Content uploaded by Willem E. van Caspel
Author content
All content in this area was uploaded by Willem E. van Caspel on Jan 11, 2024
Content may be subject to copyright.
Planetary Wave Drag:
Theory, Observation and its Role in Shaping
the General Circulation
Wim van Caspel
A thesis submitted for the degree of
Master of Science, Climate Physics
Supervisor:
Dr. A.J. van Delden
Institute for Marine and Atmospheric Research (IMAU)
Department of Physics
June 2018
Abstract
Breaking planetary waves in the wintertime stratospheric surf-zone are associated with a local
loss of angular momentum, or ’drag’ force. This drag force is, among other things, responsible
for driving the Brewer-Dobson circulation. The concept of planetary wave drag is investigated
using zonal mean quasi-geostrophic theory. In particular, quasi-geostrophic potential vorticity
is used to describe the coupled interactions between planetary waves, the polar vortex and the
stratospheric surf-zone. Theory is complemented with observation using reanalysis data. To
study the role of planetary wave drag in shaping the general circulation, a parameterization
of planetary wave drag is implemented in a zonal mean model of the atmosphere. Results
from quasi-geostrophic theory are used to interpret the model output. Model performance with
respect to the observed climatology is quantified with the use of Taylor-diagrams. Furthermore,
a selection of quasi-geostrophic results are tested and expanded upon using novel numerical
cyclo-geostrophic piecewise PV-inversion experiments with Rossby-Ertel PV-configurations.
Contents
1 A look at the stratosphere 1
1.1 Introduction....................................... 1
1.2 Wintertime dynamics and the polar vortex . . . . . . . . . . . . . . . . . . . . . 1
1.3 The quasi-geostrophic equations of motion . . . . . . . . . . . . . . . . . . . . . . 3
1.4 PV-maps and PV-inversion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2 Planetary waves 7
2.1 Shallow water β-planedynamics ........................... 7
2.2 Verticalpropagation .................................. 11
2.3 Pseudomomentum and momentum transport . . . . . . . . . . . . . . . . . . . . 13
2.4 Planetary wave breaking and critical layers . . . . . . . . . . . . . . . . . . . . . 15
3 Wave-mean flow interaction 18
3.1 Eliassen-Palm diagnostics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
3.2 The surf-zone and angular momentum loss . . . . . . . . . . . . . . . . . . . . . . 21
3.3 Sharp PV-gradients and self-sharpening jets . . . . . . . . . . . . . . . . . . . . . 23
3.4 Piecewise cyclo-geostrophic PV-inversion . . . . . . . . . . . . . . . . . . . . . . . 25
4 Planetary wave drag in the general circulation 30
4.1 A simplified model of the general circulation . . . . . . . . . . . . . . . . . . . . . 30
4.2 Quantifying model performance . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
Appendix 37
A.1 Observation: ERA5 and ERA-Interim reanalysis . . . . . . . . . . . . . . . . . . 37
A.2 Pseudomomentum and momentum . . . . . . . . . . . . . . . . . . . . . . . . . . 37
A.3 TheTaylor-identity................................... 39
A.4 Formdrag........................................ 39
A.5 PV-Gradientproof ................................... 41
A.6 Relating eddy fluxes of QGPV and Rossby-Ertel PV . . . . . . . . . . . . . . . . 43
A.7 PV-gradients of the subtropical and polar jet . . . . . . . . . . . . . . . . . . . . 45
A.8 Observation: Supplementary animations . . . . . . . . . . . . . . . . . . . . . . . 47
Chapter 1
A look at the stratosphere
1.1 Introduction
This thesis sets out to provide an overview of the theory, observation and role of planetary wave
drag in the general circulation. The theory forming the backbone of this work, largely revolves
around the quasi-geostrophic approximation. Within quasi-geostrophic theory, planetary wave
dynamics, as well as the large-scale dynamics of the stratosphere, can be studied in terms of a
Potential Vorticity (PV) budget. The description of the dynamics surrounding planetary waves
in terms of PV, lies at the heart of this work.
The contents of the first chapter are meant to serve as a general introduction to wintertime
stratospheric dynamics, as well as to develop some of the ’tools’ required for the subsequent
chapters. In Ch. 2, a few fundamental properties and characteristics of planetary waves are
discussed. These are then used in the description of planetary wave-mean flow interaction, in
Ch. 3. The contents of Ch. 2 and 3 are supplemented by observation either directly in the text,
or with reference to the animations shown in Appendix A.8. These animations are available
upon request. In Ch. 4, the role of planetary wave drag is discussed within the framework of
a zonal mean general circulation model. Model experiments are used to understand the role of
planetary wave drag in shaping the general mean circulation. The model’s parameterization of
planetary wave drag and the model’s output, is interpreted using the theory from Ch. 2 and 3.
Throughout this work, frequent comparisons of theory with observation as well as an em-
phasis on the most basic dynamics, is meant to give the reader an intuitive understanding of
the dynamics which are involved with the concept of planetary wave drag. Reference is made
to papers or books in which more rigorous derivations are given. To not obstruct the flow of
the text, lengthier derivations and concepts which do not directly contribute to the narrative of
planetary wave drag, are saved for the Appendix.
1.2 Wintertime dynamics and the polar vortex
Due to the tilt of Earth’s axis of rotation with respect to its orbit around the sun, the Earth ex-
periences seasons. The seasons are especially pronounced at high latitudes, where the difference
between summertime and wintertime insolation is greatest. During wintertime, cooling increases
the density of air, causing it to sink in accordance with hydrostatic balance. In the stratosphere,
this leads to the formation of a robust low-pressure system centered over the wintertime pole.
With radiative cooling being strongest at the pole, the low-pressure system becomes progres-
sively weaker at lower latitudes. This effectively causes pressure surfaces to bulge downwards
1
CHAPTER 1. A LOOK AT THE STRATOSPHERE 2
towards the pole, giving them the shape of a bowl. Air generally tends to flow from high to low
pressure areas, and because of the low-pressure system over the wintertime pole, air will want to
flow towards the poles. However, because of the rotation of the Earth, conservation of angular
momentum requires that the northward tendency of the wind due to the pressure gradient, is
accompanied by an eastward Coriolis deflection. When the atmosphere is in geostrophic balance,
which is generally a good approximation in the stratosphere, air will instead follow contours of
constant geopotential height (φ), defined by φ=−gz, where zis the Cartesian height of the
pressure surface and gis the acceleration due to gravity. Along a pressure surface, the geopoten-
tial height is indicative of the height of the pressure surface relative to the surface of the Earth.
For the wintertime stratosphere, the bowl-shaped pressure surfaces can viewed as a succession
of increasingly lower (circular) geopotential height contours. This causes the air to circle the
wintertime pole, forming a giant polar vortex. This vortex, referred to as the stratospheric polar
vortex, is fundamentally ’driven’ by radiative cooling.
If the polar vortex is for whatever reason slowed down, geostrophic balance will be disrupted,
and the pressure gradient force will induce poleward motion. In the presence of retrograde forces,
the stratospheric polar vortex can then be viewed as trying to ’drain’ into the North-pole’s upper
troposphere. To illustrate this metaphor, the geopotential height along the 50 hPa isobar is
shown in Fig. 1.1. The arrows in Fig. 1.1 represent the direction and magnitude of the wind,
19.0
19.2
19.4
19.6
19.8
20.0
Figure 1.1: Stereographic projection of the 2010-2016 DJF mean values of the geopotential height (black
contours) at 50 hPa, centered around the North pole. Contours are scaled by a factor of 10−5, arrows
indicate the direction and magnitude of the isobaric wind. Largest arrows correspond to a wind speed of
roughly 30 m/s. Data obtained from ECMWF ERA5 reanalysis (Appendix A.1).
with the largest arrows being on the order of 30 m/s. Towards the center of the vortex, the wind
can be seen having a component which is not tangential to the geopotential height contours.
This would not be possible if the flow is in perfect geostrophic balance, which alludes to the
presence of turbulent motion and retrograde forces. For the latter, an intuitive candidate would
be internal molecular viscous forces, but in the stratosphere these are negligible. As is hinted at
CHAPTER 1. A LOOK AT THE STRATOSPHERE 3
in the title of this work, this thesis explores the concept of breaking planetary waves providing
the retrograde force responsible for the observed systematic northward component of the wind.
The poleward tendency of the wind drives the so-called Brewer-Dobson circulation, which was
named in honour of pioneering work done by Alan Brewer and Gordon Dobson in the middle
of the 20th century. The systematic northward ’pumping action’ by breaking planetary waves
in the stratosphere, causes material tracers such as ozone to collect within the polar vortex. In
addition to this, the high winds surrounding the vortex edge, located approximately between
the 19.2e5 - 19.8e5 m2/s2geopotential height contours in Fig. 1.1, act as a barrier, creating a
marked difference between the composition of the interior and exterior of the vortex. This is
especially relevant to the formation of the wintertime stratospheric ozone hole. Much of theory
in discussed in this work, stems from research which was initiated by the need to understand
the dynamics of the ozone hole.
In contrast to wintertime, the higher latitudes receive an abundance of solar radiation during
summer. This effectively reverses the processes which lead to the formation of the wintertime
polar vortex, and hence in summer a stratospheric high-pressure system with accompanying
westward winds is observed. Due to dynamics discussed in Ch. 2 and 3, the dynamics surround-
ing planetary waves in the summertime stratosphere are however much less pronounced than
those in wintertime. This gives, for example, the Brewer-Dobson circulation a strong seasonal
character (Butchart [1]).
1.3 The quasi-geostrophic equations of motion
Conditions in the stratosphere are such that that the full equations of motion on a sphere, can
be greatly simplified. Namely: (1) there is a high level of static stability, restricting vertical
motion and making static stability only a function of height, (2) the Earth’s rotation dominates
the momentum balance, restricting meridional motion, (3) the horizontal length scale is much
smaller than the radius of the Earth. For added convenience, an often employed method is to
zonally average the entire system of equations. For any variable a(x, y, z, t) the zonal average is
denoted by an overbar, and is calculated as
¯a(y, z, t) = 1
LZL
0
a(x, y, z, t)dx, (1.3.1)
where Lis the length of the latitude circle at yand where the (x, y, z)-coordinates span a
Cartesian grid. In addition to zonally averaging, variables can be split in a zonal mean and eddy
term as follows,
a(x, y, z, t) = ¯a(y , z, t) + a0(x, y, z, t).(1.3.2)
This definition of eddies, or zonal asymmetries, therefore depends fundamentally on the defini-
tion of the zonal mean. The power of Eq. 1.3.1 lies in the fact that it allows for the ’natural’
decomposition of the general circulation in a primary zonal circulation, a secondary meridional
circulation, and eddies in the form of zonal asymmetries. Note that by the definition of Eq.
1.3.2, it follows that a0(x, y, z, t) = 0. As a final simplification, the often employed method of a
β-plane approximation is used. This approximation was first published by Carl-Gustav Rossby
in the early 20th century, and it assumes the mid-latitude background planetary vorticity gra-
dient to vary linearly. Around a certain latitude, usually taken to be y0at 45 degrees north, the
Coriolis parameter is then written as
f=f0+β(y−y0).(1.3.3)
CHAPTER 1. A LOOK AT THE STRATOSPHERE 4
Here yis the meridional coordinate and f0is the Coriolis parameter at y0. The β-plane parameter
is defined as β= 2Ωa−1cos(φ0), where Ω and aare the Earth’s angular velocity and radius,
respectively.
Geostrophic flow is non-divergent, which allows for the geostrophic components of the wind to
be written in terms of the geostrophic stream function (ψ). The geostrophic stream function ψ=
p/f0ρ0, is defined such that the geostrophic velocity field is given by (u, v)=(−∂yψ, ∂xψ), and
that the buoyancy force is given by b0=f0∂zψ. Using the quasi-geostrophic scaling arguments
on a β-plane, zonally averaging, and with the definition of the zonal mean and eddy terms, the
quasi-geostrophic equations of motion are written as1
∂¯u
∂t −f0¯va=Gx−∂
∂y u0v0(1.3.4a)
∂¯va
∂y +∂¯wa
∂z = 0 (1.3.4b)
f0
∂
∂z ¯u=−∂
∂y ¯
b(1.3.4c)
∂¯
b0
∂t + ¯waN2=B − ∂
∂y v0b0.(1.3.4d)
Here ( ¯va,¯wa) is the second order ageostrophic velocity, N2is the square of the buoyancy fre-
quency defined in QG-theory by N2=∂z¯
b, and Gxand Bare external momentum and buoyancy
forcings, respectively. The buoyancy field is defined as b=¯
b(z) + b0(x, y, z, t), where ¯
bis the
basic state buoyancy. The buoyancy force b0is defined as b0=−g(ρ0/ρ), or equivalently as
b0=g(θ0/θ), where ρand θare the density and potential temperature, respectively. From top
to bottom, Eq. 1.3.4a is the zonal momentum budget, Eq. 1.3.4b the continuity equation, Eq.
1.3.4c describes thermal wind balance and Eq. 1.3.4d is the thermodynamic equation. The
latter depends on the fact that waonly advects the basic state buoyancy ¯
b.
The perturbation terms (u0, v0)=(−∂yψ0, ∂xψ0), represent eddy terms of the first-order
geostrophic stream function. These eddy terms are independent of the ageostrophic circulation,
and by Eq. 1.3.1, need not be small. These eddy terms are therefore of an higher order than the
ageostrophic velocities that drive the residual circulation. One of the key features of Eq. 1.3.4a
- Eq. 1.3.4d, is that the time-development of the flow is completely captured by the quantity
called Quasi-Geostrophic Potential Vorticity (QGPV). The QGPV (q) is defined in terms of the
geostrophic stream function as
q=∇2ψ+f+f2
0
ρ
∂
∂z ρ
N2
∂ψ
∂z (1.3.5a)
Dgq
Dt =χ. (1.3.5b)
In Eq. 1.3.5b, Dg/Dt represents the geostrophic material derivative defined by Dg/Dt =∂t+
ug∂x+vg∂y, and χrepresents sources and sinks of q, such as friction and diabatic processes. The
geostrophic subscript on the velocity terms is often dropped, to write (ug, vg) = (u, v). When
there are no non-conservative effects, i.e. χ= 0, this corresponds to Gx=B= 0. The terms
on the right-hand side of Eq. 1.3.5a represent geostrophic relative vorticity, planetary vorticity
defined by Eq. 1.3.3, and a ’stretching’ term, respectively. Note that for the relative vorticity,
it follows from the definition of ψthat ζg=∇2ψ= (∂xv−∂yu). From Eq. 1.3.5b it follows
1A step-by-step derivation of the Quasi-Geostrophic (QG) equations of motion is available in many books on
geophysical fluid dynamics, e.g. Vallis [2] Ch.5, or in log-pressure coordinates in Andrews et al. [3] Ch.3.
CHAPTER 1. A LOOK AT THE STRATOSPHERE 5
that for conservative flows (χ= 0), qis a conserved quantity following horizontal geostrophic
motion. Splitting the variables for a conservative flow in a mean and eddy term as before, and
taking the zonal mean, reduces Eq. 1.3.5b to
∂¯q
∂t +∂
∂y v0q0= 0.(1.3.6)
This expression relies on the fact that, (1) terms containing zonal derivatives vanish with zonal
averaging, (2) zonal means of perturbation terms vanish by the definition of Eq. 1.3.2, (3)
geostrophic flow is non-divergent, such that ∂y¯
v0= 0. Normally, the non-divergent geostrophic
flow is augmented by the ageostrophic circulation to ensure mass conservation. The result of
Eq. 1.3.6 is therefore a bit counter-intuitive in the sense that the complete time development of
the zonal mean flow, which also includes the ageostrophic velocities, is determined only by the
geostrophic meridional eddy fluxes of q.
The quasi-geostrophic approximation essentially filters small-scale dynamics from the equa-
tions of motion, which is why it is so well suited for modelling the dynamics of the stratosphere.
What remains is a flow in which the pressure gradient and Coriolis force are almost exactly in
balance, with ’quasi’ referring to the inclusion of second-order inertia effects. This implies that
small-amplitude waves such as inertio-gravity and gravity waves cannot be represented by QG-
theory, but these waves typically only play a role at altitudes much higher than the stratosphere
(e.g. Holton [4]). In the context of this thesis, the most important property of QG-theory, is
that the QGPV-budget entirely governs the time development of the flow. This allows for a ’PV-
view’ of the dynamics, in the knowledge that it directly translates to the dynamics governed by
the full equations of motion given by Eq. 1.3.4a - 1.3.4d. It should be noted that QG-theory is
rather a rough approximation, with a more complete description of PV-dynamics being given by
the ’exact’ Rossby-Ertel PV, often simply referred to as PV in isentropic coordinates. However,
the qualitative insights given by QG-theory are robust, in the sense that more sophisticated
models do not have fundamentally different large-scale dynamics.
To conclude, QG-theory allows for a ’PV-view’ of the large-scale dynamics in the strato-
sphere. Rossby-Ertel PV dynamics may refine this picture, but the large-scale features are still
qualitatively similar to those from QG-theory. This notion lies at the core of the first three
chapters of this thesis, as it justifies the study of the stratosphere in terms of (QG)PV.
1.4 PV-maps and PV-inversion
Throughout this work, direct observations are often graphically displayed using isentropic Po-
tential Vorticity maps (PV-maps). These maps have a number of properties which can help
with the interpretation of the observed dynamics, as is argued in detail by Hoskins et al. [5].
There are two main properties which stand out. The first is that Rossby-Ertel PV (Z), here
referred to simply as PV, is a conserved quantity following adiabatic and frictionless motion
along isentropic surfaces. In isentropic (x, y, θ)-coordinates, Zis defined as
Z=ξθ+f
σ,(1.4.1)
where σis the isentropic density defined by σ=−g−1∂θp,fis the Coriolis parameter, and
ξθ=∇ × (uθ, vθ) is the relative isentropic vorticity. Conditions in the stratosphere are such
that, to good approximation, the flow is frictionless and adiabatic on the time scale of about
two weeks. This is roughly equivalent to the time scale of planetary wave breaking, which allows
for the use of Zas a tracer following wave breaking motion from observation.
