Hydrodynamic Equations of the Sea
3.13 Boundary Conditions in Fluid Dynamics
Hydrodynamic Equations of the Sea 119
at at at
at ax aY
w = - + u - + u - (3.120)
as the kinematic boundary condition on the vertical velocity of a free sur- face. This boundary condition is clearly nonlinear, involving products of height and velocity. However, if the fluid velocities on the surface or the slopes, &$/ax and a [ / a y , respectively, are small, the terms involving the horizontal currents can be neglected compared with
a[/&;
then the kinematic condition reduces to(3.121)
This form is useful in small-amplitude linear problems.
Beyond the kinematic condition, it is also clear that pressure differences across a free surface cannot be supported without disrupting that surface, unless a normal surface stress exists to balance the difference; intermolecu- lar forces exert such stresses via a surface tension, T ~ , which resists the pres- sure upon distension. The value of T~ for clean seawater is approximately 0.079 N m - ' (79 poise). Surface stress is found experimentally to be propor- tional to the curvature of the distorted free surface, or inversely proportion- al to the radius of curvature, R ; in two dimensions, it is the sum of the curvatures, 1/R,
+
1 /R,, that is the operative quantity. Then the pressure difference that can be supported isp - Pa = - 7 s
(t
+t>
' (3.122)where p a is the pressure above the free surface (i.e., atmospheric pressure).
The radii of curvature are counted positive when the center of curvature is above the surface, or outside the liquid (see Fig. 3.19). For the case shown, R,
<
0, and the interior pressure is thus greater than the exterior, as it must be under the influence of the surface tension. Now the curvatures are ap- proximately the second derivatives of the surface elevation, so that the pres- sure just below the surface isp = , - . ( $ + a ' E > . aY2 (3.123)
This equation is termed the dynamic boundary condition, and its imposition
P
I
Fig. 3.19 Small-scale forces at air-water interface, showing pressure difference, p - p a , balanced by surface tension, r 8 . The radius of curvature is R,, which is nega- tive i f its center is beneath the surface of the liquid. Capillary waves have restoring forces due to surface tension.
leads to equations for capillary waves, for example. It is also required as a condition between layers of two dissimilar liquids, in which event 7s clear- ly will have a numerical value different from the one cited above. Oils and other surfactants can cause considerable variation in T,, with even a thin film of oil on a surface reducing 7, to less than one-half of its clean sur- face value.
Boundary Conditions for Viscous Fluids
A viscous fluid requires additional balances of forces at boundaries, the most obvious of which is that the tangential (as well as the normal) compo- nent of fluid velocity must match that of the boundary:
If the boundary is fixed, this condition requires that the interior flow veloci- ty go to zero at the boundary or, if moving, to move in unison with it. This transition takes place through a relatively thin layer such as the surface
Ek-
man layer or the benthic boundary layer on the seafloor, whose thicknesses are counted in tens of meters. In such a region, a large velocity shear (du/dz) normal to the surface develops that may often render the boundary layer unstable, much as internal shear does (see Fig. 3.14). In the shear layer, small disturbances may grow rapidly via shear flow instability, leading to a turbu- lent boundary layer that transports fluid properties in directions normal to the flow via eddy fluxes at rates much greater than laminar flow allows. It is this process that the tensors A and K attempt to describe. Other factors such as density gradients and the presence of contaminants can also influence the rate of growth of the turbulent layer. Shear flow instability in a two-
Hydrodynamic Equations of the Sea 121
layer fluid is discussed in Chapter 5 . These examples of the use of the bound- ary conditions will be given during the process of solving for oceanic mo- tions such as waves and currents. Additionally, the field variables for thermal, acoustical, electromagnetic, and optical fields have their own sets of bound- ary conditions, and these, along with their governing equations, will be de- veloped further along the way.
For now we leave the mechanical portions of the physics of the sea and turn to thermodynamics and energy fluxes in order to understand the effects of heat and radiation in the ocean. We will return to hydrodynamics at the end of Chapter 4, where the coupled dynamical and thermodynamical equa- tions will be developed.
Bibliography Books
Batchelor, G. K., An Introduction to Fluid Dynamics, Cambridge University Press, Defant, A., Physical Oceanography, Vols. I and 11, Pergamon Press, New York, Gill, A. E., Atmosphere-Ocean Dynamics, Academic Press, New York, N.Y. (1982).
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Landau, L. D., and E. M. Lifshitz, FIuidMechanics, Pergamon Press, Oxford, Eng- Neumann, G., and W. J. Pierson, Jr., Principles of Physical Oceanography, Prentice- Prandtl, L., Essentials of Fluid Dynamics, Hafner Publishing Co., New York, N.Y.
von Arx, W. S., An Introduction to Physical Oceanography, Addison-Wesley Pub- Yih, C.-S., Fluid Mechanics, McGraw-Hill Book Co., New York, N.Y. (1969).
Journal Articles and Reports
Haxby, W. F., 1982-83 Lamont-Doherty Geological Observatory Yearbook 12, Columbia University, Palisades, N.Y. (1983).
Marsh, J . G., A. C . Brenner, B. D. Beckley, and T. V. Martin, “Global Mean Sea Surface Based Upon the Seasat Altimeter Data,” J . Geophys. Res., Vol. 91, p. 3501 (1986).
Oceanographic Atlas of the North Atlantic Ocean, Section I , U.S. Naval Oceano- graphic Office, Washington, D.C. (1969).
Schwiderski, E. W . , Global Ocean Tides, Part II, U.S. Naval Surface Weapons Center, NSWC TR-79-414, Dahlgren, Va. (1979).
Thorpe, S. A., “Experiments on the Instability of Stratified Shear Flows: Miscible Fluids,” J . Fluid Mech. Vol. 46, p. 299 (1979).
Cambridge, England (1967).
N . Y . (1961).
land (1959).
Hall, Inc., Englewood Cliffs, N.J. (1966).
(1952).
lishing Co., Reading, Mass. (1962).
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Chapter Four Thermodynamics and Energy Relations
“The heat’s on, the pressure’s great, The heat’s on, we’re in a state. ’’
Rock song, 1985