## Simple Wave Types

As examples of pure waves, we consider three types of wave motion in this section:

(a) Acoustic or sound waves in which the fluid movements are longitudinal, i.e., they move in the direction of wave propagation and the restoring force is compressibility of the fluid;

(b) Gravity waves in which the fluid movements are transverse, i.e., perpendicular to the direction of wave propagation and the restoring force is the earth's gravity; and

(c) Rossby waves which are horizontal transverse waves perpendicular to the direction of wave propagation and in which the restoring force is the variation of the Coriolis parameter with latitude.

In what follows, we discuss some aspects of their formation and propagation: (a) Acoustic or sound waves

In sect. 3.4.5, we showed how the laws of thermodynamics could be applied to deduce the Laplace's equation for the velocity of sound in air. The basic idea there was that the process of compression and rarefaction produced by the sound waves occurred so rapidly that the changes could only be termed as adiabatic. Here, we use the perturbation method to derive an expression for the velocity of sound in air. For simplicity, we assume that the oscillation is entirely longitudinal and occurs along the x-axis with no transverse component along the y or z axis. This means that in the momentum equations, v = w = 0 and there is no dependence of the perturbation variables along the y and the z axes. With these restrictions, we write the momentum, continuity and thermodynamic energy equations as du/dt +(1/p)dp/dx = 0 (15.6.1)

where u is the zonal component of the wind, p is pressure, p is density and 0 is potential temperature, and d/dt = d/dt + udu/dx.

Since 0 = (p/pR)( 1000/p)R/Cp , we can eliminate 0 from (15.6.3) and write

where y = cp/cv. Eliminating p between (15.6.4) and (15.6.2), we get

We now divide the dependent variables, u, p and p into their basic state portions (underlined) and the perturbed portions (primed) and write u(x, t) = u + u'(x, t)

Substituting (15.6.6) into (15.6.1) and (15.6.5), we get d(u + u')/dt +(u + u')d(u + u')/dx + {1/(p + p')}d(p + p')/dx = 0 d(p + p')/dt +(u + u')d(p + p')/dx + Y(p + p')d(u + u')/dx = 0

Since, by the perturbation theory, |p'/p| <<< 1, we can simplify the density term {1/(p + p')} to (1/p). Neglecting the products of the perturbation quantities and noting the basic state parameters are constant, we can write the linearized perturbation equations

Differentiating (15.6.7) with respect to x and eliminating du'/dx from (15.6.8), we obtain

The differential equation (15.6.9) is a wave equation. We assume a solution of the form p' = Aexp{ik(x - ct)} (15.6.10)

Substituting in (15.6.10), we find that the phase velocity c must satisfy the relation

Solving (15.6.11) for c, we get c = u± v^Yp/p) = u^v^yRI) (15.6.12)

Equation (15.6.10) is , therefore, a solution of (15.6.9) provided that the phase velocity is given by the relation (15.6.12) which states that the adiabatic speed of the sound wave relative to the zonal wind is ±^/[yp/p) or iy^R!). The mean zonal wind in (15.6.12) plays the role of Doppler shifting the sound wave, so that frequency of the wave is given by the relation

According to (15.6.13), the frequency of the sound wave will appear to be higher when an observer is downstream of a source than when he is upstream.

(b) The gravity wave

Most of us are familiar with the type of waves that are produced on the surface of water in a pond when, for example, a stone is thrown into it. Where the stone drops, the energy of the impact creates a set of waves at the water surface which propagate outward in all directions in ever-widening circles till the energy is dissipated by friction. This is clearly a case of gravity waves the formation of which can be understood in the following way (see a schematic in Fig. 15.3).

