Rsta 1915 0001
Rsta 1915 0001
Rsta 1915 0001
Communicated byW
. .N S h a w , Sc.D
.,of the Me
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Office.
O ur knowledge of wind eddies in the atmosphere has so far been confined to the
observations of meteorologists and aviators. The treatm ent of eddy motion in
either incompressible or compressible fluids by means of mathematics has always
been regarded as a problem of great difficulty, but this appears to be because
attention has chiefly been directed to the behaviour of eddies considered as indi
viduals rather than to the average effect of a collection of eddies. The difference
between these two aspects of the question resembles the difference between the
consideration of the action of molecule on molecule in the dynamical theory of gases,
and the consideration of the average effect, on the properties of a gas, of the motion
of its molecules.
It has been known for a long time th a t the retarding effect of the surface of
the earth on the velocity of the wind must be due, in some way, to eddy motion ;
but apparently no one has investigated the question of whether any known type
of eddy motion is capable of producing the distribution of wind velocity which has
been observed by meteorologists, and no calculations have been made to find out
how much eddy motion is necessary in order to account for this distribution. The
present paper deals with the effect of a system of eddies on the velocity of the
wind and on the temperature and humidity of the atmosphere. In a future paper
the way in which they are produced and their stability when formed will be
considered.
I t is well known th at wind velocity, temperature, and humidity vary much
more rapidly in a vertical than in any horizontal direction, and further th at the
vertical component of wind velocity is very small compared with the horizontal
velocity. I t has been assumed, therefore, th a t the average condition of the air
at any time is constant for a given height, over an area which is large compared
with the maximum height considered. If u and v represent the components of
undisturbed wind velocity parallel to horizontal axes, x and y, running from South
V O L. C C X V .---- A 5 23. B [Published January 21, 1915,
2 G. I. TAYLOR ON EDDY MOTION IN THE ATMOSPHERE.
to North and from W est to East respectively, and if T and m represent the
average temperature and the average amount of water vapour per cubic centimetre
of air, this is equivalent to assuming th a t u, , T, and m are functions of 2 , the
height, and t, the tim e ; and th a t they are independent of x and
Let us first consider the propagation of heat in a vertical direction. The ordinary
conductivity of heat by molecular agitation is so small that no sensible error will
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of the layer from which it originated, the potential tem perature at the point
a t time ts i 6 (z0}t0).
The amount of heat which passes per second across the area A is therefore
perJj wO (z0£0) dxdy. Now 0 (z0,t0) may be expressed
e (zQ, =
tQ) e
provided th a t the changes in 6 in the height and in the time t0 are small
compared with 6 . Hence
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~ P<T^ \ \ kW ( d
. (z-z^) d x d y for jj
does not vary with z if the eddy motion is uniformly distributed. Hence the rate
at which heat enters the volume ASz is
z00" J )
\ xw dxdy.
Now since mixtures which take place within this volume merely alter the
distribution of the heat contained in it without affecting its amount, this must be
d6
equal to pa — A .zSHence we obtain the equation for the propagation of heat b
ct
means of eddies in the form ^ 4- I f w (z—z0) dx dy. But 11 (z—z0) dx dy
ct cz A. J J a . A J J a
is the average value of tv (z—z0) over a horizontal area, hence it may be expressed
in the form \ ( dw,) where d is the average height through which an eddy mo
from the layer at which it was at the same temperature as its surroundings, to
the layer with which it mixes. w is defined by the relation ^ (wd) = average
value of w (z—z0)over a horizontal plane; it roughly represents the average
b 2
4 G. I. TAYLOR ON EDDY MOTION IN THE ATMOSPHERE.
vertical velocity of the air in places where it is moving upwards. The divisor 2
is inserted because the air at any given point is equally likely to be in any portion
of the path of an eddy, so th at the average value of z— should be approximately
equal to \ ( d)
.
