Fundamental Equations and Approximations

Abstract

The fundamental physical principles governing the motion of lake waters are the conservation laws of mass, momentum and energy. When diffusion processes of active or passive tracer substances are also considered, these laws must be complemented by transport equations of tracer mass. All these statements have the form of balance laws and in each of them flux terms arise, for which, in order to arrive at field equations, phenomenological postulates must be established. Hydrodynamics of lakes can be described by a Navier-Stokes-Fourier-Fick fluid or its simplifications. Its field equations are partial differential equations for the velocity vector v, the pressure p, the temperature T and, possibly, the mass concentrations c_α (α =1, ..., N) of N different tracers (i.e. a suspended sediment, phosphate, nitrate, salinity, etc.). Boundary conditions for v, p, T and c_α must also be established; in view of the fact that surfaces may deform and that evaporation may occur, these are not alltogether trivial. In fact the equations of motion of the free or of internal surtaces of density discontinuity — these are the so-called kinematic surface equations — serve as further field equations with the surface displacements as unknown boundary variables. Additional boundary conditions have to be formulated at the lake bottom and along the shore. The latter play a more significant role in physical limnology than in oceanography because for many phenomana the boundedness of the lake domain will affect the details of the processes while oceans may for the same processes be regarded as infinite or semi-infinite. This, for instance, implies that by and large wave spectra in the ocean are continuous, while they are often quantized in lakes.