Radiative forcing

Radiative processes within the atmospshere are very complicated. Nevertheless, handsome analytic approximations like "delta-Eddington" (Joseph et al., 1976) and two-stream approximations (Coakley and Chýlek, 1975) exist; for an overview see Meador and Weaver (1980). For homogeneous plane-parallel layers these approximations lead to linear differential equations with constant coefficients and, in consequence, to exponential solutions for vertical energy fluxes. The applicability of these solutions is usually limited to certain ranges of the optical properties like $AOD$, single scattering albedo and the moments of the scattering phase function. Thus, we are faced with two problems. First, we neither know the optical parameters of the aerosol cloud nor do we know their temporal change, apart from the $AOD$. Second, we cannot assume a plane-parallel layer but have to consider a spherical layer and have to deal with large zenith angles, i.e. close to $90^{\circ}$. Thus, we decide to assume an exponential behaviour as it is supposed in the most simple solutions of the radiation transfer equation and adapt it to observations. The weakening effect of a ray passing through a layer follows from the RTE as

$$N_{b}=N_{t} \exp\left\{-\int\limits_{z_{b}}^{z_{t}}\sigma_{E} \omega dz\right\}$$ (7)

with

$N_{b}$	=	radiative energy flux at the bottom of the aerosol layer,
$N_{t}$	=	radiative energy flux at the top of the aerosol layer,
$z_{t}$	=	top of the aerosol layer,
$z_{b}$	=	bottom of the aerosol layer,
$\omega$	=	ratio between the vertical thickness of the layer and the ray path, and
$\sigma_{E}(z)$	=	extinction coefficient within the aerosol layer.

Although this solution is more realistic than Lacis' et al. (1992) approximation which does neither depend on $N_{t}$ nor on the factor $\omega$, we also neglect an explicit consideration of foreward scattered radiation. Anyway, if the part of foreward scattered radiation is proprotional to the extinction we are not wrong if we use equation (7) with a calibration coefficient $p_{2}$ to be estimated from observations (see section 4). Regarding $\sigma_{E}$ as the sum of the extinction coefficients with and without volcanic aerosol, $\sigma_{v}$ and $\sigma_{0}$ respectively, we can rewrite equation (7) as

$$N_{v}=N_{0} \exp\left\{-\int\limits_{z_{b}}^{z_{t}}\sigma_{v} \omega dz\right\}$$ (8)

with $N_{0} =$ radiative energy flux at the bottom of the undisturbed layer (without any volcanic aerosol) and $N_{v}=$ radiative energy flux at the bottom of the volcanic aerosol layer. Because of the very weak dependence of $\omega$ from $z$ within the stratosphere we can further write

$$\int\limits_{z_{b}}^{z_{t}}\sigma_{v} \omega dz=AOD \cdot \omega .$$ (9)

The energy flux change $\Delta N$ with respect to volcanic stratospheric aerosol is now given as the difference between $N_{0}$ and $N_{v}$. To obtain an approximated forcing $\Delta Q$, the vertical portion of the downward energy flux change

$$\Delta N_{\perp} = N_{0,\perp}\cdot [1-\exp(-AOD \omega)]$$ (10)

is then multiplied by the planetary coalbedo $1-\alpha_{p}$. The observed volcanic $AOD$ has to be multiplied by the calibration coefficient $p_{2}$ to consider that it is a bulk property and that we neglect an explicit description of the scattered radiation. Thus we get

$$\Delta Q = N_{0,\perp} (1-\alpha_{p}) \left[1-\exp\left(-AOD\cdot\omega\cdot p_{2}\right)\right].$$ (11)

Now this equation has to be averaged over the time steps and latitude belts. If the time steps and latitude belts are chosen to be small, $\Delta Q$ depends weakly on latitude and time within the latitude belts $i$ and time steps $n$. Thus, we approximate all variables by their corresponding latitude-belt and time-step averages. Therefore $AOD$ depends on $i$ and $n$, while all other variables depend on $i$ and have an annual cycle. The mean annual cycles of $N_{0,\perp}$ and $1-\alpha_{p}$ are taken from a Fourier-Legendre-decomposition applied by North and Coakley (1979). The ratio $\omega$ between the ray path $l$ and the thickness of the layer $d$ depends on the zenith angle $\Theta$ of the incoming solar radiation. It is usually set to $\omega = 1/\cos (\Theta)$ which is applicable for $\Theta< 80^{\circ}$. For $\Theta$ close to $90^{\circ}$, i.e. at sunrise and sunset, $1/\cos(\Theta)$ diverges and therefore this parameterization would overestimate the forcing. To meet this problem, we consider the curvature of the stratospheric aerosol layer and use a more realistic parameterization of $\omega$. We consider no refraction and therefore the ray path is a straight line. According to observations (Deshler, 1993) we regard the aerosol layer as homogeneous with a thickness $d=10 km$ and a bottom height $h$ of $15 km$ above the Earth surface. Therefore the layer has a curvature with the radius $R_{b}=R+h$ at the bottom and $R_{t}=R_{b}+d$ at the top, if $R$ is the Earth radius. According to Figure 2 the ray path $l$ is given by

$$\begin{array}{lcl} l & = & a_{2}-a_{1}\\ a_{1} & = & \fr... ...R_{t}} \sin (\pi - \Theta) \right] \right\}. \end{array}$$ (12)

$\Theta$ itself is a function of latitude $\varphi$, the time within the year $t_{y}$, and the time within the day $t_{d}$ (e.g. Monin, 1986):

$$\Theta = \arccos \left(\sin\delta \sin\varphi + \cos\delta \cos\varphi\cos\phi \right).$$ (13)

It can be approximated by the formulation given by Paltridge and Platt (1976)

$$\begin{array}{ccccccc} \delta & = & .006918 & - & .399912 ... ...\cos 3 \vartheta & + & .001480 \sin 3\vartheta \end{array}$$ (14)

with

$$\begin{array}{rcl} \vartheta & = & 2 \pi t_{y}\\ \phi & = & 2 \pi (t_{d}-12\mbox{ h}). \end{array}$$ (15)

On the time scales of interest a consideration of the time dependence within the day is not necessary. Therefore, in the next step, the ratio $\omega$ is averaged over the day light period. The borders of the integral are taken from (13) under the condition $\Theta=\pi /2$ which means sunrise and sunset and depend from latitude and time within the year. Inserting equation (14) in (13) and the resulting one in (12) we find an expression of $\omega(t_{y},\varphi)$ as a function of latitude and time within the year. Using this approximation, $\omega$ is restricted to be smaller than 13 in the case of $\Theta \rightarrow \frac{\pi}{2}$ (sun position at the horizon). Although we use a crude approximation compared to detailed solutions of the RTE, we realize two important facts: the seasonal and spatial patterns of the undisturbed radiation uptake $N_{0,\perp } \cdot(1-\alpha_{p})$ given by North and Coakley (1979) and the seasonal and spatial patterns of the ratio $\omega$.