Ž .
Ž Sun and Shine, 1994 and may even have a significant impact on climate Sun and
. Shine, 1995 . Mixed-phase clouds cover the Arctic Ocean throughout a large portion of
Ž .
the year e.g., Intrieri et al., 1999 and, thus, have a significant impact on the radiative Ž
budget of the Arctic Ocean. This is quite important as many studies e.g., Curry and .
Ebert, 1990; Royer et al., 1990; Curry, 1995; Lynch et al., 1995 have illustrated the possible sensitivity of the Arctic system to alterations in cloud radiative properties,
Ž .
particularly the frequently observed low-lying Arctic stratus clouds ASC . Since
microphysical data are sparse for ASC, we use microphysical information derived from Ž
. RAMS bin microphysical simulations of ASC Harrington et al., 1999 , which compare
favorably with observations in our radiative computations. We pick cases that span a large range of liquid and ice water paths, covering the ranges observed in the Arctic.
This model information is useful since the simulated clouds behave like observed ASC and have microphysical structures similar to available observations. However, it must
always be kept in mind that this information is derived from a numerical model. Thus, our results should be viewed as an attempt to assess the possible qualitatiÕe impacts of
cloud microstructure on the Arctic radiation budget.
2. Bulk model parameterization
Modeling frameworks such as RAMS are computationally intensive and require radiation routines that are efficient yet accurate. For this reason, two-stream radiation
models are frequently used. The coupling of any two-stream radiative transfer scheme to a microphysical model requires the efficient computation of cloud optical properties.
Ž . Ž .
These consist of the single scatter albedo v , the optical depth t , and the asymmetry Ž .
parameter g . These parameters are computed for each band of the radiative transfer model and are combined from values computed for each hydrometeor type. Our current
RAMS radiation model has two band structures: a broader band structure which is based Ž
. Ž .
on Ritter and Geleyn 1992 RG; three solar and five infrared bands and a narrower
Ž . Ž
. band structure based on Fu and Liou 1992
FL; six solar and 12 infrared bands . This Ž
. Ž
. model is fully described in Harrington 1997 and briefly in Harrington et al. 1999 .
Bulk microphysical models in use today predict the evolution of a variety of hydrometeor classes and various moments of the distribution function. RAMS predicts
the evolution of seven separate hydrometeor species: cloud droplets, rain, pristine ice, Ž
. snow, aggregates, graupel, and hail Walko et al., 1995; Meyers et al., 1997 . Each class
is defined by particular growth mechanisms, and not necessarily by standard terminol- Ž
. ogy for details see Walko et al., 1995 .
Ž In RAMS and in several other microphysical modeling frameworks e.g., Ferrier,
. 1994; Mitchell, 1994 , hydrometeors are assumed to have the form of gamma distribu-
tion,
ny1
N D
1 D
t
n D s y
exp y ,
1
Ž . Ž .
ž ž
G n D
D D
Ž .
n n
n
where N is the number concentration, G is the gamma function, n is a parameter
t
describing the shape of the size spectrum, D is the diameter of the hydrometeors, and D
is the characteristic diameter of the distribution. For ice hydrometeors, which
n
typically lack spherical symmetry, D and D are replaced respectively by L and L ,
n n
where L is defined as the maximum dimension of a given crystal habit. This functional form has many desirable attributes. It is easily integrated and
frequently it can be fit to observed spectra. Furthermore, its variables have clear physical interpretations; the mean size of hydrometeors of a gamma distribution is given by
D s n D . The shape parameter, n , describes the spectral breadth of the distribution. A
n
value of n s 1 produces a broad, exponential distribution function while a larger value Ž
. of n say 15 produces a very narrow spectrum.
2.1. Bulk optical properties: liquid and ice A major difficulty with computing the optical properties for microphysical models is
Ž . that one must integrate not only over the size distribution, n D , but also over a given
Ž .
Ž .
radiative band-width D l . Thus, for the extinction Q the following integral must be
ext
Ž .
solved Slingo and Schrecker, 1982 ,
`
b s
A D Q D, m l n D d D E d lr
E d l,
Ž . Ž
. Ž .
H H H
ext ext
l l
D l
D l
E s S Solar ,
E s B l, T Infrared ,
2
Ž .
