Question

The lapse rate is the rate at which the temperature in Earth's atmosphere decreases with altitude. For example, a lapse rate of \(6.5^{\circ}\) Celsius / km means the temperature decreases at a rate of \(6.5^{\circ} \mathrm{C}\) per kilometer of altitude. The lapse rate varies with location and with other variables such as humidity. However, at a given time and location, the lapse rate is often nearly constant in the first 10 kilometers of the atmosphere. A radiosonde (weather balloon) is released from Earth's surface, and its altitude (measured in kilometers above sea level) at various times (measured in hours) is given in the table below. $$\begin{array}{lllllll} \hline \text { Time (hr) } & 0 & 0.5 & 1 & 1.5 & 2 & 2.5 \\ \text { Altitude (km) } & 0.5 & 1.2 & 1.7 & 2.1 & 2.5 & 2.9 \\ \hline \end{array}$$ a. Assuming a lapse rate of \(6.5^{\circ} \mathrm{C} / \mathrm{km},\) what is the approximate rate of change of the temperature with respect to time as the balloon rises 1.5 hours into the flight? Specify the units of your result and use a forward difference quotient when estimating the required derivative. b. How does an increase in lapse rate change your answer in part (a)? c. Is it necessary to know the actual temperature to carry out the calculation in part (a)? Explain.

Short answer

Is it necessary to know the actual temperature for the calculation in part (a)? Answer: The rate of change of temperature with respect to time at 1.5 hours into the flight is approximately -5.2ºC per hour. An increase in the lapse rate will increase the magnitude of the negative value of the rate of change of temperature with respect to time, indicating a faster decrease in temperature. No, it is not necessary to know the actual temperature for the calculation in part (a).

Step-by-step solution

Calculate the Altitude Difference at 1.5 hours

To calculate the rate of change of temperature with respect to time, we need to find the altitude difference between the times 1.5 hours and 2 hours. At 1.5 hours, the altitude is 2.1 km, and at 2 hours, the altitude is 2.5 km. So the difference in altitude is: $$\Delta \text{ Altitude} = 2.5 - 2.1 = 0.4 \, \text{km}$$

Calculate the Temperature Difference using the Lapse Rate

We have a lapse rate of \(6.5^{\circ} \mathrm{C} / \mathrm{km}\). Therefore, the temperature difference during the time interval from 1.5 to 2 hours is: $$\Delta \text{ Temperature} = \text{Lapse Rate} \times \Delta \text{ Altitude} = 6.5 \frac{\text{C}}{\text{km}} \times 0.4 \, \text{km} = 2.6^{\circ} \mathrm{C}$$ So, the temperature decreases by 2.6ºC.

Calculate the Time Difference and the Rate of Change of Temperature with respect to Time

The time difference is: $$\Delta \text{ Time} = 2 - 1.5 = 0.5 \, \text{hours}$$ Now we can calculate the rate of change of temperature with respect to time as: $$\frac{\Delta \text{ Temperature}}{\Delta \text{ Time}} = \frac{2.6^{\circ} \mathrm{C}}{0.5 \, \text{hours}} = 5.2^{\circ} \mathrm{C} \, \mathrm{per} \, \mathrm{hour}$$ Thus, the rate of change of temperature with respect to time at 1.5 hours into the flight is approximately -5.2ºC per hour. Answer to part (a): -5.2ºC per hour.

Analyze the Effect of Increasing the Lapse Rate on the Rate of Change of Temperature with respect to Time

When the lapse rate increases, the temperature will decrease more rapidly with an increase in altitude. As a result, the temperature difference during the time interval from 1.5 to 2 hours will be larger. Consequently, the rate of change of temperature with respect to time will also become more negative, indicating a faster decrease in temperature. Answer to part (b): An increase in the lapse rate will increase the magnitude of the negative value of the rate of change of temperature with respect to time.

Discuss the Necessity of Knowing the Actual Temperature to Carry Out the Calculation in part (a)

The calculation in part (a) requires the temperature difference during the time interval from 1.5 to 2 hours. To find this, we only need the lapse rate and the altitude difference. Since the actual temperature at a specific altitude is not required to determine this temperature difference, it is not necessary to know the actual temperature to carry out the calculation in part (a). Answer to part (c): No, it is not necessary to know the actual temperature for the calculation in part (a).

Key concepts

Rate of Change of Temperature

Understanding the rate of change of temperature is crucial when studying meteorological phenomena like the lapse rate. In essence, this rate measures how much the temperature decreases as you move upward in the Earth's atmosphere. It's expressed as a temperature change per unit of altitude, typically in degrees Celsius per kilometer. However, in exercises like the one given, students are often asked not just about the static rate of temperature change per altitude, but also its dynamic counterpart over time.

For instance, if the lapse rate is stated as \(6.5^\circ \mathrm{C} / \mathrm{km}\), and we wish to uncover the temperature change as the weather balloon rises, we need the altitude change over a certain time period. The altitude readings at different times allow us to calculate this dynamic rate, showing how rapidly the temperature is expected to decrease as the balloon ascends over time.

It's important to emphasize that we are approximating this rate of change over a discrete interval, not finding an exact value at a single point in time. This sort of estimate is essential in many practical applications, like forecasting weather patterns or understanding climate variations with altitude.

Forward Difference Quotient

The forward difference quotient is a method for estimating the derivative, or the rate of change, of a function at a particular point. This concept is a staple in numerical analysis, helping us to understand how a function changes without requiring the exact slope at that specific point - a piece of information often unavailable in real-world data.

In our textbook problem, the forward difference quotient is used to calculate the approximate rate of change of temperature with respect to time as the balloon rises. This calculation involves taking the temperature difference over a certain altitude change, and then dividing it by the time it took for that change. Conceptually, you can think of it like calculating the average speed but for temperature change over time rather than distance over time.

To use the forward difference quotient, you need two points: in this case, the temperature at two different times. By subtracting the first measurement from the second and dividing by the time difference, we get an approximate rate of temperature change per hour, not the actual rate at any precise moment.

Derivative Estimation

Derivative estimation is the process of finding an approximate rate of change of a function based on available data points. It's a critical tool for times when an analytic expression of the derivative is hard to come by or when we only have a discrete set of data points rather than a continuous function.

In the context of the exercise, we're essentially estimating the derivative of temperature with respect to time. However, since we only have discrete data points (temperature readings at specific times), we can't calculate the exact derivative. Instead, we use methods like the forward difference quotient as an estimation strategy.

This concept is heavily relied upon in fields such as physics, economics, and engineering. For instance, astronomers might use derivative estimation to understand the changing brightness of a star, while economists may use it to predict changes in stock market trends. Similarly, engineers might use it for analyzing the stress changes in materials under varying forces. In each of these cases, they start with known data points and work out the rates of change to project and understand the behavior of the system they're studying.