North, Gerald R., Erukhimova Tatiana L. Atmospheric Thermodynamics: Elementary Physics and Chemistry
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Problems 247
ρ density (kg m
−3
)
S,dS surface, surface element vector
v three-dimensional velocity vector field (m s
−1
)
V upper case usually indicates two-dimensional velocity
Problems
9.1 By writing out the components, prove that
d
dt
(a · b) =
da
dt
· b + a ·
db
dt
.
9.2 A horizontal wind is blowing from southwest to northeast at 30
◦
north of east. Express
it in terms of i, j, k.
9.3 Find a unit vector 45
◦
north of east. Find a unit vector perpendicular to this one in the
x–y plane.
9.4 The vectors 2i +3j,4i −k determine a plane. Find a unit vector perpendicular to this
plane.
9.5 A parcel moves along a path:
y = y
0
+ α sin ωt
x = x
0
+ α cos ωt + γ t.
If γ = 0, what is the curve that describes the motion? Describe the motion when
γ>0. What is the position vector as a function of time? What is the velocity vector?
9.6 A certain pressure field varies spatially as
p(x, y, z) = p
0
+ A sin
3
R
x cos
π
2L
y e
−z/H
.
How many maxima are there at the surface (z = 0) and at the center of the channel
(y = 0) as one goes around the circle (x goes from 0 to 2π R)? What is the total
differential of the pressure? What is the pressure gradient?
9.7 The wind has the form v = (v
0
/r
) n
θ
, n
θ
=−sin θi + cos θj. What are u, v? What
is the divergence of this wind, div
2
(v)?
9.8 A certain surface pressure field is given by
p(x, y) = a(x − x
0
)
4
+ b(y − y
0
)
2
, a, b > 0.
Describe this pressure surface near (x
0
, y
0
). What is the pressure gradient as a function
of x and y? What is the directional derivative in the direction 45
◦
north of east?
9.9 The 500 hPa level is given by
Z
500
= 5500 + 50 sin(φ − ωt) cos
πy
2L
, −L ≤ y ≤ L
where the vertical heights are in meters and the longitudinal distances are in kilometers.
L is about 30
◦
of latitude. Suppose ω = 2π radians per month. Describe this
disturbance.
248 The thermodynamic equation
9.10 Suppose the diabatic heating rate for a parcel is 10 W kg
−1
and the wind is blowing
at an angle of 60
◦
north of east at 10 m s
−1
. Also suppose the temperature gradient is
0.1Kkm
−1
and directed due east. You may assume the horizontal gradient of pressure
is zero. Find the rate of change of temperature along the motion and find the local rate
of change of the temperature at a fixed point.
9.11 Sinking air. Suppose the atmospheric profile is isothermal. A parcel is descending at
2cms
−1
and the diabatic heating rate is zero. What is the rate of temperature increase
along the motion; at a fixed altitude? (Use reasonable values for parameters not given.)
9.12 Dry air is blowing from west to east at a speed of u = 12ms
−1
. The temperature
gradient is ∇T = 2.0 ×10
−6
(Km
−1
) i. The air is being heated diabatically at a rate
1.1 × 10
−5
Kkg
−1
s
−1
. The air moves along an isobaric surface.
(a) Write an expression for the rate of change of the temperature of the air along its
path in terms of the heating rates (thermodynamic equation).
(b) Based upon (a) evaluate the material derivative.
(c) The air is passing over a fixed station. Find the rate of change of the temperature
at the station.
9.13 Show that the two forms (9.85, 9.87) of the thermodynamic equation are equivalent
by using θ = T
(
p/p
0
)
−κ
.
Appendix A
Units and numerical values of constants
The units used in atmospheric science are the Standard International (SI) units.
These are essentially the MKS units. The units of length are meters, abbreviated
m; those for mass are the kilogram, abbreviated kg; and for time the unit is the
second, abbreviated s. The units for velocity then are m s
−1
. The unit of force is
the newton (1 kg m s
−2
, abbreviated N). Tables A1–A3 show the SI units for some
basic physical quantities commonly used in atmospheric science.
