Degeneration of Basin Scale Internal Waves in Lakes

Understanding the factors leading to wave degeneration has been a major goal of limnologists. This is because internal waves ultimately lose their energy (degenerate) to dissipation (viscous frictional heating of the fluid at mm scales) and diapycnal mixing (mixing of fluid perpendicular to isopycnals or surfaces of constant density) in regions where the flow is turbulent. In turn, mixing drives biogeochemical fluxes. Turbulence is produced directly from the seiche induced currents through fluid straining in the lake interior and TBBL, and from processes that are uncoupled from seiche generation, such as surface wave breaking and inflows, which also act to disrupt the seiche motion. As a general rule, for deep lakes, the period of internal seiche decay is about 1 day per 40 m of water-column

0.5 0 0.5 Wavefunction, y

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Figure 7 (a) Wavefunction profiles for vertical modes one (m = 1), two (m = 2), and three (m = 3) supported by a constant N stratification. (b) Characteristic velocity profiles codC/dz for wavefunctions shown in panel (a).

depth. For very deep lakes, e.g., Lake Baikal where H = 1637m, this equates to a decay period of more than one month.

Field observations show the degeneration of basin scale internal waves to occur primarily as a result of turbulence production in the TBBL rather than the interior; observations of dissipation and mixing are more than ten times greater in the TBBL than the interior (see The Benthic Boundary Layer (in Rivers, Lakes, and Reservoirs)). The degeneration can occur through four possible mechanisms: (1) viscous damping of seiche currents in the TBBL, (2) the formation of shear instabilities in the interior, (3) the production of nonlinear internal waves that will break on sloping topography; and (4) the formation of internal hydraulic jumps. By calculating the timescales over which each mechanism will occur, regimes have been delineated in which a particular mechanism will dominate (Figure 8). The regimes are defined according to the inverse Wedderburn number W= / h1 and the depth of the seasonal thermocline (h1/H). Although strictly applicable to long narrow lakes that match the rectangular system used in the analysis, the regime diagram has been shown to suitably predict field observations from a variety of lakes (Table 2). The degeneration regimes are described below.

h1/H

Deep lakes/shallow thermocline Shallow lakes/deep thermocline

Figure 8 Analytical regime diagram showing the degeneration mechanisms of seiches in long rectangular lakes. The regimes are characterized in terms of the normalized initial forcing scale = fyi/h1 and the depth of the seasonal thermocline (h1 /H). Laboratory observations are also plotted (*, Kelvin-Helmholtz (K-H) billows and bore; broken undularbore; a, solitary waves; □ , steepening; o, damped linear waves). From Horn DA, Imberger J, and Ivey GN (2001) The degeneration of large-scale interfacial gravity waves in lakes. Journal of Fluid Mechanics 434:181-207.

h1/H

Deep lakes/shallow thermocline Shallow lakes/deep thermocline

Regime 1: Damped Linear Waves

Under relatively calm conditions ( W^1 < — 0.2) weak seiches develop, which are damped by viscosity in the TBBL. Seiche amplitudes and currents are not sufficient for solitary wave production, shear instability and/or supercritical flow. This regime corresponds to regime A in Figure 4. The time scale associated with viscous damping of a basin-scale seiche TD is estimated from the ratio of the seiche energy to the rate of energy dissipation in the benthic boundary layer etbbl :

Seiche energy etbbl Lake volume TBBL volume

1 to 10 d(for moderately sized lakes)

The energy dissipation in the lake interior £Intelior is neglected because observational studies show that etbbl > 10eInterior. More complex models for viscous seiche decay may be found in references below.

Regime 4: Kelvin-Helmholtz Billows

Kelvin-Helmholtz instabilities can form due to a number of mechanisms. Under strong forcing conditions and as the thermocline approaches mid-depth (> —0.8 and > —0.3), the seiche-induced currents will be sufficiently strong to overcome the stabilizing effects of stratification. Shear instabilities will occur before solitary waves can be produced and/or the seiche is damped by viscosity. Shear from other processes, such as the surface wind stress, can augment that from seiche induced currents and induce instabilities under weaker forcing conditions.

Shear instabilities manifest themselves as high-frequency internal waves that are common features through the metalimnion of lakes and oceans. The waves are the early stages of growth of the instabilities and have sinusoidal profiles, frequencies — 10~2Hz, wavelengths —10-50 m and amplitudes —1-2 m. Depending on the particulars of the stratification and velocity shear profiles, the shear instabilities may be classified according to their profile (e.g., Kelvin-Helmholtz billows, Holmboe waves or combinations thereof). Instabilities grow exponentially from small random perturbations in the flow, leading to rapid degeneration as patches of localized turbulent mixing or billowing (see Small-Scale Turbulence and Mixing: Energy Fluxes in Stratified Lakes). This process is shown schematically in Figure 9.