The equation for the propagation of heat by means of eddies may now be written
a 0_wdtf6
dt ~ 2 ......................................................... W
= #T#
StPa Sz2
Having obtained the path of the air, the next step is to find the tem perature of the
sea below it. This is a comparatively easy m atter, for a careful watch is kept by the
liners on the tem perature of the N orth Atlantic. The results of their observations are
plotted by the Meteorological Office on weekly charts, on which isothermal lines are
drawn to represent sea temperatures of 80° F., 70° F., 60° F., 50° F., and 40° F. These
charts are published on a small scale in the weekly weather report of the Meteorological
Office, but Captain Campbell Hepworth was kind enough to lend me the originals, and
on them I plotted the air paths.
One of the charts, with the air’s path marked on it, is shown in fig. l .#
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I t has been found th a t the tem perature of the air rarely differs from th a t of the
surface of the sea by more than 2° C., and usually the difference is only a fraction of
Mid. July 30
Mid J u l y 31
8 p.m. Aug 4
oon Aug 2
Path of air and sea temperature for kite ascent of August 4th.
Fig. 1.
a degree. The temperature of the base of the atmosphere a t any point along the
air’s path has, therefore, been assumed to be th at of the surface of the sea. In many
of the kite ascents the temperature of the sea, and therefore of the surface air,
increased up to a certain point along the air’s path and then began to decrease.
While the air was moving along the first part of the path its tem perature m ight be
expected to decrease with height at a rate greater than the adiabatic rate.f When
* Others are reproduced in the ‘ Report of the “ Scotia” Expedition, 1913.’
t If the temperature of the air diminishes at the adiabatic rate of 10° C. per kilometre, its potential
temperature is constant, so that no amount of eddy motion can transfer heat either upwards or
downwards.
G G. I. TAYLOR ON EDDY MOTION IN THE ATMOSPHERE.
the air entered the portion of the path along which temperature was diminishing
it might be expected th at the cooling effect of the sea would not spread upwards
instantaneously, but th at it would make its way gradually into the upper layers.
We might expect, therefore, that, if a kite were to be sent up into the air as it was
passing over the second part of its path, the temperature would increase up to a
certain h e ig h t; and that, above th a t height, it would have the tem perature gradient
which it had acquired during its passage along the first portion of its path.
If a curve be drawn to represent the tem perature of the atmosphere at different
heights a change from heating to cooling along the air’s path will give rise to a
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corresponding bend in this curve. The height of this bend above the surface of the
earth will depend partly on the interval which elapsed between the time when the air
was passing over the portion of the path where heating stopped and cooling began
HEIGHT
IN METRES
uooo
and the time of the ascent, and partly on the eddy conductivity of the atmosphere.
I f we know two of these quantities we should be able to calculate the third.
On the right hand side of fig. 2 is shown the temperature distribution at various
heights from the surface up to 1100 metres in the case of the air which had blown
along the path drawn on the chart shown in fig. 1. It will be seen th at there are
two bends in the curve. The lowest portion from the surface up to 370 metres
evidently corresponds with the cooling of the lowest strata of the atmosphere which
had been going on ever since the air turned back from the warm water of the Atlantic
towards the cold water of the Great Bank of Newfoundland.
The air explored in the ascent of August 4th turned towards the west at 8 a.m. on
August 3rd and continued blowing on to colder and colder water till the time of the
G. T. TAYLOR ON EDDY MOTION IN THE ATMOSPHERE. 7
ascent, 8 p.m., A ugust 4th. I t appears, therefore, th a t the cooling had extended
upwards through a height of 370 metres in 36 hours. An arrow has been drawn on
the base line to represent the tem perature of the sea which, as we should expect, is
slightly less than the tem perature of the air which is being cooled by it. The portion
of the tem perature curve of fig. 2 which lies between 370 metres and 770 metres is
due to the warming which the air had undergone between the evening of July 30th
and 8 a.m., August 3rd. The portion of the curve above 770 metres to which the
warming of July 30th to August 3rd had not yet reached, is due to the cooling which
the air experienced as it blew off the warm land of Canada on to the cold Arctic water
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6 = -7 = f
V 7TJz (2tvdi )~2 \ 2 i W d f X *)
(£), say
where l = z (2 w'dt)~}iand \]s (£) represents the expression in square brackets.