CHAPTER 1. A LOOK AT THE STRATOSPHERE 6
Taking into account that the background planetary vorticity (f) dominates the PV-budget, a
clear positive South-to-North PV-gradient is usually visible on PV-maps. This makes it so that
meridional displacement of air is easily distinguishable. For example, southward displacement
will lead to a intrusion of high PV air into a region of relatively low PV. In the context of
this thesis, this is particularly relevant because planetary waves tend to consistently mix PV
down-gradient, i.e. from North to South (section 2.4). The latest reanalysis product of ECWMF
(ERA5, Appendix A.1) natively interpolates to a 0.3-by-0.3 degree grid, and provides data at
hourly intervals. This results in PV-maps with a high amount of detail, as is demonstrated in
Fig. 1.2, where the PV-field on the 850K isentrope over the time span of two days is shown.
Supplementary animations of the time development of the PV-field over the entire Northern
hemisphere can be found in Appendix A.8. PV-maps, and especially animated PV-maps, can
provide valuable insight on much of the planetary wave dynamics discussed in Ch. 2 and 3.
Figure 1.2: PV-maps of the 850K isentrope over the Pacific ocean. Left panel: instantaneous PV-field
on 2015-12-28 00:00 hrs. Right panel: instantaneous PV-field on 2015-12-30 00:00 hrs. PV is expressed
in Potential Vorticity Units (1 PVU = 10−6K m2kg−2s−1). Data retrieved from ECMWF ERA5
reanalysis (Appendix A.1).
The second important property of PV-maps, ties in with a cornerstone of the ’PV-view’ dis-
cussed in section 1.3, namely that of the PV-invertibility principle. The principle of PV-inversion
states that all dynamical fields can be reconstructed from the PV-distribution, using a balanced
assumption. Thus, PV-inversion links, for example, the pressure, velocity and temperature field
to the instantaneous PV-distribution, by inverting a function of the form
E(u, v, p, θ, . . . ) = Z(x, y, θ).(1.4.2)
A conceptually powerful aspect of PV-inversion, is that it connects changes in the distribution of
PV to changes in the distribution of (angular) momentum. This is an important notion in terms
of planetary wave drag, i.e. angular momentum loss, because planetary waves are principally
investigated in terms of their induced fluxes of (QG)PV. An important aspect of PV-inversion is
that it requires appropriate boundary conditions for the inversion to be unique. The ’resolution’
of the inversion depends largely on the restrictions imposed by the balanced assumption. For
example, the quasi-geostrophic assumptions are pretty ’rough’, and hence PV-inversion with the
use of QGPV, results in relatively coarse inverted fields. More accurate methods for PV-inversion
revolve around using the ’exact’ Rossby-Ertel PV.
Chapter 2
Planetary waves
Planetary waves, often referred to as Rossby waves, are waves in the oceans and atmosphere
which exist fundamentally due to a vorticity gradient. The strong pole-to-pole vorticity gradient
imposed by the counter-clockwise rotation of the Earth, makes it so that planetary waves are
prevalent in the Earth’s ocean and atmosphere. Planetary waves tend to reside where the
vorticity gradient is strongest. Tying in with this notion, is that the stratospheric polar vortex
edge is associated with a high vorticity gradient. Planetary waves therefore prefer to reside on the
polar vortex edge. There they play an important role in governing the dynamics surrounding
the polar vortex, which also impact the dynamics of the stratosphere as a whole. Indeed,
planetary waves arguably underpin all of the most dominant dynamical features of the wintertime
stratosphere (e.g. Plumb [6] for a discussion).
Earth’s monotonically increasing South-to-North background potential vorticity distribution,
gives planetary waves a set of peculiar ’one-way’ properties. For example, planetary waves can
only propagate westward relative to the background flow, and never eastward. The one-way
nature of some of the properties of planetary waves, lies at the heart of the concept of planetary
wave drag. In this chapter, the focus lies on describing these wave properties, as well as on
describing some of the general characteristics of the waves. This will be done mostly in terms of
results from quasi-geostrophic theory. However, only in chapter 3 will quasi-geostrophic theory
be used to explicitly describe the interaction between planetary waves and the mean flow.
2.1 Shallow water β-plane dynamics
This section concerns perhaps the simplest manifestation of planetary waves, namely that of
a zonally propagating planetary wave in a shallow water β-plane channel. The main goal of
this section is to introduce the concept of planetary wave elasticity, and to show how quasi-
geostrophic planetary waves can be associated with zonal mean meridional eddy fluxes of QGPV
(q, Eq. 1.3.5a).
In a motionless atmosphere, the background planetary vorticity gradient will make it so that
PV-contours lie along latitude circles. Planetary waves are expressed as undulations of these
otherwise ’straight’ contours. As argued in Ch. 7 from Dijkstra [7], for a constant layer depth
Hand variable β-plane Coriolis parameter f, the shallow water β-channel PV (˜q) can be used
to describe the mechanism of horizontal planetary wave propagation. The expression for ˜qis
given by
˜q=ξ+f
H,(2.1.1)
where fis as in Eq. 1.3.3, and ξis the relative vorticity. Note that ξ= 0 for a motionless layer
7
CHAPTER 2. PLANETARY WAVES 8
of fluid, and that counter clock-wise rotation of fluid parcels is associated with positive ξ, and
clock-wise rotation with negative ξ. Per usual, ˜qis a conserved quantity for conservative flows
(χ= 0 as in Eq. 1.3.5b).
Consider a section of a shallow water β-plane, centered around the latitude y0. In a mo-
tionless state, the PV-contour ˜q(y0), which corresponds to the background planetary vorticity
at y0, will lie along the y0-axis, as depicted by the dotted grey line in Fig. 2.1. If a planetary
Figure 2.1: One period of a planetary wave in a shallow water β-channel with constant layer depth H,
centered around y0. The dotted line corresponds to the PV-contour ˜q(y0) for a state of rest, the black
line to the PV-contour after a planetary wave is introduced. Meaning of the symbols is explained in the
text, PV is described by Eq. 2.1.1.
wave is introduced, the PV-contour will deform, taking on the shape of the solid black line in
Fig. 2.1. With a positive background PV-gradient (∂y˜q > 0) as well as PV being conserved,
a northward undulation of the PV-contour will displace fluid with relatively low PV to a re-
gion of relatively high PV. This creates a negative PV-anomaly with respect to the background
PV-distribution. Likewise, a southward undulation will create a positive PV-anomaly. With
Eq. 2.1.1, the effects of a PV-anomaly can be understood as a change in relative vorticity: If
the fluid was initially at rest, i.e. ξ= 0, a northward displacement of a fluid parcel will induce
negative relative vorticity to compensate for the increase in f. This relation follows from the
conservedness of ˜qand the numerator of Eq. 2.1.1. Similarly, ξmust become positive wherever
the PV-contour undulates southwards. The velocity fields associated with the induced relative
vorticity fields, are drawn inside the ridge and trough of the wave in Fig. 2.1. In the middle of
Fig. 2.1, where ˜q(y0) intersects the y0-axis, the velocity fields can be seen to conspire to ’push’
the black contour down. If the wave is imagined to extend beyond the domain drawn in Fig.
2.1, it can be seen that the induced velocity fields conspire to push the contour up along the left
and rightmost y0-intersections drawn in Fig. 2.1. This mechanism effectively pushes the wave
down wherever the zonal derivative of the PV-contour is negative, and up wherever it is positive.
This combined pushing action, makes it so that the wave propagates westward. This is based
on the analysis of the induced relative vorticity fields, whose orientation and strength depends
on the background vorticity gradient. It is then a consequence of the positive South-to-North
planetary background PV-gradient, that planetary waves propagate westward. This implies that
their zonal phase speed cpis always negative, which is marked by the arrow in Fig. 2.1. The
negative definite sign of cpis reflected in the expression for the dispersion relation of planetary
waves on a motionless horizontal flow (e.g. Vallis [2] section 6.4.2), given by
ω=−βk
k2+l2,(2.1.2)
where kand lare the longitudinal and latitudinal wavenumber, respectively. With the definition
of k= 2π/λ, where λis the longitudinal wave length, and the analogous definition of l,kand l
CHAPTER 2. PLANETARY WAVES 9
are always positive. Note that the dependence of Eq. 2.1.2 on βdemonstrates that planetary
waves owe their existence to a (planetary) vorticity gradient. The zonal phase speed of waves is
defined by cp=ω/k, which for planetary waves, using Eq. 2.1.2, follows as
cp=−β
k2+l2.(2.1.3)
By the definitions of β,kand l,cpis always negative, which is in accordance with the propagation
mechanism described by Fig. 2.1. The notion of westward propagation extends to waves imposed
on a constant zonal flow. For such flows, ∂yu= 0, and hence relative vorticity is unaffected. The
described restoring mechanism, which operates in terms of the induced relative vorticity fields,
then also remains unaffected. A more general notion would therefore be that planetary waves
can only propagate westward with respect to the background zonal flow. A consequence of this
is that, if the background zonal wind is more eastward than the wave’s phase speed is westward,
an observer on the surface of the Earth will be able to see a planetary wave propagate eastward.
From the role of the relative vorticity fields in planetary wave propagation, it can be under-
stood that relative vorticity essentially ’drives’ the waves’ propagation mechanism. The strength
of the induced relative vorticity fields depends on the strength of the background (planetary) vor-
ticity gradient, in the same way that ξdepends on fin Eq. 2.1.1. Thus, on a high PV-gradient,
the ’PV restoring mechanism’ is greatest. The restoring mechanism inhibits the tendency of the
PV-contours to deform: It is as if the contours are strung tighter on a high PV-gradient. In
literature this is referred to as the elasticity of the PV-contours, or equivalently, as ’planetary
wave elasticity’ (e.g. Baldwin et al. [8]). Planetary wave elasticity is relatively high on the high
PV-gradient associated with the polar vortex edge. This increases the resilience of the polar
vortex edge, leading to the persistence of its structure. An example of numerical planetary
wave elasticity experiments, or ’barrier-penetration experiments’, can be found in section 3 of
Dritschel and McIntyre [9].
To demonstrate the connection between planetary waves and zonal mean meridional fluxes
of QGPV, consider a shallow water quasi-geostrophic wave developing from an initially straight
QGPV-contour as in Fig. 2.2. It should be noted that the single layer expression for QGPV
Figure 2.2: One period of a planetary wave in a quasi-geostrophic shallow water β-channel, centered
around y0. The dotted line corresponds to the QGPV-contour q(y0) for a state of rest, the black line to
the QGPV-contour after a planetary wave is introduced. Arrows indicate meridional displacement of the
QGPV-contour as the wave develops.
is not equivalent to Eq. 2.1.1. The single layer QGPV can be derived by applying the quasi-
geostrophic approximation to Eq. 2.1.1, and its expression will appear in an example in section
3.2. As before, the black line in Fig. 2.2 represents an undulation of the dotted grey contour
corresponding to the QGPV-distribution of an initially motionless state. Because QGPV is
CHAPTER 2. PLANETARY WAVES 10
conserved, the QGPV-contour represents a material contour. If a material contour is to be
displaced meridionally, this can only be achieved through advection by a meridional component
of the wind. In QG-theory, QGPV is only advected by the geostrophic wind. Meridional
displacement of a material contour of QGPV, can then only be achieved through a non-zero
component of the geostrophic meridional wind1,v0. If the PV-contour is displaced northward,
v0>0, and if it is displaced southward, v0<0. Because of the positive background QGPV-
gradient, negative and positive PV-anomalies will form in the ridge and trough of the wave,
respectively. Inside the ridge of the wave, qis lower than the zonal mean ¯q, and hence q0<0.
Likewise, in the trough of the wave, qis higher than the zonal mean ¯q, and hence q0>0. For a
wave developing from the initially horizontal QGPV-contour, the corresponding signs of v0and
q0are drawn inside the trough and ridge shown in Fig. 2.2. The significance of these eddy terms,
lies in the observation that the multiplication of v0with q0is always negative. This applies also
to the zonal mean of the eddy product, such that v0q0<0 when a wave develops as in Fig. 2.2.
If the solid black line drawn in Fig. 2.2 would return back to its original state, i.e. the dotted
grey-line, the sign of the v0terms will reverse. The sign of q0will however remain the same.
After all, the sign of q0corresponds to the sign of the QGPV-anomaly, which does not depend
on whether the anomaly is growing or diminishing. The sign of the eddy product v0q0will then
be everywhere positive. ’Leaving’ planetary waves, can therefore be associated with v0q0>0.
In terms of a zonal mean ’PV-view’ of the atmosphere, the relation between eddy fluxes of
v0q0and the development of planetary waves turns out to be an important notion, which will be
further elaborated upon in Ch. 3. There, eddy fluxes of v0q0will be shown to relate to a zonal
force as well as the time development of zonal planetary wave activity in a stratified atmosphere.
Adding to the pivotal role of eddy fluxes of QGPV, is that in zonal mean QG-theory, changes
in the zonal mean QGPV-distribution can only be brought about by eddy fluxes of v0q0, as is
determined by Eq. 1.3.6. To aid the discussion in section 2.4, the Taylor-identity for a quasi-
geostrophic shallow water β-plane (the ’full’ identity will be discussed in section 1.3), is written
as
v0q0=−∂u0v0
∂y .(2.1.4)
The right-hand side of Eq. 2.1.4 corresponds to a zonal force by virtue of its appearance in
the zonal momentum budget given by Eq. 1.3.4a. The undulation of the PV-contour in Fig.
2.1 can then, by the sign of the corresponding negative eddy fluxes of q, be associated with a
westward zonal force by Eq. 2.1.4. The discussion surrounding Fig. 2.1 is an informal example
of the relation between v0q0and planetary wave activity, considering only a single layer of
fluid. However, in the description surrounding Fig. 2.1, a distinction should be made between a
planetary wave entering the layer and the wave being forced within the layer. Generally speaking,
if a wave is forced somewhere, it will propagate away from this region. In addition to the
horizontal propagation mechanism discussed in this section, it will be discussed how planetary
waves can propagate vertically in section 2.2. Either way, planetary waves will generally be
leaving the region where they are forced, such that v0q0>0 is typical of planetary wave ’source’
regions.
1The meridional geostrophic velocity (v) and QGPV (q) are split in a zonal mean and eddy term as v=
¯v+v0=∂xψ+v0=v0and q= ¯q+q0.
CHAPTER 2. PLANETARY WAVES 11
2.2 Vertical propagation
Planetary waves are typically forced within the troposphere by baroclinic instability, orography
and land-sea temperature contrasts2, with the strongest forcing occurring at roughly 60 degrees
North (or 60 degrees South for the Southern Hemisphere). In the context of this thesis, the
focus lies on planetary waves in the stratosphere. It follows then that the waves are able to
propagated vertically. The conditions under which vertical propagation can occur, as well as
the manner in which the waves do so, are discussed in this section.
A classic result in wave mean flow theory is the so-called Charney-Drazin criterion, published
in 1961 by Charney and Drazin [10], which puts a constraint on when planetary waves can
propagate vertically. For a constant background zonal wind ¯u, the (quasi-geostrophic) Charney-
Drazin criterion reads
0<¯u < β
k2+l2+ (f0/2NH)2.(2.2.1)
Here Nis the buoyancy frequency, His the vertical scale height (typically taken to be 8km),
and the other terms are as before. The first condition from Eq. 2.2.1 requires that the zonal
mean winds be positive, i.e. to the east, for planetary waves to propagate vertically. This con-
dition plays an important role in shaping the difference between the summer and wintertime
stratosphere. During summer, the upper tropospheric and lower stratospheric winds are west-
ward for reasons discussed in section 1.2, such that planetary waves cannot propagate upward.
Consequently, the summer polar vortex is much more devoid of planetary waves compared to
its wintertime counterpart. The second condition from Eq. 2.2.1, implies that when the zonal
winds are too strong, upward propagation is also inhibited. The ’cut-off ’ speed is higher for
longer planetary waves, which follows from the denominator of Eq. 2.2.1. Here ’longer’ refers to
the zonal wavelength, which is the most relevant measure of planetary wavelength in the context
of this thesis. For a purely zonal wave, l= 0. Assuming the term (f0/2N H)2from Eq. 2.2.1 to
be constant, the upper bound on ¯uis only determined by k. With the definition of k= 2π/λ,
where λis the zonal wavelength, it follows that the right-most term in Eq. 2.2.1 is largest for
longer waves.
During wintertime, zonal winds generally increase with altitude. This is related to thermal
wind balance (Eq. 1.3.4c), which states that a negative meridional temperature gradient (caused
by radiative cooling) induces a positive vertical shear of the zonal wind. For all but the longest
planetary waves, this causes the ’cut-off’ zonal wind speed prescribed by Eq. 2.2.1, to be reached
at a certain altitude. This effect can be observed in the vertical isentropic cross-section of the
atmosphere between 370K and 850K, shown in Fig. 2.3. In this figure, planetary wave-numbers
can be identified by counting the number of troughs and ridges lying along a latitude circle and
sharp vorticity gradient. From top left to bottom right, (1) at 370K, there exist planetary waves
in the range of wave-number 3 to 5, (2) at 430K, wavenumber 5 has been filtered and wavenumber
3 is most distinguishable, (3) at 600K only wavenumber 2 is distinguishable, (4) at 850K there
is only a single trough centered over the Aleutian Low, implying that all but wave-number 1
has been filtered. The edge of the polar vortex coincides with the maximum of the zonal winds,
which will be discussed in terms of a ’PV-view’ in section 3.3. The strong winds along the
vortex edge, act as a ’window’ through which only the longest planetary waves can propagate,
because the shorter waves get filtered. In the Southern Hemisphere (SH), transient waves, which
generally have a relatively high wavenumber, are more important than in the NH (e.g. Scinocca
and Haynes [11]). Or rather, because of the unobstructed Southern Ocean, there is a lack of
2Orography and land-sea temperature contrasts are frequently bundled as ’topographic forcings’.
