Houze Robert A., Jr. Cloud Dynamics
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List of Symbols xix
@;
three-dimensional convergence of the eddy flux of
momentum
@;
H horizontal component of the convergence of the eddy flux
of momentum (turbulence term in equation for
VH)
F(ffi
T
)
fraction of a rain area covered by rain by rates exceeding
the threshold rain rate
ffi
T
F B buoyancy source for averaged pressure perturbation
F D dynamic source for averaged pressure perturbation
F M turbulence source for averaged pressure perturbation
Fr
Froude number
j Coriolis parameter
f fraction of a final mixture of two parcels of air constituted
by one of the two initial parcels
G antenna gain
C{j
generation of eddy kinetic energy
g magnitude of gravitational acceleration
g
In
particle mass distribution function
I'
circulation
I',
circulation of a vortex
'Y
==
-af/az,
basic-state lapse rate
'YE
==
Vj/2K
m
H a vertical distance; in Chapter 5, H represents the
half-depth of an ice-cloud outflow from cumulonimbus; in
Chapter 12, it represents the vertical thickness of a layer
of homogeneous fluid
H,
==
Pf(pg)
iI
heating rate
1 ( A
)K
'if
==
C
p
~
iI
rje
I contribution to D
OlDt
by infrared radiation
'ifL contribution to
DO
/Dt by latent heating
is
contribution to
DO
/Dt by solar radiation
iT
contribution to
DOlDt
by turbulent mixing
h a vertical distance; in Chapters 2-11, it always represents
a vertical depth of a layer of fluid; in some discussions it is
specifically the height of the top of the planetary boundary
layer; in Chapter 12 it is used to represent the height of
topography
It moist static energy
h
a
amplitude of two-dimensional topography
he depth of the Ekman layer
h,
amplitude of Fourier component of function describing
terrain height
XX List of Symbols
A
==
c.T
+ p a +
Lq.,
h height of terrain
he
a positive number dependent on Froude number and fluid
thickness that must exceed the terrain height if a
two-dimensional flow is not to be blocked
t: maximum height of a mountain
TJ
x-component of vorticity
TJ
r radar reflectivity
It(m)
rate of decrease of the number concentration of drops of
mass m as a result of their coalescence with drops of all
other sizes
hem) rate of generation of drops of mass m by coalescence of
smaller drops
i
VH
van't
Hoff factor
i,
j,
k unit vectors in the
x,
y, and z directions
Jv vertical flux of water vapor in soil
Jw vertical flux of liquid water in soil
K constant turbulent exchange coefficient
'J{ eddy kinetic energy
K
A
turbulent exchange coefficient for an arbitrary quantity
Yl
K;
collection of cloud water
K
H
turbulent exchange coefficient for q n
K; turbulent exchange coefficient for water category i
K
L
turbulent exchange coefficient for liquid water
K;
turbulent exchange coefficient for horizontal momentum
K v turbulent exchange coefficient for water vapor
K
o
turbulent exchange coefficient for
()
K~
turbulent exchange coefficient for vorticity
K
==
kl
pc
(Chapter 2); collection kernel for a particle of mass
m collecting a particle of mass m' (Chapter 3)
IKI2
a function of the complex index of refraction
k wave number in x-direction
k
B
Boltzmann's constant
k, horizontal wave number in the x-direction of a Fourier
component (denoted by s) of function describing terrain
height
K Rd/c
p
K a thermal conductivity of air
K
s
thermal conductivity of soil
i<
thermal conductivity
L latent heat of vaporization
X entrainment rate (per unit time)
L
J
latent heat of fusion
L,
latent heat of sublimation
La[
along-front length scale
List of Symbols xxi
L
ef
cross-front length scale
LDR linear depolarization ratio
LFC
level of free convection
L wind scale height
;£ entrainment rate (per unit time)
fe a constant value of ;£
I wave number in y-direction
e when used as a unit, e stands for liter; when used as a
mathematical symbol, it is the square root of the Scorer
parameter
e
2
Scorer parameter
eG cube root of grid volume
eL
e~
Scorer parameters for lower and upper layers, respectively
A entrainment rate (per unit height)
A wavelength
AR Rossby radius of deformation
A
r
slope parameter of Marshall-Palmer particle size
distribution
M
==
v +
[x,
absolute momentum
.M ==f(M - M
o
)
M g geostrophic absolute momentum
M,
molecular weight of dissolved salt
M w molecular weight of water
Mev efficiency factor for evaporation
m mass of a cloud or precipitation particle
m mass of a rising cloud element
