11/29/07 1

Turbulence, rain drops and the l1/2 number density law

S. Lovejoy1, D. Schertzer2

1Department of Physics, McGill University, 3600 University st., Montreal, Que., Canada

2CEREVE, Ecole Nationale des Ponts et Chaussées, 6-8, avenue Blaise Pascal, Cité Descartes, 77455

MARNE-LA-VALLEE Cedex, France

Abstract:

We treat rain drops as low Stokes number particles in a highly turbulent flow and we derive

equations for the number and mass densities (n, ρ) and their variance fluxes (ψ, χ). By showing

that χ, ψ are only dissipated at small scales, we conclude that ρ should follow Corrsin-Obukhov k-5/3

spectra and that n should follow a k-2 spectrum corresponding to fluctuations Δρ, Δn at scale l

scaling as l1/3 and l1/2 respectively. We directly verify these predictions as well as many of the

necessary assumptions using a unique data set of three dimensional drop positions and masses from

the HYDROP experiment. While the Corrsin-Obukhov law has never been observed in rain before,

it’s discovery is perhaps not surprising; in contrast the Δn ≈ l1/2 number density law is quite new.

The key difference between the Δρ, Δn laws is that the time scale for the transfer of χ is determined

by the strongly scale dependent turbulent velocity whereas the time scale for ψ is determined by the

weakly scale dependent drop coalescence speed. The combination of number and mass density

laws can be used to develop stochastic compound multifractal Poisson processes which are useful

new tools for studying and modeling rain.

We discuss the implications of this for the rain rate statistics including a simplified model

which can explain the observed rain rate spectra.

1. Introduction

Rain is a highly turbulent process yet there is wide gap between turbulence and precipitation

research. On the one hand turbulence is increasingly viewed as a highly intermittent, highly

1 Corresponding author: Department of Physics, McGill University, 3600 University st., Montreal, QC, Canada, H3A

2T8,

e-mail: lovejoy@physics.mcgill.ca

11/29/07 2

heterogeneous process with turbulent energy, passive scalar variance and other fluxes concentrated

into a hierarchy of increasingly sparse fractal sets; over wide ranges the fields are multifractal (see

e.g. [Anselmet, 2001] for a recent review). Furthermore, advances in high powered lidars have

produced turbulent atmospheric data sets of unparalleled space-time resolutions. Analysis of such

data from aerosols have shown that if classical Corrsin-Obukhov theory of passive scalar turbulent

advection is given multifractal extensions to account for intermittency and scaling anisotropic

extensions to account for atmospheric stratification and scaling wave-like space-time extensions to

account for wave behaviour, that these rejuvenated classical theories account remarkably well for

the statistics of passive pollutants ([Lilley, et al., 2004], [Lovejoy, et al., 2007a], [Lilley, et al.,

2007], [Radkevitch, et al., 2007]). In contrast, applications of turbulence theory either to

interpreting radar echoes, or to disdrometer experiments almost invariably assume that the

turbulence is uniform so that rain has either homogeneous Poisson rain statistics (e.g. [Marshall and

Hitschfeld, 1953], [Wallace, 1953]), or that it is only weakly heterogeneous (e.g. [Uijlenhoet, et al.,

1999], [Jameson and Kostinski, 1999]).

Combining turbulence theory with rain drop physics involves several related difficulties. First,

rain is particulate. This is usually handled by considering volumes large enough so that many

particles are present and spatial averages can be taken. However since rain is strongly coupled to

the (multifractal) wind field even these supposedly continuous average fields turn out to be

discontinuous ([Lovejoy, et al., 2003]). This means that due to the systematic, strong dependence

on the scale/resolution over which the rate is estimated, the classical treatment of the rain rate R(x,t)

as a mathematical space-time field (without explicit reference to its scale/resolution) is not valid.

Finally rain does not trivially fit into the classical turbulence framework of passive scalars: rain

simultaneously modifies the wind field while moving with speeds different than that of the ambient

air. However the treatment of particles in flows has seen great progress especially with the

pioneering work of [Maxey and Riley, 1983], [Maxey, 1987] relating particle and flow velocities.

Unfortunately with the main exception of [Falkovitch and Pumir, 2004], [Falkovitch, et al., 2006]

these particle/fluid dynamics advances have not been applied to cloud or rain physics. While the

latter authors consider the case with Stokes numbers (St) ≈1 (applicable to cloud drops), in this

paper we develop the theory for rain particles with St <<1 and put the results in the context of

turbulence theory including multifractal cascades.

[Falkovitch and Pumir, 2004], [Falkovitch, et al., 2006] have admirably attacked the full

drop/turbulence interaction at its most fundamental level, but progress has been difficult.

Fortunately, at a more phenomenological level, things have been easier. [Schertzer and Lovejoy,

11/29/07 3

1987] proposed that even if rain is not a passive scalar that it nevertheless has an associated scale-

by-scale conserved turbulent flux. This proposal was made in analogy with the energy and passive

scalar variance fluxes based on scale invariance symmetries but it lacked a more explicit, detailed

justification. It nevertheless leads to a coupled turbulence/rain cascade model which predicts

multifractal rain statistics over wide ranges of scale. Today, this cascade picture has obtained wide

empirical support, particularly with the help of recent large scale analyses of global satellite radar

reflectivities which quantitatively show that from 20,000 down to 4.3 km, that the reflectivities (and

hence presumably the rain rate) are very nearly scale by scale conserved fluxes [Lovejoy, et al.,

2007b]. These global satellite observations show that cascade behaviour is followed to within

±4.6% over nearly four orders of magnitude. Other related predictions - based essentially on the

use of scaling symmetry arguments - are that rain should have anisotropic (especially stratified)

multifractal statistics (see e.g. [Lovejoy, et al., 1987] and that rain should belong to three -parameter

universality classes ([Schertzer and Lovejoy, 1987], [Schertzer and Lovejoy, 1997]), see the review

[Lovejoy and Schertzer, 1995]). However these empirical studies have been at scales larger than

drop scales and outstanding problems include the characterization of low and zero rain rate events

and the identification of the conserved (cascaded) flux itself (see however [Lovejoy, et al., 2007b]).

In other words, up until now the connection with turbulence has remained implicit rather than

explicit.

In order to bridge the drop physics - turbulence gap and to further pursue phenomenological

approaches, data spanning the drop and turbulence scales are needed. Starting in the 1980’s,

various attempts to obtain such data have been made; this includes experiments with chemically

coated blotting paper ([Lovejoy and Schertzer, 1990]) and lidars, ([Lovejoy and Schertzer, 1991]).

