Lecture 1: Scales of motion in the ocean
Goal of the next four lectures on the
mechanical energy budget of the World
Ocean
• Lecture 1: What are the mechanical energy reservoirs in the
World Ocean? Most of the kinetic energy in the ocean is
geostrophically balanced. How do we know this?
• Lecture 2: How are the energy reservoirs maintained? What’s
the rate of forcing of the geostrophically balance flow?
• Lecture 3: Pathways to dissipation. How does energy leak from
the geostrophically balanced flow?
• Lecture 4: Relation of mechanical energy dissipation and
abyssal mixing.
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Lecture 1: Scales of motion in the ocean
Recurring themes: 1) Geostrophically
balanced flow 2) Skepticism
• Geostrophically balanced flow is of central interest.
– It is the largest reservoir.
– It is fairly clear how it is maintained.
– Unclear how it is dissipated – one of the main unsolved
problems in ocean dynamics.
• Skepticism: These are not lectures principally on the “facts” of
ocean dynamics. They are presenting and critically evaluating
a central theme of research, and critically assessing the
presentation principally by two reviews in the last 7 years
(Wunsch and Ferrari , 2004; Ferrari and Wunsch, 2009)
• Form your own opinions of what is a valuable approach to
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Lecture 1: Scales of motion in the ocean
understand the ocean.
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Lecture 1: Scales of motion in the ocean
Outline of lesson 1, Tuesday Feb 15, 2011
• The principle energy reservoirs of the World Ocean.
• How do we know this?
• What’s the value of this picture/approach?
• What’s wrong with it?
• Most of the kinetic energy in the ocean is geostrophically
balanced.
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Lecture 1: Scales of motion in the ocean
What’s valuable/wrong with reducing the
ocean to a few global statistics
• Provides the “big picture” of how the ocean works.
• The “MIT” approach is very influential in guiding research.
For example, abyssal mixing is probably the most funded
subject of research in physical oceanography.
• Important limitation is that we loose the interesting details, for
example some regions certainly exhibit large departures from
the global mean.
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Lecture 1: Scales of motion in the ocean
5-1
Lecture 1: Scales of motion in the ocean
Five main reservoirs of mechanical energy
In order of decreasing timescale:
1. General Circulation (20 × 1024 J): Long time mean.
2. Mesoscale Eddies (13 × 1018 J): Days and longer.
3. Internal Waves (1.4 × 1018 J): |f | < ω < N
4. Internal Tides (0.1 × 1018 J): Discrete frequencies mostly
semidurnal and diurnal.
5. Surface Waves and Turbulence (11 × 1018 J): Seconds to
minutes.
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Lecture 1: Scales of motion in the ocean
What do these quantities mean?
• Mostly a decomposition in terms of timescale.
• An assumption made of the corresponding physics.
• Mechanical energy has two forms – kinetic and potential. Both
are relative to an arbitrary choice of reference.
• For kinetic energy the obvious reference frame is that moving
and rotating with the Earth.
• For potential energy we need to define the zero. Very messy
complex arbitrary business.
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Lecture 1: Scales of motion in the ocean
How could we know the above statistics?
• What suitable measurements are available?
• What processing must be done, what assumptions made?
• How to form the global statistics?
• What uncertainties result in the global estimate?
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Lecture 1: Scales of motion in the ocean
Do we know the General Circulation?
• Value in (Wunsch and Ferrari , 2004; Ferrari and Wunsch,
2009) is taken from Oort et al. (1989), and is much out of date.
• What could be done now: Use hydrographic databases (e.g.
World Ocean Database 2009 http://www.nodc.noaa.gov/
OC5/SELECT/dbsearch/dbsearch.html ) contains hundreds of
thousands of CTD casts. From these one can infer the time
mean temperature and salinity fields and therefore the mass
field of the World Ocean.
• From the mass field one can infer the geostrophic velocity
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Lecture 1: Scales of motion in the ocean
10
vertical shear through the thermal wind relation:
∂ug
g
=+
∂z
f
∂vg
g
=−
∂z
f
∂ρ
∂y
∂ρ
∂x
(1)
(2)
• One must assume the velocity at a given geopotential, for
example often one assumes a “level of no motion” at one or two
thousand meters depth. Then we can integral the thermal wind
relations (1) to obtain the climatological, geostrophic velocity
everywhere.
