Atmospheric & Oceanic Applica- tions of Eulerian and Lagrangian Transport Modelling Joakim Kjellsson

Atmospheric & Oceanic Applications of Eulerian and Lagrangian
Transport Modelling
Joakim Kjellsson
Abstract
This thesis presents several ways to understand transports of air and water masses in
the atmosphere and ocean, and the transports of energy they imply. It presents work
using both various kinds of observations and computer simulations of the atmosphere
and oceans. One of the main focuses is to identify similarities and differences between
models and observations, as well as between different models.
The first half of the thesis applies Lagrangian methods to study flows in the atmosphere and oceans. Part of the work focuses on understanding how particles follow the
currents in the Baltic Sea and how they disperse. It is suggested that the commonly
used regional ocean model for the Baltic Sea, RCO, underestimates the transport and
the dispersion of the particles, which can have consequences for studies of e.g. biogeochemistry as well as for operational use. A similar methodology is used to study
how particles are transported between the tropics and mid-latitudes by the large-scale
atmospheric circulation. It is found that the mass transport associated with northbound and southbound particles can cancel in the zonally averaged circulation, and it
is proposed that the degree of cancellation depends on the method of averaging.
The latter half of the thesis focuses on Eulerian stream functions and specifically
a thermodynamic stream function that combines the zonal and meridional circulations
of the atmosphere into a single circulation. The stream function is used as a diagnostic
to study the inter-annual variability of the intensity and thermodynamic properties of
the global atmospheric circulation. A significant correlation to ENSO variability is
found both in reanalysis and the EC-Earth coupled climate model. It is also shown
that a set of models from the CMIP5 project show a slowdown of the atmospheric circulation as a result of global warming and associated changes in near-surface moisture
content and upper-level radiative cooling.
Cover image: Plume of ash from the Eyjafjallajökull volcano seen by the Envisat Medium Resolution Imaging Spectrometer on 11 May 2010. Forecasts of ash clouds are often done by Lagrangian
models. Photo: ESA
c
Joakim
Kjellsson, Stockholm 2014
ISBN 978-91-7447-823-5
Printed in Sweden by US-AB, Stockholm 2014
Distributor: Department of Meteorology, Stockholm University
Till
studera
min
mor
meteorologi
Eva
vid
dessutom
som
föreslog
Stockholms
stöttade
mig
att
jag
skulle
universitet,
hela
vägen
och
. . .
Vi satte båtarna i bäcken
såg dom flyta in i tunneln
kanske vidare mot Vättern
och kanalen ut till Nordsjön
över vågorna mot Irland
ut på havet och sen blåsa
iväg och aldrig komma tillbaka
mer till bäcken
där det började
ur “Söndermarken” av Lars Winnerbäck
List of Papers
The following papers, referred to in the text by their Roman numerals, are included in
this thesis.
PAPER I: Surface drifters and model trajectories in the Baltic Sea,
Kjellsson J. and Döös K. (2012), Bor. Env. Res., 17, 447–459.
PAPER II: Lagrangian decomposition of the Hadley and Ferrel Cells,
Kjellsson J. and Döös K. (2012), Geophys. Res. Lett., 39, L15807.
DOI: 10.1029/2012GL052420
PAPER III: The Atmospheric General Circulation in Thermodynamical Coordinates,
Kjellsson J., Döös K., Laliberté F. and Zika J. (In press.), J. Atmos. Sci.
DOI: 10.1175/JAS-D-13-0173.1
PAPER IV: Slowdown of Atmospheric General Circulation with Global Warming,
Kjellsson J. (Manuscript),
Reprints were made with permission from the publishers Boreal Environment Research Publishing Board (Paper 1), John Wiley & Sons Ltd. (Paper 2), and the American Meteorological Society (Paper 3).
Author’s contribution
Paper 1 emerged within the EU/BONUS+ project BalticWay where my task was to
validate the upper-ocean mass transport in an ocean model using observations collected by Kristofer Döös. The idea came from me and Kristofer Döös. Ocean model
integrations were provided by Markus Meier from SMHI. I processed the observational data, designed and performed all trajectory simulations and all analyses. The
paper was written by me and Kristofer Döös.
For Paper 2, the idea came from discussions between me, Kristofer Döös, Jonas Nycander and Rodrigo Caballero. I designed the trajectory simulations together with
Kristofer Döös. I performed all trajectory simulations, analyses and wrote the paper
with inputs from Kristofer Döös.
Paper 3 was an idea based on discussions with me, Kristofer Döös, Johan Nilsson,
Maxime Ballarotta, Jan Zika, and Frédéric Laliberté partly during an oceanography
workshop in Stockholm. EC-Earth integrations were provided by Laurent Brodeau.
I performed all analyses and wrote the initial draft of the paper. The final paper was
improved by me, Kristofer Döös, Jan Zika and Frédéric Laliberté.
Paper 4 was my own idea based on results from Paper 3. I collected all data, performed
all analyses, and wrote the paper with inputs from Kristofer Döös and Maxime Ballarotta.
Contents
Abstract
ii
List of Papers
v
Author’s contribution
vii
Acknowledgements
xi
1
What is Lagrangian and what is Eulerian?
13
2
The Eulerian framework
2.1 Computer models of atmosphere and ocean . . . . . . . . . . . . . .
2.2 Eulerian stream functions . . . . . . . . . . . . . . . . . . . . . . . .
17
17
19
3
The Lagrangian framework
3.1 Lagrangian observations of the world oceans . . . . . . .
3.2 TRACMASS - An algorithm for Lagrangian trajectories
3.3 Some Lagrangian statistics . . . . . . . . . . . . . . . .
3.4 Lagrangian stream functions . . . . . . . . . . . . . . .
23
23
23
26
27
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4
Transports in the Baltic Sea and environmental risks
29
5
The overturning circulation of the atmosphere
5.1 Past and present . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.2 Future . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
31
31
35
6
Outlook
39
Sammanfattning
References
xli
xliii
Acknowledgements
They say that behind every successful meteorologist stands an even more successful
oceanographer. In my case, there are two: Kristofer Döös and Jonas Nycander, who
decided to hire me when others would not. Thank you for giving me the opportunity
to pursue a PhD in Atmosphere and Ocean sciences! I’d also like to thank the people
from the Baltic Way project for support, especially the leaders Tarmo Soomere and
Ewald Quak. I’m also grateful for all the help and advice from Peter Lundberg and
Johan Nilsson over the years. Thank you! Also, thank you Tom Rossby and my ship
mates onboard the R/V Endeavour from University of Rhode Island for teaching me
how observational oceanography is done.
I’d like to thank Fred Laliberté and Jan Zika, two brilliant young scientists from
whom I learned a lot during the latter half of my PhD, and also Bror Jonsson (without
the ö. . . ) for introducing me to Princeton, being so supportive, and for all the long
discussions of science, cold beverages and politics.