CHAPTER 2. PLANETARY WAVES 12
Figure 2.3: PV-maps ranging from 370K to 850K with no particular scaling, showing a vertical cross-
section of the atmosphere on the 27th of December, 2015. Data retrieved from ECMWF ERA5 reanalysis
(Appendix A.1). Animations of the wintertime (DJF) months along eight isentropic levels between 370K
and 850K are available in Appendix A.8, Fig. A.10.
long stationary topographically forced planetary waves. Because the shorter baroclinically forced
waves cannot propagate up into the stratosphere so easily, the SH wintertime stratospheric polar
vortex is typically more devoid of planetary waves compared to its NH counterpart.
The Charney-Drazin criterion puts a constraint on when planetary waves can propagate
vertically, but it does not say anything about the manner in which the waves do so. One of
the perhaps more obvious factors affecting vertical propagation, is the exponentially decreasing
density with height. Because of this decrease in density, the wave amplitude can be expected to
grow exponentially as a function of height (e.g. Hoskins and James [12] Ch. 18.6). The linear
description of planetary waves, which is an often employed method, assumes the amplitude of
the waves to be small. If the waves propagate upward and grow in amplitude, this makes it so
that linear theory will at some point become inadequate, and that non-linear dynamics become
important. This notion is especially relevant to the discussion on wave breaking in section 2.4.
In addition to growing in amplitude with height, planetary waves also tend to bend equatorwards
with height (e.g. Matsuno [13]).
In QG-theory, the vertical eddy momentum fluxes induced by vertically propagating plane-
tary waves is, rather counter-intuitively, proportional to a meridional eddy heat flux. Vertically
propagating planetary waves induce deformations of stratification surfaces, which gives rise to
the concept of ’form stress’, discussed in Appendix A.4. The vertical planetary wave induced
momentum flux per unit mass, is given by the expression
f0
N2v0b0.(2.2.2)
The buoyancy force b0is by definition proportional to a density perturbation, or equivalently,
a temperature perturbation. Eq. 2.2.2 can therefore be expressed in terms of an eddy flux of
heat, meaning that the vertical eddy momentum flux is proportional to v0θ0, where θis potential
temperature. How this corresponds to a vertical momentum flux, follows from the analysis
in Appendix A.4. The derivation of Eq. 2.2.2 given therein, can be summarized as follows: A
vertical perturbation of a pressure surface is proportional to a buoyancy perturbation, and hence
also to a temperature perturbation. A vertical buoyancy perturbation in turn induces a zonal
CHAPTER 2. PLANETARY WAVES 13
pressure perturbation ∂xp0. This pressure perturbation is proportional to v0through geostrophic
balance, which states that ∂xp=f0ρ0v. The coupling of a vertical pressure perturbation with a
buoyancy perturbation, and of a zonal pressure perturbation with v0, makes it so that the vertical
eddy momentum flux is proportional to a meridional eddy buoyancy (or heat) flux. Note that
by the definition of f0,N2and b0, the dimension3of the expression in Eq. 2.2.2 is m2s−2. This
is equivalent to the dimension of the horizontal wave induced eddy momentum flux u0v0, which
will be discussed in section 3.1. The description of how planetary wave induced eddy momentum
fluxes tie in with planetary wave dynamics, requires the concepts of planetary wave activity and
the Eliassen-Palm vector, which will be discussed in sections 2.3 and 3.1, respectively.
2.3 Pseudomomentum and momentum transport
Pseudomomentum (M) is a property of waves, and in the context of this thesis, it relates to
the momentum transport by planetary waves. However, pseudomomentum and momentum are
not the same thing. Pseudomomentum is, by definition, a wave property which is invariant
under spatial translation of the wave in a homogeneous medium. This makes it an abstract
concept, but luckily the quasi-geostrophic (second-order) expression for pseudomomentum takes
on a relatively simple form. Namely, it is proportional to the zonal pseudomomentum density
defined in Eq. 2.3.3, which will be further elaborated on in this section. The interaction between
planetary waves and the mean flow, make it look ’as if’ the waves have momentum equal to
their pseudomomentum. At the center of this section, lies the notion that zonal planetary wave
pseudomomentum is negative definite when the background PV-gradient is positive, as is the
case on Earth. It is then ’as if ’ planetary waves can only have negative zonal momentum.
This makes it so that the waves can only transport negative zonal momentum away from their
source region, and towards the region where they dissipate. In Appendix A.2, the concept of
pseudomomentum is discussed in some more detail with a graphical illustration.
Hamiltonian dynamics revolve around the study of conserved quantities. When PV is con-
served, pseudomomentum is by definition a conserved quantity, and hence its description, or
derivation, is set within the framework of Hamiltonian dynamics. In the context of geophysical
fluids, a review of Hamiltonian dynamics is given in the chapter on Dynamical Meteorology, pub-
lished in the Encyclopedia of Atmospheric Sciences [14]. One of the results described therein, is
for the pseudomomentum of planetary waves in a barotropic flow on a β-plane. Planetary waves
are expressed as disturbances to a general ’basic state’ PV-distribution (q0). In the context of
Fig. 2.1 in section 2.1, the basic state would be the background planetary vorticity gradient,
and the wave-induced PV-anomalies would be the disturbances. For a zonally symmetric basic
state and the function Ydefined by Y(q0(y)) = y, where q0is a monotonically increasing basic
state PV, the barotropic pseudomomentum is given by (Eq. 47 p. 329 of [14])
M=ZZ (−Zq−q0
0Y(q0+ ˜q)−Y(q0)d˜q)dxdy. (2.3.1)
Here the upper integration limit q−q0represents the local deviation from the basic state PV-
distribution, which corresponds to the local magnitude of the wave-induced PV-anomalies. To
determine the sign of M, consider the following: If at some point in the (x, y)-domain q−q0<0,
the term ˜qis less than or equal to zero over its integration range, implying that also q0+ ˜q≤q0
3The dimensions of the buoyancy force, buoyancy frequency and planetary vorticity are given by b0=−gρ0/ρ
[m/s2], N2=∂z¯
b[1/s2] and f0= 2Ω sin φ0[1/s], respectively.
CHAPTER 2. PLANETARY WAVES 14
and hence Y(q0+ ˜q)≤Y(q0) over the integration range, by the definition of Y. This will make
it so that the term Y(q0+ ˜q)−Y(q0)is always less than or equal to zero over the integration
interval [0, q−q0]. If at some point in the domain q−q0>0, the reverse is true, such that ˜qis
greater or equal to zero and corollary Y(q0+ ˜q)−Y(q0)is also greater than or equal to zero.
This makes the integral over d˜qin Eq. 2.3.1 always evaluate to a non-negative value, in the
same way that for example, Ra
0xdx is positive for any non-zero real value of a. The integrand
in Eq. 2.3.1 is the negative of the integral over d˜q, thus the integrand between curly brackets
is always negative. This causes Mto be always less than or equal to zero, with it only being
zero if q=q0throughout the domain (i.e. when no disturbances, or waves, are present). Note
that this result is tied to the definition of q0being monotonically increasing, because only then
is the sign of Y(q0+ ˜q)−Y(q0)determined by the sign of q−q0.
As remarked in [14], the general basic state q0used in Eq. 2.3.1, can also be chosen to be
the zonal mean QGPV (¯q) described by Eq. 1.3.5a. Because the zonal mean equations will
then also have a vertical component, Y(·) will have a z-dependence of the form Y(¯q+q0;z). A
note here is that, by Noether’s theorem, any conservation law is associated with a symmetry.
In the case of qin a meridional plane, the conservation of pseudomomentum still relates to the
zonal symmetry of the medium (e.g. McIntyre and Shepherd [15] section 7), as it did for the
barotropic example surrounding Eq. 2.3.1. In the following, the z-dependence of Yis omitted,
with the note that the integral of Eq. 2.3.3 is instead to be taken over the (y, z)-domain. In
QG-theory, ¯qdominates the QGPV budget. This can be used as a basis for a small amplitude
assumption of the form q0¯q, where q= ¯q+q0. Using this to expand Y(¯q+q0) to second-order,
gives
Y(¯q+ ˜q)≈Y(¯q) + ˜q∂Y
∂q ¯q
+O(3).(2.3.2)
Inserting this in Eq. 2.3.1 and evaluating the integral over d˜qfrom 0 to q−¯q=q0, gives the
integrand between curly brackets as
−q02
2∂y¯q,(2.3.3)
where it was used that ∂qY¯q=∂¯qy≈(∂y¯q)−1by the definition of Y. Note that ∂y¯qis positive
because of the background planetary vorticity gradient, and hence Eq. 2.3.3 is negative definite.
Because the integrand of Eq. 2.3.1 is integrated over the entire domain to find M, the integrand
itself represents a ’density’ of pseudomomentum. As such, Eq. 2.3.3 represents the second-order
pseudomomentum density per unit mass of small-amplitude quasi-geostrophic planetary waves.
Much of the importance of Eq. 2.3.3 lies in the notion that it will appear in a wave conservation
law in Ch. 3, namely Eq. 3.1.4. There the time development of Eq. 2.3.3 will be shown to relate
to meridional eddy fluxes of q. This will formalize the relation between the sign of v0q0and the
development of planetary waves, as discussed in section 2.1.
As is discussed in Appendix A.2, when waves interact with their medium, it is often ’as
if’ the waves have momentum equal to their pseudomomentum. Such an analogy also hold
for planetary waves, as will follow from the discussion in section 1.3. There it will be shown
that the sign of planetary wave zonal pseudomomentum growth is equal to the sign of the
zonal forcing, which comes in the form of EP-vector divergence. The importance of planetary
wave pseudomomentum being negative definite, lies in the notion that the waves can then only
transport negative momentum away from their source region. Or conversely, that they can only
transport zonal momentum towards their source region and away from the region where they
propagate to (or ’break’, as will be discussed in section 2.4). This accelerates the zonal flow in
regions where they propagate away from, and decelerates the zonal flow in the regions where
CHAPTER 2. PLANETARY WAVES 15
the propagate to. By this mechanism, vertically propagating planetary waves drive the surface
westerlies and the poleward stratospheric Brewer-Dobson circulation.
2.4 Planetary wave breaking and critical layers
The effects of a planetary wave entering a layer of fluid were discussed in terms of a PV-budget
in section 2.1. It was argued that in the context of a quasi-geostrophic shallow water β-plane,
undulations of a PV-contour bring about meridional eddy fluxes of q, which are equivalent to
a negative zonal forcing through Eq. 2.1.4. If a planetary wave would enter and subsequently
leave the layer, the eddy fluxes of v0q0cancel each other out, and the net effect on the mean flow
would be zero. The initially straight PV-contour will have been deformed, and then restored
back again to its original shape. This can only happen if the initial deformation is a reversible
process. If the deformation of the PV-contour is irreversible, the undulation of the contour
becomes permanent, and corollary the negative zonal forcing associated with v0q0<0 also
becomes permanent. This irreversible deformation is defined as planetary wave breaking, which
will be discussed in this section.
Breaking of planetary waves frequently occurs in the mid-latitude stratosphere, which is
referred to as the ’stratospheric surf-zone’ (a term coined by McIntyre and Palmer [16]). This
choice of word rests on the analogy between the surf-zone of a typical beach, where ocean
gravity waves grow in amplitude and break. The stratospheric surf-zone, henceforth referred
to simply as the surf-zone, is a ’nonlinear critical layer’. A critical layer is defined as a zone
in which nonlinear wave-dynamics become important, relative to their linear dynamics. For
reference, comprehensive (analytical) discussions on the interaction between linear and non-
linear dynamics surrounding planetary wave critical layers, can be found in, for example, North
et al. [14] pp. 317-332, and Bhler [17] Ch. 7.
During wintertime, the zonal winds associated with the stratospheric polar vortex are latitu-
dinally sheared. This ties in with the general structure of the polar night jet, as can be observed
in the top right panel of Fig. A.9 in Appendix A.8. The latitudinal shear of the zonal wind
implies that the winds decrease southwards of the jet core. In section 2.2, it was mentioned
that planetary waves tend to grow in amplitude and bend equatorwards as they propagate
upwards. As they propagate vertically, planetary waves will therefore come into contact with
regions where the zonal winds become weaker. In [14] pp. 317-318, the interaction between
the shear flow and equatorwards moving planetary waves is discussed in the idealized context
of a shallow water β-plane. For linear planetary waves with zonal phase speed c, imposed on a
linear positive South-to-North shear flow U(y), the stability of the shear flow is found to break
down at the critical line where U(yc) = c. At the critical line, the solution of the waves’ zonal
phase speed cbecomes unphysical, due to a singularity in the linear equations (following the
discussion surrounding Eq. 3, p. 318 from [14]). For y > ycthe solutions are that of normal
planetary waves, and for y < ycthe waves are found to be evanescent. To resolve the singu-
larity at the critical line, nonlinear terms (right-hand side of Eq. 2, p. 318 from [14]) must be
resolved in the region surrounding the critical line. In this region, called the critical layer, the
PV-contours tend to wrap up in a typical Kelvin cat’s eye structure. This structure, together
with the aforementioned properties of the wave solution, are sketched in Fig. 2.4 for stationary
planetary waves (c= 0 such that yc= 0).
As vertically propagation planetary waves bend equatorwards and grow in amplitude on a
meridionally sheared zonal flow, their linear description will at some fail, and their PV-contours
will become wrapped up in a nonlinear critical layer. With this wrapping action, the contours
CHAPTER 2. PLANETARY WAVES 16
Figure 2.4: Illustration of a planetary wave critical layer for stationary planetary waves on a latitudinally
sheared zonal flow. Arrows on the streamlines indicate direction of the flow. Arrows oriented in the y-
direction represent meridional displacement of zonally propagating planetary waves. Figure retrieved
from North et al. [14], p. 319.
come to lie closer and closer to each other. As the wrapping progresses to smaller scales still, it
becomes irreversible and the PV in the critical layer becomes mixed. The associated Kelvin cat’s
eye pattern stems from an analytical solution of a nonlinear critical layer, which in literature
is referred to as the Stewartson-Warn-Warn solution (e.g. Haynes and McIntyre [18] for a
description). The Kelvin cat’s eye pattern of planetary wave breaking occurs frequently in the
real atmosphere, as can be seen from the observations in Appendix A.8, Fig. A.7, Fig. A.9 and
Fig. 1.2.
The meridional displacement of planetary waves, is itself influenced by the critical layer.
The critical layer is said to either, (1) absorb, then only waves ’bending’ equatorwards exist,
(2) reflect, then polewards deflected waves also exist, (3) over-reflect, the critical layer emits
planetary waves, such that there are more poleward than equatorwards propagating waves. The
dynamics of these three ’configurations’ of the critical layer, are considered in detail in Killworth
and McIntyre [19]. The equatorward and poleward moving waves are also drawn in Fig. 2.4,
represented by the meridional arrows. The exact behavior of the critical layer depends on the
characteristics of the flow inside of it. However, an intuitive example is given by a critical layer
in which the PV-gradient has been reduced to zero by breaking planetary waves. This is referred
to as a ’mature’ critical layer. Because planetary waves can only exist on a PV-gradient (section
2.1, a mature critical layer cannot sustain planetary waves, and it will thus reflect incoming
waves.
When planetary waves break, their wave character seizes to exist, and the effect that the
waves would normally have on the mean flow becomes permanent. As argued in section 2.1,
the introduction of a planetary wave is associated with downgradient zonal mean meridional
eddy fluxes of q, i.e. v0q0<0. When planetary wave breaking evolves into turbulent mixing
of q, it then ’follows’ that turbulent mixing is also associated with downgradient eddy fluxes
q. This notion is indeed in line with the theory of geostrophic turbulence, from which a result
is that diffusive quasi-geostrophic turbulence, associated with qcascading to smaller scales and
Chapter 3
Wave-mean flow interaction
In this chapter, planetary wave-mean flow interaction, the surf-zone and planetary waves on
a sharp PV-gradient, will be treated in terms of a quasi-geostrophic PV budget. In doing so,
many of the concepts described in Ch. 2 will be brought together in a unifying PV-view of
the dynamics. Core theoretical concepts are the Taylor-identity, linking the divergence of wave-
induced momentum fluxes to eddy fluxes of (QG)PV, and a wave-conservation law for planetary
waves. Furthermore, a selection of results from quasi-geostrophic theory will be tested and
expanded upon using numerical piecewise PV-inversion experiments of Rossby-Ertel PV. In
particular, the angular momentum changes associated with up and downgradient eddy fluxes of
PV will be considered.
3.1 Eliassen-Palm diagnostics
An important result in zonal mean quasi-geostrophic theory, is the so-called Taylor-identity.
This identity links (zonal mean) meridional eddy fluxes of qto the divergence of the Eliassen-
Palm (EP) vector. Divergence of the EP-vector also represents an ’implicit’ zonal force, which
is expressed through the PV-inversion principle. Furthermore, EP-vector divergence is coupled
to the growth of planetary wave pseudomomentum, discussed in section 2.3, through a wave-
conservation law. The goal of this section is to use the wave-conservation law to describe the
effect of planetary wave propagation and dissipation on the mean flow.
The Taylor-identity follows from the multiplication of q0, defined by Eq. 1.3.5a and the
separation q= ¯q+q0, with the meridional geostrophic velocity perturbation v0(Appendix
3.1). The derivation requires only the splitting of variables in a mean and eddy term, and
the identity is therefore generally valid, meaning that no small-amplitude assumption is needed.
With ∇= (∂y, ∂z), the Taylor-identity is written as
v0q0=−∂
∂y u0v0+∂
∂z f0
N2v0b0=∇ · (F(y), F (z)),(3.1.1)
where F(y)and F(z)are the meridional and vertical components of the Eliassen-Palm (EP) vector
F, respectively. Writing Eq. 3.1.1 in terms of EP-vector divergence is a matter of convention,
which was embraced shortly after the vector appeared in Transformed Eulerian Mean (TEM)
theory, first published in a paper by Andrews and McIntyre [22]. A key result of TEM-theory,
is that it interprets EP-vector divergence as a single eddy forcing term appearing in the zonal
momentum budget (TEM-equivalent of Eq. 1.3.4a). Namely, that a converging EP-vector
(v0q0<0) and diverging EP-vector (v0q0>0) are a negative and positive ’transformed’ zonal
18
CHAPTER 3. WAVE-MEAN FLOW INTERACTION 19
momentum forcing, respectively. In TEM-theory, having a single eddy term in the governing
equations is an ’improvement’ over the conventional equations given by Eq. 1.3.4a - Eq. 1.3.4d,
which contain the two separate eddy forcing terms proportional to v0b0and u0v0, respectively.