The unit of pressure is the pascal (1 N m
−2
=1 Pa). This is of special importance
in meteorology. In particular, atmospheric scientists like the millibar (abbreviated
mb), but in keeping with SI units more and more meteorologists use the hectopascal
(abbreviated hPa; 1 hPa = 100 Pa = 1 mb). The kilopascal (1 kPa = 10 hPa) is the
formal SI unit and some authors prefer it. One atmosphere (abbreviated 1 atm) of
pressure is
1 atm = 1.013 bar
= 1013.25 mb
= 1013.25 hPa
= 101.325 kPa
= 101325 Pa
= 1.01325 × 10
5
Pa (A1)
and 1 mb = 1 hectopascal = 100 Pa. Sometimes one encounters pressure in inches
of mercury (in Hg) or millimeters of mercury (mm Hg); 1 atm = 760.000 mm Hg
= 29.9213 in Hg.
The dimensions of a quantity such as density, ρ, can be constructed from the
fundamental dimensions of length, mass, time and temperature, denoted by the
letters L, M , T , θ respectively. The dimensions of density, indicated with square
brackets [ρ], are ML
−3
. In the SI system its units are kg m
−3
. Many quantities are
pure numbers and have no dimension; examples include arguments of functions
such as sine or log. The radian is really not a unit in the sense used here.
249
250 Appendix A
Table A1 Useful numerical values
Universal
gravitational constant (G) 6.673×10
−11
Nm
2
kg
−2
universal gas constant (R
∗
) 8.3145 J K
−1
mol
−1
Avogadro’s number (N
A
) [gram mole] 6.022 ×10
23
molecules mol
−1
Boltzmann’s constant (k
B
) 1.381 ×10
−23
JK
−1
molecule
−1
proton rest mass 1.673 ×10
−27
kg
electron rest mass 9.109 ×10
−31
kg
Planet Earth
equatorial radius 6378 km
polar radius 6357 km
mass of earth 5.983×10
24
kg
rotation period (24 h) 8.640×10
4
s
acceleration of gravity (at about 45
◦
N) 9.8067 m s
−2
solar constant 1370 W m
−2
Dry air
gas constant (R
d
) 287.0 J K
−1
kg
−1
molecular weight (M
d
) 28.97 g mol
−1
speed of sound at 0
◦
C, 1000 hPa 331.3 m s
−1
density at 0
◦
C and 1000 hPa 1.276 kg m
−3
specific heat at constant pressure (c
p
) 1004 J K
−1
kg
−1
specific heat at constant volume (c
v
) 717JK
−1
kg
−1
Water substance
molecular weight (M
w
) 18.015 g mol
−1
gas constant for water vapor (R
w
) 461.5 J K
−1
kg
−1
density of liquid water at 0
◦
C 1.000×10
3
kg m
−3
latent heat of vaporization at 0
◦
C 2.500×10
6
Jkg
−1
STP T = 273.16 K, p = 1013.25 hPa
Table A2 Selected physical quantities and their units
Quantity Unit Abbreviation
mass kilogram kg
length meter m
time second s
force newton N
pressure pascal Pa = Nm
−2
=0.01 hPa
energy joule J
temperature degree Celsius
◦
C
temperature degree Kelvin K
speed m s
−1
density kg m
−3
specific heat J kg
−1
K
−1
Units and numerical values of constants 251
Table A3 Selected conversions to
SI units
Quantity Conversion
energy/heat 4.186 J =1 cal
1kWh=3.6×10
6
J
pressure 1 atm =760 mm Hg
1 atm =29.9213 in Hg
distance 1 m =3.281 ft
temperature T (K) = T (
◦
C) + 273.16
Each side of an equation must have the same dimensions. This principle can
often be used to find errors in a problem solution.