The most satisfactory of these was the HYDROP experiment ([Desaulniers-Soucy., et al., 2001])

which involved stereophotography of rain drops in ≈10 m3 volumes typically capturing the position

and size of 5,000 - 20,000 drops (nominal rain rates were between 2 and 10mm/hr; see fig. 1 for an

example, and table 1 and [Lilley, et al., 2006] for information about HYDROP). Analyses to date

([Lovejoy, et al., 2003], [Lilley, et al., 2006]) have shown that at scales larger than a characteristic

scale determined by both the turbulence intensity and the drop size distribution (but typically

around 20-30 cm see e.g. fig. 2), that the liquid water content (LWC; i.e. the mass density), number

density and other statistics behave in a scaling manner as predicted by cascade theories. While

these results suggest that rain is strongly coupled to the turbulent wind field at scales larger than 20-

30 cm (potentially explaining the multifractal properties of rain observed at much larger scales), the

11/29/07 4

analyses did not find an explicit connection with standard turbulence theory. See also [Marshak, et

al., 2005] for similar results for cloud drops.

Fig. 1: Second scene from experiment f295. For clarity only the 10% largest drops are shown,

only the relative sizes and positions of the drops are correct, the colours code the size of the drops. The boundaries are defined by the flash lamps used for lighting the drops and by the depth of field of the photographs.

Perhaps the key new result of this paper concerns the fluctuations in the particle number density

Δn which we find - both theoretically and empirically - follows a new Δn ≈ l1/2 law with prefactors

depending on turbulent fluxes. This explicit link between the particle number density fluctuations

and turbulence is expected to hold under fairly general circumstances and provides the basis for

constructing compound multifractal –Poisson processes. While the traditional approach to drop

modeling (see e.g. [Srivastava and Passarelli, 1980], [Khain and Pinsky, 1995]) is to hypothesize

specific parametric forms for the drop size distribution and then to assume spatial homogeneity in

the horizontal and smooth variations in the vertical, our approach on the contrary assumes extreme

turbulent - induced variability governed by turbulent cascade processes and allows the drop size

distributions to be defined only implicitly with their essential variations being due to turbulence.

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This paper is structured as follows. In section 2 we carefully discuss the relation between drop

and wind speeds showing that rain drops typically have low Stokes numbers so that Maxey’s

relations can be used and we directly estimate the mean relaxation times, lengths and speeds from

the HYDROP experiment. In section 3 we derive the basic equations for number density, mass

density, the corresponding fluxes density (including rainrate), and the coalescence speed. In section

4 we derive the l1/3, l1/2 laws for mass and number densities and verify the results empirically on the

HYDROP data, in section 5 we conclude.

2. The connection between drops and turbulence

2.1 Drop Relaxation times, distances, velocities:

Raindrops are inertial particles moving in turbulent flows under the influence of gravity. For an

individual drop moving with velocity v in a wind field u, Newton’s second law implies:

dv

dt= g +

g

VR!v

!vVR

"

#$%

&'

(d )1

; !v = u ) v (1)

where we have taken the downward acceleration of gravity (g) as positive, and assumed a power

law drag with exponent ηd. VR(M) is the “relaxation velocity” of a drop mass M. In still air, after

an infinite time, VR is the theoretical terminal drop velocity (we use capitals for single drop

properties, lower case for various averages). The corresponding relaxation times TR(M) and

distances LR(M) are given by:

TR=VR

g

LR=VR

2

g

(2)

These are the typical time and length scales over which the drops adjust their velocities to the

ambient air flow. According to classical dimensional analysis for drag on rigid bodies, the low

Reynold’s number regime is independent of fluid density while the high Reynold’s number regime

is independent of fluid viscosity. This implies that in the first regime, the drag is linear (ηd = 1;

“Stokes flow”), while in the second, it is quadratic (ηd = 2) so that the drag coefficient (CD) is

constant. For these regimes we have:

TR=L2!

w

"v

; LR=gL

4!w

2

"v

2; V

R=gL

2!w

"v

; Re << 1 (3)a

11/29/07 6

TR=

!DL"

w

g"a

#

$%&

'(

1/2

; LR= !

D

"w

"a

#

$%&

'(L; V

R=

!DgL"

w

"a

#

$%&

'(

1/2

; Re >> 1 (3)b

where L is the drop diameter, ρw, ρa are the densities of water and air !D= " / 3C

D( ) , CD is the

standard drag coefficient and !v is the dynamic viscosity. In the Stokes flow, we have ignored

numerical factors of order unity. Since at 20oC, pressure = 1 atmosphere, !w/ !

a" 1.29X103 , for

drops with diameters 0.2 < L < 4 mm (the range detected by the HYDROP experiment discussed

below), we expect LR to be typically of the order 20 cm – 4 m.

According to theory and experiment reported in [Le Clair, et al., 1972], for spheres, the high

Re limit is obtained roughly for Re>102 where CD≈1 (although it decreases by another factor of 2-3

as Re increases to ≈ 5000). According to semi-empirical results presented in [Pruppacher, 1997],

Re ≈ 1 is attained for drops with diameter (L) ≈ 0.1mm, whereas Re = 102 is attained for drops

diameter 0.5mm. On this basis, Pruppacher proposes breaking the fall speed dynamics into three

regimes with (roughly) L <0.01 mm, 0.01<L<1 mm, L>1 mm. The small L “Stokes” regime has

ηd = 1, the second regime is intermediate and the third (high Re) regime has roughly ηd = 2 up to a

“saturation” at L ≈ 4mm. Although for rain, this regime is more complicated than for rigid spheres

due to both drop flattening and due to internal drop flow dynamics, according to data and numerics

reviewed in [Pruppacher, 1997], the basic predictions of the ηd = 2 drag law (VR! L

1/2 ,

LR! "

w/ "

a( )L ) are reasonably well respected over the range 0.5 – 4 mm. For example, using

Pruppacher’s data on LR at 700 mb and a drag coefficient of CD = 0.8, the predictions of eq. 3 are

verified to within ±25% over this range.

2.2 Empirical estimates of mean relaxation lengths using drop stereophotography:

According to the above analysis, we expect the transition from rain drop behaviour dominated by

turbulence to behaviour dominated by inertia to occur at LR; the latter being roughly a linear

function of drop size and of the order of 1 m. Before discussing other aspects of turbulence theory,

we give the first direct estimates of the average LR in rain using the HYDROP data.

Fig. 2 shows the 3D isotropic (angle averaged) spectrum of the 19 stereophotographic drop

reconstructions averaged over each of the five storms. Each storm had 3 - 7 “scenes” (from

matched stereographic triplets) with 5,000 – 20,000 drops each taken over a 15 - 45 minute period

(see table 1). The low wavenumber reference line indicates the theoretical Corrsin-Obukhov k-5/3

angle integrated spectrum. Since Gaussian white noise has constant spectral energy density, the

11/29/07 7

integrated spectrum varies as k2 and is indicated by the high wavenumber reference line. It can be

seen that the transition between passive scalar behaviour - where the drops are highly influenced by

the turbulence and the high wavenumber regime, where the drops are totally chaotic (white noise)

occurs at scales of roughly 40 – 75 cm.

