Zdunkowski W., Bott A. Dynamics of the atmosphere: A course in theoretical meteorology
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Answers to problems 701
(b) Replace θ in (22.58) by ρ and add the term (∂/∂ρ)( ˙ρ∂p/∂ρ).
(c) Lower boundary condition: Replace θ
s
in (22.59) by ρ
s
,butρ
s
= 0. Upper
boundary condition: ˙ρ = 0.
(d) Replace θ in (22.61) by ρ and add the right-hand side term ˙ρ∂p/∂ρ.The
integration limits are (0,ρ). Also multiply ∂p/∂t by −1.
22.4:
∂z
H
∂t
=−∇
h
· [v
h
(z
H
− z
s
)]
Chapter 23
23.4:
(a) q(t) = [q
0
− q
g
(0)] exp(−if
0
t) + q
g
(t) −
t
0
dq
g
(t
)
dt
exp[−if
0
(t − t
)] dt
(b) q
0
= q
g
(0) and dq
g
(t)/dt = 0or(d/dt) ∂φ/∂x = 0, (d/dt) ∂φ/∂y = 0.
Hence φ = ax + by + F (p, t), where F (p, t) is an arbitrary function.
Chapter 26
26.5: P
3
(µ) = 5µ
3
/2 − 3µ/2.
List of frequently used symbols
a
i
: Lagrangian enumeration coordinates
c: displacement speed (m s
−1
)
c
p
: specific heat at constant pressure (m
2
s
−2
K
−1
)
c
v
: specific heat at constant volume (m
2
s
−2
K
−1
)
D: divergence (s
−1
)
D: deformation part of the velocity dyadic (s
−1
)
d
g
: geometric differential
D
3
/Dt: budget operator (s
−1
)
e: specific internal energy (m
2
s
−2
)
e
i
, e
i
: covariant and contravariant unit vectors
E: unit dyadic
Eu: Euler number
f : vertical Coriolis parameter (s
−1
)
Fr: Froude number
F
R
: radiative flux (kg s
−3
)
g: acceleration due to gravity (m s
−2
)
g
ij
,g
ij
: covariant and contravariant metric fundamental quantities
√
g: functional determinant
h: specific enthalpy (m
2
s
−2
)
I
k
: phase-transition rate of substance k (kg m
−3
s
−1
)
J: viscous-stress tensor (kg m
−1
s
−2
)
J
h
: enthalpy flux (kg s
−3
)
J
h
s
: sensible enthalpy flux (kg s
−3
)
J
k
: diffusion flux (kg m
−2
s
−1
)
J
k
t
: turbulent diffusion flux of substance k (kg s
−3
)
J
θ
t
: turbulent heat flux (kg s
−3
)
K: exchange coefficient (m
2
s
−1
)
K: specific kinetic energy for relative motion (m
2
s
−2
)
K
A
: specific kinetic energy for absolute motion (m
2
s
−2
)
702
Frequently used symbols 703
k: wavenumber (m
−1
)
L: Lagrangian function (m
2
s
−2
)
l: horizontal Coriolis parameter (s
−1
)
l: mixing length (m)
L
∗
: Monin–Obukhov length (m)
M: mass (kg)
M
k
: partial mass of substance k (kg)
m
k
: concentration of substance k
p: pressure (kg m
−1
s
−2
)
P
E
: Ertel’s potential vorticity (kg
−1
m
4
s
−3
K
−1
)
P
R
: Rossby’s potential vorticity (kg
−1
msK)
Q: entropy production (kg m
−1
s
−3
K
−1
)
q: specific humidity
q
i
, q
i
: covariant and contravariant basis vectors
q
i
,q
i
: covariant and contravariant position coordinates of a vector
R: distance from the earth’s axis (m)
R
0
: gas constant for dry air (m
2
s
−2
K
−1
)
Re: Reynolds number
Ri: Richardson number
Ri
f
: flux Richardson number
Ro: Rossby number
r: position vector
S: surface (m
2
)
s: specific entropy (m
2
s
−2
K
−1
)
St: Strouhal number
T : temperature (K)
t: time (s)
T: stress vector (kg m
−1
s
−2
)
T: general stress tensor (kg m
−1