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PLOT:
Lecture
1: Scales
of motion in the ocean
10-1
Lecture 1: Scales of motion in the ocean
Gravitational Potential energy
• Why did Wunsch and Ferrari (2004) find over one million times
as much energy in the general circulation than the eddy field?
• Also all the mechanical energy in the general circulation,
according to them, is in the form of potential energy.
• Definition of gravitational potential energy:
Z
PE = gρ(x, y, z)(z − z0 ) dx dy dz
where g is Earth’s gravitational acceleration, ρ is the time
mean density of sea water at location x, y, z and z0 is arbitrary
reference depth.
• More often people talk about Available Potential Energy, APE,
which is the gravitational potential energy minus that in a
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Lecture 1: Scales of motion in the ocean
reference state. The reference state is taken as a minimum PE
state obtained by, for example, an adiabatic rearrangement of
the water.
• One encounters awkward questions like, are we allowed to move
parcels of water over a mountain in doing the adiabatic
rearrangement? The ocean never comes close to this minimum
state, so what is the relevance of this minimum state?
• Wunsch and Ferrari (2004) avoid discussion of APE, and claim
all the PE is available!
• Seem unaware that there is an arbitrary reference level even for
gravitational potential energy.
• In my view, 20 × 1024 J in the General Circulation is a
meaningless and irrelevant statement.
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Lecture 1: Scales of motion in the ocean
How do we find the energy on different
timescales?
• Used moored current meter records.
• Moored current meter records of horizontal flow, typically with
15 min to hourly sampling, and typical durations of 6 months
to 2 years, can produce “frequency power spectra”.
• Qualitatively, this is a decomposition of the energy into
different frequency bands.
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35.29 N 167.052 E 3645m
−2
10
Lecture 1: Scales of motion in the ocean
13-1
−3
u
P (m2 s−2 cpd−1)
10
−4
10
−5
10
−6
10
with tides
without tides
−7
10
−2
10
−1
10
0
10
! (cpd)
1
10
2
10
Figure 1: KE frequency spectrum for a current meter record
from the North Pacific.
Lecture 1: Scales of motion in the ocean
14
Frequency power spectra
• More quantitatively,
u(t) =
N
X
û(ωn ) exp(−iωn t)
n=1
1
∆t
Z
0
∆t
u(t)2 dt =
N
X
û(ωn )û∗ (ωn )∆ω
n=1
• In the limit of an infinitely long, continuously sampled,
stationary record can pass to the limit of a continuous
spectrum:
Z ∞
∞
X
1
û(ωn )û∗ (ωn ) + v̂(ωn )v̂ ∗ (ωn )∆ω =
E(ω)dω
KE =
2 n=0
0
Lecture 1: Scales of motion in the ocean
15
• For real records we are limited in the maximum frequency,
minimum frequency and frequency resolution. The maximum
frequency or fastest motions we can represent by the Nyquist
frequency which is 1/2 the sampling frequency:
1 2π
2 δt
where δt is the sampling frequency, e.g. 60 minutes.
max(ω) =
• For real records we are limited in the minimum frequency or
slowest motions we can represent by the length of the record.
In principle the minimum is:
2π
∆t
where ∆t is the duration of the record, e.g. 18 months. But in
practise we want reasonable statistical uncertainty and for a
record of say, 18 months, we have only one realisation of the
18-month period motions. Thus in practise we decompose a
min(ω) =
Lecture 1: Scales of motion in the ocean
record into overlapping windows of subsets of the data, e.g. for
an 24-month record we might form 8 windows of 6-month long
subsets of the data that overlap by 3-months.
• Because almost all moored deep-water current meter records
are less than 24 months long, we have very little knowledge of
subsurface motions on timescales longer than 180 days.
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Lecture 1: Scales of motion in the ocean
Geostrophic energy
• Mostly, apparently, mesoscale eddies.
• Timescales from a few days and longer.
• From Zang and Wunsch (2001) (but that’s for the North
Hemisphere only) using a wide variety of data sources including
satellite altimetry, moored current meter records, etc. Also
from the ECCO2 model (but no reference or details given).
• How one could address this: Use gridded satellite altimetry
http://www.aviso.oceanobs.com/ to obtain the near-global
field of sea surface height on spatial scales of about 50 km and
larger, and timescales of about two weeks and longer, to
estimate the global geostrophic, time varying kinetic energy.
For the vertical structure one could assume a first baroclinic
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Lecture 1: Scales of motion in the ocean
mode structure and multiple by two to obtain the barotropic
component (Wunsch, 1997).
