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• Author or Editor: Thomas F. Jordan
• Journal of Physical Oceanography
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# Vertical-Structure Functions for Time-Dependent Flow in a Well-Mixed Fluid with Turbulent Boundary Layers at the Bottom and Top

James R. Baker and Thomas F. Jordan

## Abstract

The elements of an eigenfunction expansion for time-dependent currents as a function of depth are worked out for viscosity that is given as a parabolic function of depth that goes to zero at both the bottom and top of the water. This yields currents with logarithmic behavior characteristic of turbulent boundary layers at both the bottom and top. Also, solutions are obtained for the two viscosity functions that are half a parabola, going to zero at either the bottom or top but not both. In all cases the solutions are Legendre functions. In some cases the eigenfunctions are Legendre polynomials.

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# Vertical Structure of Time-Dependent Flow Dominated by Friction in a Well-Mixed Fluid

Thomas F. Jordan and James R. Baker

## Abstract

Solutions of a linear hydrodynamic equation of motion with linear boundary conditions are obtained to describe the horizontal current, as a function of depth and time, determined by a given history of the wind force and pressure gradient up to that time, at a fixed point in the horizontal plane, in well-mixed water of finite depth. The bottom friction is assumed to be proportional to the bottom current, with zero bottom current and zero bottom friction considered as limiting cases. The general solution is established as an eigenfunction expansion when the eddy viscosity is given as a positive function of depth. Explicit formulas are worked out for viscosity functions that are constant, exponential, or varying as a power of the height from somewhere below the bottom or above the top of the water. For the latter the limit as the viscosity goes to zero at the bottom or top is considered. Numerical results are presented for viscosities that are constant, exponential, linear, or varying as the 3/4 power.

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# Vertical Structure of Time-Dependent Flow for Viscosity that Depends on Both Depth and Time

James R. Baker and Thomas F. Jordan

## Abstract

A previously developed eigenfunction expansion, that describes horizontal current as a function of depth and time, is extended to include any eddy viscosity given as a product of a function of depth and a function of time.

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