Introduction

Water parcels in lakes and reservoirs change their position at a wide range of spatial and temporal scales, describing trajectories that are the result of random and periodic displacements superimposed on more orderly or coherent patterns. It is the coherent and orderly pattern of motions with long-(seasonal) time scales and large-(basin) spatial scales, what we refer to as circulation (or currents). Random motions, in turn, acting at subseasonal and sub-basin scales are responsible for mixing and diffusion processes. Our focus, in this chapter, is on the large-scale circulation in homogeneous or unstratified water bodies (here on HB) and in the upper mixed layer (here on SML) of stratified lakes. The behavior of HB and SML are considered together, since both are homogeneous layers directly forced at the free surface by wind. The exact circulation that develops in response to wind forcing depend on the specific spatial and temporal patterns characterizing the wind field over the lake, together with morphometric characteristics of the layers themselves. In HB, currents interact with the bottom and variations in topography help create gyres and other secondary flows. Under stratified conditions, in turn, currents in the SML are influenced by the time-varying topology of the thermocline (the lower limit of the SML), which is controlled by internal or density driven motions in complex interaction with the wind field and the Earth's rotation (Cor-iolis forces). Hence, in generating the currents in the SML, both internal wave motions and wind forcing are intricately linked. In this chapter we will focus on the response of homogenous layers to wind forcing, leaving aside any discussion on the time evolution of their lower boundaries, i.e., whether they remain constant (as is the case of the bottom of the lake in HB) or they change with time, as a consequence of internal wave motions, as is the case of the SML. The temporal evolution of the thermocline, or in general, that of isopycnal surfaces is discussed in depth in other chapters.

The goal of this chapter is to introduce concepts and tools that are needed to understand the mechanisms by which circulation in these homogenous layers is generated in response to external forcing, mainly wind. Wind forcing over lakes is episodic in nature, characterized by a sequence of events of varying intensity and duration interspersed with periods of calm. To describe the response of a lake to such forcing, we will assume that, to first order, lakes behave as linear systems. Under that assumption, the state of motion in a given lake and at any given instant can be described as the result of superimposing the responses to all individual wind events that have acted over its free surface in the past (i.e., a convolution exercise). Owing to frictional losses, water bodies have 'limited' memory, and only those wind events in the 'closest' past (within a frictional adjustment time scale) will effectively determine the circulation patterns exhibited at any given point in time. Here, our focus will be on describing the currents that, according to the linear theory, will develop in an initially quiescent lake in response to a suddenly imposed wind.

Our rationale for using the linearized equations of motion as the starting point for the description of circulation (and not the full non-linear Navier-Stokes equations, governing the motion of fluids in nature) is that linear theory can accurately predict the spatial scales of the large-scale motions, along with their build up or decay time. Furthermore, considerable insight can be gained into the mechanisms involved in the generation of currents by using a simplified set of equations. However, the reader should be aware that the description of the dynamics of circulation provided by the linear theory is, at most, approximate and 'other' features appear as a consequence of the nonlinearity of the fluid motion. The nonlinearities make the problem of studying circulation patterns intractable with analytical tools. The analysis of the nonlinear dynamics of motion needs to be approached with sophisticated numerical models that simulate the hydrodynamic behavior of lakes. Much of the research in the last few years in the study of lake circulation has been in this direction. Considerable advances, also, have been done in the identification of circulation patterns in lakes, with the help of new observational technologies not previously existing, as high-resolution remote sensing (Figure 1), autonomous satellite-tracked drogues, acoustic Doppler velo-cimetry, and others. We will review some of the studies conducted in several lakes located throughout the world in which the large-scale circulation has been described and studied.

Temperature anomaly (°C)

Figure 1 ETM+ Band 6 (high gain) temperature anomaly, June 3, 2001 18:28 UTC, in Lake Tahoe. The ETM+ image was interpolated to a 90 m grid using bilinear interpolation. The satellite image-derived surface current vector field is overlaid on the image. Adapted from Steissberg TE, Hook SJ, and Schladow SG (2005). Measuring surface currents in lakes with high spatial resolution thermal infrared imagery. Geophysical Research Letters 32: L11402, doi:10.1029/2005GL022912, 2005. Copyright (2005) American Geophysical Union. Reproduced with permission from the American Geophysical Union.

Temperature anomaly (°C)

Figure 1 ETM+ Band 6 (high gain) temperature anomaly, June 3, 2001 18:28 UTC, in Lake Tahoe. The ETM+ image was interpolated to a 90 m grid using bilinear interpolation. The satellite image-derived surface current vector field is overlaid on the image. Adapted from Steissberg TE, Hook SJ, and Schladow SG (2005). Measuring surface currents in lakes with high spatial resolution thermal infrared imagery. Geophysical Research Letters 32: L11402, doi:10.1029/2005GL022912, 2005. Copyright (2005) American Geophysical Union. Reproduced with permission from the American Geophysical Union.

Fundamental Concepts Shallow Water Equations

The starting point in our analysis of currents in the SML or HB is the linear set of equations governing the motion for a shallow layer of homogeneous fluid in direct contact with the atmosphere, i.e.,

— — fv — g ——I----— x—momentum at ox p az

— — —fu — g — I---y—momentum ot oy p oz din — rf) 0 , 0 ,

Here, t is time, u and v are the velocity components along the x- and y- Cartesian directions, g is the acceleration of gravity, f is the Coriolis parameter (or inertial frequency), H(x, y) is the depth of the layer in equlibrium, n and rj denote the vertical displacement of the free surface and the bottom boundary from the equilibrium level, and h = H + q — q'. The symbols tyz and txz represent the x- and y-components of the shear stress applied to a horizontal surface at a vertical location z. The z- coordinate is here considered positive upwards. These symbols will be used consistently throughout this chapter. Equations [1] and [2] are expressions of the second Newton's Law (i.e., force equal mass times acceleration). The first term in their right-hand sides represents the Coriolis force, through which the Earth's rotation influences the dynamics of geophysical flows; the second term accounts for the effects of pressure gradients on fluid motion; and, finally, the third term represents the transfer of horizontal momentum in the vertical z direction by turbulent diffusion. It is presumed, in the derivation of eqns [1]-[3], that the vertical dimension of the layers is much smaller than their horizontal length dimension L, i.e., H << L. Consequently the ratio of vertical to horizontal velocities is small (the motions are quasihorizontal) and the distribution of pressure is hydrostatic in the layer. Furthermore, it will be presumed that q << H and q' << H. Only the components txz and tyz of the stress tensor are considered here, since they are probably the most important in determining the dynamics of the homogenous layers. It will be presumed that they can be modelled using a gradient transport relationship, i.e.,

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