I also wish to thank the Climate Research School and the Bolin Centre for Climate
Research for the great courses, the travel grants and the PhD conference. I’ve also
had a lot of help from the National Supercomputer Centre at Linköping University
and their fantastic facilities and support team. Furthermore, I thank Rune Grand Graversen et al. for a great CESM course in Stockholm.
Thanks to my travel companions over the years for making conferences such as
EGU and AGU way more fun than they seem. Thank you Maxime (and Natalie),
Magnus, Jenny, Cecilia, Marie, Cian, Abubakr, Peggy, Matthias and Henrik! And
thank you Susanne for helping me filling out the travel bills! Thanks also to the MGFs
Saeed, Laurent and Cian for all the lunches. A big thanks to the innebandy team
(a member list to long for the purposes of this paragraph) for making Thursdays my
favourite working day.
A very special thanks to my roommate Léon. After 11 years together in the
Swedish educational system we now part ways. Your hard work has been a true inspiration, and I wish you all the best of luck in your future career as a scientist, marathon
runner and family man.
Moreover my family are worthy of a special acknowledgement. I would not have
survived the undergraduate studies without the support from my mom Eva and dad
Tommy. Nor would it not have been this fun without my sister Sofia. The greatest
thanks of all goes out to my dear girlfriend Viktoria. Thank you for putting up with
me when I try to explain my work, and thank you for supporting me when things did
not go that well.
PS. Climate-model data for Paper 4 was partly gathered during a time when most
websites for US research centers were down (PCMDI, GFDL, NASA, etc.), so I’d like
to “thank” President Obama, the US Congress and Senate for making data collection
a challenge.
1. What is Lagrangian and what
is Eulerian?
When you observe a fluid in motion, for instance water in a river, there are two possible viewpoints: the Eulerian1 and Lagrangian2 view. In the Eulerian view the observer stands at a fixed point in space and observes how the flow evolves. In the
Lagrangian view the observer moves with the flow and observes its evolution while
moving. A simple analogy is the various ways of seeing the “Vasaloppet”, the 90 km
cross-country skiing race between Sälen and Mora held every year in Sweden. Bystanders will see the race from a fixed point in space, i.e. Eulerian, while TV cameras
follow the race by moving with the leader, i.e. Lagrangian.
Consider the temperature, T , at Observatoriekullen in Stockholm. The observer is
then taking the Eulerian view (fixed point in space), and the temperature in Stockholm
depends both on local effects (i.e. the sun warming the air) and advective effects
(i.e. warm air blowing in over Stockholm). However, if the observer would take the
Lagrangian view (follow the air as it moves in over Stockholm) the total effect on the
air mass would be observed. We summarise this into the equation
∂T
+ v · ∇T =
|∂t {z
}
Eulerian
dT
,
dt
|{z}
(1.1)
Lagrangian
where v = (u, v, w) is the three-dimensional wind (cf. Vallis (2006)). The Lagrangian derivative is thus a sum of the local Eulerian effect (∂ T /∂t) and the advective
Eulerian effect (v · ∇T ). The word “Lagrangian” is sometimes replaced by “material”
to indicate that the observer is following a piece of material, i.e. a particle or a parcel.
Observations of the atmosphere and oceans are often from an Eulerian view, i.e.
weather stations, satellites, moored bouys, etc. The weather forecasts presented on
television is often described using the Eulerian view, i.e. “the temperature in Stockholm will be +5◦ C” and “the wind will be 5 m s−1 westerly”. Observations from
a Lagrangian view are also used, but to a much smaller extent. An example is the
Global Drifter Program3 which uses about 1000 satellite-tracked floating buoys in the
ocean which can measure temperature and air pressure every hour while drifting with
the ocean currents. Another example is Morel & Desbois (1974) who released about
1 After
Leonhard Euler (1707-1783)
Joseph Louis Lagrange (1736-1813)
3 http://www.aoml.noaa.gov/phod/dac/index.php
2 After
13
Trajectory Forecast
Mississippi Canyon 252
NOAA/NOS/OR&R
Estimate for: 0600 CDT, Thursday, 5/06/10
Date Prepared: 1300 CDT, Wednesday, 5/05/10
This forecast is based on the NWS spot forecast from Wednesday, May 5 AM. Currents were obtained from the NOAA
Gulf of Mexico, Texas A&M/TGLO, and NAVO/NRL models and HFR measurements. The model was initialized from
satellite imagery analysis provided by NOAA/NESDIS obtained Wednesday morning and a Wednesday morning
helicopter overflight. The leading edge may contain tarballs that are not readily observable from the imagery (hence not
included in the model initialization).
Mobile
Gulfport
Bay St Louis
Mississippi Sound
Milton
Pensacola
Pascagoula
Mobile
Bay
Forecast location for oil
on 06-May-10 at 0600 CDT
Chandeleur
Sound
Breton
Sound
F
G
Mississippi Canyon 252
Incident Location
0
15
30
60
Miles
Winds are forecast to be light (5 kts) and variable continuing through Thursday morning. S/SE
winds at 10 kts are expected to resume again Thursday afternoon/evening and continue through
Friday. The Mississippi Delta, Breton Sound and Chandeleur Sound continue to be threatened by
shoreline contacts throughout the forecast period. All three ocean models show a westward
current developing south of the Delta transporting oil to the west of the Delta.
Trajectory
Uncertainty
Light
Medium
Heavy
Potential
X
beached oil
Next
Figure 1.1: Forecasted transport from the Deepwater Horizon oil leak calculated
by the
Forecast:
National Oceanic and Atmospheric Administration (NOAA) using a Lagrangian
transport
May 6th AM
model. Blue colours show the concentration of oil, and red marks show locations where
oil may reach the shore. Image courtesy of NOAA’s Office of Response and Restoration.
14
500 balloons over the Southern Hemisphere and used information about their motion
to map the winds around Antarctica. Lagrangian forecasts can also be of great importance. In 2010, the Deepwater Horizon oil ridge in the Gulf of Mexico sank causing
the largest oil leak in U.S. history with devastating consequences for the marine life
and coastal areas. The National Oceanic and Atmospheric Administration (NOAA)
produced daily forecasts of the oil spreading to warn the public and to plan operations
to recover or burn some of the oil still at sea.
15
16
2. The Eulerian framework
2.1
Computer models of atmosphere and ocean
Almost all computer models of the atmosphere and oceans are Eulerian, i.e. they give
the results from a fixed point of view. To represent the global atmosphere in a computer one must first discretise the atmosphere by dividing it into a number of boxes as
in Fig. 2.1. The method is very similar to dividing a photograph into pixels. A photograph on a computer is actually a digital representation of the view from the camera
lens. In the same way, the global atmosphere in a computer model is a digital representation of a continuous atmosphere. Similar to photography, more pixels/boxes should
generally give a picture/model that is more similar to reality. However, a higher resolution, i.e. more boxes, consumes more computer resources. As meteorologists and
climate modellers strive to produce better weather forecasts and climate predictions
some of the most powerful computers in the world are used for running atmospheric,
ocean, or climate models.