Appendix B
Notation and abbreviations
atm pressure unit, one atmosphere (Chapter 1)
a, b, etc. arbitrary vectors (Chapter 9)
a, a
x
vector acceleration and its x component (Chapter 9)
a Bohr radius (Chapter 2)
a droplet radius (m) (Chapter 5)
a, b empirical coefficients used in van der Waals equation of state
(Chapter 5)
a
∗
critical droplet radius (m) (Chapter 5)
a
x
, a
y
, a
z
the Cartesian components of vector a (Chapter 9)
A a vector field (Chapter 9)
A horizontal area of a slab (m
2
) (Chapter 6)
c speed of light in vacuum (m s
−1
) (Chapter 8)
c
v
, c
p
specific heats (heat capacity per kg) at constant volume, pressure
(J kg
−1
K
−1
) (Chapter 3)
c
v
, c
p
molar specific heats (J mol
−1
K
−1
) (Chapter 3)
c
v
, c
p
, G overbar indicates quantities expressed per mole (Chapter 4)
CAPE convective available potential energy (J kg
−1
) (Chapter 7)
CIN convective inhibition energy (J kg
−1
) (Chapter 7)
C
v
, C
p
heat capacities at constant volume, pressure (J K
−1
) (Chapter 3)
div
2
V divergence of vector field V in two dimensions (Chapter 9)
dH /dt rate of heating (Chapter 9)
dH /dt,dQ/dt time rate of change of enthalpy, heating rate (J s
−1
) (Chapter 3)
(dH )
p
change in enthalpy at constant pressure (J) (Chapter 3)
dQ
rev
infinitesimal absorption of heat, subscript indicating that the
change is reversible (J) (Chapter 4)
dQ
M
/dt rate of heating per unit mass (J kg
−1
s
−1
) (Chapter 9)
d
¯
Q,d
¯
W differentials for heat, work, bar emphasizes path dependence (J)
(Chapter 3)
dS
U ,V
infinitesimal change in entropy during which U and V are held
constant (J K
−1
) (Chapter 4)
252
Notation and abbreviations 253
D/Dt material derivative (Chapter 9)
f
H
◦
(X ) standard heat of fusion of substance X (J mol
−1
) (Chapter 3)
G
◦
change in Gibbs energy per mole, at standard conditions
(kJ mol
−1
) (Chapter 8)
V change in volume (m
3
) (Chapter 3)
vap
H
◦
(X ) standard heat of vaporization of substance X , overbar indicates
1 mol of substance, superscript o indicates at standard
conditions (usually 25
◦
C) (J mol
−1
) (Chapter 3)
x displacement in x (Chapter 3)
H
◦
change in enthalpy per mole, at standard conditions (kJ mol
−1
)
(Chapter 8)
t time interval (s) (Chapter 2)
e, e
s
vapor pressure, saturation vapor pressure (Pa) (Chapter 5)
e
vapor pressure over a solution (Pa) (Chapter 5)
E
act
activation energy (J) (Chapter 8)
E energy (J) (Chapter 8)
= M
w
/M
d
= 0.622 (dimensionless) (Chapter 5)
η mixing ratio (Chapter 8)
f frequency of electromagnetic wave (Hz) (Chapter 8)
f number of degrees of freedom of a molecule (Chapter 3)
f
x
partial derivative with respect to x (Chapter 3)
F force (N) (Chapters 2, 3)
F energy flux (J s
−1
) (Chapter 8)
F
λ
(z) flux of electromagnetic energy at wavelength λ, elevation z
(J s
−1
m
−1
) (Chapter 8)
F, F/
M
force, per unit mass (N kg
−1
) (Chapter 7)
F force (N) (Chapter 4)
g, l, aq gaseous, liquid, aqueous phase (Chapter 8)
g acceleration due to gravity (9.81 m s
−2
) (Chapter 6)
g Gibbs energy per kilogram (J kg
−1
) (Chapter 4)
g, g
z
, g
0
