MATH3216 Ocean Surface Waves Notes: weeks 9–10
2
Contents
1 Waves in a Non-Rotating Reference Frame
1.1 Surface Gravity Waves . . . . . . . . . . . . . . . .
1.1.1 The Navier-Stokes Equation . . . . . . . . .
1.1.2 Boundaries Conditions . . . . . . . . . . . .
1.1.3 The Laplace Equation . . . . . . . . . . . .
1.1.4 Solutions . . . . . . . . . . . . . . . . . . .
1.1.5 Dispersion Relation . . . . . . . . . . . . .
1.2 Linear Isotropic Dispersion . . . . . . . . . . . . .
1.2.1 An Introduction to Group Velocity . . . . .
1.2.2 Deep Water (Short Wave) Approximation .
1.2.3 Shallow Water (Long Wave) Approximation
1.3 Particle Paths . . . . . . . . . . . . . . . . . . . . .
1.4 Wave Conservation . . . . . . . . . . . . . . . . . .
1.4.1 Conservation I . . . . . . . . . . . . . . . .
1.4.2 Conservation II . . . . . . . . . . . . . . . .
1.4.3 Linear Bottom Slope . . . . . . . . . . . . .
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7
7
7
8
9
9
10
10
10
11
12
13
15
15
16
17
2 Shallow Water Waves in a Rotating Reference
2.1 Poincaré (gravity-inertia) waves . . . . . . . . .
2.1.1 Dispersion Relation . . . . . . . . . . .
2.1.2 Horizontal Particle Paths . . . . . . . .
2.1.3 Phase Speed and Group Velocity . . . .
2.2 Kelvin Waves . . . . . . . . . . . . . . . . . . .
2.3 Rossby Waves . . . . . . . . . . . . . . . . . . .
2.3.1 The Beta Plane . . . . . . . . . . . . . .
2.3.2 Governing Equation . . . . . . . . . . .
2.3.3 Dispersion Relation . . . . . . . . . . .
2.3.4 Group Velocity . . . . . . . . . . . . . .
Frame
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21
21
22
23
24
24
26
26
27
28
29
3 Waves and the El Niño-Southern Oscillation
3.1 What is ENSO? . . . . . . . . . . . . . . . . .
3.1.1 The “Normal” State . . . . . . . . . .
3.1.2 The Bjerknes Feedback . . . . . . . .
3.2 Equatorial Wave Dynamics . . . . . . . . . .
3.2.1 The Equatorial Beta Plane . . . . . .
3.2.2 The Equatorial Kelvin Wave . . . . .
3.2.3 The Equatorial Rossby Wave . . . . .
3.3 Explaining ENSO Periodicity . . . . . . . . .
3.3.1 The Delayed Action Oscillator . . . .
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31
32
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34
35
36
37
3
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4
CONTENTS
Introduction
References for these notes are:
• Gill, A. E. 1982. Atmosphere-Ocean Dynamics. Vol. 30 of International Geophysics Series.
Academic Press, 662pp + xv.
• Pedlosky, J. 1987. Geophysical Fluid Dynamics, 2nd Edition. Springer, 710pp + xiv.
• Lighthill, J. 1978. Waves in Fluids. Cambridge University Press, 504pp + xv.
• Clarke, A. J. 2008. Dynamics of El Niño & the Southern Oscillation. Academic Press, 308
+ xv.
• Philander, S. G. 1991. El Niño, La Niña, and the Southern Oscillation. Vol. 46 of the
International Geophysics Series. Academic Press, 293 + xi.
5
6
CONTENTS
Chapter 1
Waves in a Non-Rotating
Reference Frame
1.1
1.1.1
Surface Gravity Waves
The Navier-Stokes Equation
We wish to find an expression that will accurately describe the disturbed (with a small perturbation) free surface depicted in figure 1.1. We begin with the rotational, compressible Navier-Stokes
equation and continuity equation
∂ρv
+ v · (∇ρv) + 2Ω × ρv = g − ∇p + µ∇2 ρv
∂t
∂ρ
+ ∇ · ρv = 0
∂t
(1.1)
(1.2)
∂
∂
∂
where ρ is density, v = (u, v, w) is the velocity, ∇ = ( ∂x
, ∂y
, ∂z
) is the differential operator,
∗
2Ω = (0, f , f ) is the intertial term, g = (0, 0, −g) is gravity, p is pressure, and µ is the viscosity.
If we now make some assumptions.
• Constant density, ρ = ρ0 . This means that
of the differential operators as a constant.
∂ρ
∂t
= 0 and density may be taken out to the LHS
• Inviscid flow, µ = 0. The molecular friction, on large scales, can reasonably be set to zero.
• For the moment, we will assume that the earth’s rotation is unimportant, Ω = 0.
• The products of perturbation quantities are small, in other words, non-linear term is unimportant, v · ∇v = 0.
• Hydrostatic pressure, ∂p
∂z = −gρ0 – particularly as we are assuming a uniform density, it is
quite reasonable to assume that the pressure is proportional to the depth.
z = η(x, y, t)
ẑ
z = −H
Figure 1.1: The shaded region is the (constant depth) ocean floor, the dashed line indicates mean
sea level and the solid line represents the free surface. H is distance from mean sea level to the
ocean floor, and η(x, y, t) is the deviation of the free surface from mean sea level.
7
8
CHAPTER 1. WAVES IN A NON-ROTATING REFERENCE FRAME
Now, we integrate the pressure from the surface to a depth z, assuming that the pressure
vanishes, p = 0 at the surface z = η.
Z z
p=−
gρ0 dz
(1.3)
η
Z
0
=
Z
η
gρ0 Dz +
gρ0 dz
z
(1.4)
0
= −gρ0 z + gρ0 η
= p0 + p
0
⇒ p = ρ0 gη
0
(1.5)
(1.6)
at z = η
(1.7)
We can see that the first pressure term in equation (1.5) is proportional to z regardless of horizontal
position, and we call this term the equilibrium pressure, p0 , because this is the pressure if the system
were unperturbed. The second term on the other hand, is dependent on η, the deviation of the free
surface height from mean sea-level. We call this term the perturbation pressure p0 . Our continuity
and momentum equation therefore become
∇·v =0
(1.8)
0
∂p
∂u
=−
∂t
∂x
∂p0
∂v
=−
ρ0
∂t
∂y
∂w
∂p0
ρ0
=−
∂t
∂z
ρ0
1.1.2
(1.9)
(1.10)
(1.11)
Boundaries Conditions
We state that the vertical velocity must be zero at the ocean bottom (since the ocean bottom is
assumed to be a constant depth and impermiable).
w=0
at z = −H
(1.12)
The free surface boundary condition is a little more tricky. We say that a particle in the free
∂η
∂z
surface will stay there. Using the material derivative1 and noting that2 ∂z
∂t = 0, ∂z = 0 ∂x = 0,
∂z
∂z
∂z = 1 and ∂y = 0.
D(z − η)
= 0 at z = η
Dt
or in other words
∂η
∂η
∂η
+u
+v
= w at z = η
∂t
∂x
∂y
(1.13)
(1.14)
which, for small perturbations, we can ignore the multplicative terms, and equation (1.14) reduces
to
∂η
w=
at z = η
(1.15)
∂t
Since the perturbations are small, we relax this restriction by saying that this applies at z = 0
rather than z = η in order to make it easier to solve for, without introducing a large error.
w=
∂η
∂t
at z = 0
(1.16)
This approximation is known as the rigid lid approximation.
D
∂
that the material derivative: Dt
= ∂t
+ v · ∇, follows a particle of fluid, while the inertial expression
d
measures the fluid passing a point in space. Note that some authors use dt
to denote the material derivative.
2 It is extremely important to note that w = Dz . This is a statement about the rate of change of position of a
Dt
water parcel whose position depends on time. Since z 6= z(t), then it follows that ∂z
= 0. What this is saying is
∂t
that our vertical coordinate does not have a time dependency. This is a subtle, but important distinction to make.
1 Recall
1.1. SURFACE GRAVITY WAVES
1.1.3
9
The Laplace Equation
We now take the derivative with respect to x of equation (1.9), y of equation (1.10) and z of
equation (1.11). Adding them all together, and using operator commutation, we end up with a
Laplacian equation, recognising the portion of the equation that is identically zero – see equation (1.8)
∂ ∂u ∂v
∂w
∂ 2 p0
∂ 2 p0
∂ 2 p0
ρ0
+
+
=
+
+
(1.17)
∂t ∂x ∂y
∂z
∂x2
∂y 2
∂z 2
∇ 2 p0 = 0
(1.18)
Note that ∇2 ≡ ∇ · ∇.
1.1.4
Solutions
There are many possible solutions to this equation, however, we only consider the instance where
the solution p0 varies sinusoidally. Interestingly, this does not really restrict us, as Fourier’s theorem means that we can describe an arbitrary disturbance with a superposition of sinusoids. In
particular, a long crested, travelling wave has the form
η = η0 cos(k · x − ωt)
(1.19)
or
o
n
η = < η0 eiΦ̂
(1.20)
where η0 is the amplitude, k is the x component of the wavenumber, l is the y component of the
wavenumber and ω is the angular frequency. We also define the wavenumber vector and the phase
k = (k, l)
(1.21)
Φ̂ = kx + ly − ωt = k · x − ωt
(1.22)
where x = (x, y) We also define the magnitude of the wavenumber, κ.
