Before starting the discussion of theoretical models of the PBL system we note that the external parameters listed above are determined, in turn, by the processes of interaction between the atmospheric PBL and free atmosphere, and between the ocean PBL and deep ocean. But because the eddy fluxes of various substances decrease as they move away from the interface of two environments and vanish at the outer boundaries of the PBLs, then interaction between the boundary layers, the free atmosphere and the deep ocean is realized not by the eddy transfer but, rather, by processes with different origins. The ordered vertical motions and the penetration of turbulence into a neighbouring non-turbulent domain (entrainment) serve as connecting links in the system of the atmospheric PBL-free atmosphere, and the ocean PBL-deep ocean. It should also be borne in mind that the relaxation time for velocity fields in the PBL system is much less than in the free atmosphere and especially in the deep ocean. Therefore, when examining the dynamics of two PBLs their isolation from the ocean-atmosphere system and subsequent analysis without consideration of the mechanisms of feedbacks with the free atmosphere and with the deep ocean turns out to be justified in a number of cases. One discusses the known autonomy of the system of the atmospheric and ocean planetary boundary layers in these terms.
4.4.1 Theoretical models using a priori information on the magnitude and profile of the eddy viscosity coefficient
When analysing observational data on the wind and ice drift in the period of the 'Fram' drift, Nansen noticed that drifting ice in the Central Arctic deviated to the right of the wind direction by 30-40°. He explained this fact by the Earth's rotation and indicated that because the upper ocean layer is driven to motion by ice the currents induced in the ocean should deviate from the wind direction even more than 30-40°. Theoretical research on this problem was initiated by Ekman (1902), who obtained results conforming to Nansen's qualitative conclusions. Ekman (1905) presented a theoretical confirmation of the clockwise (in the Northern Hemisphere) rotation of the drift current velocity with depth and adduced some arguments in favour of using the results obtained to describe wind velocity changes with height over land. This idea was later realized by Akerblom (1908) and independently by Exner (1912). It is advisable to dwell at length on Exner's work because he was the first to combine the planetary boundary layers of the ocean and atmosphere in a single dynamic system.
Under stationary conditions and parallel and equidistant isobars the motion in the PBL is described by the following equations:
~ kMi ^ + f(vt - Vgi) = 0; ~ km p - f(Ui - Usi) = 0, (4.4.1)
d z: dz: d z: dz, where uh vt are components of the wind velocity or drift current; zi is the vertical coordinate oriented vertically upward in the atmosphere and downward in the ocean (the origin is located at the free ocean surface); the subscript i takes on values 1, 2 while i = 1 complies with the atmospheric PBL, i = 2 complies with the oceanic PBL; all other designations are the same.
Equations (4.1.1) are solved on the assumption that velocity components Vg2, Ug2 of the geostrophic current are equal to zero, coefficients km of eddy viscosity in each PBL remain constant with height and fixed, and the following conditions are fulfilled at the ocean-atmosphere interface and at considerable distance from it:
dp, , dv2 kM\Pi -¡— = ~ kuiPi -7— at Zj 2 = 0, dz1 dz2
Boundary conditions (4.4.2), as known, mean continuity of the velocity and momentum flux at the free ocean surface; conditions (4.4.3) describe the boundedness of the velocity at a sufficiently large height within the limits of the respective PBL.
The final expressions for the vertical profile of the wind and current velocity take the form
Ui = Ugi + e~aiZil(u0 - Ugi) cos atzt + (v0 - Fgi) sin a;z;], j vt = Vgi + e~at::,[(v0 - Vgi) cos a,z; - (u0 - Ugi) sin a,zj, j where u0, v0 are the current velocity components at the ocean surface defined by the equalities
Taking advantage of estimates of the eddy viscosity coefficients existing at that time Koschmider (1938) calculated the change of wind velocity in the atmosphere and of drift current velocity in the ocean. It turned out that the velocity distribution in the system of two PBLs was described by the double Ekman spiral, and the current velocity at the water-air interface was not zero and coincided in direction with the geostrophic wind (see Figure 4.2). From this, in particular, follows the conclusion about the anticlockwise rotation of the wind velocity vector with height in the next-to-surface atmospheric layer.
Further development of the theory was achieved by Shvets (1939) and Hesselberg (1954). To obtain better agreement between calculated results and experimental data they took into account the change in the atmospheric pressure gradient with height. In doing so, Hesselberg, unlike Shvets, assigned the slip condition as the boundary condition at the water-air interface. Both authors considered the eddy viscosity coefficient to be independent of height, and preassigned. Calculated changes of wind velocity with height were found to be qualitatively close to observational changes: the rapid increase of wind velocity was marked within the first several hundred metres, then between 500 and 1000 metres it fell or even terminated altogether, and at larger heights the velocity increase commenced again.
The generalization of the Exner model in the case where the ocean surface is covered with freely drifting ice was presented by Laikhtman (1958). In this case the continuity condition for the momentum flux at the water-air interface was replaced by the budget equation for forces acting upon a unit
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