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|phenomena such as spiral galaxy, hurricane, and drain in the sink (Figures 14, and 15). By neglecting the thickness of the spiral flow, Eq.(2) of the Navier-Stokes Equations in cylindrical coordinates can be expressed in the form:|
ur (ur/r) + u (ur/) = Fr ---------- (15a)
ur [(r u)/r] + u [(r u)/] = 0 ---------- (15b)
where u = /t. If we assumed further that the force acts on the fluid only in the radial direction, and the surface density is constant for the fluid, then
Figure 15 Hurricane
|Eq.(1) of the continuity equation becomes:|
(1/r) [(r ur)/r] + (1/r) [(r u)/] = 0 ---------- (16)
|A relationship between the velocity components is obtained by substituting Eq.(16) to |
ur = b u ---------- (17)
where b is a constant having the dimension of length. This formula can be integrated once more to yield:
r = a + b ---------- (18)
Figure 16 Archimedean Spiral
|where a is another constants of integration. Eq.(18) expresses the trajectory of an Archimedean spiral (see curve on the left of Figure 16). So far there is no restriction on whether the movement is to plunge inward or to expand.|
with M to be the mass of the central black hole, while the mass on the disk within r is given by M' = 2 r dr, where = 0 (r0/r) is the surface density in unit of gm/cm2, 0 is its cutoff value at the edge of the galaxy r = r0, and the third term is the centrifugal force. Thus, the radial and rotational components of the velocity can be expressed as:|
ur2 = GM / r + 2G0r0 - r2u2 ---------- (19b),
r2u2 = r (GM + 2G 0r0 r) / (b2 + r2) ---------- (19c),
Figure 17a Rotation Curve for Milky Way [view large image]
|which has the same profile of the rotation curve for the disk as shown in Figure 17a with a peak at rm (40r0b/M) b for 40r0b >> M. The observed curve takes into account of the dark matter in the halo.|
|dp/dr (which has overwhelmed the Coriolis and centrifugal forces in this case):|
ur2 / r = (1/) dp/dr ---------- (20).
The dependence of p on r is shown in Figure 17b. The observational data (for the "Charlie" type hurricane) can be approximated by the empirical formula:
p = p0 [5.5 - e-k(r - re)] ---------- (21) for r re,
where re is the distance from the center to the wall of the eye (Figure 17b), p0 = 220 mb and k is a constant related to the steepness of the pressure gradient. Thus, the pressure gradient is:
dp/dr = kp0e-k(r - re) ---------- (22) for r re
ur2 = (krp0/) e-k(r - re) ---------- (23) for r re.
At the wall of the eye, r = re, and ur2 = krep0/.
The rotation curve is given by r2u2 = (kr3p0/b2) e-k(r - re), which reproduces a profile similar to the curve in Figure 17b with a maximum at rm = 3/k. |
This kind of analysis is also applicable to the drain in the sink although on a much smaller scale.
|Thus the radial distance changes by about 125 km covering about 2/3 of the average hurricane size when turns one cycle. There is no tightly wound arms. This model seems to represent the actual system quite well. Meanwhile, the calculated radial and circular velocities at the wall of the eye are ur 4.6 m/h and reu 2.3 m/h respectively indicating a calm region there.|
Figure 17c is a vivid example from a small scale hurricane (a thunderstorm supercell) over Montana, USA showing the wall, as well as the dark cloud spiralling inward.
Figure 17c Thunderstorm Supercell [view large image]