A1 4
APFENDIX
3.
At vertical incidence the angle between the wave normal and the ray direction is o< , which is
given by (Davies 196 6 Sec. 2.5.2 ) tan o< -
1 ....2!±.
- µ d0
e
= angle between the wave normal and the magneticfield (2 2 ° at Birdling's Flat). ignoring collisions (z = o) gives
Using eg_uation tan 0< = 2 (µ -1) YT YL +
---=-
-
J
Y.! + 4 (1-X)2 Y 2 T Lwhere the + and - signs refer to the ordinary and extraordinary rays respectively.
A1 5
APPENDIX
4.
REFLECTIONS FROM A SINUSOIDAL LAYER .
The effects of specular reflections from a sinusoidal layer have been discussed by Bramley (1 953) and Austin (1967) who were concerned with direction finding and amplitude variations respectively. This
appendix considers the changes in range as the re�cting ripple moves overhead.
Let the layer be at �0�gst average height h
and the undulation have the form
where
fi
= a sin (k >C X - wt) a = amplitudeA = 2�/k = horizontal wavelength
Provided h is very much greater than the amplitude (which in this case means a S 10 km) the range to the specular point is given by
(A4.1)
so that where tan
e
= x/h = -d9'/d X cos y = -A (y + wt) y = k X - w L 2 2 A = 1/47C
ah A16 (A4i2)x. can be f'ound by a graphical solution of' the trans cidental e4uation A4.2 and so r can be f'ound f'rom A4.1.
The phase path is assumed to be P = 2r.
It is noted that two specular ref'lections will occur when the line t = -A (y + wt ) just touches the
curve t = cos y (i.e. A = 1). Since the greatest value of' the slope of' cos y is one if' A is less than one there will be at least 3 specular points f'or some
values of' (wt) .
The range changes f'or dif'f'erent values of' the parameter are shown plotted in Fig.4.
A1 7
APPENDIX 5.
SOME ASPECTS OF GRAVITY WAVE THEORY.
In previous chapters it has been suggested that a class of atmospheric motions known as internal gravity waves are an important feature of the atmos
phere. It has been argued that some of the observations made in the present experiment are well explained by
the properties of these waves, although many of the discussions were of a qualitative nature. This appendix attempts to present the theory of gravity
wave propagation in a more rigerous fashion with
special emphasis on those features which the author feels are more important for the ionosphere. For other works on internal gravity waves, the reader is referred to Hines (196 0) and Midgley and Liemohn
(
196 6). The wider aspects of hydrodynamic wave motion in general have been discussed in a fairly
abstract sense by Ecaart (196 0) but a good understanding of these waves may be obtained from an excellent paper by Tolstoy 196 4( ).
Ao 6 .1 NOTATIOK.
x,z g C H y w ''t' V u , u X Z X w B u u x, z V ,V X Z A1 8 Cartesian co-ordinates, z vertical upwards Perturbed atmospheric density and pressure. P1 = Pz (1 + p) where pz is the equilibrium pressure and p is the fractional perturbation
(6P/p) produced by a superimposed wave motion, similarly for P1 = Pz (1 + P)
(o,o,-g)
2
Velocity of sound, c = vp /I Z Z p
Scale height c2/yg Ratio of specific heat
Steady background wind in x direction Operation
:t
+ U0 :xAngular wave frequency and period
x component of wave number, assumed real Doppler shifted wave number = w-kx U0
Horizontal and vertical particle perturbation velocities
Velocity divergence = -2£ + --L
au au
ax a
zVaisala Brunt frequency
Horizontal and vertical group velocities,
Horizontal and vertical phase velocities, v/kx' v /kz
A1 9
A.5.2 BASIC EQUATIONS.
