Physical Oceanography, Vol. 18, No. 3, 2008
EFFECT OF VERTICALLY LOGARITHMIC STEADY CURRENTS
ON SHALLOW SURFACE WAVES
S. Patil and V. P. Singh
The combined wave-current flow has been solved by researchers by assuming wave over either
depthwise constant or linear current profile. Some complicated nonlinear current profiles have
also been considered to simulate various shear currents. We consider a nonlinear current verti-
cally logarithmic in nature and examine its interaction with a periodic surface wave. The Na-
vier-Stokes equations for incompressible flow are solved for the current part and by using peri-
odic boundary conditions. The effect of logarithmic current on wave components is assessed.
The corresponding celerity and dispersion equation yields a close-form solution for the shallow-
wave approximation. Several comparative trends between wave-only, wave with log current,
and wave with constant current for the wave following/opposing these currents have been dis-
cussed. The flow properties of the first order are presented which can be applicable to the real
inland and coastal flows, where progressive waves are ubiquitous over a depthwise logarithmic
current. The work is further extended to the second-order semiempirical wave component by
using past experimental data on the wave spectrum of combined flow.
Several working groups, like atmospheric sciences, oceanography, including waves, and remote sensing
have investigated the benefits of coastal and estuarine hydrodynamics, where the topic of combine wave-current
flow is of immense importance. When a surface wave meets a steady-current underneath, it undergoes changes
in the amplitude during the adaptation process  to reach to a steady-state after which the combined profile re-
mains permanent over space and time. The interaction shows that waves are strained and refracted by currents
and the exchange of impulse, momentum, and energy occurs between the waves and mean flow. This causes
changes in the magnitude of current at the bottom and surface  and also changes in the flow properties, such
as the horizontal and vertical velocities, turbulent characteristics, pressure and energy distributions, etc . Mix-
ing due to waves and currents greatly enhances the transfer of chemical and biological elements, especially in the
coastal zone. The wave-induced orbital velocity causes the erosion of sediments, which are advected by the
steady current . The wave flumes or the offshore basins in hydrodynamic laboratory are, therefore, designed
especially to provide precise scaled versions of various wave-current combinations. In the experiments on
wave-current flows, a sheared current is used with vertical profile almost linear near the free surface but very
much curved near the bottom [5–7]. This current profile can be simulated by well-known logarithmic law .
Examples of combined wave-over-depthwise-logarithmic-current flows (hereto denoted as wave-log currents,
1) are: uniform flows in open channels with surface waves due to the ubiquity of wind shear or long-
period tides in coastal zones. The vertical current profile in these flows is nearly logarithmic. Accurate model-
ing of this flow pattern is important for dispersion and diffusion studies, geophysical understanding, and coastal
construction because any approximation to the existing wave-log current may lead to numerous biases in the re-
sults. In the present work, we give the formulation and properties of the wave-log current flow. A brief litera-
ture review on wave-current interaction can be found in the next section.
Biological and Agricultural Engineering, Texas A&M University, College Station, Texas, USA.
Published in Morskoi Gidrofizicheskii Zhurnal, No.
20–40, May–June, 2008. Original article submitted December 19, 2006.
0928-5105/08/1803–0133 © 2008 Springer Science+Business Media, Inc. 133