Fundamentally, a computer model of the atmosphere is a computer program that
solves a series of equations for each box (Fig. 2.1). The output from an atmospheric
model is thus temperature, wind, humidity, pressure, etc. in each box. An ocean
model typically outputs current velocity, temperature, salinity, density, etc. A “climate” model is a coupled model, where models of e.g. atmosphere, ocean, sea-ice,
land, air chemistry, marine ecosystem, etc., run at the same time and feed back on
each other. It therefore outputs a wide range of data.
A model of the atmosphere, like the Integrated Forecasting System (IFS) model
from the European Centre for Medium-range Weather Forecasts (ECMWF), solves an
equation similar to
∂T
∂T
κTv ω
+ v · ∇T + η̇
−
= PT + KT ,
∂t
∂ η (1 + (δ − 1)q)p | {z }
|
{z
}
physics
(2.1)
dynamics
in each grid box. Similar equations for wind, humidity, etc. can be found in e.g.
Kalnay (2003). In this equation, T is the temperature and the terms on the left-hand
side correspond to: local heating/cooling, horizontal transport, vertical transport and
adiabatic expansion. These terms are often called the “dynamics” of a model since
they represent the evolution of the flow using basic fluid dynamics. The two terms
on the right-hand side, PT and KT , are the “parameterisation” and “diffusion” terms,
often called the “physics” of the model. These include the effect of processes that are
not resolved in computer models.
17
Part IV: Physical Processes
Chapter 1
Overview
Table of contents
1.1
1.2
Introduction
Overview of the code
1.1
INTRODUCTION
a)
top of atm.
z ≈ 40 km, p = 0 hPa
z
b)
The physical processes associated with radiative transfer, turbulent mixing, subgrid-scale orographic drag,
moist convection, clouds and surface/soil processes have a strong impact on the large scale flow of the
atmosphere. However, these mechanisms are often active at scales smaller than the horizontal grid size.
Parametrization schemes are then necessary in order to properly describe the impact of these subgridscale mechanisms on the large scale flow of the atmosphere. In other words the ensemble e↵ect of the
subgrid-scale processes has to be formulated in terms of the resolved grid-scale variables. Furthermore,
forecast weather parameters, such as two-metre temperature, precipitation and cloud cover, are computed
by the physical parametrization part of the model.
This part (Part IV ‘Physical processes’) of the IFS documentation describes only the physical
parametrization package. After all the explicit dynamical computations per time-step are performed,
the physics parametrization package is called by the IFS. The physics computations are performed only
in the vertical. The input information for the physics consists of the values of the mean prognostic
variables (wind components, temperature, specific humidity, liquid/ice water content and cloud fraction),
the provisional dynamical tendencies for the same variables and various surface fields, both fixed and
variable.
c)
p
surface
z = 0 km, p ≈ 1000 hPa
Figure 1.1 Schematic diagram of the di↵erent physical processes represented in the IFS model.
Figure 2.1: The “grid” of a computer model of the atmosphere. a) The grid as seen from
grid for a given square in a). Each box is3 called a “grid box”
and represents a region enclosed between two longitudes, two latitudes and two vertical
levels. c) Each grid box includes a set of processes that are too small to be resolved and
are instead “parameterised”. Figure c) courtesy of ECMWF.
IFS Documentation
– Cy31r1
above. b)
The vertical
18
The letter “P” in PT denotes “Parameterisation” which means adding the effect of
a process rather than the process itself. A few examples of “parameterised” processes
in the IFS atmospheric model from ECMWF is shown in Fig. 2.1c. For instance,
clouds are not captured by any global weather forecasting model, but since we know
the effect they have we can add a parameterisation that explicitly changes the humidity, temperature, etc. in the model in accordance with our knowledge of clouds. The
parameterisations, PT , also includes heating from short-wave radiation from the Sun,
latent heat release, absorption by greenhouse gases, and more. The term KT represents
mixing not captured by the model, also known as “diffusion”. In general, atmosphere
and ocean models include more diffusion than what would be realistic in order to keep
the flow smooth.
Similar to the equation for temperature (eq. (2.1)) an atmospheric model also includes equations for e.g. wind, pressure and humidity, while an ocean model includes
equations for e.g. currents and salinity. These equations also have some specific parameterisations as shown in Fig. 2.1c. For instance, atmosphere and ocean models
always parameterise the effect of sub-grid turbulence to some extent. By sub-grid turbulence, we mean motions that are too small to be resolved by the model, i.e. smaller
than the grid boxes. An example is the friction caused when air blows over the Earth’s
surface. Sub-grid turbulence is especially important in the ocean since geostrophic
eddies1 can not be resolved by most global ocean models and must be parameterised.
The earliest ocean models did not include such a parameterisation which meant that
they had severe difficulties in producing a realistic global climate. Gent & McWilliams
(1990) were the first to include this which greatly improved the realism of their ocean
model. This parameterisation is nowadays known simply as “GM” and is still frequently used.
2.2
Eulerian stream functions
There a many ways to analyse data given in the Eulerian framework, e.g. maps of
temperature and winds in the atmosphere or currents in the ocean. A common method
to analyse the Eulerian transports of mass and energy in the atmosphere and oceans
is to use stream functions. Mathematically, a stream function is a two-dimensional
function, Ψ(x, y), defined as
∂Ψ
= v,
∂x
−
∂Ψ
= u,
∂y
(2.2)
where u and v are the east–west and north–south velocities respectively. If the flow
is two-dimensional and does not change in time, then the stream lines in the stream
function Ψ are equal to Lagrangian trajectories. However, the atmosphere and oceans
are three-dimensional flows that vary in time, and must therefore be reduced to two
1 similar
to the low- and high-pressure systems in the atmosphere
19
dimensions. A barotropic stream function, Ψ(x, y), can be obtained by integrating
vertically and averaging over time,
∂Ψ
1
=
∂x
t1 − t0
Z t1 Z z
−
v dz dt,
t0
0
∂Ψ
1
=
∂y
t1 − t0
Z t1 Z z
u dz dt.
(2.3)
0
t0
The unit of Ψ(x, y) is [m3 s−1 ] which means a transport of volume. The density of
sea water is fairly constant in the ocean so a volume transport of 1 m3 s−1 is a mass
transport of ≈ 1000 kg s−1 . For this reason, the unit Sverdrup 1 Sv = 106 m3 s−1
is often taken as a volume transport in the ocean but as a mass transport of 1 Sv =
109 kg s−1 in the atmosphere1 . The case for the atmosphere is discussed further below.
The depth-integrated barotropic stream function, Ψ(x, y), represents the total transport projected onto the longitude–latitude plane. We can instead integrate over longitudes in which case we flow reduces to latitude–depth coordinates,
1
∂Ψ
=
∂z
t1 − t0
Z t1 Z xE
v dx dt,
t0
xW
−
1
∂Ψ
=
∂y
t1 − t0
Z t1 Z xE
w dx dt,
t0
(2.4)
xW
where xE and xW are the eastern and western integral bounds. The stream function Ψ(y, z), also known as the meridional overturning stream function, is a commonly
used tool to study the north–south transports of mass and energy in the atmosphere and
ocean. Calculating the meridional overturning stream function for the Atlantic ocean
shows the Atlantic Meridional Overturning Circulation (AMOC) and how warm water
masses are transported from the tropical Atlantic to the North Atlantic (cf. Kuhlbrodt
et al. (2007)) which plays a large role in controlling European climate.
