A Semi-Implicit, Three-Dimensional Model for Estuarine ... - USGS

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160 A Semi-Implicit, Three-Dimensional Model for Estuarine Circulation Appendix A - Coriolis Acceleration This section presents a derivation of the Coriolis acceleration term appearing in the equation of motion. A general treatment of the Coriolis acceleration for particle motion in a rotating reference frame can be found in many standard textbooks on mechanics (for example, Beer and Johnston, 1962). The derivation here is specifically for tidally-driven flows in estuaries and discusses the form of the Coriolis term that is in common use. Vector notation is used in this section, although not elsewhere in this report, because it is an especially natural and compact notation for deriving the Coriolis acceleration. Vectors are identified with bold- face type. In particular, heavy use is made of the vector cross product which is defined in any standard textbook on vector analysis, such as Wills (1958). Consider the earth’s center as the origin of a reference frame Ex’y’z’ (fig. A.1) that is fixed in space (neglecting the motion of the earth’s center as it travels through space). The estuarine reference frame Oxyz is fixed to the surface of the earth and is rotating with angular velocity Ω about the earth’s axis. Let P be the position of a material element of fluid that is moving with the flow in the estuary. The position of P is defined at any instant by the vector r in the rotating frame and the vector r′ in the fixed frame. By simple vector addition one can write r′ = R + r, (A.1) where R is the position vector of the point O at the origin of the rotating frame and has a magnitude equal to the earth’s radius. The absolute velocity v′ of the point P relative to the fixed frame is obtained by differentiating A.1, v′ dr′ dRdr = ------- = ------- + ----- . (A.2) dt dt dt The first term on the right side of this expression is the velocity of the origin of the rotating reference frame Oxyz. From elementary mechanics, the velocity of rigid-body rotation about a fixed axis is defined by the vector cross product dR ------- = Ω × R . (A.3) dt The second term on the right side of A.2 is the velocity of P relative to a nonrotating frame having the same orientation as Ex’y’z’, but with the origin at the point O. In this case, the velocity is defined by dr dr ----- = Ω × r + ⎛------ ⎞ , (A.4) dt ⎝ dt⎠r where (dr/dt) r is the rate of change of r with respect to the rotating frame Oxyz and Ω × r is induced by the rotation. Substituting A.3 and A.4 into A.2 gives the absolute velocity as dr v′ = Ω × R + Ω × r + ⎛------ ⎞ . (A.5) ⎝ dt⎠r Differentiating this expression again using A.3 and the operator (d/dt) defined by A.4 gives an expression for the absolute acceleration
Earth’s center x ‘ z ‘ N E Ω � Axis of rotation R r ‘ y (north) Estuary (size greatly exaggerated) Equator O r P Appendix A - Coriolis Acceleration 161 Earth’s surface z x-axis is east (into page) Figure A.1. Estuarine reference frame Oxyz shown fixed to the earth’s surface at latitude φ and rotating about the earth’s axis with angular velocity Ω. The earth’s center is the origin of a reference frame Ex′y′z′ that is considered to be fixed in space. y ‘
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In cooperation with the Interagency
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U.S. Department of the Interior Dir
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vi 4.2 Semi-Implicit One-Dimensiona
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viii Figure 5.6. Graph showing solu
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x Tables Table 2.1. Dimensionless n
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2 A Semi-Implicit, Three-Dimensiona
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2.4.1.2 Dynamic Surface Condition 2
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Water surface Water surface Level p
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h 1 h 2 h 3 h 4 h 5 h 6 z 1 ⁄ 2 z
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A Semi-Implicit, Three-Dimensional Model for Estuarine ... - USGS

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