In a continuously stratified flow, the stability behavior is governed by the Taylor-Goldstein equation, from which it can be shown that a gradient Richardson number

Table 2 Comparison between predicted regime and observations in several lakes

Lake

Dates

Observations

W (W~1)

Regime

Source

Loch Ness

Oct. 2-3, 1971

'Pronounced front or surge'

3 (0.3)

2

1

Lake of Zurich

Sept. 11-14, 1978

'Steep fronted solitary wave'

3 (0.3)

2

2

Windermere

Aug. 14-20, 1951

'Damped harmonic oscillations'

5 (0.2)

1-2

3

Sept. 13-17, 1951

'Oscillatory waves' with some steepening

3 (0.3)

1-2

Babine Lake

July 5-10, 1973

'Surges'

2 (0.5)

2

4

Aug. 12-15, 1973

'Steep shock front'

3 (0.3)

2

Oct. 2-7, 1973

'Surges' and 'solitary waves'

2 (0.5)

2

Seneca Lake

Oct. 14-21, 1968

'Surges' consisting of trains of 'solitons'

2 (0.5)

2

5

Kootenay Lake

July 13-Aug. 17,

'Surges' consisting of waves resembling 'solitons'

2 (0.5)

2

6

1976

Balderggersee

Nov. 1-15, 1978

'Asymetrical waves'

9(0.1)

2

7

Nov. 16-22, 1978

'Steepened wave front.. .described as an internal surge'

2 (0.5)

2

Lake Biwa

Sept. 4-13, 1993

'Undular bores and solitary waves'

1 (1)

2

8

Most observations fall In regime 2 > ~ 0.3) and the Internal seiche will degenerate Into nonlinear Internal waves. For Windermere, the baroclinlc tilting Is weaker < ~ 0.3) and the predominant response is a damped seiche. After Horn DA, Imberger J, and Ivey GN (2001) The degeneration of large-scale interfacial gravity waves in lakes. Journal of Fluid Mechanics 434: 181-207. Sources

1. Thorpe SA, Hall A, and Crofts I (1972) The internal surge in Loch Ness. Nature 237: 96-98.

2. Mortimer CH and Horn W (1982) Internal wave dynamics and their implications for plankton biology in the Lake of Zurich. Vierteljahresschr. Naturforsch. Ges. Zurich 127(4): 299-318.

3. Heaps NS and Ramsbottom AE (1966) Wind effects on the water in a narrow two-layered lake. Philosophical Transactions of the Royal Society of London A 259: 391-430.

4. Farmer DM (1978) Observations of long nonlinear internal waves in a lake. Journal of Physical Oceanography 8: 63-73.

5. Hunkins K and Fliegel M (1973) Internal undular surges in Seneca Lake: A natural occurrence of solitons. Journal of Geophysical Research 78: 539-548.

6. Wiegand RC and Carmack E (1986) The climatology of internal waves in a deep temperate lake. Journal of Geophysical Research 91: 3951-3958.

7. Lemmin U (1987) The structure and dynamics of internal waves in Baldeggersee. Limnology and Oceanography 32: 43-61.

8. Saggio A and Imberger J (1998) Internal wave weather in a stratified lake. Limnology and Oceanography 43: 1780-1795.

Most observations fall In regime 2 > ~ 0.3) and the Internal seiche will degenerate Into nonlinear Internal waves. For Windermere, the baroclinlc tilting Is weaker < ~ 0.3) and the predominant response is a damped seiche. After Horn DA, Imberger J, and Ivey GN (2001) The degeneration of large-scale interfacial gravity waves in lakes. Journal of Fluid Mechanics 434: 181-207. Sources

1. Thorpe SA, Hall A, and Crofts I (1972) The internal surge in Loch Ness. Nature 237: 96-98.

2. Mortimer CH and Horn W (1982) Internal wave dynamics and their implications for plankton biology in the Lake of Zurich. Vierteljahresschr. Naturforsch. Ges. Zurich 127(4): 299-318.

3. Heaps NS and Ramsbottom AE (1966) Wind effects on the water in a narrow two-layered lake. Philosophical Transactions of the Royal Society of London A 259: 391-430.

4. Farmer DM (1978) Observations of long nonlinear internal waves in a lake. Journal of Physical Oceanography 8: 63-73.

5. Hunkins K and Fliegel M (1973) Internal undular surges in Seneca Lake: A natural occurrence of solitons. Journal of Geophysical Research 78: 539-548.