The theory of hydrodynamic wave motions usually assumes that the waves are of perturbation magnitude only, second order effects being negligible. For present purposes the basic assumption is made
that the motions are taking place in a fluid which is moving with a constant velocity and the only
external force is that due to gravity acting vertically
downwards. The neglect of the Coriolis force restricts the discussion to motions with periods a few hours or less. Since the only preferred direction is imposed by gravity the wave motions can be discussed in a two dimensional co-ordinate system, the axis x being horizontal and that of z vertical positive upwards. The wave motions are then described by the following equations
Dy + 2-._ VP _ �
Dt Pz 1 Pz g = 0 ( 1 )
(2)
A20 These are the linearized Etulerian equations of motion, continuity and energy (Lamb 1 945). The basic assumption also made that the atmosphere is in hydrostatic equilibrium
oP
--A = - p g
a
zz
From equation (4) and using the relationship
p .(: z and Pz 'tA. Hence Pz = t and Pz = -fd
/H
e z1
H e z -f d/H
- p /H z (1 + -pz H H')(4)
Where a prime indicates differentiation with respect to z. Rearranging equations ( 1 ) to (3 ) such
P1 'o1 are replaced by terms in p , Pz and P,P z assuming that the wave,propagate harmonically the x direction and with respect to time such
,-< i (wt - k x) u , u , p, p "" e x X Z that and in that
the equations (1 ) to
(3)
reduce to ivp ivU z -U � 1 H = k gHp X = g (p-Hp I ')-gp = y (i vp -u
z X =-
(i vp -u
z (1 + H 1 � H (1 + H 1 ) H A21 ( 5) (6) (7) (8)Eliminating U ,P, P X from equations (5) and t7) gives
the equations (v2 - k 2 c2 ) k 2 u1 = X X + X g
u
z V 2 y2 z 1 gk 2 2 2 k 2 X' =(- -
-L ) X - (v - g X ) H c2 y2Eliminating U from (9) and ( 1 0 ) gives
z
2
C X ' ' + (H' ; 1 )x'
+ (kx2 (wB2 - v2) +�:)x
= o C (9) ( 1 0 ) ( 11 )The term � is the Vais�l� Brunt frequency and is the frequency at which a small parcel of air will oscillate at if it is displaced fro m its equilibrium position
A22 (Tolstoy 1963). In the notation used above
= W:k 1 + Y -11
WB
gr
y HJ
which is directly equivalent to (6.2) provided only
changes in the velocity of sound (c2 oe. T) are con sidered.
A5.3 PROPAGATION OF ATMOSPHERIC WAVES.
Using equation (11) it is now possible to discuss the propagation of waves in the vertical direction. So far the effects of temperature vari-
ations have been included but if these are neglected, the atmosphere assumed isothermal, the coefficients in the equations are constant which simplify the discussion considerably.
this assumption is made.
As a first approximation
(i) ISOTEERMAL ATMOSPHERE. Equation (11) is of the form
X'' + a (z) X ' + b (z) X + 0 ( 1 2 )
gives where r{ ' X = TJ e z/2H + k 2 z T} = 0 k 2 = k2 ( °13 2 - V 2 ) , 2 A23 ( 13 ) + v 2 w a 2 c2 ( 1 4 )
and �a = C = acoustic resonant frequency
2H
-(Tolstoy 1963 ) and is the highest frequency at which the atmosplhere will oscillate as a whole. The first thing that should be noticed is that X is proportional to exp (z/2H) so that a wave propagating upwards will
experience a growth in amplitude by this amount. The physical reason is that the knetic energy of the waves is ½ pz
u
2, where U is the perturbation velocity , and so the amplification ensures that the kinetic energy remains constant as the atmospheric den�ity decreases.There is a corresponding amplitude decrease for wave• propagating downwards. (13 ) is the usual form of the
wave equation and has an oscillating solution when k z 2 � o, so that k can be regarded as the vertical z wave number. The wave will propagate in the vertical direction with wavelength � = (z 2�/k )z . When k z 2 < o
the wave will not propagate vertically but merely decay by a factor exp \2�/ kz\ •
If kz2 > o the waves are usually called internal (or cellular) and if k 2 < o these are referred to as z
A24
evanescent (or external or non-cellular). Examination of (14) show that the waves will be internal provided
2 2 2 2
v > wa or v < wB . These are classed as acoustic and gravity modes respectively. The division between
the classes can best be seen by referring to what is usually termed the diagnostic which shows k z 2 curves
plotted in (w, kx) space - Fig. 28. w is the low a frequency cut-off for the internal acoustic modes and wB is the corresponding high frequency cut-off for internal gravity modes. It is not proposed to discuss acoustic modes further but to concentrate on gravity waves.
2
V
For most purposes
- i
2c2
is much less than
k 2 X
('"B
2 - v2)which is equivalent to saying that2
V 2 << X C • 2 In the atmosphere for heights up to 100 km
this restricts the discussion to waves with V x � 1 0Om/ sec.
2 w 2 - v2 2
k Z = B --- k X
y2 ( 1 5 )