κ2 = k 2 + l 2
(1.23)
This quantity is the magnitude of the wavenumber in the direction of propagation, while k and l
are the wavenumber in the x and y directions respectively3 . Therefore, the wavelength is 2π/κ.
The phase velocity (that is, the velocity of lines of constant Φ̂) and phase speed are, respectively
are
ω ω ,
(1.24)
c=
k l
ω
c=
.
(1.25)
|κ|
Since we have assumed that p0 = ρ0 gη in equation (1.5), we can substitute our solution, equation (1.19) into the Laplacian, equation (1.18), and get the following relation.
∂ 2 p0
− κ2 p0 = 0
∂z 2
(1.26)
Solutions to this equation must be a sum of exponentials or hyperbolics.
p0 = Aeκz + Beκz
0
or p = A cosh(κz) + B sinh(κz)
or p0 = A cosh[κ(z + B)]
(1.27)
(1.28)
(1.29)
where A and B are constants. In order to satisfy the bottom boundary condition, the most
suitable expression is the last, equation (1.29). Our bottom boundary condition, equation (1.12),
in combination with the vertical accelleration expression, equation (1.11), it is trivial to see that
3 If an observer was measuring the wavelength in the x direction, they would find it to be 2π/k and similarly
2π/l in the y direction, while in the direction of propagation, they would find it to be 2π/κ.
10
CHAPTER 1. WAVES IN A NON-ROTATING REFERENCE FRAME
∂p0
∂z
= 0 at z = −H. Also recall that sinh(φ) = 0 uniquely for φ = 0. Substituting this boundary
condition into equation (1.29) and solve for B, we see that
∂p0
= Aκ sinh[κ(z + B)]
∂z
0 = Aκ sinh[κ(−H + B)]
at z = −H
⇒B=H
(1.30)
Now we wish to find A by recalling our surface boundary condition, equation (1.16).
∂ 2 p0
− κ2 p0 = 0
∂z 2
κ2 A cosh[κ(0 + H)] = κ2 ρ0 g at z = 0
ρ0 g
⇒A=
cosh(κH)
(1.31)
Therefore, our solution has the form
p0 =
ρ0 gη0 cos(k · x − ωt) cosh[κ(z + H)]
cosh(κH)
(1.32)
Substituting this equation into equation (1.11) we get the vertical velocity
w=
1.1.5
κgη0 sin(k · x − ωt) sinh[κ(z + H)]
ω cosh(κH)
(1.33)
Dispersion Relation
It remains to satisfy equation (1.16). Taking the time derivative of our assumed solution for η,
equation (1.19), and stating our expression for the vertical velocity, equation (1.33) at z = 0.
∂η
= ωη0 sin(k · x − ωt)
∂t
κgη0 sin(k · x − ωt) sinh(κH)
w=
ω cosh(κH)
(1.34)
at z = 0
(1.35)
Now subsituting into equation (1.16), we find our dispersion relation.
κgη0 sin(k · x − ωt) sinh(κH)
ω cosh(κH)
sinh(κH)
⇒ ω 2 = gκ
cosh(κH)
ωη0 sin(k · x − ωt) =
ω 2 = gκ tanh(κH)
1.2
(1.36)
(1.37)
Linear Isotropic Dispersion
In the coming sections, we will investigate the disperion properties of linear surface gravity waves
(recall from the previous section that we have assumed that the non-linear terms are small and
therefore unimportant) in which the phase speed, c = ω/κ, is invariant with direction. Here, we
briefly review what exactly a dispersion relation is and what the consequences are for waves we
observe in the ocean. We can see from the dispersion relation in equation (1.37) that the only
thing that the frequency is independent of direction, but depends only on the wavenumber. Waves
with this property are known as dispersive waves (the reason for this will become clear soon). Conversely, waves that do not have a wave speed dependent on wavenumber are called non-dispersive.
1.2.1
An Introduction to Group Velocity
In the real world we never have a purely sinusoidal perturbation, but instead, we describe perturbations as a superposition of many sinusoids. Since dispersive waves have varying phase speed
1.2. LINEAR ISOTROPIC DISPERSION
11
ẑ
Figure 1.2: The solid line represents a wave train. In this instance, the wave train consists of
two superimposed sinusoids of different wavelength (and therefore different phase velocity c. The
dashed line represents the envelope, the peaks and troughts of which travel at the group velocity
cg . The group velocity is the speed at which the energy of the wave train travels – hence its
importance in the study of wave dynamics.
depending on frequency, we will observe that if we observe waves produce by a disturbance (for
example, a storm) they will tend to spread out the longer that they travel. Thus, waves that are
generated by a distant storm tend to be very regular (as the waves have spread out, with long
wave length waves arriving first, followed by short wave length waves). On windy days, waves tend
to be much less regular as the wind has generated the waves, and the frequencies have not had
sufficient time to spread out. In fact, the difference in arrival times of different wave lengths is a
way that we can calculate the distance to a storm far away.
For dispersive waves, we will introduce a variable known as the group velocity cg , a vector
quantity. This variable describes the speed of the crests of wave packets, as illustrated in figure 1.2. The group velocity and the phase velocity are not the same for dispersive waves, and can
even be in opposite directions.
The simplest example that we can use to illustrate this is given by the superposition of two plane
waves in one dimension, with equal amplitude, but slightly different wavelength and frequency –
also recalling that cos(−φ) = cos(φ).
η = η0 cos[(k + δk)x − (ω + δω)t] + η0 cos[(k − δk)x − (ω − δω)t]
(1.38)
η = 2η0 cos(δkx − δωt) cos(kx − ωt)
(1.39)
Where δ has the usual meaning of a small but finite change. The first trigonometric function in
equation (1.39) represents the “wave envelope”, while the second term can be thought of as the
“carrier wave.” If we take the wave envelope term and do a small approximation, we can see a
term that can respresent its speed.
dω
(1.40)
2η0 cos(δkx − δωt) ≈ 2η0 cos δk x − t
dk
By inspection, we see that the envelope must be travelling at a speed of cg = dω
dk . These arguments
can be generalised to two dimensions, which will give us the vector quantity for the group velocity.
∂ω ∂ω
cg =
,
(1.41)
∂k ∂l
1.2.2
Deep Water (Short Wave) Approximation
If we now investigate the the special limiting case of the dispersion relation, equation (1.37) in
which the depth of the ocean is much greater than the wavelength, that is, H 2π/κ and recall
that as limφ→∞ tanh(φ) = 1.
lim ω 2 = lim gκ tanh(κH)
κH→∞
κH→∞
⇒ ω 2 = gκ
(1.42)
(1.43)
12
CHAPTER 1. WAVES IN A NON-ROTATING REFERENCE FRAME
This also allows us to approximate our expression for the phase speed.
c2 =
g
κ
(1.44)
Note that in this limit, the phase speed is still dependent on the wavenumber, and is therefore
dispersive.
We also have an expression for the perturbation pressure, p0 in equation (1.32), recalling that
sinh(φ)
cosh(φ + ψ) = cosh(φ) cosh(ψ) + sinh(φ) sinh(ψ), cosh(φ)
= tanh(φ) and sinh(φ) + cosh(φ) = eφ .
p0 = ρ0 gη0 cos(k · x − ωt)eκz
(1.45)
Notice that the depth of the ocean, H does not appear in any of the equations. This is because
these waves do not feel the effects of the bottom (hence the name deep water wave). Another
very important thing to note as a consequence of this, is the depth dependence of the perturbation pressure (the exponential term). This means that the wave attenuates proportionally to the
wavelength. We can do a simple calculation to find the e-folding depth (e−1 ∼ 0.37), which gives
us a measure of the depth of influence of a surface wave.
e−1 = eκz
−1 = κz
z=−
λ
2π
(1.46)
where λ is the wavelength. If we use similar arguments, except setting e−π , noting that e−π ∼ 0.04,
we can see that the influence of the surface wave on the subsurface is up to a depth of approximately half a wavelength (that is, at a depth that is equal to half a wavelength, the perturbation
pressure is only ∼ 4% the perturbation pressure at the surface).
Now, we can also find the group velocity using the dispersion relation, equation (1.43). We
assume that the medium is isotropic (that is, k = l) and as such, the phase velocity will be
independent of direction (for brevity we only consider k, but the result applies equally to l).
∂ω
∂k
1
√ ∂
= g (k) 2
∂k
r
1 g
=
2 k
1
⇒ cg = c
2
cg =
(1.47)
(1.48)
A typical short wave has a period 2πω −1 = 10s, and thus it has a wavelength 2πκ−1 = 105m,
an e-folding depth of 25m and a phase speed of c = 15ms−1 .
1.2.3
Shallow Water (Long Wave) Approximation
On the other hand, the other limiting case is where H 2π/κ, where the effects of the bottom
are very important. Opposite to the short wave approximation, we take the limiting case in which
the depth of the ocean is much shorter than the wavelength, and recall that limφ→0 tanh(φ) = φ.
lim ω 2 = lim gκ tanh(κH)
κH→0
κH→0
2
⇒ ω 2 = gκ H
(1.49)
(1.50)
We can find our phase speed from this.
c2 = gH
(1.51)
Importantly, note that these waves are non-dispersive, as there is no dependence of the velocity
on the wave number – thus, the velocity of the wave energy is the same as the phase velocity.