m in some discussions, m represents a wave number in the
z-direction; in other discussions it is the angular
momentum about the axis of a cylindrical coordinate system
m, mass of a crystal of type k
m,
==
Vel - k
2
m, mass of dissolved salt (Chapter 3); vertical wave number
of a Fourier mode (denoted by
s) of the function
describing the vertical velocity of the air flowing over
two-dimensional topography (Chapter 12)
rit time rate of change of m
rit
col
time rate of change of m as a result of collection
rit
eond
time rate of change of mass of a drop of mass m as a
result of condensation of vapor
rit
dif
time rate of change of m as a result of vapor diffusion
rit
me1
time rate of change of mass of an ice particle as a result of
melting
(Amh
mass of air detrained to the environment
(Am)e mass of air entrained from environment
/Lf vertical mass flux in a
jet
at height z
xxii List of Symbols
net rate of production of drops of mass m by breakup
f.L/ Gibbs free energy of a liquid molecule
f.Lu
==
Vk
2
-
e~
f.Lv Gibbs free energy of a vapor molecule
{L2
==
-m
2
N in discussion of dynamics, N is buoyancy frequency of the
mean state; in discussions of cloud microphysics and radar
meteorology,
N is the particle size distribution function
(number density)
N consumption of eddy kinetic energy
N,
number of ice nuclei per liter of air
9( i time rate of change of the concentration of drops per unit
size interval in size range
i resulting from nucleation from
the vapor phase
particle size distribution function for crystal type
k
buoyancy frequency of reference state, except in Chapter
3, where No represents the intercept parameter of
Marshall-Palmer particle size distribution
total concentration of drops per unit volume
change in the number density of drops of mass m by
stochastic collection
~)COI
~)bre
IV
buoyancy frequency in pseudoheight coordinate system
n coordinate normal to a streamline
n mass of a cloud condensation nucleus
n, number of molecules per unit volume of ice
n, number of water molecules per unit volume of liquid
v frequency
n angular speed of the earth's rotation
W vorticity
W
a
==
lJi +
~j
+ ({
+f)k,
absolute vorticity
WH
horizontal vorticity
Wag
absolute geostrophic vorticity
P Ertel's potential vorticity, except in Chapter 4 where P
represents a probability density function
PB(m) probability that a drop of mass m breaks up per unit time
P e equivalent potential vorticity
I!P
g geostrophic potential vorticity in geostrophic space
Pg geostrophic potential vorticity in physical space
I!P
m
==
I!P
eg if air is saturated,
I!P
g if unsaturated
P t transmitted power
I!P
eg geostrophic equivalent potential vorticity in geostrophic
space
Peg
geostrophic equivalent potential vorticity
List of Symbols xxiii
PRF
pulse-repetition frequency
P probability
that
a
drop
of
mass m will collect a drop
of
mass
m'
in time interval
Ii
t
P
r
average returned
power
P pressure
of
air
Pc central
pressure
of
a hurricane
p reference
pressure,
usually representative
of
conditions
near
the
earth's
surface, often taken to be 1000 mb
p~
pressure
perturbation associated with buoyancy field
Pi>
pressure perturbation associated with wind field
<I>
geopotential
<i>
==
<I>
+
v~/2
cf>
latitude
cf>'
velocity potential for perturbation flow
cf>r
phase
of
reflected
radar
waves
</l
ratio
of
mass
of
water
evaporated by entrainment to the
total mass
of
water
of
a rising parcel
of
air
II
Exner
function
7T
==
P*/Po
.ps
soil moisture potential
.p,
'1',
'1"
stream
functions
Q Q-vector
Qh
Q2 components
of
Q-vector
Q; Q-vector
component
in geostrophic
(X)
space
Q} integrated effect
of
turbulence in a volume containing a
hydraulic
jump
QB(m' .m)
dm
number
of
drops
of
mass m to m + dm formed by the
breakup
of
one
drop
of
mass
m'
Qc
rate
at which hailstone loses
heat
to air by conduction
o,
rate
at which
heat
is gained as a result
of
the riming of a
hailstone
Qs
rate
at which a hailstone gains heat by deposition
Q
==
liH
in a homogeneous fluid
qc mass
of
cloud liquid
water
per
unit mass
of
air
qd mass
of
drizzle
per
unit mass
of
air
qg mass
of
graupel
per
unit mass
of
air
a» total mass
of
liquid
water
and/or
ice per unit mass
of
air
(hydrometeor mixing ratio)
qh mass
of
hail
per
unit mass
of
air
q/ mass
of
cloud ice
per
unit mass
of
air
q, mass
of
water
of
type i
per
unit mass
of
air (mixing ratio
of
a given
type
of
water
substance)
qL mass
of
liquid
water
per
unit mass of air
qm' mass
of
