## Txz PK and tyz pK

where p is water density and K is an eddy momentum diffusivity of (or kinematic viscosity), which in the most general case is dependent on the vertical z coordinate. Boundary conditions at the free surface and the bottom of the homogenous layers are t I — tw

where (r£, ty) are the x- and y- components of the stresses applied to the bottom surface of the layer t and (tW, tj) represent the x- and y- components of the

### Lateral and Vertical Circulations

Consistent with the linear approach and for the clarity of presentation, the circulation is considered as the result of superimposing a depth-integrated

(or 'lateral') circulation and a vertically-varying horizontal motion (vertical or overturning circulation). The vertical distribution of horizontal velocities can be estimated by solving the momentum equations (eqns [1] and [2]), once a suitable approximation to a distribution of K(z) has been adopted and using the gradients of the free-surface elevation at any (x, y) location as external inputs. This problem is referred to as the 'local problem'. The remaining problem (referred to as the 'global' problem) is to calculate the basin-wide distribution of pressure or lake level (see wind setup). This, in turn, involves solving the set of equations that result from integrating in depth eqns [1]-[3]. The depth-integrated form of eqs. [1]-[3] are also referred to as the 'transport' equations, and they are posed in terms of the horizontal transports U = f_H u dz and V = H v dz.

### Rotational Effects in Shallow Layers

Rotational effects need to be taken into account when analysing the circulation of lakes with horizontal dimensions larger than the Rossby radius of deformation R0 and when considering processes with time scales which are on the order of f_1or larger. The Rossby radius of deformation is estimated as the ratio between c, a characteristic speed at which information is propagated, and f i.e., R0 = c/f. For example the speed at which long- surface waves presure perturbations in the free surface of shallow layers travel can be estimated as

For typical values of f = 10~4 s_1 and H = 10 m, R0 for long surface waves is approximately 100 km, larger than the dimensions of most lakes around the world. Hence, the rotational effects can be safely ignored in most lakes when analysing free surface dynamics. Internal waves (which modify rj in eqn [3]) however, have periods which are on the order of the inertial period (2p/f) and their Rossby radius (referred to as the internal Rossby radius) is of order 1000 m. Moreover, surface currents are typically on the order of 10_1 ms_1, and R0 is also k 1000 m). Hence, it is likely that circulation is affected by rotational effects in most lakes. Only in very small or narrow lakes, with widths b << R0 can the influence of the Earth rotation be safely neglected and the motion of water can be described using the same governing eqns. [1]-[3] but setting f = 0.

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