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Chapter1.C: Equation of motion in a rotating frame of reference (Earth)
The equation of motion (1.A-4) derived in Chapter 1.A, copied here:
Du/Dt = -∇p/ρ - gk + Ffric(1.C-1)
is an accurate application of Newton’s law in a fixed (inertial) frame of reference. However, we are earth-bound, and therefore observe motion in a rotating frame of refrence. In this frame, Coriolis acceleration and other features unique to rotating fluid motions can arise, and they are most easily described in the rotating frame. Since Newton’s law applies only to the fixed frame, our goal is then to transform (1.C-1) to one that is valid for a rotating frame. Before doing that, let us study a simple case to see how Coriolis acceleration can arise,
The Coriolis Acceleration & Geostrophic Balance
To see how Coriolis acceleration arises, consider a circular dishpan half-filled with water of constant density ρ placed on a turntable rotating at a constant rate = Ω (radians per second) counter-clockwise (Fig.A-1).
Fig.1.C-1. Illustrating the effects of rotation.
After many, many rotations (which may take hours) and for a sufficiently fast Ω, the water level is then raised around the rim wall of the dishpan, and is symmetrical around the center of the circular pan. The centrifugal force per unit volume directed outward on a fluid particle is then balanced by an inward pressure gradient:
Vθ2/r = (∂p/∂r)/ρ = g∂h/∂r,(1.C-2)
whereVθ = azimuthal velocity at radius r, positive counter-clockwise (i.e. same sense as Ω), h = depth of water, and we have used the hydrostatic relation that p = ρg(h-z) with z=0 at bottom of dishpan. If we define vθ to be the azimuthal velocity relative to the rotating dishpan, then:
Vθ = vθ + Ωr(1.C-3)
Then,(vθ + Ωr)2/r = g∂h/∂r. Simplifying:
vθ2/r + 2Ωvθ = g∂ηp/∂r,(1.C-4a)
whereηp = h - Ω2r2/(2g)(1.C-4b)
is the water surface referenced to the parabolic surface Ω2r2/(2g). The term "2Ωvθ" is the Coriolis acceleration term. The "2Ωvθ = g∂ηp/∂r" in fact is the geostrophic relation valid for large-scale flows. The centrifugal acceleration term " vθ2/r" becomes important, e.g., in a tornado which is "small" and "fast". One may think of the rotating turntable as the "earth," vθ = zonal velocity measured on earth, and Vθ = zonal velocity measured in an absolute fixed frame.
Suppose the dishpan has a tiny hole in the center so that water is drained through the hole. Then a particle placed at the rim initially has zero velocity but begins to move inward. As it does so it conserves angular momentum:
Vθ.r = Ω.r12 (the initial angular momentum at the rim; where r1 = dishpan's radius).
Then(vθ + Ωr).r = Ω.r12;i.e. vθ = Ω(r12 - r2)/r(1.C-5)
Thus the particle spirals inward in an anticlockwise sense (same sense as Ω) into the center of the dishpan, and its azimuthal speed increases as the particle nears the center (Fig.1.C-2).
Fig.1.C-2. Laboratory experiments: effects of rotation on the trajectory of a particle placed initially near the rim of the rotating circular dishpan for slow (left; GFD3_slow_rotation.mpeg) and fast (right; GFD3_fast_rotation.mpeg) rotations.
Transforming the Equation of Motion to the Rotating Frame of Reference
From the rotating dishpan idea, eqn(1.C-3), we write more generally:
uin = urot + Ω × r(1.C-6)
In this equation, uin is the velocity as viewed from the fixed (inertial) frame – by someone in the outer space looking down onto the earth. The observer sees uin as the sum of 2 vectors. The first is urot which is the velocity in the torating frame, and the second one is Ω × r which is the velocity of the fixed vector r as it is being “carried” by the rotating frame. Consider any vector A = (Ax, Ay, Az):
A = iAx + jAy + kAz(1.C-7)
defined in the rotating frame. Here, (i, j, k) are the 3 unit vectors fixed in therotating frame– i.e. they are just unit vectors defined on our usual (x, y, z) coordinate axes here on earth. The material derivative of Aviewed from the rotating frame is then (since (i, j, k) are fixed):
(DA/Dt)rot = iDAx/Dt + j DAy/Dt + kDAz/Dt(1.C-8)
But, to the fixed-frame, outer-space observer, (i, j, k) are rotating and their velocities are:
(Di/Dt)in = Ω × i,(Dj/Dt)in = Ω × j,(Dk/Dt)in = Ω × k(1.C-9)
Therefore,
(DA/Dt)in = (DA/Dt)rot + Ω × A(1.C-10)
Fig.1.C-3. The left panel shows the velocityuin(e.g. of a fluid parcel on earth) as seen from the fixed-frame, outer-space observer. The right panel shows the same velocity uin which now comprises of 2 parts: the urot as seen (and measured) by us on earth, plusthe velocity Ω × rcaused by the fact that the parcel (which is located at r) is also being carried by the solid rotation of the earth. (Note that i=, j=, and k=).
Putting A = r in (1.C-10) gives (1.C-6), since
uin = (Dr/Dt)inandurot = (Dr/Dt)rot(1.C-11)
Putting A = uin, we get:
(Duin/Dt)in= [(D/Dt)rot + Ω ×]uin= [(D/Dt)rot + Ω ×] (urot + Ω × r)
= (Durot/Dt)rot + 2Ω × urot + Ω × Ω × r(1.C-12)
Finally, then, the equation of motion in inertial frame, eqn(1.C-1) becomes:
(Durot/Dt)rot= -2Ω × urot-Ω × Ω × r-p/ -kg + Ffric(1.C-13)
Homework:
1.C-1: (a) By considering first the direction of the vector (Ω × r), show (explain) that the vector Ω×Ω×r is directed in the opposite direction to the direction of r – hence -Ω×Ω×r is in the same direction of r (see sketch below).
(b) Then show also that the magnitude of -Ω×Ω×r is |Ω|2r.
(c) Finally, show that -Ω×Ω×r = (|Ω|2r2/2).
1.C-2: What is the value of Ω on earth? What is the radius of the earth?
1.C-3: The vector -Ω×Ω×r of problem 1.C-1 is in fact the centrifugal acceleration due to the rotating frame Ω. Calculate the centrifugal acceleration of a particle fixed to the earth at the equator, and compare it with the acceleration due to gravity g, is it much larger or smaller than g?
References
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Mellor, G.L., 2004: POM User Guide.
Mellor G.L. and Yamada, T., 1982: Review of Turbulence Models.
Pedlosky, J., 1979: Geophysical Fluid Dynamics.
Roach, P., 1972: Computational Fluid Dynamics.
G.G. Stokes (1847). "On the theory of oscillatory waves". Transactions of the Cambridge Philosophical Society 8: 441–455. Reprinted in: G.G. Stokes (1880).Mathematical and Physical Papers, Volume I.Cambridge University Press. pp.197–229.
Van Dyke, M., 1964: Perturbation Methods in Fluid Mechanics.
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