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<h2 class="hd hd-2 unit-title">Introduction to oceanic circulation</h2>
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<h2>Oceans and climate</h2>
<p>While many discussions on the Earth's climate focus on the atmosphere, the ocean also plays a very important role in the climate system. A vast, massive, and moving fluid, the ocean absorbs and transports heat and carbon, and exhibits an important coupling with the atmosphere. This module will describe the mechanisms by which the ocean interacts with the atmosphere, and how it affects the coupled system on time scales of climate.</p>
<p><i>Instructor: Kerry Emanuel</i></p>
<h2>Module outline</h2>
<p>In this module, we will see an overview of the general circulation of the ocean, its interactions with the atmosphere, and its overall role in the climate system. Specifically, the module covers the following topics:</p>
<ul>
<li>Dynamical differences and similarities between ocean and atmosphere</li>
<li>The importance of the ocean as a thermal reservoir</li>
<li>Description of global ocean surface currents and the mechanisms by which they arise</li>
<li>Deep ocean currents and their importance to climate</li>
<li>The equations of ocean motion</li>
<li>Physical description of atmosphere-ocean coupling</li>
<li>Observed and projected roles of oceans in global climate change</li>
</ul>
<h2>Supplemental reading</h2>
<p>The circulation and dynamics of the ocean is another expansive topic about which entire semesters' worth of material could be taught. Again, since we are only interested in understanding roles of the ocean and its circulation in the global climate system, this week's material will serve as more of an overview of ocean dynamics than an detailed exploration of this subject. Students interested in delving into the rigorous particulars of ocean dynamics are encouraged to read the notes from <a href="http://ocw.mit.edu/resources/res-12-000-evolution-of-physical-oceanography-spring-2007/index.htm" target="_blank">MIT 12.800 on OCW</a>, in particular <a href="http://ocw.mit.edu/resources/res-12-000-evolution-of-physical-oceanography-spring-2007/part-1/wunsch_chapter5.pdf" target="_blank">Chapter 5</a>.</p>
<h2>Lecture Slides</h2>
<p>A PDF of this week's lecture slides can be found <a href="/assets/courseware/v1/8d04739431cf2c25652f939ef84417e8/asset-v1:MITx+12.340x+1T2020+type@asset+block/Oceans_Climate.pdf" target="[object Object]">here</a>.</p>
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<h2 class="hd hd-2 unit-title">Importance of the ocean</h2>
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<p><b><a href="http://svs.gsfc.nasa.gov/vis/a000000/a003800/a003827/prepetual_ocean_1080p30.mp4" target="_blank">Link to "Perpetual Ocean" GCM video</a></b></p>
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Inertia of ocean and atmospheric currents
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<p>Which has more momentum per unit volume: an ocean current with speed 1 cm/s, or an atmospheric wind near the surface of speed 10 m/s? </p>
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<text>Ocean current</text>
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<text>Atmospheric wind</text>
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<text>Roughly the same</text>
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Adiabatic flows in the ocean
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<p>If parcels of ocean water do not exchange energy with their environment, they will roughly conserve their temperature. Which of the following statements about the ocean circulation is true? </p>
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<text>In the deep ocean, water parcels rapidly change their temperature as they move from the polar regions towards the tropics</text>
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<text>Deep ocean flows are approximately adiabatic</text>
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Wind stress curl
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<p>Given the meridional profile of wind pictured in the figure above, what is the direction of the vertically integrated meridional transport by the ocean at the points A and B according to the Sverdrup relation? </p>
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Ocean and land heat content
