Nansen`s Fram expedition and the theory of the Ekman layer

Nansen’s Fram expedition
and the theory of the Ekman layer
by Karima Khusnutdinova
Department of Mathematical Sciences, Loughborough University, UK
Mathematical Reviews Seminar, 15 May 2012
*****
“If nature were not beautiful, it would not be worth studying it.
And life would not be worth living.”
Henry Poincaré
“Everything should be made as simple as possible, but not simpler.”
Albert Einstein
(Epigraphs to “Fluid Mechanics” by Kundu and Cohen.)
Introduction
Fram (“forward” from Norwegian, built in 1892) is the most famous ship
in the history of polar exploration.
“The story of the Fram is a modern Norse saga, a story of unimaginable
hardship and intelligent striving ... The prime mover behind the Fram,
the brilliant and moody scientist-explorer who commissioned its
construction and led its insanely dangerous maiden voyage into the polar
mists, remains a national patriarch. His name is Fridtjof Nansen, and
although today he is not as well-known outside Norway as other marquee
polar adventurers – Peary, Scott, and Amundsen – he should be. For
Nansen was quite simply the father of modern polar exploration; all
others were, in a very real sense, his acolytes.”
(“1,000 Days in the Ice” by Hampton Sides, National Geographic,
January 2009)
Overview
I
Fridtjof Nansen (10 October 1861 - 13 May 1930): a Norwegian
academic, adventurer, polar explorer, humanitarian (Nobel Peace
Prize 1922).
I
The great adventure: Nansen’s Fram expedition (1893 - 1896).
I
Vagn Walfrid Ekman (3 May 1874 - 9 March 1954): a Swedish
oceanographer (student of Bjerknes).
I
Nansen’s observation and the Ekman spiral.
I
I
I
Response of the upper ocean to winds.
Thin layer on a rotating sphere: f -plane model.
Ekman layer at a free surface.
I
Remarks.
I
References.
1. Fridtjof Nansen (10 October 1861 - 13 May 1930)
LINK : http : //en.wikipedia.org /wiki/Fridtjof− Nansen
1. Fridtjof Nansen (10 October 1861 - 13 May 1930)
“Nansen was a strapping blond man, fair complected, with a frosty stare
and a truculent face that seemed slightly at odds with the refinements of
his intellect. Nansen stood apart from the quixotic glory hounds who
characterised much of polar exploration’s golden age. Call him a
Renaissance Viking: He was a gifted writer, a sought-after lecturer, a
first-rate zoologist, and a prominent statesman. Fluent in at least five
languages, adroit with a camera, he made beautiful maps and
illustrations, kept up a voluminous scientific correspondence, and brought
an element of cerebral precision to all his explorations. A contemporary
German scientist said of Nansen that he “knew how to handle the
microscope as well as the ice axe and skis,” and his scientific
achievements were notable, including a groundbreaking paper on the
nature of the central nervous system.”
(“1,000 Days in the Ice” by Hampton Sides, National Geographic,
January 2009)
2. The great adventure: Fram expedition (1893 - 1896)
I
1879 American ship U.S.S. Jeannette became locked in the ice and
drifted in the Arctic for 21 months, before being crashed by the ice
in 1881. More than half of the 33 crew members have died.
Artifacts were found washed up on the coast of Greenland.
I
This led to a belief that a current over the Arctic could be used to
get to the North Pole – or close to it. Idea: to build a strong ship
which will become locked in the ice and will withstand the pressure.
I
The ship was built by Colin Archer, a naval architect of Scottish
descent. The vessel had a rounded hull, was well insulated, had a
windmill to produce electricity, and a good library.
I
The ship was called Fram and it left Oslo in the summer of 1893
with a crew of 13 and food provisions for 5 years. It became locked
in the ice in September and started drifting. However, it became
clear that Fram would not reach the Pole.
2. The great adventure: Fram expedition (1893 - 1896)
I
In March 1895 Nansen and one companion, Hjalmar Johansen, left
the Fram and headed North on the skies, with three sleds, two
kayaks, and 28 dogs.
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By April they had traveled as far north as they could – 860 140 N.
They were still 226 miles shy of the Pole, but they had ventured
farther north than any human ever had.
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Then Nansen turned the expedition around. They aimed for the
archipelago of Franz Josef Land, some 600 miles to the south.
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“For a quarter of a year we have been wandering in this desert of
ice... I lie awake at night by the hour racking my brain to find a way
out of our difficulties.” (Nansen)
2. The great adventure: Fram expedition (1893 - 1896)
I
In August they reached an island. Hunting, they regained their
strength. They built an improvised lair, where they stayed for the
next 9 months.
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”We didn’t quarrel...The only thing was that I have a bad habit of
snoring ... and Nansen used to kick me in the back.” (Johansen)
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When the spring came, they continued their journey. On June 17,
1896 Nansen suddenly heard a dog barking. ”Suddenly I thought I
heard a shout from a human ... How my heart beat, and the blood
rushed to my brain ... I hallooed with all the strength of my lungs.”
