2, we now specify when and how often the storms occur. Each storm is described by two basic timescales, t_load and t_storm. t_storm is the time over which the cloud column rises from the cloud base to its maximum height and t_load is the time between these rising cloud excitation events. A storm is triggered at t = 0 and then each subsequent storm occurs at a later time randomly sampled between t_load,max and t_load,min.

Each time a storm occurs, the energy available to the modes is the power from the cloud column(s) with a specified radius multiplied by t_storm. We assume energy equipartition where half of the available energy is transferred to thermal dissipation or bulk motions, or as Ingersoll et al. (2000) conclude, is returned to the South Equatorial Belt or Great Red Spot, and the other half is equally split between all the modes. However, the energy partitioned to a particular mode is not always added to the energy already contained within that mode. For any given mode excited with nonzero energy, we prescribe a phase, ϕ, to it that oscillates sinusoidally in time between 0 and 2π at the mode's eigenfrequency. So when a storm erupts, if E_m is the energy partitioned to a particular mode, then the additional energy added to the mode is ${E}_{m}\cos \phi$. This could result in modes being damped by some storms rather than amplified, thereby converting mode energy into thermal energy. This is a way to model the effect of "pushing the swing" at the wrong time. If at any point the mode's energy becomes negative it is reset to zero energy and the phase is reset to zero.

We also run a suite of simulations for a distribution of storm energies. We create a Gaussian distribution with a mean of the minimum possible storm energy and a standard deviation equal to one-third of the maximum possible storm energy such that the maximum storm energy, or largest storm, will be 3σ away from the mean. When a storm erupts in the simulation, its energy is randomly chosen based on this distribution and only considers values at or larger than the mean.

To account for energy dissipation, we make use of the quality factor, Q, which parametrizes how well a particular mode "rings" within Jupiter. A strongly frequency-dependent Q is considered as well as a constant one. Following the work of Mosser (1995), we use an approximate exponential relation given by

$$\begin{eqnarray}&&{\mathrm{log}}_{10}Q=9-2\nu ,\end{eqnarray} \tag{ 7 }$$

in the range of ν = [0, 3] mHz, where ν is the temporal eigenfrequency of the mode. For example, Q ≈ 10⁷ for a 1 mHz mode. We also consider simulations where we simply fix Q at a constant value of 10⁵ for frequencies below 3 mHz as suggested by Stevenson (1983). The Jovian atmospheric acoustic cutoff frequency has been estimated to be around 3.5 mHz, but the low-degree modes we consider are below this value.

Once values for Q are determined for each mode, its energy loss over one oscillation cycle is given by

$$\begin{eqnarray}&&Q=\displaystyle \frac{E}{{\rm{\Delta }}E},\end{eqnarray} \tag{ 8 }$$

where E is its energy and ΔE is the energy lost. Q follows from either Equation (7) or the constant relationship.

This process allows us to create a time-dependent map of the growth and/or decay of mode energies, mode amplitudes, and the overall surface displacement of the planet (i.e., the sum of the mode amplitudes). We then embed the code in a probabilistic distribution (see below) simulation, which will run the entire process for a specified number of times and record the maximum energy achieved by each mode per run, the maximum amplitude of each mode per run, and the maximum planetary surface displacement per run. We run each simulation for 5000 days, which is a more than adequate amount of time for the oscillations to find a quasi-equilibrium based on various tests that were performed. Each simulation is run 200 times to build the probability distribution of maximum achieved mode energies and amplitudes. The final results allow us to thus analyze the mean of this stochastic process.

Our overall strategy puts constraints on moist convection storms based largely upon theoretical estimates and observational limits of mode amplitudes and energies. In general, we consider the radial ℓ = 0, n = 1 mode as a proxy for all modes since it has the largest energy because its mode mass is the largest. If its theoretical energy prediction of 10²⁶–10²⁷ erg is failed to be met (Vorontsov & Zharkov 1981), then we conclude the specific parameters in that moist convection scenario to be unable to successfully drive modes. In conjunction, the cumulative surface displacement of the planet from all the modes shall be no smaller than 10 m, a lower limit commensurate with the observed velocity amplitude of Gaulme et al. (2011).

In the analysis, we investigate four distinct moist convection scenarios, each of which reveals something about the possible nature of these storms and their relation to the modes.