The curve (a) in fig. 3 represents the values of \Js for values of £ ranging from 0 to
1'2. It will be seen that when £ = '8 the value of \Js is -1]0th of its value at the
surface, where £ = 0.
* See ‘ F ourier 's Series and Integrals,’ H. S. Carslaw , p. 238.
G. I. TAYLOR ON EDDY MOTION IN THE ATMOSPHERE. 9
where y(£) represents the expression in brackets and £ has the same meaning as
before.
The curve ( b)in fig. 3 represents the values of x (£) for different values of £
will be seen th a t when £ = 1*2 the tem perature is about TV h of the surface
tem perature 0O. In actual cases it is not easy to say whether (a) or (b) is a better
representation of the changes in tem perature along the air’s path. In most cases
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probably (a) is the best, but, in one case, th a t of the ascent of May 3rd, the bend in
the tem perature-height curve was due to the passage of the air across a sharply
defined boundary between the warm waters of the Gulf Stream and the cold arctic
water over the great Bank of Newfoundland,* and then one m ight expect (b) to be
a truer representation of the vertical tem perature distribution. In either case we
shall not be far wrong if we assume th a t the height to which the new conditions
have reached a t time t is given by £ = 1*0 or
z2 = 2 w
........ (2)
I f we can measure z, the height of the bend in the tem perature-height curve, and
if we know t, the interval which has elapsed between the time a t which the rate of
change of surface tem perature along the air’s path suddenly altered its value and the
time of the ascent, the equation (2) enables us to calculate or |-( ) . The error
in this result may be as great as 30 per cent., but it does at any rate give a good
idea of the magnitude of the coefficient of eddy conductivity and of the amount
of eddy motion which is necessary in order to produce the vertical temperature
distributions which have been observed.
In some of the cases the potential tem perature before the change which caused the
bend in the tem perature-height curve was not constant at all heights. In the case of
the upper bend in the curve shown in fig. 2, for instance, the potential tem perature
increased with height before the warming which produced the upper band occurred.
This, however, makes no difference to the rate at which the bend is propagated
upwards. I t is evident th at if 6>x and 02 be two solutions of ^ then
ct 2 dzJ
is also a solution. If the initial potential tem perature before the change were
0 = T0+ az, and if the surface temperature were to change suddenly to Tx at time
t = 0, the temperature at height z at a subsequent time t would be
0o = To+«*+T(T1- T o) x (O.
I t is evident th at the term T0+ az does not affect the rate at which the bend in the
temperature-height curve is propagated upwards.
* See ‘ Reports of the “ Scotia” Expedition, 1913.’
VOL. CCXV.— A. c
10 G. I. TAYLOR ON EDDY MOTION IN THE ATMOSPHERE.
T a ble I.
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&2 Average
Date of observation. 2, t, if, wind force
metres. hours.
in C.G.S. units. (Beaufort scale).
and 3 '4 x l0 3; and th a t for July 17th and July 29th, when the wind force was
about 2, the values were very much lower, being 0'57 x 10:! and 1’3 x 103 respectively.
The fact th a t these figures are so consistent, although t varies from 11 hours to
7 days and z from 140 metres to 770 metres, seems to indicate th a t the eddy motion
does not diminish to any great extent in the first 770 metres above the surface.
In the first part of this paper the vertical transference of heat by means of eddies
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has been discussed. For this purpose it was necessary to consider only the vertical
component of eddy velocity, but in the questions which are treated in the succeeding
pages it is no longer possible to leave the horizontal components out of the calculations.
