Chapter 13
Appendices
13.1
Derivations
13.1.1
The Planck function
A black body is a theoretical construct that absorbs 100% of the radiation
that hits it. Therefore it reflects no radiation and appears perfectly black.
It is also a perfect emitter of radiation. Planck showed that the power per
unit area, per unit solid angle, per unit wavelength, emitted by a black body
is given by:
Bλ (T ) =
2hc2
¡
£ hc ¤
¢
λ5 exp λkT
−1
(13.1)
where h is Planck’s constant, c is the speed of light, k is Boltzmann’s constant
and λ is the wavelength of the radiation. Figs.2.2 and 2.3 are plots of Bλ (T )
against wavelength for various T ’s.
If Bλ is integrated over all wavelengths, one obtains the black-body radiance:
Z∞
Bλ (T )dλ =
σ 4
T
π
0
5 4
2π k
where σ = 15h
3 c2 is the Stefan-Boltzmann constant. The above can be written
as an integral over ln λ thus:
487
488
CHAPTER 13. APPENDICES
−4
T
Z∞
σ
π
λBλ (T )d ln λ =
(13.2)
0
−4
So if T λBλ is plotted against ln λ then the area under the curve is independent of T. This is the form plotted in Fig.2.6. For more details see Andrews
(2000).
13.1.2
Computation of available potential energy
Consider the two-layer fluid shown in Fig.8.9 in which the interface is given
by:
1
h = H + γy .
2
H
The potential energy of the system is (noting that γ < 2L
i.e. the interface
does not intersect the upper or lower boundaries)
P =
Z
H
0
= g
Z
Z
L
L
−L
dy
−L
= gρ1
Z
gρz dy dz
"Z
h(y)
ρ2 z dz +
dy
Z
H
ρ1 z dz
h(y)
0
L
Z
H
Z
L
z dz + g ∆ρ
dy
−L
0
−L
Z
∆ρ L 2
2
= gH Lρ1 + g
h (y) dy .
2 −L
Substituting for h we have
Z
Z L
2
h (y) dy =
h(y)
z dz
0
¸
1 2
2 2
H + λy + λ y dy
4
−L
1 2
2
=
H L + λ2 L3 ,
2
3
−L
and so
Z
#
L
·
µ
¶
∆ρ
∆ρ 2 3
+g
γ L
P = gH L ρ1 +
4
3
2
which, when expressed in terms of reduced gravity, g 0 = g (ρ1ρ−ρ2 ) , is Eq.(8.9).
1
13.2. MATHEMATICAL DEFINITIONS AND NOTATION
13.1.3
489
Internal energy for a compressible atmosphere
Internal energy for a perfect gas is defined as in Eq.(8.12) of Section 8.3.4:
Z
Z
Z ∞
Z
cv
cv
IE = cv ρT dV =
p dV =
dA
p dz ,
R
R
zs
where the ideal gas law has been used, dA is an area element such that
dV = dA dz, and where zs is the height of the Earth’s surface. If we neglect
surface topography, so that zs = 0, then, integrating by parts,
Z ∞
Z ∞
∂p
∞
p dz = [zp]0 −
z dz .
∂z
0
0
Since we saw in Chapter 3 that pressure decays approximately exponentially
with height, (zp) → 0 as z → ∞. Therefore, using hydrostatic balance, we
have
Z ∞
Z ∞
p dz = g
ρz dz ,
0
0
and so the internal energy is
cv
IE = g
R
Z
zρ dV .
This is the form given in Eq.(8.12).
13.2
Mathematical definitions and notation
13.2.1
Taylor expansion
We assume that the function r(x) can be expressed as an expansion at the
point xA in terms of an infinite series thus:
r(x) = c0 + c1 (x − xA ) + c2 (x − xA )2 + c3 (x − xA )3 + ....,
where the c’s are constants. To find c0 we set x = xA to yield c0 = r(xA ).
To find c1 we differentiate once wrt x and then set x = xA to yield c1 =
(dr/dx)A where the subscript zero indicates that dr/dx is evaluated at x =
xA . Carrying on we see that we can write (dm r/dxm )A = m!cm and hence:
490
CHAPTER 13. APPENDICES
µ
dr
r(x) = r(xA ) + (x − xA )
dx
¶
(x − xA )2
+
2!
A
µ
d2 r
dx2
¶
+ ....