1.3. PARTICLE PATHS
13
Ẑ0
η0
Z0 = 0
η0 eκZ0
X = X0 , Y = Y0
Figure 1.3: This shows the decreasing radius of circular particle paths with depth. The verticle
dotted line indicates X0 and Y0 , while horizontal dotted line indicates Z0 = 0, the mean sea surface
height. The solid lines are the particle paths of particles with mean position (X0 , Y0 , Z0 ), while
the dashed line is the envelope of decreasing particle path radii.
The speed of these waves is also highly influenced by the bottom depth, with waves travelling in
shallow water (for example, the continental shelf, where a typical speed might be 20ms−1 ) much
slower than in deep water (for example, the open ocean, where a typical speed might be 200ms−1 ).
We can now also find our perturbation pressure, recalling that tanh(0) = 0 and cosh(0) = 1.
We need to bear in mind that κz ' 0 since we are assuming that κ−1 H and by definition
z ≤ H (see figure 1.1).
p0 = ρ0 gη0 cos(k · x − ωt)
(1.52)
Importantly, and contrasted to the deep water waves, the pressure perturbation of the shallow
water waves does not attenuate with depth.
1.3
Particle Paths
Recall our continuity and momentum equations (1.8) – (1.11). For convenience, we rewrite the
momentum equations in a compact form.
1
∂v
= − ∇p0
∂t
ρ0
(1.53)
We have, up until now, discussed two kinds of velocities, phase velocity c (which is the propagation
of waves with a constant Φ̂) and group velocity cg (which is the propagation of the energy of “wave
trains”). Neither of these velocities is indicative of the velocity of a particle floating in the water
(for example, a bottle on the surface of the ocean, or perhaps a phytoplankton).
For illustrative purposes, we will go through in detail the way that we find the particle paths
for the deep water approximation derived in section 1.2.2. We begin by finding the velocity
by substituting the equation for the perturbation pressure, equation (1.45) into the momentum
equation, taking the spatial derivatives of the perturbation pressure and integrating with respect
to time to find the velocity v.
gη0 k
cos(k · x − ωt)eκz
ω
gη0 l
v=
cos(k · x − ωt)eκz
ω
gη0 κ
w=−
sin(k · x − ωt)eκz
ω
u=
(1.54)
(1.55)
(1.56)
While the velocity tells us how fast the particles might be travelling, what we really want to know
is their position. If we take a Lagrangian perspective and track the position of individual particles,
X = (X, Y, Z) labelling them as X0 = (X0 , Y0 , Z0 ), where we define X0 as the initial particle
position (which cooincidentally, turns out to be the mean position). We are now ready to define
14
CHAPTER 1. WAVES IN A NON-ROTATING REFERENCE FRAME
c
Ẑ0
Z0 = 0
Figure 1.4: The dotted line is the mean sea surface height (which is Z0 = 0), the solid line is the
sea surface height at an instant, while the dashed lines are the particle paths with the direction of
travel of the particle is indicated by the arrow. Note the phase velocity and that all circles have
mean position on the mean sea surface height, and therefore have radius η0 .
equations for the position of our particles.
DX
= U (X0 , t)
Dt
DY
= V (X0 , t)
Dt
DZ
= W (X0 , t)
Dt
(1.57)
(1.58)
(1.59)
Where X = X(X0 , t), Y = Y (X0 , t), Z = Z(X0 , t). We now have a system of six equations, and the
difficulty is how to link them and solve for the particle positions. If we examine equations (1.54)
to (1.56) closely, we see that they are oscillatory in all three directions, with the amplitude of the
oscillations decaying exponentially with depth. We can exploit this fact and solve for the particle
paths as deviations from the mean position. Also recall that the dispersion relation in the deep
water approximation is ω 2 = gκ.
gη0 k
DX
=
cos(k · X0 − ωt)eκZ0
Dt
ω
gη0 k
⇒ X = X0 − 2 sin(k · X0 − ωt)eκZ0
ω
gη0 l
DY
=
cos(k · X0 − ωt)eκZ0
Dt
ω
gη0 l
⇒ Y = Y0 − 2 sin(k · X0 − ωt)eκZ0
ω
gη0 κ
DZ
=−
sin(k · X0 − ωt)eκZ0
Dt
ω
⇒ Z = Z0 + η0 cos(k · X0 − ωt)eκZ0
(1.60)
(1.61)
(1.62)
We can combine these equations into a form that is much more simple to interpret, recalling that
κ2 = k 2 + l2 , sin2 φ + cos2 φ = 1 and again that ω 2 = gκ.
gη 2
0
sin2 (k · X0 − ωt)e2κZ0 (k 2 + l2 )
(X − X0 )2 + (Y − Y0 )2 + (Z − Z0 )2 =
ω2
+ η02 cos2 (k · X0 − ωt)e2κZ0
(1.63)
⇒ (X − X0 )2 + (Y − Y0 )2 + (Z − Z0 )2 =η02 e2κZ0
(1.64)
What this shows is that the particle paths in the deep water approximation are circles, with the
radius of the circles being η0 at the surface, with the radius of the circles decaying exponentially
with the mean depth of the particle, given by η0 eκZ0 . This is illustrated in figure 1.3 and figure 1.4.
Using similar arguments, it can be shown that for the shallow water approximation, the particle
motion is horizontal lines and that for the general case the particle motion is elliptical. An elliptical
particle path makes perfect sense, as this is somewhere between a circle and a horizontal line. If
we condier how the eccentricity of the ellipses would evolve if we were to move from one extrema
to the other we would find that the eccentricity would be zero, = 0 for the deep water extremum,
gradually increasing to one, = 1 at the shallow water extremum.
1.4. WAVE CONSERVATION
1.4
15
Wave Conservation
Now that we have examined waves in the deep ocean, and waves in a uniformly shallow ocean. This
begs the question, what if the ocean is not uniformly deep, but gradually becomes more shallow
(for example, waves going from the deep ocean to the continental shelf)?
1.4.1
Conservation I
For brevity, we will restrict the following discussion to two dimensions, however, it can be reasonably easily generalised to three dimensions. We begin by recalling the complex form of notation.
η = η0 eiΦ̂
(1.65)
Where Φ̂ = Φ̂(x, t) is the phase parameter defined in equation (1.22). If we now allow the wavenumber and frequency to vary slowly that is k = k(x, t) and ω = ω(x, t). Next, we note some helpful
relations.
∂ Φ̂
∂x
∂ Φ̂
ω≡−
∂t
k≡
(1.66)
(1.67)
If we also allow slow variations in the amplitude, we get the following solution for the surface
height.
η(x, t) = η0 (x, t)eiΦ̂(x,t)
(1.68)
Now, if we cross differentiate equations (1.66) and (1.67), recalling the dispersoin relation in equation (1.50) and add them together, we get the following relation.
∂k ∂ω
+
=0
∂t
∂x
(1.69)
Furthermore, if we recall the definition of the group velocity in equation (1.41) and exploit the
chain rule.
∂k ∂ω
∂k ∂ω ∂k
+
=
+
∂t
∂x
∂t
∂k ∂x
∂k
∂k
=
+ cg
∂t
∂x
Dk
⇒
=0
Dt
(1.70)
∂k
∂k
where Dk
Dt ≡ ∂t + cg ∂x which is a Lagrangian representation, but instead of following a fluid parcel,
we are following the local wave energy. If we use the shallow water approximation and recall the
dispersion relation in equation (1.50) and is ω 2 = gHk 2 in two dimensions only. Also recall the
phase velocity in equation (1.51), which is c2 = gH. Now, recall the group velocity, defined in
equation (1.41), and we see that the phase speed is the same as the group velocity.
∂ω
∂k
p
= gH
⇒ cg = c
cg =
(1.71)
We are now in a position to relax the uniform depth constraint. If we consider the situation
where the depth varies with space, H = H(x) we should be able to tell something about the
properties of the solution to the situation depicted
in figure 1.5. As previously η = η0 eiΦ̂ , k =
p
∂k
∂ω
∂ Φ̂/∂x, ω = ∂ Φ̂/∂t and ∂t + ∂x = 0, ω = gH(x)k, recognising now that ω = ω(k, H(x)). So,
using the chain rule we get a relationship for the wavelength.
∂k ∂ω ∂k
∂ω ∂H
+
+
=0
∂t
∂k ∂x ∂H ∂x
r
∂k
∂k
1 g ∂H
⇒
+ cg
=−
k
∂t
∂x
2 H ∂x
(1.72)
(1.73)
16
CHAPTER 1. WAVES IN A NON-ROTATING REFERENCE FRAME
ẑ
x̂
z = −H(x)
Figure 1.5: This diagram shows the situation where the depth H (the shaded region indicates the
bottom) varies with x with a free surface (solid line). The dotted line idicates the average the mean
surface. Note that the shape of the free surface is indicative only, see the text for the solution.
Furthermore, application of the chain rule to ∂ω
∂t , (noting that
in a similar expresson for ω.
∂ω
∂ω
+ cg
=0
∂t
∂x
∂H
∂t
= 0 and
∂k
∂t
= − ∂ω
∂x ) results
(1.74)
We can investigate the steady state solution of these equations without having to go to the
trouble of attempting to derive some explicit solution. When we say a steady state solution, we
∂
≡ 0). In the present example, this would
are assuming that all time derivative terms are zero ( ∂t
be a good assumption to make if you were observing waves coming onshore on a relatively windless
∂
day (a strong wind will provide a non-trivial forcing, which means that ∂t
6= 0).
