15.1 Introduction

In Table 11.1, we referred to a few types of atmospheric waves which concern meteorologists most of the time because of their apparent relations with the formations of weather and climate. However, in the atmosphere as well as the ocean, depending upon the fluctuations in pressure, temperature and wind, several types of waves and oscillations may be excited.

A fluid parcel when displaced from its position of equilibrium in a stable atmosphere by an external force will tend to return to its original state by a series of movements which usually take the form of oscillations or wave motions. For example, a perturbation in the pressure field will generate sound waves which vibrate longitudinally and propagate at normal temperature and pressure with a velocity of ~ 330ms-1 in the direction of vibration. Here the restoring force is pressure as per the compressibility of the fluid. A displacement in the density field produces gravity waves which are transverse waves and oscillate in a direction at right angles to the direction of propagation. They also move fast at an average speed of ~ 200ms-1. In the case of the gravity waves, the restoring force is the pull of the Earth's gravity. On the other hand, in a barotropic atmosphere certain types of slow-moving planetary waves are excited by the variation of the Coriolis parameter with latitude. These are called the Rossby waves. Pure Rossby waves oscillate in the horizontal plane and usually move westward relative to the mean wind with a phase velocity which is generally much smaller( ~ 5-10ms-1 ) than that of the sound or the gravity waves. Of these, the Rossby waves are the ones which are of the greatest meteorological interest because of their association with synoptic-scale weather.

The above-mentioned waves and oscillations which are perturbations in the mean motion of the atmosphere can be derived from the general equations of motion only after making some simplifying assumptions, since the equations are nonlinear and cannot be solved by any known method. A perturbation technique has been devised to linearize them.

However, before we describe this technique and apply it to the case of atmospheric waves, we describe some of the general properties of harmonic motion, such as amplitude, frequency, period, wave-length, phase velocity, etc., starting with the working of a simple pendulum which is an example of a non-propagating simple harmonic motion.

In a simple pendulum, a round metallic bob of mass m is suspended from a support with a weightless string of length about l m. In its rest position, the string hangs vertically under the force of gravity, mg (see Fig. 15.1).

The bob is then displaced to one side through a small angle 9 to a position P where it is let go to oscillate freely. Passing through the vertical at O, the bob will move to the other side to a position P' and then move back to continue the oscillation. At P, the equilibrium is reached when the force of displacement is balanced by the opposing force of gravity and the relevant equation is m l d29/dt2 = -m gsin9 (15.2.1)

where l denotes the length of the pendulum. Since 9 is small, sin 9 differs little from 9. So, (15.2.1) may be written d2 9/dt2 = —(g/l)9 = —|2 9 (15.2.2)

The general solution of the differential equation (15.2.2) may be written

where A, B, 90 and a are all constants which can be determined from initial conditions.

Thus the solution is a periodic oscillation with an amplitude 90 (the angle of maximum displacement) and a frequency | (or a period 2n/|). The angle (|t-a) is called the phase angle which varies linearly in time by a factor of 2n radians for every period of oscillation, and a is the phase difference between two oscillations.

In the case of an oscillation or wave propagating in space, say along the x-axis with a velocity c, another term is to be taken into consideration to determine the phase, and that is the wave number k which represents the number of waves of a particular wavelength X around a latitude circle. Thus, k being equal to 2n/X, the phase of the travelling wave is kx- (|t-a), or, (kx - |t+a). For an observer moving with the wave, the phase speed c is constant and given by the relation, c = |/k.

Though atmospheric waves are never purely sinusoidal, a wave or perturbation can be represented as a function of longitude by a Fourier series in terms of a zonal mean plus a series of sine and cosine terms. Thus, a function f(x) may be written as f(x)= X (Amsinkmx + Bmcoskmx) (15.3.1)

where km = 2nm/L is the wave number of the mth wave (m being the number of waves and L the length of a latitude circle) and Am and Bm are the amplitudes of the sine and the cosine parts respectively of the mth wave. Here, the value of the zonal mean of the function is assumed to be zero.

The co-efficients Am and Bm are calculated by first multiplying both sides of (15.3.1) by sin knx and then integrating them around a latitude circle and applying the orthogonality relationships

f sin(2nmx/L) sin(2nnx/L) dx = 0 or L/2, according as m = n or m = n, o

f cos(2nmx/L) sin(2nnx/L) dx = 0, for any m,n combination o

The calculation yields, Am = (2/L) y f(x) sin(2nmx/L) dx (15.3.3)

A similar calculation resulting from multiplication of both sides of (15.3.1) by cos kn x and integration around a latitude circle and application of orthogonality relationships gives

where fm (x) is called the mth harmonic of the function f(x) and Am and Bm are called the Fourier co-efficients of that harmonic.