acceleration due to gravity, its value as a function of altitude, its
value at the surface (m s
−2
) (Chapter 1)
g
l
, g
g
specific Gibbs energy for liquid, gas (J kg
−1
) (Chapter 5)
g
v
, g
w
specific Gibbs energy for vapor, liquid water (J kg
−1
) (Chapter 5)
G gravitational constant (Chapter 1, Table A1)
G Gibbs energy (J) (Chapters 4, 8)
G Gibbs energy per mole (J mol
−1
) (Chapter 4)
γ ratio of specific heats c
p
/c
v
(dimensionless) (Chapter 3)
254 Appendix B
d
,
m
,
e
lapse rate, −dT /dz of dry air ascending adiabatically, of moist
adiabat, of the environment (K m
−1
) (Chapter 6)
∇T ∇T = (∂T /∂x)i +···, gradient of T (Chapter 9)
h height (Chapter 1)
h height above a reference level (Chapter 6)
h specific enthalpy (J kg
−1
) (Chapter 5)
h Planck’s constant (J s) (Chapters 2, 8)
h, u specific enthalpy, internal energy (J kg
−1
) (Chapter 7)
h
0
initial height (Chapter 2)
H enthalpy (J) (Chapters 3, 4)
H scale height (Chapter 6)
H (r) flux of heat crossing the surface of a sphere (J s
−1
) (Chapter 5)
H
a
scale height of the atmosphere (Chapter 3)
H
w
a scale height for water vapor (m or km) (Chapter 6)
H
◦
standard enthalpy (kJ mol
−1
) (Chapter 8)
i, j, k the Cartesian unit vectors (Chapter 9)
J photodissociation coefficient (s
−1
) (Chapter 8)
k reaction coefficient (Chapter 8)
k
B
Boltzmann’s constant (Chapters 1, 2, 3, Table A1)
K, K
p
equilibrium constant (Chapter 8)
K
H
constant in Henry’s Law (Chapter 8)
K
kinetic energy (J) (Chapter 7)
κ = R/c
p
(dimensionless) (Chapters 3, 4, 6) (0.286 for dry air)
κ
H
thermal conductivity coefficient (J m
−1
s
−1
K
−1
) (Chapters 3, 5, 9)
LCL lifting condensation level (Chapter 5)
L dimension length (m) (Chapter 1)
L enthalpy of evaporation (latent heat) (Chapter 9)
L length of box edge (m) (Chapter 2)
λ mean free path (m) (Chapter 2)
λ wavelength of an electromagnetic wave (m) (Chapter 8)
L = H
vap
the enthalpy (latent heat) of evaporation (J kg
−1
) (Chapters 5, 6)
L
evap
latent heat of evaporation (J kg
−1
) (Chapter 7)
mb pressure unit, one millibar = 1 hPa (Chapter 1)
m
e
electron mass (kg) (Chapter 2)
m
0
mass of a molecule (kg) (Chapter 2)
M dimension mass (kg) (Chapter 1)
M
eff
effective molecular weight (g mol
−1
) (Chapter 2)
M
G
gram molecular weight of a gas (g mol
−1
) (Chapter 2)
M
v
, M
d
, M
e
gram molecular weight of vapor, dry air and effective (g mol
−1
)
(Chapter 5)
Notation and abbreviations 255
M mass (kg) (Chapters 2, 3, 4)
M mass of an object or system (Chapter 1)
M
E
mass of the Earth (Chapter 1)
M
l
, M
g
bulk mass of liquid, gas (kg) (Chapter 5)
n unit vector (Chapter 9)
n
θ
unit vector in the theta direction (polar coordinates) (Chapter 9)
n
s
number density of vapor molecules at saturation (molecules m
−3
)
(Chapter 5)
n
sat
number density of vapor molecules at saturation (molecules m
−3
)
(Chapter 5)
n
w
number density of water molecules in vapor (molecules m
−3
)
(Chapter 5)
n
0
(z) molecular density as function of height (molecules m
−3
)
(Chapter 2)
n
0
number density of photons (photons/m
−3
) (Chapter 8)
n
0
number density (molecules m
−3