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<p>Which component of the climate system has sequestered the most energy during the past half century of climate change?</p>
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<text>The upper mixed layer of the ocean (top 100 meters)</text>
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<text>The ice sheets</text>
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<text>The terrestrial biosphere</text>
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<text>The deep ocean</text>
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<h2 class="hd hd-2 unit-title">Problem Set: Oceans and climate</h2>
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<h2>The Ocean and its circulation</h2><p>This problem set is aimed at testing your understanding of the material presented in the videos of this sequence. You should now be familiar with the basic structure of the ocean surface currents. You should be familiar with gyres and western boundary currents, and the concept of Sverdrup balance. You should also have some knowledge of the oceans deep circulation, including where water tends to sink, and where it upwells. This problem set contains two problems, each of which has multiple parts.</p><p>Problems may ask you to choose the correct answer between a list of alternatives, or may ask you to enter an answer in directly. Be sure to follow the instructions for each problem carefully to ensure you receive full credit for your responses.</p><p>In general you will be allowed two attempts at any multiple choice questions, and three attempts for symbolic or numerical responses, but this will be clearly indicated to you on each problem. Many problems have parts that are dependent on previous answers, and you are encouraged to check that these are correct before you proceed.</p>
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<h2 class="hd hd-2 unit-title">Problem 1: Timescales of the ocean</h2>
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<h2>Problem 1: Timescales of the ocean</h2><p>In this problem we consider the different timescales of the ocean, and how they affect the planet's response to radiative perturbations.</p>
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1a
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<p>In this problem we consider the time scale over which the upper region of the ocean, sometimes known as the "mixed layer", responds to radiative perturbations. The mixed layer is the region of the ocean that is affected by mechanical mixing driven by the atmospheric winds, as well as potentially convective mixing driven by cooling of the surface over the diurnal or seasonal cycle. A typical depth of the mixed layer is around 50 meters. </p>
<p>Here we construct a very simple model of how the mixed layer temperature evolves toward a steady state after being perturbed. We assume the layer is mixed sufficiently rapidly that it is effectively homogenized in temperature. We denote the mixed layer temperature [mathjaxinline]T_O[/mathjaxinline]. We consider the simple case in which the only mechanism for the ocean to lose energy is via radiation. The ocean radiates upward as a black body, so that the equation for the ocean temperature may be written,
[mathjax]
\rho c_w H \frac{dT_O}{dt} = F_{in} - \sigma T_O^4.
[/mathjax]
Here [mathjaxinline]c_w = 4.200 \times 10^3 [/mathjaxinline] J/kg/K is the specific heat capacity of water, [mathjaxinline]\rho = 1.000 \times 10^3 [/mathjaxinline] kg/m<sup>3</sup> is the density of water and [mathjaxinline]H[/mathjaxinline] is the depth of the mixed layer in meters. [mathjaxinline]F_{in}[/mathjaxinline] is the downward radiation (both long- and short-wave) incident on the ocean, measured in Watts per meter squared.</p>
<p>We may define the equilibrium temperature [mathjaxinline]T_{eq}[/mathjaxinline] as the temperature that the mixed layer would need to have to be in energy balance. That is, we define [mathjaxinline]T_{eq}[/mathjaxinline] such that,
[mathjax]
\sigma T_{eq}^4 = F_{in}.
[/mathjax]
we may then write the governing equation for the mixed layer temperature as,
[mathjax]
\rho c_w H \frac{dT_O}{dt} = \sigma T_{eq}^4 - \sigma T_O^4.
[/mathjax]
If we define [mathjaxinline]\Delta T = T_O - T_{eq}[/mathjaxinline], and assume [mathjaxinline]T_{eq}[/mathjaxinline] does not vary with time, we may write,
[mathjax]
\rho c_w H \frac{d \Delta T}{dt} = \sigma T_{eq}^4 - \sigma \left(T_{eq}+\Delta T\right)^4.