In the atmosphere, the meridional overturning stream function is often shown using pressure instead of height as a vertical coordinate, i.e. Ψ(y, p). Using pressure
coordinates and multiplying the velocity by 1/g where g = 9.81 m s−2 is the gravitational acceleration gives the stream function in units [kg s−1 ]. It is thus a mass
transport rather than a volume transport as in the ocean. Several recent studies have
generalised the meridional overturning stream function so that any variable can be
used as a vertical coordinate but still yield the result as a mass transport (Döös & Nilsson, 2011; Pauluis et al., 2008). The meridional overturning stream function with a
generalised vertical coordinate, Ψ(y, χ), is
1
∂ Ψ(y, χ)
=
∂χ
t1 − t0
Z t1 I Z ps
t0
x 0
δ [χ − χ 0 (x, y, p)]g−1 v d p dx dt,
(2.5)
where the Dirac function, δ [χ − χ 0 ] is a function where δ = 1 if χ 0 = χ and δ = 0
otherwise. Note that we integrate over all longitudes. Multiplying the velocity by the
Dirac function and integrating is thus a way to “search” all longitudes and pressures
at y and only “select” velocities where χ 0 (x, y, p) = χ. The solution for Ψ(y, χ) is
1 Density
20
of sea water is ρ ≈ 1000 kg m−3 so 1 m3 s−1 ≈ 1000 kg s−1
1
Ψ(y, χ) =
t1 − t0
Z t1 I Z ps
t0
x 0
µ[χ − χ 0 (x, y, p0 ,t)]g−1 v d p dx dt,
(2.6)
where µ is a Heaviside function which is the integral of the Dirac function and
thus µ = 1 when χ 0 ≤ χ and µ = 0 otherwise. Examples of meridional overturning
stream functions calculated using eq. (2.6) are shown in Fig. 5.2.
A further generalisation of eq. (2.6) is to use two generalised coordinates. This has
been done recently to study the atmospheric circulation from a purely thermodynamic
perspective (Kjellsson et al., 2013; Laliberté et al., 2013). This generalisation is done
by taking eq. (2.5) and using λ as a second general coordinate.
∂ Ψ(λ , χ)
=
∂χ
Z
Ω
δ [χ − χ 0 (x, y, p)]δ [λ − λ 0 (x, y, p)]g−1~v dΩ,
(2.7)
where integrating over Ω is equal to integrating over the full three-dimensional
global atmosphere. Multiplying by two Dirac functions allows for a “selection” of
the three-dimensional velocities where λ 0 = λ and χ 0 = χ. To calculate the stream
function, Ψ(λ , χ) and taking temporal variations into account, we get
Ψ(λ , χ) =
1
t1 − t0
Z
Ω
δ [λ − λ 0 (x, y, p,t)]µ[χ − χ 0 (x, y, p,t)]g−1~v dΩ.
(2.8)
This method was used in Papers 3 and 4 of this thesis to define the hydrothermal
stream function which uses latent heat and dry static energy as coordinates. The discretised form of eq. (2.8) is shown in Fig 11 of Paper 3 of this thesis.
21
22
3. The Lagrangian framework
3.1
Lagrangian observations of the world oceans
As briefly mentioned in the first chapter, the Global Drifter Program gathers data from
some 1000 satellite-tracked buoys in the world oceans. These buoys float in the uppermost layer of the ocean and drift freely with the horizontal currents. Hence, the
name surface drifters. The WOCE1 standard design of a surface drifter is a float at
the surface containing a battery, GPS chip and in some cases a thermometer and/or
a barometer (Sybrandy et al., 2009). Attached to the float is a 12 meter tether line
that connects to a hollow drogue. The drogue is 6 meter long and the drogue thus sits
between 12 and 18 m depth. This design ensures that the float at the surface actually
follows the currents at 12 − 18 m depth to a high accuracy. An overview of the design
is given by Niiler et al. (1995) and Lumpkin & Pazos (2007).
Each surface drifter transmits its position and, if possible, sea-surface temperature
(SST) and air pressure once every hour to a satellite. The positions can then be used
to calculate the velocity of near-surface currents in the world oceans. Several studies have compared the surface-drifter data to ocean-model simulations to evaluate the
realism of the models. Some studies have used the drifter data to obtain gridded Eulerian fields of velocity, eddy kinetic energy and other metrics (Garraffo et al., 2001;
Rupolo, 2007). Others have used an algorithm to simulate Lagrangian trajectories using the ocean model velocity fields (Döös et al., 2011; Lumpkin et al., 2002; McClean
et al., 2002) and compared the trajectories and their statistics.
In Paper 1 of this thesis, surface drifter data from the Baltic Sea is used to evaluate the realism of the near-surface currents in the widely used Rossby Centre regional
Ocean climate (RCO) model for the first time ever.
3.2
TRACMASS - An algorithm for Lagrangian trajectories
Paper 1 and Paper 2 of this thesis applies the TRACMASS Lagrangian trajectory code
to solve atmospheric and oceanic problems. A description of the TRACMASS algorithm and other trajectory codes is therefore presented here.
1 World
Ocean Circulation Experiment
23
straightforward when using a C-grid model. Although B-grid velocities just need to be
projected on the C-grid by making a meridional average of two zonal velocities (uC
i,j,k =
B
0.5(uB
i,j,k + ui,j
viB 1,j,k ))
1,k ))
C
B +
and a zonal average of two meridional velocities (vi,j,k
= 0.5(vi,j,k
for each grid box.
vi,j
yj
Latitude
(x1,y1)
ui-1,j
ui,j
(x0,y0)
yj-1
xi-1
vi,j-1
xi
Longitude
Figure 3.1: The trajectory of a TRACMASS particle passing through a grid box similar
to those
in Fig. of
2.1.
grid box
is seen
above one
where
v j and
aremodel
the
Figure
1.2:depicted
Illustration
a The
trajectory
[x(t),
y(t)]from
through
grid
box.v j−1
The
meridional
at the
the northern
velocities
arevelocities
defined at
walls of and
the southern
grid box.walls, and ui and ui−1 are the velocities
at the eastern and western walls.
In a finite difference model there is no information of scales below the grid size. The
tracers are regarded as homogeneous within each grid box and the velocities are only
To calculate
the side
pathwalls.
of a particle,
e.g. as possible
done by toNOAA
Fig.
1.1, we
must
defined
on the grid
It is, however,
defineinthe
velocity
inside
a
know the flow velocity and the origin of the particle. The path along which the particle
moves is called a Lagrangian trajectory. The principle is to use the fact that velocity
is the time-derivative of the position, i.e.
grid box by interpolating linearly between the discretised velocity values of the opposite
dx
,
dt
which means that the position, x, of a particle at time t can be found by
u=
x = x0 +
(3.1)
Z t
u dt.