I t seems natural to suppose th a t eddies will transfer not only the heat and water
vapour, but also the momentum of the layer in which they originated to the layer
with which they mix. In this way there will be an interchange of momentum
between the different layers. I f IL and V* represent the average horizontal
components of wind velocity a t height 0 parallel to perpendicular co-ordinates x and
y, and if v!,v', tv'represent components of eddy velocity so th at the three
of velocity are U z+ u', V z + v' and id, then the rate at which as-momentum is trans
m itted across any horizontal area is
disturbance has arisen from dynamical instability, or from disturbances transm itted
from the surface of the earth. The rate at which ^-momentum leaves a layer of
thickness Sz is
d
j j*/>(U* + u')dy Sz.
dzJ
But U z is constant over the plane xy and since there is no resultant flow of fluid
across a horizontal plane j*j pU zw'dx dy = XL j * j = 0.
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_ ilL — twice the vorticity of the fluid at the point (x, y> z).
cz ox
And since every portion of the fluid retains its vorticity throughout the motion, this
must be equal to twice the vorticity which the fluid at the point (x, y, z) had before
the disturbance set in. This is equal to the value of at the height, z0,# of the
layer from which the fluid at the point (x, ,y z) originat
by the symbol [dXJJdz]ZQwe see th at the dynamical equations of fluid motion lead to
the equation
dXJz dy/ _ dw' dUz
(«)
dz dz dx dz
f o /
Substituting in (4) tbe values of ~ and given by (5) and (
cz cz
f du , , d\Jz
—U - --- h » V +iv 'dU. —w -~ )
cx zo dz ) J
d (y/2—u' dU, d ll
P f\ \tP dx dy. (?)
dx dz •
The first term integrates and vanishes when a large area is considered; but the
second term does not vanish.
* z0 ia evidently a function of x and z when the motion is confined to two dimensions.
G. I. TAYLOR ON EDDY MOTION IN THE ATMOSPHERE. 13
So far nothing has been said about the m agnitude of the disturbance ; (8) is true
even if the disturbance be large. Let us now suppose th a t the height, z —z0> through
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which any portion of the fluid has moved from its undisturbed position is of such a
magnitude th a t the change in zdjJXni th at height is small compare
itself. In th a t case (8) becomes
d?_U,
I j"| tv'(z^—z) dx
dz2
P z —Zq) dx dy.
temperature. W hether this result is true when the disturbance takes place in Yhree
dimensions, I have been unable to discover.
If it is true, there is a relation k/ p<
t=
and pthe eddy viscosity; if any method of deducing from meteorological observa
tions could be found, it would be possible to verify the relation numerically.
We may expect to discern the effect of eddy viscosity in cases where the wind
velocity changes with altitude, and where the force due to eddy viscosity prevents the
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wind from attaining the velocity which we should expect on account of the pressure
distribution. These conditions arise near the surface of the earth. The velocity and
direction which we should expect on account of the pressure distribution, are called the
gradient velocity and the gradient direction. In general, the wind near the ground falls
short of the gradient velocity by about 40 per cent., and the direction near the ground
is about 20 degrees from the gradient direction. A t a height which varies on land
from 200 to 1000 metres the wind becomes equal both in velocity and in direction to
the gradient wind.
Let us consider the motion of air over the earth’s surface under the action of a
constant pressure gradient G acting in the direction of the axis of y. The equations
of motions of an incompressible* viscous fluid aret-
EE^ — X — -Q p + u
Dt p dx p
= y - - 5 £ + ^ v 2v > (9)
Dt pdy
Dt
= Z- E + p£
pdz i
d 2u
2oaVsin A + - , .. , ( 10 )
p
0 • . G , d2v
— 2 co u sin
A ------h - — ( 11)
p p dz
dz4
Taking into consideration the fact th at v does not become infinite for infinite values
of z ,th e solution of this is
v = A2e-Bz sin Bz + A4e~Bz cos Bz..................
(J/ajdz dvjdz
-=o.
.(14)
Where the square brackets are intended to show th at the values of the quantities
contained in them are to be taken at the surface of the ground, z = 0.