• One would have to check for consistency with numerical model
data, checked against available moored current meter records.
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Lecture 1: Scales of motion in the ocean
Frequency dependence of geostrophic
motions
• Almost all moorings are inadequate to describe the frequency
dependence of motions longer than a year.
• Only records we have longer than a year or two are (besides a
handful of repeat mooring deployments) the global satellite
altimetry fields.
• No one has exploited this rich data source to explore the
longer-time scale variability in the World Ocean.
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Lecture 1: Scales of motion in the ocean
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Power spectra: 55 S
2.5
u2
v2
h2
[m2/s2/cpd] or [m2/cpd]
2
1.5
1
0.5
0 −4
10
−3
10
−2
10
−1
10
cycles per day
Figure 2: Frequency spectrum of zonal velocity (black), meridional
velocity (red), and SSH (blue) variance from 17 years of satellite
altimetry. 55◦ S
Lecture 1: Scales of motion in the ocean
20-1
Power spectra: 55 N
3.5
u2
v2
h2
[m2/s2/cpd] or [m2/cpd]
3
2.5
2
1.5
1
0.5
0 −4
10
−3
10
cycles per day
Figure 3: 55◦ N
−2
10
−1
10
Lecture 1: Scales of motion in the ocean
High Latitudes
• Most SSH variability is at low frequency, spectrum still raising
at lowest frequency (17-years period).
• Velocity spectra level-off at between 1000 and 1500 days period
(3 to 5 years).
• Strong seasonal cycle in SSH in both hemispheres, but only in
velocity in the Northern Hemisphere.
• More zonal than meridional velocity variance at frequency
lower than annual cycle.
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Lecture 1: Scales of motion in the ocean
21-1
Power spectra: 25 S
1.8
u2
v2
h2
[m2/s2/cpd] or [m2/cpd]
1.6
1.4
1.2
1
0.8
0.6
0.4
0.2
0 −4
10
−3
10
cycles per day
Figure 4: 25◦ S
−2
10
−1
10
Lecture 1: Scales of motion in the ocean
21-2
Power spectra: 25 N
4.5
u2
v2
h2
[m2/s2/cpd] or [m2/cpd]
4
3.5
3
2.5
2
1.5
1
0.5
0 −4
10
−3
10
cycles per day
Figure 5: 25◦ N
−2
10
−1
10
Lecture 1: Scales of motion in the ocean
22
Subtropics
• Most SSH variability is at low frequency, spectrum still raising
at lowest frequency for SH but not NH.
• Zonal velocity spectra weakly peaks around 1000 to 2000 days
period, and appears to fall off at lower frequency.
• Meridional velocity spectra peaks around 100 days period, and
falls off rapidly at lower frequency.
• Strong seasonal cycle in SSH in both hemispheres, but only in
velocity in the Northern Hemisphere.
• More zonal than meridional velocity variance at frequency
lower than annual cycle.
Lecture 1: Scales of motion in the ocean
22-1
Power spectra: 15 S
3.5
u2
v2
h2
[m2/s2/cpd] or [m2/cpd]
3
2.5
2
1.5
1
0.5
0 −4
10
−3
10
cycles per day
Figure 6: 25◦ S
−2
10
−1
10
Lecture 1: Scales of motion in the ocean
22-2
Power spectra: 15 N
6
u2
v2
h2
[m2/s2/cpd] or [m2/cpd]
5
4
3
2
1
0 −4
10
−3
10
cycles per day
Figure 7: 25◦ N
−2
10
−1
10
Lecture 1: Scales of motion in the ocean
23
Tropics
• Strong seasonal cycle in SSH and, unlike subtropics, also for
velocities in both hemispheres.
• Otherwise Tropics are qualitatively similar to subtropics:
• Strong difference between zonal and meridional velocity spectra
(as in subtropics).
• Most SSH variability is at low frequency, spectrum still raising
at lowest frequency for SH but not NH (as in subtropics).
• Zonal velocity spectra weakly peaks around 1000 to 2000 days
period, and appears to fall off at lower frequency (as in
subtropics).
• Meridional velocity spectra strongly peaks less than 100 days
Lecture 1: Scales of motion in the ocean
period, and falls off rapidly at lower frequency (strong
difference from zonal spectra, as in subtropics).
