L

The effect of buoyancy can be seen by decreasing the density difference between the two layers resulting in a decrease in g' and corresponding increase in ^.

Surface seiche (V0H1)

Internal seiche (V1H1)

"-Wind begins

Wind Begins

Continues Stops

Wind Begins

Continues Stops

Internal seiche (V1H1)

"-Wind begins

(b)

Internal seiche (V2H1)

fpz Wind steady Wind stops

Internal seiche (V2H1)

fpz Wind steady Wind stops t

No flow

Figure 2 Movement caused by steady moderate wind stress on a hypothetical layered lake and subsequent internal seiche motion neglecting damping. (a) Horizontal mode one surface seiche in a homogeneous one-layered system, (b) horizontal mode one vertical mode on internal seiche in a two-layered system, both adapted from Mortimer CH (1952) Water movements in lakes during summer stratification: Evidence from the distribution of temperature in Windermere. Proceedings of the Royal Society of London Series B. 236: 355-404 and (c) horizontal mode one vertical mode two internal seiche in a three-layered system. Arrows denote distribution and magnitude of water particle velocities. At t = 0, (1/2)T1, T1, (3/2)71, etc. the wave energy is purely in the potential form, isotherms are at their maximum tilt and there is no seiche induced flow, while at t = (1/4)71, (3/4)71, (5/4)71, etc. the energy is purely kinetic, giving rise to strong horizontal currents within the lake-basin and horizontal isotherms.

No flow

Figure 2 Movement caused by steady moderate wind stress on a hypothetical layered lake and subsequent internal seiche motion neglecting damping. (a) Horizontal mode one surface seiche in a homogeneous one-layered system, (b) horizontal mode one vertical mode on internal seiche in a two-layered system, both adapted from Mortimer CH (1952) Water movements in lakes during summer stratification: Evidence from the distribution of temperature in Windermere. Proceedings of the Royal Society of London Series B. 236: 355-404 and (c) horizontal mode one vertical mode two internal seiche in a three-layered system. Arrows denote distribution and magnitude of water particle velocities. At t = 0, (1/2)T1, T1, (3/2)71, etc. the wave energy is purely in the potential form, isotherms are at their maximum tilt and there is no seiche induced flow, while at t = (1/4)71, (3/4)71, (5/4)71, etc. the energy is purely kinetic, giving rise to strong horizontal currents within the lake-basin and horizontal isotherms.

A simple comparison qs/q, ~ (Ap/po)(h1/H) shows that for weakly stratified deep systems, internal displacements 1-100 m) may be more than an order of magnitude greater than their surface counterparts

0.01-1 m); for example in Lake Baikal qs/q, - 0.11m/75m - 10~3.

The available potential energy (APE) embodied in the tilted interface is readily calculated for a two-layer system by integrating the interfacial displacement over the basin length

^2(x,t)dx which may be integrated for the initial condition of a uniformly tilted thermocline

After the thermocline tilt has reached steady state, work done by continued winds is either dissipated t

Figure 3 Schematic showing the response of a stratified lake to a surface wind stress. (a-c) show the stages of development of a steady state thermocline tilt. The hypolomnion is shaded and the arrows show the relative speed and direction of the flow. (d) Isotherm distribution and temperatures in Lake Windermere, northern basin, after a steady wind for 12 h. Reprinted from Mortimer CH (1954) Models of the flow-pattern in lakes. Weather 9: 177-184.

Figure 3 Schematic showing the response of a stratified lake to a surface wind stress. (a-c) show the stages of development of a steady state thermocline tilt. The hypolomnion is shaded and the arrows show the relative speed and direction of the flow. (d) Isotherm distribution and temperatures in Lake Windermere, northern basin, after a steady wind for 12 h. Reprinted from Mortimer CH (1954) Models of the flow-pattern in lakes. Weather 9: 177-184.

