Homework 1-EESS 146B/246B–Due April 16

Homework 1-EESS 146B/246B–Due April 16
Please turn in the Ocean Data View exercise seperately from the problem set.
Ocean Data View Exercise
Use ODV to make figures similar to figure 1 showing profiles of the in situ density, potential
temperature, salinity, potential density, and the Brunt Vaisala (a.k.a buoyancy) frequency
from the WOCE CTD data set, which can be downloaded at
http://www.ewoce.org/data/index.html#WHP CTD Data
Plot profiles from the following latitude ranges: the tropics: 0 < |latitude| < 20, subtropics:
20 < |latitude| < 40, subpolar regions: 40 < |latitude| < 60, and the polar
regions: 60 < |latitude|. Make eight figures, i.e. two profiles for each region.
For each figure,
• Identify the pycnocline and a mixed layer if present. If a mixed layer is present
estimate its depth.
• Calculate the change in salinity ∆S, potential temperature ∆θ, and potential density
∆σ across the pycnocline. Estimate the change in density across the pycnocline associated
with the salinity, ∆σS = ρrefβS∆S, and the temperature, ∆σθ = −ρrefαT ∆θ.
Recall that while βS varies very little, the thermal expansion coefficient is a strong
function of temperature. Use the values of αT (θ) listed in the table 1 to estimate
∆σθ.
Compare the near surface values of the salinity and potential temperature and the strength
of the stratification in the pycnocline for the four regions. Comment on the relative contributions
of salinity and temperature in setting the change in density across the pycnocline
in the low versus high latitudes.
Problem Set
1. Equation of state of seawater. Figure 2 shows profiles of density, temperature,
and salinity that were made at two stations (labeled A and B) in the Gulf Stream. At
station A, the potential temperature and salinity on the σθ = 26.27 kg m −3 density
surface is θ = 19.0 ◦ C and S = 36.7 psu, respectively. Use the profiles to determine
the depth where the σθ = 26.27 kg m −3 density surface sits at station B. What is
the salinity on the σθ = 26.27 kg m −3 density surface at this station? Given this
information, estimate the potential temperature at station B on this density surface?
Assume an equation of state of the form: σ = σo + ρref(−αT [θ − θo] + βS[S − So]),
where αT = 2.3 × 10 −4 ◦ C −1 and βS = 7.5 × 10 −4 psu −1 . Given the difference in
temperature and salinity of the water on the σθ = 26.27 kg m −3 isopycnal at stations
A and B, did the water at station B originate north or south of the water at station
A? State the reasoning for your answer.
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2. Buoyancy and stratification. Imagine you were to push a parcel of water from
the σθ = 26.27 density surface at station A down 50 m. Using the observations
shown in figure 2, estimate the force (per unit mass) that you would need to exert
to keep the parcel at that depth.
3. Heat contained in the ocean. Hurricanes get their energy from the heat contained
in the tropical oceans. A typical hurricane extracts heat from the ocean at a rate of
5 × 10 19 Joules day −1 . Assuming that a hurricane sits over an area of the ocean of
1 × 10 12 m 2 for a period of two days, how much will the mixed layer deepen by this
heat extraction if during deepening the profiles of temperature can be modeled as
T1 =
To
To + dT
dz
T o
T2 =
z > −H
,
(z + H) z < −H
z > −D
,
(z + D) z < −D
T o + dT
dz
where T1 and T2 are the temperature profiles before and after the passage of the
hurricane, T o = To + dT
dz (H − D), and D > H. Assume that T1 = 28 C, H = 50 m,
and dT/dz = 0.05 C m−1 for your calculation. By how much does the sea surface
temperature drop after the passage of the hurricane?
4. Physical Interpretation of the Coriolis force. A ball is at rest on the Earth’s
approximately ellipsoidal surface (figure 3) in the Northern Hemisphere. Viewed
from space the ball is spinning about the Earth’s axis at the angular velocity Ω of
the Earth.
• Describe the force balance on the ball from the perspective of a stationary
observer looking at the ball from space.
• If the ball were given a kick to the east, how would the force balance (again
as viewed from the stationary frame) change? What does this change in force
balance imply about the north-south movement of the ball?
• If the ball had instead been given a kick to the north, describe how conservation
of angular momentum would affect the motion of the ball.
• Reanswer the questions above for the case when the ball is in the Southern
Hemisphere.
• Using your answers to these questions, describe the physics behind the Coriolis
force on the rotating Earth.
Assume that the ball stays on the surface of the Earth and that friction is not
important.
5. Geostrophic balance. Figure 4 shows the trajectory of a “champion” surface
drifter, which made one and a half loops around Antarctica between March, 1995,
and March, 2000 (courtesy of Nikolai Maximenko). Red dots mark the position of
the float at 30 day intervals.
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• Compute the mean speed of the drifter over the 5 years.
• Assuming that this near surface mean current is in geostrophic balance, given
this value of the speed estimate the tilt in the sea surface that must accompany
the flow. Using this estimate, how large is the change in sea surface height (SSH)
across the 600 km-wide Drake Passage? How does your estimate compare to
the observed drop in SSH seen in the satellite altimetry data (e.g. figure 9.19
in the textbook)?
• Assuming that the mean zonal current at the bottom of the ocean is zero, use the
thermal wind relation to compute the depth-averaged density gradient across
the Antarctic Circumpolar Current, then estimate the mean density contrast
across the Drake Passage. Using ODV and the WOCE data set, compare this
estimate to the observed density contrast across the Drake Passage.
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Figure 1: Profiles of in situ density, potential temperature, salinity, potential density,
and the buoyancy (Brunt-Vaisala) frequency from one station from the WOCE
CTD data, plotted using Ocean Data View. An xview file that can be used
to make a figure similar to this can be downloaded from the class website at:
http://pangea.stanford.edu/courses/EESS146Bweb/Homework 1.xview
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Figure 2: Profiles of potential density (left) , salinity (middle), and potential temperature
(right) at stations A (gray curves) and B (black curves) in the Gulf Stream. The vertical
location, salinity, and temperature on the σθ = 26.27 kg m −3 density surface at station A
is indicated by the asterisk.
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Figure 3: A ball is confined to move on the approximately ellipsoidal surface of the Earth.
The Earth rotates about its axis at an angular velocity Ω.
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Figure 4: The trajectory of a surface drifter which made one and a half loops around
Antarctica between March, 1995, and March, 2000 (courtesy of Nikolai Maximenko). Red
dots mark the position of the float in 30 day intervals. The float is moving in a clockwise
direction.
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Table 1: The temperature dependence of the thermal expansion coefficient αT .
θ ( ◦ C) αT × 1 × 10 4 ( ◦ C −1 )
-1.5 0.4
1.5 0.8
4.5 1.1
7.5 1.4
10.5 1.7
13.5 2.0
16.5 2.3
19.5 2.6
22.5 2.8
25.5 3.0
28.5 3.2
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