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Lecture 1: Scales of motion in the ocean
Summary of quick look at
frequency-dependence of geostrophic flow
• SSH always has more energy at lower frequencies (a “ red
spectrum”).
• Zonal velocity spectra are red at high latitudes, but show a
hint of falling off at the lower frequencies (17-year period).
• Zonal velocity at middle and low latitudes peak and more
clearly fall-off at lower frequencies.
• There’s a slight suggestion, more clear at mid and low
latitudes, that the spectra are not white – there is perhaps a
maximum around 1000 to 1500 days, and with less energy at
lower frequencies.
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Lecture 1: Scales of motion in the ocean
• Strong difference between zonal and meridional velocity spectra
at all latitudes, but stronger contrast at lower latitudes.
• Meridonal velocities show definite peak, and clearly fall off with
lower frequency.
• Clear evidence of annual cycle, and its higher harmonics, in the
low latitudes and Northern Hemisphere high latitudes.
• We have no knowledge of how energy is exchanged between
these different timescales, though it’s easy to calculate.
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Lecture 1: Scales of motion in the ocean
Where do we have moored current meter
records?
• Most comprehensive global archive of historical moored current
meter records contains only about 6660 deep-ocean records:
http:
//stockage.univ-brest.fr/~scott/GMACMD/updates.html.
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Lecture 1: Scales of motion in the ocean
27-1
Lecture 1: Scales of motion in the ocean
Energy on inertial and faster timescales
• Energy in internal waves: 1.4 EJ (Munk , 1981).
• For inertial waves: 0.7 EJ “. . . based on N. Atlantic current
meter records”. No further details are given.
• For internal tides: 0.05–0.3 EJ Brian Arbic pers. comm. (so
based upon Arbic’s internal tide simulation, (Arbic et al.,
2004)).
• Surface waves and turbulence: 11 EJ (Lefevre and Cotton,
2001).
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Lecture 1: Scales of motion in the ocean
Myth of the universal GM spectrum
• It is so often claimed that there is a universal spectrum for the
internal gravity wave energy.
• FW09 cite (Munk , 1981). But in that study there is no
systematic comparison of current meter records and the
supposed universal GM spectrum.
• Kurt Polzin has called the GM spectrum nothing more than a
curve fit to the winter time conditions as site D.
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Lecture 1: Scales of motion in the ocean
Frequency power spectra: hours to
months
• Ferrari and Wunsch (2009) show 3 frequency spectra of
deep-ocean midlatitude (near 15◦ N, 30◦ N, and 50◦ S) spectra.
and claim these are typical.
• What studies have analyzed large numbers of moorings to
provide statistics of frequency spectra? Here’s an incomplete
sample of what I have found: (Fu, 1981; Nowlin et al., 1986;
Schmitz and Luyten, 1991; Wunsch, 1997; Alford and
Whitmont, 2007).
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Lecture 1: Scales of motion in the ocean
31
Summary
• Reservoirs of mechanical energy in the ocean can be defined
based primarily upon timescale.
• Generally the longer timescales have larger energy, with the
only exception of the fastest timescales (surface waves and
turbulence) having intermediate energy levels, and possibly the
general circulation if we focus only on KE.
• There’s an arbitrary reference level in the gravitational
potential energy, PE, that makes it meaningless to discuss the
total PE. Sometimes available potential energy is used, but
even then there is some ambiguity in defining the reference
level.
• Many of the estimates in the most recent review (FW2009) are
Lecture 1: Scales of motion in the ocean
20 to 30 years old, and thus being made prior to the explosion
of data during the WOCE decade, are grossly out-of-date.
• There might be more up-to-date studies then used by the
FW2009 and WF2004 reviews.
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Lecture 1: Scales of motion in the ocean
For Tuesday, Feb. 22
• How the energy reservoirs are maintained.
• References:
– Wind forcing of geostrophic flow (Scott and Xu, 2009, and
references therein).
– Wind forcing of the near-inertial band (Alford , 2001, 2003;
Watanabe and Hibiya, 2002; Plueddemann and Farrar ,
2006).
– Tidal forcing and conversion from barotropic to baroclinic
tides, especially in the deep ocean (Egbert and Ray, 2003;
Nycander , 2005; Jayne and St Laurent, 2001; Arbic et al.,
2004; Egbert et al., 2004).
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Lecture 1: Scales of motion in the ocean
References
Alford, M. H. (2001), Internal swell generation: the spatial
distribution of energy ux from the wind to mixed-layer
near-inertial motions, J. Phys. Oceanogr., 31, 2359–2368.
