The Beta Effect
So, having got that out of the way, let me meander a bit into some background, on the way to the beta effect - which actually explains the northwestwards (NW) motion of cyclones in the Northern Hemisphere (NH), which causes cyclones to avoid India's West Coast - and prefer to hit the East Coast (to disastrous effect).
Without going into the derivation (which could vary in
length depending upon the level of rigour), the Coriolis force, which arises in
a rotating frame, such as the Earth itself, is given by:
F
= 2m w ´ v
where the Coriolis force F results from the vector cross-product of
the angular velocity w
and the linear velocity v, all three vectors pointing along 3 perpendicular
axes (as given by the right hand rule, w along the thumb, v along the forefinger and F along the middle finger). For the right-hand rule, see the website:
http://phys420.phas.ubc.ca/p420_12/tony/Coriolis_Force/Home.html
For the Earth, the angular velocity vector points through
the poles, the velocity is along the surface of the Earth at some given
latitude j, and
the force F deflects the cyclone in a direction perpendicular to its motion
(v). For horizontal motion, the magnitude of the Coriolis force is given by:
F
= 2m w v sin(j)
and so its magnitude is exactly zero at the Equator.
Since the Coriolis force acts to deflect any moving object
perpendicular to its motion, that motion is likely to become a vortex or a
circle. Specifically, the motion is clockwise (CW) in the Northern Hemisphere
(NH) and counterclockwise (CCW) in the Southern Hemisphere (SH).
Equating the Coriolis force to the centripetal force mv2/r,
the radius of the inertial circle is:
R
= v/(2wsin(j))
The latitude of Kanyakumari is 8.1°, of Mangalore is 12.9°
and of Mumbai is 19.1°. Corresponding values of sin(j): 0.141, 0.223 and 0.327. The
angular velocity of the Earth is w
= 7.29x10-5 rad/sec and so the radius of the inertial circles at
these points, assuming that the cyclone speed is 50 m/s (180 km/hr) is: 2,437
kms (KK), 1,537 kms (Man), and 1048 kms (Mum).
Tabulated:
|
|
Latitude
|
Sin(l)
|
Radius of inertial circle (kms)
|
|
Kanyakumari
|
8.1°
|
0.141
|
2,437
|
|
Mangalore
|
12.9°
|
0.223
|
1,537
|
|
Mumbai
|
19.1°
|
0.327
|
1,048
|
Due to the latitudinal variation, the radius of the inertial
circle is much greater closer to the Equator.
The fact that the Coriolis force deflects westwards is not
obvious – considering that the motion in the NH is clockwise, and
circular motion along inertial circles should just keep on regularly returning
the moving mass back to its pre-existing path - and the explanation is the beta effect (see below for a sketchy description).
The equations that deal with this are discussed in the
website quoted below, for a particle moving with a velocity vector (V0
cos(q), -V0 sin(q ), 0 ) with z = 0 along
the surface of the Earth, and q
= 0 as the reference direction, pointing
north:
The end result of solving the equations of motion, assuming
that w is small, is:
Vx = V0 cos(q + 2wt sin(l))
And
Vy = -V0 sin(q + 2wt sin(l))
Which means that the angle q
changes at the rate:
(dq/dt) = 2w
sin(l)
Since this quantity is positive, the angle q increases, and the mass
moves in a clockwise direction (in the NH)..
The magnitude of dq/dt
is 4.66x10-5 rad/sec
(assuming sin(l) =
0.32), or 2.67x10-3 deg/sec.
In other words, the cyclone would get deflected by 9.6
deg/hr, or 230 deg/day.
According to a course document on “Inertial Oscillations” by
Thompson (Ocean420) in Winter 2005 in a book by
Susan Hautala, LuAnne Thompson, and Kathryn Kelly:
The time period of the inertial oscillation
is given by:
Tin = TE/[2 sin (l)]
Where TE = Earth’s rotation period (24 hrs), and
gives some values at different latitudes: 69 hrs at 10°, 24 hrs at 30°
(obviously), and 16.9 hrs at 45°. The radius of the inertial circle is also much greater near the Equator (as shown above).
The UTexas website makes a number of other things clear:
a) In the Northern Hemisphere, cool winds from the
North (that move towards the Equator to replace hot air that rises), are
deflected in a clockwise direction, giving rise to the trade winds which blow
towards the southwest (SW).
b b) Cyclones originate because winds that blow from
a high pressure area to a low pressure area are deflected clockwise in the NH
(as seen in the Figure below, taken from the UTexas website), and this sets up
the cyclonic rotation. Note that the winds blowing towards the South are
deflected westwards, while the wind blowing to the North is deflected towards
the East.
c c) The related point – not mentioned by Thompson
(in this document) – is that the Coriolis force is too weak near the Equator to
set up the cyclonic rotation, which accounts for the fact that cyclones mostly
originate at latitudes with l
>7°.
In the North Indian Ocean, a tropical cyclone usually lasts 5-6
days, and they remain at hurricane intensity for 2-4 days (compared to a global
average of 6 days).
Another important quantity to evaluate is the dimensionless Rossby number Ro, which
is the ratio of inertial to Coriolis forces as mentioned in wiki:
Ro = V/(2w sin(l)L)
Where L is the spatial scale of the system, in this case a
cyclone. Only if Ro £ 1,
is the effect of Coriolis force significant relative to the inertial force.
A cyclone with a wind speed of 10-14 km/hr is slow-moving,
15-25 km/hr is a moderate cyclone, and for >25 km/hr it is a fast-moving
cyclone.
The size of a cyclone in Indian seas varies between 50 and
2000 kms, but most of them are in the size range of 300-600 kms.
