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Interfacial Waves in a Layered Stratification

Horizontal modes When steady winds cease and the surface stress condition is relaxed, the gravitational restoring force associated with the tilted interface (water surface or thermocline) becomes unbalanced. The available potential energy embodied in the tilt is released under the action of gravity and converted to kinetic energy as the interface oscillates in the form of a sinusoidal standing waves or seiche. Antinodes are found at the basin end walls and nodal points in the basin interior (Figures 2 and 5). Seiches are commonly called linear waves because the evolving wave-field is well described in space and time by the linear wave equation d^n dt2 ''

where ^(x,t) is the interfacial displacement and co the linear shallow water phase speed (speed at which the crests/troughs propagate). This equation is equally applicable to interfacial waves travelling on the free-surface or thermocline by applying the appropriate form of co = y^gff or co = \Jg' h1h2/H, for the cases of surface and internal seiches, respectively. Due to the reduced effect of gravity across the thermocline relative to the free surface (g' ^ g), surface waves travel at ~50 times the speed of internal waves.

The familiar standing wave patterns associated with seiches forms as symmetric progressive waves of equal amplitude and wavelength, but opposite sign, propagate from the upwelled and downwelled fluid volumes at the opposite ends of the basin (Figure 2). These waves are most commonly represented with cosine functions (Figure 5), which have central node(s) and antinodes at the basin walls. Summing cosine o c

Antinode

X/2 Node

Antinode

Antinode

X/2 Node

Antinode

Antinode

Antinode

Antinode

Antinode

Antinode

Antinode

Antinode

Antinode

Antinode

Antinode

Figure 5 Schematic diagram showing the first three horizontal interfacial seiche modes: horizontal mode one (n = 1), mode two (n = 2), and mode three (n = 3). Arrows denote direction of water particle velocities. Solid and dashed lines denote the interfacial displacement at one-half period intervals. Upper layer velocities for the baroclinic case are not shown and can be inferred from symmetry.

Figure 5 Schematic diagram showing the first three horizontal interfacial seiche modes: horizontal mode one (n = 1), mode two (n = 2), and mode three (n = 3). Arrows denote direction of water particle velocities. Solid and dashed lines denote the interfacial displacement at one-half period intervals. Upper layer velocities for the baroclinic case are not shown and can be inferred from symmetry.

equations for waves propagating in opposite directions, gives the equation for the horizontal mode one (H1) standing wave pattern (Figure 5(a))

y(x, t) = a cos (kx + at) + a cos (kx — at) = 2a cos kx cos at

The component wave amplitude a = dy/2 or dys/2 depending on the interface under consideration, the angular frequency a = cok, where k = 2n/X is the wavenumber and l the wavelength and T = 2p/a the wave period. For an enclosed basin, there is one half wavelength of an H1 seiche in a lake, giving l = 2L and a period of

T n nco where n = 1 for a H1 seiche is the number of nodal points or half wavelengths in the horizontal direction.

The layer-averaged horizontal velocities associated with the H1 seiche are maximum the centre of the basin and are given by

g HI/2t

These velocities are zero at the vertical boundaries, where the motion is purely vertical (Figure 5). Similarly for the surface seiche, the mid-lake depth-averaged velocity is

The oscillatory seiche currents are low-period and quasi-steady. Observations from a variety of lakes and reservoirs show the surface and internal seiche

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Longitudinal current velocity ulong [cm/s]

Figure 6 Near-bed velocity profiles in a small lake showing the oscillatory nature and no-slip boundary associated with seiche currents. Observations are over one-half of the seiche period taken at times as indicated. The profiles are offset and are all plotted with the given velocity scale. From Lorke A, Umlauf L, Jonas T, and Wuest A (2002) Dynamics of turbulence in low-speed oscillating bottom boundary layers of stratified basins. Environmental Fluid Mechanics 2: 291-313.

currents to have a typical range of 0.02-0.20 m s-1, with a maximum ~0.2ms-1 during storms and to approach zero at the no-slip sediment boundary where the flow is impeded by friction (Figure 6).