) (Chapters 3, 5)
N newtons (Chapter 2)
N total number of molecules (Chapter 2)
N
A
Avogadro’s number (molecules mol
−1
) (Chapters 1, 2, 5, Table A1)
N (z) concentration of absorbers at level z (absorbers m
−3
) (Chapter 8)
ν, ν
A
, ν
B
number of moles, number of moles of species A, B (Chapters 3, 4)
ω, f angular frequency (rad s
−1
), frequency (cycles s
−1
≡ Hz)
(Chapter 6)
p pressure (Pa) (all chapters)
p, p
G
pressure, partial pressure for gas G (Chapter 2)
p, p(z), p
0
pressure, as a function of z, at a reference level (hPa) (Chapter 6)
p(V ) pressure as a function of volume; expression for a curve in the p–V
plane (Chapter 4)
Pa unit of pressure, 1 Pa = Nm
−2
(Chapter 1)
P(z) probability density function (Chapter 2)
P(v
x
, v
y
, v
z
) joint probability density function for velocity components
(Chapter 2)
(z),
1
,
2
geopotential height as a function of height, at two levels (meters,
on charts often in decameters, dm) (Chapter 6)
q specific humidity (kg water vapor/kg of moist air) (Chapter 5)
rr= xi + yj + zk position vector (Chapter 9)
r unit vector in the r direction (polar coordinates) (Chapter 9)
r relative humidity (Chapter 5)
256 Appendix B
r
0
effective molecular radius (m) (Chapter 2)
R gas constant for a particular gas (J kg
−1
K
−1
)
R
d
gas constant for dry air (J kg
−1
K
−1
) (Chapter 2, Table A1)
R
eff
effective gas constant for a mixture of species (J kg
−1
K
−1
)
(Chapters 2, 5)
R
w
the gas constant for water vapor (Chapters 2, 5, Table A1)
R
∗
universal gas constant (J mol
−1
K
−1
) (Table 1.1)
ρ density (kg m
−3
) (all chapters)
ρ, ρ
0
, ρ
e
density, at a reference level, of the environment (kg m
−3
)
(Chapter 6)
ρ
a
, ρ
e
density for adiabat and environment (kg m
−3
) (Chapter 7)
s entropy per unit mass (lower case indicates per unit mass)
(J K
−1
kg
−1
) (Chapter 4)
s
l
, s
g
specific entropy for liquid, gas (J K
−1
kg
−1
) (Chapter 5)
SI Standard International system of units (Chapter 1)
S,dS surface, surface element vector (Chapter 9)
S entropy (J K
−1
) (Chapter 5)
S, S
A
, S
B
entropy, entropy of state A, state B (J K
−1
) (Chapter 4)
S
sys
, S
surr
, S
universe
entropy for the system, surroundings, universe (sys+surr)
(Chapter 4)
σ standard deviation (Chapter 2)
σ surface tension (J m
−2
) (Chapter 5)
σ
2
variance (Chapter 2)
σ (λ) absorption cross-section (m
2
) (Chapter 8)
σ
c
collision cross-section (m
2
,nm
2
) (Chapter 2)
t
0
, t
c
both stand for lifetime (s) (Chapter 8)
t
1/2
half life (s) (Chapter 8)
T Kelvin temperature (K) (Chapters 2, 3, 5)
T
C
temperature measured in degrees Celsius (
◦
C) (Chapter 5)
T
D
dew point temperature (K) (Chapter 5)
T
v
virtual temperature (K) (Chapter 5)
T
w
wet-bulb temperature (K) (Chapter 5)
T dimension time, also period of a repeating process, and
temperature (Chapter 1)
Temp temperature dimension (Chapter 1)
T , T (z), T
0
temperature, as a function of z, at a reference level (K)
(Chapter 6)
T
e
(z), T
a
(z) temperature of environment, of an adiabat (K) (Chapter 6)
T vertical average temperature in a layer of air (K) (Chapter 6)