6. Wiegand RC and Carmack E (1986) The climatology of internal waves in a deep temperate lake. Journal of Geophysical Research 91: 3951-3958.

7. Lemmin U (1987) The structure and dynamics of internal waves in Baldeggersee. Limnology and Oceanography 32: 43-61.

8. Saggio A and Imberger J (1998) Internal wave weather in a stratified lake. Limnology and Oceanography 43: 1780-1795.

is a necessary but not sufficient condition for instability (Miles-Howard criterion). Billowing occurs along thin layers in the flow (-10 cm thick) where Rig is low and there are interfaces with sharp density gradients. Through billowing, the interfaces become more diffuse and are replaced by shear layers of thickness d — 0.3(AU)2/g', where AU is the velocity jump over the interface. As a result of billowing, the flow becomes stable unless AU increases (e.g., due to increasing wind stress) or d decreases (e.g., due to mixed layer deepening). A spectrum of growing instabilities are theoretically possible, but the most unstable mode wave will have a wavelength A = 2p/k - 7d.

Application of the Taylor-Goldstein equation to field observations shows that the frequency of the most unstable mode is just below the maximum buoyancy frequency through the metalimnion. This is because a fluid parcel displaced vertically from its equilibrium density position, as occurs during the growth of an instability, will be subjected to buoyancy forces arising from the sudden density anomaly with respect to its surroundings. The fluid parcel will oscillate as a wave at frequency N until the motion is frictionally damped by viscosity or evolves into a billow and collapses into turbulence. Fluid parcels will not naturally oscillate at frequencies greater than N, and so waves will not propagate at these frequencies; N is thus the limiting high-frequency cut-off for internal wave motions.

Internal seiches generate substantial vertical shear due to the baroclinic flow reversals across layer interfaces (Figures 2-5). The magnitude of this shear is periodic and has a maximum value when the interfaces are horizontal (e.g., T1/4) and all wave energy is in the kinetic form. A bulk Ri may be applied over the interfaces separating discrete layers

to predict the formation of instabilities. If the background flow is time-variable, either the condition Ri = 1/4 must be maintained for longer than the growth and billowing period of the instability Th - 20(AU)/g' or Ri must be <<1/4. Values of Th in lakes are of the order of minutes or less.

Shear instabilities can occur at the nodal locations in lakes, where the vertical shear is greatest, at the base of the surface layer under strong wind conditions, on the upper and lower surfaces of thermocline

Figure 9 Schematic showing the growth and turbulent degeneration of a Kelvin-Helmholtz shear instability leading to diapycnal mixing of the stratified fluid. The condition is shown as velocity (u) and density (p) profiles in (a) and as isopycnal surfaces in (b), where the arrows denote the flow direction. In (b-j), A and B are fixed points in the flow and the lines represent surfaces of constant density (isopycnals). Adapted from Mortimer CH (1974). Lake hydrodynamics. Mitteilungen Int Ver Limnol 20: 124-197, after Thorpe SA (1987) Transitional phenomena and the development of turbulence in stratified fluids: A review. Journal of Geophysical Research 92: 5231-5248.

jets that result from vertical mode-two compression of the metalimnion, near river influents and reservoir withdrawal layers, and in regions where there is flow over rough topography.

Regime 3: Supercritical Flow

Internal hydraulic jumps occur in stratified flows at the transition from supercritical (Fr2 + Fr| > 1) to subcritical (F^2 + Fr^ < 1) flow conditions, where the upper and lower layer Froude numbers are defined as Fr2 = U2 / g'h1 and Fr^ = U22 / g'h2. Although they are more commonly observed in the ocean as a result of tidal flow over topographic features (e.g., Knight Inlet sill); internal hydraulic jumps can occur in lakes.

Progressive jumps form when supercritical flow (resulting from a gravity current, inflow, wind event or thermocline jet) propagates into an undisturbed region. Flow over a topographic feature can lead to a stationary jump in the lee of the obstacle. Localized energy dissipation and mixing occur near jumps and waves radiate from the critical point where Fr = 1.

The impacts and distribution of hydraulic phenomena in lakes are not well understood.

Regime 2: Solitary Waves

In small- to medium-sized lakes subjected to moderate forcing (0.3 < W-1 < 1.0), nonlinearities become significant and the linear wave equation no longer completely describes wave evolution. In addition to the linear standing wave (composed of symmetric cosine components combined in a standing wave pattern), asymmetrical nonlinear wave components are generated from the wind induced thermocline tilt. The asymmetric components combine into a progressive internal wave pattern. The downwelled fluid becomes a dispersive packet of sub-basin scale internal waves of depression (Figure 10) called nonlinear internal waves (NLIWs), while upwelled fluid evolves into a progressive nonlinear basin-scale wave, referred to as a rarefaction or internal surge.