In the conventional Eulerian mean, the eddy terms force a system which is coupled by the free
variables ¯u, ¯va, ¯waand ¯
b. However, because of thermal wind balance (Eq. 1.3.4c), the effect
of the eddy momentum and eddy buoyancy forcings cannot be clearly separated. The appeal
of TEM-theory, is that this ambiguity in the eddy forcing is removed. However, TEM-theory
requires an alternative definition of the zonal mean ageostrophic circulation. This complicates
its physical interpretation, especially when it is compared to observations from a conventional
Eulerian mean picture of the atmosphere. In this chapter, the Taylor-identity will be interpreted
solely in terms of the meridional eddy flux of q, avoiding TEM-theory altogether.
In Eq. 3.1.1, the divergence of the eddy momentum term u0v0relates to a zonal force by
virtue of its appearance in Eq. 1.3.4a. As was mentioned in section 2.2, the divergence of the
eddy buoyancy term in Eq. 3.1.1 also relates to a zonal force. This is through the mechanism of
’form drag’, and it relates to the force exerted by wave-induced deformation of pressure surfaces
in a stratified fluid (Appendix A.4). With EP-vector divergence being the sum of two zonal
force terms, divergence of the EP-vector itself also relates zonal force. The complication of this,
lies in the notion that the EP-vector does not by itself appear in the zonal momentum budget
of Eq. 1.3.4a. The zonal forcing is instead ’implicitly’ governed by the eddy term v0q0. By the
PV-inversion principle, qrelates to the momentum field through a balanced assumption. That
way, changes in the distribution of q, relate to changes in the angular momentum distribution.
The significance of Eq. 3.1.1, then lies in the observation that in zonal mean QG-theory, the
distribution of qcan only be altered by fluxes of v0q0(Eq. 1.3.6). How exactly a redistribution of
qaffects the angular momentum budget, will be considered in an example in section 3.2. There,
conserved downgradient transport of q, equivalent to a converging EP-vector, will be directly
associated with angular momentum loss (i.e. a retrograde, westward force).
The interpretation of EP-vector divergence reaches beyond that of a zonal forcing. This
was already hinted at by the discussion in section 2.1, where the introduction of a planetary
wave was associated with negative eddy fluxes of q. Indeed, as discussed in Edmon Jr et al.
[23] sections band d, the EP-vector can also be used to study the growth, propagation and
dissipation of planetary waves. This is motivated by the appearance of EP-vector divergence in
a wave-conservation law, which will now be derived. The first step is to linearize the equation
for the time-development of q, given by Eq. 1.3.5b, around a perturbation q0. Expanding the
terms in a mean and perturbation part with the assumption that q0¯q, and zonally averaging,
yields
∂¯
q0
∂t +v0∂¯q
∂y =χ0.(3.1.2)
Multiplying Eq. 3.1.2 by q0, dividing by ∂y¯qand writing q0∂tq0=∂t(q0)2/2, gives
∂¯q
∂y −1∂(¯
q0)2
2∂t +v0q0=∂¯q
∂y −1
q0χ0.(3.1.3)
Using the assumption that ∂y¯qis independent of t, Eq. 3.1.3 can be written as
∂A
∂t +∇ · F=D, (3.1.4)
where the wave-activity A= (q0)2/2∂y¯qis defined as the negative of the planetary wave pseu-
CHAPTER 3. WAVE-MEAN FLOW INTERACTION 20
domomentum density (Eq. 2.3.3), and the sources and sinks of non-conservative wave effects1
is defined as D=q0χ0/∂y¯q. Eq. 3.1.4 is a conservation law for small-amplitude planetary
waves, referred to as the ’Eliassen-Palm relation’ (e.g. Vallis [2] section 10.2.1). For steady
small-amplitude waves (∂tA= 0) in adiabatic and frictionless conditions (D= 0), it follows
from Eq. 3.1.4 that ∇ · F= 0. A non-divergent EP-vector in turn corresponds to v0q0= 0,
which by Eq. 1.3.6 implies that the zonal mean time development of qis also zero (∂t¯q= 0).
This in turn relates to the time development of the full flow, as is dictated by PV-inversion.
With the assumption that the presence of eddies does not affect the boundary conditions for
PV-inversion, ∂t¯q= 0 implies that the flow is steady. In terms of the full equations of motion, a
steady flow implies that in Eq. 1.3.4a and Eq. 1.3.4d, ∂t¯uand ∂t¯
bare zero. Eddies are however
still present, namely in the form of steady small-amplitude planetary waves. A non-divergent
EP-flux then implies that the eddy terms in 1.3.4a and Eq. 1.3.4d, are exactly balanced by
the ageostrophic circulation, such that ∂t¯u=∂t¯
b= 0. This is known as the non-acceleration
theorem for steady small-amplitude planetary waves, where ’non-acceleration’ refers to the mean
flow being unaffected by the eddies.
The orientation of the EP-vector in the meridional plane, holds information on where plan-
etary waves are propagating to. By its definition, the orientation of the EP-vector represents
the magnitude of the eddy momentum and eddy buoyancy terms. However, for conservative
flows (χ= 0, and corollary D= 0), Eq. 3.1.4 takes on the form ∂tA+∇ · F= 0. Wherever F
converges, Aincreases, and wherever Fdiverges, Adecreases. This makes Fa measure of where
planetary wave activity is propagating to. By this mechanism, Eq. 3.1.4 describes the interac-
tion between the momentum of the mean flow and the pseudomomentum of planetary waves.
The notion of Fbeing a measure of where planetary waves are propagating to, can be made
explicit when the waves are assumed to be governed by linear planetary wave theory, including
the assumption that the medium varies slowly in time (e.g. Vallis [2] section 10.2.2). With
these assumptions, the EP-vector can be written as F= (cy
gA, cz
gA), where cy
gand cz
gare the
meridional and vertical component of the planetary wave group velocity vector cg, respectively.
With Fparallel to the group velocity of the waves, its orientation in the meridional plane gives
a sense of where planetary waves are propagating to following their group velocity.
Eliassen-Palm cross sections of the troposphere and lower stratosphere are a frequently used
diagnostic tool for planetary waves and their mean flow interaction. In the meridional plane,
the orientation of Findicates the direction in which planetary waves are propagating. The
convergence of F(v0q0<0) shows where planetary waves dissipate, or break, and lead to angular
momentum loss (section 3.2). Regions in which Fdiverges (v0q0>0), show where planetary
waves propagate away from their source region and accelerate the mean flow. Generally speaking,
χ(Eq. 1.3.4a) and corollary D(Eq. 3.1.4) are non-zero in planetary wave source regions.
Put differently, χand Dare non-zero in regions where wave-activity (A) is generated. This
complicates the description of planetary wave source regions in terms of a ’PV-view’, as PV is
not conserved when χis non-zero. A more detailed description of planetary wave dynamics with
respect to their source region, falls outside the scope of this thesis. The idea is that once wave-
activity is generated in a source region, planetary waves accelerate the mean flow by propagating
away from this source region. This notion is in line with the waves’ pseudomomentum being
negative definite, as is discussed in section 2.3.
1In regions where quasi-geostrophic theory is a good approximation, D < 0 is associated with wave dissipation
by Rayleigh friction and Newtonian cooling and ∂tA > 0 with growing waves (Andrews [24], section 2).
CHAPTER 3. WAVE-MEAN FLOW INTERACTION 21
3.2 The surf-zone and angular momentum loss
In section 1.3, it was discussed that the zonal forcing of EP-vector convergence is ’implicit’, in
the sense that a redistribution of QGPV relates to the angular momentum budget only through
PV-inversion. In previous sections it was also discussed that the zonal mean QGPV (q) can
exclusively be transported by the term v0q0, and that planetary wave breaking is associated with
downgradient mixing of q(v0q0<0). In this section, the effects of such downgradient transport
on the zonal momentum budget will be demonstrated using an idealized ’mixing-zone’. This
mixing-zone is represented by a region in which downgradient transport of qby planetary wave
breaking, has reduced the QGPV-gradient to zero.
The following example2takes place in the context of an unbounded shallow water quasi-
geostrophic β-plane. In shallow water QG-theory, the zonal angular momentum invariant relates
to the zonal symmetry of the medium. This abstract notion is tied to the same type of analysis
used to derive the zonal pseudomomentum in section 2.3 (e.g. North et al. North et al., pp. 324
- 331). The expression for the absolute zonal angular momentum per unit horizontal area (M)
in a shallow water β-channel, is given by M=ρ0H(y)u(y)−fy, which is the one-dimensional
unit mass version of Eq. 29 from North et al. p. 328. Here fis the Coriolis parameter as in
Eq. 1.3.3, H(y) is the local layer depth, u(y) is the local zonal velocity and yis the meridional
coordinate on a β-plane. The layer depth H(y) can be written as H(y) = H0+h(y), where H0is
scaled as H0=f2
0L2
dg−1. Here Ldis the Rossby radius of deformation defined as Ld=√gH0/f0,
and gthe acceleration due to gravity. The expression for Mcan then be written as
M=ρ0H0+h(y)u(y)−f0y,(3.2.1)
where the flow (i.e. the region in which δM is non-zero) is constrained to a region where
f≈f0. If h(y) and u(y) are chosen to be perturbation terms δu(y) and δh(y) of an initially
motionless state (h(y) = u(y) = 0), the quasi-geostrophic expression for δM, using Eq. 3.2.1,
becomes δM =ρ0H0δu(y)−ρ0f0yδh(y). Note that in QG-theory, terms involving products
of perturbations are ignored. In the context of total angular momentum change, δM over the
entire domain is the quantity of interest. In integral form, this is written as
δM =ρ0Z∞
−∞
[H0δu(y)−f0yδh(y)]dy, (3.2.2)
where it was used that δu(y) = −∂yδψ and δh(y) = f0δψg−1by the definition of the stream-
function ψ, defined in section 1.3. The next step is to write δM in terms of a perturbation of
shallow water QGPV (δq). The expression for shallow water QGPV is (e.g. Vallis [2] Eq. 5.9),
q=βy +∇2ψ−L−2
dψ. (3.2.3)
The first order (geostrophic) perturbation of Eq. 3.2.3, is δq =∂2
yδψ −L−2
dδψ. Note that for a
zonal mean flow, ∇2=∂2
y. Using the definition of h(y) and H0,δh(y) can also be written as
δh(y) = f−1
0H0L−2
dδψ. Inserting this, as well as the definition of δu(y) in terms of ψ, in Eq.
3.2.2, yields
δM =ρ0H0Z∞
−∞
[−∂yδψ −L−2
dyδψ]dy
=ρ0H0Z∞
−∞
y[∂2
yδψ −L−2
dδψ]dy
=ρ0H0Z∞
−∞
yδq(y)dy. (3.2.4)
2This is an adaptation of the example from section 7 in Dritschel and McIntyre [9].
CHAPTER 3. WAVE-MEAN FLOW INTERACTION 22
In the second to last step, it was used that ∂y(y∂yδψ) = ∂yδψ +y∂2
yδψ. Integrating the left-
hand side of this expression with respect to yamounts to zero, because the induced velocity
field u(y) = −∂yψ(and corollary δu) is 0 infinitely far away from the PV-anomaly, and hence
−R∂yδψ =Ry∂2
yδψ.
Eq. 3.2.4 links a change in absolute zonal momentum to a change in the QGPV-distribution.
This relation works both ways, such that for a given δq(y), Eq. 3.2.4 can be used to calculate
δM . For example, consider a QGPV-distribution representing a mixing-zone as in panel (a) of
Fig. 3.1. This figure represents a region in which downgradient mixing of qby planetary wave
breaking, has completely homogenized the QGPV-gradient in the mixing-zone. The resulting
anomaly δq(y) is N-shaped, as in panel (b) of Fig. 3.1. For this mixing-zone configuration,
Rδq(y)dy = 0, highlighting the notion that qis conserved over the course of a planetary wave
breaking event. The mixing-zone is centered over y0= 0, and is bounded by |y| ≤ b. The dotted
Figure 3.1: Panel (a): shallow water QGPV gradient q(y) on an infinitely large β-plane, with a com-
pletely mixed surf-zone centered around y0. Panel (b): local deviation δq(y) from the background poten-
tial vorticity gradient.
grey line in panel (a) of Fig. 3.1, corresponds to a motionless atmosphere where ∂yf=β, with
fas in Eq. 1.3.3. By inspection of panel (b), the relation between βand δq(y) is given by
δq(y) = −ytan(β) for |y| ≤ b, and δq(y) = 0 otherwise. Inserting this expression for δq(y) in
Eq. 3.2.4, gives
δM =ρ0H0Zb
−b−y2tan(β)dy
≈ −2
3ρ0H0βb3,(3.2.5)
where the small angle approximation tan(β)≈βwas used. Eq. 3.2.5 represents the net momen-
tum change due to QGPV-mixing in the highly idealized mixing-zone of Fig. 3.1. All the factors
in Eq. 3.2.5 are positive, and hence δM is negative. This signifies that the angular momentum
change is negative when conserved QGPV is mixed down-gradient. The association between
downgradient mixing of QGPV and angular momentum loss, generalizes to more sophisticated
’staircase’ configurations (with Fig. 3.1 representing a single step), as is discussed in Wood
and McIntyre [25]. In section 5 and appendix A of [25], two separate proofs are given for the
sign of δM being negative for any downgradient re-configurations of stratified (or multi-layered)
CHAPTER 3. WAVE-MEAN FLOW INTERACTION 23
QGPV, provided that the background QGPV-gradient is monotonically increasing. The proofs
themselves are rather technical, but nonetheless, the essence is captured by the highly idealized
example of Eq. 3.2.5.
The idealized mixing-zone in Fig. 3.1, represents the PV-configuration that occurs after
breaking planetary waves mix qdowngradient through negative eddy fluxes of v0q0. The corre-
sponding downgradient mixing of q, causes sharp vorticity gradients to form along the edges of
the mixing-zone, as can be seen in panel (a) of Fig. 3.1. The stratospheric surf-zone, where plan-
etary wave breaking is commonplace, is then effectively a large mixing-zone. In reality however,
the sharp equatorward gradient of Fig. 3.1 is less pronounced than the poleward gradient, as
can be seen in Appendix A.8 Fig. A.11. This is because the PV-gradient is not shaped only by
downgradient mixing by planetary wave breaking, but also by diabatic processes. The dominant
PV-gradient which is associated with the edge of the polar vortex, is however still ’sharpened’
by planetary wave breaking. This sharpening process plays an important role in the dynamics
that help to maintain the polar night jet and surf-zone structure, as will be discussed in section
3.3.
On the timescale of planetary wave breaking events, conditions in the stratosphere are ap-
proximately adiabatic and frictionless. This implies that χ= 0 in Eq. 1.3.5a, and corollary
D= 0 in the wave-conservation law given by Eq. 3.1.4. The wave-conservation law then reduces
to ∂tA+∇ · F= 0, where ∇ · F=v0q0as in Eq. 3.1.1. Using the wave-conservation law,
negative eddy fluxes of v0q0correspond to an increase in planetary wave-activity (∂tA > 0). But
as discussed in section 2.4, v0q0<0 also relates to planetary wave breaking. The difference
between the two interpretations of v0q0lies in the fact that the wave-conservation law assumes
small-amplitude linear waves, whereas wave breaking is associated with irreversible non-linear
effects, where waves lose their wavelike characteristics. This makes planetary wave breaking a
sink of planetary wave-activity. For this reason, planetary wave activity is said to converge into
the surf-zone (e.g. North et al. [14], p. 382).
3.3 Sharp PV-gradients and self-sharpening jets
By comparing Fig. A.11 and the top right panel of Fig. A.9 in Appendix A.8, it can be
seen that the location of the strong PV-gradient associated with the edge of the polar vortex,
corresponds to the location of the strongest zonal winds. This ties in with the notion that sharp
PV-gradients induce jet structures, or localized strong currents. This notion will be examined
in this section using the simplest example of a sharp PV-gradient, namely that of a shallow
water quasi-geostrophic ’PV-step’. Part of the motivation of this section, lies in the observation
that in Fig. 3.1 in section 3.2, downgradient mixing of QGPV is associated with the formation
of sharp QGPV-gradients along the edges of the mixing-zone. This implies that, even though
downgradient mixing of QGPV by planetary wave breaking leads to angular momentum loss, the
jet(s) on the flanks of the mixing-zone accelerate. The mixing itself is facilitated by planetary
wave breaking, which ties in with the existence of critical layers, as discussed in section 2.4.
The inherent existence of critical lines for planetary waves on a sharp PV-gradient, will also be
discussed in this section.
A sharp zonal mean shallow water QGPV-gradient is constructed as follows, (1) the large-
scale flow is assumed to be adiabatic and frictionless, such that qis conserved, (2) the QGPV-
distribution is assumed to be piecewise uniform, with two uniform regions being separated by
a step of height ∆q, (3) the jet is assumed to be narrow such that f≈f0, fixing the Rossby
radius of deformation (Ld). Under these conditions, the zonal mean QGPV-distribution centered
CHAPTER 3. WAVE-MEAN FLOW INTERACTION 24
around y= 0 can be written as
Q(y) = f0+∆q
2sgn(y),(3.3.1)
where sgn(y) returns the sign of y. The expression for the zonal mean shallow water QGPV
(Eq. 3.2.3) can be written as q(y) = f0+∂2
yψ−L2
dψ. Setting q(y) to be the QGPV-distribution
represented by Q(y), results in
∆q
2sgn(y) = ∂2ψ
∂y2−ψ
L2
d
,(3.3.2)
which is an inhomogeneous Helmholtz equation. ψcan be solved separately for y < 0 and
y > 0, using the boundary condition that the induced velocity field is zero at ±∞. Glueing the
solutions together at y= 0, then yield a single expression. The zonal wind profile (−∂yψ) which
arises is (e.g. Harvey et al. [26] Eq. 2.4),
U(y) = ∆q
2Lde−|y|/Ld.(3.3.3)
The character of the solution of U(y), is such that the strength of the zonal jet is proportional to
magnitude of the PV-step and the Rossby radius of deformation Ld, with the latter also affecting
the meridional extent of the jet. For constant Ld, it can then be understood that an increase of
the QGPV-gradient along the edge of a mixing-zone, attributed to downgradient QGPV-mixing
by breaking planetary waves, leads to an acceleration of the jet(s) adjacent to the mixing zone.