Let us take a vertical section along the x-axis through a pond and assume that a stone falls at a point O on the water surface where it creates a depression of the surface to a certain depth. The removal or divergence of water from O lowers the hydrostatic pressure at O but increases it at the side point A where water is diverted

Fig. 15.3 Vertical section illustrating the formation of water waves and their propagation Fig. 15.3 Vertical section illustrating the formation of water waves and their propagation to and converges. The gradient of pressure then drives the water back to O thereby raising the pressure at O due to convergence but lowering it at A due to divergence from there. As a result of this movement, the water level rises at O but falls at A. The fall of pressure at A then in turn leads to convergence at A but divergence at the side point B. In this way, the pattern of divergence and convergence set in motion by the impact of the stone propagates outward in the form of a pressure or gravity wave. It should be noted that pressure at individual points increases (decreases) as the water level rises (falls) as the wave passes.

In what follows, we use a two-layer model to derive an expression for the phase velocity of a gravity wave moving along the interface between two layers of fluids of different densities. We assume that the layers are homogeneous incompressible fluids of constant densities p2 in the upper layer and p1 in the lower, with p1 > p2. The stratification is, therefore, vertically stable. In a geophysical fluid, however, the assumption of a uniform density in a medium, such as the ocean or the atmosphere, is seldom valid, though across their interface, sharp differences in their density do occur. So, the treatment given here would apply to such an interface between two media with sharp differences in densities. The working of the model is shown schematically in Fig. 15.4, which is a vertical section in the x-z plane through the interface. Fig. 15.4 Gravity wave in a two-layer model

For simplicity, we assume that there is no horizontal pressure gradient in the upper layer. We can get the horizontal pressure gradient in the lower layer by vertical integration of the hydrostatic equation. Thus, between the points B and A in Fig. 15.4 we have the pressure difference

§p = pB -pA = g(p1 - p2)(dh/dx)5x where dh/dx is the slope of the interface and 5x is the distance between the points. In the limit 5x ^ 0, the pressure gradient = g(Ap)dh/dx, where Ap = p1 - p2. We use this value of the pressure gradient in the x- momentum equation and obtain du/dt + udu/dx + wdu/dz = -g(Ap/p1)dh/dx (15.6.14)

The continuity equation in this case is du/dx + dw/dz = 0 (15.6.15)

Since the pressure gradient is independent of z, u is also independent of z, provided that u = u (z) initially. This makes du/dz = 0 in (15.6.14). On vertical integration of (15.6.15) from z = 0 to z = h, therefore, we get h w(h) - w(0) = -/du/dx5z = -hdu/dx 0

If we assume the lower boundary to be a level surface, w(0) = 0. Further, since w(h) = dh/dt = dh/dt + udh/dx, we can write the continuity equation (15.6.15) as dh/dt + d(hu)/dx = 0 (15.6.16)

The Eqs. (15.6.14) and (15.6.16) are a closed set in variables u and h and can be solved by the perturbation method. We let u = u + u';and h = H + h'

where u is the constant basic state zonal velocity and H is the mean depth of the lower layer with H > h'. The perturbation forms of (15.6.14) and (15.6.16), after neglecting the product of the perturbation variables, are then dh/dt + udh'/dx + g(Ap/px)dh'/dx = 0 (15.6.17)

Eliminating u' between (15.6.17) and (15.6.18), we obtain

Equation (15.6.19) is a wave equation in h'. We, therefore, seek a wave-type solution h' = Aexp{ik(x — ct)}

Substituting in (15.6.19), we find that the assumed solution satisfies the equation, only if c = u ± (g HAp/pj)1/2 (15.6.20)

If the two layers are air and water, Ap/px ~ 1, and (15.6.20) simplifies to c = u ± (gH)1/2

The expression (gH)1/2 is called the shallow water wave speed for the simple reason that it can only be valid for a fluid in which the depth H is much smaller than the wave length, so that the vertical velocity remains small enough for the hydrostatic approximation to be valid. For an average ocean depth of 4 Km, the phase velocity of the surface gravity waves works out to be 200 m s—1.