liquid
water
contained in drops of mass m' per
mass
of
air
xxiv List of Symbols
qr mass of rainwater per unit mass of air
qs mass of snow per unit mass of air
q
s vapor-mixing ratio of soil
q T mass of total water substance per unit mass of air
qv mass of water vapor per unit mass of air (mixing ratio of
water vapor in air)
q vs saturation mixing ratio
R radius of a spherical particle (Chapter 3); radius of a cloud
(Chapter 7); radius of a downdraft (Chapter 8)
7ft
radiative heat flux in the vertical (positive upward)
ffi
rainfall rate
R;
critical radius of a drop
R
d
gas constant for dry air
R,
radius of curvature of a streamline
R v gas constant for a unit mass of water vapor
ffi
area
area-integrated rain rate
Rei critical radius of an inscribed sphere used to express the
volume of an ice particle
il1
T
threshold rain rate
RH
a
surface relative humidity in the vicinity of but not within a
hurricane
Ra Rayleigh number
RH relative humidity
Ri Richardson number
R
=0
NLlU
o
R a constant representing the e-folding radius of a turbulent
jet
(ffi)o
average rain rate in some region
(ffi)T
average rain rate within regions where the threshold rain
rate
ffi
T
is exceeded
r radial coordinate (called range in the context of radar
meteorology)
r a radius of the outer boundary of a hurricane
r
c
radius of a circle described by the intersection of the
cone with a level surface at a fixed altitude above a
radar
r. radius of a jump in fluid depth at the initial time (Chapter
2); radius of the inner region of a vortex (Chapter 8)
r max maximum range at which a target can be detected by a
radar
P density
Pa
density of the air, including gaseous components only
PL density of liquid water
Ps density of soil
List of Symbols xxv
p; vapor density (mass of water vapor per unit volume of air)
Pvs
saturation vapor density over a plane surface of water
Pvsfc
vapor density at surface of a particle
p density of a constant reference state representative of
conditions near the earth's surface
S
==
aa
z
(ui + vj), vertical shear of the mean horizontal wind
S surface surrounding volume
"If
Sc sources plus sinks of cloud water associated with
ice-phase microphysical processes
S
g sources plus sinks of graupel associated with ice-phase
microphysical processes
S
J sources plus sinks of cloud ice associated with ice-phase
microphysical processes
S
i sources plus sinks of a particular category of water
S
r sources plus sinks of rainwater associated with ice-phase
microphysical processes
S
s sources plus sinks of snow associated with ice-phase
microphysical processes
S
v sources plus sinks of water vapor associated with
ice-phase microphysical processes
SST sea-surface temperature
S(V
R) Doppler velocity spectrum
9'(ffi
T
)
proportionality factor relating
(~mo
and F(ffi
T
)
S
==
e(00)/es(
()()
-1,
ambient supersaturation with respect to a
plane surface of liquid water
Si
==
e(00)/e
siC
00)
-1,
ambient supersaturation with respect to a
plane surface of ice
S saturated moist entropy
(T
effective radar backscatter cross section of one particle
(Til
free energy of an ice-liquid interface
(T
vi surface energy (surface tension) of a liquid-vapor interface
T temperature
TB temperature at the top of the boundary layer
TG temperature at which a particular crystal habit grows
To
temperature of outflow from hurricane eyewall
;if s temperature of the soil
T, temperature at which a parcel of air at temperature T,
water vapor mixing ratio
qv,
and pressure p would become
saturated by lowering its pressure dry adiabatically
Tv virtual temperature
Tw temperature of a drop of water
to mean temperature of outflow from hurricane eyewall
t time
XXVI
List of Symbols
T A vertical eddy flux of
.stl
T e time scale of evaporation
T m time scale of mixing
T p duration of an emitted radar pulse
T x vertical flux of momentum in x-direction
T y vertical flux of momentum in y-direction
e azimuth angle of a cylindrical coordinate system
()
potential temperature
() a apparent potential temperature perturbation
()e equivalent potential temperature
() H horizontal beamwidth angle
()v virtual potential temperature
()ec equivalent potential temperature at the center of a hurricane
()es saturation equivalent potential temperature
() cv cloud virtual potential temperature
o potential temperature of a constant reference state
representative of conditions near the earth's surface
()
v vertical beamwidth angle