(3.2)
t0
Numerically, the velocity at time step n, un , is used to calculate the displacement
from xn to xn+1 ,
xn+1 = xn + un ∆t,
(3.3)
where ∆t is the time step, and n is the time index. This method is often called Euler’s method and is common for solving Ordinary Differential Equations. The method
is however non-centered in time and is only first order accurate. The accuracy can be
doubled by using a second order Runge-Kutta scheme.
xn+1
u
n+1/2
= xn + un+1/2 ∆t,
(3.4)
= u (x + 0.5u ∆t,t + 0.5∆t) ,
n
n
n
(3.5)
where un+1/2 is the velocity between time step n and n + 1. A scheme similar to
this is used by NOAA’s HYSPLIT model (Draxler & Hess, 1998). A popular scheme
to use is the 4th order Runge-Kutta scheme where the accuracy is again doubled
24
u1
1
= xn + ∆t(u1 + 2u2 + 2u3 + u4 ),
6
= un ,
u2
= u(x + 0.5u1 ∆t,t + 0.5∆t),
(3.8)
u3
= u(xn + 0.5u2 ∆t,t n + 0.5∆t),
(3.9)
u4
= u(x + u3 ∆t,t + ∆t).
xn+1
n
n
n
n
(3.6)
(3.7)
(3.10)
This scheme, sometimes denoted RK4, makes several approximations of the velocity at time step n, n + 1/2 and n + 1 to calculate the next position xn+1 . The TOMCAT/SLIMCAT Chemical Transport Model (Chipperfield, 2006) as well as the model
used by Bowman & Carrie (2002) use an RK4 scheme to calculate the trajectories of
particles.
The TRACMASS1 code (Blanke & Raynaud, 1997; Döös, 1995; Döös et al., 2013)
is based on equations that look quite different. Start with the mass flux in a grid box
Ui = ui ∆y∆z.
(3.11)
The continuous mass flux U(x) is then found by linear interpolation
U(r) = Ui−1 + (r − ri−1 )(Ui +Ui−1 ),
(3.12)
where the position x is exchanged for r = x/∆x. Eq. (3.12) can be set up as a
differential equation
dr
+ β r + δ = 0,
ds
(3.13)
where β = Ui−1 − Ui and δ = −Ui−1 − β ri−1 and time t is exchanged for s =
t/(∆x∆y∆z). Eq. (3.13) has the solution to r and s,
δ
δ
e−β (s−s0 ) −
r0 +
β
β
1
r1 + δ /β
= s0 − log
,
β
r0 + δ /β
r(s) =
s1
(3.14)
(3.15)
where r(s) is the position as a function of the time coordinate s, and s1 − s0 is the
time it takes to travel from r0 to r1 . The TRACMASS algorithm thus gives the position
and time for each trajectory by evaluating eqs. (3.14) and (3.15) which are exact for a
stationary flow. de Vries & Döös (2001) developed an exact solution for TRACMASS
trajectories in time-varying flows. This is opposed to Runge-Kutta methods where the
position is approximated using e.g. eqs. (3.6)-(3.9).
1 TRACing
the water MASSes of the North Atlantic and the Mediterranean
25
3.3
Some Lagrangian statistics
Paper 1 focuses on calculating and comparing Lagrangian statistics for trajectories
of surface drifters and model-simulated (“synthetic”) trajectories in the Baltic Sea to
evaluate the realism of the model.
There exists several kinds of statistical metrics to diagnose the trajectories of particles in a flow. An overview was given by LaCasce (2008). Two commonly used
metrics are absolute and relative dispersion. Absolute dispersion, denoted D2A (t), is a
measure of the displacement from the origin,
D2A (t) =
1 M
∑ |x(t) − x(0)|2 ,
M m=1
(3.16)
where M is the number of particles. Absolute dispersion is often denoted as a
squared distance with units [m2 ]. It can be calculated for a single particle and is thus
sometimes called single-particle dispersion.
Relative dispersion, denoted D2R (t), is a measure of the distance between two particles at a given time,
D2R (t) =
1
|xi (t) − x j (t)|2 ,
N p i6∑
=j
(3.17)
where N p is the number of pairs of particles such that particle i is never the same as
particle j. Relative dispersion is often calculated for pairs of particles that are initially
very close to study how they separate. This gives information about the diffusivity of
the flow, i.e. rate of mixing (Klocker et al., 2012; LaCasce & Bower, 2000; Sallée
et al., 2008).
The variability of the flow can be diagnosed by calculating the auto-correlation,
R(τ), of the “eddy” velocity, u0 , defined as the total velocity minus the time-mean
velocity, u0 (t) = u(x) − u. The velocity auto-correlation at time lag τ is the correlation
of the eddy velocity time series u0 (t) and the eddy velocity time series u0 (t + τ) which
has been shifted in time by τ. Hence,
1
T →∞ σ 2 T
V
R(τ) = lim
Z T
0
u0 (t + τ) · u0 (t) dt,
(3.18)
where σV2 is the variance of the eddy velocity, u0 . The velocity auto-correlation,
R(τ), can be said to hold the “memory” of the particle. The Lagrangian velocity time
scale, TL , can be seen as a measure of how long time a particle “remembers” when it
has been and is defined as
TL =
Z ∞
R(τ) dτ.
(3.19)
0
If the flow is highly turbulent the velocity of a particle will de-correlate quickly
and TL becomes short. If, on the other hand, the flow is less variable, the velocity will
26
be auto-correlated for a much longer time and TL will be long. When calculating TL it
is common to filter out variations on short time scales e.g. inertial oscillations or tides
since they tend to dominate R(τ) otherwise. It is often impractical to evaluate the full
integral in eq. (3.19). Lumpkin et al. (2002) noted that R(τ) often becomes noisy at
large τ and presented a few methods of circumventing this problem. In Paper 1 of this
thesis the integral is truncated at the lowest τ where R(τ) = 0.
Using TL we can define the velocity time scale Tv and acceleration time scale Ta
(Nilsson et al., 2013; Rupolo, 2007) as
TL +
Tv
Ta
=
r
σ2
TL2 − 4 σV2
A
r2
TL −
=
(3.20)
σ2
TL2 − 4 σV2
A
(3.21)
2
Rupolo (2007) calculated both the auto-correlation time scale of the velocity, Tv ,
as well as the auto-correlation of acceleration, Ta . The ratio of these
γR =
Ta
,
Tv
(3.22)
has been denoted the “Rupolo ratio” where a high γR generally indicates a highly
turbulent flow (Nilsson et al., 2013). The results in Paper 1 showed that this ratio
generally is very low for the Baltic Sea.
3.4
Lagrangian stream functions
The concept of Lagrangian stream functions was introduced by Blanke et al. (1999).