internally as heat or acts to mix the water column by further deepening the surface layer. Surface layer deepening has been characterized into four distinct regimes based on the strength of the stratification and winds (Figure 4). For strong stratification and weak winds (Regime A) the thermocline set-up is small, internal seiches persist for long times, mixing is weak and the thermocline remains sharp. If the stratification is weaker or the winds stronger (Regimes B and C), seiche amplitudes increase and become a predominant feature, shear instabilities (e.g., Kelvin-Helmholtz billows) form leading to entrainment of the metalimnion into the epilimnion, enhanced mixing and causing rapid damping of the internal seiches. For weak stratification and strong winds (Regime D), shear instabilities are strong; the thermocline becomes diffuse with a steep slope and rapidly deepens toward the lake bed. This creates a sharp downwind interface and a broad upwelling at the upwind shore. The upwelled fluid creates a longitudinal temperature gradient, which subsequently mixes the lake horizontally. The colder upwelled water is nutrient rich and as it mixes rapid fluctuations in temperature and biogeo-chemistry result. In deeper lakes, stratification can be strong during summer and upwelling of metalimnetic (partial upwelling) or hypolimnetic (full upwelling) water is unlikely. In these lakes upwelling is favored just after spring turnover or prior to fall turnover when the thermal stratification is weak or near the surface.

Wedderburn and Lake Numbers

The degree of tilt of the base of the surface layer resulting from an applied wind stress may be quantified using the dimensionless Wedderburn number W, which as the ratio of the wind disturbance force to the gravitational baroclinic restoring force is given by

Here, g is evaluated across the base of the surface layer. The cases of W » 1, W — 1 and W < 1 correspond to Regimes A, B/C, and D, respectively.

For idealized laboratory studies and back-of-the-envelope calculations, substitution of ^ into the equation for W leads to a leads to two-layer form

dV, where d-q, is the steady wind induced vertical displacement of the seasonal/diurnal thermocline measured at the boundary. Due to the order of magnitude scaling, the factor of 2 has been dropped as is commonly found in the scientific literature. Moreover, the somewhat counterintuitive nature of W as d^ ! 0, leads to frequent use of the inverse form of the Wedderburn number ( W— d^ /hi).

For lakes which are not well approximated using a two-layer stratification, W has been generalized into the Lake Number, LN (see Density Stratification and Stability). This accounts for the depth

Weak stratification Strong stratification severe storm weak winds

Regime

Figure 4 Schematic showing the mixed layer deepening response of a lake to wind stress. (a) Regime A: internal waves; (b) Regime B: internal waves and slight billowing; (c) Regime C: strong billowing and partial upwelling; (d) Regime D: intense billowing and full upwelling. Adapted From Fischer HB, List EJ, Koh RCY, Imberger J, and Brooks NH (1979). Mixing in Inland and Coastal Waters. San Diego, CA: Academic Press. After Spigel RH, and Imberger J (1980) The classification of mixed layer dynamics in lakes of small to medium size. Journal of Physical Oceanography 10: 1104-1121.

Figure 4 Schematic showing the mixed layer deepening response of a lake to wind stress. (a) Regime A: internal waves; (b) Regime B: internal waves and slight billowing; (c) Regime C: strong billowing and partial upwelling; (d) Regime D: intense billowing and full upwelling. Adapted From Fischer HB, List EJ, Koh RCY, Imberger J, and Brooks NH (1979). Mixing in Inland and Coastal Waters. San Diego, CA: Academic Press. After Spigel RH, and Imberger J (1980) The classification of mixed layer dynamics in lakes of small to medium size. Journal of Physical Oceanography 10: 1104-1121.

dependence of stratification and horizontal area. For a constant wind stress over a lake with an arbitrary basin shape and stratification

Here Ao is the surface area of the lake and hv is the height from the lake-bed to the centre of volume of the lake. The stability of the lake

(z — hv )A(z)p(z)d z incorporates the variable stratification p(z) and irregular bathymetric area A(z). For large Lake numbers, the stratification will be severe and dominate the forces introduced by the wind stress. The isotherms will be horizontal, with little or no seiching and associated turbulent mixing in the benthic boundary layer and interior. Changes in St with latitude and season cause Ln to vary spatially and temporally around the globe (see Mixing Dynamics in Lakes Across Climatic Zones). Under comparable wind conditions LN is maximal in the mid-latitudes during summer.

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