Wind Set-Up of the Free Surface

The action of the wind across the lake surface results in frictional momentum transfer from the wind to the water. This transfer occurs in the form of a stress (N m~2) applied at the free surface. The stress may be parameterized as

T = C0PlU2w where CD is the drag coefficient, pa = 1.2 kg m~3 is the air density, and Ui0 the wind speed measured at 10 m above the water surface. Typically CD = 1.3 x 10~3, but this value may vary by ±40% depending upon the wind speed, water depth, and relative temperature a. ®

N = constant

Figure 1 Characteristic continuous water-column stratifications as found in lakes and typical layer approximations. (a) Homogeneous water-column of constant density. (b) Two-layer approximation of the continuous stratification, where the layer separation occurs at the thermocline. (c) Three-layer approximation of the continuous stratification, where the layer separation occurs at the diurnal and seasonal thermoclines. (d) Three-layer approximation of the continuous stratification, where the layer separation occurs at the upper and lower surfaces of the metalimnion. (e) Continuous stratification throughout the water column with constant buoyancy frequency. (f) Continuous stratification where the hypolimnion is characterized by a constant buoyancy frequency.

N = constant

Figure 1 Characteristic continuous water-column stratifications as found in lakes and typical layer approximations. (a) Homogeneous water-column of constant density. (b) Two-layer approximation of the continuous stratification, where the layer separation occurs at the thermocline. (c) Three-layer approximation of the continuous stratification, where the layer separation occurs at the diurnal and seasonal thermoclines. (d) Three-layer approximation of the continuous stratification, where the layer separation occurs at the upper and lower surfaces of the metalimnion. (e) Continuous stratification throughout the water column with constant buoyancy frequency. (f) Continuous stratification where the hypolimnion is characterized by a constant buoyancy frequency.

difference between water surface and adjacent air column.

The momentum transfer associated with steady winds will push the surface water to the leeward shore, causing a displacement of the free surface due to the presence of the solid boundary (Figure 2a); for long and shallow lakes this may be as large as several meters (e.g., ~2m in Lake Erie) (see Currents in the Upper Mixed Layer and in Unstratified Water Bodies).

This displacement is called wind set-up. If the wind stress is applied for sufficient time (one quarter of the fundamental seiche period as defined below), a steady-state tilt of the free surface will occur where there is a balance between the applied wind force (t x surface area) and the hydrostatic pressure force due to the desire of the free surface to return to gravitational equilibrium. Balancing these forces at steady state, given the equation for the slope of the free surface dx gH

where u* = \Jt/po is the surface wind shear velocity, %(x, t) is the interfacial (surface) displacement from the equilibrium position, and x is the longitudinal coordinate. The equation for the free-surface slope may be integrated to give the maximum interfacial displacement, as measured along the vertical boundary u2 L

Wind Set-Up of the Internal Stratification

In a manner analogous to the set-up at the free surface, wind induced displacements can also occur along the thermocline. Consider a simple two-layered lake. Water piled-up at the leeward shore by windward drift simultaneously pushes down the thermo-cline while pushing up the free surface (Figure 2(b), Figure 3). The free surface remains nearly horizontal owing to a return flow that develops in the hypolom-nion, leading to vertical velocity shear through the metalimnion. A corresponding upwelling occurs at the windward shore (Figure 3). The steady-state slope of the free surface is given by a balance between the baroclinic gravitational pressure force from the tilted thermocline and the force due to the wind-stress acting through the epilimnion

dx ¿hi where g = g(p2 — pi)/p2 is the reduced gravity across the interface (thermocline) (see Currents in Stratified Water Bodies 1: Density-Driven Flows). The equation for the thermocline slope may be obtained through integration over the basin length rç,- 0 = 0, x = 0, L) = ±

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