More details on the HYDROP experiment can be found in [Desaulniers-Soucy., et al., 2001] and

for these data sets see [Lilley, et al., 2006]. The main differences between the part of the data

analyzed in the latter and below is that in the latter, a very conservative choice of sampling volume

was used. By using only data from the region best lit and most sharply in focus, >90% of the drops

with diameter >0.2mm were identified. In this study, we sought to get a somewhat wider a range of

scales with as many drops as possible by exploiting the fact that fourier techniques are very

insensitive (except at the lowest wavenumbers) to slow falloffs in the sensitivity (and hence drop

concentrations) near the edges of the scene (this is a kind of empirically produced spectral

“windowing” close to the numerical filters used to reduce leakage in spectral estimates). We

therefore took all the data in a roughly rectangular region about 4.4X4.4X9.2 m in size (geometric

mean = 5.6 m), and then analyzed only the spectrum for wavenumbers k ≥ 2 (i.e. spatial scales ≤ 2.3

m). This somewhat larger scene size with respect to the previous HYDROP analyses yielded an

increase in the range of scales by a factor of about 2.8 and about 2-3 times more drops.

11/29/07 8

Fig. 2: This shows the 3D isotropic (angle integrated) spectrum of the 19 stereophotographic drop

reconstructions, for ρ, the particle mass density. Each of the five storms had 3 - 7 “scenes” (from matched stereographic triplets) with ≈ 5,000 – 40,000 drops (see table 2) each taken over a 15 - 30 minute period (orange = f207, yellow = f295, green = f229, blue green = f142, cyan = f145). The data were taken from regions with roughly 4.4 m X 4.4 m X 9.2 m in extent (slight changes in the geometry were made between storms). The region was broken into 1283 cells (3.4 cm X 3.4 cm X 7.2 cm); we use the approximation that the extreme low wavenumber (log10k=0) corresponds to the geometric mean, i.e. 5.6 m, the minima correspond to about 40 - 70 cm; see table 2). The single lowest wavenumbers (k = 1) are not shown since the largest scales are nonuniform due to poor lighting and focus on the edges. The reference lines have slopes -5/3, +2, the theoretical values for the Corrsin-Obukhov (l1/3) law and white noise respectively.

11/29/07 9

Storm 207 295 229 142 145 Number of Triplets/scenes

7 3 3 3 3

Wind speed at 300m (in m/s)

27.5 10 17.5 2.5 22.5

Nominal Rain rate (mm/hr)

6-10 1.4-2.2 2-4 2-4 1.4-2.2

Lmin (m) 0.49±0.08 0.53±0.09 0.44±0.06 0.75±0.11 0.59±0.07

L (mm) 1.29±0.12 1.24±0.06 1.01±0.02 0.98±0.09 1.36±0.07 Number of drops

19300±13100 21400±1400 33000±8200 34800±9800 13000±2500

mean LWC at 70cm scale (g/m3)

0.50±0.63 0.44±0.42 0.39±0.57 2.40±1.35 0.44±0.58

maximum LWC at 70cm scale (g/m3)

3.89 1.65 2.48 5.93 2.16

Wind at 300 m height (m/s)

27.5 10 17.5 2.5 22.5

Mean vR,n

at 70cm resolution in m/s

1.26±0.12 1.32±0.03 1.23±0.06 1.10±0.04 1.17±0.09

Mean vR,! at

70cm resolution in m/s

1.34±0.13 1.40±0.04 1.33±0.03 1.55±0.004 1.27±0.09

Table 1: Comparison of scale of the spectral minimum (Lmin) estimated from the drop mass spectra with the corresponding mean drop diameters; this is an estimate of the mean relaxation length. Fig. 3a shows the individual values, fig. 3b shows the generally small effect of weighting the spectrum to smaller or to larger drops. The spread is the storm-to-storm variability based on 3 triplets per storm (the exception being storm 207 for which there were 7 triplets). As expected from theory, the relaxation time is not dependent of the wind/turbulence level which varied considerably (the data at 300 m altitude are from a UHF windsonde 250 m south of the experiment). Mean (of the 19 triplets) diameter = 1.19±.17 mm (i.e. ±14%; spread is triplet to triplet variation, not standard deviation for each triplet), mean relaxation distance =0.56±0.13 m (i.e. ±23%), mean number of drops 23200 ± 11800 (i.e. ±51%). These numbers are larger than those in [Lilley, et al., 2006] since all the reconstituted drops were used, not only those in the more reliable central region. The LWC statistics is for the well-lit central region only, averaged at 70cm scale (roughly the relaxation scale). To obtain the ratio of LWC to air density, divide by ρa = 1290 g/m3 for air.

11/29/07 10

Fig. 3a: The mean drop diameter L versus the ratio of the mean (mass density based, η = 1) relaxation length lmin (estimated by the spectral minimum), to L for the 19 triplets. The overall mean ratio l

min/ L =

2.8±0.8X102. This result shows that if lmin is used to estimate the mean relaxation length: lR ≈ lmin then the

theoretical formula (eq. 3b) lR!

"3C

D

#

$%&

'()w

)a

#

$%&

'(L is valid to within ±28% for the 19 triplets with CD=2.90.

Fig. 3b: For each storm, this shows the mean distance (in m) corresponding to the minimum of the

spectrum of ρη as a function of the order of moment η (from top to bottom on the right hand side: 142, 145, 295, 207, 229). Since higher η values weight the spectra to the larger drops so that there is a slow increase of

11/29/07 11

Lmin with increasing η. The main exception is the 142 case which is the most dominated by small drops and shows a near doubling in Lmin when comparing the number density (η=0) with the mass density (η = 1).

2.3 Relations between Drop and wind velocities:

In regimes of “weak turbulence” where Du

Dt<< g , and power law drag (ηd >0) we may follow

[Maxey, 1987], [Maxey and Riley, 1983], [Falkovitch and Pumir, 2004], [Falkovitch, et al., 2006]

(who considered ηd = 1), and obtain an expansion for the drop velocity v in terms of the wind

velocity u:

v ! u + TR g "Du

Dt

#$%

&'(+O TR

2D2u

Dt2

#$%

&'(

(4)

although he only investigated the η d = 1 case in detail [Maxey, 1987] pointed out the insensitivity of

this result to the exact form of the drag law (up to first order, it is independent of ηd; note that we

use the notation D / Dt = ! / !t + u "# for the Lagrangian derivative of the wind). We now consider

the limits of validity of this approximation in a turbulent flow when it is applied to eddies at the

mean relaxation scale lR. We should note that this equation implies that even if the wind field is

incompressible, that the rain velocity field is compressible:

! " v =VR

g! " u "!u( ) =

VR

g

# 2

2$ s2

%&'

()*

(5)

where ω is the vorticity, and s is the strain rate [Maxey, 1987].

