Perturbation Technique in 3D Cloud Optics: Theory and Results Igor N. Polonsky1 , Anthony B. Davis1 and Michael A. Box2 1
2
Los Alamos National Laboratory, Space and Remote Sensing Sciences Group (ISR-2), Los Alamos, NM 87545, USA
[email protected] School of Physics, Univ of New South Wales, Sydney, NSW, 2052 Australia
[email protected]
1 Introduction It is well known that generally to simulate accurately radiative transfer through a realistic cloudy atmosphere one should use numerical approaches such as Monte Carlo [12], or SHDOM [3]. However, it is usually required too much time to make a simulation which is inconvenient when just we need an answer on a simple question like how significant the 3D effects are for a given problem. The perturbation method is what comes to mind first if we need to go further into modelling of the radiative transfer through cloud atmosphere starting from the simplest framework of one dimensional radiative transfer [2, 8]. Recently, two perturbation approaches have been used. One is somewhat orthodox [14] and based on assuming that some term in the radiative transfer equation is small enough to be considered as a small parameter to construct a perturbation series [4,5,9–11]. A different type of perturbation approach may be formulated on the basis of the joint consideration of both the direct and corresponding adjoint problem [1, 7, 13, 15, 18]. This approach can be derived using a variational principle [17] allowing one to obtain a required solution practically without effort. The goal of this paper is to formulate a variational principle to derive the perturbation approach specifically for the problem of cloud optics. We will also demonstrate how it can be used to explain some effects in cloud optics.
2 Definition of the Problem Let us consider radiation propagation through a cloud which has a shape of a slab and is illuminated by a steady, uniform, and collimated beam (e.g., sunlight). We introduce a Cartesian coordinate system with the origin on the upper slab surface with the z-axis directed toward the inner normal. The direction is defined by the Euler polar, θ and azimuth φ angles. The radiance I(r, vn) at the point r in the direction n can be calculated using the framework of the radiative transfer equation (RTE) [e.g, 6]
166
I.N. Polonsky et al.
n · I(r, n) + σe (r)I(r, n) =
σs (r) 4π
P (r, n, n )I(r, n )dn + S(r, n) . (1)
4π
Here n and r are the vectors which define the direction and position, respectively, I(r, n) is the radiance at the point r in the direction n, σe (r) and σs (r) are the extinction and scattering coefficients, respectively, and P (r, n, n ) is the phase function normalized as 1 P (r, n, n )dn = 1 4π 4π
S(r, n) is the source function which in the case of the sunbeam has the form S(r, n) = S0 δ(n − n0 )δ(z − ε) . Here S0 is the Sun flux at the top of the cloud, and ε is infinitesimal small and has been introduced for convenience. Certainly, (1) has to be complimented with the boundary conditions I(r, n)|z=0 = 0, µ > 0 , 1 I(r, n)|z=H = A(r, n, n )|z=H I(r, n )|z=H |dn , µ < 0 . π
(2) (3)
µ>0
Here A(r, n, n ) describes the reflection properties of the underlying surface if one exists.
3 Variational Principe to Derive the Radiative Transfer Equation We may assume that the most measurements in cloud optics can be describe by integration of the radiance with some receiver function R(r, n) P = R(r, n)I(r, n)dndr (4) Ξ
where Ξ denotes the region of interest in the position-direction space. According [17] we may introduce a functional ˜ n)S(r, n) dndr ˜ R(r, n)I(r, n) + I(r, J = I(r, n)LI(r, n)dndr − Ξ
Ξ
+
˜ n)I(r, n)dndr I(r,
z=0,µ>0
− z=H,µ0
A(r, n, n )I(r, n )dn dndr .
(5)
Perturbation Technique in 3D Cloud Optics: Theory and Results
167
Here to simplify notation the operator L = n · + σe (r) −
σs (r) 4π
dn P (r, n, n )⊗ ,
(6)
4π
has been introduced. Note that the notation ⊗ is used to indicate that the final term is an integral operator, not merely a definite integral. This func˜ n) results in RTE tional has a clear property that the variation of it over I(r, (1) with boundary condition (2). The variation of the functional over I(r, n) ˜ n) results in the equation for I(r, ˜ n) = σs (r) P (r, n , n )I(r, ˜ n )dn + R(r, n) (7) ˜ n) + σe (r)I(r, −n · I(r, 4π 4π
with boundary conditions ˜ n)|z=0 = 0, µ < 0, I(r, 1 ˜ n )|z=H |dn , µ > 0 . ˜ I(r, n)|z=H = A(r, n , n)|z=H I(r, π
(8) (9)
µ0
z=H,µ