(Nansen)
I
Nansen’s rescuer was a British explorer Frederick George Jackson.
Jackson had sailed his ship, Windward, to Franz Josef Land. He was
not looking for Nansen, but he knew that Nansen might be there.
I
Conincidentally, the Fram has returned safely in the same month
(August of 1896).
2. The great adventure: Fram expedition (1893 - 1896)
Fram (left), Jackson meets Nansen (right, the photo is taken after
the actual meeting):
LINK : http : //en.wikipedia.org /wiki/Nansen0 s− Fram− expedition
2. The great adventure: Fram expedition (1893 - 1896)
Theoretical (left) and actual (right) routes of the expedition:
70
80
70
LINK : http : //en.wikipedia.org /wiki/Nansen0 s− Fram− expedition
3. Vagn Walfrid Ekman (3 May 1874 - 9 March 1954)
Ekman (left) and Bjerknes (his supervisor, right):
LINK : http : //en.wikipedia.org /wiki/Vagn− Walfrid− Ekman
3. Vagn Walfrid Ekman (3 May 1874 - 9 March 1954)
I
Ekman became committed to oceanography while studying physics
at the University of Uppsala and, in particular, on hearing Bjerknes
lectures on fluid dynamics.
I
On completing his doctorate in Uppsala in 1902, Ekman joined the
International Laboratory for Oceanographic Research, Oslo.
I
From 1910 to 1939 he continued his theoretical and experimental
work at the University of Lund, as professor of mechanics and
mathematical physics.
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Ekman was elected a member of the Royal Swedish Academy of
Sciences in 1935.
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He was a gifted amateur bass singer, pianist, and composer.
4. Nansen’s observation and the Ekman layer
Introduction
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Fridtjof Nansen observed that ice drifts at an angle of 200 − 400 to
the right of the prevailing wind direction while on an Arctic
expedition aboard the Fram.
I
Nansen asked Bjerknes to study the problem. Bjerknes encouraged
Ekman (his student), who presented his results in 1902 as his
doctoral thesis.
I
The Ekman layer is a frictional layer near the free surface of the
ocean, which is acted on by a wind stress τ . The flow is the result of
a balance between pressure gradient, Coriolis and viscous forces.
4.1 Response of the upper ocean to winds
Ekman spiral:
LINK : http : //en.wikipedia.org /wiki/Ekman− layer
4.1 Thin layer on a rotating sphere: f -plane model
Navier-Stokes equations in a reference frame rotating at an angular
velocity Ω with respect to the “fixed stars”:
∇p
ρ
Du
=−
+ g + ν∇2 u − 2Ω × u.
Dt
ρ0
ρ0
Here, u is the velocity vector, p is pressure, ρ is density, ν is viscosity,
∂
D
=
+u·∇
Dt
∂t
is the material derivative, g is an effective gravitational force (it includes
the centrifugal force), −2Ω × u is the Coriolis force.
Continuity equation:
1 Dρ
+ ∇ · u = 0.
ρ Dt
In the Boussinesq approximation (incompressibility condition):
Dρ
=0
Dt
and ∇ · u = 0.
4.1 Thin layer on a rotating sphere: f -plane model
Approximations:
The ocean is a thin layer in which the depth scale of flow is a few km,
while the horizontal scale is of order 100 − 1000km.
Let u = (u, v , w ), and u, v ∼ U, w ∼ W . Let H be the depth scale and
L the horizontal scale.
From continuity equation
u x + vy + w z = 0
=⇒
W ∼
UH
,
L
where H is the depth scale and L is the horizontal scale. Thus, for such
flows, W U, V .
Large-scale geophysical flows are solved using spherical coordinates.
If, however, L R (= 6371km), the motion can be studied in a local
Cartesian system on a tangent plane.
4.1 Thin layer on a rotating sphere: f -plane model
Local Cartesian system on a tangent plane (φ is the latitude):
x- eastward, y - northward, z - upward
rad
Ω = 2π day
= 0.73 × 10−4 s −1 - angular velocity of the Earth.
Components of the angular velocity in the local system:
Ωx = 0,
Ωy = Ω cos φ,
Ωz = Ω sin φ.
4.1 Thin layer on a rotating sphere: f -plane model
Coriolis force:
−2Ω × u
i
= 0
u
j
2Ω cos φ
v
k
2Ω sin φ
w
= −2Ω[(w cos φ − v sin φ)i + u sin φj − u cos φk].
Approximation:
w cos φ v sin φ (w v ).
Then,
−2Ω × u = fv i − fu j + (2Ω cos φ)u k,
where f = 2Ω sin φ is the Coriolis parameter.
f > 0 in N hemisphere, and f < 0 in S hemisphere.
f = ±1.45 × 10−4 at the poles, zero at the equator.
Next approximation: 2Ω cos φ u g ρ pz
ρ0 , ρ0 .