A
So, if x = xA + δx, where δx is a small increment in x, then, to first order in
δx, we may write:
µ ¶
dr
r(xA + δx) ' r(xA ) + δx
.
dx A
Taylor expansions were used often in Chapters 4 (in the derivation of
vertical stability criteria) and in Chapter 6 in the derivation of the equations
that govern fluid motion.
13.2.2
Vector identities
Cartesian coordinates are best used for getting our ideas straight, but occasionally we also make use of polar (sometimes called cylindrical) and spherical
polar coordinates (see Section 13.2.3).
The Cartesian coordinate system is defined by three axes at right angles
to each other. The horizontal axes are labeled x and y, and the vertical axis
b, y
b, b
is labeled z, with associated unit vectors x
z (respectively). To specify
a particular point we specify the x coordinate first (abscissa), followed by
the y coordinate (ordinate), followed by the z coordinate, to form an ordered
triplet (x, y, z).
If φ is a scalar field and a a 3-dimensional vector field thus:
b + ay y
b + az b
a = ax x
z
where ax , ay and az are magnitudes of the projections of a along the three
axes, then we have the following definitions:
b ∂φ
b ∂φ
+y
+b
z ∂φ
[the "gradient of φ" or "grad φ", a vector]
I. ∇φ = x
∂x
∂y
∂z
∂ay
∂ax
∂az
II. ∇ · a = ∂x + ∂y + ∂z
[the "divergence of a" or "div a", a scalar]
³
´
³
´
³
´
∂ay
∂ay
∂a×
∂az
∂ax
z
b ∂a
b
b
III. ∇ × a = x
+
y
+
z
[the "curl
−
−
−
∂y
∂z
∂z
∂x
∂x
∂y
of a", a vector]
Inspection of I, II, and III shows that we can define the operator ∇
uniquely as:
∂
∂
∂
b ∂x
b ∂y
IV. ∇ ≡ x
+y
+b
z ∂z
13.2. MATHEMATICAL DEFINITIONS AND NOTATION
491
The quantity ∇ · (∇φ) is:
∇ · (∇φ) =
∂ 2φ ∂ 2φ ∂ 2φ
+
+ 2 = ∇2 φ
∂x2 ∂y 2
∂z
V. a · b = a¯ x bx + ay by +¯az bz [the scalar product of a and b, a scalar]
¯ x
b b
z ¯¯
¯ b y
¯
b (ay bz − az by )+b
VI a×b = ¯ ax ay az ¯¯ = x
y (az bx − ax bz )+b
z (ax by − ay bx )
¯ bx by bz ¯
[the vector
of a and b, a vector].
¯
¯ product
¯
¯
¯ is the determinant. Thus, for example, in Eq.(7.17),
Here ¯¯
¯
¯
¯
¯ x
µ
¶
µ ¶
b b
z ¯¯
¯ b y
∂σ
∂σ
¯
¯
b
b −
b
z × ∇σ= ¯ 0 0 1 ¯ = x
+y
,
∂y
∂x
¯ ∂σ ∂σ ∂σ ¯
∂x
∂y
∂z
g
z × ∇σ yields the component of the thermal wind
and so, du
= − f ρg b
dz
ref
equation, Eq.(7.16).
Some useful vector identities are:
1. ∇ · (∇ × a) = 0
2. ∇ × (∇φ) = 0
3. ∇ · (φa) = φ∇ · a + a · ∇φ
4. ∇ × (φa) = φ∇ × a + ∇φ × a
5. ∇ · (a × b) = b · (∇ × a) − a · (∇ × b)
6. ∇ × (a × b) = a (∇ · b) − b (∇ · a) + (b · ∇) a − (a · ∇) b
7. ∇ (a · b) = (a · ∇) b + (b · ∇) a + a × (∇ × b) + b × (∇ × a)
8. ∇ × (∇ × a) = ∇ (∇ · a) − ∇2 a
b∇2 ax + y
b∇2 ay + b
In (8), ∇2 a = x
z∇2 az .
An important special case of (7) arises when a = b :
492
CHAPTER 13. APPENDICES
µ
¶
1
(a · ∇) a = ∇
a · a − a × (∇ · a) .
2
These relations can be verified by the arduous procedure of applying the
definitions of ∇, ∇·, ∇× (and a · b, a × b as necessary) to all the terms
involved.