However, according to (15.6.20), the phase velocity of the gravity waves along the interface between two layers very much depends upon the value of the ratio, Ap/pj, besides the depth of the lower layer. The ratio can be approximated to 1 only as long as p2 is negligible compared to pj, as in the case between air and water. Gravity waves can also form inside the ocean where there is a slight density difference between the upper mixed layer and the lower and denser deep ocean along an interface called the thermocline. If we assume a value of 0.001 for the ratio AP/P

1 , it follows from (15.6.20) that the phase speed of the internal gravity waves that will occur in this case and travel along the thermocline will be only one-tenth of that of the surface gravity waves. The internal gravity waves in the ocean can travel both horizontally and vertically, but those travelling vertically are reflected from the upper and the lower boundaries to remain trapped inside the ocean as stationary waves.

(c) Rossby waves

The wave type that is most important in connection with the large-scale meteorological processes and directly related to observed weather and climate is the Rossby or planetary wave. Under barotropic conditions, the Rossby wave is an absolute vor-ticity conserving motion and caused by the variation of the Coriolis parameter with latitude, or what is called the P-effect.

We can qualitatively understand its propagation in the atmosphere in the following way (see Fig. 15.5). Here, we assume the atmosphere to be barotropic.

Let a chain of fluid parcels be initially located along a latitude circle where the Coriolis parameter is fo at time to. A fluid parcel is then displaced meridionally so as to reach latitude with Coriolis parameter f1 after time t1 after covering a distance 5 y. Since the absolute vorticity n is conserved following motion in a barotropic atmosphere(13.7.4) and it is given by the relation, n = Z + f, where Z is the relative vorticity of the parcel, we can write down the relation between the vorticities before and after displacement as follows:

(Z + f)t1 — ft0 = 0, since the parcel has no vorticity at time to.

where p = df/dy is assumed to be constant.

The meaning of (15.6.21) is clear. The latitudinal displacement of the parcel generates perturbation relative vorticity, negative for northward displacement and positive for southward displacement from the initial latitude, as shown in Fig. 15.5.

The westward gradient of perturbation relative vorticity thus generated by the parcel oscillation forces the Rossby wave to move westward.

The dispersion relationship and the phase velocity of the barotropic Rossby waves can be derived formally by finding wave solutions of the linearized vorticity equation as follows: Since the absolute vorticity is conserved following the motion, we write its vertical component as

For simplicity, we assume that the motion consists of a basic state zonal velocity u and a horizontal perturbation (u', v') with perturbation relative vorticity Z(= dv'/dx — du'/dy). Further, we define a perturbation stream function y so that u' = —dy/dy, and v' = dy/dx, and Z' = V2 y.

The perturbation form of (15.6.22) is then

where p = df/dy and we have neglected the products of the perturbation terms.

We assume a solution of (15.6.23) of the form y = Re{Aexp(i^)}

where ^ = kx + ly - |t, k and l being the wave numbers in the zonal and the meridional directions respectively and | the frequency.

Substitution of (15.6.24) into (15.6.23) yields

-(-| + ku)(k2 + l2)+k p = 0 which we solve for | and obtain

Since, cx = |/k, we have finally cx - u = -p/(k2 +12) (15.6.26)

Thus, the Rossby waves propagate westward relative to the background zonal wind with a phase velocity which depends upon the wave numbers and hence increases with wavelengths. They are, therefore, dispersive waves. For very long waves, their westward phase velocity may at times be large enough to equal the basic state mean zonal wind velocity with the result that the waves become stationary with respect to the ground. If the wavelength exceeds the critical value for stationariness, the Rossby waves may actually retrogress. For a typical midlatitude disturbance with a zonal wavelength of 6,000 km and latitudinal width of 3,000 km, the Rossby wave speed relative to the zonal wind, computed from (15.6.26) is approximately -6ms-1.

Pure Rossby waves, however, are rare occurrences in the real atmosphere where other types of waves such as the gravity and sound waves are also excited. It is possible to use the linearized versions of the full primitive equations to study atmospheric waves following a procedure similar to that for the barotropic vorticity equation, though the procedure is rather involved. It is found that the free oscillations occurring in a hydrostatic gravitationally stable atmosphere consist of both westward and eastward moving gravity waves somewhat modified by the rotation of the earth and westward-propagating Rossby waves which are slightly modified by gravitational stability. These free oscillations are the normal modes of the atmosphere and they are continually excited by the various forces acting in the atmosphere.

