Assuming the ball, until being caught, moves with a constant forward velocity (horizontal component) of 15 m s-1, determine the lateral deflection of the ball from a straight line due to the Coriolis effect. [Neglect friction and any wind or other aerodynamic effects.]
5. Imagine that Concorde is (was) flying at speed u from New York to London along a latitude circle. The deflecting force due to the Coriolis effect is toward the south. By lowering the left wing ever so slightly, the pilot (or perhaps more conveniently the computer on board) can balance this deflection. Draw a diagram of the forces—gravity, uplift normal to the wings, and Coriolis—and use it to deduce that the angle of tilt, y, of the aircraft from the horizontal required to balance the Coriolis force is tan y ■■
where Q is the Earth's rotation, the latitude is v and gravity is g. If u = 600 m s-1, insert typical numbers to compute the angle. What analogies can you draw with atmospheric circulation? [Hint: cf. Eq. 7-8.]
Consider horizontal flow in circular geometry in a system rotating about a vertical axis with a steady angular velocity Q. Starting from Eq. 6-29, show that the equation of motion for the azimuthal flow in this geometry is, in the rotating frame (neglecting friction and assuming 2-dimensional flow)
Dvg dVg dVg
Dt dt dr
where (vr, vg) are the components of velocity in the (r, g) = (radial, azimuthal) directions (see Fig. 6.8). [Hint: write out Eq. 6-29 in cylindrical coordinates (see Appendix A.2.3) noting that vr = Dr/Dt; vg = rDg/Dt, and that the gradient operator is V = (d/dr,1/r 3/36).]
Assume that the flow is axisymmetric (i.e., all variables are independent of 6). For such flow, angular momentum (relative to an inertial frame) is conserved. This means, since the angular momentum per unit mass is
Show that Eqs. 6-45 and 6-47 are mutually consistent for axisymmetric flow.
When water flows down the drain from a basin or a bath tub, it usually forms a vortex. It is often said that this vortex is anticlockwise in the northern hemisphere, and clockwise in the southern hemisphere. Test this saying by carrying out the following experiment.
Fill a basin or a bath tub (preferably the latter—the bigger the better) to a depth of at least 10 cm; let it stand for a minute or two, and then let it drain. When a vortex forms,14 estimate as well as you can its angular velocity, direction, and radius (use small floats, such as pencil shavings, to help you see the flow). Hence
14A clear vortex (with a ''hollow'' center, as in Fig. 6.7) may not form. As long as there is an identifiable swirling motion, however, you will be able to proceed; if not, try repeating the experiment.
calculate the angular momentum per unit mass of the vortex.
Now, suppose that, at the instant you opened the drain, there was no motion (relative to the rotating Earth). If only the vertical component of the Earth's rotation matters, calculate the angular momentum density due to the Earth's rotation at the perimeter of the bath tub or basin. [Your tub or basin will almost certainly not be circular, but assume it is, with an effective radius R such that the area of your tub or basin is nR2 to determine m.]
(c) Since angular momentum should be conserved, then if there was indeed no motion at the instant you pulled the plug, the maximum possible angular momentum per unit mass in the drain vortex should be the same as that at the perimeter at the initial instant (since that is where the angular momentum was greatest). Compare your answers and comment on the importance of the Earth's rotation for the drain vortex. Hence comment on the validity of the saying mentioned in (b).
(d) In view of your answer to (c), what are your thoughts on Perrot's experiment, GFD Lab VI?
dimensional, inviscid (F = 0) flow of a homogeneous fluid of density pref thus:
Dt pref dx
Dt Pref dy where D/Dt = d/dt + ud/dx + vd/dy and the continuity equation is du dv
(a) By eliminating the pressure gradient term between the two momentum equations and making use of the continuity equation, show that the quantity (dv/dx - du/dy + f) is conserved following the motion, so that
(b) Convince yourself that z.V x u dv dx du dy
(see Appendix A.2.2), that is dv/dx - du/dy is the vertical component of a vector quantity known as the vorticity, Vx u, the curl of the velocity field.
The quantity dv/dx - du/dy + f is known as the ''absolute'' vorticity, and is made up of ''relative'' vorticity (due to motion relative to the rotating planet) and ''planetary'' vorticity, f, due to the rotation of the planet itself.
By computing the ''circulation'' —the line integral of u about the rectangular element in the x, y plane shown in Fig. 6.22—show that:
circulation _ average normal area enclosed component of vorticity
Hence deduce that if the fluid element is in solid body rotation,
then the average vorticity is equal to twice the angular velocity of its rotation.
(d) If the tangential velocity in a hurricane varies like v = 106/r ms-1, where r is the radius, calculate the average vorticity between an inner circle of radius 300 km and an outer circle of radius 500 km. Express your answer in units of planetary vorticity f evaluated at 20o N. What is the average vorticity within the inner circle?
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