There are three main things that we can easily see from these equations.
p g ∂H
∂k
= − 12 H
1. cg ∂x
∂x k by equation (1.73). As the depth of the ocean is decreasing with
increasing x, thus, ∂H
∂x < 0. Therefore, we see that the wavenumber must increase with
increasing x. Since the wavelength is reciprocal to the wavenumber, the wavelength must
shorten as the waves travel.
= 0 ⇒ ∂ω
∂x = 0 by equation (1.74). This means that as the travelling waves get closer to
shore (with decreasing H), the frequency of the waves will not change.
√
3. cg = gH implies that the wave velocity will decrease as the wave progresses.
2.
∂ω
∂t
1.4.2
Conservation II
Recall from the shallow water approximation that the perturbation pressure, p0 , is independent of
depth and is proportional to the sea surface height, η – cf. equations (1.7) and (1.52). Thus, the
horizontal velocity is independent of depth (i.e. the velocity at any horizontal point is the same at
all depths). We shall now exploit this fact, along with continuity, equation (1.8) to find solutions
for a situtation where we have a non-uniform depth.
Let us first restate the horizontal equations of motion and the continuity equation.
∂u
1 ∂p0
=−
∂t
ρ0 ∂x
∂η
= −g
∂x
∂v
1 ∂p0
=−
∂t
ρ0 ∂z
∂η
= −g
∂z
∇·v =0
Next, we need to state our boundary conditions. We assume for our surface boundary condition,
that the vertical velocity at the surface is the same as the rate of change of sea surface height – cf.
1.4. WAVE CONSERVATION
17
equation (1.14) and the discussion on the surface boundary condition in section 1.1.4.
w=
∂η
∂t
at z = 0
(1.75)
For our bottom boundary condition, we state that there can be no flow normal to the bottom.
Now, recall from vector calculus that the normal vector of a surface is described by its gradient
when the surface has been given implicitly as a set of points F (x, y, z) = 0. The surface of the
ocean floor is described by z = −H(x, y) or alternatively, z + H(x, y) = 0. Thus its normal vector
∂H
is given by n = ∇(z + H) = ( ∂H
∂x , ∂y , 1). Also recall from vector calculus that the dot product of
two vectors describes how aligned they are (i.e. orthogonal vectors have a zero dot product). If
we want to ensure that there is no normal flow (which is a reasonable condition to impose as the
ocean floor is impermiable), we take the dot product of the velocity and the normal vector and
we can see that this must be zero. This is really the same boundary condition as what we have
used previously, except we must make its mathematical description more general, as previously
the normal vector for the bottom was always vertically aligned – cf. equation (1.12).
v·n=0
at z = −H(x, y)
(1.76)
⇒ v · ∇(z + H) = 0 at z = −H(x, y)
∂H
∂H
⇒u
+v
+ w = 0 at z = −H(x, y)
∂x
∂y
(1.77)
(1.78)
If we now depth integrate the continuity and exploit the fact that our bottom boundary equation
gives us an expression for the vertical velocity at z = −H, we get a neat differential equation.
Remember that both u and v are independent of z, and our surface boundary condition, equation (1.75).
Z
0
∂u ∂v
∂w
+
+
dz = 0
∂y
∂z
−H ∂x
Z 0
Z 0
∂w
∂u ∂v
+
Dz +
dz = 0
∂x
∂y
−H ∂z
−H
∂u ∂v
+
+ [w(0) − {w(−H)}] = 0
0 − −H
∂x ∂y
∂η
∂H
∂H
∂u ∂v
+
=0
+
+ u
+v
H
∂x ∂y
∂t
∂x
∂y
∂η
∂
∂
⇒
+
(Hu) +
(Hy) = 0
∂t
∂x
∂y
(1.79)
(1.80)
(1.81)
(1.82)
(1.83)
We get from equation (1.82) to equation (1.83) by use of the chain rule. Now, if we differentiate
equation (1.83) with respect to t and then recall our horizontal equations of motion stated earlier
in this section, we come up with a single equation for the free surface. We are assuming operator
commutation and we remember that H is independent of time.
∂2η
∂
∂u
∂
∂v
+
H
+
H
=0
(1.84)
∂t2
∂x
∂t
∂y
∂t
1 ∂2η
∂
∂η
∂
∂η
+
H
+
H
=0
(1.85)
2
g ∂t
∂x
∂x
∂y
∂y
The major implicit assumption in all of the above is that the vertical accelleration much smaller
than gravity ∂w
∂t g. This means that the relationship we have now derived will not hold for
steep slopes as continuity will dictate that ∂w
∂t ∼ g.
1.4.3
Linear Bottom Slope
Let us consider a two dimensional system (in x and z) where we have a bottom slope described by
H = H(x). Let us now assume that our solution is a simple harmonic time dependent solution.
η = Z(x)e−iωt
(1.86)
18
CHAPTER 1. WAVES IN A NON-ROTATING REFERENCE FRAME
Substituing this into equation (1.85), we get an ordinary differential equation for Z(x).
ω2
d
− Z−
g
dx
dZ
H
dx
=0
(1.87)
This is now an ordinary differential equation with a variable coefficient H. We can get exact
analytic solutions for certain forms of H. One of those forms is where H linearly increases with x,
see figure 1.6. We will now investigate this form.
H = αx where
α>0
(1.88)
Recall that α cannot be too large as we have assumed that the vertical accelleration must be much
smaller than gravity, dw
dt g. Thus, equation (1.87) becomes
d
dx
x
+
ω2
Z = 0.
gα
dZ
dx
r
x
gα
(1.89)
This equation has an exact solution.
Z(x) = AJ0 2ω
+ BY0 2ω
r
x
gα
(1.90)
Where J0 is the Bessel function of the first kind of order 0 and Y0 is the Bessel function4 of the
second kind of order 0 and A & B are constants of integration. Now recall that the Bessel function
of the second kind, Y0 (x) → −∞ as x → 0. We require a non-singular solution at that point, thus,
we must set B = 0. Now our solution is finite and we have a form for Z.
r x
Z(x) = AJ0 2ω
(1.91)
gα
r x
⇒ η(x, t) = AJ0 2ω
e−iωt
(1.92)
gα
Now, if we wish to find the associated velocity, we need to recall from section 1.4.2 that
∂η
∂u
= −g
∂t
∂x
(1.93)
Similarly to the surface height, we assume a particular form, and subsitute into equation (1.93) to
0 (x)
find our solution for u(x, t). We also need to recall that DJDx
= −J1 (x).
u(x, t) = U (x)e−iωt
r
r 1 1
x
√
J0 2ω
⇒ −iωU = −gA2ω
gα 2 x
gα
r
r g −1
x
2
U (x) = Ai
x J1 2ω
α
gα
r
r g −1
x
⇒ u(x, t) = Ai
x 2 J1 2ω
e−iωt
α
gα
(1.94)
(1.95)
(1.96)
(1.97)
While this may seem like a complicated result, we can use it in relatively simple ways to tell us
information about the amplitude of waves near the coast where the coast is reasonably straight
and the sea floor is gently sloping, similarly to the situation described in figure 1.6.
Given a measured amplitude of η = ηL at the edge of the sloping topography, x = L, we
can relatively simply find an expression that tells us the amplitude of the sea surface waves for
` x ´2m+n
P
J (x) cos(nπ)−J−n (x)
(−1)m
that Jn (x) = ∞
and Yn (x) = n
where Γ is the generalised
m=0 m!Γ(m+n+1) 2
sin(nπ)
factorial function (to include non-ingeters). These definitions of Bessel functions are included for completeness and
we do not need to overly concern ourselves with their exact form – however, we will rely on their properties to gather
information about constants of integration as well as properties of solutions.
4 Recall
1.4. WAVE CONSERVATION
19
η = +ηL
η
η = −ηL
x=0
x=L
Figure 1.6: Schematic of a linear bottom slope. We can find an exact solution for the wave
amplitude in this situation.
0 ≤ x ≤ L. We substitute this into equation (1.90) to solve for A.
Z(L) = ηL
(1.98)
s
⇒ ηL = AJ0
2ω
!
L
gα
q x
J0 2ω gα
⇒ η(x, t) = ηL q e−iωt
L
J0 2ω gα
The maximum amplification of the wave is going to occur at x = 0 is going to be Z(0) =
(1.99)
(1.100)
“ ηL
”.
q
L
J0 2ω gα
Thus, if we look at the amplification factor at x = 0 as this will give us an idea about the signficance of this amplification.
An example: Let’s plug in some reasonable numbers and see what we find. Say L = 10km=
104 m, α = 0.1 (this represents a decrease in depth of 1m for every 10m horizontally), g = 10ms−2
2π
and let the period T , be in hours, such that the frequency is given by ω = 60×60T
.
s
2ω
2 × 2π
L
=
gα
3600T
∼
1
3T
r
104
10 × 0.1
(1.101)
(1.102)
If we
choose a long period wave (low frequency) wave with a period of about 12 hours, we get
1
J0 36
' 0.999. This indicates that the amplification is insignificant as the amplitude at x = 0 is
virtually the same as that at x = L.
Next, if we choose a short period (high frequency) wave with a period of about 10 minutes
(= 1/6 hour), we find that J0 (2) ' 0.224. Thus the amplitude of the wave at the shore, x = 0, is
1
0.224 ' 4.5 times greater than at the edge of the slope, x = L.
