Vertical reflectivity profiles are usually described in terms of a ratio or difference from the reflectivity at the “reference level” (section 2.2.1). In stratiform precipitation, where there is little vertical motion, the VPR shows a characteristic layered structure on three levels: the melting layer “bright band”, frozen precipitation above it, and the rain at the “reference level” and below.
The key features of stratiform VPRs are conceptually simple, and have been described both here and in section 1.5.3. The following subsections present further detail on the variations in κ, D and N(D) within the three layers of stratiform rain, and how these affect the atmospheric reflectivity profile. Notation wise, the properties of liquid drops are indicated by the subscriptw, frozen bys(snow) ori(ice), and melting hydrometeors by the subscriptm.
Hydrometeor formation and growth
In stratiform precipitation ice crystals form, often by deposition or contact nucleation, at a generating level, and grow by vapour deposition. Sinceκi is constant, the change in
reflectivity with height near the top of the profile is a function of the increasing number of ice particles,N, and the growth in diameterD.
As the atmospheric temperature rises at lower levels, ice crystals begin to aggregate and form snowflakes. Aggregation causes a steady increase in reflectivity, as the decrease in number concentration with aggregation is outweighed by the impact of increasing diameter. Once aggregation begins the dielectric factorκsis no longer constant, but varies
in proportion to snow density (Sauvageot, 1992, chapter 2, pg 97-98). SinceZ ∝ |κ|2D6
(equation 1.7), it follows that the reflectivity of each individual snowflake is proportional to the square of its mass, and that the total reflectivity is related to the overall mass of precipitation per unit volume.
Given the majority of VPR modelling literature is focused on the bright band, there are few models which include a fully parameterised reflectivity profile shape in the ice layer. The microphysically-based correction scheme of Kirstetter et al. (2013) uses a parameterisation for ice-level reflectivity in which N and D increase linearly from the precipitation top until the onset of melting, and κs is defined by the matrix-inclusion
model of Boudevillain and Andrieu (2003) (discussed further below). This model has the benefit of simplicity, but neglects the difference between the vapour deposition and aggregation layers, and has a tendency to overestimate the reflectivity gradient above the melting layer.
A more common element of modelling studies is the inverse relationship between snowflake density and diameter, which has the general empirical formρs =γsDsys (whereys <0).
The chosen form can be combined with mass conservation and a standard assumption of no aggregation or breakup during melting to derive direct analytical relationships between reflectivities immediately above and below the melting layer (Hardaker et al., 1995; Szyrmer and Zawadzki, 1999; Wood et al., 2015). These studies typically aim toward developing detailed, predictive models of the bright band reflectivity profile.
Modelling the bright band
The most prominent feature of radar VPRs is the reflectivity enhancement associated with the stratiform melting layer, or “bright band”. The impacts of uncorrected bright band in radar QPEs include visible ring-shaped artefacts containing order of magnitude overestimation errors, which through a combination of freezing level height and beam broadening can affect the majority of the radar’s domain. A desire to mitigate these im- pacts has led to the large number of studies aiming to characterise bright band behaviour and correct for the effects in radar precipitation estimates.
The origin of the bright band can be easily understood with reference to the definition of reflectivity and its three contributing variables. Immediately above the 0oC isotherm,
stratiform precipitation is made up of a population of aggregate snowflakes. The presence of air inclusions means that snow aggregates at this level are much larger than their melted counterparts (Ds>> Dw). At the onset of melting, liquid water begins to collect in air
inclusions and on the surface of aggregate snowflakes. This leads to a rapid increase in κ towards that of liquid water. Since κw ≈ 5×κi and κi > κs due to the density
relation (section 1.2.1), the immediate increase in reflectivity can be extremely large. Matrosov et al. (2007) observe typical bright band enhancements in PPI scans of 5-7 dB, which is likely to be smaller than the true enhancement due to the vertical smoothing caused by broadening of the radar beam. As melting continues, the increase in κ is balanced by decreases in both drop diameter (through melting), and the decrease in drop concentrationN through increasing fall velocity (from 1 m s−1 in snow to 5 m s−1 in rain, Mittermaier et al., 2004). This causes a decrease in reflectivity between the initial peak and completion of melting, where precipitation is recognisable as rain and which defines the standard “reference level”.
Despite its conceptual simplicity, the details of the melting layer bright band are ex- tremely difficult to model. The size and depth of the observed bright band peak depend strongly on how each snowflake melts: specifically where the liquid water collects, as in- clusions or on the surface of the melting snowflake (Fabry and Szyrmer, 1999, discussed below). The modelled reflectivity is therefore extremely sensitive to the chosen melting model, which may or may not reflect the realities of melting in any given bright band. The difficulties of accurately modelling stratiform melting are exemplified by the early attempt of Hardaker et al. (1995) to characterise the bright band shape in detail for VPR correction. This paper introduces a physically-based one-dimensional bright band simulator, which uses a model of melting snowflakes as ice with air inclusions and an adiabatic lapse rate of 6oC km−1 above the melting layer to calculate a full analytical profile of reflectivity with height. Modelled VPRs are then compared with radar observa- tions. The peak bright band reflectivities generated by the Hardaker et al. (1995) model
were found to be correlated with surface rainfall rate over the range 0-7 mm h−1, which
is consistent with both the vertically-pointing radar observations of Fabry and Zawadzki (1995) and the bright band area relation of Kitchen et al. (1994). However in general, despite the careful consideration given to every detail of this model, the features of the simulated bright band did not agree well with observed VPRs.