We now check that eq. 4 is indeed justified for raindrops. According to fig. 2, for l>lR, the

spectrum is nearly that of the theoretical Corrsin-Obukhov passive scalar form so that the standard

turbulence estimate of derivatives at scale l can be used:

Du

Dtl

n

! "1/2#e

1/2$n! %u

l#e

$n; #

e= l

2 /3"$1/3 (6)

where τe is the eddy turn over time (i.e. eddy lifetime) for eddies of size l. Applying this in eq. 4

we obtain:

v ! u + VR

!z " St#u

l( ) +O St2( )#ul ; St =

TR

$e,R

(7)

11/29/07 12

where St is the Stokes number evaluated at the eddy turn over time: !e,R

= LR

2 /3"#1/3 . We see that as

long as St<1, the series converges. To see how strong a constraint this is, we can use the measured

mean relaxation distances from section 2.2 and eqs. 2, 6 to obtain:

St =!R

" e=

LR

g

#$%

&'(

1/2

l)2 /3*1/3 (8)

Note that due to the weak LR dependence above and (roughly linear) dependence of LR on drop

diameter L (c.f. eq. 3b) that the use of a single average/effective lR for all drops may be a reasonable

approximation.

The criterion St<1 also ensures that the eddy scale turbulent accelerations are much less than the

acceleration of gravity; this can be seen by using:

1

g

Du

Dt

!"#

$%&lR

' g(1)1/2* R(1/2

= St( )3/2 (9)

In section 3.3 we discuss direct evidence from drop sondes that the vertical wind accelerations at

scales 5 - 10 m rarely exceed 2% of g. The critical Stokes number St = 1 is thus obtained for eddies

of critical size lc:

lc =LR

g

!"#

$%&

3/4

'1/2 (10)

Using the overall mean value lR ≈ 0.56 m, (TR = 0.24 s, vR = 2.34 m/s; these are the means of

VR(M), ΤR(M)) and the atmospheric mean ε (which is often estimated at 10-4m2s-3), we find lc≈1mm.

According to theoretical considerations in [Wang and Maxey, 1993] and experimental results in

[Fessler, et al., 1994], the Stokes number which has the maximum effect in causing preferential

concentration of particles occurs at St = 1, we thus conclude that at the drop collision scales (≈1mm

here) that the drops are quite decoupled from the wind field, a fact that we use here.

In the model developed below, we use eq. 4 for drop velocities only down to LR, this will be

justified only if St << 1 at that scale. Applying eq. 10 to eddies at the relaxation length, (setting

LR=lc) we obtain:

St =!R

" e,R

= #1/3g$2 /3

!R

$1/3= #

1/3g$1/2LR$1/6 (11)

Hence in terms of the measured mean lR:

St < 1! " < g3/2lR1/2 (12)

Using the above value for the mean relaxation length (inferred from the spectra) lR≈0.56m, we

obtain a limit ε < 25 m2 s-3, which is a very large value for atmospheric turbulence. For example,

11/29/07 13

using standard estimates for multifractal intermittency parameters (e.g. [Schmitt, et al., 1996]), and

taking the scale ratio 107 (planetary scale/ relaxation scale), we find that the probability of

exceeding this “weak turbulence” limit is of the order 10-9). Alternatively, taking the estimate 10-4

m2 s-3 for the mean ε, we find that the eddy turn over time at the mean relaxation length scale is ≈ 15

s so that St ≈ 1.6X10-2.

3. The basic equations:

3.1 The drop mass distribution function N :

Having established that at the relaxation scales, the Stokes number is typically small we can

systematically use eq. 4 to relate the turbulent wind field to the drop velocities. To do this, we first

introduce N = N M,x,t( ) which is the drop mass distribution function i.e. the number of drops with

mass between M and M+dM per unit volume of space at location x, time t. Ignoring condensation,

the usual coalescence (Smolukowski) equation (used for example in cloud and rain modeling

[Srivastava and Passarelli, 1980]) can be written:

!N

!t= N " N (13)

The right hand side term is the coalescence operator in compact notation (see [Lovejoy, et al.,

2004]), N1! N2 :

N1! N2 =

1

2! M " #M ,M,x,t( )N

1M " #M ,x,t( )N

2#M ,x,t( )d #M

0

M

$

-N1

M,x,t( ) ! #M ,M,x,t( )N2

#M ,x,t( )d #M

0

%

$

(14)

If we now assume that the drop-drop collision mechanism is space-time independent then the full

coalescence kernel H can be been factored H !M ,M,x,t( ) = ! x,t( )H !M ,M( ) so that, space-time

variations in the coalescence rate are accounted for by ϕ which is the coalescence speed; we return

to this in section 3.4. The budget equation for N , is thus:

!N

!t= "# $ Nv( ) + ! x,t( ) N H N (15)

The validity of this turbulence-drop coalescence equation can be verified by integrating over a

volume; the left hand side is the change in the volume of particles between M and M+dM, the first

right hand side term is the flux of such particles across the bounding surface, the second is the

change in the number within the volume due to coalescence. In this equation, v(M,x,t) is the

11/29/07 14

velocity of a particle mass M. While this equation was invoked in [Falkovitch and Pumir, 2004],

only the static v=0 case was studied. Eq. 15 is also used in conventional drop size distribution

modeling but typically with the additional assumption of horizontal homogeneity (only smooth

vertical variability is considered).

3.2 The mass and number densities and fluxes:

We now consider the first two moments of the turbulence-drop coalescence eq. (15); the number

density (n) and drop mass density (ρ), as well as their fluxes: the mass flux r and the number flux

! :

n x,t( ) = N M , x,t( )dM

0

!

" ; # x,t( ) = N M , x,t( )MdM0

!

" (16)

! = v M , x,t( )N M , x,t( )dM! (17)

r = Mv M , x,t( )N M , x,t( )dM! (18)

Note that the vertical component of r is the usual rain rate; R=rz. (see section 3.4).

With r, ! we can obtain the budget equations for the particle number and mass densities, by

respectively integrating eq. 15 with respect to M and by multiplying eq. 15 by M and then by

integrating to obtain:

!n

!t= "# $! +% N H N (19)

!"

!t= #$ % r (20)

where we have used the fact that the coalescence operator conserves drop mass:

N H NM = 0 (21)

but not number density n. We see that for the ρ equation (20), coalescence is not directly relevant

whereas it is relevant for n.