13.2.3
Polar and Speherical Coordinates
Polar coordinates
Any point is specified by the distance r (radius vector pointing outwards)
from the origin and θ (vectorial angle) measured from a reference line (θ is
positive if measured counterclockwise and negative if measured clockwise),
as shown in Fig.6.8.
x = r cos θ; y = r sin θ
b θ = (vr , vθ ) where vr = Dr and vθ = r Dθ
The velocity vector is u = b
rvr + θv
Dt
Dt
b
are the radial and azimuthal velocities, respectively, and b
r, θ are unit vectors.
In polar
coordinates:
¡
¢
∇φ = ∂φ
, 1 ∂φ
∂r r ∂θ
∂
∇ · u = 1r ∂r
(ru) + 1r ∂v
¡
¢ ∂θ
2
∂
∇2 φ = 1r ∂r
r ∂φ
+ r12 ∂∂θφ2
∂r
Spherical polar coordinates
In spherical coordinates a point is specified by coordinates (λ, ϕ, z) where,
as shown in Fig.6.19, z is radial distance, ϕ is latitude and λ is longitude.
Velocity components (u, v, w) are associated with the coordinates (λ, ϕ, z)
i.e. u = z cos ϕ Dλ
is in the direction of increasing λ (eastward), v = z Dϕ
is
Dt
Dt
in the direction of increasing ϕ (northward) and w = Dz
is
in
the
direction
Dt
of increasing z (upward, in the direction opposite to gravity).
In spherical
coordinates:´
³
1
, 1 ∂φ , ∂φ
∇φ = z cos ϕ ∂φ
∂λ z ∂ϕ ∂z
∂u
1
∇ · u = z cos
+
ϕ ∂λ
2
∇ φ=
∂2φ
1
z 2 cos2 ϕ ∂λ2
∂v cos ϕ
1
+ ∂w
z cos ϕ ∂ϕ ³ ∂z
+
1
∂
z 2 cos ϕ ∂ϕ
∂φ
cos ϕ ∂ϕ
´
+
1 ∂
z 2 ∂z
¡ 2 ∂φ ¢
z ∂z
13.3. USE OF FORAMINIFERA SHELLS IN PALEO CLIMATE
13.3
493
Use of foraminifera shells in paleo climate
A key proxy record of climate is δ 18 O, the ratio of 18 O to 16 O in the shells of
surface (planktonic) and bottom (benthic) dwelling foraminifera which are
made of calcium carbonate, CaCO3 . The δ 18 O in the shell, measured and
reported as
³ 18 ´
³ 18 ´
O
− 16 O
16 O
O
smpl
std
¡ 18 O ¢
δ18 O =
× 1000o /oo ,
16 O
where
³ 18 ´
O
16 O
smpl
is the ratio of
18
std
O to
16
O in the sample and
³ 18 ´
O
16 O
std
is the
ratio in a standard reference, is controlled by the δ 18 O value of the water
and the temperature at which the shell formed. It turns out that 18 O is
increasingly enriched in CaCO3 as the calcification temperature decreases,
with a 1 ◦ C decrease in temperature resulting in a 0.2o /oo increase in δ 18 O.
However, in order to use oxygen isotope ratios in foraminifera as an indicator
of temperature, one must take into account the isotopic signal associated with
ice volume stored on the continents and local precipitation-evaporation-runoff
conditions, as we now describe.
18
16
The vapor pressure of H16
2 O is higher than that of H2 O, and so H2 O
becomes concentrated in the vapor phase (atmosphere) during evaporation
of seawater. The residual water therefore becomes enriched in 18 O relative
to water vapor. Similar physics (that of Rayleigh distillation) results in
the condensate (rain or snow) from clouds being depleted in 18 O relative
to 16 O. Therefore as an air mass loses water, for example as it is advected
to higher latitudes or altitudes, or transported into a continental interior,
it becomes progressively depleted in 18 O. Precipitation in polar latitudes
is commonly 20 − 40o /oo depleted in 18 O relative to average ocean water.
During glacial periods, when perhaps 120 m of sea level equivalent water
is stored in polar ice sheets as highly isotopically depleted ice, the entire
ocean becomes enriched in 18 O by approximately 1o /oo . This whole-ocean
change in δ 18 O has been a powerful tool for reconstructing ice age cycles
and paleoclimatology in general. The same principal results in regions of the
ocean that receive continental runoff having low δ 18 O values in their surface
water, and those regions of the ocean that experience an excess of evaporation
over precipitation to have high surface water δ 18 O values.