Higher horizontal mode seiches (n > 1) are also observed in lakes. These are described by a more general solution of the linear wave equation, where the initial condition is that of a uniformly tilted interface, made up of the superposition of all higher horizontal modes. The general solution is n = 1 v /

n = 1 for the horizontal mode-one (H1) seiche, n = 2 for the horizontal mode-two (H2) seiche, n = 3 for the horizontal mode-three (H3) seiche (Figure 5). An infinite number of modes are possible, each with decreasing amplitude and energy as the modal number increases. The fundamental solution is composed only of odd modes (i.e., -q(x, t) = 0 when n is even) as is intuitively expected because only odd odes have a nodal point at the mid-basin location where there is zero displacement associated with an initial uniform initial tilt (Figures 2 and 3). By calculating the APE associated with each mode, it can be shown that that more than 98% of the wave energy is contained in the H1 mode, but the energy distribution between modes may be significantly affected by resonant forcing and basin shape.

Examples of surface and internal seiche periods for various horizontal modes are given in Table 1. Energy will pass between potential and kinetic forms as the wave periodically oscillates with time (Figure 2). At t = 0, (1/2)71 T1, (3/2)T1, etc. the wave energy is purely in the potential form, while at t = (1/4)T1, (3/4)T1, (5/4)T1, etc. the energy is purely kinetic, giving rise to horizontal currents within the lake-basin (Figure 6). For a non dissipative system, the modal energy distributions represent the sum of kinetic and potential energy and are independent of time. Dissipative processes will lead to a decrease in wave amplitude, but not period, with time (Table 1); unless there is sufficient mixing across the thermo-cline to cause a change in the stratification and hence co. For surface seiches the decay in amplitude with each successive period can range from 3% (Lake Geneva) to 32% (Lake Erie).

Vertical modes When the vertical density structure may be approximated with three or more fluid layers (Figure 1c,d), in addition to vertical mode-one, horizontal mode-one (V1H1) seiches (Figure 2b), higher vertical mode seiches are supported; for example V2H1, etc. (Figure 2c). For a three-layer system co becomes

u 2H where g = (1 - pjp2)hlh2 + (1 - Pi /p3)hlh3 + (1 - p2 M)h2h3 and a = h1 h2 h3 (1 - Pl = P2 )( 1 - P2/P3):

Substitution into the equation for the wave period Tn = 2L/nco gives the period of a vertical mode-two

Table 1 Observed surface and internal seiche periods from a variety of lakes and the associated amplitude decay

Lake and location T (h) T2 (h) T3 (h) T4 (h) T5 (h) Fractional decrease in amplitude with each successive period

Surface seiches Tanganyika (Africa)1 Loch Earn (Scotland)1 Yamanaka (Japan)1,2 Garda (Italy)1'2

Geneva (Switzerland-France)1,2 Vattern (Sweden)1,2 Baikal (Russia)1 Michigan (Canada-USA)1 Erie (Canada-USA)1'2 Internal seiches Baldegg (Switzerland)3 Lugano (Switzerland-Italy)4 Windermere (England)3 Zurich (Switzerland)5 Loch Ness (Scotland)3 Geneva (Switzerland-France)3