We now introduce the average relaxation velocities vR,n

, vR,! weighted by number density

and mass density respectively:

vR,n

=1

nVRM( )N M( )dM! (22)

vR,! =

1

!VRM( )N M( )MdM" (23)

11/29/07 15

We obtain:

! = n!n; !

n= u + v

R,n

"z "

1

g

Du

Dt

#$%

&'(

(24)

r = !"!; "! = u + vR,!!z #

1

g

Du

Dt

$%&

'()

(25)

so that αn, αρ are the effective velocities of the drops with respect to the wind.

Fig. 4 This shows scatterplots for the 70 cm resolution estimates of v

R,! versus vR,n

for the five storms (yellow =207, green =295, blue =229, purple =142, red = 145). The points clustered along the bisectrix have a single drop in the averaging volumes.

It is of interest to consider the spatial variability of vR,! , v

R,n and also how they are related

to each other. Fig. 4 shows the scatter plots obtained from the HYDROP data by using the high Re

theoretical relaxation formula (eq. 3b) with the empirically determined mean drag coefficient CD =

2.90 (fig. 3a). Table 1 shows the storm by storm mean as well as the spread between scenes within

the same storm, averaging over a 70 cm resolution, i.e. at about the relaxation scale. From the

table, we can see that the mean over each storm is fairly stable, whereas from fig. 4 we see that the

11/29/07 16

variability is very large. Although there is clearly no one to one relation between vR,! and v

R,n the

following relation is roughly satisfied: vR,n

= avR,!

b with a = 1.17±0.09, b = 0.70±0.10 (the standard

deviations being the storm to to storm spread in values). This shows that vR,! and v

R,n cannot be

taken to be equal. To get an idea of the spatial variability (important below), we also calculated the

mean horizontal spectra ofvR,! , v

R,n (fig. 5a) although the range of scales is small, the figure shows

that there is evidence that the latter has a roughly Corrsin-Obukhov spectrum at low wavenumbers.

While this result follows if ρ has a Corrsin-Obukhov spectrum (since the vR’s are powers of ρ; see

section 4.4, fig. 8c), note that it is contrary to the usual assumption that vR

is horizontally

homogeneous (being determined by a spatially homogeneous drop size distribution).

To investigate the variability in the vertical, we determined the spectra of the vertical

gradients of vR,! , v

R,n (using finite differences estimates of !v

R,n/ !z , !v

R," / !z ), these are needed

in section 3.3) and are shown in fig. 6a, b. We see that in both cases the spectrum shows a slight

tendency to rise at the higher wavenumbers; if the spectrum is E(kz) ≈ kz-β, then roughly β ≈ -0.3.

Recall that that whenever β<1, that the variance is dominated by the small scales/large

wavenumbers (an “ultra violet catastrophe”). Since the spectrum of !vR/ !z is kz

2 times the

spectrum of vR

, this is consistent with a near Kolmogorov value β ≈ 1.7 in the vertical. Finally, we

find that the spectrum of !vR

2/ !z which has nearly the same β as !v

R,n/ !z (the small differences

are due to intermittency corrections).

Fig. 5a: Horizontal Spectra of the number weighted mean relaxation velocities vR,n with highest wavenumber = (70 cm)-1 showing a roughly k-5/3 (Kolmogorov) spectrum at low wavenumbers (see black reference line), flattening out near the mean relaxation scale at the right. Yellow=207,

Fig. 5a: Same but for the density weighted mean relaxation velocities vR,ρ.

11/29/07 17

green=295, blue=229, purple=142, red= 145.

Fig. 6a: Spectra of the vertical gradients of the number weighted mean relaxation velocities !v

R,n/ !z , showing their slowly increasing

character up to the mean relaxation scale (the maximum at the far right). Yellow=207, green=295, blue=229, purple=142, red= 145.

Fig. 6a: Same but for the density weighted mean relaxation velocities !v

R," / !z .

3.3 The mass and number fluxes:

To put the mass flux vector into a more useful form, we can appeal to the dynamical fluid

equations:

!a

Du

Dt" g = "#p +$#2

u + Fr

# %u = 0

(26)

Where we have used the incompressible continuity equation for u adequate for our purposes, (see

[Pruppacher, 1997]) and Fr is the reaction force of the rain on the wind. The reaction force per

volume of the rain on the air is:

Fr= ! g "

dv

dt

#$%

&'() ! g "

Du

Dt

#$%

&'(+ !

vR,!

g

D2u

Dt2

(27)

plus higher order corrections. We therefore obtain:

g !Du

Dt=

1

" + "a

#p !$#2u ! "

vR,"

g

D2u

Dt2

%&'

()*

(28)

11/29/07 18

We can now note that the mean relaxation scale lR is typically much greater than the turbulence

dissipation scale ldiss so that at the scale lR, we can neglect the viscous term. In addition, since

St<<1 at lR, we can neglect the D2u

Dt2

term. We therefore obtain the approximation:

g !Du

Dt"

1

# + #a

$p (29)

and hence using this in eqs. 24, 25 we obtain:

!n" u +

vR,n

g # + #a( )$p (30)

!"# u +

vR,"

g " + "a( )$p (31)

Finally if we make the hydrostatic approximation ( !p = g " + "

a( )!z ), we see that this reduces to

a simple form:

!

n" u + v

R,n

!z (32)

!

"# u + v

R,"

!z (33)

i.e. the effective velocities are equal to the wind velocities plus the appropriate mean relaxation

speeds. To judge the accuracy of the hydrostatic approximation, we can consider the dimensionless

difference 1! 1

g"a

#p

#z which in non precipitating air at scales above the viscous scale, should be

equal to the dimensionless vertical acceleration: !1

g

Dw

Dt. Although we did not have measurements

during HYDROP, this difference has been evaluated empirically using state of the art drop sondes

with roughly 5 - 10 m resolutions in the vertical during the “NOAA Winter Storms 04” experiment

(north Pacific ocean). It was found that the typical mean value at the surface was about 0.02 with

fluctuations of the order ±0.005 so that, even at 10 m scales we may use the hydrostatic

approximation to reasonable accuracy. In addition, at the higher wavenumbers, the vertical

spectrum of 1! 1

g"a

#p

#z$1

g

Dw

Dt (fig. 7) increases nearly linearly with wavenumber showing its strong

dependence on the small scales. This suggests that the (notoriously difficult to measure) vertical

velocity (the integral of the acceleration) has a spectrum k2 times smaller, i.e. roughly k-1 (see also

the discussion on the vertical velocity in [Lovejoy, et al., 2007a] where similar conclusions are

reached).

11/29/07 19

Fig. 7: This shows the quadratically detrended vertical acceleration compensated by multiplying by k-1,

bottom 4km, 10 m resolution. The flat spectrum corresponding to scales smaller than about 400m corresponds to Ea(k) ≈ k+1 behaviour. The data are average spectra from 16 drop sondes during the Pacific 2004 experiment over the north Pacific Ocean.

