Rotationally Constrained Convection in the Sun: Applicable to Planetary Atmospheres?

Convection in the Sun is characterized by two distinct length scales: granules (with diameters of ≈1 Mm), and supergranules (with diameters of ≈30 Mm). Numerical convection models, including radiative losses near the surface, have been remarkably successful in replicating the observed characteristics of solar granules (e.g., Stein & Nordlund 1998). However, efforts to replicate the supergranule scale were much less successful for 40+ years (see Rincon & Rieutord 2018).

Featherstone & Hindman (2016): hereafter FH16) suggested including the effects of solar rotation in solar convection models. Coriolis forces can dominate gas flow on sufficiently large length scales. FH16 modeled solar convection in two distinct contexts: non-rotating, and rotating. In non-rotating models, large-scale convection cells were identifiable with sizes up to "global" length-scales. However, in rotating models, the largest scale convection cells were absent. FH16 concluded that Coriolis forces suppress the largest convective features. Also, because of suppression of large-scale modes, the convective power spectrum in rotating models contains a new peak which has no analog in non-rotating models. In one particular rotating model, the peak was found to lie at spherical harmonic degree l  = 71, corresponding to a linear diameter of ≈60 Mm on the Sun.

FH16 proposed that the supergranule scale in the Sun is controlled by a Rossby number Ro defined as the ratio between the nonlinear term U.gradU ≈ U²/L and the Coriolis term 2ΩxU in the Navier–Stokes equation. Thus, Ro ≈ const*U/LΩ. If L exceeds a critical value L_crit, then Ro becomes so small that convection cells are suppressed by rotation. With supergranule (horizontal) velocities in the Sun of order 10⁴ cm s⁻¹ and Ω of order 3 × 10⁻⁶ rad s⁻¹, the value of L_crit (Sun) = U/Ω is of order 30 Mm. Thus, in supergranules with length-scales of 30 Mm, the value of Ro is of order the "const" ≈1.

A follow-up calculation (Vasil et al. 2021), including a detailed analysis of the various terms which contribute to deep convection in a rotating star, found a peak in the convective spectrum at a length-scale of ≈30 Mm, i.e., the observed supergranule diameter. Thus, rotational suppression of convection may be key to determining the supergranule scale.

This raises the question: are there other examples in astrophysics where rotation interferes with convective flows? I consider two examples.

Thermal convection in the Earth's atmosphere contributes to the formation of clouds on a range of scales. Using aircraft and spacecraft data, Wood & Field (2011) have reported on the (horizontal) size distribution of clouds between L  = 0.1 km and L  = 10 Mm. The distribution is a power law (with index −1.66 ± 0.04) for sizes from 0.1 up to ≈1500 km. The well-defined spectral index over 4 orders of magnitude in length-scales suggests that a single physical process (plausibly, convection) dominates the formation of these clouds.

Remarkably, when Wood and Field examined the number of clouds with sizes larger than 1–2 Mm, the cloud numbers fell significantly below the power law extrapolation, i.e., clouds with L > 1–2 Mm are definitely less abundant than expected from extrapolating data on smaller clouds. A graphical illustration of a cut-off in cloud sizes may be seen every day by perusing real-time snapshots of Earth obtained by the DSCVR satellite (see https://epic.gsfc.nasa.gov). Inspection of videos on the website indicate that clouds rarely, if ever, are large enough to obscure all surface features: Earth's clouds really are scarcer at scales L~>~1–2 Mm.

Wood and Field discuss some possibilities to explain the suppression of larger clouds. One possibility concerns Rossby waves in the Earth's atmosphere: these waves contribute to the phenomenon of "semi-permanent centers of action" on weather charts (e.g., the "Bermuda high", the "Aleutian low"). In Rossby waves, the Coriolis force contributes significantly to the dynamics of gas flow. In view of FH16, I wonder: might the Rossby number Ro in the Earth's atmosphere have fallen below a critical value when the cloud length scales have values of >1–2 Mm?

Compared to the sizes of supergranules (30 Mm), the cut-off of cloud sizes in Earth's atmosphere occurs at a length-scale which is smaller by 15–30. Note that the angular velocity of Earth's rotation is ≈30 times larger than the Sun's. As a result, the value of L_crit (Earth) = U/Ω is smaller than the solar value by a factor of order 30, provided that the values of U do not differ by significant factors on Earth and in the Sun.

With regard to the values of U, the question is: how do wind speeds on Earth compare with the observed supergranule flow 0.1 km s⁻¹ ≈ 360 km hr⁻¹) on the Sun's surface?

In this regard, the largest systematic wind patterns on Earth occur in two pairs of meandering global jet streams, one pair at latitudes of 50°–60° (the polar jet) and the other pair at 30° (the subtropical jet) in both N and S hemispheres. The wind speeds in the jet streams can reach values of 70 m s⁻¹ ≈ 250 km hr⁻¹ (Koch et al. 2006). Thus, the speeds U of global jet streams on Earth are within ≈30% of the supergranule flow speeds on the Sun.

On smaller-than-global length scales, but still on length scales where Coriolis forces are dominant, the features with the fastest wind flows on Earth are hurricanes. Typical wind speeds are as high as U = 160–240 km hr⁻¹, comparable to the fastest jet stream speeds. In some stronger hurricanes, U may exceed 320 km hr⁻¹ (Sigda 1977): these speeds are within ≈10%–20% of supergranule flow speeds on the Sun. Thus, winds on Earth in the largest observed cloud formations (with sizes up to ≈1 Mm) can have speeds which approach the speeds associated with supergranule flows in the Sun.

As a result, it seems at least plausible that the existence of a cut-off scale-size for clouds on Earth at L≈ 1.5–2 Mm may be related to analogous effects of rotational suppression (with Ro ≈ 1) as those which give rise to supergranule sizes on the Sun. The ratio of critical horizontal scale-sizes on Sun and Earth (30–1) can be attributed mainly to the fact that Earth's rotation period is some 30 times shorter than the Sun's rotation period.

On Venus, the axial rotation period (243 Earth days) is ≈8 times longer than the Sun's rotation period. As a result, L_crit (Venus) ≈ U/Ω in the atmosphere of Venus will be ≈8 times larger than L_crit (Sun), provided that the wind speeds U are not greatly different from those on the Sun. How fast are the wind speeds on Venus? Kouyama et al. (2013) report that the wind speeds at the cloud tops can be as large as 0.1 km s⁻¹, i.e., 300–400 km hr⁻¹ (Kouyama et al. 2013). These overlap with supergranule U values. Therefore, Coriolis forces on Venus lead to convective suppression only when the cloud length-scales become ≈8 times larger than supergranules. Thus, rotational suppression of convection in Venus occurs only when cloud sizes are larger than L_crit (Venus) ≈ 8 × 30 Mm = 240 Mm.

However, since the diameter of Venus is only 12 Mm, i.e., ≪L_crit, rotational suppression of convection on Venus is insignificant: cloud sizes can in principle be as large as the entire planet. This is consistent with the fact that on Venus, clouds cover the entire surface.

FH16 and Vasil et al. (2021) suggest that rotational interaction with solar convection on large length-scales provides an explanation of the characteristic length-scale (≈30 Mm) associated with supergranules. Here, I suggest that an analogous rotational interaction with convection may explain why the size distribution of clouds in the Earth's atmosphere has a cut-off on length-scales L > 1–2 Mm. And I also suggest that the lack of rotational suppression in the atmosphere of Venus can explain why clouds cover the entire surface of Venus.