Ž . Ž
. Ž .
l l
l s
Ž . where A D is the cross-sectional area of any hydrometeor, m is the complex index of
refraction, E is the solarrinfrared energy density, and T s 273 K is the reference value
l s
Ž . used for the Planck function. Eq. 2 is known as thin averaging, which works quite well
for extinction but tends to underestimate v in broad-band models. In order to reduce this over-absorption, the thick averaging of Edwards and Slingo, 1996 is used to
determine the band-averaged v,
4 r
`
v s ,
3
Ž .
2 2
1 q r y g 1 y r
Ž .
Ž .
` `
Ž .
where r is the reflectance of an infinitely thick layer see Edwards and Slingo, 1996 ,
`
Ž . and r
and g are averaged similarly to Q in Eq. 2 . We use thick averaging for
` ext
liquid drops and thin averaging for ice crystals as this produces excellent broad-band Ž
. accuracy in comparison to a 220 band model Edwards and Slingo, 1996 .
2.1.1. Liquid phase An analytical solution to the scattering problem of water drops exists in the
Lorenz–Mie theory. This approach, however, is not practical for computing the optical properties since the solution is numerically cumbersome and it cannot be integrated
analytically over the size distribution given above. An attractive and efficient alternative
Ž .
is Anomalous Diffraction Theory ADT , but this theory can produce relatively large Ž
. Ž
. errors Mitchell, 2000 . Fortunately, Mitchell 2000 has derived a method which vastly
Ž .
improves ADT the Modified ADT, or MADT
by parameterizing the ‘‘missing physics’’ associated with internal reflectionrrefraction, resonance tunneling, and edge
effects that are not accounted for in ADT. We derive the optical properties of water Ž
drops with this theory since the errors associated with it are quite small see Mitchell, .
2000 . Modified ADT, like ADT, gives the extinction and absorption coefficients of
spherical drops. The ADT optical properties are modified in the following manner in Ž
. MADT see Appendix A for definitions ,
C
res
Q D, l, m s 1 q
Q q Q
, 4
Ž .
Ž .
ext , m ext
edge
ž
2 Q
D, l, m s 1 q C q C Q
, 5
Ž .
Ž .
Ž .
abs , m ir
res abs
where Q and Q
are the ADT extinction and absorption coefficients, C is the
ext abs
res
modification for resonance tunneling, C is the modification for internal reflectionrre-
ir
fraction, and Q is the modification for edge effects. These functions allow the
edge
Ž . Ž
. integral over the size distribution in Eq. 2 to be evaluated analytically. Mitchell 2000
has already solved this problem for a form of the gamma distribution function. However, with a view to putting this solution in a form usable in RAMS, we have solved the
Ž . integral for the generalized gamma distribution given by Eq. 1 . The extinction and
absorption integrated over size, defined respectively as b and b
, become,
ext abs
b s A Q
q Q q Q
, 6
Ž .
ext ext
res ,e edge
b s A Q
q Q q Q ,
7
Ž .
abs abs
ir res
where A is the integrated cross-sectional area of the distribution, and the other terms are the integrated ADT extinction and absorption, and the integrated MADT correction
Ž .
terms see Appendix A . The above solution explicitly shows the impact of each additional term in MADT on the total extinction and absorption, and is more succinct
Ž .
than the solution given in Mitchell 2000 . These functions are easily coded and require little computation time compared to an approach based on Lorenz–Mie theory. When
integrated over a radiation model bandwidth, errors are reduced even further. As an example, Fig. 1 shows a comparison between MADT, ADT and Lorenz–Mie theory for
Q
as a function of D . The extinction shows only small errors near characteristic
ext n
diameters of about 10 mm. While MADT gives a fairly accurate representation of Q
and v, it does not give
ext
information about the asymmetry parameter. Because of the time-consuming computa- tions involved, a dataset was constructed of values for one distribution shape, n s 6, for
D values ranging over 1–1000 mm, and for each radiation band. These tabulated values
n
are then used in the parameterization described in Section 2.1.3 below.
Fig. 1. Computations of Q using MADT, Lorenz–Mie theory, and ADT for the 8.3–9.0 mm band of the
ext
radiation model.
2.1.2. Ice phase Since ice crystals have edges, the accurate computation of ice crystal optical
properties is not as straight-forward as it is for liquid drops. Many methods exist for ice optical property computations with perhaps the simplest being the use of equivalent
surface area or volume spheres. However, such methods may not be appropriate ŽGrenfell and Warren, 1999; Mitchell and Arnott, 1994; Stackhouse and Stephens, 1991;
. Ž
. Wielicki et al., 1990 . For example, Stackhouse and Stephens 1991 found that the
measured albedo of cirrus clouds was significantly larger than that predicted with ice Ž
spheres. Ray-tracing results and the parameterizations based on them e.g., Takano and .
