Gravity Waves

Of the two classes of periodic motions outlined earlier, gravity waves are the most well-studied and best understood in inland waters. We will consider only linear waves, that is, motions where the amplitude of the oscillations of the thermocline is small compared with the depth of the surface and bottom layer. This is not a major restriction on the analysis, as the inclusion of nonlinear effects has been shown in most cases to require only a minor correction to the linear approximation. In this article, we focus on cases where the Burger number is ~1, such that rotational effects can be expected. For surface (barotropic) waves, this would typically require lakes of more than 300 km width, of which there are very few. For baroclinic motions, where the phase speed c is far less than for barotropic motions, there are many lakes in which the Burger number is ~1. For typical values of the baroclinic internal wave phase speed (0.05-0.4 ms-1), the internal Rossby radius is ~1-5 km (Figure 1), indicating that internal gravity waves in lakes of this scale (or larger) should experience the rotational effects of the earth. Note also that unlike the barotropic phase speed, which depends on water depth alone, the baroclinic phase speed varies as a function of stratification and so changes through the year. Rotation may therefore play a more important role in the internal wave dynamics during the strongly stratified period when the internal Rossby radius (and therefore the Burger number) are minimal than at other times of the year.

c = 0.4 m/s c=0.3m/s c = 0.2 m/s c = 0.1 m/s c = 0.05 m/s

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0 10 20 30 40 50 60 70 80 90 Latitude

Figure 1 Internal Rossby radius as a function of latitude for several internal wave speeds. For horizontal length scales similar to or greater than the internal Rossby radius, rotational effects should be observed.

c = 0.4 m/s c=0.3m/s c = 0.2 m/s c = 0.1 m/s c = 0.05 m/s

10-1

0 10 20 30 40 50 60 70 80 90 Latitude

Figure 1 Internal Rossby radius as a function of latitude for several internal wave speeds. For horizontal length scales similar to or greater than the internal Rossby radius, rotational effects should be observed.

To understand the form of these motions in inland waters, it is instructive to build up our knowledge from simpler systems. We begin in a rotating system without boundaries, such as in the middle of the ocean far from the coast, where the classic gravity wave solutions are called plane progressive Poincare waves. The amplitude (q) and velocity structure (u,v) associated with these waves can be described by q — q0 cos(kx — at)

where u is the velocity in the direction of propagation of the wave, v is the velocity in the transverse direction, q0 is the maximum amplitude, k is the wave number (—2p/l, where l is the wavelength), a is the wave frequency (—2p/T, where T is the wave period), H is the water depth (or equivalent water depth He for an internal wave), and f is the inertial frequency. The fluid particle trajectories (in plan view, as the vertical motion is small due to the linear wave assumption) are ellipses with major axes in the direction of propagation, with the ratio of the ellipse axes equal to a/f and the direction of rotation anticy-clonic (i.e., opposite to the direction of rotation of the earth). For short waves, which have high frequency, a/f is large and so the trajectory ellipses are long and thin. Long waves are the opposite, with a/f approaching one as the wave frequency is low and therefore the particle tracks are circular and trace out the well-known 'inertial circles' in the ocean (Figure 2). The radius of these circular tracks is U/f, which can be reformulated using eqns. [11] and [12] as q0/kH.

An important aspect of the dynamics of internal waves influenced by the earth's rotation is that energy is generally not equally split between kinetic and potential forms. For the plane progressive Poincare wave (Figure 2), the mean kinetic energy per unit area is

where p is the water density, and the potential energy per unit area is

such that the ratio of potential to kinetic energy is

Poincare Wave

Figure 2 A long plane progressive Poincaré wave, in an infinite ocean, where a~f. Note the rotation of the current vectors is opposite to the direction of the earth's rotation. Reproduced from Mortimer CH (1974) Lake hydrodynamics. Mitt. Int. Ver. Theor. Angew. Limnol. 20: 124-197, with permission from E. Schweizerbart (http://www.schweizerbart.de/).

Figure 2 A long plane progressive Poincaré wave, in an infinite ocean, where a~f. Note the rotation of the current vectors is opposite to the direction of the earth's rotation. Reproduced from Mortimer CH (1974) Lake hydrodynamics. Mitt. Int. Ver. Theor. Angew. Limnol. 20: 124-197, with permission from E. Schweizerbart (http://www.schweizerbart.de/).

to kinetic ratio approaching one (as in the nonrotating case where f!0), and that waves close to the inertial frequency will have close to zero potential energy signal (such as the wave shown in Figure 2). This has implications for measurement of these waves, as they will only generally be observed by current measurements (a measure of kinetic energy) and not by fluctuations in stratification (a measure of potential energy variation).