[/mathjax]
If the temperature is not too far from its equilibrium value, we may expand this using a <a href="https://www.khanacademy.org/math/calculus/sequences_series_approx_calc/maclaurin_taylor/v/maclauren-and-taylor-series-intuition" target="_blank">Taylor series</a> so that,
[mathjax]
\rho c_w H \frac{d\Delta T}{dt} = -\lambda(\Delta T)
[/mathjax]
with [mathjaxinline]\lambda = 4\sigma T_{eq}^3[/mathjaxinline]. This last step uses a little applied mathematics where we approximate the 4th power dependence of radiation with a straight line. This is valid if the temperature difference [mathjaxinline]\Delta T[/mathjaxinline] is small compared to [mathjaxinline]T_{eq}[/mathjaxinline]; for Earthlike temperatures, this works pretty well even for values of [mathjaxinline]\Delta T[/mathjaxinline] as large as 10<sup>o</sup>C.</p>
<p> We can think of this equation as governing the approach to equilibrium of the ocean mixed layer after a temperature perturbation of [mathjaxinline]\Delta T[/mathjaxinline]. For an initial temperature perturbation [mathjaxinline]\Delta T_i[/mathjaxinline], estimate the time it would take for the temperature of the ocean mixed layer to get half-way to its equilibrium value. That is, esimate the time at which [mathjaxinline]\Delta T /\Delta T_i = 1/2[/mathjaxinline]. Express your answer in days.</p>
<p>You may assume the mixed-layer depth is 50.00 meters, and that [mathjaxinline]T_{eq} = 290.0 [/mathjaxinline] K.</p>
<p>Enter your answer below as a number of days. <b>Do not include units in your answer.</b></p>
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<p>As we saw in the videos, the sub-tropical ocean surface currents are made up of gyres, moving water slowly equatorward in the open ocean, and more rapidly poleward near the western boundary of the basin. Ocean water tends to move toward the center of these gyres where it downwells. The <a href="http://en.wikipedia.org/wiki/Great_Pacific_garbage_patch" target="_blank">Great Pacific Garbage Patch</a> is a result of ocean surface currents converging at the center of the gyre in the North Pacific ocean, and bringing debris with them.</p>
<p>The flow speed of currents in the open ocean is relatively slow, between 1 and 10 cm/s. Suppose an ocean parcel moved around the North Pacific gyre at a roughly constant speed of 3 cm/s. How long would it take the parcel to move around the Pacific gyre once? Idealize the gyre as circular, with a radius of 3000 km. Express your answer in years.</p>
<p>Enter your answer below as a number of years. <b>Do not include units in your answer.</b></p>
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1c
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<p>The <b>thermohaline circulation</b> is a deep ocean circulation that is driven by density contrasts that result from variations in the temperature and salinity of sea water. It is often idealized as following a path in which dense water sinks in certain regions in the high Northern latitudes, then travels at depth Southward to the Southern ocean, where it upwells. In reality, the ocean is much more complex than this, but we can use this simplified picture to understand the timescale of the deep ocean circulation.</p>
<p> Suppose the deep water in the ocean travels at a speed of roughly 1 mm/s. Estimate how long it would take deep water sinking just South of the coast of Greenland to upwell near Antarctica. You may assume the water takes the most direct route. Express your answer in years.</p>
<p>Enter your answer below as a number of years. <b>Do not include units in your answer.</b></p>
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<h2 class="hd hd-2 unit-title">Problem 2: Sverdrup balance</h2>
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<h2>Problem 2: Sverdrup balance</h2>
In this problem we examine Sverdrup balance in a closed ocean basin. Sverdrup balance may be formulated in many ways. In the lecture, we derived it through the <i>vorticity</i> balance of the flow. Regardless of how it is formulated, one must remember that the balance of terms Sverdrup elucidated ignores certain processes that are not thought to be important in the open ocean. But there are circumstances in which these assumptions break down, as we shall see below.
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2a
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<p>Consider the case of a closed rectangular ocean basin extending from [mathjaxinline]x = 0[/mathjaxinline] to [mathjaxinline]x = L_x[/mathjaxinline] and [mathjaxinline]y = 0[/mathjaxinline] to [mathjaxinline]y = L_y[/mathjaxinline], and having a depth [mathjaxinline]H[/mathjaxinline] . The atmospheric zonal wind profile [mathjaxinline]u_a[/mathjaxinline] that exists above this ocean basin is zonally uniform (does not depend on [mathjaxinline]x[/mathjaxinline]) and given by,
[mathjax]
u_a = \frac{U_0}{2}\left(1 - \cos\left(\frac{2\pi y}{L_y}\right)\right).