15.7 Internal Gravity (or Buoyancy) Waves in the Atmosphere

Unlike the ocean which has a top boundary, the atmosphere has no real upper boundary. So, the case for the occurrence of internal gravity waves in the atmosphere is different from that in the ocean. In fact, the internal gravity waves in the atmosphere are nothing but buoyancy oscillations that form only in a stably stratified atmosphere and can propagate both horizontally and vertically. They are known to form on the leeside of mountains, which are called lee or mountain waves. They are believed to be an important mechanism for transporting momentum and energy to higher levels of the atmosphere. Clear air turbulence experienced by aircraft at high altitudes is also believed to be caused by vertically propagating internal gravity waves. In the equatorial stratosphere, they are believed to be responsible for eastward-propagating Kelvin waves and westward-propagating inertia-gravity waves, as well as the well-known quasi-biennial oscillation.

15.7.1 Internal Gravity (Buoyancy) Waves - General Considerations

Let us limit our discussion of internal gravity waves to an x-z plane in which the phase lines 8s (^ = const) of the parcel oscillations in the propagating wave are tilted at an angle a to the vertical (see Fig. 15.6).

It was shown in (3.4.12) that in the case of buoyancy oscillations in a stably stratified atmosphere, the buoyancy force acting on the parcel is given by the expression, - N28z, where N = {(g/9)(d9/dz)}1/2 is the frequency of the oscillations and z is the vertical co-ordinate . In the present case, the parcel oscillations occur in a direction which is not vertical but inclined to the vertical at an angle a. So, since 8z = 8s cos a, and its projection along 8 s is 8 zcos a, the buoyancy force on the oscillating particle is -N2(8scos a) cos a. The momentum equation of the parcel oscillation along 8 s may, therefore, be written d2 (8s) /d t2 = - (Ncos a)2 8s (15.7.1)

The general solution of (15.7.1) is Fig. 15.6 Tilt of the path of parcel oscillations (constant phase p - heavy lines) to the vertical by angle a. The blunt arrow shows the direction of phase propagation

Thus when the phase lines are inclined to the vertical by an angle a, the frequency of the buoyancy oscillations is given by N cos a. This heuristic result can be verified by considering the linearized equations for internal gravity waves in the x-z plane in a stably stratified incompressible atmosphere. We simplify the equations by making the Bossiness approximation in which the density is treated as constant except where it is coupled to gravity and the vertical scale of the motions is less than the scale height H(8 km).

We write the basic equations, neglecting rotation, as follows: The momentum equations:

du/dt + udu/dx + Wdu/dz + (1/p) dp/dx = 0 dw/dt + udw/dx + wdw/dz +(1/p) dp/dz + g = 0

The continuity equation:

du/dx + dw/dz = 0 The thermodynamic energy equation:

d9/dt + ud9/dx + wd9/dz = 0 where the potential temperature 9 is given by

where ps is surface pressure and k = R/cp.

We now linearize the above equations by assuming a motionless basic state with constant density p0 and putting p = p(z)+p'; p = p0 + p'; 9 = 0(z) + 0'; u = u'; w = w' The basic state pressure p(z) must then satisfy the relation dp/dz = -pog and the basic state temperature may be expressed as ln 0 = Y-1 lnp - ln p0 + const

The linearized equations are then obtained by the perturbation technique, i.e., by substituting from (15.7.7) in Eqs. (15.7.2-15.7.5) and neglecting products of the perturbation variables. For example, we simplify the last two terms in (15.7.3) as

~ (1/P0)(1 - p'/p0)dp/dz +(1/p0)dp'/dz + g = (1/P0)dp'/dz +(P'/P0)g (15.7.10)

where (15.7.8) has been used to eliminate p.