• 1.  
  It is likely that perhaps moist convective storms on Jupiter are much more frequent than we think. Small storms could be erupting from the cloud base all the time, yet they simply do not have enough buoyancy and upward momentum to reach observable cloud heights. We therefore investigate what temporal storm frequency is necessary for a 1 km radius storm with 100 columns per storm with a t_storm of 135 minutes (the time required for the cloud to reach maximum altitude from the cloud base) to achieve theoretical mode energy and amplitude estimates. This may also be thought of as 100 1 km radius storms erupting simultaneously across the planet.
• 2.  
  What if the largest storms were primarily responsible for driving the modes? We thus also investigate the storm frequency necessary to achieve the theoretical mode energies and amplitudes for the largest physically feasible storms in our simulations, the 25 km radius cloud column with 200 columns per storm with a t_storm of 45 minutes.
• 3.  
  We then examine the scenario where the energy in a storm contributing to the modes is actually much greater than just the kinetic energy that our cloud models suggest. Utilizing the storm energy found by Gierasch et al. (2000), we find the storm frequency necessary to achieve theoretical mode estimates for storms of 5 × 10²² erg s⁻¹.
• 4.  
  Finally, we consider the probable case whereby a distribution of storm sizes can erupt. Using the one-sided Gaussian distribution of storms described in Section 4.2, we find what distribution of storm energies can reproduce theoretical mode energies and amplitudes.

The values we use for t_storm are derived from the vertical velocities shown in Figure 2. For example, Hueso et al. (2002) show that a rising storm cloud column can have a radius no larger than 25 km, but that a moist convective storm can have up to 200 of these updrafts. The 25 km radius column has an average updraft velocity of 30 m s⁻¹ over the 80 km range between the cloud base and maximum cloud height, resulting in t_storm = 45 minutes. From Table 2, we thus use an input power of 200 × 1.64 × 10²⁰ erg s⁻¹ = 3.28 × 10²² erg s⁻¹. In addition, Hueso et al. (2002) allude to a storm frequency of about 12 days, thus we begin testing our simulations with the minimum and maximum values of t_load to be 9 and 15 days, respectively. The value for t_load is randomly sampled between its maximum and minimum values, that is, once a storm concludes the time until the subsequent storm erupts is randomly selected to be between 9 and 15 days, for example. This temporal range is then adjusted between simulations to find which storm eruption frequencies produce the mode energies that agree with theoretical estimates and observed amplitudes.

In the scenario in which only the smallest storms contribute to driving Jovian modes, we find that for a 1 km radius storm with 100 columns per storm with a t_storm of 135 minutes, there is no temporal spacing between storm eruptions that reproduces the desired mode energies given the predicted kinetic energy available in these columns. If one of these storms erupts every 1–2 days, the l = 0, n = 1 mode has an energy of 2 × 10²¹ (1.5 × 10²¹) erg and Jupiter has an overall surface displacement of 10.5 (5.3) cm for the frequency-dependent (constant) quality factor. In order to achieve theoretical energies, it would require an eruption of ∼10⁷ single column storms or ∼10⁵ 100 column storms each day on Jupiter. This would imply an average surface density of one storm column per 6423 km².

When examining the case of the largest realistic storm frequency, we reach a similar conclusion to Case 1. For a storm of 200 columns with each column having a radius of 25 km, having one of these storms erupt every 1–2 days, the l = 0, n = 1 mode has an energy of 1.4 × 10²⁵ (9 × 10²⁴) erg and Jupiter has an overall surface displacement of 7.3 (4.1) m for the frequency-dependent (constant) quality factor. Theoretical estimates would require ∼70 of these storms to erupt per day. However, given that Hueso et al. (2002) allude to a ∼12 day spacing of these storms, coupled with the observations made by Gierasch et al. (2000), which indicate an average abundance of ∼43 of these storms being present on Jupiter at any given time (not all of which are as large as the one observed; see Section 3.2), we conclude that there are simply too few storms of this size to drive oscillatory modes.

Perhaps our derivations of the available storm energy are incorrect. Perhaps there is actually much more storm energy readily available to the modes than just the kinetic energy we have considered. Or maybe there is additional energy associated with Jovian storms that can also contribute to mode driving. As an upper limit, we consider the entirety of the storm energy observed by Gierasch et al. (2000). They observed a storm with a power of 5 × 10²² erg s⁻¹. Using their estimated storm surface density of 0.66 × 10⁻⁹ km⁻², we find that storms like this would supply ∼3.25 × 10³⁰ erg day⁻¹ to the modes (found by $\tfrac{5\times {10}^{22}\,\mathrm{erg}\,{{\rm{s}}}^{-1}}{{10}^{6}\,{\mathrm{km}}^{2}}\div0.66\times {10}^{-9}\,{\mathrm{km}}^{-2}$ converted to erg day⁻¹ and halved as we assume only half the available storm energy drives modes). When utilizing this energy input in our simulations, we find the mode energies and amplitudes are much too high, resulting in an energy of 3.5 × 10²⁹ erg for the l = 0, n = 1 mode and an overall surface amplitude of 775 m.