The principle is that any Lagrangian particle trajectory is a transport of mass from
one point to another. Similarly to the Eulerian case, a meridional overturning stream
function can then be defined for the atmosphere as
∂ Ψ(y, p)
=
∂p
I
x
g−1 v dx,
∂ Ψ(y, p)
=−
∂y
I
x
g−1 ω dx,
(3.23)
where v and ω are the meridional and vertical velocities of the Lagrangian particle at latitude y and pressure p. If there are more than one Lagrangian trajectory
the velocities can be gridded onto a three-dimensional field and a Lagrangian stream
function representing all particles can be calculated using eq. (3.23). In this case a
stream line in the Lagrangian stream function will represent the particle path averaged
over several particles.
A Lagrangian stream function can be calculated for a selection of particle trajectories. By sorting them into different classes and calculating the Lagrangian stream
27
function for each class the total stream function can be decomposed into different
kinds of motion. Decomposition of Lagrangian stream functions similar to those in
eq. (3.23) were used by Jönsson et al. (2011) in the Baltic Sea and by Döös et al.
(2008) in the Southern Ocean to understand the transports of different water masses.
The method was applied to the atmosphere in Paper 2 of this thesis to study the transport of air masses to and from the tropics.
28
4. Transports in the Baltic Sea
and environmental risks
The Baltic Sea is a relatively small sea enclosed by a large population. The only connection to the world oceans are the narrow Danish straits which makes the Baltic Sea
particularly sensitive to pollution. Shipping and near-shore activities can thus pose
an environmental risk for the life in and around the Baltic Sea. Although oil spills
are uncommon, oil tankers carrying 100000-150000 tonnes1 of oil are continuously
crossing the sea (HELCOM, 2010). In the case of an accident, Lagrangian trajectories
provide a valuable tool to model the transport of pollutants. An example was given
in Fig. 1.1 showing the forecasted destinations of oil slicks in the Gulf of Mexico. In
the Baltic Sea, forecasts of Lagrangian particles are produced by the Swedish Meteorological and Hydrological Institute (SMHI) using a particle tracking model and the
HIgh Resolution Operational Model for the Baltic sea (HIROMB) model (Funkquist,
2001). Naturally, Lagrangian trajectories can also be calculated backward in time to
find where the pollution came from.
HIROMB is an operational model that continuously produces 48-hour forecasts.
For studying the circulation of the Baltic Sea it is however more common to use ocean
models that run for a considerably longer time. One such model is the Rossby Centre regional Ocean climate (RCO) model (Meier et al., 1999) which has been used
to produce hindcasts as far back as the 1960’s (Meier et al., 2003). Data from the
RCO model and TRACMASS trajectories were used by Döös et al. (2004) to study
the large-scale circulation of the Baltic Sea and by Jönsson et al. (2004) and Jönsson
et al. (2011) to understand the transport in and out of the Gdansk Bay and the Gulf of
Finland. Corell et al. (2012) used the RCO model with TRACMASS trajectories to
study the transport and spreading of larvae and how Marine Protected Areas (MPAs)
are connected. Soomere et al. (2010) and Soomere et al. (2011) used similar methods
to calculate the transport of future oil spills in the Gulf of Finland and used that information to find the shipping routes that would represent the smallest environmental
risk in the event of an accident.
In Paper 1 we find that TRACMASS trajectories driven by RCO model output
have less absolute and relative dispersion (eqs. (3.16) and (3.17)) than the trajectories
of observed SVP drifters. This implies that the currents in the RCO model are slower
and less variable than those observed, which is a result that will have impacts on future
transport modelling in the Baltic Sea.
11
tonne = 103 kg
29
30
5. The overturning circulation of
the atmosphere
5.1
Past and present
One of the key features of the atmospheric circulation is the way air masses are transported between the warm tropics and the cold polar regions. Fig. 5.1 shows three of
the early conceptual models of the equator-to-pole flows in the atmosphere. Halley
(1686) (finder of “Halley’s” comet) and Hadley (1735) proposed that warm, moist air
can rise in the tropics because it is lighter than the surrounding air. As it rises the
moisture condenses, releases latent heat and forms clouds and rain. At high altitudes,
air masses flow towards the poles and radiates some of their energy to space. Hence,
near the poles the air sinks as it becomes cold and dense. The loop is then closed by
air near the surface flowing towards the equator. In this “Halley-Hadley” model the
atmosphere acts as a heat engine that transports air masses from the warm tropics to
the cold polar regions.
The Halley-Hadley model as shown in Fig. 5.1a is, however, not correct. The
Halley-Hadley model predicts equatorward winds at the surface in the midlatitudes
(∼ 60◦ N/S) while measurements found them to be poleward on average. Ferrel (1859)
and Thomson (1912) therefore, independently, “corrected” the Halley-Hadley idea by
adding a cell in midlatitudes. Thomson (1912) added the cell beneath the “Hadley”
cell (Fig. 5.1b) while Ferrel (1859) suggested a three-cell structure in each hemisphere (Fig. 5.1c). The “Ferrel” cell was based on the idea of an imbalance between
air pressure and the Earth’s rotation causing surface winds in midlatitudes to be southwesterly on average.
The most prominent east–west flow is the Walker circulation, which comprises
several cells, of which the most well known is the Walker cell over the tropical Pacific
Ocean (cf. Peixoto & Oort (1992)). Air rises over the western Pacific and sinks over
the east Pacific. The cell is closed by a westerly flow at high altitude and easterly flow
near the ocean surface. It was discovered by Walker (1924) who described a surface
air pressure gradient over the tropical Pacific Ocean and noted that the rainfall over
the west Pacific increased when the pressure gradient was strong. Bjerknes (1969) realised that the pressure difference was associated with a zonal overturning cell which
he named the Walker cell. The Walker cell is linked to the sea-surface temperature
(SST) gradient over the tropical Pacific Ocean (Neelin et al., 1998; Philander, 1983;
Rasmusson & Carpenter, 1982). Periods of small SST gradients, known as El Niño
events, are associated with a weaker Walker cell and periods of large SST gradients,
31
height
a) Halley-Hadley model
H
90°S
60°S
H
0°
30°S
30°N
height
90°N
b) Thomson model
H
F
90°S
60°S
H
30°S
0°
F
30°N
height
60°N
90°N
c) Ferrel model
P
90°S
60°N
F
60°S
30°S
0°
P
F
H
H
30°N
60°N
90°N
Figure 5.1: The equator-to-pole motion of air masses as imagined by a) Halley (1686)
and Hadley (1735) and b) Ferrel (1859) and Thomson (1912). “H”, “F”, and “P”, denote
“Hadley”, “Ferrel”, and “Polar” cells. The Halley-Hadley model comprises two large
“Hadley” cells that span the Northern and Southern hemispheres respectively. The Thomson model includes a shallow “Ferrel” cell in the mid-latitudes, while the Ferrel model has
a three-cell structure in each hemisphere.
32
a)
b)
c)
d)
Figure 5.2: Meridional overturning circulation of the atmosphere using various vertical coordinates. Positive values indicate clockwise circulation. Crosses mark the maxima/minima of the stream functions, and dashed lines show the mean surface values. Units
in Sverdrup (1 Sv = 109 kg s−1 ). Adapted from Kjellsson et al. (2013).
known as La Niña events, are associated with a more intense Walker cell.
