Stratospheric transport of volcanic aerosol

Transport processes in the stratosphere seem to be very complicated and are not known in detail. Therefore, usually observed $AOD$ patterns are used to drive GCMs (Stenchikov et al. 1997). Because the volcanic aerosol reaches mainly the lower stratosphere with a maximum concentration at an altitude about 21 km in the tropics and about 10 km in the polar regions (Hitchmann et al., 1994), one may try to use stratospheric circulation models to simulate aerosol distributions. According to Plumb (1996), however, there is enough information available to create realistic formulations of the stratospheric transport without the necessity to solve the primitive equations. We perform this with a non-local diffusion formalism as it is introduced in boundary layer meteorology by Stull (1984). Considering $m$ equal-area latitude belts, each containing an aerosol amount $a_{i}(n)$ (given as mass concentration) at the time $n$, we can define a transilient $m\times m$ matrix $\bf A_{1} \rm$ to describe the temporal evolution of a spatial pattern by

$$\bf a\rm (n+1) = \bf A_{1} a\rm (n)$$ (4)

where the vector $\bf a\rm (n)$ is given as $\bf a\rm (n)=(a_{1}(n),a_{2}(n),...,a_{m}(n))$. The initial conditions are the aerosol loading $a_{i}^{*}(0)=a^{*}$ in the latitude belt of eruption $i=i^{*}$ at the time of eruption $n=0$. The transilient matrix does not need to be constant in time. Instead, the matrix rather depends on the annual cycle. Therefore we rewrite equation (4) to introduce a dependence on the season $s$:

$$\bf a\rm (n+1) = \bf A_{s} a\rm (n).$$ (5)

Note that the time step $n$ is independent from the season. Now we consider 16 latitude belts and four seasons. So the transilient matrix consists of $16\times 16 \times 4= 1024$ coefficients. This seems to be a compromise between the desired spatial resolution and the deficiency of information about the transport processes. To decrease the very high degree of freedom we introduce the following assumptions:

• If we assume symmetric seasons, there are only two different ones: an extreme one (a winter- and a summer hemisphere) and a moderate one without hemispheric differences. This reduces the amount of coefficients in the transilient matrix to $16\times 16 +16\times 8= 384$.
• If we consider very short time steps, we can apply a local (non-isotropic) exchange. This means that only the diagonal and the first subdiagonals of the matrix are filled with non-zero elements. Therefore we have to know only $2\times 16 + 4\times 15= 92$ elements. Taking also into account the conservation of mass (i.e. the transport is without sources and sinks) the sum over any column or row of the matrix has to be unity. This leads to 30 independent coefficients.
• According to Hitchmann et al. (1994) we can distinguish between tropics (bordered by a pronounced mean aerosol gradient at about $20^{\circ}$N and $20^{\circ}$S), extratropics, and polar regions (poleward of about $60^{\circ}$N and $60^{\circ}$S), see Figure 1. By using 16 equal-area latitude belts, there are 6 belts between $\pm 22^{\circ}$ representing the tropics, 8 belts that represent the extratropics and 2 representing the polar region ( $61^{\circ}-90^{\circ}$).
• Now we consider an isotropic transport within the tropics, anisotropic exchange between tropics and extratropics, isotropic transport within the extratropics (two seasons) and isotropic exchange between extratropics and polar regions in the summer- and winterhemisphere as well as during the moderate seasons. Together with the condition of conservation of mass this leads to only 8 independent coefficients (Table 3), which have to be taken from or estimated from literature.

We assume that the mixing within the tropical region leads to a nearly homogeneous distribution (less than 10% deviation from the mean value) within 3 months after an eruption that occurs between $\pm 7^{\circ}$ latitude. This is realized by an exchange coefficient of 91% per month. Volk et al. (1996) calculated an entrainment rate from extratropics into tropics of about 7% from observations. Because this value does not change between an altitude of about 16 to 21 km we assume that this is adequate for our purpose. The same authors found an average detrainment rate of 5% to 35% with a pronounced altitudinal dependence. An analysis by Waugh (1996) leads to a transport rate from the tropics to the northern hemisphere extratropics of about 8 to 10% of the tropical mass per month. Because the uncertainty of these estimations is about 50%, the results are in reasonable agreement. Assuming a homogeneous aerosol distribution within the six tropical latitude belts considered in our approximation, the exchange coefficients from the tropical border to the extratropics follow to be three times the transport rate per tropical mass. This leads to about 24% to 30% per month. Considering that the distribution of aerosol within the tropics is not exactly homogeneous we use the upper estimate as the exchange coefficient. Boering et al. (1994) give an extratropical mixing time of about 2 to 3 months. According to this we use an extratropical exchange coefficient of 90% per month. Then a stratospheric aerosol amount entering the extratropics is distributed nearly homogeneously (with $\pm 5\%$ deviation of the average value) within the extratropics after 80 days. Because circulation is more vigorous in winter and spring than in other seasons, Hitchmann et al. (1994) found that extratropical radiation extinction in winter-spring and summer-fall hemispheres differ by about 20 to 50%. To consider the lower exchange in summer and spring we assume the extratropical exchange coefficients in these seasons to be 45% per month which is half of the winter/spring value. The exchange coefficient between the extratropics and the summer polar region is supposed to be the same as the extratropical exchange coefficient in summer. In winter the polar vortex surpresses most of the exchange. Nevertheless, since the mean aerosol cloud in this latitude region is mainly in a height of about 8 to 16 km (Hitchmann et al., 1994) where the polar vortex is not tight, we assume an exchange coefficient of about 10% per month in winter time. For spring and fall (April/May and October/November) we use an average exchange coefficient of 23% per month. Given the exchange coefficients between latitude belt $i$ and $k$ in per cent per month $m_{i,k}$, it is easy to get the exchange coefficients $a_{i,k}$ in per cent per time step as

$$(1-a_{i,k})^{\mu}=1-m_{i,k}$$ (6)

if one month equals $\mu$ time steps. For practical calculations we take 180 time steps per year to fill the transilient matrix and use $\bf A_{s}\rm ^{3}$ for calculations with 60 time steps per year. Note that $\bf A_{s}\rm ^{3}$ is now a non-local transilient matrix.