Liou, 1989; Fu and Liou, 1992; Ebert and Curry, 1992 have shown that ice crystals scatter more and absorb less than equivalent volume spheres. This result has led to
Ž .
successful methods like Grenfell and Warren’s 1999 in which each crystal is modeled as a collection of spheres that have the same total volume and surface area as a single
crystal. Even though more successful scattering methods are being developed for ice Ž
. Ž
. crystals, the work of Doutriaux-Boucher et al. 2000 and Labonnote et al. 2000 cast
doubt on the use of pure-ice hexagonal crystals for the characterization of ice cloud optical properties. In their studies of cirrus clouds, the best retrievals were obtained
using hexagonal ice crystals with inhomogeneous inclusions of air bubbles. Even though this is the case, it is difficult to know in advance the percentage of air inclusions in a
population of crystals and, therefore, we ignore this factor in our studies. In addition, most of the above approaches for calculating ice cloud optical properties are tied to
particular band structures and ice classes while the method of Grenfell and Warren Ž
. 1999 is roughly similar to our method, which is described below.
Ž .
We use the approach described in Mitchell and Arnott 1994 and Mitchell et al. Ž
. Ž
. 1996 , which builds upon the results of Takano and Liou 1989 to develop a version of
ADT for ice hydrometeors versatile enough to be implemented in a variety of micro- physical frameworks. Additionally, this technique compares well to ray-tracing results.
In this method, the ADT absorption for large spheres is modified by including the Ž
. internal reflectionrrefraction term C
and by replacing the distance that a ray passes
ir
Ž .
through a sphere with the effective distance d that a ray passes through an ice crystal.
e
This gives, 4p n
i
Q s 1 q C
d 1 y exp y
d ,
8
Ž .
Ž .
abs ,i ir
e e
ž
l Ž
. Ž
. where C
is defined in Eq. A4 . We use formulae from Mitchell and Arnott 1994 ,
ir
Ž .
Ž .
Mitchell et al. 1990 , and Auer and Veal 1970 to compute values of d for three ice
e
classes: hexagonal plates, hexagonal columns, and five-branch bullet rosettes. Fig. 2 Ž .
shows d as a function of the maximum dimension of the crystal L . As expected, the
e
effective distance is much shorter for non-spherical ice than for spheres and this reduces the absorption by crystals.
Ž . In order to integrate Eq. 8 analytically over the size distribution, we follow Mitchell
Ž .
and Arnott 1994 and fit d as a linear function of L. Since d is obviously non-linear
e e
in L, we use linear fits over the following five ranges of L: 1–30, 30–100, 100–500, 500–2000, and 2000–10000 mm. Such a breakdown produces excellent accuracy in the
Ž .
Ž . fits to within 5 but requires the integral over size in Eq. 2 to be evaluated over a set
of truncated size ranges,
L
h
b L , L
s P L Q
L, m, l n L d L, 9
Ž .
Ž . Ž
. Ž . Ž .
H
abs ,i l
h abs ,i
L
l
where L and L are the lower and upper limits defined by the piece-wise linear
l h
Ž . endpoints, and P L is the projected crystal area that is parameterized as in Mitchell and
Ž .
Fig. 2. Comparison of d for various ice habits. Included in the figures are spheres dash-dotted line ,
e
Ž .
Ž .
Ž .
hexagonal columns solid line , hexagonal plates short dashed line , and rosettes long dashed line .
Ž .
Ž .
Arnott 1994 . We recast the form of the solution given in Mitchell et al. 1996 for the generalized gamma distribution as,
b L , L
s P L , L Q
L , L y C
L , L ,
10
Ž .
Ž .
Ž .
Ž .
Ž .
abs ,i l
h l
h abs ,i
l h
ir l
h
where the limits of integration are explicitly shown. The three terms above show the dependence of the absorption on the average projected area, the ADT absorption
coefficient, and the internal reflectionrrefraction term, all of which are given in Appendix A.4. The total absorption is then determined by summing the integral
solutions for the five d size-ranges,
e
5
b s
b L
, L .
11
Ž .
Ž .
Ý
abs ,i abs ,i
l , j h , j
js1
Ž .
For extinction, we use equivalent d spheres as in Mitchell et al. 1996 , except that
e
Ž .
we use MADT instead of ADT to compute. Mitchell et al. 1996 compute the mass and number median sizes for equivalent d sphere distributions. These sizes are used to
e
derive new N, D , and n for use in the extinction computations. Instead of following
n
this approach, which can lead to numerical problems for very narrow distributions Ž
. Harrington, 1997 , we hold the total mass, number, and n constant, and derive a new
Ž .