0.075 |
0.038 |
0.028 |

0.24 |
0.14 |
0.10 |

0.26 |
0.18 |
0.09 |

0.72 |
0.48 |
0.37 |

1.2 |
0.59 | |

3.0 |
1.6 |
1.3 |

4.6 | ||

9.1 |
5.2 |
3.7 |

14.3 |
9.0 |
5.9 |

9.3 |
4.6 |
3.1 |

24 |
12 |
8.0 |

24 |
13 |
9 |

45 |
24 |
17 |

57 |
27 |
18 |

74 |
46 |
0.099 14 11 22 18 1Wilson W (1972) Seiches. In ChowVT(ed.) Advances in Hydroscience 8: 1-94. 2Wüest AJ and Farmer DM (2003) Seiches. In McGraw-Hill Encyclopedia of Science and Technology, 9th Edition. New York: McGraw-Hill. 3Lemmin U and Mortimer CH (1986) Tests of an extension to internal seiches of Defant's procedure for determination of surface seiche characteristics in real lakes. Limnology and Oceanography 31:1207-1231. 4Hütter K, Salvade G, and Schwab DJ (1983) On internal wave dynamics in the northern basin of the Lake of Lugano. Geophysical and Astrophysical Fluid Dynamics 27: 299-336. 5Horn W, Mortimer CH and Schwab DJ (1986) Wind-indüced internal seiches in Lake Zürich observed and modeled. Limnology and Oceanography 31: 1232-1254. wave, where the horizontal modal structure is defined by n. Vertical mode-two seiches can be generated when there is an asymmetry in the tilting of upper and lower interfaces of the stratifying layers (the diurnal and seasonal thermoclines or the upper and lower boundaries of the metalimnion). Laboratory and limited field data shows that such asymmetries are introduced from the compression and expansion of the metalimnion that occurs along the downwind and upwind shores, respectively, under upwelling conditions (e.g., Figure 3(d)). The strength of the vertical mode two response has been hypothesized to depend upon the relative values of the Wedderburn and Lake Numbers. A V2 response occurs for small W and large LN (strong tilting of the upper interface, large shear across the base of the surface layer and a relatively undisturbed lower interface); whereas a V1 response occurs for small Wand large LN (comparable tilts of both interfaces and a strong velocity in the hypolimnion). Higher vertical mode basin-scale internal waves have been observed in several lakes, generally after a sudden wind pulse has excited an initial V1H1 response, which then evolves into a V2 seiche (e.g., Wood Lake, Upper Mystic Lake, Lake Constance). Resonance with the wind forcing (e.g., Lake Alpnach), sloping basin topography and unequal density differences between stratifying layers can cause this preferential excitation of higher vertical modes. ## Internal Modes in a Continuous StratificationThe two-layer assumption for the stratification in lakes is inappropriate for the many shallow lakes (H < ~ 15 m; e.g., Frains Lake) and in the hypolom-nion of lakes that are very deep (e.g., Lake Baikal) or where the stratification has a significant chemical (saline) component (e.g., Mono Lake). These systems are better modeled using a continuous stratification where the tilting of the isopycnals due to wind set-up is captured by LN. Upon relaxation of a wind stress, a continuous stratification will support a spectrum of vertical basin-scale modes. The frequency associated with each mode is given by which is dependent upon the basin geometry and may be used to calculate the wave period T = 2p/o. The structure of each vertical mode m is described by the wavefunction Cm, which is obtained from the linear long-wave equation with constant N m = 1, 2, 3, and is a measure of the wave-induced vertical displacement of the internal strata at a particular depth. For example, the V1 wave has a maximum internal displacement at mid-depth, whereas the V2 wave has a positive displacement at 1/4H and a negative displacement at 3/4H (Figure 7a); showing the characteristic opening of the strata. The wavefunction contains no knowledge of the actual wave amplitude and consequently is often normalized — 1 < Cm < 1. The utility of Cm is that it may be numerically calculated from the Taylor-Goldstein equation for an arbitrary N(z) profile, thus determining the wave modes that are supported by a particular water column stratification. The basin-scale vertical modal structure, either a single mode or combination of modes, may be calculated when the wave amplitude is known -q(z, t) « Cm(z)a(t). The vertical modal structure can then be projected in space and time throughout the basin by assuming a wave profile ■q(x, z, t) « Cm(z)v(x, t); this is typically a cosine for seiches or sech2 function for solitary waves (described later). The horizontal velocity profile induced by the wave motion scales with the gradient of the wavefunction codCm/dz (Figure 7(b)). Analytical expressions for the velocity field can be found in stratified flow texts. The continuous velocity profiles in Figure 7 are consistent with the depth averaged currents presented in Figure 2. In both models, although technically incorrect (Figure 6), a free-slip bottom boundary has been assumed. Accordingly the velocities from these models are not representative of flow near the bottom boundary. |

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