20
CHAPTER 1. WAVES IN A NON-ROTATING REFERENCE FRAME
Chapter 2
Shallow Water Waves in a
Rotating Reference Frame
The effects of rotation are not significant for many classes of water waves. However, for many
aspects of global climate, planetary scale waves that are affected by the Earth’s rotation are very
important. In this chapter we will build on the theory that we have developed in the previous
chapter by relaxing the assumption that the effects of rotation are unimportant.
We will examine a number of types of important planetary scale waves. In the next chapter,
we will examine how some of these waves play an extremely important role in one of the most
globally dominant climatic features, the El Niño-Southern Oscillation.
2.1
Poincaré (gravity-inertia) waves
Thus far in our discussion on waves, we have ignored the effects of rotation, and indeed, we have
not even specified the form of rotation (see section 1.1.1). To breifly recap, the rotation vector,
Ω = 21 (0, f ∗ , f ) has two components: a horizontal component and a vertical component. For the
vast majority of problems (including those considered here), we can ignore the horizontal component, f ∗ = 0. The vertical component is the familiar coriolis parameter, f = 2Ω sin(φ), where φ is
the latitude and Ω is the Earth’s angular velocity.
If we now remove the irrotational assumption, which is true for waves with period O(hours),
we arrive at new equations of motion.
∂η
∂u
− f v = −g
∂t
∂x
∂v
∂η
+ f u = −g
∂t
∂y
p = ρg(η − z)
(2.1)
(2.2)
(2.3)
We now assume that the velocity is of the form u(x, y, t) = F (x, y)e−iωt and v(x, y, t) = G(x, y)e−iωt
and substitute this into the equations for the horizontal velocities.
∂η
∂x
∂η
−iωv + f u = −g
∂y
−iωu − f v = −g
(2.4)
(2.5)
The equations are now in a relatively easy form to solve for the velocities by cross substitution.
u=g
∂η
−iω ∂x
+ f ∂η
∂y
v = −g
ω2 − f 2
∂η
iω ∂η
∂y + f ∂x
ω2 − f 2
21
(2.6)
(2.7)
22
CHAPTER 2. SHALLOW WATER WAVES IN A ROTATING REFERENCE FRAME
ω
f
κ
Figure 2.1: The dispersion relation, ω 2 = f 2 + gHκ2 , for Poincaré waves (a.k.a. inertia-gravity
waves). Note that the frequency has a lower bound, being ω 2 > f 2 .
If we now assume that these waves do not have a large meridional extent, we can consider φ to
be constant and therefore f to be constant when taking spatial derivatives. We also assume that
the frequency does not change significantly with space and we may also consider this a constant.
Finally, we also assume that the time dependent portion of the sea surface height, η has the form
e−iωt .
Combining these assumptions with the results from the previous chapter, recalling the continuity equation for shallow water waves with an arbitrary bathymetry (but not allowing it to get
too steep), equation (1.83). For completeness, we restate equation (1.83) first.
∂η
∂
∂
+
(Hu) +
(Hy) = 0
∂t
∂x
∂y
∂
∂η
∂η
g
∂η
∂
∂η
+f
+f
−iωη + 2
H −iω
−
H iω
=0
ω − f 2 ∂x
∂x
∂y
∂y
∂y
∂x
(2.8)
(2.9)
If we now assume a uniform depth (i.e. constant H), we end up with a much easier expression to
solve.
∂2η
∂2η
∂2η
∂2η
gH
−iω
+
f
−
iω
+
f
=0
(2.10)
−iωη + 2
ω − f2
∂x2
∂x∂y
∂y 2
∂y∂x
gH
∂2η
∂2η
−iωη + 2
−iω
−
iω
=0
(2.11)
ω − f2
∂x2
∂y 2
ω2 − f 2
∂2η
∂2η
η+ 2 + 2 =0
(2.12)
gH
∂x
∂y
2.1.1
Dispersion Relation
By inspection of equation (2.12), we can see that η(x, y, t) = η0 ei(k·x−ωt) is a solution to this
equation on the condition that
−(k 2 + l2 ) +
ω2 − f 2
=0
gH
⇒ ω 2 = f 2 + gHκ2 .
(2.13)
(2.14)
Recall that κ2 = k 2 + l2 . We reach this dispersion relation by simply substituting our solution into
the partial differential equation in equation (2.12) and then dividing through by η.
We can see by inspection of relation (2.14) that ω must always be greater than the coriolis
parameter (aka, the inertial frequency), ω 2 > f 2 . The dispersion relation is show in figure 2.1.
2
f
We can see that for small f or large k (i.e. k 2 gH
), this class of waves becomes the normal,
surface gravity wave in the shallow water limit, described in section 1.2.3.
2.1. POINCARÉ (GRAVITY-INERTIA) WAVES
23
y
(u, v)
b
x
a
Figure 2.2: The time evolution of the horizontal velocity asssociated with a Poincaré wave take
η0 gf k
the form of an ellipse with the length a = ωη02 gωk
−f 2 being the major axis and b = ω 2 −f 2 being the
minor axis.
2.1.2
Horizontal Particle Paths
Let us now substitute the solution for sea surface height, η(x, y, t) = η0 ei(kx+ly−ωt) , into the
expressions for velocity, equations (2.6) and (2.7).
ωk + ilf
η
ω2 − f 2
−ωl + if k
η
v=g 2
ω − f2
u=g
(2.15)
(2.16)
To keep things simple, lets now assume that we have a wave propagating in the x direction
only (i.e. l = 0). We also follow the convention that the physical solution is the real part of the
resultant complex expression, recalling that eix = cos(x) + i sin(x).
ωk
u=< g 2
η0 [cos(kx − ωt) + i sin(kx − ωt)]
ω − f2
η0 gωk
⇒u= 2
cos(kx − ωt)
ω − f2
if k
v=< g 2
η
[cos(kx
−
ωt)
+
i
sin(kx
−
ωt)]
0
ω − f2
η0 gf k
sin(kx − ωt)
⇒v=− 2
ω − f2
(2.17)
(2.18)
(2.19)
(2.20)
We can thus see that there is a component of the velocity which is perpendicular in the horizontal
to the direction of travel of the wave. If we square the velocities and add them together, recalling
that cos2 (x) + sin2 (x) = 1, we get a parametric equation in the form of an ellipse.
u2 (ω 2 − f 2 )2
v 2 (ω 2 − f 2 )2
+
=1
η02 g 2 ω 2 k 2
η02 g 2 f 2 k 2
u2
v2
+
=1
a2
b2
We can see that the major axis of the ellipse (which is in the x direction) has a length of a =
ad a minor axis in the y direction of b =
η0 gf k
ω 2 −f 2
(2.21)
(2.22)
η0 gωk
ω 2 −f 2
and is shown schamatically in figure 2.2.
Notice that
a
ω
=
b
f
⇒ b ' a for ω ' f
⇒ b ' 0 for ω f.
(2.23)
(2.24)
(2.25)
24
2.1.3
CHAPTER 2. SHALLOW WATER WAVES IN A ROTATING REFERENCE FRAME
Phase Speed and Group Velocity
Recall the expression for phase speed from equation (1.25). We can easily find the phase speed of
a Poincaré wave by recalling the dispersion relation in equation (2.14).
ω2
κ2
2
f
+ κ2 gH
c2 =
2
r κ
f2
⇒c=
+ gH
κ2
c2 =
(2.26)
(2.27)
(2.28)
We can see that this merely reduces to the phase speed for a surface gravity, equation (1.51) wave
when f = 0, which is non-dispersive (phase speed does not depend on wave number). However,
Poincaré waves are dispersive as there is a component of the phase speed which depends on the
wave number!
So, unlike the surface gravity waves, the group velocity for Poincaré waves will be different to
the phase speed. As we are considering an isotropic medium, we will only find the group velocity
for the x direction without loss of generality. Also, recall the definition of group velocity from
equation (1.41)
∂ω
∂ p 2
f + gHk 2
=
∂k
∂k
1
2kgH
= p
2 f 2 + gHk 2
kgH
kc
k
⇒ cg = gH
ω
=
2.2
(2.29)
(2.30)
(2.31)
(2.32)
Kelvin Waves
The Kelvin wave requires some kind of wave guide. To begin with, we will consider a lateral
boundary and the dynamics of the “coastally trapped Kelvin wave.”
As we are considering something with a boundary, we assert that there can be no normal flow
accross the boundary (we are applying the same condition to our lateral boundary, as we applied
to our bottom boundary in section 1.4.2). If we take a simple geometry in which we have a lateral
boundary at y = const. Indeed, in the instance where v · n = 0 at y = const. As a result of
this condition, there is, in fact, no associated meridional velocity anywhere. Our other boundary
condition is that (u, v) → 0 as y → ∞. We can now write down our equations of motion.
v ≡ 0 ∀ x, y
∂η
∂u
= −g
∂t
∂x
∂η
∂u
+H
=0
∂t
∂x
∂η
f u = −g .
∂y
(2.33)
(2.34)
(2.35)
(2.36)
A solution for the sea surface height which satisfies our boundary conditions both at the lateral
boundary and at infinity, is
η = η0 ei(kx−ωt) e−y/R .
(2.37)
We can see from equation (2.36) that our expression for u is:
u=
g
η0 ei(kx−ωt) e−y/R
fR
(2.38)
2.2. KELVIN WAVES
(a)f > 0
y
25
(b)f < 0
decaying η
c
decaying η
y
c
x
x
Figure 2.3: For both subfigures, dashed lines represent travelling wave crests with the direction of
travel indicated. (a) A Kelvin wave travelling eastward along a southern boundary in the northern
hemisphere; (b) A Kelvin wave travelling along a southern boundary in the southern hemisphere.
