Cura CHIVA

Improving our understanding of the DSD and its small-scale variability will lead to better understanding of the physical processes at play in rainfall, reduced uncertainty in precipitation measurements, and more accurate numerical weather prediction (NWP). In this thesis we present new techniques for the measurement and stochastic simulation of the DSD, and we use them to quantify DSD variability and test DSD-retrieval techniques in Mediterranean rainfall. This introductory chapter sets the scene for the rest of the thesis, by briefly introducing the main topics and outlining the current state of the art. In Section 1.1 the DSD and its related bulk variables are introduced in more detail. Measurement and estimation of the DSD are discussed in Section 1.2. The effects of DSD variability and the change of support problem are discussed in Section 1.3. The bulk of the data used in this thesis were collected in Ardèche, France, a region that experiences heavy Mediterranean rainfall. The meteorological processes of this region are briefly introduced in Section 1.4. In Section 1.5, the outline of the rest of the thesis is shown.

1.1 The raindrop size distribution

Let us imagine a rainstorm, frozen in time. We take one cubic metre of space within the storm, and within this space we collect all the raindrops, count them and measure their sizes.

On average, there would be about 103falling raindrops in this cubic metre (Uijlenhoet and

Sempere Torres, 2006). Most of the drops would be small, between about 0.1 and 1 mm in diameter, and close to spherical (Pruppacher and Klett, 2000). There would also be some larger drops, which would be affected by air resistance, and thus not spherical. The bottom of these drops flattens, giving them an oblate shape that can be predicted as a function of the drop’s volume (e.g. Beard and Chuang, 1987; Andsager et al., 1999; Pruppacher and Klett, 2000; Thurai and Bringi, 2005; Thurai et al., 2007). Because not all raindrops are spherical, we speak of their size in terms of their equivolume diameter: the diameter of a sphere that contains the same amount of water as the drop. The great majority of raindrops have equivolume diameters between 0.1 and 6 mm (Uijlenhoet and Sempere Torres, 2006).

Now in our imaginary situation, let us allow the system to fall into motion. Each drop falls at a velocity that depends on its mass plus atmospheric conditions. The still-air terminal velocity of raindrops can be accurately predicted (e.g. Atlas et al., 1973; Beard, 1976; Brandes

et al., 2002) and ranges from 0.1 to more than 9 m s−1(Uijlenhoet and Sempere Torres, 2006;

Roe, 2005). As the rain falls, the number and sizes of the drops in our cubic metre of air are constantly changing. Evaporation causes the loss of some (mostly small) drops (e.g. Rosenfeld and Ulbrich, 2003). Drops collide with each other, with smaller drops joining together and coalescing to form larger drops, and larger drops breaking up (e.g. Pruppacher and Klett, 2000). Raindrops can only reach a certain size – about 10 mm – before they break up into smaller drops solely due to aerodynamic forces (Pruppacher and Klett, 2000), and it is for this reason that there are always more small drops than larger ones.

written N (D) [mm−1m−3] , is the number of raindrops with equivolume diameter in the range [D, D + δ) mm, per unit volume of air (Marshall and Palmer, 1948). The DSD describes the microstructure of liquid precipitation. Integral parameters of rainfall, also known as bulk rainfall variables, can be derived as weighted moments of the DSD (e.g. Ulbrich, 1983; Testud et al., 2001). Any bulk variable P can be expressed as

P = aP

∞

Z

0

wPDpN (D)d D, (1.1)

where p and apare constants and wpis a weight that may depend on D (Ulbrich, 1985). In

this section the most commonly used bulk variables are briefly defined in increasing moment order (for a detailed review, see e.g. Bringi and Chandrasekar, 2001).

The zeroth moment of the DSD is the total drop concentration Nt[m−3], defined simply as

Nt= ∞

Z

0

N (D)d D. (1.2)

The DSD can be expressed as the total drop concentration multiplied by a probability density

function f (D) [mm−1], such that N (D) = Ntf (D). The liquid water content, W [g m−3] is

related to the third moment of the DSD:

W =π10 −3_ρω 6 ∞ Z 0 D3N (D)d D, (1.3)

whereρ_ω[g cm−3] is the density of water. The flux of rainwater at a surface is expressed by the

rain rate R [mm h−1], defined as

R = 6π10−4

∞

Z

0

N (D)v(D)D3d D, (1.4)

where v(D) [m s−1] is the still-air fall velocity for a drop with equivolume diameter D. For the

work presented in this thesis, we used the model of Beard (1976) to calculate v(D). R is the variable that is of most interest to hydrologists.