Alford, M. H. (2003), Improved global maps and 54-year history of
wind-work on ocean inertial motions, Geophys. Res. Lett., 30,
art. no. 1424.
Alford, M. H., and M. Whitmont (2007), Seasonal and spatial
variability of near-inertial kinetic energy from historical moored
velocity records, J. Phys. Oceanogr., 37 (8), 2022–2037,
doi:{10.1175/JPO3106.1}.
Arbic, B., S. Garner, R. Hallberg, and H. Simmons (2004), The
accuracy of surface elevations in forward global barotropic and
baroclinic tide models, Deep-Sea Res., 51 (25-26), 3069–3101,
34
Lecture 1: Scales of motion in the ocean
doi:{10.1016/j.dsr2.2004.09.014}, Joint Assembly of the
EGS/AGU/EUG, Nice, FRANCE, APR 06-11, 2003.
Egbert, G., and R. Ray (2003), Semi-diurnal and diurnal tidal
dissipation fromTOPEX/Poseidon altimetry, Geophys. Res.
Lett., 30 (17), doi:10.1029/2003GL017676.
Egbert, G., R. Ray, and B. Bills (2004), Numerical modeling of the
global semidiurnal tide in the present day and in the last glacial
maximum, J. Geophys. Res., 109, C03,003,
doi:doi:10.1029/2003JC001973.
Ferrari, R., and C. Wunsch (2009), Ocean Circulation Kinetic
Energy: Reservoirs, Sources, and Sinks, Ann. Rev. Fluid Mech.,
41, 253–282.
Fu, L. (1981), Observations and models of inertial waves in the
deep ocean, Rev. Geophys., 19 (1), 141–170.
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Lecture 1: Scales of motion in the ocean
Jayne, S., and L. St Laurent (2001), Parameterizing tidal
dissipation over rough topography, Geophys. Res. Lett., 28 (5),
811–814.
Lefevre, J.-M., and P. D. Cotton (2001), Ocean surface waves, in
Satellite Altimetry and Earth Sciences: A handbook of techniques
and applications, edited by L.-L. Fu and A. Cazenave, pp.
305–328, Academic press.
Munk, W. (1981), Internal waves and small-scale processes, in
Evolution of Physical Oceanography. Scientific Surveys in Honor
of Henry Stommel, edited by B. A. Warren and C. Wunsch, pp.
264–291, MIT press.
Nowlin, W. D., J. S. Bottero, and R. D. Pillsbury (1986),
Observations of internal and near-inertial oscillations at drake
passage, J. Phys. Oceanogr., 16, 87–108.
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Lecture 1: Scales of motion in the ocean
Nycander, J. (2005), Generation of internal waves in the deep
ocean by tides, J. Geophys. Res., 110 (C10028),
doi:10.1029/2004JC002,487.
Oort, A. H., S. C. Ascher, S. Levitus, and J. P. Peixóto (1989),
New estimates of the available potential energy in the world
ocean, J. Geophys. Res., 94 (C3), 3187–3200.
Plueddemann, A., and J. T. Farrar (2006), Observations and
models of the energy flux from the wind to mixed-layer inertial
currents, Deep-Sea Res. II, 53, 5–30.
Schmitz, W., and J. Luyten (1991), Spectral time scales for
midlatitude eddies, J. Mar. Res., 49 (1), 75–107.
Scott, R. B., and Y. Xu (2009), An update on the wind power
input to the surface geostrophic flow of the world ocean,
Deep-Sea Res. I, 56 (3), 295–304.
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Lecture 1: Scales of motion in the ocean
Watanabe, M., and T. Hibiya (2002), Global estimates of the
wind-induced energy flux to inertial motions in the surface mixed
layer, Geophys. Res. Lett., 29, 10.1029/2001GL014,422.
Wunsch, C. (1997), The vertical partition of oceanic horizontal
kinetic energy and the spectrum of global variability, J. Phys.
Oceanogr., 27, 1770–1794.
Wunsch, C., and R. Ferrari (2004), Vertical mixing, energy, and the
general circulation of the oceans, Ann. Rev. Fluid Mech., 36,
281–314.
Zang, X., and C. Wunsch (2001), Spectral description of
low-frequency oceanic variability, J. Phys. Oceanogr., 31,
3073–3095.
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