Using the other two equations, we get two expressions which allows us to solve for R.
ω = f Rk
gHk
ω=
fR
gH
⇒ R2 = 2
f
(2.39)
(2.40)
(2.41)
We should recognise |R| as being the Rossby radius of deformation, the length scale for which
rotation becomes important. Due to the fact that our expression for R is a square, it therefore has
two roots. If we now substitute the positive root of R into equation (2.39), we find the dispersion
relation.
p
ω = gHk
(2.42)
From the dispersion relation, we find the phase velocity. We begin, however, by using equation (2.39) – the reason for which will become obvious soon.
ω
k
= fR
p
= gH
c=
(2.43)
(2.44)
The most obvious thing of note about the phase velocity is that the Kelvin wave is nondispersive. There is, however, something of equal importance that is somewhat more subtle. Since
R appears as a square, it thus has two roots. The sign of the root of R that we choose depends
on the situation. We always want η to decay to zero far away from the boundary, and either
the positive or negative root of R should be chosen to satisfy this boundary condition. Note
from the expression for phase velocity in equation (2.43) that it is the combination of the signs of
R and f that dictates the direction of propagation. To illustrate, we will now look at two examples.
Example – easterly propagating wave in the northern hemisphere: see figure 2.3(a). For this
example, we should choose the positive root of R to ensure that the sea surface height decays far
from the boundary. Thus, since both f and R are positive, c is also positive.
Example – westerly propagating wave in the southern hemisphere: see figure 2.3(b). For this
example, we should choose the positive root of R to ensure that the sea surface height decays far
from the boundary. Since we have f < 0 and R > 0, we thus have c < 0.
To generalise this discussion, we note that an observer following a Kelvin wave will always
have the boundary on their right in the northern hemisphere and on their left in the southern
hemisphere. Thus, at meridional boundaries, Kelvin waves travel poleward at an eastern boundary and equatorward at a western boundary. Furthermore, if we had an enclosed basin entirely in
the northern hemisphere, Kelvin waves would always travel anti-clockwise around the boundaries.
Conversely, in the southern hemisphere, they would always travel clockwise.
26
CHAPTER 2. SHALLOW WATER WAVES IN A ROTATING REFERENCE FRAME
f
2Ω
-90◦
-60◦
-30◦
0◦
30◦
60◦
90◦
φ
−2Ω
Figure 2.4: The f plane (solid line) is shown with a beta plane (dashed line) centred at 45◦ . Note
how the beta plane closely approximates the f plane between 30◦ and 60◦ , however, particularly
at high latitudes, rapidly diverges from the f plane.
2.3
Rossby Waves
Rossby waves owe their existance to the conservation of potential vorticity. As you will recall from
previous lectures, vorticity is conserved (see notes on planetary vorticity and vortex stretching).
There are two instances where these dynamics will become important, one is when we have a large
meridional extent (and are no longer able to assume that f is locally constant). The other is over
variations in bottom topography.
In order to begin our discussion on Rossby waves, we must first investigate the beta plane
approximation.
2.3.1
The Beta Plane
Recall that the coriolis parameter is
f = 2Ω cos(φ).
(2.45)
where Ω is the angular speed of rotation of the earth and φ is the latitude. Recalling that y = re φ
where re is the earth’s radius, we can rewrite this expression as a variation from a particular
latitude.
f = 2Ω sin(φo + y/re )
(2.46)
We can now do a Taylor expansion about this point, only taking the first two terms.
f ' 2Ω sin(φ0 ) + 2Ω
' f0 + βy
y
cos(φ0 )
re
(2.47)
(2.48)
We can see that f0 = 2Ω sin(φ0 ), and β = 2Ω cos(φ0 )/re . The beta plane is illustrated in figure 2.4,
and as can be seen, it is not valid at high latitudes, where the rate of change of the gradient of f
D2 f
is fast (i.e. Dφ
2 is large).
At mid-latitudes, the beta plane approximation is valid only if βy is small compared with f0 . If
the meridional length scale is L, then βL f0 . There is a form of the beta plane which is valid at
the equator and is important for equatorial wave dynamics, however, we will cover this in chapter 3.
2.3. ROSSBY WAVES
2.3.2
27
Governing Equation
Let us now restate the equations of motion with a constant depth, H.
∂u
∂η
− f v = −g
∂t
∂x
∂v
∂η
+ f u = −g
∂t
∂y
∂u ∂v
∂η
+H
+
=0
∂t
∂x ∂y
(2.49)
(2.50)
(2.51)
As usual, we assume that the solution for sea surface height and our horizontal velocities are of
the form F (x, y)e−iωt . Next, we substitute this into the three equations of motion.
∂η
∂x
∂η
−iωv + f u = −g
∂y
∂u ∂v
+
−iωη + H
=0
∂x ∂y
−iωu − f v = −g
(2.52)
(2.53)
(2.54)
In order to solve for these three variables, we start by rearranging equation (2.52) in terms of u
and equation (2.53) in terms of η and then taking its derivative with respect to y.
fv
g ∂η
−
iω iω ∂x H ∂u ∂v
η=
+
iω ∂x ∂y
2
∂η
H
∂ u
∂2v
⇒
=
+
∂y
iω ∂y∂x ∂y 2
u=
(2.55)
(2.56)
Now, we are able to substitute equations (2.55) and (2.56) into equation (2.53) and rearrange.
2
∂ u
f
∂η
gH
∂v
−iωv −
fv − g
=−
+ 2
iω
∂y
iω ∂x∂y ∂y
2
∂ v
∂η
∂2u
(ω 2 − f 2 )v + gH 2 = −g f
+H
(2.57)
∂y
∂x
∂x∂y
In order to simplify this equation, we need to do some heavy algebra.
We begin by taking the derivative with respect to y of equation (2.52) and with respect to x of
equation (2.53). First, we must note that using the beta plane approximation from equation (2.48)
∂
∂v
∂v
∂v
and the product rule, ∂y
(f v) = f0 ∂y
+ βv + βy ∂y
= f ∂y
+ βv.
−iω
∂u
∂v
∂2η
−f
+ βv = −g
∂y
∂y
∂y∂x
∂v
∂u
∂2η
−iω
+f
= −g
∂x
∂x
∂x∂y
(2.58)
(2.59)
Next we subtract equation (2.58) from equaton (2.59) and assume operator commutation.
∂v
∂u
∂u ∂v
−iω
−
+f
+
+ βv = 0
(2.60)
∂x ∂y
∂x ∂y
Recognising that we have seen the second bracketed term before, we then substitute equation (2.54)
into equation (2.60).
∂v
∂u
f iω
−iω
−
η + βv = 0
+
∂x ∂y
H
iω
∂u
∂v
fη + H
= iω
− βv
(2.61)
H
∂y
∂x
28
CHAPTER 2. SHALLOW WATER WAVES IN A ROTATING REFERENCE FRAME
Next, we differentiate this last equation with respect the x, and then multiply both sides by
−gH/iω.
∂η
∂2u
−g f
+H
∂x
∂x∂y
= −gH
∂2v
gHβ ∂v
+
∂x2
iω ∂x
(2.62)
Notice that the left hand side of this equation is exactly the right hand side of equation (2.57), the
equation that we have set out to simplify. We are now able to eliminate η from these equations.
2
ω − f2
∂2v
∂2v
∂v
−iω
v+ 2 + 2 +β
=0
(2.63)
gH
∂x
∂y
∂x
Note, that with a little more manipulation, and setting β = 0 and f = f0 we can regain equation (2.12), our governing equation for Poincaré and Kelvin waves. This is nice, as it gives us
confidence in our result.
Let us now consider waves that are very low frequency waves, ω 2 f 2 . We can then further
simplify equation (2.63) using this approximation. Let us also replace all f 2 terms with f02 by
assuming that f0 βy f02 and (βy)2 f02 , which is not unreasonable for mid-latitudes, as we
have stated in section 2.3.1 that the beta plane approximation is only valid for small values of βy
compared to f0 .
2
∂ v
∂2v
f02
∂v
−iω
+
−
v
+β
=0
(2.64)
∂x2
∂y 2
gH
∂x
2.3.3
Dispersion Relation
As usual, we seek solutions of the form v = Aei(kx+ly−ωt) . We now substitute this into equation (2.64) to find the dispersion relation.
f2
−iω −k 2 − l2 − 0 + ikβ = 0
gH
−kβ
⇒ω= 2
κ + R−2
(2.65)
(2.66)
2
2
2
We recognise R2 = gH
f 2 as the Rossby radius and κ = k + l . Let us now look at the phase
velocity, c = (cx , cy ), in the x direction – cf. equation (1.24).
cx =
⇒ cx =
ω
k
−β
κ2 + R−2
(2.67)
Note that the phase velocity in the x direction is always negative. As we take east to be our positive x direction, we can therefore deduce that Rossby waves always travel in a westward direction.