A Fourier harmonic can also be expressed more conveniently and in a more compact form by using complex exponential notation as in the Euler formula eix = cos x + isin x, where i = y/( — 1) Thus, we can write, fm(x)= Re{Cmexp(i kmx)} (15.3.6)

where Re denotes the real part of { }, and Cm is a complex co-efficient. Comparison of (15.3.6) with (15.3.5) shows that

The computation of Fourier co-efficients enables us to identify the wave components which contribute to an observed perturbation of a meteorological variable, such as the height of an isobaric surface along a latitudinal circle. Sometimes one or more dominant wave components may be identified which account for most of the observed variation. If only a qualitative representation is desired, then in most cases a limited number of components may suffice.

In the case of a harmonic oscillator, the frequency of the oscillation depends only on the physical characteristics of the oscillator and not on the motion itself. For a propagating wave, however, the frequency depends on the wave number as well as the physical characteristics of the medium. Thus, the phase velocity depends on the wave number, since c = |/k, unless, of course, | is a function of k. Waves in which the phase speed varies with the wave number are called dispersive waves and the formula relating the frequency with the wave number is called the dispersion relation.

However, not all waves in the atmosphere are dispersive. For example, acoustic waves are non-dispersive. Their phase speed is constant, regardless of the wave number. When two waves of the same amplitude but differing slightly in frequency and wave number are superposed on each other in the course of their propagation in the same direction, the amplitude of the resulting wave fluctuates as the interacting waves get in and out of phase. Amplification occurs where they get in phase and energy gets concentrated, whereas the amplitude dies down where they get out of phase. The occurrence of such periodic ups and downs of amplitude in acoustics is known as 'beats'.The resulting wave thus moves as a wave group with its amplitude fluctuating in time (see Fig. 15.2).

In what follows, we derive an expression for the propagation of a wave group. Let two waves of equal amplitude but differing in frequency by 2A^ and wave number 2Ak be superimposed on each other.

Then, the amplitude of the resulting wave may be written as

A(x, t) = exp[i{(k + Ak)x - + A^)t}]+exp[i{(k - Ak)x - (|i - A^)t}]

Re-arranging, we get A(x, t) = [exp {i(Ak x - A^ t)} + exp { - i(Ak x - A^ t)}] exp {i(kx - ^t)}

Or, A(x, t) = 2cos(Ak x - A(i t) exp {i(kx - ^t)} (15.4.1)

The meaning of (15.4.1) is as follows: It represents a high-frequency carrier wave of wavelength 2n/k and phase velocity ^/k, whose amplitude is modulated by a wave-like periodic variation, or a wave-group, of wavelength 2n/Ak and phase speed A^/Ak. Thus, the amplitude of the wave-group which is indicated by an envelope in the lower part of Fig. 15.2 travels at a speed which is different from the speed of the carrier wave. If Cg denotes the group velocity, its relationship with the phase speed c of the carrier wave may be found as follows:

At the limit when Ak ^ 0, Cg = d^/dk. Since, c = ^/k and k = 2n/X, where X is the wavelength, we find

It may be noted that when the wave-group is non-dispersive (i.e., dc/dX = 0), it travels at the same speed as the carrier wave.

15.5 The Perturbation Technique

The basic assumptions of this technique are the following:

Assumption 1. A meteorological variable, such as pressure, temperature, wind or density, can be split into two states: a basic state that is assumed to remain invariant with time (t) and the co-ordinate axis along which it is measured (say, x), and a perturbed state that changes with these variables. For example, if we apply the technique to the zonal component of the wind u, it can be written as u(x, t)=u + u'(x, t), (15.5.1)

where u ( u underlined) denotes the mean basic state of u, and u' (primed) the perturbation.

Assumption 2. The basic state variables must on their own satisfy the governing equations when there is no perturbation. This obviously requires that the perturbation field must remain small enough for the product of the perturbation variables to be neglected. This requires that |u'/u| C 1, and, for the nonlinear term udu/dx of the governing equations, for example, we can write:

But, since u d u'/dx > u'du'/dx, the assumption leads to the result udu/dx = u du'/dx (15.5.2)

The technique, therefore, enables us to reduce the nonlinear differential equations to linear differential equations in perturbation variables in which the basic state variables appear as constant co-efficients to the perturbation terms. The linearized equations can then be solved by known techniques to yield information regarding the properties of the perturbation. Since, in most cases, a perturbation is assumed to have a sinusoidal form, the solution enables us to find the various characteristics of the wave form such as its frequency, wavelength, phase velocity, etc.

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