The introduction of a boundary allows for the existence of Kelvin waves. The classical Kelvin wave solution is one in which the velocity perpendicular to the shore is considered to be zero (Figure 3). These waves propagate parallel to the boundary with the maximum amplitude at the shore, where the waves crests to the right (in the Northern Hemisphere) when looking along the direction of propagation. The amplitude decreases exponentially offshore at a rate equal to the Rossby radius of deformation R, q — q0e y/Rcos(kx — at)

This indicates that waves with a frequency much greater than the inertial frequency will have a potential

where x is both the alongshore direction and the direction of propagation, and y is the offshore direction (Figure 3). Note that the phase speed of the wave is c = y/gH, the same as for a wave in a nonrotating system. Current vectors, by definition, are rectilinear and oscillate in the alongshore direction only. As with waves in a nonrotating frame, the ratio of potential to kinetic energy is unity. For internal Kelvin waves, the dynamics are the same, except that the baroclinic phase speed applies and the wave amplitude decreases

Internal Rossby Radius

Figure 3 A long Kelvin wave progressing in the x-positive direction, with the shore located at y = 0. Channel walls can be placed vertically at any point of constant y, for example indicated by the planes AB and CD. Reproduced from Mortimer CH (1974) Lake hydrodynamics. Mitt. Int. Ver. Theor. Angew. Limnol. 20: 124-197, with permission from • • • (http://www.schweizerbart.de/).

Figure 3 A long Kelvin wave progressing in the x-positive direction, with the shore located at y = 0. Channel walls can be placed vertically at any point of constant y, for example indicated by the planes AB and CD. Reproduced from Mortimer CH (1974) Lake hydrodynamics. Mitt. Int. Ver. Theor. Angew. Limnol. 20: 124-197, with permission from • • • (http://www.schweizerbart.de/).

exponentially offshore with the internal Rossby radius of deformation Ri.

The introduction of a second boundary significantly complicates the waves supported in a rotating system. A channel (defined as two parallel walls with open ends) is able to support progressive Poincare waves, made up of an obliquely incident plane progressive Poincare wave with its reflection, and standing Poincare waves, consisting of two progressive Poincare waves traveling in opposite directions. These waves consist of cells similar to those presented in Figures 2 and 3, however the velocity at the border of each cell approaches zero. Closing a basin, and therefore creating a 'lake,' significantly complicates the wave field. For a rectangular basin, due to the complexity of the corners, an incident plus a reflected Kelvin wave is required along with an infinite number of Poincare waves of the same frequency to satisfy the boundary conditions. Far simpler solutions can be found by assuming lakes to be represented by circular or elliptical basins of uniform depth, which we will use in the following discussion. As outlined earlier, a key nondimensional parameter controlling this response is the Burger number S. Based on this parameter, it is possible to determine the wave frequency for both circular and elliptic basins, the ratio of potential to kinetic energy in the wave response, and the response of a basin to external forcing. We will also rely on the simplified Kelvin and plane progressive Poincare waves described earlier to assist in interpreting the results.

To understand the spatial structure of the waves (and the currents they induce) in a rotating system, it is helpful to consider the two end points: strong rotation (S!0) and no rotation (Sn). For inland waters, these might also be considered the 'large lake' and 'small lake' case. For S^0, in the lake interior, we might expect plane progressive Poincare waves to be present as outlined earlier (Figure 2), where the frequency approaches the inertial frequency, the current vectors rotate anticyclonically and the majority of energy is in the kinetic form. At the lake boundary, we might expect the classical Kelvin wave solution, where the offshore decay in amplitude is exponential at a rate R, the velocity at the boundary is parallel to the shore, the ratio of potential to kinetic energy is unity and the frequency approaches zero (Figure 3). Data collected from the North American Great Lakes support this conceptual model, with motion in the interior dominated by near-inertial frequencies and motion at the boundary appearing in the form of 'coastal jets,' which are the manifestation of the Kelvin wave solution. As the lake gets smaller (i.e., for Sn), we should expect waves that are similar to the nonrotating case, where the ratio of potential to kinetic energy is unity, and the offshore decay of amplitude is no longer exponential, and the current vectors become rectilinear (i.e., the ellipses become long and thin).