Using the hydrostatic approximation we therefore obtain the following equations for the

number and mass densities:

!n!t

+ vR,n

!z + u( ) "#$ = %

n

2

! vR,n( )

2

!z+$v

R,n

g

1

2& 2 % s2'

()*+,%- N H N (34)

!"!t

+ vR,"!z + u( ) #$" = %

"2

! vR,"( )

2

!z+"v

R,"

g

1

2& 2 % s2'

()*+,

(35)

where we note the additional coalescence term in the equation for n.

3.4 The rain rate:

In a similar manner, we can obtain the z component of the mass flux r which is the rain rate

R:

11/29/07 20

R = r( )z= ! w + v

R,!( ) (36)

where for the vertical wind we have used the notation w = u( )z. Although the full ramifications of

this equation for rain will be developed elsewhere, it should be noted that this approximation will

break down in regions where the particle number density n (strongly correlated with ρ) is so small

that there is low probability of finding a drop (see the discussion of the necessary compound

cascade/Poisson model in section 4.3, 4.4 and eq. 61). In this way zero rain rate regions simply

correspond to regions with very small n.

To understand how this may affect the rain statistics, consider the following simple model for

this. Since n and ρ are highly correlated, we may set ρ to zero whenever it is below some threshold

ρt. If for the moment we ignore the vR,ρ term the model is:

R = !tw; !

t=! ! > t

0 ! < t (37)

We now take ρ to have Corrsin-Obukhov statistics (Δρ ≈ lH with H = 1/3 corresponding to a

spectrum k-β ; ignoring intermittency corrections, β = 1+2H= 5/3 as found in the HYDROP

experiment), and w to have H ≈ 0 statistics (β ≈ 1; as inferred indirectly from the drop sonde data,

fig. 7). We seek the spectrum of R. First consider the effect of the threshold. At large scales it

imposes a fractal support which flattens the spectrum; at high wavenumbers it remains smoother

than w so that the product ρtw will have high wavenmber statistics dominated by the vertical wind ≈

k-1, while at low wavenumbers, it will be much shallower (depending somewhat on the threshold

and the value of H). In addition if we confine our rain analyses to regions with no zeroes, then the

break disappears and we expect the roughly k-1 statistics. If we now consider the (neglected) !vR,!

in eq. 36, we see that that it will have the much smoother, nearly Corrsin-Obukhov (β ≈ 5/3)

statistics; its contribution will to the spectrum will thus be dominated by the vertical wind term.

These conclusions have been substantiated by numerical simulations and will be the subject of a

future paper.

This simple model may well be sufficient to explain observations of rain from high

(temporal) resolution raingauges: several studies have indeed found corresponding ω-0.5 spectra at

long times but ω-1 spectra at short times (see e.g [Fraedrich and Larnder, 1993] with transitions

occurring at scales of 2-3 hours (see [de Montera, et al., 2007]).

11/29/07 21

3.5 Energy flux, Number and mass variance fluxes:

In fig. 1, we noted that at scales l>lR, ρ accurately obeyed the Corrsin-Obhukov law for passive

scalars: Eρ(k)≈k-5/3. The reason is that the variance flux ! = "#$2

#t is “conserved” by the nonlinear

terms and can therefore only be dissipated at small scales. To see this, multiply eq. 24 by -2ρ; to

obtain:

! = "#$2

#t= 2$% & r = % & $2'$( ) + $2% &' $ (38)

Integrating over a volume and using the divergence theorem, we see that only the !2" #$!

represents a dissipation of the variance flux. We therefore obtain:

!diss " #2$ %& # " #2'vR,#'z

+vR,#

g( 2/ 2 ) s2( )

*+,

-./

(39)

(with the hydrostatic approximation). These terms are only important at small scales. This is true

of the vorticity/shear term since Eω(k) = Es(k) = k2Eu(k) (the standard result in isotropic turbulence

which is a consequence of the fact that ω, s are derivatives of the velocity). In Kolmogorov

turbulence therefore both ω2 and s2 have roughly k1/3 spectrum. Furthermore, we have seen form

fig. 6 that the spectrum of !vR!z

is also dominated by the small scales and also has a near k1/3

spectrum so that we may expect both dissipation terms to be important. Unlike the variance flux

dissipation in passive scalar advection which is due purely to molecular diffusion (and hence

depends only on the Laplacian of the velocity (the curl of the vorticity), here the dissipation

depends on the field itself (ρ2) but also on the local relaxation velocity, its vertical derivative, the

vorticity and shear (the latter actually reduces the dissipation).

Similarly, for the dissipative part of the number variance flux:

!diss

= "#n2

#t= n

2$ %&n+ 2n' N H N (40)

so we have:

! diss " n2

#vR,n#z

+vR,n

g$ 2/ 2 % s2( )

&'(

)*++ 2n, N H N (41)

(again using the hydrostatic approximation). The dissipation of number density flux thus has a first

term of the same form as the mass density flux, but an addition coalescence term which is actually

cubic in n and which does not depend on spatial gradients. We therefore expect it to contribute to

11/29/07 22

dissipation of ψ over a wide range of scales with intensity strongly dependent on the local drop

concentration. The dissipation mechanism is thus quite different than for ρ.

3.6 Coalescence:

We now consider in more detail the coalescence term:

N ! M, "M ,x,t( ) N = ! x,t( ) N H M, "M( ) N (42)

where we have factored the general kernel ! M, "M ,x,t( ) into a purely mass dependent

H M, !M( ) and a

time-space varying speed term ϕ. The H M, !M( ) kernel describes the basic collision mechanism,

whereas the ϕ determines the speed. Fairly generally, we may write:

! M, "M ,x,t( ) = E M, "M( ) L + "L( )

2#v M, "M ,x,t( ) (43)

where E is the collision efficiency (typically taken as a power law of the mass ratios, c.f. [Beard

and Ochs III, 1995]), and Δv is the difference in velocities of the colliding drops. Following the

empirical spectrum which indicates a rapid transition from drops following the flow at scales larger

than LR to drops with apparently white noise statistics at smaller scales, it is natural to consider the

simplified model that the particles follow the flow down to LR (roughly independent of particle

mass) and then that they are independent at smaller scales. When two particles collide, we may

thus assume that since they decoupled from the flow a distance LR away they have followed roughly

linear trajectories. More precisely, denoting the corresponding positions by x, x’ and masses M, M’.

This velocity difference is therefore:

!v = v M , x( ) " v #M , x#( ) = u x( ) " u x#( ) " TRM( )

Du x( )

Dt" T

R#M( )Du x#( )Dt

$

%

&&&

'

(

))); x " x# = L

R

(44)

where we evaluate the velocity differences at scale LR. If we take M>M’, so that TRM( )>T

R!M( )

we see that the second term is bounded by:

TRM( )

Du x( )

Dt! T

R"M( )Du x"( )Dt

< TRM( )#a; #a =

Du x( )

Dt!