s
−2
)
u: velocity component in x-direction (m s
−1
)
u
∗
: frictional velocity (m s
−1
)
V : volume (m
3
)
v: velocity component in y-direction (m s
−1
)
v: relative velocity (m s
−1
)
v
A
: absolute velocity (m s
−1
)
v
D
: deformation velocity (m s
−1
)
v
g
: velocity of the geostrophic wind (m s
−1
)
v
h
: velocity of the horizontal wind (m s
−1
)
v
P
: velocity of the point (m s
−1
)
v
T
: translatory velocity (m s
−1
)
704 Frequently used symbols
v
'
: rotational velocity (m s
−1
)
w: velocity component in z-direction (m s
−1
)
W
i
,W
i
: covariant and contravariant measure numbers of v
P
α: specific volume (m
3
kg
−1
)
β: Rossby parameter (m
−1
s
−1
)
-
ij k
,-
k
ij
: Christoffel symbols of the first and the second kind
γ : lapse rate (K m
−1
)
/: energy dissipation (kg m
−1
s
−3
)
ζ : vorticity (s
−1
)
η: absolute vorticity (s
−1
)
λ: degrees of longitude
µ
k
: specific chemical potential of substance k (m
2
s
−2
)
ρ: mass density (kg m
−3
)
ρ
k
: partial density of substance k (kg m
−3
)
: surface Hamilton operator
σ
0
: static stability (kg
−2
m
4
s
2
)
τ : stress vector (kg m
−1
s
−2
)
φ: geopotential (m
2
s
−2
)
φ
a
: gravitational potential (m
2
s
−2
)
φ
z
: centrifugal potential (m
2
s
−2
)
8
: general dyadic
8
: symmetric dyadic
8
: antisymmetric dyadic
8
a
: adjoint dyadic
8
×
: vector of a dyadic
8
I
: first scalar of a dyadic
8
II
: second scalar of a dyadic
8
III
: third scalar of a dyadic
8
−1
: inverse dyadic
ϕ: degrees of latitude
χ: velocity potential (m
2
s
−1
)
ψ: stream function (m
2
s
−1
)
{ψ}: jump of the field function ψ
Ω: angular velocity of the earth (s
−1
)
': rotational part of the velocity dyadic (s
−1
)
ω: angular speed (s
−1
)
ω: generalized vertical speed (kg m
−1
s
−3
)
Frequently used symbols 705
List of constants
β: Rossby parameter (55
◦
latitude) (1.313 × 10
−11
m
−1
s
−1
)
c
L
: speed of sound at 273 K (331 m s
−1
)
γ : gravitational constant (6.672 ×10
−11
Nm
2
kg
−2
)
C
p,0
: specific heat at constant pressure, dry air (1005 J kg
−1
K
−1
)
C
p,1
: specific heat at constant pressure, water vapor (1847 J kg
−1
K
−1
)
C
v,0
: specific heat at constant volume, dry air (718 J kg
−1
K
−1
)
C
v,1
: specific heat at constant volume, water vapor (1386 J kg
−1
K
−1
)
c
2
: specific heat of liquid water (4190 J kg
−1
K
−1
)
c
3
: specific heat of ice (2090 J kg
−1
K
−1
)
f : vertical Coriolis parameter (55
◦
latitude) (1.195 × 10
−4
s
−1
)
l: horizontal Coriolis parameter (55
◦
latitude) (0.836 × 10
−4
s
−1
)
l
21
: latent heat of vaporization (2.5 × 10
6
Jkg
−1
)
l
31
: latent heat of sublimation (2.834 × 10
6
Jkg
−1
)
l
32
: latent heat of melting (0.334 × 10
6
Jkg
−1
)
M
E
: mass of the earth (5.973 × 10
24
kg)
µ:Lam
´
e coefficient (293 K) (1.815 ×10
−5
kg m
−1
s
−1
)
R
∗
: universal gas constant (8.314 32 J mol
−1
K
−1
)
R
0
: gas constant of dry air (287.05 J kg
−1
K
−1
)
R
1
: gas constant of water vapor (461.51 J kg
−1
K
−1
)
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