Although this does not reproduce the mode estimates we expect, it is promising because it does reveal that enough associated storm energy is indeed present to excite Jovian modes via moist convection. By varying the daily energy input from the storms, we find that in order to excite the modes to their theoretical energies and observed amplitudes (10²⁶–10²⁷ erg for the l = 0, n = 1 mode and a 10–100 m cumulative surface amplitude), we find that the modes require, on average, a daily storm eruption that provides between 5 × 10²⁷ and 10²⁸ erg day⁻¹. If all the observed energy in the storm were available to modes, then this energy requirement could be achieved by an average surface abundance of approximately 43 Jovian storms of average surface area 764 km², or approximately 33 storms with average surface area 1000 km². It is worth noting that even though these storm abundances are commensurate with those in Section 3.2, these storms utilize the total thermal surface power density as that observed by Gierasch et al. (2000), not the kinetic power.

Using a Gaussian distribution of storm energies as described in Section 4.2, we investigate what distribution of storm energies can excite modes to predicted levels of energy and amplitude. If the minimum energy of the distribution (recall that the minimum energy is the mean because it is a one-sided Gaussian) matches our smallest considered storm energy of 8.1 × 10²⁰ erg with a standard deviation one-third the maximum considered storm energy of 1.35 × 10²⁶ erg (so that the largest storms are three standard deviations from the mean), we find the l = 0, n = 1 mode energy to have 1.5 × 10²⁵ (6 × 10²⁴) erg for the frequency-dependent (independent) quality factor and an overall surface amplitude of 7.6 (3.8) m. So for a distribution of storm energies contained within the kinetic energy limits of our cloud models, we find that there is not enough energy to supply to the modes.

However, we do find two distributions that can successfully reproduce the mode predictions. If both the mean and standard deviation are 100 times larger, such that the minimum energy is 8.1 × 10²² erg and 3σ from the mean is 1.35 × 10²⁸ erg, then the overall surface amplitude of Jupiter becomes 38 m and the l = 0, n = 1 mode has an energy of 8 × 10²⁶ erg. The storm cadence for this simulation was an eruption every 1–2 days, although increasing this temporal range only slightly decreases the resultant amplitudes and energies due to the large quality factor.

The results of the second simulation are shown in Figure 4. For this energy distribution, the minimum energy is still that of our smallest storm (8.1 × 10²⁰ erg), but with a standard deviation 100 times larger than our initial estimate (1.35 × 10²⁸ erg). For a storm cadence of one eruption every 1–2 (5–10) days the l = 0, n = 1 mode contains an average energy of 8.2 × 10²⁶ (3.2 × 10²⁶) erg with an overall surface displacement of 38 (29) m. The spread in energies and amplitudes in Figure 4 is typical of these simulations.

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As a final note, it is worth examining the assumption made in Section 2 regarding the water abundance on Jupiter. Although we do not quantitatively examine the effect of a varying atmospheric water abundance, an abundance variation does indeed affect the moist convective process and therefore our results. If the water abundance in Jupiter's atmosphere is decreased, the water cloud base would form at a lower pressure. For example, if the water abundance was 1% that of solar, as suggested by Bjoraker (1985), the cloud base would form around 2.5 bars of pressure (Stoker 1986). This would result in lower cloud column momentum, and thus less kinetic energy available to drive oscillations.

Conversely, if the water abundance is greater than three times solar, we would expect to see more energy available to the modes for reasons opposite that described above, until an enrichment of 9.9 times solar is reached. Leconte et al. (2017) and Guillot (1995) show that for a water enrichment greater than 9.9 times solar, the atmosphere becomes more stabilized against convection, thereby inhibiting storms. The result is a more time-dependent moist convection. Although this may explain the quasi-periodic nature of Saturn's moist convective storms as described in Li & Ingersoll (2015), Jupiter's moist convection is more stochastic, thereby limiting the water abundance estimate to no more than 10 times solar. Since we assumed an enrichment of 3 times solar for this work and the range of Jupiter's water abundance is approximately limited between 1 and 10 times solar, it is safe to say that the kinetic energy available to modes from moist convection is on the conservative estimate of the water abundance spectrum, thereby making it more likely that moist convection can indeed drive Jupiter's oscillations. We did not run simulations with a varying water abundance as the kinetic energy is so low relative to the storm energy observed by Gierasch et al. (2000) that even an increase of an order of magnitude of available kinetic energy would still fall short of the required mode excitation energy.