Fig. 5.2a shows the meridional overturning stream function in pressure coordinates, Ψ(y, p), (eq. (2.6)) calculated from ERA-Interim reanalysis data covering
1979-2009. The picture shows three distinct cells on each hemisphere, similarly to the
picture depicted by Ferrel (1859). The three cells are commonly known as the Hadley,
Ferrel and Polar cells (Grotjahn, 1993).
Fig. 5.2b shows the meridional overturning stream function calculated using latent
heat, l [J kg−1 ], as vertical coordinate, i.e. Ψ(y, l). The latent heat of an air parcel is
the heat that would be released if all the water vapour would condense. It can be
calculated as l = Lv q [J kg−1 ] where Lv is the latent heat of condensation and q is the
specific humidity of the air. The stream function Ψ(y, l) shows that the Hadley cells
have an equatorward flow at high l and a poleward flow at low l as also found by
Döös & Nilsson (2011). The opposite is true for the Ferrel cells. This indicates that
the Hadley cells transport moisture from the subtropics into the tropics, and the Ferrel
cells transport moisture from the subtropics into the midlatitudes. Furthermore, the
Ferrel cells have a much higher amplitude than the Hadley cells in Ψ(y, l) indicating
that they facilitate higher moisture transport.
33
1
2
c)
d)
3
Figure 5.3: The hydrothermal stream function, Ψ(l, s), calculated from ERA-Interim reanalysis 1979-2009. Negative values (blue colours) indicate anti-clockwise circulation.
Units are in Sverdrup, 1 Sv = 109 kg s−1 . The dashed line is the 345 kJ kg−1 MSE line.
The double-dashed line is the Clausius-Clapeyron relation for saturated near-surface air
(eq. (5.1)). From Paper 3 of this thesis with permission from AMS.
Fig. 5.2c shows the meridional overturning stream function where dry static energy is used as vertical coordinate. Dry static energy is the total energy that the air
would have if it was at rest (i.e. no wind) and dry (i.e. no humidity). It comprises
two parts; heat, c p T , and potential energy, gz. Dry static energy, s = c p T + gz, is
approximately conserved for adiabatic processes 1 . The result of Ψ(y, s) is that the
Hadley cells and Ferrel cells merge to form one circulation in each hemisphere with
an equatorward flow at low s and a poleward flow at high s, similar to e.g. Townsend &
Johnson (1985) and Held & Schneider (1999). This implies that the dry static energy
transport is poleward at all latitudes.
Fig. 5.2d shows the meridional overturning stream function using moist static energy as a vertical coordinate. Moist static energy, h = s + l [J kg−1 ], is the total energy
of air at rest 2 . The stream function with moist static energy as a vertical coordinate,
Ψ(y, h), shows a single circulation in each hemisphere with poleward flow at high h
and equatorward flow at low h, similar to Pauluis et al. (2008) and Döös & Nilsson
(2011). Note that the picture is very similar to the Halley-Hadley model in Fig. 5.1a.
Fig. 5.2d implies that the moist static energy transport is poleward at all latitudes. As
shown by e.g. Döös & Nilsson (2011), the poleward energy transport attains its maximum of ∼ 5 PW in the midlatitudes, in close agreement with estimates by Trenberth
& Caron (2001).
Paper 3 and 4 of this thesis describe the global atmospheric motions from a thermodynamic perspective by calculating a stream function where latent heat and dry
static energy act as coordinates, i.e. Ψ(l, s), as formulated in eq. (2.8). This is denoted
the hydrothermal stream function and the result when using ERA-Interim reanalysis
data 1979-2009 (same as in Fig. 5.2a-d) is shown in Fig. 5.3. The hydrothermal
stream function is the sum of all mass transports in the atmosphere that are associated
1 Dry
static energy, s, is similar to potential temperature, θ , so that s ≈ c p θ .
static energy, h, is similar to equivalent potential temperature, θe , so that h ≈ c p θe .
2 Moist
34
2000
RCP4.5
RCP6
RCP8.5
Hist.
Pre-ind.
1800
CO2 concentration [ppmv]
1600
1400
1200
1000
800
600
400
200
1800
1900
2000
2100
2200
Year
2300
2400
2500
Figure 5.4: CO2 concentrations for climate-model simulations of the period 1765-2100
taken from Meinshausen et al. (2011). Values are shown for the different CMIP5 experiments, “pre-industrial” (1765-1850), “historical” (1765-2005), “RCP4.5” (1765-2100),
“RCP6” (1765-2100) and “RCP8.5” (1765-2100).
with a change in either latent heat or dry static energy. Following the outer stream line
in Fig. 5.3 shows that the global atmosphere comprises three distinct processes which
are shown as arrows. The flow marked as “1” is a flow where latent heat is converted
into dry static energy, which is similar to precipitation as in the rising branch of the
Hadley cells (Figs. 5.1 and 5.2). The flow marked as “2” is dry air where the dry
static energy decreases, which indicates radiative cooling at high altitude. Lastly, “3”
marks a flow where latent heat and dry static energy increase. This could be due to
moistening and warming of air near the surface or by mixing with moist air in shallow
convective cells. The hydrothermal stream function combines the thermodynamic effects of the Walker circulation, the Hadley circulations and the midlatitude low- and
high-pressure systems into a single circulation in l–s space (Kjellsson et al., 2013). It
thus provides a convenient measure to study the variability of the global atmospheric
circulation.
5.2
Future
Observed CO2 concentrations and those projected by various “emission scenarios”
(Meinshausen et al., 2011) of the 21st century are shown in Fig. 5.4. Emissions of
greenhouse gases (GHG) and tiny particles known as “aerosols” have been shown to
have an effect on the global climate system (IPCC, 2013) by e.g. increasing the surface
air temperatures globally (Estrada et al., 2013; Hansen et al., 2005, 2013). The global
warming and its effects have been extensively studied using climate models, recently
in the Coupled Model Inter-comparison Project phase 5, CMIP5 (Taylor et al., 2012).
It has been found that warming at the Earth’s surface will in turn cause changes to
e.g. the global transports of mass and energy in the atmosphere and oceans (Caballero
35
present
future
convection
radiative cooling
P2 = P1 ⇤ (1 + 0.02 ⇤ T )
M2 = M1 ⇤ (1 0.05 ⇤ T )
P1
s ≈ 310 kJ/kg
M1
moistening
q1
q2 = q1 ⇤ (1 + 0.07 ⇤
surface
T)
surface
Figure 5.5: Schematic description of the thermodynamic processes controlling the intensity of the atmospheric circulation and the change from present-day to the end of the 21st
century according to Held & Soden (2006). P is precipitation, M is convective mass flux
and q is specific humidity, related by P = Mq. Subscript 1 denotes present and 2 denotes
future. s = 310 kJ kg−1 is a dry static energy surface that does not intersect the ground.