The interesting properties of a Rossby wave do not finish there. We can rewrite the dispersion
relation, equation (2.66) to describe a circle1 in (k, l) space.
kβ
k2 +
+ l2 + R−2 = 0
ω
"
2 #
2
2kβ
β
β
2
k +
+
+ l2 + R−2 =
2ω
2ω
2ω
2
2
β
1
β
+ l2 =
− 2
k+
2ω
2ω
R
(2.68)
(2.69)
(2.70)
This dispersion relation thus describes a cicrle in (k, l) space centred on (−β/2ω, 0) with radius
p
(β/2ω)2 − R−2 . This relation is shown in figure 2.5. For this relation to exist, the radius of the
1 recall,
the form of a circle, centred on x0 , y0 with radius r is (x − x0 )2 + (y − y0 )2 = r2
2.3. ROSSBY WAVES
29
circle must be bigger than zero, and we thus find there must be an upper limit on ω.
0<
2.3.4
β
2ω
2
β
2ω
2
−
1
<
R2
ω2 <
1
(βR)2
4
1
R2
(2.71)
(2.72)
(2.73)
Group Velocity
Recall from equation (1.41) that group velocity is defined as cg = (∂ω/∂k, ∂ω/∂l). Geometrically,
in our (k, l) plane, this means that the group velocity is orthogonal to the lines of constant ω.
β
f02
2
2
, 2kl
(2.74)
cg = 2 k − l −
gH
f02
k 2 + l2 + gH
In order to find which direction the group velocity travels, we set its components to zero, which
will give us the dividing line. We begin with the k component.
k 2 − l2 −
f02
=0
gH
(2.75)
⇒ k 2 = l2 +
f02
gH
(2.76)
This hyperbola describes the dividing line between eastward and westward travelling group velocities, with westward travelling group velocities to the right, and eastward travelling group velocities
to the left. This is a very interesting result, as we have westward propagating waves, but the wave
energy can travel eastwards.
Next, we examine the group velocity in the y direction.
∂ω
=
∂l
κ2 +
β
f02
gH
2 2kl
(2.77)
2 2
f0
β 2 and κ2 + gH
are always positive, while k is always negative. Thus, the group velocity always
has the opposite sign of l. That is, waves with a northward phase velocity have a southward group
velocity and waves with a southward phase velocity have a northward phase velocity. The group
velocity is indicated by the arrows on figure 2.5.
2 As
we can see from figure 2.4 the gradient of f is always positive.
30
CHAPTER 2. SHALLOW WATER WAVES IN A ROTATING REFERENCE FRAME
k 2 = l2 +
f02
gH
l
k
Figure 2.5: The dispersion relation for Rossby waves. The dotted line is the line that separates
the eastward and westward group velocity with the hyperbolic relationship between k and l. The
dashed circles are lines of constant frequency, ω = constant. The arrows represent the direction of
the group velocity cg .
Chapter 3
Waves and the El Niño-Southern
Oscillation
Why would one want to study ENSO? ENSO is the most dominant interannual climate signal
on Earth. Both ENSO warm (aka El Niño) and cold (aka La Niña) events have major economic
and social impacts on people. ENSO warm events can cause droughts in the western Pacific from
India to Indonesia to Australia. In the eastern Pacific, one of the most productive fisheries in the
world, have very poor years. It is not just those who live in the shadow of the Pacific ocean who
are affected. ENSO “teleconnections” have been shown to affect the amount of snowfall in North
America as well as the number of Atlantic hurricanes in a given season to name only a couple. A
map of ENSO relationships are shown in figure 3.1.
(a) ENSO warm events
(b) ENSO cold events
Figure 3.1: Illustrative map of ENSO regional relationships. Images courtesy of NOAA.
As it turns out, equatorial wave dynamics plays an extremely important role in the dynamics
of ENSO, and holds the key to unlocking the mystery of ENSO. In this chapter, we will apply
what we have learned so far about wave dynamics to this important topic in climate science. It is
an area where wave dynamics has taught us so much, but at the same time, still has much to tell
us. ENSO is a very active area of research, and is exteremly complex. Presented here is a simple
explanation of a very complex process.
31
32
3.1
CHAPTER 3. WAVES AND THE EL NIÑO-SOUTHERN OSCILLATION
What is ENSO?
ENSO warm and cold events1 refer to changes in the average temperatures of particular parts of
the sea surface temperature of the tropical Pacific. The reason for these changes in average SST,
and the concequences of these changes are what is of most interest to us.
3.1.1
The “Normal” State
The waters of the western tropical Pacific are some of the warmest in the world, with temperatures
exceeding 28◦ C. At the same time, we have the persistent easterly trade winds. Since we are at the
equator, geostrophy is not strong, and these winds “drag” the very warm surface waters westward,
“piling” them up in the western Pacific. This region of very warm water is known as the western
Pacific warm pool (WPWP). The WPWP is clearly evident in figure 3.2. The WPWP is defined
as the region enclosed by the 28.5◦ C isotherm. At the same time, the temperature gradient causes
these easterly winds. We thus have a positive feedback between the winds easterly winds and the
zonal temperature gradient. This is important and important fact, and we will return to this later.
Figure 3.2: Mean equatorial Pacific Sea Surface Temperature. Note the eastern Pacific cold tongue
that persists to a greater or lesser extent throughout an “average” year. Image couresty of NOAA.
The very high surface temperatures of the western Pacific causes atmospheric convection. There
is an upper atmospheric westerly return wind, which then sinks when it reaches the eastern Pacific.
This is called the “Walker circulation” after Sir Gilbert Walker, who first proposed the circulation.
The Walker circulation is illustrated in figure 3.3.
1 The term ENSO warm event and ENSO cold event is becoming the preferred terminology of climate scientists,
over the more commonly used El Niño and La Niña. The reason for this is because, historically, El Niño (translates
to “the Child Jesus”) referrs to a southward flowing current that appears off the coast of Peru around Christmas
(hence the name bestowed upon this current by the fisherman of the town of Paita in Peru). Having said that El
Niño and ENSO warm event, as well as La Niña and ENSO cold event are used interchangeably in these notes.
3.2. EQUATORIAL WAVE DYNAMICS
33
Figure 3.3: An illustrative “normal” sea surface temperature, thermocline depth and atmospheric
convection and tradewind pattern. Image courtesy of NOAA.
The water along the equator, travelling from east to west, causes deep water in the eastern
Pacific to be upwelled. It is this cold, nutrient rich water, which makes the eastern Pacific fishing
grounds so fertile and productive. This cold, upwelled water joins the movement of surface water
westward, slowly warming as it goes. This cold water is referred to as the “eastern Pacific cold
tongue.” This is evident in figure 3.2.
3.1.2
The Bjerknes Feedback
The situation described in the previous section involves a very closely coupled atmosphere-ocean
interaction. Even a slight perturbation to this system, under favourable circumstances, can cause
a negative feedback loop. A slackening of the trade winds, weakens the zonal equatorial temperature gradient, which in turn weakens the easterly winds, which in turn slackens the tradewinds
further, and so on and so forth. This feedback is called the Bjerkens feedback after the scientist
Jacob Bjernes, who first proposed this feedback in 1969 to describe how ENSO warm events are
instigated and grow.
Obviously, there must be some sort of “turnabout” mechanism at work, otherwise, we would
be in a constantly increasing El Niño. It was not for nearly another 20 years before a viable
mechanism was proposed. In order to understand this mechanism, we must apply what we have
learned about waves to the equator.
3.2
3.2.1
Equatorial Wave Dynamics
The Equatorial Beta Plane
The equator tends to act as a wave guide, due to the fact that the sign of f changes. This wave
guide tends to “trap” waves between approximately 7◦ S and 7◦ N (although, this depends somewhat on the type of wave we are considering).
Recalling equation (2.48), for the beta plane approximation, we note that f0 = 0 at the equator.
If we look at figure 2.4, we see however, that the beta plane should be a good approximation there,
even though in the vicinity of the equator f0 6 βy, in the vicinity of the equator, the beta plane
approximation is still appropriate.
f = βy
(3.1)
34
CHAPTER 3. WAVES AND THE EL NIÑO-SOUTHERN OSCILLATION
(a) El Niño
(b) La Niña
Figure 3.4: Illustrative sea surface temperature, thermocline depth and atmospheric convection and
tradewind pattern. Compare these images with the “normal” case in figure 3.3. Images courtesy
of NOAA.
Another important thing to note in the vicinity of the equator, is that the normal expression for
our Rossby radius of deformation, (see section 2.3.3) does not hold (as it has f in the denominator,
and at the equator, would be infinite). Instead, we use the equatorial Rossby radius of deformation,
Re .
√
c
gH
2
≡
(3.2)
Re =
β
β
In the equatorial beta plane, we can now state our equations of motion for a constant depth
ocean.
∂u
∂η
− βyv = −g
∂t
∂x
∂v
∂η
+ βyu = −g
∂t
∂y
∂η
∂u ∂v
+H
+
=0
∂t
∂x ∂y
3.2.2
(3.3)
(3.4)
(3.5)
The Equatorial Kelvin Wave
The equatorial Kelvin wave is so called, because it very closely resembles the coastally trapped
Kelvin wave disucssed in section 2.2. The main difference is that instead of havnig a solid boundary on one side, with exponential decay far away, the equator acts as a wave guide and we have
exponetial decay at y → ±∞.
Our decay length scale, instead of being the Rossby radius of deformation, R, like in the
coastally trapped Kelvin wave, is instead the equatorial Rossby Radius of deformation, Re .
2
η = η0 e−y /2Re cos(kx − ωt)
r
g −y2 /2Re
e
cos(kx − ωt)
u=
H
v=0
(3.6)
(3.7)
(3.8)
The dispersion relation for the equatorial Kelvin wave is the same as for the coastally trapped
Kelvin wave.