The peculiarity of this, lies in the notion that downgradient mixing by breaking planetary waves
leads to net angular momentum loss, as is discussed in section 3.2, whereas the jet(s) along the
edges of the mixing-zone accelerate3.
A sharp PV-front also by itself supports planetary waves4. Following the discussion in
[26], the position of material PV-contours in a horizontal plane, is represented by the function
η(x, y, t). A kinematic argument links the total time derivative of ηto advection of the contour
by the meridional wind v, through
∂
∂t +u∂
∂x η(x, y, t) = v. (3.3.4)
Undulations of the contours are assumed to be small-amplitude disturbances of the form η(x, y, t)
= ˆη(y) exp [ik(x−ct)]. The function ˆη(y) represents meridional meanders of the contours. Other
variables are split as q=Q(y) + q0,u=U(y) + u0,v=v0, where the perturbation terms are
small and where Q(y) and U(y) are as in Eq. 3.3.1 and Eq. 3.3.3, respectively. In terms of
the streamfunction ψ, the meridional velocity perturbation can be written as v0=∂yψ0. The
streamfunction perturbation itself can be calculated by inverting an equation similar to that of
Eq. 3.3.2, but with the left-hand side as −η(x, 0, t)∆qsgn(y). This yields (e.g. Harvey et al.
[26] Eq. 2.7 and 2.8),
ψ0=φ(y, k) ˆη(0)eik(x−ct),(3.3.5)
where
φ(y, k) = ∆q
2κLde−κ|y|/Ld.(3.3.6)
The function φ(y, k) represents meridional meanders of the streamfunction, and in Eq. 3.3.6,
the ’effective wavenumber’ is defined as κ2= 1 + k2L2
d. Using the expressions for ψ0,U(y) and
3A more comprehensive and analytical discussion of this concept, is given by ’The simplest jet-resharpening
problem’ in Wood and McIntyre [25] pp. 1269-1270.
4The shallow water QG dispersion relation for planetary waves on a sharp PV-gradient, is different from the
’regular’ shallow water dispersion relation given by Eq. 2.1.2.
CHAPTER 3. WAVE-MEAN FLOW INTERACTION 25
Q(y) to linearize Eq. 3.3.4 around the QGPV-step, gives
(U(y)−c)ˆη(y) = φ(y, k)ˆη(0).(3.3.7)
The solution for the meridional meanders of the PV-contours is, using Eq. 3.3.7,
ˆη(y) = ˆη(0) φ(y)
U(y)−c,(3.3.8)
where the zonal phase speed (c) of the waves is given by the sum of the wave’s phase speed and
the velocity of the background zonal wind U(y). The waves represented by Eq. 3.3.7 reside on
the QGPV-step centered at y= 0, such that c=U(0) −φ(0, k). Inserting the definition of U(y)
and φ(y, k) in the expression for c, gives
c=∆q
2Ld1−1
κ.(3.3.9)
The importance of Eq. 3.3.9 lies in the notion that for U(y) = c, a singularity exists in Eq. 3.3.8.
Following the discussion in section 2.4, the the singularity is located on the critical line yc. In the
region surrounding yc, the so-called critical layer, non-linear effects come into effect, facilitating
planetary wave breaking. A qualitative insight given by Eq. 3.3.8, is that the existence of
critical layers is inherent to jet-structures. For the PV-structure associated with jets, planetary
wave elasticity is highest on the sharp PV-gradient, and lowest in the adjacent ’mixing-zone’,
where planetary wave breaking is favoured. In the critical layer, irreversible deformation of
PV-contours leads to downgradient mixing of PV. This sharpens the PV-gradient along the
mixing-zone, which in turn accelerates the jet core and increases its resilience, or planetary wave
elasticity. In literature, this phenomenon is referred to as a self-sharpening jet, and it plays an
important role in the jet’s ability to ’efficiently’ maintain its structure. A more comprehensive
description of the dynamics of a self-sharpening jet, can be found in, e.g., Dritschel and Scott
[27], Harvey et al. [26], Wood and McIntyre [25] and McIntyre [28].
3.4 Piecewise cyclo-geostrophic PV-inversion
The connection between a downgradient reconfiguration, or flux, of QGPV (q) and angular
momentum loss, was generalized in a paper by Wood and McIntyre [25]. In their paper, it
is mentioned that equivalent theorems for downgradient fluxes of isentropic Rossby-Ertel PV
(hereafter referred to simply as PV), have not yet been found. They argue that the difficulty lies
in the nonlinearity of the ’cyclostrophic’ PV-inversion operator. This operator is required to per-
form PV-inversion on scales smaller than the large (geostrophic) scale considered by QG-theory.
Cyclostrophic balance refers to the balance between the pressure gradient and centripetal force.
If cyclostrophic balance is extended to also include the Coriolis force, this is referred to as ’cyclo-
geostrophic’ balance, or gradient wind balance. Cyclo-geostrophic is a suitable balanced assump-
tion for both large and small scale flows, or in other words, it is a suitable assumption for flows
of any Rossby number. In this section, numerical cyclo-geostrophic PV-inversion experiments
are performed with idealized up and downgradient ’flux’ configurations of PV. The experiments
are motivated by the role of EP-vector divergence (and thus transport of QGPV by eddies) in
the transfer of angular momentum between the troposphere and stratosphere. Furthermore, it
can be shown that isentropic eddy fluxes of PV are approximately equal to quasi-geostrophic
eddy fluxes of QGPV. This makes eddy fluxes of PV approximately equal to quasi-geostrophic
EP-vector divergence, which suggests that the effects on the angular momentum budget of up
CHAPTER 3. WAVE-MEAN FLOW INTERACTION 26
and downgradient PV transport, is similar in nature to that of QGPV-transport. The relation
between isentropic eddy fluxes of PV (v0Z0) and eddy fluxes of QGPV (v0q0) is discussed from a
theoretical and observational point of view in Appendix A.5 and A.6, respectively.
The experiments are constructed as follows, (1) numerical cyclo-geostrophic PV-inversion is
performed in a meridional plane, with potential temperature as height coordinate, (2) potential
temperature levels are spaced 2 Kelvin (K) apart and the meridional step size is 0.1 degrees, (3)
changes in the PV-distribution are superimposed on a reference PV-distribution corresponding
to a motionless atmosphere, (4) conditions are assumed to be adiabatic and frictionless, such
that total PV is conserved in any redistribution of PV, (5) idealized up and downgradient
configurations are constructed in a manner analogous to the mixing-zone shown in Fig. 3.1
of section 3.2. The cyclo-geostrophic PV-inversion is performed with the method described in
Van Delden and Hinssen [29]. A notable assumption used is that the edges of the domain are
sufficiently far away from the induced velocity field for the boundary condition u= 0 to be valid.
The downgradient configuration shown in the left panel of Fig. 3.2, is analogous to the
’mixing-zone’ shown in panel (a) of Fig. 3.1. The upgradient configuration in the right panel
of Fig. 3.2 is constructed in a manner analogous to the opposite of δq, where δq represents a
downgradient redistribution of QGPV as in panel (b) of Fig. 3.1. From a quasi-geostrophic zonal
Figure 3.2: Idealized up and downgradient transport of cyclo-geostrophic PV along the 736K isentrope.
Dotted grey line represents the reference PV-distribution, which corresponds to a motionless atmosphere.
Left panel: Conserved downgradient transport between 40 and 60 degrees North. Right panel: Conserved
upgradient transport between 40 and 60 degrees North. 1 PVU = 10−6K m2kg−1s−1.
mean perspective, the opposite of δq would imply that QGPV is transported up its gradient
through positive fluxes of v0q0. The region where up and downgradient configurations of PV are
imposed, is centered along the 736K isentrope and 50 degrees North. 736K corresponds to the
middle of the domain in terms of its vertical extent, and likewise, 50 degrees North corresponds
to the middle of the domain in terms of its latitudinal extent. The dotted grey lines in Fig. 3.2
represent the reference PV-distribution, corresponding to a motionless atmosphere. As discussed
in Hinssen et al. [30], it is the deviation from this reference distribution which induces a zonal
wind.
To demonstrate the character of cyclo-geostrophic PV-inversion, seven-layer thick up and
downgradient configurations are inverted. Here seven-layer thick configurations are chosen over
single-layer ones, because then the inverted fields are more pronounced. They are however
qualitatively the same. Centered around 736K, the PV-anomalies are then vertically bounded by
the 730K and 742K isentrope. The resulting zonal velocity profiles for the down and upgradient
CHAPTER 3. WAVE-MEAN FLOW INTERACTION 27
configurations, are shown in the left and right panel of Fig. 3.3, respectively. By comparing the
Figure 3.3: Induced zonal velocity fields by seven-layer up and downgradient transport of PV centered
around 736K, as determined by cyclo-geostrophic PV-inversion. Left panel: Downgradient transport with
each layer being as in the left panel of Fig. 3.2. Right panel: Upgradient transport with each layer being
as in the right panel of Fig. 3.2.
left panel of Fig. 3.3 and Fig. 3.2, it can be seen that the sharp gradients along the edge of
the ’mixing-zone’, induce eastward zonal jets. This is in accordance with the quasi-geostrophic
relation between sharp PV-gradients and jet structures, as discussed in section 3.3. The eastward
jets along the edge of the mixing-zone and westward jet on the interior of the mixing-zone, are
both in qualitative agreement with the zonal velocity field induced by a shallow water quasi-
geostrophic mixing-zone (e.g. Dritschel and McIntyre [9] Fig. 7 panel 2). The same qualitative
picture holds for the wind field induced by the upgradient PV-configuration. The eastward jet
in the right panel of Fig. 3.3 coincides with the sharp PV-gradient shown in the right panel of
Fig. 3.2, and the interior of the adjacent homogenized zones induce westward winds.
To quantify how up and downgradient configurations of PV affect the angular momentum
budget, the absolute angular momentum per unit mass ( ˆ
M) is considered. This quantity takes
into account changes in angular momentum attributed to the angular momentum of the induced
velocity field, as well as angular momentum changes due to a mass shift. With the assumption
r≈a, where ais the radius of the Earth and ris the distance from the Earth’s center, the
absolute angular momentum per unit mass is defined as (Holton and Hakim [31] p. 331)
ˆ
M= (Ωacos φ+u)acos φ, (3.4.1)
where the hat signifies the distinction between Mdefined by Eq. 3.2.1. Using ˆ
M, the total
angular momentum can be calculated for the reference atmosphere and for the state of the at-
mosphere given by PV-inversion of the configurations shown in Fig. 3.2, over the entire domain.
The resulting change in absolute angular momentum per unit mass (δˆ
M) is indicative of the
effect of PV-transport on the total angular momentum budget. For a single layer downgradient
configuration as in the left panel of Fig. 3.2, δˆ
Mis plotted in the left panel of Fig. 3.4 as
a function of the isentropic level on which the downgradient configuration is imposed. Note
that single layer configurations are chosen in order to match the theoretical example given in
section 3.2. From Fig. 3.4, it can be seen that downgradient transport of PV leads to a net
reduction of angular momentum. This is in line with the result from quasi-geostrophic theory,
where downgradient transport of QGPV is associated with angular momentum loss. From Fig.
CHAPTER 3. WAVE-MEAN FLOW INTERACTION 28
Figure 3.4: Change in absolute angular momentum per unit mass (δˆ
M), as determined by cyclo-
geostrophic PV inversion of conserved redistributions of the PV distribution corresponding to a motionless
atmosphere. Left panel: δˆ
Mfor single layer downgradient transport of PV, with each layer being as in
the left panel of Fig. 3.2. Right panel: δˆ
Mfor single layer upgradient transport of PV, with each layer
being as in the left panel of Fig. 3.2.
3.4, it can be seen that δˆ
Mis higher for lower isentropic levels. This can be attributed to
the notion that the mass bounded between equally spaced (in Kelvin) isentropes, is higher for
lower isentropes, as density decreases with height. Noting that potential temperature increases
with height whereas pressure decreases with height, the interval of 710K to 754K falls roughly
between 17.5 and 15.0 hPa. The result of this, is that the density along the isentropes shown
in Fig. 3.4 decreases approximately linearly, which is reflected in the linear curve of δˆ
M. At
lower isentropic levels, a reconfiguration of PV induces the displacement of relatively more mass,
giving rise to higher absolute values of δˆ
M. The same can be said for δˆ
Mresulting from the
upgradient configurations, shown in the right panel of Fig. 3.4. There the now positive val-
ues of δˆ
Mare higher on lower isentropic levels. The choppiness of the curves shown in Fig.
3.4, can most likely be attributed to the coarseness of the grid as well as the approximations
made in calculating δˆ
Mfrom the inverted fields. A notable difference between δˆ
Mfor the up
and downgradient configurations, is that they are not each others opposite. Given the anti-
symmetry of their respective re-distributions of PV, one might expect their respective δˆ
Mto
also be anti-symmetric. The observed disparity between the negative and positive δˆ
M, is com-
plicated by the physical interpretation of the upgradient configuration being less clear-cut than
that of the downgradient configuration, which can ’naturally’ be associated with mixing-zones.
An upgradient re-configuration of PV could also be argued to be the ’inverse’ of the downgradi-
ent configuration shown in the left panel of Fig. 3.2, in the sense that upgradient fluxes would
restore the PV-distribution back to its initial state. The resulting δˆ
Mwould then indeed be the
opposite of those resulting from the downgradient configurations. But regardless of the disparity
between δˆ
Mfor the up and downgradient configurations, the character of their respective net
angular momentum changes is in line with those anticipated from QG-theory.
In section 3.2, an analytical example was given of the angular momentum loss associated with
downgradient transport of QGPV. In particular, it was shown that in QG-theory, the change in
absolute angular momentum is proportional to δM =−2/3ρ0H0βb3(Eq. 3.2.5), where bis half
of the meridional extent of the anti-symmetric ’mixing-zone’ shown in Fig. 3.1. This theoretical
result is investigated using cyclo-geostrophic PV-inversion, with a mixing-zone as in the left
CHAPTER 3. WAVE-MEAN FLOW INTERACTION 29
panel of Fig. 3.2. For the meridional extent of the mixing-zone, measured from its center at 50
degrees North, the parameter ˆ
bis introduced to distinguishes from bused in Fig. 3.1. In the
left panel of Fig. 3.5, δˆ
Mas a function of the total meridional extent (2ˆ
b) of the downgradient
configuration, or mixing-zone, is plotted. δˆ
Mcan be seen to increase sharply as a function of
Figure 3.5: Change in absolute angular momentum per unit mass (δˆ
M), as determined by cyclo-
geostrophic PV inversion, for increasingly wider downgradient configurations, or ’mixing-zones’ as in
section 3.2. Left panel: single layer mixing-zone at 736K with meridional extent 2b, with bas in Fig. 3.1,
centered around 50 degrees North. Right panel: Cube root of the absolute value of δˆ
M, as a function of
half the mixing-zone width (b).
the width of the mixing-zone. In the right panel of Fig. 3.5, the cube root of the absolute
value of δˆ
Mis plotted as a function of ˆ
b. The fact that this curve is linear, at least between
ˆ
b= 3 and ˆ
b= 15 degrees, shows that δˆ
Mis proportional to ˆ
b3. This is in agreement with the
quasi-geostrophic result for δM , given by Eq. 3.2.5. The deviation from the linear trend at
ˆ
b≤2 degrees, could be attributed to (1) numerical issues with the coarseness of the grid and the
methods used to calculate δˆ
M, in particular with the jets not being resolved properly for small
ˆ
b, (2) the nature of δˆ
Mchanges when ˆ
bbecomes smaller than the Rossby radius of deformation,
where the flow shifts from geostrophic to cyclostrophic order. Although point (2) would raise
interesting questions, point (1) is by far the most likely candidate, judging by the morphology
of the resolved wind-fields for ˆ
b≤2 degrees (not shown here).
Chapter 4
Planetary wave drag in the general
circulation
Planetary waves play an important role in shaping the general circulation. Not least, plane-
tary waves are involved in driving the observed surface westerlies and wintertime stratospheric
Brewer-Dobson circulation. To elucidate the role of planetary waves in the general circulation,
the transfer of angular momentum by planetary waves has been parameterized in a simplified
zonal mean model of the atmosphere, described in Van Delden [32]. In section 4.1, this model is
used to perform model experiments. The model’s parameterization of planetary waves and the
model’s output, will be interpreted based on the theory described in Ch. 2 and 3. In section
4.2, model performance will be quantified with the use of Taylor-diagrams. Taylor-diagrams are
a graphical measure of the correspondence between the modelled and observed atmosphere, and
are commonly used in climate science.
4.1 A simplified model of the general circulation
The model is a simplified zonal mean primitive equations model, that was constructed for the
purpose of gaining insight into the fundamental dynamical processes that shape the general cir-
culation. The model features a simplified representation of (1) the seasonal insolation cycle, (2)
greenhouse gasses, namely CO2, O3and H2O, (3) a hydrological cycle, including an Intertropical
Convergence Zone (ITCZ), (4) planetary wave drag in the stratosphere. These dynamics can
be individually turned on or off, in order to study their dynamical effects on the zonal mean
atmosphere. In the context of this thesis, planetary wave drag is of particular interest and will
correspondingly be emphasized throughout this section. This section will be concluded with
suggestions for possible improvements of the model. For a more comprehensive description of
the model, the reader is referred to Van Delden [32].
A key aspect of the model is that it does not inherently support planetary waves. The model
does support horizontally propagating acoustic Lamb waves, but these have negligible physical
significance. In the current iteration of the model, planetary waves are instead parameterized by
prescribing a retrograde ’drag’ force in the wintertime stratospheric surf-zone. This parameteri-
zation is meant to capture the effect of angular momentum loss due to planetary wave breaking.
