On the vertical structure of the Rhine region of freshwater influence

Abstract

An idealised three-dimensional numerical model of the Rhine region of fresh water influence (ROFI) was set up to explore the effect of stratification on the vertical structure of the tidal currents. Prandle’s dynamic Ekman layer model, in the case of zero-depth-averaged, cross-shore velocities, was first used to validate the response of the numerical model in the case of barotropic tidal flow. Prandle’s model predicted rectilinear tidal currents with an ellipse veering of up to 2%. The behaviour of the Rhine ROFI in response to both a neap and a spring tide was then investigated. For the given numerical specifications, the Rhine plume region was well mixed over the vertical on spring tide and stratified on neap tide. During spring conditions, rectilinear tidal surface currents were found along the Dutch coast. In contrast, during neap conditions, significant cross-shore currents and tidal straining were observed. Prandle’s model predicted ellipse veering of 50%, and was found to be a good indicator of ellipticity magnitude as a function of bulk vertical eddy viscosity. The modelled tidal ellipses showed that surface currents rotated anti-cyclonically whereas bottom currents rotated cyclonically. This caused a semi-diurnal cross-shore velocity shearing which was 90° out of phase with the alongshore currents. This cross-shore shear subsequently acted on the horizontal density gradient in the plume, thereby causing a semi-diurnal stratification pattern, with maximum stratification around high water. The same behaviour was exhibited in simulations of a complete spring–neap tidal cycle. This showed a pattern of recurring stratification on neaps and de-stratification on springs, in accordance with observations collected from field campaigns in the 1990’s. To understand the increase in ellipticities to 30% during neaps and the precise shape of the vertical ellipse structure, stratification has to be taken into account. Here, a full three-dimensional numerical model was employed, and was found to represent the effect of de-coupling of the upper and lower layers due to a reduction of mixing at the pycnocline.

Appendix

Prandle (1982a,b) obtained vertical velocity profiles and associated depth variations of the tidal ellipse properties using the following simplified model:

$$\frac{{\partial u}} {{\partial t}} - fv = - g\frac{{\partial \zeta }} {{\partial x}} + \frac{\partial } {{\partial z}}E\frac{{\partial u}} {{\partial z}}$$

(A1)

$$ \frac{{\partial v}} {{\partial t}} + fu = - g\frac{{\partial \zeta }} {{\partial y}} + \frac{\partial } {{\partial z}}E\frac{{\partial v}} {{\partial z}} $$

(A2)

$$ \begin{array}{*{20}l} {{z = 0:} \hfill} & {{\frac{{\partial u}} {{\partial z}} = 0,} \hfill} & {{\frac{{\partial u}} {{\partial z}} = 0} \hfill} \\ \end{array} $$

(A3)

$$ \begin{array}{*{20}l} {{z = - D:} \hfill} & {{\frac{{\partial u}} {{\partial z}} = su,} \hfill} & {{\frac{{\partial v}} {{\partial z}} = sv} \hfill} \\ \end{array} $$

(A4)

where x is the alongshore axis, y is the cross-shore axis, z is the vertical axis, t is the time, u is the alongshore velocity, v is the cross-shore velocity, f is the Coriolis parameter, ζ is the water level elevation and E is the vertical eddy viscosity. The linearised bottom friction s is defined as $s = {8kU} \mathord{\left/ {\vphantom {{8kU} {3\pi E}}} \right. \kern-\nulldelimiterspace} {3\pi E}$, in which U is the depth-averaged velocity and k is a friction factor introduced by Prandle (1982b). In this model, vertical acceleration, convective and density terms are neglected. The equations can be solved assuming zero frictional stress at the surface and applying a vertically constant value of the eddy viscosity E.

The equations can be solved by rephrasing the current and pressure gradient vectors as the sum of two phasors, $R^{ \pm }$ and $G^{ \pm }$, counter-rotating in the complex plane as follows:

$$R = R^{ + } + R^{ - } = u + iv$$

(A5)

$$ G = G^{ + } + G^{ - } = \frac{{\partial \zeta }} {{\partial x}} + i\frac{{\partial \zeta }} {{\partial y}} $$

(A6)

where $R^{ \pm }$ (and analogously $G^{ \pm }$) are defined in the complex plane as

$$R^{ \pm } = {\left| {R^{ \pm } } \right|}\exp {\left( {i\varphi _{ \pm } } \right)}\exp {\left( { \pm i\omega t} \right)}$$

(A7)

where ω is the angular frequency and φ is the phase in the complex plane. Substituting $R^{ \pm }$ in the equations of motion (Eqs. A1 and A2) yields:

$$ i{\left( {f + \omega } \right)}R^{ + } = G^{ + } + \frac{\partial } {{\partial z}}E\frac{{\partial R^{ + } }} {{\partial z}} $$

(A8)

$$ i{\left( {f - \omega } \right)}R^{ - } = G^{ - } + \frac{\partial } {{\partial z}}E\frac{{\partial R^{ - } }} {{\partial z}} $$

(A9)