[/mathjax]
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<p>Sverdrup balance relates the depth integrated ocean circulation to the curl of the wind stress. We assume the wind stress is proportional to the wind speed, and in the same direction, i.e.,
[mathjax]
\mathbf{\tau_s} = k\mathbf{u_a},
[/mathjax]
where [mathjaxinline]\mathbf{u_a}[/mathjaxinline] is the vector wind near the surface, and [mathjaxinline]k[/mathjaxinline] is a constant. We then may write the Sverdrup relation as,
[mathjax]
\rho \beta \int_0^H v_O dz = -k \frac{\partial u_a }{\partial y},
[/mathjax]
where [mathjaxinline]v_O[/mathjaxinline] is the meridional velocity of the ocean, and [mathjaxinline]\rho[/mathjaxinline] is the density of water, roughly 1.0 x 10<sup>3</sup> kg/m<sup>3</sup>. Assuming the ocean current is constant with depth, find an expression for the velocity of the meridional current in the ocean basin using Sverdrup balance.</p>
<p>Express your answer in terms of [mathjaxinline]\beta[/mathjaxinline] ("beta"), [mathjaxinline]U_0[/mathjaxinline] ("U_0"), [mathjaxinline]k[/mathjaxinline] ("k"), [mathjaxinline]H[/mathjaxinline] ("H"), [mathjaxinline]L_y[/mathjaxinline] ("L_y"), [mathjaxinline]\rho[/mathjaxinline] ("rho"), the co-ordinates [mathjaxinline]x[/mathjaxinline] ("x") and [mathjaxinline]y[/mathjaxinline] ("y"), and the constant [mathjaxinline]\pi[/mathjaxinline] ("pi"). Explicitly indicate any mathematical operations with the appropriate symbol (e.g. '+', '-', '*', '/', '^'). Trigonometric functions of an argument [mathjaxinline]A[/mathjaxinline] can be entered as "sin(A)", "cos(A)", "tan(A)", "csc(A)", "sec(A)", and "cot(A)".</p>
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<span>[mathjaxinline]v_O [/mathjaxinline] =</span>
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2b
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<p>For an ocean depth of 2.0 km and a basin extending meridionally 45 degrees of latitude, calculate the <b>magnitude</b> of the maximum northward velocity of the ocean current according to the solution of the previous problem.</p>
<p>You may assume
[mathjax]
U_0 = 10.0 \mbox{ m/s}, \\
\beta = 1.2\times 10^{-11} \mbox{ m}^{-1}\mbox{s}^{-1},\\
k = 0.010 \mbox{ kg/(m}^2\cdot\mbox{s)}.
[/mathjax]
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<p>Remember that you may convert degrees of latitude into distances using the radius of the Earth. Express your answer in meters per second.</p>
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2c
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<p>The Sverdrup solution for a closed basin with the wind stress formulation given above predicts a transport that is in the same direction at all longitudes (all values of [mathjaxinline]x[/mathjaxinline]). But this violates conservation of mass; one cannot have the entire ocean moving northwards at a given latitude! </p>
<p>Which of the following is the fundamental reason of this failure of the Sverdrup relation.</p>
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<text>The atmospheric wind is never perfectly zonally symmetric.</text>
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<text>We have neglected friction when deriving the Sverdrup relation.</text>
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<text>We have treated the ocean as an incompressible fluid, but in reality it has some compressibility.</text>
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<text>We have neglected variations in the gradient in the Corioilis parameter ([mathjaxinline]\beta[/mathjaxinline]).</text>
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2d
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<p>Henry Stommel showed that the return circulation required to allow the Sverdrup solution to conserve mass occurs in a boundary current on the <b>western edge</b> of ocean basins. Suppose this boundary current is roughly 2.0 x 10<sup>2</sup> km wide, in the same order of magnitude as the width of the Gulf Stream and Kuroshio current on Earth. If the ocean basin has a width of 1.0 x 10<sup>4</sup> km, estimate the <b>magnitude</b> (absolute value) of the maximum meridional velocity of this western boundary current, in m/s. You may assume the boundary current contains all the return flow required for mass conservation of the Sverdrup solution from part B, and that the magnitude of the boundary current does not depend on depth.</p>
<p>Enter your answer below as a number of meters per second. <b>Do not include units in your answer.</b></p>
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2e
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<p>Which of the above diagrams showing streamlines of the ocean currents best represents the solution calculated in the previous parts of the problem (that is, the Sverdrup solution with Stommel's western boundary current included)?</p>
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<text>A</text>
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<text>B</text>
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<text>C</text>
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<text>D</text>
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