The planetary wave drag (D) is parameterized as
D=D0B(φ)L(z),(4.1.1)
30
CHAPTER 4. PLANETARY WAVE DRAG IN THE GENERAL CIRCULATION 31
where
B(φ) = sin (2|φ|) (4.1.2)
and
L(z) =
sin πz−z0
z1−z0if z0< z < z0
0 otherwise,
(4.1.3)
with z1= 25km, z0= 10km and D0=−5·10−5m/s2. In the expression for B,φis the
latitude coordinate in degrees. The region in which Dis defined roughly corresponds the shape
of the wintertime stratospheric surf-zone. The parameterization of Dtakes into account the
Charney-Drazin condition for upward planetary wave propagation discussed in section 2.2, by
setting D0= 0 if the zonal wind u < 0 at any level below the region in which Dwould otherwise
be prescribed.
Three different configurations of the model are compared against the observed climatology in
Fig. 4.1. Model run 2a, shown in the top left panel of Fig. 4.1, includes only a yearly insolation
Figure 4.1: Comparison of model output with climatological observation for January. Model output
is interpolated to ERA-Interim pressure levels. Red, magenta and blue lines mark isentropes of the
over, middle and underworld, respectively. Solid black lines mark positive zonal velocities, dotted black
lines negative zonal velocities. Thick green line marks the 2 PVU dynamical tropopause. Red and blue
shading represent positive and negative meridional wind, respectively, with 0.5 m/s spacing. Top left:
Model run 2a, featuring only an insolation cycle. Bottom left: Model run 3a, featuring an insolation
and hydrological cycle. Top right: Model run 4a, featuring an insolation cycle, hydrological cycle, and
planetary wave drag. Bottom right: Climatological ERA-Interim observation.
cycle and longwave radiation absorption by CO2. The resulting zonal mean atmosphere has
CHAPTER 4. PLANETARY WAVE DRAG IN THE GENERAL CIRCULATION 32
the properties that (1) the isentropes bulge downward towards the equator everywhere, (2)
year round hemisphere-wide zonal jets exist on both hemispheres, (3) no significant meridional
component of the wind is present, (4) the dynamical tropopause marked by the 2 PVU contour,
decreases inversely with latitude and is symmetric with respect to the equator. Model run 3a,
shown in the bottom left panel of Fig. 4.1, extends model 2a with the addition of a hydrological
cycle and shortwave absorption by H2O and O3. Differences in the model output with respect to
model 2a are, (1) the hydrological cycle’s transfer of energy affects the position of the isentropes
of the middle and underworld, (2) updraft induced by latent heat release in the ITCZ, as
well as updraft induced by the position of the Hadley cell in January, causes a reversal of the
temperature gradient close to the equator, inducing westward winds there, (3) sharpening of the
PV-gradient of the dynamical tropopause between 30 and 40 degrees latitude in the summer-
hemisphere. Sharpening of PV-gradients is associated with an acceleration of the associated jet
core (section 3.3), which alludes to a role of the ITCZ in ’driving’ the subtropical jet. More
specifically, that latent heat release in the ITCZ helps to maintain the strong temperature
gradient, or corollary pressure gradient, which induces the subtropical jet (e.g. Persson [33]).
In the winter-hemisphere of model 3a, evaporation leads to a more rapid cooling of the surface,
because one-fifth of the evaporation is set to convergence into the ITCZ. This effectively makes
evaporation a mechanism for equatorward energy transport. Through thermal wind balance, the
corresponding increase in the temperature gradient in the winter-hemisphere induces a stronger
zonal jet. Model run 4a, shown in the bottom left panel of Fig. 4.1, includes the parameterization
of planetary wave drag. Notable improvements over model 3a, and also with respect to the
observed climatalogical zonal mean plotted in the bottom right panel, are (1) the existence of
a cold-point tropopause, created by upwelling induced by the stratospheric meridional winds
which are in turn induced by planetary wave drag, (2) reversal of the stratospheric winds, with
the winds now blowing westward in the summer-hemisphere, (3) separation of the hemisphere-
wide yet into a subtropical and polar night jet, (4) a sharp PV-gradient along the dynamical
tropopause PV-gradient between 60 and 30 degrees North and South. Some aspects of reality
which are captured less well by model 4a, are (1) the magnitude and location of the meridional
wind, (2) the temperature of the lower atmosphere, which is too cold, (3) magnitude of the zonal
wind, with both the westward and eastward winds generally being too weak. Between model 4a
and 3a, the westward winds within the equatorial region nearly completely vanish. A possible
cause of this could be an increased spreading of heat by more homogeneous planetary wave drag-
induced upwelling. During the winter to summertime transition, planetary wave drag brings the
zonal winds close to a full stop. When the zonal winds are sufficiently weak during the onset of
summer, the positive equator-to-pole temperature gradient caused by absorption of shortwave
radiation by ozone, induces a westward wind through thermal wind balance. The upwelling
induced by planetary wave drag, also appears to play a role in ’inverting’ the temperature
gradient in the upper regions of the atmosphere. Without planetary wave drag, the eastward
winds are too strong for the westward ’signal’ of the wind to express itself.
The parameterization of planetary wave drag given by Eq. 4.1.1, can be considered a
’metaphor’ for a more complex set of interactions. Namely, a metaphor for nearly all the
dynamics discussed in Ch. 2 and 3, with planetary wave breaking and the resulting planetary
wave drag being only a piece of the puzzle, so to speak. Dynamics that cannot be captured by
the model due to planetary waves not being resolved, are (1) the ’elasticity’ of planetary waves
on the sharp PV-gradient of the polar night jet (or polar vortex edge), adding to the resilience
of the polar night jet, (2) a critical layer or ’surf-zone’, which by itself influences the behavior
of planetary waves (i.e. the surf-zone can reflect, absorb or over-reflect planetary waves), (3)
CHAPTER 4. PLANETARY WAVE DRAG IN THE GENERAL CIRCULATION 33
self-sharpening of the polar night jet’s PV-gradient as a result of planetary wave breaking, ac-
celerating the core of the jet and further increasing its planetary wave elasticity, (4) a source
region of planetary waves, in which the zonal flow is accelerated and from which planetary waves
propagate upwards into the stratosphere. The interplay between the aforementioned (planetary
wave) dynamics, ’naturally’ lead to the establishment of a polar night jet, or polar vortex1. Be-
cause the dynamics surrounding the polar night jet rely to such a large extent on the existence of
planetary waves, an isolated polar night jet does not by itself appear in the model, as is evident
from model 3a (bottom left panel of Fig. 4.1).
The parameterization of planetary wave drag introduced in model 4a, effectively ’pushes’ the
hemisphere-wide zonal jet apart into two separate jets. In the winter-hemisphere, the resulting
zonal wind pattern resembles that of the subtropical and polar night jet. Separation of the oth-
erwise hemisphere-wide yet into a subtropical and polar night jet solely by applying planetary
wave drag, does however not fit in with the description of the ’wave-turbulence jigsaw puzzle’.
Within the framework of the jigsaw puzzle (hereby referring to a self-sharpening jet and the
dynamics discussed in Ch. 2 and 3), the resilience and self-sharpening properties of the polar
night jet play an important role. They in part cause the wintertime stratosphere to efficiently
organize into a jet and surf-zone (PV-)structure, with ’efficiently’ referring to the amount of
planetary wave breaking (or planetary wave drag) involved in maintaining a clear distinction
between the low PV-gradient of the surf-zone and high PV-gradient of the polar night jet. A
measure of the amount of planetary wave breaking involved in maintaining the stratosphere’s
PV-structure, is the strength of the meridional wind. When a westward force is applied, the at-
mosphere is brought out of thermal wind balance. To restore thermal wind balance, a meridional
component of the wind is induced, which correspondingly weakens the temperature gradient by
transporting heat polewards. In addition to this, a westward force corresponds to a converging
EP-vector, which is equivalent to an equatorwards flux of (QG)PV2(section 3.1). As such, the
zonal mean meridional component of the wind is an indirect measure of the amount of down-
gradient PV-mixing by breaking planetary waves. Downgradient PV-mixing in turn reduces the
PV-gradient of the surf-zone and sharpens the PV-gradient of the polar night jet, and hence
it helps to ’maintain’ the wintertime stratospheric PV-structure. With the assumption that
the polar night jet does not vary much in strength during a climatological winter month, the
meridional wind becomes a measure of the amount of planetary wave breaking, or planetary
wave drag.
From observation, typical zonal mean meridional winds throughout the surf-zone are on the
order of 5 cm/s. In model 4a, shown in the top right panel of Fig. 4.1, the zonal mean meridional
winds are on the order of 50 cm/s. That the modelled meridional wind is an order of magnitude
higher than the observed wind, indicates that the real atmosphere is more efficient at organizing
the otherwise hemisphere-wide jet into a subtropical and polar night jet, compared to the ’brute
force’ split by planetary wave drag in model 4a. In the opinion of the author, this result is in
agreement with the framework set by the wave-turbulence jigsaw puzzle (in particular, that of a
self-sharpening jet). Further support of this view, is that the winds of the observed subtropical
1The interplay between these dynamics and the resulting ’emergent’ jet structure, is described in full as the
’wave-turbulence jigsaw puzzle’ by McIntyre [28]. The self-organizing of a stratified and rotating fluid into ’PV-
steps’ (i.e. jets and mixing-zones), can also be observed in idealized model experiments, e.g. Dritschel and Scott
[27][34].
2In isentropic coordinates, a poleward mass flux leads to dilution of PV-substance over the pole, which cor-
responds to an equatorward Rossby-Ertel PV-flux. With this, a (zonal mean) meridional wind can also be
understood as an equatorward PV-flux, without invoking planetary wave dynamics.
CHAPTER 4. PLANETARY WAVE DRAG IN THE GENERAL CIRCULATION 34
and polar night jet are significantly stronger than those modelled3. If the parameterization of
planetary wave drag is indeed less efficient at forming two separate jets, a relatively large amount
of angular momentum is lost in the process, which could result in the modelled jets being weaker
than the observed jets.
A ’wave-turbulence jigsaw’ description of the subtropical jet is beyond the scope of this
thesis. It does however seem likely that dynamics similar to that of the polar night jet apply,
as the subtropical jet features (externally forced) planetary waves on a sharp PV-gradient. The
observed zonal winds of the subtropical jet are higher than those modelled, which too alludes to
an ’efficient’ manner in which the subtropical jet is formed and sustained. It should be noted that
the subtropical and polar night jet can to a certain extent be considered as two separate entities.
Although the parameterization of planetary wave drag indeed splits the otherwise hemisphere-
wide jet into two jets, the sharp PV-gradients that form along the surf-zone when planetary waves
break in the real atmosphere, do not correspond to the sharp PV-gradients of the subtropical
and polar night jet. Downgradient PV-mixing by planetary wave breaking typically occurs
along isentropic surfaces, because conditions in the stratosphere are approximately adiabatic
and frictionless on the timescale of planetary wave breaking events. However, the subtropical
and polar night jet lie along completely separate isentropes, as is discussed in more detail in
Appendix A.7.
Even though the parameterization of planetary wave drag in model 4a remains a metaphor
for a broader set of interactions, model 4a does yield a qualitatively correct picture of the
atmosphere’s middleworld. Exceptions are the zonal winds which are too weak, and the Brewer-
Dobson circulation which is too strong. But regardless, model 4a produces interesting insights,
and the particular parameterization of planetary wave drag fits the framework of a simplified
zonal mean model. A future model improvement could be to also include the eastward zonal
force associated with planetary wave source regions. Indeed, if planetary waves are to be pa-
rameterized in the form of a body force, it would only be fair to include both the waves’ positive
and negative zonal forcings. A notion which supports this view, is that on average Earth’s
rotation speed, or angular momentum, does not change. If planetary waves are parameterized
only by a westward force, this would create a persistent reduction of angular momentum of the
atmosphere, and hence also of that of the Earth. Including a ’surface’ eastward force whose
positive torque is balanced by the negative torque in the region spanned by D(Eq. 4.1.1),
would perhaps be the simplest option. Such a parameterization corresponds to some extent
with the parameterization scheme proposed by Hitchman and Brasseur [35], in which total plan-
etary wave activity (A, Eq. 2.3.3) is conserved, with Abeing produced in planetary wave source
regions and dissipated in the ’surf-zone’. Other possible ’planetary wave’ extensions, but which
are not necessarily in line with the model’s philosophy (i.e. they are rather complex), could be
(1) a parameterization scheme in which planetary wave activity is conserved, but including the
dynamics of a single resolved planetary wave, as proposed by Garcia [36], (2) a dynamical zonal
mean surf-zone created by the inclusion of a single zonal harmonic, as proposed by Haynes and
McIntyre [37].
3It could also be that the modelled zonal winds are too weak for other reasons. The model’s zonal winds can
be increased by decreasing the heat capacity of the surface, which causes higher temperature gradients and thus,
through thermal wind balance, higher zonal velocities. For a separate subtropical and polar night jet to form,
planetary wave drag (D0) correspondingly has to be adjusted. If the modelled zonal winds are configured to
match the order of the observed winds, the meridional winds induced by planetary wave drag are roughly 30 to
40 times larger than those observed. This effectively rules out the model’s zonal winds being too weak for reasons
other than having being slowed down by planetary wave drag.
CHAPTER 4. PLANETARY WAVE DRAG IN THE GENERAL CIRCULATION 35
4.2 Quantifying model performance
Taylor-diagrams, first described in a paper by Taylor [38], are an often employed method for
quantifying model performance. A Taylor-diagram provides a graphical summary of how closely
a modelled zonal mean pattern rmatches the observed zonal mean pattern f, where rand
frepresent the same parameter, e.g. the zonal wind. In a Taylor-diagram, the correlation
coefficient R, centered root-mean-square difference E0and standard deviation of the model σr
and observation σf, are plotted in a two-dimensional plane. This is possible because the three
statistical measures are related through a geometric cosine-rule, by
E02=σ2
f+σ2
r−2σfσrR. (4.2.1)
For a dataset with Nuniform grid-points, the correlation coefficient, centered root-mean-square
error and standard deviations are calculated as
R=
1
NPN
n=1 fn−¯
f(rn−¯r)
σfσr
(4.2.2a)
E02=1
N
N
X
n=1 hfn−¯
f−(rn−¯r)i2(4.2.2b)
σ2
r=1
N
N
X
n=1
(rn−¯r)2(4.2.2c)
σ2
f=1
N
N
X
n=1 fn−¯
f2,(4.2.2d)
where the mean of the respective fields is denoted by an overbar. Eq. 4.2.2a-4.2.2d show that
the mean value of the fields is subtracted in each statistical measure, and hence Taylor-diagrams
are unable to quantify an overall bias in the data.
In the context of the model experiments described in the discussion surrounding Fig. 4.1,
the ’pattern’ of most interest is arguably the zonal wind. This is motivated by the notion
that the pattern of the zonal wind between model 3a and 4a greatly improves with respect to
the observed climatology. Furthermore, following the discussion in section 4.1, the pattern of
the meridional wind is expected to be unrealistic due to the manner in which planetary waves
are parameterized. The pattern of the zonal mean wind is most striking during the respective
wintertime months on the Northern and Southern hemisphere. During wintertime, stratospheric
winds in the summer hemisphere are westward and in the winter hemisphere a subtropical and
polar night jet are present. For the months January and June, the skill of the models with respect
to the observed climatological zonal wind is quantified by the Taylor-diagrams shown in Fig.
4.2. In the Taylor-diagrams, the proximity of the markers for model 2a, 3a and 4a to the red dot
marked ’observation’, is a measure of skill of the models with respect to reproducing the observed
climatological pattern of the zonal wind. It can be seen that model 3a is an improvement over
model 2a, but they both perform poorly. Model 4a is a significant improvement over model 3a,
in particular in terms of their correlation coefficient. The large difference between the standard
deviation of model 3a and 4a, can be attributed to planetary wave drag having reduced the
spread in the magnitude of the zonal wind by having reduced the strength of the eastward zonal
wind. There appear to be no notable differences between the Taylor-diagrams of the Northern
and Southern hemisphere wintertime month(s).
Taylor-diagrams comparing the model’s respective potential temperature patterns with the
observed climatology, are generally very good, in the sense that all statistical measures lie close
CHAPTER 4. PLANETARY WAVE DRAG IN THE GENERAL CIRCULATION 36
Figure 4.2: Taylor-diagrams comparing model 2a, 3a and 4a to the observed climatology. Left and right
panel show January and June mean zonal mean zonal wind, respectively. Green, blue and grey curves
mark lines of constant root-mean-square error, correlation coefficient and standard deviation, respectively.
For each month of the year, Taylor-diagrams comparing the zonal wind, meridional wind and potential
temperature distribution to the observed climatology are available upon request.
to observation. Although the appearance of the cold-point tropopause in model 4a is a clear
improvement over model 3a, this improvement does not manifest itself in the Taylor-diagrams
(not shown here). This could be attributed to the patterns of the potential temperature distri-
bution always being in close correspondence with the observed climatology, because potential
temperature increases monotonically with height. What may also play a part, is the close hori-
zontal packing of high valued isentropes in the stratosphere, which could dominate the statistical
’signal’.
Appendix
A.1 Observation: ERA5 and ERA-Interim reanalysis
All observational data used within this work comes either from the ECMWF ERA5 or ERA-
Interim reanalysis product. ECMWF uses extensive data assimilation system together with
numerical weather prediction models, to produce a reanalysis dataset of the atmosphere, based
on observations (Dee et al. [39]). For ERA-Interim, the dataset spans the period from 1979
until now, with data being available at 6 hourly time intervals. ERA-interim interpolates its
Gaussian grid natively to a 0.75-by-0.75 degree NetCDF grid.
ERA5 is ECMWF’s latest reanalysis product, which is currently in the process of being rolled
out (https://www.ecmwf.int/era5). As of now, ERA5 only goes as far back as 2008, making it
unsuitable for climatological analysis. Its hourly availability of data and native NetCDF grid of
0.3-by-0.3 degrees, gives it a greatly improved temporal and spatial resolution over ERA-Interim.