By assuming a solution of the form $R^{ \pm }\kern-4pt =\kern-4pt [a\exp {\left( { - \alpha z} \right)} + b\exp {\left( {\alpha z} \right)} + c] \cdot \exp {\left( { \pm i\omega t} \right)}$, and dividing by the expression for the depth-averaged current ${\left\langle u \right\rangle } = {\int\limits_0^D {u{\left( z \right)}dz} }$ (neglecting the integral over D±ζ), we obtain the following solution:

$$ \frac{{R^{ \pm } }} {{{\left\langle {R^{ \pm } } \right\rangle }}} = \frac{{\cosh {\left( {\alpha ^{ \pm } {\left[ {z - D} \right]}} \right)} - {\left[ {\cosh {\left( {\alpha ^{ \pm } D} \right)}} \right]} - \frac{{\alpha ^{ \pm } }} {s} \cdot {\left[ {\sinh {\left( {\alpha ^{ \pm } D} \right)}} \right]}}} {{ - {\left[ {\cosh {\left( {\alpha ^{ \pm } D} \right)}} \right]} + {\left( {\frac{1} {{\alpha ^{ \pm } D}} - \frac{{\alpha ^{ \pm } }} {s}} \right)} \cdot {\left[ {\sinh {\left( {\alpha ^{ \pm } D} \right)}} \right]}}} $$

(A10)

where $\alpha ^{ \pm } = {\left( {1 + i} \right)}{\sqrt {{{\left( {f \pm \omega } \right)}} \mathord{\left/ {\vphantom {{{\left( {f \pm \omega } \right)}} {2E}}} \right. \kern-\nulldelimiterspace} {2E}} } \equiv 1 \mathord{\left/ {\vphantom {1 {\delta ^{ \pm } }}} \right. \kern-\nulldelimiterspace} {\delta ^{ \pm } }$ is the inverse of the boundary layer height. From the solutions $R^{ \pm }$ the ellipse properties can be evaluated as follows:

$$ \begin{array}{*{20}c} {{A_{{major}} }} & {{\text{ = }}} & {{{\left| {R^{ + } } \right|} + {\left| {R^{ - } } \right|}}} & {{{\text{Major axis}}}} \\ {{A_{{minor}} }} & {{\text{ = }}} & {{{\left| {R^{ + } } \right|} - {\left| {R^{ - } } \right|}}} & {{{\text{Minor axis [positive in }}\exp {\left( { + i\omega t} \right)}{\text{direction] }}}} \\ {\psi } & {{\text{ = }}} & {{\tfrac{{\text{1}}} {{\text{2}}}{\left( {\phi _{ - } + \phi _{ + } } \right)}}} & {{{\text{Inclination}}}} \\ {\phi } & {{\text{ = }}} & {{\tfrac{{\text{1}}} {{\text{2}}}{\left( {\phi _{ - } - \phi _{ + } } \right)}}} & {{{\text{Phase}}}} \\ {{\text{E}}} & {{\text{ = }}} & {{\frac{{A_{{minor}} }} {{A_{{major}} }} = \frac{{{\left| {R^{ + } } \right|} - {\left| {R^{ - } } \right|}}} {{{\left| {R^{ + } } \right|} + {\left| {R^{ - } } \right|}}}}} & {{{\text{Ellipticity [positive in }}\exp {\left( { + i\omega t} \right)}{\text{direction]}}}} \\ \end{array} $$

(A11)

Note that the ellipticity used here differs from the eccentricity that is generally used in mathematics. Following Souza and Simpson (1996), the ellipticity is defined as the signed ratio of the minor axis to the major axis. The ellipticity is therefore 0 for a degenerate ellipse, 1 for a cyclonic circular path and −1 for an anti-cyclonic circular path.

For comparison to numerical results, which yield the u and v velocity components, the ellipse properties can also be calculated as follows:

$$ A_{{major}} = \tfrac{1} {2}{\left| {{\left( {a + d} \right)} + i{\left( {c - b} \right)}} \right|} + \tfrac{1} {2}{\left| {{\left( {a - d} \right)} + i{\left( {c + b} \right)}} \right|} $$

(A12)

$$ A_{{minor}} = \tfrac{1} {2}{\left| {{\left( {a + d} \right)} + i{\left( {c - b} \right)}} \right|} - \tfrac{1} {2}{\left| {{\left( {a - d} \right)} + i{\left( {c + b} \right)}} \right|} $$

(A13)

$$ \psi = \tfrac{1} {2}\arg {\left( {\tfrac{1} {2}{\left[ {{\left( {a + d} \right)} + i{\left( {c - b} \right)}} \right]}} \right)} + \tfrac{1} {2}\arg {\left( {\tfrac{1} {2}{\left[ {{\left( {a - d} \right)} + i{\left( {c + b} \right)}} \right]}} \right)} $$

(A14)

$$ \phi = \tfrac{1} {2}\arg {\left( {\tfrac{1} {2}{\left[ {{\left( {a + d} \right)} + i{\left( {c - b} \right)}} \right]}} \right)} - \tfrac{1} {2}\arg {\left( {\tfrac{1} {2}{\left[ {{\left( {a - d} \right)} + i{\left( {c + b} \right)}} \right]}} \right)} $$

(A15)

when coefficients a to d are defined, as in

$$u = a\cos {\left( {\omega t} \right)} + b\sin {\left( {\omega t} \right)}$$

(A16)

$$v = c\cos {\left( {\omega t} \right)} + d\sin {\left( {\omega t} \right)}$$

(A17)