4.1 Thin layer on a rotating sphere: f -plane model
Equations of motion:
1
px + νH (uxx + uyy ) + νV uzz ,
ρ0
1
vt + uvx + vvy + wvz + fu = − py + νH (vxx + vyy ) + νV vzz ,
ρ0
1
ρ
wt + uwx + vwy + wwz = − pz − g
+ νH (wxx + wyy ) + νV wzz ,
ρ0
ρ0
ρt + uρx + v ρy + w ρz = 0,
ut + uux + vuy + wuz − fv = −
ux + vy + wz = 0.
f -plane model: f = f0 = 2Ω sin φ0 = const.
β-plane model: f = f0 + βy , where β =
R = 6371km - radius of the Earth.
h
df
dy
i
=
φ0
h
df dφ
dφ dy
i
φ0
=
2Ω cos φ0
.
R
4.1 Thin layer on a rotating sphere: f -plane model
Rossby number:
Ro =
Nonlin. acceleration
U 2 /L
U
∼
= .
Coriolis force
fU
fL
Reynolds number:
Re =
UL
Inertia force
∼
.
Viscous force
ν
Ekman number:
E=
Viscous force
ρνU/L2
ν
∼
= 2.
Coriolis force
ρfU
fL
Large-scale quasi-steady motions away from boundaries are described by
the geostrophic balance:
−fv = −
1
px ,
ρ0
fu = −
1
py .
ρ0
In such a flow (ui + v j) · ∇p = 0, i.e. the horizontal velocity is along the
lines of constant pressure.
4.2 Ekman layer at a free surface (Ekman,1905)
This is a neutrally stable frictional boundary layer near the free surface of
the ocean, which is acted on by a wind stress τ in the horizontal
direction. We look for a steady solution.
Problem formulation (for u and v ):
d 2u
px
+ νv 2 ,
ρ0
dz
d 2v
py
fu = − + νv 2 ,
ρ0
dz
du
ρ0 νv
= τ x at z = 0,
dz
dv
ρ0 νv
= τ y at z = 0,
dz
u, v → ug , vg as z → −∞.
−fv = −
where ug , vg is the geostrophic flow. Here, ρ0 = const.
4.2 Ekman layer at a free surface
Solution:
u + iv = ug + ivg +
τ x + iτ y z +i( z − π )
√
eδ δ 4 .
ρ f νv
For f > 0 (in N hemisphere):
u
v
1
z
π
z
π
= ug + √
e z/δ [τ x cos( − ) − τ y sin( − )],
δ
4
δ
4
ρ f νv
1
z
π
z
π
= vg + √
e z/δ [τ x sin( − ) + τ y cos( − )].
δ
4
δ
4
ρ f νv
The surface velocity if deflected 450 to the right (in N hemisphere) of
the applied wind stress. The velocity vector rotates with depth, forming
the Ekman spiral and the magnitude exponentially decays with an
e-folding scale of δ, which is called the thickness of the Ekman layer:
r
2νv
δ=
.
f
5. Remarks
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Similar frictional layers: the boundary layer over the ocean bottom
and the atmospheric boundary layer over the solid Earth (Ekman
layer on a rigid surface).
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Pure Ekman spirals are not observed, mainly because of the
assumptions of constant viscosity and steadiness. The theory
accurately describes the flow averaged over many days (http :
//oceanworld.tamu.edu/resources/ocng− textbook/chapter 09/)
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This is an important theoretical framework, which allows one to
explain the observed phenomena, and to understand the important
consequences. Ekman transport plays a fundamental role in the
theory of wind-driven ocean circulation.
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Related phenomenon: tea leaf paradox (solved by Einstein in 1926)
Link : http : //en.wikipedia.org /wiki/Tea− leaf− paradox
6. References
1. “1,000 Days in the Ice” by Hampton Sides, National Geographic,
January 2009.
2. P.K. Kundu, I.M. Cohen, Fluid Mechanics, Academic Press, 3ed.,
2004.
3. A.E. Gill, Atmosphere - Ocean Dynamics, Academic Press, 1982.
4. http : //en.wikipedia.org /wiki/Fridtjof− Nansen
5. http : //en.wikipedia.org /wiki/Nansen0 s− Fram− expedition
6. http : //en.wikipedia.org /wiki/Vagn− Walfrid− Ekman
7. http : //en.wikipedia.org /wiki/Ekman− layer
8. http : //en.wikipedia.org /wiki/Tea− leaf− paradox
9. http : //oceanworld.tamu.edu/resources/ocng− textbook/chapter 09
One more coincidence:
The photo of Martin D. Kruskal
(LINK : http : //en.wikipedia.org /wiki/Martin− David− Kruskal )
“I am most grateful to ... Professor Martin D. Kruskal, whose use of
mathematics to solve difficult physical problems was developed to a high
art form and reminds me of a Vivaldi trumpet concerto. His codification
of rules of applied limit processes into the principles of “Asymptotology”
remains with me today as a way to view problems.”
Ira M. Cohen, In the Preface to Second Edition of “Fluid Mechanics” by
Kundu and Cohen.