Volcanic lightning reveals umbrella cloud dynamics of the 15 January 2022 Hunga volcano eruption, Tonga

Abstract

The 15 January 2022 eruption of Hunga volcano, Tonga, significantly impacted the Kingdom of Tonga as well as the wider Pacific region. The eruption column attained a maximum height of 58 km whilst the umbrella cloud reached a diameter approaching 600 km within about 3 h. The frequency of volcanic lightning generated during the eruption was also unprecedented, with the Vaisala Global Lightning Database (GLD360) recording over 3 × 10⁵ strikes over a 2-h period. We have combined Himawari-8 satellite imagery with the spatiotemporal distribution of lightning strikes to constrain the dynamics of umbrella spreading and infer a timeline of events for the climactic phase of the eruption. Lightning was initially concentrated directly above Hunga, with an areal extent that grew with the observed eruption cloud. However, about 20 min after the eruption onset, radial structure appeared in the lightning spatial distribution, with strikes clustered both directly above Hunga and in an annulus of radius ~ 50 km. Comparison with satellite imagery shows that this annulus coincided with the umbrella cloud front. The lightning annulus and umbrella front grew synchronously to a radius of ~ 150 km before the umbrella cloud growth rate decreased whilst the annulus itself contracted to a smaller radius of about 50 km again. We interpret that the lightning annulus resulted from an enhanced rate of particle collisions and subsequent triboelectrification due to enhanced vorticity in the umbrella cloud head. Our results demonstrate that volcanic lightning observations can provide insights into the internal dynamics of umbrella clouds and should motivate more quantitative models of umbrella spreading.

Appendix. Parallax correction

Objects at altitude above the Earth’s surface appear at erroneous spatial locations when viewed by satellites due to a parallax projection (Fig. 7). In the following, we use S to denote the location of satellite, P to denote the true location of an object at altitude and P′ the apparent projected location of P on the Earth’s surface. The Earth itself is approximated as a sphere with a radius of $R$ = 6378.1 km. Since Himawari-8 is geostationary and Hunga is in the tropics, this is a reasonable assumption. We also use both spherical polar and Cartesian coordinate systems, before transforming to latitude and longitude coordinates at the end. The polar coordinate $\varphi$ is measured with respect to polar north, the azimuthal angle $\theta$ westward with respect to the prime meridian and the radial coordinate $r$ with respect to the Earth’s centre. The corresponding Cartesian system is defined as

$$x=r{\sin}\varphi {\text{cos}}\theta ,$$

(5)

$$y=r{\text{sin}}\varphi {\text{sin}}\theta ,$$

(6)

and

$$z=r{\text{cos}}\varphi .$$

(7)

Thus, in Cartesian coordinates, the position vector of S is given by

$${{\varvec{x}}}_{{\text{s}}}=\left({r}_{{\text{s}}}{\text{cos}}{\theta }_{{\text{s}}}, {r}_{{\text{s}}}{\text{sin}}{\theta }_{{\text{s}}},0\right)=(R+A)({\text{cos}}{\theta }_{{\text{s}}},{\text{sin}}{\theta }_{s},0),$$

(8)

where ${r}_{{\text{s}}}=R+A$ is the radial position of S, $A$ = 35793 km is the satellite altitude and ${\theta }_{{\text{s}}}$ is the azimuthal position of S. Similarly, the position vector of S′ is given by

$${{\varvec{x}}}{'}=\left(r{'}{\text{sin}}\varphi {'}{\text{cos}}{\theta }{'},{r}{'}{\text{sin}}{\varphi }{'}{\text{sin}}{\theta }{'},r{'}{\text{cos}}\varphi {'}\right)=R\left({\text{sin}}\varphi {'}{\text{cos}}{\theta }{'},{\text{sin}}{\varphi }{'}{\text{sin}}{\theta }{'},{\text{cos}}\varphi {'}\right),$$

(9)

where $r{'}$, $\varphi {'}$ and $\theta$’ are the radial, polar and azimuthal positions of P′, respectively.

Fig. 7

Schematic showing the geometry leading to the parallax effect with a view showing the North (N) and South (S) poles and b a cross-section through the Earth’s equator, showing west (W) and east (E) directions. S denotes the location of the satellite, P the top location of the plume and P′ the projected location of the plume on the Earth’s surface. ${\theta }_{{\text{s}}}=140.7^\circ$ is the azimuthal position of S, ${\theta }_{{\text{p}}}$ is the azimuthal position of P and $\theta {'}$ is the azimuthal position of P′. ${\varphi }_{{\text{p}}}$ and $\varphi {'}$ are the polar positions of P and P′, respectively

In order to determine the coordinates of P, we define the location of the line connecting S and P′ as

$${\varvec{L}}\left({\varvec{s}}\right)={{\varvec{x}}}_{{\text{s}}}+s\left({{\varvec{x}}}{'}-{{\varvec{x}}}_{{\text{s}}}\right),$$