Similarly, the perturbation form of (15.7.9) may be obtained by noting that ln {9( 1 + 979)} = Y-1 ln {p( 1 + p'/p)}— ln {po( 1 + p'/po)} + const (15.7.11) which, with the aid of (15.7.9), can be simplified to

where cs = v^(Yp/p0) is the velocity of sound waves.

For buoyancy wave motions, the density fluctuations due to pressure change are much smaller than those due to temperature change, i.e., |p'/c2| C |p09'/9 Hence, (15.7.12) may be approximated to

Using (15.7.10) and (15.7.13), the linearized equations may now be written as du'/dt +(1/p0)dp'/dx = 0 (15.7.14)

We eliminate p' from (15.7.14) and (15.7.15) by subtracting d(15.7.14)/dz from d(15.7.15)/dx and get the relation d(dw'/dx - dv!/dz)/dt - (g/9)d9'/dx = 0 (15.7.18)

We now use (15.7.16) and (15.7.17) to eliminate u' and 9' and obtain d2(d2w'/dx2 + d2w'/dz2)/dt2 + N2d2w'/dx2 = 0 (15.7.19)

where N2 = g d(ln 9)/dz is the square of the buoyancy frequency and is assumed to be constant.

Equation (15.7.19) is a wave equation in w'. We seek its wave solution of the form w' = Re{Aexp(i^)} (15.7.20)

where ^ = kx + mz - |t is the phase in which k and m may be regarded as the components of a vector(k,m) along the x- and the z-axis respectively and | is the frequency of the wave. In terms of wavelengths, k = 2n/^x and m = 2n/^z, where and are the wavelengths along the respective axis.

By substituting (15.7.20) in (15.7.19) we obtain the dispersion relatioship p = ±Nk/ (k2 + m2 )1/2 (15.7.21)

In terms of wavelengths, p = ±NXz/(% + Xz2 )1/2 = ±N cos a (15.7.22)

where a is the angle between the phase line ^ = const and the vertical (Fig. 15.6).

Obviously, (15.7.22) is in agreement with the parcel oscillation frequency heuris-tically derived earlier (15.7.1). Thus, the tilt of the phase lines of the internal gravity waves depends only on the ratio of the wave frequency to the buoyancy frequency (p/N) and is independent of the wavelength.

If we let k > 0 and m < 0, it follows that for the phase ^ = kx + mz to remain constant, an increase of x must be accompanied by an increase of z. This means that in this case the phase lines tilt eastward with height, as shown in Fig. 15.7.

The choice of p positive in (15.7.21), therefore, implies that the phase propagation is eastward and downward in this case with the horizontal and the vertical components of the phase velocity being given respectively by: cx = p/k and cz = p/m. The components of the group velocity Cg are:

Since the energy propagates with the group velocity, it follows from (15.7.23) that for internal gravity waves downward phase propagation is accompanied by upward energy propagation.

Fig. 15.7 Idealized Jl cross-section in the x-z plane showing the phases of the pressure, temperature and velocity perturbations for an internal gravity wave. Thin i?

arrows show the perturbation x velocity field and and the blunt arrows the phase velocity (Reproduced from

Wallace and Kousky, 1968, with permission of American

Meteorological Society). 15.7.2 Mountain Lee Waves

A well-known example of internal gravity waves in the atmosphere are lee waves which form on the leeside of mountains when air parcels are forced to rise against a north-south oriented mountain range in a thermally stable atmosphere. If the vertical motions associated with these waves are strong enough and the air is sufficiently moist, condensation may occur in the rising parts of the waves which become visible in the form of rows of lee clouds. Such cloud formations are common occurrences on the leeside of mountains such as the French Alps (Gerbier and Berenger, 1961) and several other mountain ranges of the world.