In addition to this, a few other notable improvements over ERA-Interim are, (1) information
on the quality of the data, (2) much improved troposphere, (3) improved representation of
tropical cyclones, (4) better global balance of precipitation and evaporation. In the context of
this thesis, the most important benefit of ERA5 over ERA-Interim, is its higher spatial and
temporal resolution. This makes it possible to observe the time-development of planetary wave
breaking events in high detail.
A.2 Pseudomomentum and momentum
Following the discussion ’On the wave momentum Myth’ published by McIntyre [40], the distinc-
tion between pseudomomentum and momentum is explicitly addressed here. Pseudomomentum
is a property of waves, which by definition is invariant under spatial translation of the wave
in a homogeneous medium. Momentum is a vector quantity in the sense of Newton’s first law.
For waves in a medium, confusion can arise due to pseudomomentum often behaving ’as if’ the
waves themselves have momentum equal to their pseudomomentum, whereas physically, waves
are associated with fluxes of momentum.
In the most general sense, momentum is a conserved quantity of a physical system under
spatial translation of that system. For example, the notion of conservation of angular momentum
is tied to the spatial translation of rotation. The wave property called pseudomomentum lends
its name to a similar symmetry operation. Namely, that it is a conserved quantity under spatial
translation of the wave in space, in a homogeneous medium. To illustrate the difference between
momentum and pseudomomentum, consider a one-dimensional homogeneous rope4initially tied
between x= 0 and x=L, carrying a disturbance as in Fig. A.3. The symmetry of this system
lies in the x-direction. If the entire rope is displaced by a distance ∆x, as in panel (a), the
4This example is an adaption of the ’rope dynamics’ described by Stone [41].
37
APPENDIX 38
conserved quantity under consideration is momentum. If however the rope is held fixed and
only the disturbance is displaced a distance ∆x, as in panel (b), the conserved quantity under
consideration is pseudomomentum, given that the properties of the rope are homogeneous. If
the rope would become progressively thinner towards x= 0, the disturbance would grow in
size as it approaches this point. Spatial translation of the wave within the medium would then
not be a symmetry operation, and the conservation of pseudomomentum would not apply. Put
differently, if the rope is not homogeneous, a translation of the wave in space within this medium,
changes the properties of the wave.
Figure A.3: Panel (a): Configuration of a rope initially tied between x= 0 and x=Land carrying a
disturbance (dashed line). The entire system is displaced by a distance ∆xin the negative x-direction
(solid line). Panel (b): Initial configuration as in panel (a). Now the disturbance is displaced by a
distance ∆x, with the rope’s ends fixed.
As an extension of the previous example, imagine a motionless vertically oriented metal
plate being attached to the string at x= 0. If the disturbance from panel (b) of Fig. A.3
would propagate towards x= 0, it will eventually reach the plate, strike it and reflect back
off it. If the plate is free to move, the impinging wave will give it a momentum impulse,
and the reflected wave will be damped. If the rope is a non-dispersive medium, the incoming
and reflected wave will have the same propagation speed, and thus the damping will reduce the
amplitude of the wave. What results, is in an increase of momentum of the plate and decrease in
pseudomomentum of the wave. For the metal plate, a change in momentum is in turn equivalent
to having been subjected to a force. This thought-experiment highlights the notion that, under
many circumstances, pseudomomentum determines the force when a wave interacts with matter
(e.g. McIntyre [40]).
The exact description of how pseudomomentum and the momentum of the medium interact,
depends on the particular problem at hand, and is often far from trivial. In the context of
quasi-geostrophic planetary wave theory, the planetary wave’s pseudomomentum and mean flow
interaction is fully described by two equations: (1) the relation between wave induced fluxes of
momentum and the mean flow is, without loss of generality, described by the Taylor-identity
(section 3.1 and Appendix A.3), (2) the second-order wave-conservation law (Eq. 3.1.4), coupling
the time development of zonal pseudomomentum to a zonal force. More detailed analysis of the
properties of pseudomomentum, as well as examples which provide more context, can be found
in Bhler [17] Ch. 4, Stone [41] and McIntyre [40]. In the context of momentum transport by
geophysical waves, Vallis [2] Ch. 10 and Ch. 15 are also rich in examples.
APPENDIX 39
A.3 The Taylor-identity
In this section, the zonal mean product of v0with q0is derived. Here q0is the quasi-geostrophic
vorticity eddy term defined by the separation q= ¯q+q0, and v0is the eddy component of the
geostrophic meridional wind. In zonal mean form, where there are no purely zonal derivatives,
the eddy term of Eq. 1.3.5a is written as
q0=∂2ψ0
∂y2+∂
∂z f0b0
N2.(A.3.1)
With the definition of v0=∂xψ0, the multiplication of q0with v0follows as
v0q0=∂ψ0
∂x
∂2ψ0
∂y2+∂ψ0
∂x
∂
∂z f0b0
N2.(A.3.2)
Expanding the first term on the right-hand side, gives
∂ψ0
∂x
∂2ψ0
∂y2=∂
∂y ∂ψ0
∂x
∂ψ0
∂y −∂ψ0
∂x
∂2ψ0
∂x∂y =−∂
∂y u0v0.(A.3.3)
Here it was used that ∂xψ0∂xyψ0=1
2∂x(∂2
yψ0) = 0, and that (u0, v0) = (−∂yψ0, ∂xψ0). The second
term on the right-hand side of Eq. A.3.2 can be written as
∂ψ0
∂x
∂
∂z f0b0
N2=∂
∂z f0b0
N2
∂ψ0
∂x −∂2ψ
∂x∂z
f0b0
N2
=∂
∂z f0b0
N2
∂ψ0
∂x −f2
0
N2
∂
∂x 1
2
∂ψ
∂z 2
=∂
∂z f0
N2v0b0.(A.3.4a)
On the second line it was used that ∂zψ=b0/f0by the definition of b0. Inserting the expressions
of Eq. A.3.4a and Eq. A.3.3 in the expression of Eq. A.3.2, yields
v0q0=−∂
∂y u0v0+∂
∂z f0
N2v0b0.(A.3.5)
This expression for v0q0is equivalent to the quasi-geostrophic zonal mean divergence of the
Eliassen-Palm vector, and is often referred to as the Taylor-identity. Special attention goes out
to the notion that the derivation of Eq. A.3.5 required only the separation of qand vin a mean
and eddy term. Hence no small-amplitude assumption is needed, such that the Taylor-identity
is generally valid.
A.4 Form drag
Form drag is a mechanism for the vertical transfer of momentum in a stratified fluid. It takes
place through deformation of interfaces by which layers in a fluid are bound, to which the
term ’form’ in ’form drag’ owes its existence. The deformation of interfaces gives rise to a
pressure force acting upon the layer bounded by these interfaces. To derive a general zonal
mean expression for form drag, consider a layer bounded by two interfaces h1(x, y) and h2(x, y)
on a domain periodic in x(Fig. A.4). Note that the interfaces are two-dimensional, and are
hence independent of z. The force resulting from a pressure gradient opposes the direction of
APPENDIX 40
Figure A.4: Volume of fluid bounded by the surfaces h1(x, y) and h2(x, y), which at y=y0are separated
by a mean vertical distance ∆¯
h. The domain is periodic in x, and the shaded region between x= 0 and
x=Lrepresents the area under consideration in a zonal mean analysis of form drag between the interfaces
at y0.
the gradient, as is evident from the equations of motion, which generally are written with a
negative pressure gradient term on the right-hand side and a positive acceleration term on the
left-hand side. By inspection of the shaded region in Fig. A.4, the zonal mean force resulting
from the zonal pressure gradient is then given by
¯
Fx=−1
LZL
0Zh1
h2
∂p
∂x dxdz. (A.4.1)
Integrating Eq. A.4.1 by parts with respect to z, yields the integrand
Zh1
h2
∂p
∂x dz =∂p
∂x zh1
h2−Zh1
h2
∂
∂z
∂p
∂x dz =∂p
∂x zh1
h2
.(A.4.2)
Here it was used that ∂x∂zp= 0 by hydrostasy, i.e. the vertical pressure gradient does not
depend on the horizontal position within the layer. Further evaluation of Eq. A.4.1 gives
¯
Fx=−1
LZL
0∂p
∂x zh1
h2
dx =−h1
∂p1
∂x +h2
∂p2
∂x ,(A.4.3)
where ∂xpirepresents the zonal pressure gradient along the surface hi. The terms on the right-
hand side of Eq. A.4.3 are the form stresses associated with the interfaces h1and h2, acting on
the layer. These stress terms can be written as τ1and −τ2, respectively, such that ¯
Fx=τ1−τ2,
where τi≡ −hi∂xpi. If τ1and τ2are not equal, there will be a momentum change within the
layer due to the pressure force, to which the term ’drag’ in ’form drag’ refers. A pressure force,
and hence form stress, is a force per unit area. ¯
Fxresults from form stresses acting over a
vertical distance, and it is therefore proportional to a force applied to a volume. With the mean
layer depth being ∆¯
h, it follows that per unit volume, ¯
Fx= (τ1−τ2)/∆¯
h. When the surfaces
are chosen to be infinitesimally close to each other, this expression is equivalent to the definition
of a derivative of τiwith respect to ¯
h. Thus for ∆¯
h→0, ¯
Fx=∂¯
hτi. Noting the equivalence
between ∆¯
hand ∆z, and dropping the subscript i, the zonal ’form drag’ force per unit volume
is written as
¯
Fx=∂τ
∂z .(A.4.4)
Of particular interest is the quasi-geostrophic form of Eq. A.4.4. As before, hrefers to
an interface h(x, y) bounded by two layers of fluid. In quasi-geostrophic theory, has well as
APPENDIX 41
pressure surfaces (p(x, y)) are taken to be quasi-horizontal. Using the separation h=¯
h+h0and
p= ¯p+p0, and applying quasi-geostrophic scaling to Eq. A.4.4, τfollows as
τ=−h0∂p0
∂x .(A.4.5)
One can imagine an interface perturbation (h0) being proportional to a pressure perturbation
(p0). The latter is in turn proportional to a buoyancy perturbation, or using the ideal gas law,
a temperature perturbation. The next step is to formalize this relation. Consider an interface
hwith mean value ¯
hand a buoyancy surface (or, equivalently, potential temperature surface) b
with mean value ¯
b. These surfaces can be expressed in terms of each other to quasi-geostrophic
accuracy, i.e. they can be written as h(b) and b(h). Using a first order Taylor expansion of h(b)
around ¯
b, yields
h(b) = h(¯
b) + ∂h
∂b b=¯
b
[¯
b−b]≈h(¯
b)−∂¯
h
∂b b0,(A.4.6)
where it was used that b0=b−¯
b, and that to first order ∂bhb=¯
b≈∂b¯
h. Using h0=h(b)−h(¯
b),
the approximation ∂b¯
h≈(∂h¯
b)−1and the equivalence between ∂hand ∂z, Eq. A.4.6 can be
written as
h0=−b0 ∂¯
b
∂z !−1
.(A.4.7)
In the quasi-geostrophic limit, ∂z¯
b=N2, where Nis the buoyancy frequency defined in section
1.3. Lastly, by geostrophic balance, ∂xp0is related to v0through ∂xp0=f0ρ0v0. Inserting the
expression for h0and ∂xp0in Eq. A.4.5, gives
τ=f0ρ0
N2v0b0.(A.4.8)
In the analysis leading up to Eq. A.4.4, the zonal form drag per unit volume was identified as
the vertical derivative of τ. Taking the vertical derivative of Eq. A.4.8 and dividing by ρ0gives
the quasi-geostrophic zonal ’form drag’ force per unit mass,
¯
Fx=f0
N2
∂
∂z v0b0=∂
∂z F(z),(A.4.9)
where F(z)is the vertical component of the Eliassen-Palm vector discussed in section 3.1. A
buoyancy perturbation b0is proportional to a negative density perturbation, but equivalently
also to a positive temperature perturbation θ0. From Eq. A.4.8, the poleward eddy heat flux
can then be interpreted as a deformation of a pressure surface, giving rise to form stress, and
corollary, vertical transport of zonal momentum. Through Eq. A.4.9, the vertical divergence of
the form stress determines the local zonal forcing per unit mass. At first glance this relation
might appear peculiar, but the dependence of Eq. A.4.9 on v0and b0is fundamentally due to
geostrophic balance and the fact that a perturbation of a stratified surface is proportional to a
buoyancy, or temperature, perturbation.
A.5 PV-Gradient proof
Following the example of this proof by Andrews et al. [3], the vertical log-pressure coordinate z=
−Hln (p/p0) is adopted, where His the scale height and p0the surface pressure. Starting with
the definition of Ertel potential vorticity Z=ξaσ−1(Eq. 1.4.1), a coordinate transformation
APPENDIX 42
from a derivative of Zin isentropic θ-coordinates to one in z-coordinates, is applied. This is
done using the identity for coordinate transformations (e.g. Charney and Stern [42], Eq. 2.28),
∂
∂s θ
=∂
∂s z−∂
∂s z
θ∂θ
∂z −1∂
∂z ,(A.5.1)
to write: ∂Z
∂s θ
=Zs−θsθ−1
zZz.(A.5.2)
Here scan be, among other parameters which span a surface, xor yon a β-plane. The subscripts
sand zdenote partial derivatives at constant z. Assuming horizontal variations in θzare small,
such that θzs = 0, the first term on the right-hand side can be written as θz(Zθ−1
z)s. Together
with the identity (Z−1)z=Z−2Zz, Eq. A.5.2 can then be written as
∂Z
∂s θ
=θzZθ−1
zs+θ−1
zZ2Z−1θsz.(A.5.3)
Note that derivatives with respect to sand zare interchangeable, since they are both taken
with respect to constant z. Per unit coordinate, density in isentropic coordinates (σ) is related
to density (˜ρ) in (x, y, z)-coordinates by
σdθ = ˜ρdz. (A.5.4)
Here the tilde is used to signify that ˜ρrepresents density in log-pressure coordinates. Using
Eq. A.5.4, isentropic density can to first-order be expressed as σ= ˜ρθ−1
z, and quasi-geostrophic
scaling assumes small horizontal gradients, such that ˜ρ= ˜ρ0(z). Dropping the z-dependence for
convenience, Zcan then be written as Z= ˜ρ−1
0θzξa. Inserting this in Eq. A.5.3, gives
∂Z
∂s θ
=θz
˜ρ0"(ξa)s+ξ2
a
˜ρ0˜ρ0θs
ξaθzz#.(A.5.5)
In z-coordinates, quasi-geostrophic potential vorticity is defined as (e.g. Andrews et al. [3], Eq.
3.2.15)
q=ξg+f0
˜ρ0˜ρ0θe
θ0zz
,(A.5.6)
where θe=θ−θ0(z), and where ξgis the geostrophic absolute vorticity. With the first-order
approximations ξa≈f0,ξs≈ξgs and θz≈θ0z(with θ0only a function of z), the term in square
brackets on the right-hand side of Eq. A.5.5 can be written as
∂
∂s ξg+∂
∂s
f0
˜ρ0˜ρ0θ
θ0zz
=θz
˜ρ0
∂q
∂s .(A.5.7)
Here it has been used that all terms except for θare independent of s, and that θs=θes .
The assumption ξa≈f0requires that the vorticity balance is dominated by planetary vorticity.
Substituting Eq. A.5.7 in Eq. A.5.5 and taking sto be y(the meridional coordinate in a
beta-plane approximation), gives
∂Z
∂y θ≈θ0z
˜ρ0
∂q
∂y z
.(A.5.8)
In Eq. A.5.8, the left-hand side corresponds to the isentropic meridional gradient of Ertel
potential vorticity and the right-hand side to a scaled isobaric meridional gradient of quasi-
geostrophic potential vorticity in z-coordinates. The relation of Eq. A.5.8 also holds if zis
taken to be the Cartesian height coordinate, but then scaling factor of the QGPV-gradient will
be different.
APPENDIX 43
A.6 Relating eddy fluxes of QGPV and Rossby-Ertel PV
In Appendix A.5 it was discussed how in a stratified atmosphere, the gradients of isentropic
Rossby-Ertel PV (Z) and isobaric quasi-geostrophic PV (q) are related. This relationship ap-
pears to carry over to the zonal mean meridional eddy fluxes of Zand q, as is remarked by
McIntyre [43] p. 48. The relation between these two eddy fluxes of PV, alludes to the no-
tion that by the analysis of section 3.1, isentropic zonal mean meridional eddy fluxes of Zare
approximately equal to (quasi-geostrophic) EP-vector divergence. Theory describing concepts
equivalent to that of quasi-geostrophic EP-vector divergence, are however not worked out for
isentropic Rossby-Ertel PV, at least to the author’s knowledge. The numerical PV-inversion
experiments in section 3.4, as well as the contents of this section, do however suggest that isen-
tropic eddy fluxes of Zplay a role in the angular momentum budget similar to that of the
quasi-geostrophic eddy flux term v0q0. To supplement the PV-inversion experiments from 3.4, in
this section, an example from observation is used to demonstrate the relation between the eddy
fluxes of isentropic Zand isobaric q.
The argument linking Zto qis based on the relation between their respective quasi-geostrophic
gradients,
∂Z
∂y θ≈α∂q
∂y p
,(A.6.1)
where αis a constant, yis the meridional coordinate in a beta-plane approximation, and where
the derivative of Zis isentropic and the derivative of qis isobaric (see Appendix A.5). The
scaling arguments leading to Eq. A.6.1, require that the isentropic and isobaric surfaces are
both quasi-horizontal, which suggests that the horizontal velocity fields along the isentropic
and isobaric surfaces are also related. This in turn suggests that the zonal mean meridional
eddy fluxes of Zand qare related, i.e. that v0Z0θ∝v0q0p, where the θand psubscripts
denote the vertical coordinate. The term v0Z0θcan be directly calculated from observation.