Substituting for u, v, dujdz and zdjv,and putting z = 0, equa
A 1 VT
A2+ A 4\ = = Aa+ Qc (15)
A2—A J A4 A4
gradient wind in such a way th a t if one stands facing the surface wind the gradient
wind will be coming from the right if a be positive.
Then
tan a — — (16)
z =0 A2+ Q (
\ / | y + o , =o = \( a 2+ q g)2
It is interesting to compare the value given by (17) for the ratio of Qs to QG with
the value, cos a, given by G u l d b e r g and M o h n # for the same ratio, and with the
most recent observations of wind velocity at different altitudes above the surface of
the earth.
Mr. G. M. B. D o b so n of the Central Flying School at Upavon has recently
publishedt the results of a number of observations made by means of pilot balloons
over Salisbury Plain, which is an excellent place for such observations on account of
its open situation. He finds th at a is smaller for light winds than for strong winds,
and he accordingly divides up his ascents into three classes, those which took place in
light* winds, when the velocity of the wind at a height of 650 metres is below 4'5
metres per second, those in moderate winds between 4'5 and 13 metres per second,
and those in strong winds above 13 metres per second.
The comparison is shown in Table II. I t will be seen that the observed deviation
of the surface wind from the gradient direction agrees well with the theory we have
been considering, but not with the theory of G u l d b e r g and M o h n .
The agreement between theory and observation is, however, more striking in
another respect. The deviation of the direction of the wind at any height from the
gradient direction is due to the retarding of the wind velocity below the gradient
velocity by friction or by viscosity. One m ight expect, therefore, th a t the wind
would attain the gradient direction a t the same height as the gradient velocity. This
would, in fact, follow from the theory of G u l d b e r g and M o h n . Most observations
have failed to give reliable information on this point, partly because irregularities on
the surface of the earth have introduced complicated conditions, which cannot be
taken account of, and partly because the observations have not been grouped according
to the wind velocity.* N either of these objections applies to Mr. D o b s o n ’s observa
tions. Salisbury Plain, though inferior to the sea, is as good a place for wind
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T able II.
observations near the surface as one could find on la n d ; and as has been explained
already, his results are grouped according to wind velocity. Mr. D o b so n finds th at
the gradient direction is not attained till a height is reached which is more than twice
the height at which the gradient velocity is first attained. He remarks, in fact, th at
the gradient velocity is usually attained at a height of 300 metres, though the gradient
direction is not found till a height of 800 metres has been attained. This is a most
remarkable result, but it might have been expected from the equations (12) and (13).
The height at which the gradient direction is attained is given hy putting — 0 in
(12). If H x be the height in question
A2sin BHi + A4cos BHj = 0
* Owing to the fact that n /p depends on the wind force we should evidently expect more consistent
results when the observations are grouped according to wind velocity.
VOL. COXY.----A. D
18 G. I. TAYLOR ON EDDY MOTION IN THE ATMOSPHERE.
so th at
tan BH4 = — ™
A2
Substituting for A2 and A4 their values in terms of a
Since a is positive and less than ^ the smallest positive value of Hi is given by
BH, = ^ + a .....................................................(19)
4
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The height H 2 at which the wind velocity first becomes equal to the gradient
velocity is given by u2+ v2 = QG2. This reduces to
e-Bii2 _ (1 + tan a)cos BH2—(1 —tan a )sin BH
tan a
Equation (20) can be solved so as to give tan a in terms of BH2, and when several
corresponding values of a and BH2 have been obtained BH2 can be obtained by
interpolation in terms of a. In Table III. are shown the values of BHj and BH2 and
H 1/H 2 corresponding to values of a from 0 to 45 degrees.
T able III.
a. b h 2. BHi. Hi
h 2‘
|
30 degrees 1-20 2-88 2-4
__________
45 degrees 1-44 3-15 2-2
In order more easily to compare the theoretical results with the observations the
curves shown in fig 4. have been prepared. Fig. 4 shows the way in which wind
Fig. 4.
Calculated Curves
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20* IO°
WIND VELOCITY IN FRACTIONS OF GRADIENT W IN D DIRECTION
2400
Fig. 5.
