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

Appendix A - Coriolis Acceleration 163
where the advective acceleration (u • ∇)u is introduced by the Eulerian specification of the flow. Equation A.9 states that the
equation of motion in the rotating frame is equivalent to that in an absolute frame so long as it is assumed that a pseudo body force
per unit mass represented by (–2Ω × u) 54 acts upon the fluid. This pseudo body force is known as the Coriolis acceleration
(force/unit mass) and is often significant in estuarine modeling.
In Cartesian coordinates using u = ui + vj + wk and Ω = Ω x i + Ω y j + Ω z k, the acceleration terms are
-----
∂u
u
∂t
∂u
----- v
∂x
∂u
----- w
∂y
∂u
+ + + ----- – 2Ω
∂z zv + 2Ωyw , (A.10)
∂v
----- u
∂t
∂v
----- v
∂x
∂v
----- w
∂y
∂v
+ + + ----- + 2Ω
∂z zu – 2Ωxw , and (A.11)
∂w
------ u
∂t
∂w
------ v
∂x
∂w
------ w
∂y
∂w
+ + + ------ + 2Ω
∂z xv – 2Ωyu .
(A.12)
For a reference frame chosen using the z-axis vertically-upward, but using the horizontal axes rotated by an angle θ, the components
of Ω are dependent on both the latitude φ and the rotation θ by
Ωx = Ω cosφsinθ,
Ωy = Ω cosφcosθ
, and (A.13)
Ωz = Ω sinφ.
When the x-axis is toward the east and the y-axis is toward the north (θ = 0°), these reduce to
Ω x
= 0 ,
Ωy = Ω cosφ,
and (A.14)
Ωz = Ω sinφ
.
In a shallow estuary, the vertical velocity component w is typically very small, and therefore the two Coriolis terms involving w in
the horizontal momentum equations (A.10 and A.11) are usually neglected. Frequently the term (Ω xv – Ω yu) in the z-momentum
equation is also neglected because it is considered to be small in comparison with g. Pond and Pickard (1986, p. 40) point out,
however, that, although (Ω xv – Ω yu) may be small compared with g, it may not always be small compared with the difference
between the pressure term in the z-momentum equation and g. Evces and Raney (1990) formulated a 2-D vertically-averaged
hydrodynamic model that included the z-component Coriolis term and estimated it may correct the total Coriolis term for a high
tidal range estuary near latitude φ = 45° by10 percent. The correction will be most pronounced at high latitudes (φ > 45°) and in
some instances may warrant inclusion in a model applied at lower latitudes. For the model discussed herein, however, the term is
neglected, as is customary. In this case, the form of the acceleration terms in the rotating reference frame are most commonly
expressed as
∂u
----- u
∂t
∂u
----- v
∂x
∂u
----- w
∂y
∂u
+ + + ----- – fv
(A.15)
∂z
54 Using the subscript notation from Chapter 2 of this report, the Coriolis acceleration 2Ω × u is written as 2εijkΩ ju k.