au
user-chosen weighting function that assures homogeneity
of the dimensions in the integral computed in the retrieval
of thermal properties of the air from Doppler radar
observations and decides the relative weight to be ascribed
to vertical and horizontal gradients in the integral
_ tV
aV
g
u, = 7 az + u;
U,
speed of movement of the leading edge of a gravity current
V 0 mean wind at the ground
Ue
along-front velocity scale
V cf cross-front velocity scale
VI,
V 2 horizontal velocity components of parcels in upper and
lower levels of a fluid, respectively
u radial velocity component
U individual wind component in x-direction
Us wind speed in the layer above a lower-tropospheric layer
in which wind speed increases with height
V terminal fall speed (>0)
"If volume
Va
magnitude of the horizontal wind velocity component in
the azimuthal direction from a radar
VF ventilation factor
V
k
fall speed of crystal of type k
Vp volume of a rising parcel of air
VR radar radial velocity (component of velocity of target
along the beam of a radar); special case of v
V
s
wind speed at distance
R,
from the center of a vortex
List of Symbols xxvii
VT fall speed of targets affecting radial velocity detected by
Doppler radar
V
Fc
ventilation factor for conduction
VF.I ventilation factor for sublimation
Vice fall speed scale of ice crystals and snow
Vmax magnitude of the maximum unambiguous Doppler radar
radial velocity
"V
res radar resolution volume
V
c
= uci + vcj cloud or cell motion relative to the ground
V
p
storm propagation velocity component resulting from new
cell development
V.I
velocity of a multicell thunderstorm
V
c
velocity of an individual cell
"V
e
volume of air entrained into a rising parcel of air
V mass-weighted fall speed
VH mass-weighted mean-particle fall speed for ice-particle
hydrometeors
v three-dimensional velocity of a parcel of air
v wind component in y-direction
v tangential velocity component
Va
==
(u - ug)i + (v - vg)j, ageostrophic wind
v
g
==
Ugi + Vgj, geostrophic wind
VH
horizontal wind vector
vn wind component normal to a boundary
v
t wind component tangent to a boundary
v
max
maximum mean tangential velocity in a vortex
W
==
wfl'ag
OW'
creation of 'J{ by pressure-velocity correlation
W wind component in z-direction
W B background vertical velocity
We
entrainment velocity
Web
entrainment velocity at cloud base
Wet
entrainment velocity at cloud top
WI,
W
u
vertical velocities in lower and upper layers, respectively
w deviation from background vertical velocity
WB
(Chapter
5); amplitude of sinusoidally varying vertical velocity
(Chapter 12)
w
==
Dal
Dt,
vertical velocity in pseudoheight coordinate system
8W difference in the vertical velocities of two parcels of fluid
separated by a horizontal radial distance
X
==
x + vglf, geostrophic coordinate
XA
constant used in the parameterization of the vertical eddy
flux of
sIJ
in a cloud
x horizontal coordinate
g y-component of vorticity
xxviii
List of Symbols
a
+w-
az
a
_a
..»
_a
==-+u-+v-+w-
at
ax
ay
sz
a a a
==
at
+ u
g
ax
+ v
g
ay
a a a
==
at
+ (u
g
+ u
a
)
ax
+ v
g
ay
a a a
==
aT
+ U
g
ax + v
g
ay
horizontal coordinate
radar reflectivity factor, except in
alaZ
(see below)
equivalent radar reflectivity factor
differential reflectivity
radar reflectivity factor that is horizontally transmitted and
horizontally received
radar reflectivity factor that is horizontally transmitted and
vertically received
radar reflectivity factor received in circular polarization of
the opposite sense as that transmitted (orthogonal
component)
radar reflectivity factor received in circular polarization of
the same sense as that transmitted (parallel component)
radar reflectivity factor that is vertically transmitted and
vertically received
Z
height'
13
pseudoheight
Zb height of the base of an ice-cloud outflow from
cumulonimbus
Zf
height of top of vortex funnel
Zm height of the middle of an ice-cloud outflow from
cumulonimbus
zo
height of lower tip of vortex funnel
z. height of the top of a lower-tropospheric layer in which
wind speed increases with height
ZT height at which
()
=
"0
z, height of the top of an ice-cloud outflow from
cumulonimbus
z-component of vorticity
vertical component of geostrophic vorticity
vertical component of geostrophic absolute vorticity
derivative operator for a variable that is a function of time
only
==
:t
+ v· V, total derivative
D
g
Dt
D
A
Dt
1)
1)T
Zpar
Zvv
Zorth