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154 6 Rotational EffectsThe solution of (6.67) is (e.g., Cushman-Roisin, 1994):( )r − ro − Rh(r) = H 1 expR(6.69)where r o is the initial radius of the low-density patch with r o >> R. The internalRossby radius of deformation gives an estimate of the frontal width. The solutionfor geostrophic f ow in the surface layer (northern hemisphere) follows from (6.65)and is given by:v 1 (r) =− √ ( )r − ro − Rg ′ H 1 expR(6.70)The f ow direction is reversed for the southern hemisphere. In this analyticalsolution, frontal fl ws attain the swiftest speeds at the location where the densityinterface outcrops at the sea surface, whereas there are no fl ws just outside thisfront. Such a discontinuity cannot exist in the real world. Instead of this, lateralfriction produces a transition zone across the front and frontal fl w speeds tend tobe smaller than predicted by theory. Interestingly, although the steady-state frontalfl w is purely geostrophic, its magnitude is independent of the Coriolis parameter. Itshould also be noted that the maximum frontal speed equals the phase speed of longinternal waves. Typical oceanic values of v 1 and R, respectively, are 0.1–0.5 m/s and1–5 km. Geostrophic adjustment can be expected to occur on a time scale exceedingseveral inertial periods.An isolated layer of dense water on the seafloo also becomes subject to thegeostrophic adjustment process (Fig. 6.22). Under the assumptions that there are nofl ws outside this layer and absence of frictional effects, the steady-state momentumequation for the bottom layer can be written as:f v 2 =−g ′ ∂h∂rwhere h is the downward displacement of the density interface with reference to theinitial thickness H 2 . Conservation of potential vorticity can be expressed as:Fig. 6.22 Illustration of geostrophic adjustment of a dense bottom layer