How the characteristics of these waves vary as a function of Burger number and aspect ratio of the lake is graphically presented in Figure 4. The nondi-mensional frequency o/f and the ratio of potential to kinetic energy are presented for analytical solutions to the circular and elliptical basin case for cyclonic (Kelvin-type) and anticyclonic (Poincare-type) waves. Note that the potential to kinetic energy ratios are integrated over the entire lake, and do not represent the character at a particular point in space. We first consider waves in a circular basin (where the aspect

0

0.6 0.4 0.2 0

Burger number (R/L)

Burger number (R/L)

-e-

Cyclonic (1:1)

-A-

Cyclonic (2:3)

■ ■□■

Cyclonic (1:3)

-*-

Anti-cyclonic (1:1)

-v- Anti-cyclonic (2:3)

0

Anti-cyclonic (1:3)

Figure 4 Nondimensional frequency (upper panel) and ratio of potential to kinetic energy (lower panel) as a function of wave type and aspect ratio, where the numbers in parentheses refer to the aspect ratio. The absolute value of the nondimensional frequency is presented as the results are independent of hemisphere. Reproduced from Antenucci JP and Imberger J (2001) Energetics of long internal gravity waves in large lakes. Limnology and Oceanography 46: 1760-1773, with permission from the American Society for Limnology and Oceanography.

Figure 4 Nondimensional frequency (upper panel) and ratio of potential to kinetic energy (lower panel) as a function of wave type and aspect ratio, where the numbers in parentheses refer to the aspect ratio. The absolute value of the nondimensional frequency is presented as the results are independent of hemisphere. Reproduced from Antenucci JP and Imberger J (2001) Energetics of long internal gravity waves in large lakes. Limnology and Oceanography 46: 1760-1773, with permission from the American Society for Limnology and Oceanography.

ratio by definition is 1:1). In the strong rotation case (S!0), the cyclonic wave frequency goes to zero and the energy ratio approaches unity. This is the Kelvin wave limit for a semi-infinite boundary outlined earlier (Figure 3). For the anticyclonic wave, the frequency approaches the inertial frequency and the energy is predominantly kinetic - the plane progressive Poincare wave solution outlined earlier (Figure 2). As S increases, the wave frequency for both types of waves increases and slowly converges as the lake gets smaller. The energy ratio for the anticyclonic wave also increases and asymptotically approaches unity, the nonrotation limit (Sn). For the cyclonic wave, the energy ratio increases to a maximum of ~1.5, before asymptotically approaching unity as Sn. Importantly as Sn these two solutions have the same characteristics (frequency, energy ratio, cross-basin structure), except that they rotate in opposite directions. They will thus manifest themselves at high S as a standing wave.

The distribution of potential (i.e., thermocline oscillations) and kinetic energy (i.e., currents) in the basin also changes as the importance of rotation changes (Figure 5). For the strong rotation case (S!0), the cross-shore potential energy structure of the cyclonic waves has the exponential decay associated with Kelvin waves propagating along a shoreline, where we can rewrite eqn. [14] q = q0e~3=SL so that for small S the exponential decay is rapid relative to the lake width (Figure 5(a)). The kinetic energy is also predominantly located close to the shore (Figure 5(c)), hence the term 'coastal jet' being applied to these motions in the North American Great Lakes. As the importance of rotation decreases (S increases), there is a stronger signal of the cyclonic waves present in the interior. It is important to note that for this case, the currents are not parallel to the boundary everywhere in the lake - next to the shoreline they remain parallel as in Figure 3; however, towards the interior, the current ellipses become more circular and actually rotate in a cyclonic direction. For the anticyclonic (Poincare-type) waves, the structure changes very little as the importance of rotation changes (Figure 5(b) and 5(d)). Note that as Sn, the distribution of the potential and kinetic energy in the cyclonic wave approaches that of the anticyclonic wave.

We now consider the impact of changing the aspect ratio by moving towards elliptical basins from a circular basin shape. Note that the Rossby radius is defined in the elliptical basin based on the length of the major axis, not the minor axis, in Figure 4. The effect of decreasing the aspect ratio is that the system approaches the nonrotating case for lower values of S. Unlike the case of the circular basin, the frequencies of

Rossby Gravity WaveRossby Gravity Wave Rossby Gravity Wave

Figure 5 Radial structure of cyclonic and anticyclonic wave energy distribution as a function of Burger number for a circular lake for the lowest frequency motion (fundamental mode). The cyclonic wave structure is shown in panels (a) and (c). The anticyclonic wave structure is shown in panels (b) and (d). The radial structure for each Burger number has been normalized by its maximum value. Note that not all Burger numbers are shown in panels (b) and (d). Note the exponential decay in (a) is the same as that represented in Figure 3 and eqn. [16]. Reproduced from Antenucci JP and Imberger J (2001) Energetics of long internal gravity waves in large lakes. Limnology and Oceanography 46: 1760-1773, with permission from the American Society for Limnology and Oceanography.

each cyclonic and anticyclonic wave pair diverge rather than converge. In the limit of Sn, the cyclonic waves transform into longitudinal seiches, whereas the anticyclonic waves become the transverse seiche solution. It is for this reason that wind-forcing in the transverse direction has been observed to more easily generate anticyclonic, Poincare-type, waves.

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