& Langen, 2005; Hwang & Frierson, 2010; O’Gorman & Schneider, 2008; Vecchi
& Soden, 2007; Weaver et al., 2012). For instance, warming of the surface air temperatures are projected to warm and freshen the surface waters in the North Atlantic,
which would slow down the Atlantic Meridional Overturning Circulation (AMOC).
In the atmosphere, warmer air can hold more water vapour which will lead to a more
moist atmosphere in the global mean. As a result of global warming, global-mean precipitation is projected to increase by about 1 − 2 % for each degree of warming while
the atmospheric circulation is projected to slow down (Held & Soden, 2006; Vecchi &
Soden, 2007).
It is possible to estimate the impact of global warming on the atmospheric circulation using some thermodynamic arguments. The Clausius-Clapeyron (Clapeyron,
1834; Clausius, 1850; Wallace & Hobbs, 2006) relation gives the maximum amount
of water vapour that air of a certain temperature can hold.
dqs
qs Lv
=
,
dT
Rv T 2
(5.1)
where qs [kg kg−1 ] is the saturation specific humidity, i.e. the amount of kilograms
of water vapour that 1 kg air of temperature T holds when the relative humidity is
100%. Eq. (5.1) can be simplified by letting Lv , the latent heat of vapourisation, be
constant and evaluating how qs changes for a given amount of warming.
∆qs
Lv
= α(T )∆T =
∆T,
qs
Rv T 2
(5.2)
where T = 273 K yields α ≈ 7 % K−1 . Eq. (5.2) thus means that the global-mean
specific humidity, q, will increase with about 7% for each degree of global warming,
assuming that relative humidity is unchanged (which Held & Soden (2000) found it
to approximately be). Held & Soden (2006) found that the climate models used for
36
the 4th Assessment Report of the IPCC (Solomon et al., 2007) showed an increase
in column-integrated q of 7.5% K−1 . They set up a simple scaling argument where
the precipitation, P, is equal to the upward flux of moisture which is the product of
upward mass flux of air, M, and the specific humidity of that air, q, i.e.
P = Mq.
(5.3)
∆P/P = ∆M/M + ∆q/q
(5.4)
Differencing,
The last term is equal to the left hand side of eq. (5.2) under the assumption that
relative humidity is constant. Climate-model simulations have shown that precipitation increases by 2−3% per degree of global warming (Allen & Ingram, 2002; Held &
Soden, 2006). Inserting numbers into eq. (5.4) thus gives: If precipitation increases by
2% K−1 and specific humidity increases by 7% K−1 , then the upward mass flux must
decrease by 5% K−1 . A schematic representation of the processes involved are shown
in Fig. 5.5. Hence, global warming could cause the atmospheric overturning circulations to slow down. This has been found to be the case in most climate models used in
the IPCC reports (Bony et al., 2013; Vecchi et al., 2006). Knutson & Manabe (1995)
also suggested that the atmospheric circulation could slow down since the warming of
the upper troposphere (about 15 km height) must balance radiative cooling.
37
38
6. Outlook
The most advanced atmosphere/ocean/climate models continue to increase in complexity as they include more and more of the processes in the climate system. The
need for tools to evaluate the differences between the models and the real world will
therefore also grow. This thesis has presented a set of new tools for doing just that.
In the Baltic Sea, a new regional ocean model is being introduced which has yet to
be fully validated against surface-drifter observations. Knowledge of its ability to reproduce the transport and spreading of particles will have impacts for e.g. studies of
biogeochemistry as well as operational use. The number of surface-drifter observations is very limited and any additional deployed drifters may prove highly beneficial
for evaluating the currents in ocean models.
For atmospheric purposes, this thesis has also shown that the atmospheric circulation can be very different in different coupled climate models and reanalysis. It
has been shown that the models and reanalysis respond to climate variability such as
ENSO as well as global warming in a similar way. However, the differences between
the models have not been discussed in any greater detail. Specific model simulations
where parameterisations or other conditions are changed one at a time could give better insight. To fully understand the origin of these differences the hydrothermal stream
function as presented in Papers 3 and 4 could be included in an atmospheric model as
an “online”-diagnostic, i.e. be calculated at run-time. This could highlight the impact
of various parameterisations, e.g. entrainment, rain auto conversion, turbulent mixing
in the planetary boundary layer, etc.
The decomposition of the meridional mass fluxes done in Paper 2 can be extended
to include other models and a comparison of future and present-day climates. The
meridional transport of sensible and latent heat is projected to increase with global
warming (Paper 4), but the question remains to what extent this is due to increased
mass fluxes or merely an increase in heat content of the air masses. Furthermore, the
dependence on spatial and temporal resolution for the conclusions in Papers 3 and 4
remains an open question.
39
40
Sammanfattning
Denna avhandling presenterar olika metoder för att studera rörelser av luft- och vattenmassor i atmosfären och haven samt den transport av energi detta innebär. Resultat
kommer dels från olika typer av observationer och dels från datormodeller av atmosfären, haven, och det kopplade klimatsystemet. Ett övergripande tema i avhandlingen
är att dels jämföra datormodeller med varandra, och dels att jämföra dem med observationer av verkligheten.
Avhandlingens första hälft använder det “Lagrangeska” synsättet där ett flöde betraktas som summan av flera individuella partiklar i rörelse. En del handlar om att
förstå hur partiklar i Östersjön följer strömmarna nära havsytan och hur partiklarna
sprider sig. Genom att sjösätta flöten som kan spåras med hjälp av satelliter inhämtas information om strömmarnas hastigheter vilket sedan jämförs med resultat från
en datormodell. Resultaten indikerar att datormodeller tenderar att underskatta dels
strömmarnas hastigheter och dels hur snabbt partiklar sprids, något som kan ha implikationer för såväl studier av biogeokemiska kretslopp som operationella insatser.
Liknande metodik appliceras därefter på atmosfärens storskaliga cirkulation för att
studera hur luftmassor rör sig mellan tropikerna och mellanbredderna (ca. 45◦ N/S).
Det visas att masstransport mot polerna kan tas ut av masstransport mot ekvatorn så
att ett medelvärde endast visar mycket små rörelser. Det föreslås att medelvärdet därför kan variera starkt beroende på hur medelvärdet beräknas.
Avhandlingens andra hälft fokuserar på det “Eulerska” synsättet där ett flöde ses
som en hastighet som varierar i tid och rum. Mer specifikt används s.k. strömfunktioner och en framförallt en termodynamisk strömfunktion där atmosfärens rörelser
analyseras utifrån energin i luftmassorna istället för longitud, latitud och höjd/tryck.
Denna strömfunktion sammanfattar atmosfären som en värme- och fuktcykel som beskriver bl. a. nederbörd, strålningsavkylning och omblandning i det planetära gränsskiktet. Cykeln visas ha tydlig koppling till El Niño-fenomenet i Stilla havet samt till
global uppvärmning. Kopplingen innebär bl.a. att atmosfärens globala cirkulation blir
långsammare men att nederbörden ökar. Denna koppling återfinns i flera olika klimatmodeller vilket indikerar att resultaten är robusta.
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