Although their outcomes differ, there are certain consistent assumptions which underpin the majority of microphysical bright band studies, and lead to certain standard results. Typical models of the melting layer are one-dimensional, and assume no aggregation or breakup of partially melted droplets (Hardaker et al., 1995; Szyrmer and Zawadzki, 1999; Heyraud et al., 2008; Kirstetter et al., 2013). The requirement for mass flux conservation then applies on an individual particle basis, which implies the constraint:
Ds3= ρwD
3
w
ρs
(2.1) on the diameter of any given snowflake. Combining this with a snow density-diameter relation makes it possible to model the diameter of a snowflake immediately above the melting layer in terms of the diameter of its melted counterpart. Similarly, the behaviour ofDm with melted mass fractionf (as a function of height) is governed by mass conser-
vation: Dm(Dw, f) =Dw ρw ρs (1−f) +f 13 (2.2) The initial assumption of no aggregation or breakup implies that knowledge of the rain drop size distributionN(Dw) provides constraints on both N(Dm) at all heights within
the melting layer, and on N(Ds) immediately above it. A further standard assumption
that all hydrometeors fall at their terminal velocity Vt, which varies with diameter and
phase (Heyraud et al., 2008), rendersN(Dw) sufficient to describe fully the changes in
particle size distribution with height up to the top of the melting layer (Hardaker et al., 1995). In practise however, N(Dw) itself is a parameterised approximation, and the
combination of many empirical relations and assumptions means such models cannot yet reproduce with reliable accuracy the shape of observed VPRs.
Apart from changes in diameter, reflectivity behaviour in the bright band is also influ- enced by changes in the dielectric factor. The dielectric factor of melting hydrometeors,
κm, can be calculated using a modified form of the equations of Boudevillain and Andrieu
they define: m2s = 1−fa 1−(1−β)fa m2i + βfa 1−(1−β)fa m2a (2.3) whereβ = 2m 2 i m2 a−m2i m2 a m2 a−m2i log m2 a m2i −1 (2.4)
wherefais the volume fraction of air inclusions within the snowflake andκsrelates toms
as defined in equation 1.5. Replacingmi with the complex refractive index of the chosen
matrix andma by that of the inclusions, these formulae can be extended to describe the
mixtures of water, ice and air that comprise melting snow.
Since equation 2.3 is asymmetrical, the exact value of κm depends on the nature of
the three-level matrix-inclusion model used to describe melting snowflakes. Fabry and Szyrmer (1999) evaluate the following six different microphysical models for melting snow:
1. A snow core (ice inclusions in an air matrix) with a water shell 2. Snow (ice inclusions in air) inclusions in a water matrix
3. Air inclusions in a melting snow (ice inclusions in water) matrix 4. Melting snow (ice inclusions in water) inclusions in an air matrix
5. As model 4, but with a density discontinuity (higher density core surrounded by lower density shell)
6. Inner core represented by model 3, with an outer shell represented by model 4 Each of these is used to simulate the melting layer reflectivity profile, and the results compared with observations to infer the model that best represents reality. Simulated bright band reflectivities were found to be extremely sensitive to the melting model. The final, most complex model best reproduced the observations available to this study; models 1-3 overestimated the magnitude of the bright band, whilst models 4 and 5 underestimated the peak. On this basis model 6 is adopted by Kirstetter et al. (2013) in developing their microphysically-based VPR determination scheme.
A key conclusion arising from Fabry and Szyrmer (1999) echoes that of earlier work (Hardaker et al., 1995, and others): that modelling the bright band is extremely complex and sensitive to small details of the underlying microphysical situation. It follows that detailed microphysical simulations are unlikely ever to provide a practical solution to VPR determination in real time. Approximations and parameterisations remain necessary in estimating the VPR from real time data, in order to make the corrections required for operational QPE.
Reflectivities below the reference level
The major microphysical processes governing cold stratiform rain occur above the melting layer. Low-level growth, aggregation and breakup of liquid drops are generally assumed to be negligible, implying no change inDworN(Dw) with height, and sinceκwis constant
reflectivity is not expected to change significantly below the reference level (at the base of the melting layer). The exceptions to this are warm rain, which is not dealt with here, and contributions from orographic processes such as the feeder-seeder mechanism (Carruthers and Choularton, 1983). Kirstetter et al. (2013) parameterise low-level processes with a linear reflectivity gradient from the base of the melting layer, whilst Kitchen (1997) account for contributions from the feeder-seeder mechanism using a model estimate of orographic enhancement, which is applied to the lowest 1.5 km of the VPR in complex terrain. Changes in reflectivity with height in the rain layer are not considered further as part of this thesis.