In the Atlantic Ocean off the coast of Newfoundland, Canada,...

In the Atlantic Ocean off the coast of Newfoundland, Canada, the temperature and salinity (saltiness) vary throughout the year. Figure 17.10 shows a parametric curve giving the average temperature, $T$ (in $^{\circ} \mathrm{C}$ ) and salinity (in grams of salt per kg of water) for $t$ in months, with $t=1$ corresponding to mid-January $^{1}$. (a) Why does the parameterized curve form a loop? (b) When is the water temperature highest? (c) When is the water saltiest? (d) Estimate $d T / d t$ at $t=6,$ and give the units. What is the meaning of your answer for seawater?

In the Atlantic Ocean off the coast of Newfoundland, Canada, the temperature and salinity (saltiness) vary throughout the year. Figure 17.10 shows a parametric curve giving the average temperature, $T$ (in $^{\circ} \mathrm{C}$ ) and salinity (in grams of salt per kg of water) for $t$ in months, with $t=1$ corresponding to mid-January $^{1}$.
(a) Why does the parameterized curve form a loop?
(b) When is the water temperature highest?
(c) When is the water saltiest?
(d) Estimate $d T / d t$ at $t=6,$ and give the units. What is the meaning of your answer for seawater?

Key Concepts

Parametric Curves

A parametric curve represents a relationship between two or more quantities by expressing each variable as a separate function of an independent parameter, often time. This approach is particularly useful in modeling trajectories, cyclical processes, and other phenomena where variables change in a coordinated way over a shared parameter.

Periodic Functions

Periodic functions are those that repeat their values at regular intervals, a property which is essential in modeling natural processes that follow seasonal or cyclic patterns. Recognizing periodicity helps in understanding and predicting repeated patterns in phenomena such as environmental temperature and salinity.

Hysteresis and Loop Formation

A hysteresis loop occurs when a system's response follows a different path during increasing conditions than during decreasing ones, resulting in a closed curve. This concept is important when the state of a process, like temperature or salinity, not only depends on the current input but also on the history of the system, leading to looped behavior in parametric representations.

Derivatives and Rate of Change

The derivative, such as dT/dt, represents the instantaneous rate of change of one quantity with respect to another. In the context of parametric equations, it provides insights into how fast a particular variable, like temperature, is changing at a specific moment, thereby informing us about the dynamics and responsiveness of the system.

Units and Physical Interpretation

When computing derivatives, clearly understanding the units is crucial for interpreting the results physically. For instance, a derivative of temperature with respect to time expressed in degrees Celsius per month indicates how quickly the temperature is changing over a monthly period, providing a tangible sense of the system's dynamics.

Transcript

00:01 So we have a graph here of the salinity in the atlantic ocean off the coast of newfoundland, and we're claiming the temperature and salinity vary throughout the year, so we're graphing that here on this plot.

00:14 Temperature is on the horizontal axis, the x -axis so to speak, and the salinity is on the vertical axis.

00:23 With all these things, t equals 1 corresponding to some point in mid -january.

00:28 That footnote is just citing the source for this data.

00:33 Question a, why does the parameterized curve form a loop? well, we're assuming that it is the same every year, right? it's the same every year.

00:45 So in order for it to be the same every year, at the same point, say in mid -january, we need it to be...

00:55 Excuse me, in order for it to be the same every year, we need for it to get back where it started at the end of the year, which means it has to form a loop.

01:07 It's periodic, and the way that you get a periodic expression in this case, it is going to be a loop.

01:18 January of, say, 2011 and january 2012 should be the same, which means we have to get back to where we started, and that's a loop.

01:31 So that's great, that's an explanation.

01:34 When is the water temperature highest? well, the water temperature looks like it's highest over here.

01:41 If you follow this here, t is 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12.

01:49 Each dot is the middle of a month, so i forgot to count what this one was.

01:54 1, 2, 3, 4, 5, 6, 7, 8.

01:57 So that's t equals 8, which should correspond to mid -august.

02:03 If each point, if each t is in the middle of the year, in the middle of the given month, the 8th month is august, so the water temperature is highest in mid -august, which makes sense, because in the northern hemisphere, august is near the end of summer.

02:16 It's had the most summertime heat coming at it.

02:20 When is the water saltiest? well, it looks like that's this point here, which you can count time is january, february, march, april.

02:30 So this is apparently saltiest in mid -april.

02:34 That is the fourth month.

02:37 I mean, estimate the temperature, the derivative of temperature with respect to time at t equals 6, and give the units.

02:45 Let's see, so we'll clear this stuff out.

02:48 T is 6...