Du x"( )Dt

(45)

11/29/07 23

However; taking !u = u x( ) " u x#( ) = $lR

1/3LR

1/3 and using the estimate (see eq. 6) !a = !u / "e,R

, we

see that the second term is of the order TR/ !

e,R= St times the first. However since the Stokes

number St is <<1 at LR, this can be neglected with respect to the turbulent velocity gradient at the

relaxation scale.

Using the average relaxation scale lR, we therefore obtain the estimate:

!v = !u = "lR

1/3lR

1/3 (46)

which is independent of the particle mass. Using this in the interaction kernel (eq. 43), we obtain:

! x,t( ) N H N = " x,t( )lR

1/3lR

1/3N #

3

4#

!

"####

$

%&&&&

1/3

E M, 'M( ) M1/3 + 'M 1/3( )2N (47)

or:

H M, !M( ) =3!2

4

"

#

$$$$

%

&

'''''

1/3

E M, !M( ) M1/3 + !M 1/3( )2

" x,t( ) = # x,t( )lR

1/3lR

1/3

(48)

i.e. the coalescence speed ϕ is equal to the turbulent velocity gradient at the mean relaxation scale

(the subscript on the integral denotes spatial averaging at the relaxation scale).

Although ε(x,t) and hence the speed ϕ(x,t) is highly variable, it defines characteristic velocities

for the coalescence process: !lR

1/3lR

1/3 . We note that this approximation implies that the interdrop

collision rate is determined by the turbulence and not by the mass dependent relaxation velocities

(as is usually assumed).

4. The scaling laws:

4.1 The Corrsin-Obukhov law:

We have seen in the previous section that except at small dissipation scales, the mass density

variance flux (χ), number density variance flux (ψ) and energy flux (ε) are conserved:

! = !"u

2

"t; " = !

"#2

"t; $ = !

"n2

"t (49)

The standard Kolmogorov and Corrsin-Obukhov laws are obtained by assuming that there is a

quasi constant injection of the passive scalar variance flux χ and energy flux ε fluxes from large

scales and that this is thus transferred without significant loss to the small scales where it is

11/29/07 24

dissipated. Between an outer injection scale and the dissipation scale, there is no characteristic

scale, hence at any intermediate scale l one expects:

!l!"u

l

2

"e,l

; #l!"$

l

2

"e,l

(50)

where Δul is the typical shear across an l sized eddy, and Δρl is the corresponding typical gradient of

passive scalar; the τe,l’s are the corresponding transfer times (the “eddy turnover time”). For both ε,

χ the time scale is determined by the turbulent velocity at the length scale l:

!l

=l

!ul

(51)

This is the time scale for Δρl as well as Δul since neither process is affected by coalescence. This

leads to:

!l!"u

l

3

l; "

l!"#

l

2"u

l

l (52)

Solving for Δul, Δρl, we obtain the classical Kolmogorov and Corrsin Obukhov laws:

!u

l= !

l

1/3l1/3 (53)

!!

l= "

l

1/2#l

"1/6l1/3 (54)

Finally, we might add that by invoking a third property of the equations – that they are “local” in

fourier space, i.e. interactions are strongest between neighbouring scales, we obtain the standard

cascade phenomenology, the basis of cascade models and multifractal intermittency.

4.2 The new number density scaling law

We have noted (section 3.6) a key difference between the mass density and the number density

equations: due to the conservation of mass in coalescence processes, the mass density equation was

independent of coalescence processes while on the contrary the number density was dependent on

the latter.

Whereas the speed of variance flux transfer for ε, χ is the strongly scale dependent Δul the

corresponding speed for the number density variance flux ψ is determined by the weakly scale

dependent speed of the coalescence processes at the scale l:

!

l= "

lR

1/3lR

1/3( )l

! "l

1/3 "#

1/3lR

1/3; # = l / lR

(55)

11/29/07 25

the subscript on the bracketed term indicates spatial averaging at the scale l. We have used the

multiplicative property of the ε cascade so that it factors into a product of a cascade from the

external scale down to scale l with another smaller scale cascade over the range l to lR , i.e. over the

ratio λ. We then use the approximation that at scales below l, that spatial averaging may be

replaced by ensemble averaging and that lR is roughly constant. For such cascades, we have

!"

q = "K q( ) where K(q) is the moment scaling exponent characterizing the statistics of the cascade,

hence !"

1/3 = "K 1/3( )

= l / lR

( )K 1/3( )

; empirically in atmospheric turbulence K(1/3) ≈ -0.07 so that

this “intermittency correction” is small (see e.g. [Schmitt, et al., 1992], [Schmitt, et al., 1996]). For

the coalescence variance flux transfer ψ we therefore obtain:

!l!"n

l

2

"l,R

; "l,R

=l

#l

(56)

so that the ψ cascade is determined by the weakly scale dependent coalescence processes not by the

strongly scale dependent turbulent processes. We thus obtain:

!l="n

l

2

#l

="n

l

2

l$l

1/3l

lR

%

&'(

)*

K 1/3( )

lR

1/3 (57)

Hence for the scaling of the fluctuations in number density at scale l:

!nl"#

l

1/2$l

%1/6lR

%1/6l1/2 (58)

where we have ignored the small intermittency correction with exponent –K(1/3)/2 (which

empirically is of the order 0.03). This l1/2 law (eq. 58) is the key result of this section, in fourier

space (ignoring multifractal intermittency corrections), this implies En(k)≈ k-2 whereas for ρ, we

have the classical Corrsin-Obukhov result: Eρ(k)≈ k-5/3.

4.3 Coupled χ, ψ cascades and the l1/2 law:

If we combine the l1/3 law from Δρ, and the l1/2 law for Δn we obtain simple relation:

!!l

="

l

#l

"

#

$$$$

%

&

'''''

1/2

lR

l

"

#

$$$$

%

&

'''''

1/6

!nl (59)

which has no explicit dependence on the energy flux ε and shows that mass density fluctuations

tend to damp out with respect to the number density fluctuations at larger and larger scales.

11/29/07 26

Eq. 59 raises the question of the statistical couplings between the ε, χ, ψ cascades. While the link

between ε and χ, ψ is not obvious, χ, ψ are indirectly linked through the number size distribution N,

so that they cannot be statistically independent of each other. Indeed, the ratio χ/ψ has dimensions

of mass2 and it seems clear that at least at large enough scales the two should be related by the

(ensemble, i.e. climatological) drop mass variance M

2 :

!

