RMPh 2023, Експеримент — Задача 1. Experimental studies in rotating reference frames

Part A

In this part of the problem, all processes will be considered in a frame $Oxyz$, which is rotating with angular speed $\vec{\omega} = \omega\hat{z}$ with respect to some inertial reference frame. The rotation axis passes through the origin $O$. You must give all answers with respect to the rotating frame.

For the beginning, we will treat the simple case of a non-interacting particle ($\vec{F}=\vec{0}$) with mass $m$. Suppose that, at $t=0$, this point mass is located at $x=0$, $y=R$, and $z=0$, and has velocity $\vec{v_0}=-v_0\hat{y}$. In this case, the equation of motion has the solution

A1 Check that this solution satisfies equation (1). [1 т.]

At small times, close to the beginning of motion, and for a certain range of initial speeds, we may neglect one of the two fictitious forces, so that the trajectory becomes a circle.

A2 1. Which of the two fictitious forces should you neglect to obtain a circular trajectory? 2. Show that in this limit, the two above expressions reduce to a circular trajectory, the radius and center of which you will find. 3. For what range of values (high speeds vs. low speeds) of the initial speed is this approximation suitable? [2 т.]

You will now use this knowledge to solve a puzzle.

You can take videos (of length $1\ s$ and frame rate $240\ fps$) in a rotating reference frame, whose rotation speed is unknown. The camera is rotating together with the reference frame. The length scale of these videos is also unknown: you can measure distances on the frames only in arbitrary units of length (a.u.l.). To find these two unknowns, you can perform the following experiment: from a fixed point, at distance $R = 1\ a.u.l.$ from the coordinate system origin, you launch a bead (blue on the video) with speed $v_0$ oriented towards the origin. The origin is marked by a red dot on the videos, and the axis of rotation passes through it, perpendicular to the image plane.

You can launch the bead using either of two cannons. Cannon 1 has adjustable initial speed (in cm/s), in the range [5, 30] cm/s, in steps of 0.1 cm/s. Cannon 2 shoots beads at random, unknown speeds, but these speeds are in a higher range than those achieved by cannon 1 (the order of magnitude of these speeds is 150 cm/s).

You will use the RMPh22 app to obtain these videos. To use the app, input the password Coriolis. Please refer to the attached technical guide for details. Then, you can choose which cannon you will use. If you use cannon 1, you will also get a choice of the launching speed. Make sure to use a point and not a comma as a decimal separator. Finally, you will get to choose the name of your output file. No need to add an extension to the filename, the app will automatically save it with the right extension.

Note that, to reflect real-life lab conditions, the app takes a few minutes for each run to complete. Be patient and do not open the output file before the app closes by itself.

To analyze the videos, you will use the Tracker software. To understand how you should use it, carefully read the attached Technical guide. The main function you will use is object tracking: Tracker will follow the bead, frame by frame, and will give you the bead position as a function of time. You will then be able to extract the $(t,x,y)$ data set and analyze it further by the means of your choice.

We will begin our analysis with trajectories that are close-to-circular.

A3 1. What cannon should you use to achieve almost circular trajectories? Justify. 2. Explain how, by launching beads with this cannon, you can collect data that will allow you to find the angular frequency $\omega$ of the rotating frame. You might find Appendix 1 useful. [1,5 т.]

A4 Using 5 different launching speeds, find the angular frequency $\omega$ of the rotating frame. Present significant values in your data set, as well as important steps in your analysis. Give a mean value and an error bar for $\omega$. [2 т.]

In order to receive credit for your experimental results, you need to submit the full data sets which you refer to in your analysis. If your analysis produces graphs or plots, you can either sketch them using millimeter paper or by using your computer. In the latter case, please submit an additional .pdf file containing numbered graphs, with appropriate captions. In your analysis, refer to the graphs by their number.

Now, choose a cannon that will allow you to find the conversion between a.u.l. and SI length units.

A5 Which cannon will you choose? Justify your choice. [0,5 т.]

If you choose the right cannon, you will notice that the bead speed changes significantly during the given time frame. Thus, trying to measure the initial bead speed in a.u.l./s will be very noisy and prone to errors, so we advise against it. You will need a more elaborate method for full credit. Hint: you might find the solution of equation (1) useful.

A6 Describe a method that will allow you to find the conversion rate a.u.l./cm. [1 т.]

A7 Using 5 different launching speeds, find the conversion rate between a.u.l. and cm. Present significant values in your data set, as well as important steps in your analysis. Show graphs, if applicable. Give a mean value and an error bar for your result. [2 т.]

Part B

Wind is the generic name given to any air displacement movement in the lower atmosphere. We can model its behavior using fluid dynamics combined with thermodynamic equations and conservation laws. However, in certain cases, we may use classical mechanics to model simple winds. Notice that, for our study, the size of the air parcel is irrelevant, meaning that using accelerations instead of forces would be easier. In this part, we assume the following are true:

• The air flow is stationary (no change in time) and plane (we are not concerned with vertical motion);
• There are four relevant forces in our study:

  • Pressure-gradient force, giving the following acceleration: $\overrightarrow{a_p} = -\frac{1}{\rho}\mathrm{grad}\,P$. Here $\mathrm{grad}\,P$ is the pressure gradient and can be represented geometrically at a certain point through the line of fastest increase in the atmospheric pressure. If the pressure varies only on radial direction, $\mathrm{grad}\,P = \frac{\mathrm{d}P}{\mathrm{d}r}\hat{e}_r$, where $\hat{e}_r$ is the unit vector of the radial direction.
  • Friction force with the terrestrial surface, giving an acceleration $\overrightarrow{a_f} = -k\vec{v}$.
  • Horizontal centrifugal force, giving an acceleration $\overrightarrow{a_{cf}} = \omega^2\vec{r}$, where $\omega$ is the angular velocity of the air parcel with respect to a given point and $\vec{r}$ is the position vector with respect to that point.
  • Coriolis force, giving the acceleration $\overrightarrow{a_c} = 2\vec{\Omega}\times\overrightarrow{v_r}$, where $\vec{\Omega}$ is the angular velocity of the Earth and $\overrightarrow{v_r}$ is the relative velocity of wind with respect to the Earth. Throughout the following tasks, we will neglect the latitudinal extension of the phenomena, in order to easily compute this force.