Since lee waves are observed to remain stationary with respect to the ground under prevailing wind conditions, it follows that to an observer moving with the westerly wind, the constant phase lines of the lee waves will appear to be gradually moving westward. This means that the constant phase lines will tilt westward with height, allowing phase to move downward and wave energy to be transported upward. This westward tilt of the phase lines is shown in Fig. 15.6. To an observer moving with the mean zonal wind u, the frequency of the lee waves generated by a sinusoidal-shaped mountain is given by: = —u/k.

We can calculate the value of m from (15.7.24) for given values of u, k and N to determine the tilt of the phase lines with height. Since waves propagate vertically, we must have m2 > 0 (that is, m is positive). With a positive m, it follows from (15.7.24) that u must be less than N/k.

From this, we conclude that conditions which are favourable for the formation of lee waves are generally: a wide mountain range (small k), strong thermal stability (large N), and light mean zonal wind (small u) in the vertical. At times, when the atmosphere above the mountain height is stably stratified under a strong thermal inversion, the amplitudes of the lee waves can be quite powerful to cause strong downslope winds and intense clear air turbulence on the ground behind the mountain.

### 15.8 Dynamics of Shallow Water Gravity Waves

A problem of fluid dynamics of great geophysical interest is how mass and velocity fields adjust to each other in a rotating fluid the depth of which is much smaller than the horizontal scale of the perturbation of the flow. In a series of papers, Rossby (1937, 1938a, b; 1940) addressed this problem and threw light on several of the flow characteristics. In this section, we give a brief review of some of his findings, following a treatment by Gill (1982).

15.8.1 The Adjustment Problem - Shallow Water Equations in a Rotating Frame

Let us consider a rotating fluid that is initially at rest relative to a frame of reference that is rotating with a uniform angular velocity f/2 (where f is the Coriolis parameter) about a vertical axis. The fluid motion is considered relative to this frame and is supposed to be a small perturbation of height h from the state of relative rest at all times. The horizontal scale of the perturbation is assumed to be large compared to the depth, so that the hydrostatic approximation (p' = -pgh, where p' is perturbation pressure and p is fluid density) can be applied

The momentum equations, after making the hydrostatic approximation, are:

where (u, v) are the velocity components along the x, y axes respectively and are independent of depth.

The continuity equation is:

where H is the undisturbed depth of the fluid.

Taking the divergence of the momentum equations (i.e., taking d/dx of (15.8.1) and d/dy of (15.8.2) and adding) and substituting from (15.8.3), we obtain d2h/dt2 - c2(d2h/dx2 + d2h/dy2) + f HZ = 0 (15.8.4)

where we have put c2 = gH, and Z = (dv/dx - du/dy).

Since Z is the relative vorticity of the fluid, we need to know how it changes with time in order to solve (15.8.4). In this regard, the principle of conservation of potential vorticity (13.5.6) is of fundamental importance. By taking the curl of the momentum equations (15.8.1-15.8.2) and eliminating divergence with the aid of the continuity equation (15.8.3), we obtain the relation d(Z/f - h/H)/dt = 0 (15.8.5)

which states that potential vorticity is invariant with time. We now define a quantity Q', given by

and call it the perturbation potential vorticity. Since (15.8.5) expresses the fact that Q' retains its initial value at each point at all times, we have the relation

Equation (15.8.7) signifies an infinite memory of an inviscid rotating fluid to retain its initial perturbation potential vorticity and can be exploited to find an equilibrium solution for a particular initial state without considering details of the transient motion at finite times in between.

Following Rossby (1938a), we consider a case for which Q' is nonzero. The particular initial conditions that we consider are: u = v = 0 and the surface elevation h is given by the expression, h = -h0 sgn(x), (15.8.8)

where sgn(x) is the sign function (sign of x) defined by sgn(x) = 1 or -1, according as x < 0 or x > 0 The integral of (15.8.5) in this case is

Substitution of (15.8.9) in (15.8.4) gives an equation in h alone, viz., d2h/dt2 -c2(d2h/dx2 + d2h/dy2) + f2h = -f H2Q'(x, y,0) = -f2h0 sgn(x)