To determine v0q0pfrom observation is more complicated, because qis a derived quantity. In
pressure coordinates, qis defined by (Holton and Hakim [31] Eq. 6.25)
q=1
f0∇2φ+f+∂
∂p f0
σ
∂φ
∂p ,(A.6.2)
where the interpretation of the terms on the right-hand side is equivalent to those on the right-
hand side of Eq. 1.3.4a. In the expression for qgiven by Eq. A.6.2, 1/f0∇2φis the geostrophic
relative vorticity, fis as in Eq. 1.3.3, φis the geopotential height and σis defined by (Holton
and Hakim [31] Eq. 6.13a)
σ=−RT0
p
dln θ0
dp .(A.6.3)
In the definition of σ,T0(p) and θ0(p) correspond to the temperature and potential temperature
of the standard (reference) atmosphere, respectively. Potential temperature is defined in terms
of temperature, pressure and surface pressure (p0=1000 hPa), by
θ=Tp0
pR/cp
.(A.6.4)
For the observed isobaric relative vorticity fields ξobs, the assumption ξobs ≈ξg= 1/f0∇2φis
made, justified by the fact that geostrophic balance is generally a good approximation in the
stratosphere. The ’stretching’ term on the right-hand side of Eq. A.6.2, can be calculated using
the temperature values defined by the standard atmosphere and the observed geopotential φ.
APPENDIX 44
In the following analysis, the terms v0q0pand v0Z0θwill be compared along the 50 hPa
isobar and 530K isentrope, using ERA5 reanalysis data. Following the definition of potential
temperature, the 50 hPa isobar and 530K isentrope correspond to approximately to the same
height. To determine qat 50 hPa, observational data from the 30, 50 and 70 hPa is used.
Within this region, the standard atmosphere temperature is constant, such that T0can be taken
to be independent of pin calculating σ. The expression for qcan then be expanded, using the
definition of σ, as
q=ξg+f0+β(y−y0) + f0
σ
∂φ
∂p 1
p−∂ln θ0
∂p −1∂2ln θ0
∂p2!+f0
σ
∂2φ
∂p2.(A.6.5)
The derivatives in Eq. A.6.5 are either of first or second order. Their respective (second-
order accurate) central finite difference approximations, can be fully expressed in terms of three
equidistant pressure levels. The finite difference expressions for the first and second-order deriva-
tives for a parameter x(p) evaluated at p1, are given by
∂x
∂p p1
=xp0−xp2
2∆p(A.6.6a)
∂2x
∂p2p1
=xp0−2xp1+xp2
(∆p)2.(A.6.6b)
With these finite difference expressions, qcan be evaluated at the ’middle level’ (p1= 50 hPa)
using measurements from the level below (p2= 70 hPa), the level above (p0= 30 hPa), and
from the middle level itself.
For comparing v0q0pand v0Z0θ, two 3-day intervals over the course of a significant wave
breaking event are chosen. The time frame of 3 days ensures that adiabatic and frictionless
conditions apply. The first interval is chosen at the start of a wave breaking event occurring at
530K on the 22th of December, 2015, with the interval covering the 22th to 25th of December.
The second interval is chosen at the terminating phase of that wave breaking event, covering
the 27th to 30th of December, 2015. The respective eddy fluxes of PV over the two intervals are
shown in Fig. A.5, including their correlation coefficients. The first phase of the wave breaking
event (top panel) is marked by a bulge of positive (northward) eddy potential vorticity fluxes,
centered over approximately 65 degrees North. For planetary wave breaking events, it appears
to be typical that the sign of the eddy vorticity fluxes changes about halfway through the wave
breaking event, as can be seen from the animation shown in Appendix A.8 Fig. A.9. This can
also be observed in the bottom panel of Fig. A.5, where the eddy fluxes of PV are approximately
the reverse of those in the top panel. In the terminating phase of the wave breaking event, the
bulge of negative eddy fluxes is however shifted north by approximately 10 degrees. Note that
the net eddy PV flux over an entire wave breaking event is negative, or downgradient (this
notion is however not part of the consideration in this section). Some uncertainties related to
the methods used to construct Fig. A.5, are: (1) the observed eddy fluxes of v0Z0θare not
constrained by quasi-geostrophic scaling, but this is however something which is assumed in the
proof of Eq. A.6.1, (2) the β-plane approximation becomes progressively worse further away
from y0, leading to possibly erroneous values of qclose to the pole and equator, (3) the numerical
schemes used to calculate qeach have at least some numerical error. With these uncertainties
in mind, the link between v0Z0θand v0q0pdoes however still appears to be clearly demonstrated
by Fig. A.5.
APPENDIX 45
Figure A.5: Comparison of two 3-day mean intervals of v0q0
p=50hP a and v0Z0θ=530K. Correlation
coefficient between the two respective eddy fluxes is plotted in the top left corner of the top and bottom
panels. Top panel: 22th to 25th of December ’15 mean zonal mean eddy fluxes of PV. Bottom panel:
27th to 30th of December ’15 mean zonal mean eddy fluxes of PV.
A.7 PV-gradients of the subtropical and polar jet
Sharp PV-gradients can be seen to form along the edges of the idealized ’mixing-zone’ shown in
Fig. 3.1. Because the mixing-zone represents downgradient mixing of PV by breaking planetary
waves, this might suggest that such a mixing-zone ’PV-structure’ can be observed along isen-
tropes intersecting the stratospheric surf-zone. As was mentioned in section 3.2 however, this
is not the case. The diabatic processes ’driving’ and the polar vortex and the resilience of the
polar vortex structure, appear to play a more dominant role in shaping the stratospheric PV-
gradient. This is apparent from the single sharp PV-gradient observed along the 600K isentrope
and between 65 and 75 degrees North, shown in the top panel of Fig. A.6. In the lower panel of
Fig. A.6, the sharp PV-gradient along the 350K isentrope and between 25 and 35 degrees North,
is associated with the subtropical jet. In the middle panel of Fig. A.6, the PV-gradient along
the 395K isentrope is plotted. The 395K isentrope intersects the region between the subtropical
and polar night jet, as can be seen in the bottom right panel of Fig. 4.1. Correspondingly, no
sharp PV-gradient is observed along the 395K isentrope. What Fig. A.6 illustrates, is that the
sharp PV-gradients resulting from (isentropic) downgradient mixing of PV by breaking plane-
tary waves, should not be associated with the sharp PV-gradients of the subtropical and polar
night jet. The sharp PV-gradients of the subtropical and polar night jet do not lie along the
same isentrope, and they are arguably maintained by different (diabatic) dynamics. Planetary
wave breaking can however still lead to the sharpening of the PV-gradients associated with these
jets.
APPENDIX 46
30 40 50 60 70 80 90
0
50
100
150
PVU
(a)
December '15 mean zonal mean PV ([
Z
]) and reference PV ([
Zref
]) at 600K (a), 395K (b) and 350K (c)
[
Z
]
[
Zref
]
30 40 50 60 70 80 90
2.5
5.0
7.5
10.0
12.5
15.0
PVU
(b)
[
Z
]
[
Zref
]
30 40 50 60 70 80 90
Latitude (degrees)
2
4
6
8
PVU
(c)
[
Z
]
[
Zref
]
Figure A.6: December 2015 mean zonal mean Rossby-Ertel PV (Z) along the 600K (top), 395K (mid-
dle) and 350K (bottom) isentropes. Dotted grey line represents the reference PV-distribution (Zref),
corresponding to a motionless atmosphere. 1 PVU = 10−6K m2kg−1s−1. Data obtained from ECMWF
ERA5 reanalysis (Appendix A.1).
APPENDIX 47
A.8 Observation: Supplementary animations
Figure A.7: Stereographic PV-map of the 850K isentrope, centered over the North Pole. Animations
span the wintertime months (DJF). Data obtained from ECMWF ERA5 reanalysis (Appendix A.1).
Figure A.8: Equirectangular PV-map of the 850K isentrope. Animations span the wintertime months
(DJF). 1 PVU = 10−6K m2kg−1s−1. Data obtained from ECMWF ERA5 reanalysis (Appendix A.1).
APPENDIX
Figure A.9: Top left: PV-map showing the time development of the flow. Top right: Instantaneous
zonal mean zonal velocity (dashed line), and DJF mean zonal mean zonal velocity (solid grey line).
Bottom left: Arrows indicate instantaneous isentropic (u0, v0) field. Red and blue shading corresponds to
v0Z0, where Zis the isentropic Rossby-Ertel PV. Bottom right: Instantaneous zonal mean v0Z0(dashed
line), and DJF mean zonal mean v0Z0(solid grey line). Animations span the wintertime months (DJF).
Data obtained from ECMWF ERA5 reanalysis (Appendix A.1).
Figure A.10: Vertical cross-section of the atmosphere, showing the time-development of the PV-field
between 370K (top left) to 850K (bottom right). Animations span the wintertime months (DJF). Data
obtained from ECMWF ERA5 reanalysis (Appendix A.1).
APPENDIX
Figure A.11: Time-development of the zonal mean Rossby-Ertel PV (Z) along the 600K isentrope.
Dotted grey line: Reference PV-distribution as defined in Hinssen et al. [30], corresponding to a motion-
less atmosphere. Black line: DJF zonal mean PV-distribution. Green line: Instantaneous zonal mean
PV-distribution. The red dot indicates when large-scale planetary wave breaking is going on, as seen
from observation. Animations available for 600K, 700K and 850K isentropes over the wintertime months
(DJF). Data obtained from ECMWF ERA5 reanalysis (Appendix A.1).
Bibliography
[1] N. Butchart, “The brewer-dobson circulation,” Reviews of geophysics, vol. 52, no. 2,
pp. 157–184, 2014.
[2] G. K. Vallis, Atmospheric and oceanic fluid dynamics. Cambridge University Press, 2017.
[3] D. G. Andrews, J. R. Holton, and C. B. Leovy, Middle atmosphere dynamics, 40. Academic
press, 1987.
[4] J. R. Holton, “The role of gravity wave induced drag and diffusion in the momentum
budget of the mesosphere,” Journal of the Atmospheric Sciences, vol. 39, no. 4, pp. 791–
799, 1982.
[5] B. J. Hoskins, M. McIntyre, and A. W. Robertson, “On the use and significance of isen-
tropic potential vorticity maps,” Quarterly Journal of the Royal Meteorological Society,
vol. 111, no. 470, pp. 877–946, 1985.
[6] R. A. Plumb, “Planetary waves and the extratropical winter stratosphere,” The Strato-
sphere, Dynamics, Transport and Chemistry, Geophys. Monogr, vol. 190, pp. 23–41, 2010.
[7] H. A. Dijkstra, Dynamical oceanography. Springer Science & Business Media, 2008.
[8] M. P. Baldwin, P. B. Rhines, H.-P. Huang, and M. E. McIntyre, “The jet-stream conun-
drum,” Science, vol. 315, no. 5811, pp. 467–468, 2007.
[9] D. Dritschel and M. McIntyre, “Multiple jets as pv staircases: The phillips effect and the
resilience of eddy-transport barriers,” Journal of the Atmospheric Sciences, vol. 65, no. 3,
pp. 855–874, 2008.
[10] J. G. Charney and P. G. Drazin, “Propagation of planetary-scale disturbances from the
lower into the upper atmosphere,” Journal of Geophysical Research, vol. 66, no. 1, pp. 83–
109, 1961.
[11] J. Scinocca and P. Haynes, “Dynamical forcing of stratospheric planetary waves by tropo-
spheric baroclinic eddies,” Journal of the atmospheric sciences, vol. 55, no. 14, pp. 2361–
2392, 1998.
[12] B. J. Hoskins and I. N. James, Fluid Dynamics of the Mid-Latitude Atmosphere. John
Wiley & Sons, 2014.
[13] T. Matsuno, “Vertical propagation of stationary planetary waves in the winter northern
hemisphere,” Journal of the atmospheric sciences, vol. 27, no. 6, pp. 871–883, 1970.
[14] G. R. North, J. A. Pyle, and F. Zhang, Encyclopedia of atmospheric sciences. Elsevier,
2014, vol. 1.
[15] M. McIntyre and T. Shepherd, “An exact local conservation theorem for finite-amplitude
disturbances to non-parallel shear flows, with remarks on hamiltonian structure and on
arnol’d’s stability theorems,” Journal of Fluid Mechanics, vol. 181, pp. 527–565, 1987.
BIBLIOGRAPHY
[16] M. McIntyre and T. Palmer, “The surf zone in the stratosphere,” Journal of atmospheric
and terrestrial physics, vol. 46, no. 9, pp. 825–849, 1984.
[17] O. Bhler, Waves and mean flows. Cambridge University Press, 2014.
[18] P. Haynes and M. McIntyre, “On the evolution of vorticity and potential vorticity in the
presence of diabatic heating and frictional or other forces,” Journal of the Atmospheric
Sciences, vol. 44, no. 5, pp. 828–841, 1987.
[19] P. D. Killworth and M. E. McIntyre, “Do rossby-wave critical layers absorb, reflect, or
over-reflect?” Journal of Fluid Mechanics, vol. 161, pp. 449–492, 1985.
[20] P. B. Rhines, “Geostrophic turbulence,” Annual Review of Fluid Mechanics, vol. 11, no.
1, pp. 401–441, 1979.
[21] I. M. Held, “Momentum transport by quasi-geostrophic eddies,” Journal of the Atmo-
spheric Sciences, vol. 32, no. 7, pp. 1494–1497, 1975.
[22] D. Andrews and M. E. McIntyre, “Planetary waves in horizontal and vertical shear: The
generalized eliassen-palm relation and the mean zonal acceleration,” Journal of the Atmo-
spheric Sciences, vol. 33, no. 11, pp. 2031–2048, 1976.
[23] H. Edmon Jr, B. Hoskins, and M. McIntyre, “Eliassen-palm cross sections for the tropo-
sphere,” Journal of the Atmospheric Sciences, vol. 37, no. 12, pp. 2600–2616, 1980.
[24] D. G. Andrews, “On the interpretation of the eliassen-palm flux divergence,” Quarterly
Journal of the Royal Meteorological Society, vol. 113, no. 475, pp. 323–338, 1987.
[25] R. B. Wood and M. E. McIntyre, “A general theorem on angular-momentum changes due
to potential vorticity mixing and on potential-energy changes due to buoyancy mixing,”
Journal of the Atmospheric Sciences, vol. 67, no. 4, pp. 1261–1274, 2010.
[26] B. J. Harvey, J. Methven, and M. H. Ambaum, “Rossby wave propagation on potential
vorticity fronts with finite width,” Journal of Fluid Mechanics, vol. 794, pp. 775–797, 2016.
[27] D. Dritschel and R. Scott, “Jet sharpening by turbulent mixing,” Philosophical Transac-
tions of the Royal Society of London A: Mathematical, Physical and Engineering Sciences,
vol. 369, no. 1937, pp. 754–770, 2011.
[28] M. E. McIntyre, “The atmospheric wave–turbulence jigsaw,” in Rotation and Momentum
Transport in Magnetized Plasmas, World Scientific, 2015, pp. 1–43.
[29] A. J. Van Delden and Y. B. Hinssen, “Pv-θview of the zonal mean state of the atmo-
sphere,” Tellus A: Dynamic Meteorology and Oceanography, vol. 64, no. 1, p. 18 710, 2012.
[30] Y. Hinssen, A. v. Delden, T. Opsteegh, and W. d. Geus, “Stratospheric impact on tropo-
spheric winds deduced from potential vorticity inversion in relation to the arctic oscilla-
tion,” Quarterly Journal of the Royal Meteorological Society, vol. 136, no. 646, pp. 20–29,
2010.
[31] J. R. Holton and G. J. Hakim, An introduction to dynamic meteorology. Academic press,
2012, vol. 88.
[32] A. J. Van Delden, “Pv-θview of diabatic–dynamical interaction in the general circulation,”
Tellus A: Dynamic Meteorology and Oceanography, vol. 66, no. 1, p. 24 880, 2014.
[33] A. Persson, “The coriolis force and the subtropical jet stream,” Weather, vol. 58, no. 6,
pp. 244–246, 2003.
BIBLIOGRAPHY
[34] R. K. Scott and D. G. Dritschel, “The structure of zonal jets in geostrophic turbulence,”
Journal of Fluid Mechanics, vol. 711, pp. 576–598, 2012.
[35] M. H. Hitchman and G. Brasseur, “Rossby wave activity in a two-dimensional model:
Closure for wave driving and meridional eddy diffusivity,” Journal of Geophysical Research:
Atmospheres, vol. 93, no. D8, pp. 9405–9417, 1988.
[36] R. R. Garcia, “Parameterization of planetary wave breaking in the middle atmosphere,”
Journal of the Atmospheric Sciences, vol. 48, no. 11, pp. 1405–1419, 1991.
[37] P. Haynes and M. McIntyre, “On the representation of rossby wave critical layers and wave
breaking in zonally truncated models,” Journal of the atmospheric sciences, vol. 44, no.
17, pp. 2359–2382, 1987.
[38] K. E. Taylor, “Summarizing multiple aspects of model performance in a single diagram,”
Journal of Geophysical Research: Atmospheres, vol. 106, no. D7, pp. 7183–7192, 2001.
[39] D. P. Dee, S. Uppala, A. Simmons, P. Berrisford, P. Poli, S. Kobayashi, U. Andrae, M.
Balmaseda, G. Balsamo, d. P. Bauer, et al., “The era-interim reanalysis: Configuration and
performance of the data assimilation system,” Quarterly Journal of the royal meteorological
society, vol. 137, no. 656, pp. 553–597, 2011.
[40] M. McIntyre, “On the wave momentummyth,” Journal of Fluid Mechanics, vol. 106,
pp. 331–347, 1981.
[41] M. Stone, “Phonons and forces: Momentum versus pseudomomentum in moving fluids,”
in Artificial Black Holes, World Scientific, 2002, pp. 335–363.
[42] J. G. Charney and M. Stern, “On the stability of internal baroclinic jets in a rotating
atmosphere,” Journal of the Atmospheric Sciences, vol. 19, no. 2, pp. 159–172, 1962.
[43] M. E. McIntyre, “How well do we understand the dynamics of stratospheric warmings?”
Journal of the Meteorological Society of Japan. Ser. II, vol. 60, no. 1, pp. 37–65, 1982.