Lext

! "L

ext

M2 (60)

where Lext is the external scale of the cascades. A “strongly coupled” model would take

!

l! "

lM

2 , i.e. it would assume this relation to hold at all scales l, but this is likely to be too

strong an assumption to be realistic.

In order to exploit these statistical relations in order to make stochastic processes with the

corresponding statistics, we may use standard multifractal simulation techniques to simulate ε, χ, ψ

and from them (by fractional integration) to obtain the extra l1/3, l1/2 scalings, the, u, ρ, n fields.

However the n field determines the probability per unit volume of finding a drop; in other words it

should control a compound Poisson-multifractal process. In a future paper, we show how to make

such processes which produces stochastic realizations respecting all the above the statistics, and

which determines implicitly the drop size distributions. For the moment, we note that the

particulate nature of rain in fact imposes a length scale equal to the mean interdrop distance:

linter

= n!1/3 (61)

(this is the actual number density, not its fluctuation Δn). Therefore, wherever linter>lR , lR must be

replaced by linter. This will occur at low rain rates and will modify the low rain rate statistics. This

behaviour in fact provides a natural “cutoff” mechanism for the transition from rain to no rain; in

effect for lower and lower rain rates, the inner scale of the cascade becomes larger and larger. We

can study this using compound cascade Poisson processes.

4.4 The empirical number density law:

In order to empirically test the l1/2 law for the number density, we replaced each drop mass by the

indicator function and produced the spectra shown in fig. 8a.

11/29/07 27

Fig. 8a: Same as previous but for n, the particle number density (calculated using an indicator function

on a 1283 grid. The reference lines have slopes -2, +2, the theoretical values for the l1/2 law and white noise respectively.

We see that the convergence to the low wavenumber theoretical k-2 behaviour (straight line)

occurs at slightly smaller scales than for the k-5/3 behaviour of ρ since n is less variable (smoother)

than ρ. In fig. 8b we show the ratio of the ensemble spectra (all 19 triplets) for Eρ(k), En(k); this

shows that the number density field really is smoother (by about k1/3) than the corresponding

spectrum for the mass density. Because we have taken the ratio of the spectra, the y axis is “blown

up” with respect to the previous; this is quite a sensitive indicator that the basic theory is roughly

correct. Finally, in fig. 8c, we show the exponents of spectra of ρη; η = 0, 1 are the number and

mass density spectra whereas η = 1/6, 7/6 are important for vRn, vR,ρ respectively.

11/29/07 28

Fig. 8b: This shows the ratio of the ensemble spectra Eρ(k)/En(k); for each of the 5 storms, and the overall

ensemble (purple), with theoretical reference line slope -1/3. This is a sensitive test of the prediction of eqs. 54, 58.

11/29/07 29

Fig. 8 c: These are mean β values for the different η’s with the bars indicating storm to storm differences.

The fits were for wavenumbers < (70cm)-1 (the turbulent regime). Note that for η = 0 (number density) and η =1 (mass density), the theoretical predictions (2, 5/3; ignoring intermittency corrections) are well verified since β = 2.02±0.11, 1.74±0.12 respectively.

5. Conclusions:

Cloud and rain physics have been divorced from turbulence theory for too long. While attempts

to unify them at the most fundamental level are very difficult, phenomenological turbulent cascade

models provide an attractive alternative. This is because they have proved to be not only highly

accurate but even indispensable for understanding and modelling both rain and turbulence; they

thus provide the natural framework for explicitly marrying the two. In this paper, we benefited

from two key advances: the general and elegant relation between particle and wind velocities

([Maxey, 1987]) and a new and unique source of data spanning the drop/turbulence scales: the

HYDROP experiment. The latter provides the positions and masses of rain drops in a roughly 10

m3 volume; with Maxey’s relations, they enabled us to convincingly formulate dynamical equations

for both liquid water mass and particle number densities and to empirically verify many of the

assumptions. With the exception of a mean vertical velocity, the liquid water mass equations were

11/29/07 30

identical to those of standard passive scalar advection and therefore predicted that the rain should

follow the Corrsin-Obukhov (l1/3 , k-5/3 law) of passive scalar advection. However, the l1/2 (i.e. k-2

spectrum) number density law was quite new and arises from the fact that while coalescence

processes conserve total mass, they do not conserve total particle numbers. This allows the (weakly

scale dependent) coalescence speed to determine the rate of scale by scale transfer of number

variance flux whereas the speed of the corresponding mass density variance flux depends on the

strongly scale dependent turbulent velocity. In both cases, the HYDROP data was able to verify

that both the l1/3 and l1/2 laws were obeyed with reasonable accuracy.

Although in this paper we only touch on it, an important consequence concerns the rain rate field

R which is the vertical component of the liquid water mass flux vector. Within the hydrostatic

approximation (which is of acceptable accuracy in this context), we showed that R is the product of

the mass density and the sum of the vertical velocity and mean relaxation speed. However, when

the number density (n) field (which is correlated with the ρ field) is sufficiently close to zero, it will

imply zero rain rates since the probability of finding a drop becomes very low (a full study of this

effect must be done using compound Poisson-multifractal processes discussed below). In this way,

a natural rain / no-rain mechanism emerges and we argue that it can quantitatively explain

observations of rain rate spectra.

The conventional methods of modelling the evolution of raindrops (e.g. [Srivastava and

Passarelli, 1980], [Khain and Pinsky, 1995], [Khairoutdinav and Kogan, 1999]) give turbulence at

most a minor (highly “parameterized”) role: the atmosphere is considered homogeneous and the

spatial variability of the drop size distribution arises primarily due to complex drop interactions.

Our approach is in many ways the opposite: down to a small relaxation scale (below which a kind

of white noise “drop chaos” exists), the drops are shown to be strongly coupled with the highly

variable (multifractal) wind field. The coalescence speed is then determined by the turbulent

velocity gradient at this decoupling scale. By considering the zeroth and first moments of the drop

size distribution (n, ρ), and showing how they are related to scale by scale conserved turbulent

fluxes, we obtain strong (but implicit) constraints on the drop size distribution which we expect to

be highly variable simply because it is controlled by turbulent cascades. This idea can be taken

further: if we are given a turbulent number density field, it can be used to control a compound

Poisson process which determines the locations of the drops. Once the positions are determined,

the corresponding ρ field can then be used to attribute masses to the particles. In this way we

obtain a three dimensional model of particle positions and masses which implicitly determines the

highly variable drop size distribution in the vicinity of each point and which respects the turbulence

11/29/07 31

laws and is compatible with the microphysics (see [Lovejoy and Schertzer, 2006] for a preliminary

model). Such models can be used to solve longstanding observers problems in radar meteorology,

satellite rain algorithms, rain amount estimation and cloud physics.

6. Acknowledgements:

This research was carried for purely scientific purposes, it was unfunded.

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