Observed Curves
■ 1600
• 400
-410-----1
12
■14*
WIND VELOCITY IN METRES PER SECOND W IND DIRECTION
velocity and direction vary with height in the theoretical case we have been con
sidering when a = 20 degrees. Fig. 5 is reproduced by permission of Mr. D obson. It
d 2
20 G. I. TAYLOR ON EDDY MOTION IN THE ATMOSPHERE.
represents the observed velocity and direction of strong winds at different heights.
In each of the figures the curve on the right represents deviations from the gradient
direction, which is shown as a vertical line. The curve on the left represents wind
velocity at different heights.
I t will be seen th a t there is good agreement between the two sets of curves.
Strong winds have been chosen for the comparison in preference to light winds,
because it is less likely th a t heat-convection currents will persist through such a
distance before mixing takes place, as to prevent the resistance, due to eddy motion,
from obeying the ordinary laws of viscosity. The observed curves for light winds,
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however, agree as well with the theoretical curves as those for strong winds.
Besides the various points of resemblance already noticed between theory and
observation, an inspection of the curves in figs. 4 and 5 reveals yet another.
Above the height at which the gradient direction is attained the wind goes on veering
slightly up to a certain height, when it begins to return again to the gradient
direction. The wind is again blowing along the gradient direction at a height
slightly less than twice the height at which it first attained it. Nearly all the curves
in Mr. D o b so n ’s paper have this characteristic sinuosity, but they are not the only
ones which show it. Mr. J. S. D i n e s , in his Third Report to the Advisory Committee
for Aeronautics (1912), has published a number of curves which exhibit the same
sort of sinuosity. The theoretical curve, fig 4, has the same characteristic. The
successive heights H l5 H 'l5 H'^, ... at which the wind blows exactly along the gradient
direction are given by the solutions of equation (18).
37T
We have already obtained the first solution, namely BHX= ~^ +a.
W e have seen th at
If* therefore, we can measure a and H x we can calculate B. The commonest value
for a on land is 20 degrees, in fact, for all except light winds, it is near to 20 degrees.
In the kite ascents on the “ Scotia ” the wind usually veered two points (22|- degrees)
in the first 100 or 200 metres and after th a t remained constant in direction at greater
heights. I t appears, therefore, th a t on the sea also a is about 20 degrees. Assuming
then th a t a = 20 degrees, we see from Table III. th a t B H X= 2*7.
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sin X
P (2’7)2
But co, the angular velocity of the earth, is 0*000073 ; and in latitude 50 degrees N.,
which is the latitude of the South of England and also of the northern portions of the
Bank of Newfoundland, sin X = 0*77.
At sea,t in the regions to which the “ Scotia’s ” cruises were confined, H! commonly
lay between 100 metres and 300 metres so th at p\p lay between 0*77 x 103 and
6*9 x 103.
Except for the kite ascent of July 17th, 1913, the values of which, as was shown
on p. 14, should be equal to p/p, lie between these values.* I t is unf
lack of skilled assistance in flying the kites from the “ Scotia ” prevented me in most
cases from being able to get simultaneous values of and For the kite ascent
of August 2nd, however, I have the following observations :—A t 350 feet the wind had
veered one point from the surface wind. A t 770 feet the wind had veered two points
from the surface wind, and at all greater heights the veer was two points. I t seems,
therefore, th at at 770 feet, i.e.,230 metres, or at some less hei
attained the gradient direction, so the p/p lay between 0’77 x (23,000)2x 10~5 or
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4'0x 103 and 077 x 103. On referring to Table I. it will be seen th a t the value of
k/ pct on th at occasion was 2 '5 x l0 3. These results certainly tend to confirm the
theoretical deduction th a t k/ pa- = p/p, but more evidence is wanted before the point
can be regarded as finally settled.