(10)

where ${\varvec{L}}$ denotes the position of points on the line and $s$ is a parameter indicating distance along the line. Combining Eqs. 8 and 9 with Eq. 10, we can show that the radial coordinate of each point on the line ${r}_{{\text{L}}}=|{\varvec{L}}|$ is given by

$${r}_{{\text{L}}}={\left\{{s}^{2}\left[{R}^{2}+{\left(R+A\right)}^{2}-2R\left(R+A\right){\text{sin}}\varphi {'}{\text{cos}}\left({\theta }{'}-{\theta }_{{\text{s}}}\right)\right]+s\left[2R\left(R+A\right){\text{sin}}\varphi {'}{\text{cos}}\left({\theta }{'}-{\theta }_{{\text{s}}}\right)-2{(R+A)}^{2}\right]+{\left(R+A\right)}^{2}\right\}}^{1/2}.$$

(11)

Next, we know that at P, ${r}_{{\text{L}}}=R+h$, where $h$ is the altitude of P above the Earth’s surface. So, defining ${s}_{{\text{p}}}$ as the value of $s$ corresponding to the location of P, we can use Eq. 11 to derive a quadratic equation for ${s}_{{\text{p}}}$

$${s}_{{\text{p}}}^{2}\left[{R}^{2}+{\left(R+A\right)}^{2}-2R\left(R+A\right){\text{sin}}{\varphi }{'}{\text{cos}}\left({\theta }{'}-{\theta }_{{\text{s}}}\right)\right]+2\left(R+A\right){s}_{{\text{p}}}\left[{R \sin}{\varphi }{'}{\text{cos}}\left({\theta }{'}-{\theta }_{{\text{s}}}\right)-\left(R+A\right)\right]+{(R+A)}^{2}-{\left(R+H\right)}^{2}=0.$$

(12)

Solving Eq. 12 produces two roots, the smallest of which corresponds to the position of P (the larger is a location on the opposite side of the Earth). Once the equation is solved, the position of P in Cartesian coordinates is given by

$${{\varvec{x}}}_{{\text{p}}}=\left({x}_{{\text{p}}},{y}_{{\text{p}}},{z}_{{\text{p}}}\right)={\varvec{L}}(s= {s}_{{\text{p}})}.$$

(13)

These Cartesian coordinates are then converted back to spherical polar equivalents using

$${\varphi }_{{\text{p}}}=90^\circ +{{\text{sin}}}^{-1}\left(\frac{{z}_{{\text{p}}}}{R+h}\right),$$

(14)

and

$${\theta }_{{\text{p}}}=\left\{\begin{array}{c}\begin{array}{c}{{\text{tan}}}^{-1}\left({y}_{{\text{p}}}/{x}_{{\text{p}}}\right) \quad \text{if} \quad {x}_{{\text{p}}},{y}_{{\text{p}}}>0 \\ {180^\circ -{\text{tan}}}^{-1}\left({-y}_{{\text{p}}}/{x}_{{\text{p}}}\right) \quad \text{if} \quad {x}_{{\text{p}}}<0,{y}_{{\text{p}}}>0\\ {180^\circ +{\text{tan}}}^{-1}\left({y}_{{\text{p}}}/{x}_{{\text{p}}}\right) \quad \text{if} \quad {x}_{{\text{p}}},{y}_{{\text{p}}}<0 \end{array}\\ {360^\circ -{\text{tan}}}^{-1}\left({-y}_{{\text{p}}}/{x}_{{\text{p}}}\right) \quad \text{if} \quad {x}_{{\text{p}}}>0,{y}_{{\text{p}}}<0\\ 90^\circ \quad \text{if} \quad {x}_{{\text{p}}}=0,{y}_{{\text{p}}}>0\\ 270^\circ \quad \text{if} \quad {x}_{{\text{p}}}=0,{y}_{{\text{p}}}<0\\ 0 \quad \text{if} \quad {x}_{{\text{p}}}>0,{y}_{{\text{p}}}=0\\ 180^\circ \quad \text{if} \quad {x}_{{\text{p}}}<0,{y}_{{\text{p}}}=0\end{array} \right.,$$

(15)

where ${\varphi }_{{\text{p}}}$ and ${\theta }_{{\text{p}}}$ are the polar and azimuthal coordinates of P. Additionally, in Eq. 14, we have used the fact that, at all times, the umbrella cloud is in the southern hemisphere.

Finally, we convert these spherical polar coordinates back to latitude ${\lambda }_{{\text{lat}}}$ and longitude ${\lambda }_{{\text{long}}}$ using

$${\lambda }_{{\text{lat}}}={\varphi }_{{\text{p}}}-90^\circ ,$$

(16)

and

$${\lambda }_{{\text{long}}}=\left\{\begin{array}{c}360^\circ -{\theta }_{{\text{p}}} \quad \text{if} \quad {\theta }_{{\text{p}}}>180^\circ \\ -{\theta }_{{\text{p}} } \quad \text{if} \quad {\theta }_{{\text{p}}}<180^\circ \end{array}.\right.$$

(17)