From Stratodynamics


Kensuke Naito1), Norihiro Izumi2), Miwa Yokokawa3) and Tomohito Yamada2)

1) Graduate School of Engineering, Hokkaido University. kensuke.g.naito@gmail.com

2) Faculty of Engineering, Hokkaido University

3) Faculty of Information Science and Technology, Osaka Institute of Technology


Boundary waves, such as antidunes and cyclic steps (Parker & Izumi 2000) have been known to be formed on river beds or ocean floors by currents. It has also been found that the step-like topographies are formed on the ice surfac, such as the surface of glaciers and the surface of polar ice caps on Mars (Smith & Holt 2010) as well as on the Earth. Because these topographies are formed perpendicular to the direction of the currents, they are assumed to be boundary waves. In the case of polar ice caps, the currents are considered to be density airflow, i.e. katabatic wind (Howard et al 2000). Although the formation of boundary waves on river beds or ocean floors has been studied by a great number of researchers, their formation on the ice surface has hardly ever been studied. In this study, we performed a series of laboratory experiments, and proposed a mathematical model of the formation of boundary waves on the ice surface created by currents.

The experiments were conducted with the use of a flume which has 1.8m in length, 2cm in width and 8cm ice layer on the bottom. We controlled the flow conditions (flow discharge, slope of the flume) and temperature conditions (temperature of the ambient air, fluid and ice), and then we ran water on the ice layer. Boundary waves were formed when the Froude number of the flow was higher than 1.15 (Fr > 1.15), and temperature of the ambient air was higher than the freezing point of water (0°C). We also found that the temperature distribution is a factor which determines the direction of migration of the waves. When the temperature of the ambient air was higher than the freezing point of water, boundary waves were formed. In addition, the bedwaves migrated in the upstream direction. Meanwhile, when the temperature of the ambient air was lower than the freezing point, no boundary waves were formed. In this case, however, once we made a hollow on the flat ice surface, boundary waves were formed. Then, the waves migrated in the downstream direction. Based on the experimental results, we proposed a mathematical model by use of the Reynolds-averaged Narvier-Stokes equation, heat transfer equations for flow and ice, and a heat balance equation at the flow-ice interface as follows.

<flow: Reynolds averaged N-S equations>

<heat transfer in the flow: heat convection equation>

<heat transfer in the ice: heat conduction equation>

<heat balance at the ice surface for solidification process and for melting process>

Where variables with “-” are time-averaged, x and z are the streamwise and vertical coordinates respectively, and u, w, p, g, , f and  denote velocity in x direction, velocity in z direction, pressure, gravitational acceleration, density of fluid, ice surface elevation and kinematic viscosity. kf, ks, T, Tm, Tf, Ts are heat conduction coefficient of fluid and ice, temperature, temperature at the ice surface, averaged temperature of fluid and ice. cf, hsl, h are specific heat of fluid, latent heat and convective heat transfer coefficient. Because it is assumed that the time variation of the flow is much faster than that of ice surface evolution, time derivative terms in the flow equations are neglected. This is a quasi-steady assumption. The Reynolds stress and the turbulent heat flux may be expressed by the use of the eddy diffusivity of moment and heat ;

The eddy viscosity of momentum T can be expressed by mixing length l;

according to Prandtl’s mixing length hypothesis, the mixing length l is in proportional to z as


and k is Karman constant (=0.4). And the eddy viscosity of heat is assumed to be expressed

In order to solve the velocity and temperature distribution in the flow, we employed the Prandtl-Taylor analogy.