In the first tasks, we will not consider the rotation of the air parcel in the horizontal plane, so we will disregard the centrifugal force.

B1 Express the stationarity condition and make a schematic figure showing the orientation of the three accelerations. [1 т.]

B2 Derive a relation between friction force and Coriolis force in terms of $\alpha$, the angle between the direction of velocity and the direction of the pressure-gradient force. Obtain from this relation $k$, the coefficient of friction. [1 т.]

The maps available in the Maps1 folder plot isobars (lines of equal atmospheric pressure) as well as wind (white lines), considered at 0 elevation about the ground level. The pressure difference between two isobars is $\Delta P = 2\ \mathrm{mBar}$. The name of each file is the latitude corresponding to the pinned point.

B3 Use the maps provided in Maps1 to measure the angle $\alpha$ at ground level and calculate for each of them the coefficient $k$. [2 т.]

NOTE: It is easy to import each image in the Tracker software used in the previous tasks and use the protractor to measure the angles. Please refer to the technical guide for any of the aspects referring to the use of this software.

Some of the above maps depict land regions (those where human settlements are labeled), while others depict seas or oceans.

B4 Draw a conclusion upon the relation between the coefficients above land and water from the investigated images. [0,5 т.]

Let us now introduce through those observations Buys Ballot's law, as named by meteorologists. Imagine you are standing with your back towards the wind (it tends to push you forward).

B5 Where will the high-pressure region be situated, with respect to your body? [0,5 т.]

At higher altitudes, where the surface no longer plays a role in the atmospheric dynamics, for linear isobars the pressure gradient force is balanced by Coriolis force, making the air parcel move parallel to the isobars. We are approximating here that the radius of curvature of the isobars is infinite, which translates in practice to some lines that can be considered as parallel for a much larger distance than the distance between them. Under this approximation we were able to neglect the centrifugal force. This situation is called geostrophic wind and represents one of the most important approximations in atmospheric physics.

B6 Considering all aspects discussed in this part, try to explain why meteorologists sometimes use data taken above the ground level (around 600 m). [0,5 т.]

The accuracy of the above approximation can be numerically described by a parameter called Rossby number. The smallness of this number characterizes the validity of the geostrophic approximation. The formula for computing this number is $\mathrm{Ro} = \frac{U}{fL}$. Here $U$ is the characteristic velocity scale of the system, which we will replace with wind speed, $L$ is the characteristic length scale of the system, which we will replace with the approximate radius of curvature of the isobars in the given point, and $f = 2\Omega\sin\phi$. To calculate the radius of curvature in a given point, approximate its vicinity as circular and apply the methods in Appendix 1. Another method is to consider tangent lines to the curve and intersect their corresponding normal lines in the center of curvature. Calibrate the ruler in Tracker using the distance depicted on the map.

B7 Use the maps provided in Maps2 folder and information from the following table to evaluate the order of magnitude for the Rossby number in the flows at the marked points (red spots). Based on the description of the geostrophic approximation, infer whether it is valid for these Rossby numbers or not and derive a general condition for the Rossby number where the geostrophic approximation can be used. [2 т.]

On a baric map such those we use, areas of high atmospheric pressure are denoted by H and called anticyclones, while those of low pressure are denoted by L and called cyclones. These areas can be compared with mountains and valleys, so we can imagine that isobars are closed lines in the vicinity of cyclones and anticyclones, similar to level curves on a topographic map.

B8 Establish the direction of this motion (clockwise or anticlockwise) for cyclones in the Southern Hemisphere. Explain your answer! [0,5 т.]

Tropical cyclones are one of the most concerning atmospheric phenomena, especially in the light of climate change. Depending on the region, they are better known as hurricanes, typhoons etc. In Maps3 folder, you can find a map on which a hurricane is present. The center of rotation in the storm is known as "eye". The purpose of this task is to analyze the relevance of geostrophic approximation in studying tropical cyclones and, if needed, replace it with a more suitable one.

The latitude of the hurricane is $\phi \approx 33^{\circ}$. In order to measure the radius of the hurricane, we will use an approximation. Calibrate the ruler in Tracker with the width (distance from the leftmost to the rightmost point) of Hispaniola Island (the island on which the Dominican Republic and Point-du-Prince are situated), which is known to be of approximate 650 km.

B9 Estimate the Rossby number for the hurricane, taking for velocity scale the velocity depicted in the image and for distance scale the distance from the pinned point to the eye of the hurricane. [1 т.]

B10 Compare the Rossby number for the hurricane to the numbers obtained in B6 and comment about the validity of geostrophic approximation. If necessary, use the definition of Rossby number and the balance of forces to obtain a better approximation for flows at high enough altitude to disregard friction with the terrestrial surface. [1 т.]

The maps used in Part B are obtained from Windy.com.