494
CHAPTER 13. APPENDICES
Ω (rad/s)
0
0.10 0.25 0.5
1
1.5
2
f = 2Ω (rad/s) 0
0.21
0.5
1
2
3
4
τ (s)
∞ 10. 46 25.1 12.6 6.3 4.2 3.14
rpm
0
1
2.4 4.7 9.5 14.3 19.1
Table 13.1: Various measure of rotation rates. If Ω is the rate of rotation of the
tank in radians per second, then the period of rotation is τ tank = 2π
s. Thus if
Ω
Ω = 1, τ tank = 2π s.
13.4
Laboratory experiments
13.4.1
Rotating tables
The experiments described throughout this text come to life when they are
carried out live, either in demonstration mode in front of a class or in a laboratory setting in which the students are actively involved. Indeed a subset
of the experiments form the basis of laboratory-based, hands-on teaching at
both undergraduate and graduate level at MIT. As mentioned in Section 0.1,
the experiments have been chosen not only for their relevance to the concepts
under discussion, but also for their transparency and simplicity. They are
not difficult to carry out and ‘work’ most, if not every!, time. The key piece
of equipment required is a turntable (capable of rotating at speeds between
1 and 20 rpm) on which the experiment is placed and viewed from above by
a co-rotating camera. We have found it convenient to place the turntable on
a mobile cart – see Fig.13.1 – equipped with a water tank and assorted
materials such as ice-buckets, cans, beakers, dyes etc. The cart can be used
to transport the equipment and as a platform to carry out the experiment.
Measurement of table rotation rates
The rotation rate can be expressed in terms of period, revolutions per minute,
or units of ‘f ’, as described below and set out in Table 13.1:
1. The angular velocity of the tank, Ω , in radians per second
2. The Coriolis parameter (f ) defined as f = 2Ω
3. The period of one revolution of the tank is τ = 2π
Ω
4. Revolutions per minute, rpm = 60
τ
13.4. LABORATORY EXPERIMENTS
495
Figure 13.1: A turntable on a mobile cart equipped with a water storage tank and
pump, power supply and monitor. A tank filled with water can be seen with an
ice bucket in the middle, seated on the turntable and viewed through a co-rotating
overhead camera.
496
13.4.2
CHAPTER 13. APPENDICES
List of laboratory experiments
GFD 0: Rigidity imparted to rotating fluids – Section 0.2.2.
GFD I: Cloud formation on adiabatic expansion – Section 1.3.3.
GFD II: Convection – Section 4.2.4.
GFD III: Radial inflow – Section 6.6.1.
GFD IV: Parabolic surfaces – Section 6.6.4.
GFD V: Inertial Circles – Section 6.6.4.
GFD VI: Perrot’s bathtub experiment – Section 6.6.6.
GFD VII: TaylorColumns – Section 7.2.1.
GFD VIII: The Hadley Circulation and thermal wind – Section 7.3.1.
13.5. FIGURES AND ACCESS TO DATA OVER THE WEB
497
GFD IX: Cylinder ‘collapse’ – Section 7.3.3.
GFD X: Ekman layers – Section 7.4.1.
GFD XI: Baroclinic instability – Section 8.2.2.
GFD XII: Ekman pumping and suction – Section 10.1.2.
GFD XIII: Wind-driven ocean gyres – Section 10.2.4.
GFD XIV: Abyssal ocean circulation – Section 11.3.2.
GFD XV: Source-sink flow – Section 11.3.4.
More details about the equipment required to carry out these experiments, including turntables and fluid carts, can be obtained from Professor
John Marshall at MIT.
13.5
Figures and access to data over the web
The vast majority of the data and figures presented in this book were accessed
and plotted using the Climate Data Library of the International Research
Institute of the Lamont-Doherty Earth Observatory of Columbia University.
The library contains numerous datasets from a variety of earth science disciplines and climate-related topics which are accessible free over the web – see
498
CHAPTER 13. APPENDICES
http://iridl.ldeo.columbia.edu/. Web-based tools permit access to data sets,
analysis software for manipulation of the data and graphical presentation,
together with the ability to download the data in numerous formats.