On p. 13 it was shown th at p/p = | - ( dw.) The size of the
effects we have been considering, are evidently governed by W e may say roughly
th at d is less than the average diameter of an eddy ; if therefore we could measure ,
we should be able to determine the size of the eddies. Now Mr. J. S. D in e s has made
a large number of observations of small vertical gusts with tethered balloons. On
p. 216 of the Technical Report of the Advisory Committee for Aeronautics is shown a
trace which represents the vertical component of the wind velocity at any time during
a certain interval of five minutes, on January 19th, 1912. The average wind velocity
during the interval was 7 metres per second ; and I find from the trace, which Mr. D in e s
says is typical, th at the average deviation from the mean vertical velocity (the mean
wind was not quite horizontal) was 25 cm. per second. W e may take this as w.
Assuming th at the gradient direction was attained at a height of 800 metres the
value of ^(wd) would be 50 x 103 or wd — 105 approximately.
Hence
105
d = —— = 4 x 1 03 cm. = 40 metres.
25
The wind was blowing with velocity 7 metres per second so th at it would cover
7 x 60 = 420 metres, or about 10 times d, in a minute. If the vertical and horizontal
dimensions of an eddy are about the same, this would mean (since d is less than the
diameter of an eddy) th at rather less than 10 eddies would pass a given spot in a
minute. On examining Mr. D i n e s ’ trace it will be found th at there are roughly
about 6 peaks per minute on the curve representing vertical velocity.
These calculations are very rough, but they do show at any rate, that actual
observations of eddy motion do not involve anything th at is contrary to the
assumptions on which the theory contained in this paper is based.
* See Table I.
G. I. TAYLOR ON EDDY MOTION IN THE ATMOSPHERE. 23
The equation (8) throws a new light on the much discussed question of the stability
of the laminar motion of an inviscid fluid.
Lord R a yleigh has considered the stability of a fluid moving in such a way th a t IT,
the undisturbed velocity, is parallel to the axis of x and is a function of z. His method
is to impose a small disturbing velocity of a type which is simple harmonic with respect
to x,satisfies the equations of motion, and contains a factor He then discusses
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the conditions under which n may be complex. If is not complex the motion is
stable ; if ni
s complex the motion is exponentially unstable.
Perhaps the most important result of Lord R ayleigh ’s investigation is the conclu
sion he arrives at th a t if d?TJfdz2does not change sign in the space, bet
bounding planes, unstable motion is impossible. A particular case of laminar motion
in which d2XJfdz2has the same sign throughout the fluid is th a t of an inviscid fluid
flowing with the same velocity as a viscous liquid moving under pressure between two
parallel planes. In this case, therefore, unstable motion should be impossible.
O sborne R eynolds, however, working in an entirely different way, has come to the
conclusion th at a viscous fluid moving between parallel planes is unstable if the
coefficient of viscosity is less than a certain value which depends on the distance between
the planes and on the velocity of the fluid. R eynold’s result is in accordance with
our experimental knowledge of the behaviour of actual fluids.
I t is evident th a t there is a fundamental disagreement between the two results for,
according to R eynolds, the more nearly inviscid the fluid, the more unstable it is
likely to b e ; while according to R ayleigh instability is impossible when the fluid is
quite inviscid.
Various attem pts have been made to find the cause of the disagreement, but none
of them appear to have been very successful.
The object of this note is to show th at equation (8) may be used to prove the tru th
of Lord R ayleigh ’s result for the case of a general disturbance, not necessarily harmonic
with respect to x ; and to show also th at it may be used to assign a reason for the
difference between R ayleigh ’s and R eynolds’ results.
Starting from the principle th at when an inviscid fluid in laminar motion is disturbed
by dynamical instability, each portion of it retains the vorticity of the layer from which
it started, it was shown* th at the rate at which momentum parallel to the axis of x
flows into a slab of area A and thickness Sz is :—
This expression is true for all disturbances, however large, but when the distance
z0— zis small the first term only is of importance. Now it is evident th a t w is related
to z0—z by the relations
S w=o> S (z)= w-
Hence
-IV,
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or
( l j + u) ^ ( z 0- z ) + w ^ ( z 0- z ) + = -w .
