Question

Ntroduction The flow of geophysical fluids (i.e., the Earth’s ocean and atmosphere, and the atmospheres of gas giants planets such as Jupiter and Saturn) is complicated, involving a vast number of processes and interactions among them on scales ranging from centimeters to the planet’s size, and timescales going from seconds to millennia. Two effects mainly constrain the flow of geophysical fluids: the planet’s rotation and stratification. In this lab we will deal with the first of the aforementioned effects. We will learn how the unusual properties of rotating fluids manifest themselves in, and profoundly influence, the circulation of the Earth’s ocean and planetary atmospheres. The planet’s rotation makes these fluids more similar than one might expect. On what scales might the atmosphere, ocean, or our laboratory experiment, “feel” the effect of rotation? Suppose that U is a typical horizontal current speed, and the typical distance over which the currents varies is L. Then the timescale of the motion (Tmotion) is L/U. Compare this with the period of rotation Trot, define a nondimensional number (the Rossby number):

Ro := Trot/Tmotion = Trot × U/L. If Ro is much greater than one, then the timescale of motion is short relative to a rotation period, and rotation will not significantly influence the motion. If Ro is much less than one, then the motion will be aware of rotation. Let us estimate Ro for large-scale flow in the atmosphere and ocean.

• Amosphere: L ∼ 5000 Km, U ∼ 10 m/s, and T = 1 day, giving Ro = 0.2, which suggest the rotation will be important.

• Ocean: L ∼ 1000 Km, U ∼ 0.1 m/s, giving Ro = 0.01, and rotation will be a controlling factor. Pre Lab 1.

It is clear from the Ro estimations above, that rotation is very important in shaping the patterns of air and ocean currents on sufficiently large scales. How can we study this effect on an small rotating tank (L ∼ 30cm)? If we generate a current in the tank of U ∼ 0.1 cm/s, what would be an appropriate rotation period?

Answer 1

Step-by-step explanation:

To study the effect of rotation on a small rotating tank, we want the Rossby number to be much less than one, so that rotation will be a controlling factor in shaping the patterns of flow.

Based on the given information, the length scale of the tank is L = 30 cm and the current speed is U = 0.1 cm/s. To calculate the timescale of the motion, Tmotion, we can use the formula Tmotion = L/U, which gives us:

Tmotion = 30 cm / 0.1 cm/s = 300 s

Next, we need to estimate an appropriate rotation period, Trot, so that the Rossby number Ro = Trot / Tmotion will be much less than one. We can use the formula Ro = Trot * U / L, rearrange it to solve for Trot:

Trot = Ro * L / U

If we take Ro to be 0.1 (for example), then we have:

Trot = 0.1 * 30 cm / 0.1 cm/s = 30 s

So, with a rotation period of 30 s and a current speed of 0.1 cm/s, we should expect the rotation to have a significant influence on the patterns of flow in the small rotating tank.