Wave trapping and flow around an irregular near circular island in a stratified sea

Abstract

Wave trapping and induced flow around an island is examined. The exactly circular island solutions are reprised and the solutions extended, and shown to apply to a stratified sea. The homogeneous solutions are then used to deduce the wave trapping and flow around a near circular island. It turns out that the cotidal pattern for a perfectly circular island is relatively immune to variations in geometry and radially dependent depth variations. This helps explain the similarity in the behaviour of the tides around various islands (the Pribilof Islands near Alaska, Oahu in Hawaii, Cook Island off north west Australia, Bermuda off the eastern coast of the USA, and Bear Island in the Norwegian Sea). The dominant steady drift and its rate of decay off-shore is also calculated.

Appendix A

The derivation of Eq. (38) is non-trivial for the less mathematically adept, so it is given here for information. Let ${\bf \hat{n}}$ and ${\bf \hat{s}}$ denote directions normal and parallel to the curved or irregular coastline. The condition of zero flow normal to the coast is expressed as:

$$\left(i\sigma\frac{\partial}{\partial n}+f\frac{\partial}{\partial s}\right)\zeta=0$$

(44)

on the boundary. For the boundary considered in this paper, $r=a(1+\xi(\theta))$ where $\xi(\theta)$ is a differentiable function of $\theta$. It is also a stationary random function, but this is irrelevant to this appendix. In order to determine ${\bf \hat{n}},$ first put

$$G=r-a(1+\xi(\theta))$$

(45)

then

$${\bf \hat{n}}={{{\bf\nabla}G}\over {|{\bf\nabla}G|}}={{1}\over {\sqrt{r^2+a^2\xi_\theta^2}}}\left(r\cos\theta +a\xi_\theta\sin\theta,r\sin\theta-a\xi_\theta\cos\theta\right)$$

(46)

and using

$$\mbox{\boldmath$\nabla$}\equiv\left(\cos\theta\frac{\partial}{\partial r}-\frac{\sin\theta}{r}\frac{\partial}{\partial\theta},\sin\theta\frac{\partial}{\partial r}+\frac{\cos\theta}{r}\frac{\partial}{\partial\theta}\right)$$

(47)

gives immediately that

$$\frac{\partial}{\partial n}=\mathbf{\hat{n}}\cdot\mbox{\boldmath$\nabla$}=\frac{1}{\sqrt{r^2+a^2\xi_\theta^2}}\left\{r\frac{\partial}{\partial r}-\frac{a\xi_\theta}{r}\frac{\partial}{\partial\theta}\right\}$$

(48)

Using

$$\mathbf{\hat{s}}=\frac{d\mathbf{r}/d\theta}{|{d\mathbf{r}/d\theta}|}$$

(49)

and that

$$\frac{\partial}{\partial s}=\mathbf{\hat{s}}\cdot\mbox{\boldmath$\nabla$}$$

(50)

it follows that

$$ \mathbf{\hat{s}}=\frac{1}{\sqrt{r^2+a^2\xi_\theta^2}}\left(a\xi_\theta\mathbf{\hat{r}}+r\hat{\mbox{\boldmath{$\theta$}}}\right) $$

(51)

and since

$$ \mbox{\boldmath$\nabla$}=\left(\mathbf{\hat{r}}\frac{\partial}{\partial r}+\hat{\mbox{\boldmath{$\theta$}}} \frac{1}{r}\frac{\partial}{\partial\theta}\right)$$

(52)

it is now also immediate that

$$ \frac{\partial}{\partial s}=\mathbf{\hat{s}}\cdot\mbox{\boldmath$\nabla$}=\frac{1}{\sqrt{r^2+a^2\xi_\theta^2}}\left(a\xi_\theta\frac{\partial}{\partial r}+\frac{\partial}{\partial\theta}\right).$$

(53)

The boundary condition can now be expressed as

$$ i\sigma\left\{r\frac{\partial\zeta}{\partial r}-\frac{a\xi_\theta}{r}\frac{\partial\zeta}{\partial\theta}\right\}+f\left\{a\xi_\theta\frac{\partial\zeta}{\partial r}+\frac{\partial\zeta}{\partial\theta}\right\}=0 $$

(54)

at r=a(1+ξ(θ)), which is Eq. (38).