ω = kc
(3.9)
Kelvin waves tend to travel very fast, with a typical speed being 2.8ms−1 . With this speed, it
would take roughly two months for a Kelvin wave generated off the coast of New Guinea to reach
the coast of South America.
3.2. EQUATORIAL WAVE DYNAMICS
y
35
decaying η
c
x
EQ
c
decaying η
Figure 3.5: The equatorial Kelvin wave. The dashed lines represent wave crests, the phase velocity
is indicated and is always eastward. The equator, EQ, is indicated and is equivalent to y = 0.
The sea surface height anomaly η decays away from the equator, scaling like the equatorial Rossby
radius of deformation, Re . All particle motion is in the x direction (not shown).
3.2.3
The Equatorial Rossby Wave
If we recall our original equation for Rossby waves, equation (2.63), and seek solutions of the form
ei(kx−ωt) . We begin by restating that equation for brevity, and remembering that f = βy.
2
∂2v
∂2v
∂v
ω − f2
v+ 2 + 2 +β
=0
(3.10)
−iω
gH
∂x
∂y
∂x
2
ω
(βy)2
∂2v
−iω
v−
v − k 2 v + 2 + ikβv = 0
(3.11)
gH
gH
∂y
2
ω
(βy)2
kβ
∂2v
2
+
−
−
k
−
v=0
(3.12)
∂y 2
gH
gH
ω
It turns out that there is a solution to such an equation.
v = Dn (ξ) cos(kx − ωt)
(3.13)
2
Dn (ξ) = 2−n/2 eξ /2 Hn (ξ) is the Hermite function (sometimes also known as the parabolic cylinder
function), Hn (ξ) is the nth Hermite polynomial2 and ξ = y/Re . The Hermite function should
be familiar to the physicists amongst you, as it is a solution to the Schrödinger equation for a
harmonic oscillator in quantum mechanics. The first six Hermite functions are shown in figure 3.6.
We can, with some effort (that we shall not concern ourselves with here), also find expressions for
the zonal velocity, u and sea surface height, η.
√
√
p
nDn−1 (ξ)
n + 1Dn+1 (ξ)
u = i 2βei(kx−ωt)
+
(3.14)
ω + ck
ω − ck
√
√
p
nDn−1 (ξ)
n + 1Dn+1 (ξ)
i(kx−ωt)
η = − 2βe
+
(3.15)
ω + ck
ω − ck
Note that the meridional velocity (v) is proportional to the Hermite function, and so is symmetric
for even n, and anti-symmetric for odd n. Conversely the zonal velocity u, and the sea surface
2 Not
to be confused with H, the depth of the ocean. The Hermite polynomial is defined as Hn (ξ) =
2 Dn
−ξ2 . As with the Bessel function considered in section 1.4.3, we will not be needing to concern
(−1)n eξ Dξ
ne
ourselves too much with the Hermite polynomial, but we will be exploiting some of its properties. For reference,
the first 6 Hermite polynomials (starting with n = 0) are H0 = 1, H1 = 2ξ, H2 = 4ξ 2 − 2, H3 = 8ξ 3 − 12ξ,
n
= 2nHn−1 and
H4 = 16ξ 4 − 48ξ 2 + 12 and H5 = 32ξ 5 − 160ξ 3 + 120ξ. Another couple of handy identites are DH
Dξ
ξHn = nHn−1 + 0.5Hn+1
36
CHAPTER 3. WAVES AND THE EL NIÑO-SOUTHERN OSCILLATION
(a) Even Hermite Functions
(b) Odd Hermite Functions
Figure 3.6: Scaled Hermite functions from n = 0 to n = 5. These are only indicative, and the
numbers on the axes have no physically based meaning.
height, η are linear combinations of symmetric Hermite functions for odd n and linear combinations
of anti-symmetric Hermite functions for even n.
The corresponding dispersion relation is.
ω2
βk
β
− k2 −
= (2n + 1) √
gH
ω
gH
(3.16)
Note that there are two solutions for each n ≥ 1 as this dispersion relation is quadratic in ω. If
we assume that the term βk
ω is small (i.e. large ω, for high frequency), then we get a dispersion
relation for equatorially trapped Poincarè waves.
ω 2 ' (2n + 1)β
p
gH + k 2 gH
(3.17)
This dispersion relation corresponds to relatively high frequency waves and is the upper branches
of the dispersion relation in figure 3.7. Equatorially trapped Poincarè waves are unimportant in
ENSO dynamics and we will not consider them further.
The other branch assumes low frequency, and that
ω'−
k
√
2
ω2
gH
is small.
√
βk gH
gH + (2n + 1)β
(3.18)
Note, that as with the Rossby wave discussed in section 2.3, the only direction of propagation is
westward.
The case of n = 0 is a special case, and is often referred to as the mixed Poincaré-Rossby (or
mixed gravity-planetary or Yanai) wave. This wave is also not important in the study of ENSO,
and will not be considered here. The dispersion relation for all four types of equatorial waves
(Kelvin, n = −1; mixed, n = 0; Poincaré, n ≥ 1; and Rossby, n ≥ 1) is shown in figure 3.7.
3.3
Explaining ENSO Periodicity
Armed with our knowledge of eastward propagating equatorial Kelvin waves and westward propagating equatorially trapped Rossby waves, we are now ready to investigate the “turnabout”
mechanism discussed in section 3.1.2. Firstly, we introduce the various “Nino” regions, shown in
figure 3.8.
3.3. EXPLAINING ENSO PERIODICITY
37
Figure 3.7: Dispersion diagram for equatorially trapped waves, including the first two Poincaré
(intertia-gravity) waves, the first two Rossby (planetary) waves, the equatorially trapped Kelvin
wave and the mixed gravity-Rossby wave (also sometimes known as a Yanai wave).
Figure 3.8: A map indicating the Nino 1, 2, 3 and 3-4 regions. Note that there are also Nino 5 and
6 regions, but these are not commonly used and omitted in this map. Image courtesy of NOAA.
3.3.1
The Delayed Action Oscillator
In order to understand the Delayed Action Oscillator (DAO), we must firstly recall the definition
of the western Pacific warm pool (WPWP) in section 3.1.1. As an ENSO warm event is devloping,
the eastern edge of the WPWP creeps eastward, altering the region of atmospheric convection,
and also affecting the surface winds. Due to this fact, the DAO considers that the region of most
importance (that is, the region of strongest atmosphere-ocean coupling) is the central-western Pacific (approximately, around the date-line).
We wish to establish a function that describes this anomalous displacement of the eastern edge
of the WPWP, which is defined as the 28.5◦ C isotherm, between 4S and 4N. We begin by denoting
its anomalous displacement.
x = x(t)
(3.19)
The anomalous displacement must be caused by anomalous currents, which we denote as u0 , which
is a linear combination of three parts u01 , u02 and u03 .
Dx
= u0 (t) ≡ u01 (t) + u02 (t) + u03 (t)
Dt
(3.20)
38
CHAPTER 3. WAVES AND THE EL NIÑO-SOUTHERN OSCILLATION
The first part, u01 is essentially proportional to the local anomalous zonal wind stress. As metioned
above, a small displacement of the eastern edge of the WPWP will cause atmospheric deep convection to shift eastward, which causes anomalous westerly winds (see figure 3.4(a)) and anomalous
westerly ocean surface currents.
u01 (t) = ax(t)
(3.21)
This essentially describes the Bjerknes feedback, and if this were the end of the story, we would
have an exponentially growing function. Thus, there must be some negative feedback mechanisms
that cause the turnabout.
The negative feedback is a result of the anomalous westerly winds generating equatorial Rossby
waves, which propagate westward, reflect at the western boundary and return as eastward propagating equatorial Kelvin waves. The returning Kelvin waves arrive back in the central-west Pacific
some time, ∆ later. These Kelvin waves generate westward surface currents, which act to return
the WPWP edge westward, providing the necessary negative feedback to cause a turnaround.
u02 (t) = −bx(t − ∆)
(3.22)
Due to the fact that ∆, the return time of the Kelvin wave, is related to the distance that the
Kelvin wave must travel (and therefore the edge of the WPWP), some more complex DAO theories
include this effect and make ∆ = ∆(x), which introduces a non-linearity. We will not consider this
case.
The third anomalous velocity u03 results from the generation of eastward propagating Kelvin
waves by the same winds that generated the Rossby waves. Most versions of the DAO theory,
however, states that the reflection of the Kelvin waves at the eastern boundary is unimportant, or
is qualitatively the same effect as for u02 and is incorporated in this effect.
u03 = 0
(3.23)
Dx
= ax − bx(t − ∆)
Dt
(3.24)
We now have a differential equation.
This equation has a solution of the form
x(t) = Aeσt cos(ωt).
(3.25)
As it turns out, if we use some realistic numbers for a, b and ∆ we can get a solution which
matches the periodicity of ENSO reasonably well (given the simplicity of this model), with oscillations between ENSO warm, cold and normal states. By varying ∆, we can find some very
interesting properties. For instance, if we make ∆ sufficiently large, we get solutions where the
instability grows large before the negative feedback has enough time to provide the turnabout, and
we have a permanent El Niño.
Conversely, if ∆ is small, the negative feed back reacts quickly, and the instability does not
have time to grow and we are in a permanent “normal” state. This is of critical importance in
understanding why ENSO is unique to the Pacific, and does not occur in the Atlantic or the Indian
oceans. In order for the instability to grow, we require the tropical section of an ocean to have a
minimum width.