Ž characteristic size
D for the equivalent d
sphere distribution see Harrington,
n,s e
. Ž .
1997 . This value of D is then used in Eq. 6 to compute the total extinction at a
n,s
particular wavelength. Numerical tests show that this procedure produces the same Ž
. values as the method of Mitchell et al. 1996 , but requires fewer transformations.
Since little information is available regarding the asymmetry parameter for crystals at infrared wavelengths, we calculate g using spheres with sizes equal to d for a given ice
e
crystal. This technique reduces g as compared to equivalent volume spheres, which is Ž
desirable as g for spheres is uniformly larger than it is for crystals Takano and Liou, .
1989; Grenfell and Warren, 1999 . These computations also compare well with the
Fig. 3. Single-scatter for various ice crystal habits computed for the 8.33–9.0 mm band. Single-scatter for ice spheres computed with Lorenz–Mie theory is shown for reference.
Ž .
reduced-g method of Sun and Shine 1994 . Additionally, since g variation has a small influence on diabatic processes, this should be a tollerable approximation.
This method, coupled with thin averaging, is used to derive the band-averaged optical properties for the three ice habits. Fig. 3 shows v for columns, plates, and bullet rosette
crystals for the 8.33–9.0 mm band of the radiation model. In comparison with equivalent volume spheres, the reduced effective distance leads to greater scattering and less
Ž .
Ž .
Fig. 4. Comparison between computations of Q and v using MADT points and fits solid lines for bands
ext
Ž .
1, 4, and 6 1.53–4.64, 20–104, and 8.33–9 mm, respectively .
Ž .
absorption by the crystals. As shown by Mitchell et al. 1996 , on the cloud scale this leads to comparatively more reflection and less absorption.
2.1.3. General parameterization Even though the above methods are computationally expedient, the integral over
Ž . radiation band-width is still too costly to compute repeatedly at run-time. Thus, Eq. 2
needs to be parameterized in some way. This is accomplished by fitting the optical Ž .
property computations for each band over the effective radius r with an exponential-
e
sum fit, s s a q a e
b
1
r
e
q a e
b
2
r
e
, 12
Ž .
opt 1
2
where s stands for either Q
, v, or g. Fig. 4 shows fits of Q and v for three
opt ext
ext
bands of the RG radiation band-structure. These bands were chosen as examples because they span the simplest and most difficult curve fits. A distribution shape of n s 6 was
Ž .
used for these computations and is characteristic of a weakly drizzling liquid-only Ž
. ASC Olsson et al., 1998 . Note the accuracy of the fits, particularly at small r where
e
the curves are highly non-linear. Errors never exceed 4 for any of the fits. Similar Ž .
accuracy was obtained for the asymmetry parameter g , and for the optical properties of ice crystals.
It should be noted that the decision to fit the optical properties as a function of r is
e
made with some foresight. As discussed above, all bulk microphysical models require an Ž
. a priori choice for the shape of the size spectrum i.e. n for a gamma distribution . Thus,
the numerical fits must be recreated anytime n is changed. This is not a difficulty for
Ž .
Fig. 5. Asymmetry parameter plotted as a function of r for different distribution shapes, n s 2 solid line , 6
e
Ž .
Ž .
dashed line and 15 long dashed line . Results shown for the 1.53–4.0 mm band.
Q or v as these fits can be quickly recomputed. However, this is a problem for the
ext
computation of g, which requires the use of Lorenz–Mie theory. Ž
. We circumvent this problem by adopting the approach of Hu and Stamnes 1993 .
This work showed that when optical properties for varying n were plotted against r ,
e
the resultant curves were fairly independent of n . This result is found to hold in Fig. 5 and suggests that one can compute the optical properties as functions of r for a given n
e
and then use that information for any gamma distribution. Ž
. Of course, there are errors associated with this method when size ; l . For example,
solar bands show the greatest errors at small r , below those typically observed in
e
Ž clouds. However, errors of the same magnitude occur in the infrared near r s 10 not
e
. Ž
. shown , which is within the range typical of marine stratocumulus Stephens, 1978 .
Since g has a smaller influence on diabatic processes than Q or v, this error should
ext
be tolerable. In fact, calculations using this method for ASC with differing n are only slightly affected by using this method for computing g.
3. Bin model parameterization