The weakly nonlinear Korteweg-de-Vries (KdV) equation mathematically describes the generation and unidirectional progression of NLIW from the wind-induced thermocline setup

where the nonlinear coefficient a = (3co/ 2)(h1 - h2)/ h1h2 and the dispersive coefficient b = co h1h2/6. Initially, the internal surge propagates under a balance between the unsteady (Orç/ 01) and nonlinear arç(0rç/ Ox) terms. As nonlinearities become more apparent as the waveform steepens and the wavefront approaches vertical (Figure 10c). This occurs at the steepening time scale

Wave steepening causes the dispersive term brç(03rç/ Ox3) to become significant, eventually balancing nonlinear steepening at t = Ts, leading to the production of high-frequency NLIWs (Figure 10d).

In many lakes, NLIWs have a wave profile that matches a particular solution to the KdV equation t) = asech2 ^

x ct

These are called solitary waves. The maximum amplitude of the solitary wave a, the solitary wave speed c, and characteristic horizontal length-scale 1 are given by c = co H— aa and 12 = 12 — 3 aa

Wind-induced thermocline depression

Progressive internal wave

Progressive internal wave

Snowboard Typen
NLIWs propagate toward slope

Figure 10 Schematic showing the evolution of a NLIW packet. (a) Initial wind induced thermocline depression, (b-c) formation of a progressive surge through nonlinear steepening, (d-e) evolution of a dispersive NLIW packet at t = Ts. For the case shown, the wind had been blowing for less than T1/4 and a steady state tilt of the entire thermocline was not achieved (i.e., upwelling did not occur). This is a common occurrence in long (—100 km) narrow lakes (e.g., Seneca Lake, Babine Lake, etc.).

Figure 10 Schematic showing the evolution of a NLIW packet. (a) Initial wind induced thermocline depression, (b-c) formation of a progressive surge through nonlinear steepening, (d-e) evolution of a dispersive NLIW packet at t = Ts. For the case shown, the wind had been blowing for less than T1/4 and a steady state tilt of the entire thermocline was not achieved (i.e., upwelling did not occur). This is a common occurrence in long (—100 km) narrow lakes (e.g., Seneca Lake, Babine Lake, etc.).

The KdV equation reveals some interesting characteristics of solitary waves. In a two-layer system they will always protrude into the thicker layer and so are generally observed as waves of depression upon the thermocline. If the interface occurs at mid-depth, a! 0; thus preventing nonlinear steepening and subsequent solitary wave generation. Moreover, the dependence of a on h — h2 demonstrates that the degree of nonlinearity depends not only on the magnitude of the interfacial displacement, but also on the relative heights of the stratifying layers.

The dispersive nature of the wave packet is evident from the relationship between wave amplitude a and wave speed c; a spectrum of waves in a particular packet will be rank ordered according to amplitude (Figure 10e) and will disperse with time as they propagate. An estimate of the number of solitary waves and their amplitudes, while beyond the scope of this article, may be obtained from the Schrodinger wave equation.

Surface soliton

Internal soliton

Surface soliton

Internal soliton

Figure 11 Schematic showing the passage of an internal solitary wave in a two-layer stratified fluid. Dashed lines are contours of water particle speed (isotachs) and arrows indicate the magnitude and direction of the flow. A small surface solitary wave of amplitude — (p2 — p1 )a accompanies the solitary wave and causes the rip or surface slick. From Osborne AR and Burch TL (1980) Internal solitons in the Andaman Sea. Science 208 (4443): 451-460.

Figure 11 Schematic showing the passage of an internal solitary wave in a two-layer stratified fluid. Dashed lines are contours of water particle speed (isotachs) and arrows indicate the magnitude and direction of the flow. A small surface solitary wave of amplitude — (p2 — p1 )a accompanies the solitary wave and causes the rip or surface slick. From Osborne AR and Burch TL (1980) Internal solitons in the Andaman Sea. Science 208 (4443): 451-460.

Laboratory experiments show that NLIWs can contain as much as 25% of the energy (APE) introduced to the internal wave field by the winds. In large lakes they may have amplitudes and wavelengths as large as —20 m and —50-1000 m, respectively, and travel at c — 0.5-0.75 ms—1. The velocity field associated with large amplitude NLIWs, commonly found in the ocean, will form a slick (rip) on the water surface (Figure 11) allowing them to be located and tracked using shore-based, aerial or satellite imagery.

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