The evolution of the flow in a tidal current is described by the velocity vector and the pressure in the following governing equations:
∇ ·V = 0 (3.1)
Du Dt +1
ρ
∂p
∂x −f v=Fx (3.2)
Dv Dt + 1
ρ
∂p
∂y +f u=Fy (3.3)
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1 ρ
∂p
∂z +g = 0 (3.4)
where V is the velocity vector, u and v are the x- and y-component of the velocity,pis the pressure,fis the Coriolis parameter,FxandFyrepresent the frictional forces in x- and y-direction respectively and g is the gravitational acceleration [Marshall and Plumb, 2008].
3.1 is the continuity equation, while 3.2, 3.3 and 3.4 are the x, y and z components of the momentum equation, also referred to as the Navier-Stokes equations [Tu et al., 2008]. These four equations in addition to the boundary conditions give a closed system of equations which describes the flow of the tidal current.
In addition to the four variables mentioned, also temperature and salinity are variables describing an ocean flow. These variables are however not considered to be crucial for the flow features of a tidal current, and their contribution to the flow pattern has been neglected. The fluid is considered to have a uniform density.
The general continuity equation is derived from mass conservation. Given a control volume dV the change in mass over a time dt has to equal the mass going out of or entering the control volume through the control surfaces dS [Garg, 1998]. Water can, for most dynamical purposes, be considered as an incompressible fluid and the continuity equation is reduced to the one given in 3.1 [Marshall and Plumb, 2008].
The equations for motion, 3.2, 3.3 and 3.4, are derived from Newton’s second law [Tu et al., 2008]. The sum of forces working on a fluid parcel has to equal the acceleration of the parcel times its mass. Forces working on a fluid are often divided between forces working on the whole body of the parcel, referred to as body forces, and forces working on the surface of the fluid parcel, referred to as surface forces. The body forces are due to gravitation, Corilolis effect and the centrifugal effect while the surface forces are frictional forces and forces due to pressure gradients [Tu et al., 2008].
For a flow of sea water one can neglect the frictional force everywhere, except close to the boundaries [Marshall and Plumb, 2008]. Along the sea bottom irregularities will increase the rate of momentum diffusion. At the sea surface the wind will stir up the water surface and create turbulence and enhance the momentum exchange between sea and air [Marshall and Plumb, 2008].
3.1 The governing equations 21 The friction along the boundaries is included in the friction termsFx andFy which also will include the friction associated with turbines.
The pressure force working on each surface area of a fluid parcel is given as the pressure times the area of the surface [White, 2011]. The pressure gradient is the difference in pressure on two opposing surfaces, and is given as components for the x, y and z- directions in the three momentum equations.
The gravitational acceleration only has a component in z-direction, and will therefore only contribute to the z-component of the momentum equation [Marshall and Plumb, 2008]. In Rystraumen the sea floor gradient will be very small, hence there will only be a small vertical acceleration. The accel-eration term of the vertical velocityw, including the advection termV · ∇w, is zero, hence also the z-component of the friction. The z-component of the momentum equation is therefore reduced to hydrostatic pressure.
For Rystraumen, it might be discussed whether or not it is realistic to neglect the acceleration of the vertical velocity. For a large scale oceanic system the vertical motion is small [Marshall and Plumb, 2008]. However in Rystrau-men the steep walls on each side might create a vertical motion large enough to affect the flow. On the other hand, the main direction of the fluid flow is along the channel, and not across it, and therefore in this study it is reason-able to assume hydrostatic pressure.
The fluid is described in a coordinate system rotating with an angular velocity Ω. When Newton’s Second Law is considered, the acceleration term has to include both the acceleration of the fluid element relative to the earth and the acceleration due to the earth’s rotation. The latter gives rise to the centrifugal effect and the Coriolis effect. As will be explained, these effects play a major role in the dynamics of the large oceans, however on smaller scale these effects are of less influence [Marshall and Plumb, 2008].
The centrifugal effect results in an additional acceleration pointing outwards from the axis of the rotation and modifies the gravitational acceleration.
The centrifugal acceleration ac, which can be included in the gravitational acceleration, is given as
ac= Ω2Rcos(ϕ) (3.5) where Ω is the angular velocity of the earth, R is the radius and ϕ is the latitude [Marshall and Plumb, 2008]. As the cosine function will equal one at
the equator, the centrifugal force will be strongest here and decreasing toward the poles. The earth rotates 2π radians over an period of 86 400 seconds, this gives an angular velocity Ω = 7.27·10−5rad/s. R is the sum of the radius of the earthrand the depth of the seaD[Marshall and Plumb, 2008].
However, as the depth of the sea is much smaller than the radius of the earth, it is reasonable to assume that R ≈ r, where r = 6371km. At the longitudeϕ= 69◦, where Rystraumen is located, the centrifugal acceleration ac = 1.2· 10−2ms−2. This is about one hundred times smaller than the gravitational acceleration and therefore the centrifugal acceleration of the flow in Rystraumen can be neglected.
The Coriolis parameterf is given by:
f = 2Ωsin(ϕ) (3.6)
where the variables Ω and ϕ are given as for the centrifugal acceleration [Marshall and Plumb, 2008]. To decide whether or not the Coriolis term is of importance for the evolution of the flow the Rossby number can give an indication. The Rossby number is given as the ratio of the advection term on the Coriolis term:
Ro= |V|
Lf (3.7)
where|V|is the magnitude of the velocity vector,Lis the length of the chan-nel andf is the Coriolis parameter given in 3.6 [White, 2011]. In Rystraumen L= 2000m, the velocity at its maximum|V|= 3m/sandf = 1.36·10−4s−1, the Rossby number equals 11. The advection term is therefore ten times larger than the Coriolis term, so the Coriolis acceleration in Rystraumen can be neglected.