1.1. The raindrop size distribution

The median-volume drop diameter, D0[mm], is the diameter that divides the DSD into two

portions of equal water volume. More commonly used as a characteristic drop diameter,

however, is the mass-weighted mean drop diameter, Dm [mm]. It is defined as the fourth

divided by the third moment of the DSD, such that

Dm= ∞ Z 0 D4N (D)d D ∞ Z 0 D3N (D)d D . (1.5)

Weather radars emit electromagnetic radiation and measure what is reflected back off hydrom- eteors in the atmosphere. Radar reflectivity, the quantity measured by conventional radars, can be derived from the DSD (Marshall and Palmer, 1948). When the particles scattering the radiation are much smaller than the radar wavelength, the scattering properties are governed by the Rayleigh regime, and radar reflectivity Z [dBZ] is equal to the sixth DSD moment (Marshall and Palmer, 1948; Bringi and Chandrasekar, 2001), such that

Z = 10log10   Dmax Z Dmin N (D)D6d D  . (1.6)

It is often the case, however, that the particles are similar in size to the wavelength. In this case the reflectivity occurs in the Mie regime, and it can be calculated from the DSD using

Z = 10log10   106λ4 π5_|Kω|2 ∞ Z 0 σb(D)N (D)d D  , (1.7)

whereλ [cm] is the radar wavelength, |K_ω|2[-] is the dielectric factor of water, andσb(D) [cm2]

is the back-scattering cross-section for a drop with equivolume diameter D (e.g. Bringi and Chandrasekar, 2001). The scattering properties of water droplets can be calculated using the T-matrix codes of Mishchenko and Travis (1998).

In the case of polarimetric radars, in which the electromagnetic waves are horizontally and

vertically polarised, the horizontal reflectivity ZH[dBZ] is calculated by replacingσb(D) in

Equation 1.7 withσb H(D) [cm2], the back-scattering cross section in horizontal polarisation.

[cm2], the back-scattering cross-section in vertical polarisation. It is common practice to refer to radar reflectivity in dBZ as defined above. At times it is also used, however, in its linear units,

in which case we have horizontal reflectivity Zh[mm6m−3] and vertical reflectivity Zv[mm6

m−3_{], defined as Z}

h= 10ZH/10and Zv= 10ZV/10respectively. Differential reflectivity, ZDR[dB],

defined as ZH− ZV, is the ratio of horizontal to vertical reflectivity. ZDRis a useful variable in

rain, because with large drops being more oblate it is related to drop size (Seliga and Bringi, 1976).

In this section, all the integrals have been written assuming a continuous DSD function and drop sizes ranging from zero to infinity. This is idealised, because in reality, not only are drops finite in size, but they are usually measured in discrete classes of equivolume drop diameter. When working using measured data, therefore, the integrals in calculations of bulk variables

convert to sums over the drop size classes from Dmin[mm] to Dmax[mm], the smallest and

largest considered class-centre drop sizes. d D becomes the width of each class, and D is the centre diameter of each class. Studies on DSD truncation and the calculation of bulk variables have concluded that the effects of truncation are negligible as long as the included range of

diameters is large enough around D0(Willis, 1984; Ulbrich, 1985; Vivekanandan et al., 2004).

It is common for the DSD to be summarised using a functional form defined by only a few parameters. The first proposed form was the exponential function of Marshall and Palmer (1948). The Gamma DSD (Ulbrich, 1983) is an extension of the exponential form that is more appropriate for instantaneous measurements of the DSD. Other compact forms of the DSD include the normalised DSD of Willis (1984), and normalisation approaches in which the DSD is expressed using one or more of its statistical moments and a normalised DSD function that describes the shape of the distribution (e.g. Sempere-Torres et al., 1994; Testud et al., 2001; Lee et al., 2004). In this thesis we provide a DSD interpolation method that requires no functional form (Chapter 3), show results of using it to test areal rainfall retrieval functions that do use a DSD model (Chapter 4), and study the spatial invariance of a normalised DSD function (Chapter 5).