- w = | ( * 0- * ) + u £ (* 0- 4
Hence
Now when a large area is considered j j ^ ( z 0—z) integrates out and vanishes.
Hence
jj^w(z0- 2 -JJ
)dxdy- £^jj
\ ^ { z ^ - z )2d x d y =
! d2U , d
Jj dxd
*p d ? Szdt
Integrating with respect to t we find th at the difference between the momentum
in the slab A after and before the disturbance set in is
d2U
■IP ~dz?Sz dx dy'
has removed it, increases with time. This evidently includes the case of exponentially
unstable simple harmonic waves.
Hence the rate a t which x-momentum enters the slab A is positive or negative
according as d2J J / d z 2is positive or negative. In an unstable disturbance of a
which d2J
zd/J2 is everywhere positive the momentum of every layer must increase. But
if there is perfect slipping a t the boundaries no momentum can be communicated by
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them. Hence, as there is no other possible source from which the momentum can be
derived, instability cannot possibly occur. The argum ent applies equally well if
d2XJI dz2 is everywhere negative. Lord R a y l e i g h s result is therefore proved for a
generalised disturbance. In a case where d2JJ/dz2 changes sign a t some point in the
fluid any disturbance reduces the x-momentum in a layer where d2XJ/dz2 is negative,
while it increases the x-momentum in layers in which d2XJfdz2 is positive. A type of
disturbance which removes x-momentum from places where d2XJ/dz2 is negative and
replaces it in regions where d2TJ/dz2 is positive, so th at there is no necessity for the
boundaries to contribute, may be unstable.
Now consider what modifications must be made in the conditions in order th a t
instability may be possible in the case where is of the same sign throughout
(say negative). Suppose th a t instability is set up so th a t x-momentum flows outwards
from the central regions as the disturbance increases. The amount of x-momentum
crossing outwards towards the walls through a plane perpendicular to the axis of 0 ,
increases as the walls are approached. In order th a t instability may be set up this
momentum must be absorbed by the walls. There seems to be no particular reason
why an infinitesimal amount of viscosity should not cause a finite amount of momentum
to be absorbed by the walls.
In connection with this two points should be noticed. Firstly, the momentum is
only communicated to the walls while the disturbance is being produced. The time
necessary to produce a given disturbance may increase as the viscosity diminishes.
Experimental evidence, however, does not suggest that this is the case.
The second point is suggested by the conclusion arrived at on pp. 11-22, th at a
very large amount of momentum is communicated by means of eddies from the
atmosphere to the ground. This momentum must ultim ately pass from the eddies
to the ground by means of the almost infinitesimal viscosity of the air. The actual
value of the viscosity of the air does not affect the rate at which momentum is
communicated to the ground, although it is the agent by means of which the
transference is effected.
In any case it is obvious th at there is a finite difference, in regard to slipping at
the walls, between a perfectly inviscid fluid and one which has an infinitestimal
viscosity. The distribution of velocity acquired by a viscous fluid flowing between
VOL. CCXV.----A. E
26 G. T. TAYLOR ON EDDY MOTION IN THE ATMOSPHERE.
parallel planes at which there is no slipping is possible for an inviscid fluid when
there is perfect slipping, but is impossible as a steady state for an infinitesimally
viscous fluid which slips at the boundaries.
The finite loss of momentum at the walls due to an infinitesimal viscosity may be
compared with the finite loss of energy due to an infinitesimal viscosity at a surface
of discontinuity in a gas.#
I f these views are correct we should expect th at Lord R ayleigh ’s result would
not apply when there are no bounding planes and space is filled with a fluid in which
d2\J/dz2 is everywhere positive; for, in th at case, there would be nothing to prevent
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* See “ Conditions Necessary for Discontinuous Motion in Gases,” T aylor , ‘Roy. Soc. Proc.,’ 1910,
A, vol. 84, p. 371.
t ‘ Phil. Mag.,’ vol. 26, 1913, p. 1002.