For a cyclone with V = 10 m/s (36 km/hr) in the Bay of
Bengal or the Arabian Sea, with a spatial dimension of 500 kms, the dimensionless
Rossby number Ro becomes:
Ro
= 10/[(2)(7.29x10-5)(0.32)(5x105)] = 0.43
It seems that the Ro number will be even lower for a larger
cyclone (say 1000 kms) or a slower moving cyclone at a higher latitude.
Note that the highly contentious case of water draining out
in spiral fashion from a kitchen sink: does it go CW in the NH? The answer is
that, whatever happens, it’s not due to the Coriolis force. The Rossby factor
for this case is (roughly), assuming L = 0.1 m, V = 1 m/s and in India:
Ro
= 1/[(2) )(7.29x10-5)(0.32)(0.1)] = 2x105
So the Coriolis effect doesn’t have a chance compared to
inertial effects!
“Cyclones that form over the Bay of Bengal are either those
develop in situ over southeast Bay of Bengal and adjoining Andaman Sea or
remnants of typhoons over Northwest Pacific and move across south China sea to
Indian Seas. As the frequency of typhoons over Northwest Pacific is quite high
(about 35 % of the global annual average), the Bay of Bengal also gets its
increased quota.
The cyclones over the Arabian Sea either originate in situ
over southeast Arabian Sea (which includes Lakshadweep area also) or remnants
of cyclones from the Bay of Bengal that move across south peninsula. As the
majority of Cyclones over the Bay of Bengal weaken over land after landfall,
the frequency of migration into Arabian Sea is low.
In addition to all the above the Arabian Sea is
relatively colder than Bay of Bengal and hence inhibits the formation and
intensification of the system.”
I am currently reading Amitava Ghosh’s “The Great
Derangement: Climate Change and the Unthinkable“ (Penguin Random House India,
2016) and he observes that (possibly due to global warming) for the first time,
in 2015, the number of cyclones originating in the Arabian Sea was known to be greater in number originating in the
Bay of Bengal (p.58). He also notes that: “The cyclones that have struck the
west coast of Indiain the past have all traveled upwards on a northeasterly
tack, from the southern quadrant of the Arabian Sea” (p.66). He is extremely
worried that such a cyclone may hit the highly populated, low-lying coastal
megacity of Mumbai, with lethal consequences. With global warming, the
intensity of cyclones has been observed to have increased, even though the
frequency may not have.
According to Persson, since inertia circles have a lower
diameter at higher latitudes (than at lower latitudes), the inertia circles are
actually spirals transporting mass westwards, (Anders O.Persson,
History of Meteorology 2 (2005)3). Elsewhere in this paper, Persson refers to
the phenomenon of beta drift, which
explains the northwestward movement (in the Northern Hemisphere) of cyclones.
This is complicated but I will summarise what I got out of the flash simulation
in the following website:
For an axisymmetric cyclone, the vorticity is
conserved (under some reasonable conditions).
The vorticity (vector) is defined as: W
= curl(v), where v is the velocity vector. Anyway, there are two components of
vorticity: the local vorticity due to the spin of the cyclone around its
central axis and the vorticity f due to
the spin of the Earth around its axis. The latter is given by:
f = 2sin(l)
it is
zero at the Equator, and it increases as the latitude increases.
If an air parcel in or near the cyclone moves
Northwards, its Earth vorticity increases, and since the total vorticity is
conserved, its local vorticity decreases. Similarly, an air parcel that moves
South, finds its Earth vorticity decrease and its local vorticity increase. Air
parcels that move East or West do not change their latitude or Earth vorticity.
These increases and decreases in local
vorticity cause the formation of two secondary (beta) gyres (see the Figure below,
from the above website) that rotate in opposite directions: the local vorticity has
a minimum that is NE (CW rotation) of the main cyclone vortex (CW rotation),
and a maximum SW (rotating CCW) of the
main vortex. These two gyres are much weaker (by orders of magnitude) than the
main vortex – and they are not visible in satellite pictures of cyclones.
At this point, the website unabashedly declares
that ‘numerical simulation’ shows that these two gyres displace the main vortex
of the cyclone in a NW direction (in the NH), and a speed of, at most, a few
metres/sec. Note that the beta effect will displace the cyclone in the NW
direction even if it is embedded in ‘calm winds’ (a slow or almost static’
cyclone).
For me, this constitutes a ‘ne plus ultra’ –
because I am not about to get embroiled in numerical simulation of meteorology!
About 10-20% of the storm’s motion arises from
the beta effect.
Please
note that you can't observe these circulations on satellite loops because their
orders of magnitude are so much smaller than the hurricane's circulation.
Nonetheless, these circulations associated with the Beta effect are sufficiently large to cause a westward-moving hurricane to drift northwestward. Moreover, the Beta effect is the reason why tropical cyclones flirting with crossing the equator swerve to the northeast before it's too late.
Nonetheless, these circulations associated with the Beta effect are sufficiently large to cause a westward-moving hurricane to drift northwestward. Moreover, the Beta effect is the reason why tropical cyclones flirting with crossing the equator swerve to the northeast before it's too late.
The Earth vorticity parameter f arises due to the Coriolis force, and the beta effect arises due to the variation of the vorticity f with latitude (X.Liang and J.C.L.Chan J.Atmosph.Soc. (Oct.2005) p.3825)
Bottomline: The beta effect does cause the
cyclone to move away from the West Coast of India (although it also causes
cyclones to move towards the East Coast), while the frequency of cyclones in
the Arabian Sea seems to have gone up in 2015, above that in the Bay of Bengal
– according to Amitava Ghosh.
