3.1.1. Representation of the equilibrium configuration

For any vortex filament $\boldsymbol {X}(\theta,t)$ whose motion is governed by (3.1), we regard it as an equilibrium form if it moves without change of form in uniform translation. Thus, we look for vortex equilibrium form $\boldsymbol {X}(\theta, t)$ that satisfies

(3.3)\begin{equation} \exists \alpha \in \mathbb{R},\quad \boldsymbol{C}\in \mathbb{R}^3 \Longrightarrow \boldsymbol{X}(\theta,t) = \boldsymbol{X}(\theta + \alpha t, 0) + \boldsymbol{C} t, \quad \forall \, \theta \in \mathbb{R} , \end{equation}

where $\alpha$ is an arbitrary constant and $\boldsymbol {C}$ is a constant translational velocity. This condition basically states that there is an (moving) inertial reference frame in which $\boldsymbol {X}(\theta,t)$ remains unchanged. As a special case, for example, the equilibrium configuration of the helical vortex in an infinite fluid is represented by (2.3) as $\boldsymbol {X}_0(\theta,t) = (\varphi, a\cos \varphi, a\sin \varphi - h ) + ( C_0,0,0) t$, with $\varphi = \theta + \varOmega _0 t$ and $C_0 = U_0 - \varOmega _0$.

The kinematic equation (3.1) is a nonlinear integro-differential equation and, in general, it is difficult to obtain an exact solution $\boldsymbol {X}(\theta,t)$ that satisfies both (3.1) and (3.3). In this study, we seek approximate equilibrium solutions that are moderately perturbed from the helical vortex $\boldsymbol {X}_0$. We express the general equilibrium form of the vortex as

(3.4)\begin{equation} \boldsymbol{X}(\theta,t) =\boldsymbol{X}_0(\theta,t)+ \hat{\boldsymbol{r}}(\varphi) + \hat{\boldsymbol{u}} t, \quad \varphi \equiv \theta + \varOmega t,\ \varOmega \equiv \varOmega_0 + \hat{\omega}, \end{equation}

where $\hat {\boldsymbol {r}}(\varphi )$ represents the geometric variation in the vortex filament, and $\hat {\boldsymbol {u}}$ and $\hat {\omega }$ are the changes in the translational velocity and rotational speed of the vortex, respectively, due to the presence of the (free) surface. In general $\hat {\boldsymbol {u}} = (\hat {u}, \hat {v}, \hat {w})$ can have three non-zero components. It will be shown in later sections that the induced velocity in the vertical direction $\hat {w}$ is negligibly small at the leading-order compared with $\hat {u}$ and $\hat {v}$. We thus assume $\hat {w} = 0$ and consider the vortex submergence $h$ to be fixed in the stability analysis. Like the geometric configuration of $\boldsymbol {X}_0$, $\hat {\boldsymbol {r}}(\varphi )$ is periodic in $\varphi$. We expand $\hat {\boldsymbol {r}}= (\hat {x}, \hat {y}, \hat {z})$ in Fourier series

(3.5)\begin{equation} \hat{\boldsymbol{r}}(\varphi) = \sum_{m{\neq}0} \hat{\boldsymbol{r}}_{m} \, {\rm e}^{{\rm j} m \varphi} = \sum_{m{\neq}0} ( \hat{x}_{m}, \hat{y}_{m}, \hat{z}_{m} ) \, {\rm e}^{{\rm j} m \varphi}, \end{equation}

where ${\rm j}$ is the imaginary unit and $\hat {\boldsymbol {r}}_{m} = (\hat {x}_{m}, \hat {y}_{m}, \hat {z}_{m})$ represents the amplitude vector of the $m$th perturbation mode. Here the zeroth ($m = 0$) mode is not taken into account because it simply represents a shift of origin of the coordinate system. As $\hat {\boldsymbol {r}}(\varphi )$ is real, we have $\hat {\boldsymbol {r}}_{-m} = \overline {\hat {\boldsymbol {r}}_{m}}$ for all $m\in \mathbb {N}$, where $\overline {(\ )}$ represents complex conjugate.

We exclude the trivial solutions of (3.1) by restricting the average modifications in both the radial and angular directions to be zero. The former constraint guarantees the helix radius $a$ to be unaltered because any change in $a$ would be considered in the same stability analysis with different helix radius. The latter constraint excludes a phase shift along the helix tangent since it does not modify the geometric configuration of the helical vortex. These two constraints can be expressed in mathematical form as

(3.6)\begin{equation} \left\{\begin{array}{@{}l}\langle \hat{y} \cos\varphi + \hat{z} \sin\varphi \rangle = 0 \\ \langle \hat{y} \sin\varphi - \hat{z} \cos\varphi \rangle = 0 \end{array}\right. \quad \text{or} \quad \langle ( \hat{y} + {\rm j} \hat{z} ) \, {\rm e}^{-{\rm j} \varphi} \rangle = 0, \end{equation}

where $\langle\, f \rangle$ denotes the mean value of the periodic function $f(\varphi )$ in the interval $[0, 2{\rm \pi} ]$.

For convenience in the analysis and description, we introduce the complex variable $Y + {\rm j} Z$ to represent the $y$- and $z$-coordinates of the deformed helix $\boldsymbol {X}(\theta,t)=(X, Y, Z)(\theta,t)$. With this, we can express $\boldsymbol {X}(\theta,t)$ in a $2\times 1$ matrix form:

(3.7)\begin{equation} \begin{bmatrix} X \\ Y + {\rm j} Z \end{bmatrix} = \begin{bmatrix} \theta + U_0 t \\ a \, {\rm e}^{{\rm j} \varphi} - {\rm j} h \end{bmatrix} + \begin{bmatrix} \hat{u} \\ \hat{v} \end{bmatrix} t + \sum_{m\in\mathbb{Z}} \begin{bmatrix} - {\rm j} \hat{\chi}_{m} \\ \hat{\xi}_{m} \, {\rm e}^{{\rm j} \varphi} \end{bmatrix} \, {\rm e}^{{\rm j} m \varphi}, \end{equation}

with $\varphi = \theta + \varOmega t$ and $\varOmega = \varOmega _0 + \hat {\omega }$. Here $\mathbb {Z}$ represents integers, $\hat {\chi }_{m} = {\rm j} \hat {x}_{m}$ and $\hat {\xi }_{m} = \hat {y}_{m+1} + {\rm j} \hat {z}_{m+1}$. In terms of $\hat {\chi }_{m}$ and $\hat {\xi }_{m}$, the constraints in (3.6) and the assumptions made on the vortex configuration $\boldsymbol {X}(\theta,t)$ can be expressed as

(3.8a–c)\begin{equation} \hat{\chi}_{{-}m} ={-}\overline{\hat{\chi}_{m}} , \quad \hat{\chi}_{0} = \hat{\xi}_{0} = 0, \quad \hat{\xi}_{{-}1} = 0 . \end{equation}

3.1.2. Determination of the equilibrium configuration

We deduce here the governing equations for the unknown variables in the equilibrium form (3.7). To do this, we consider the leading-order terms in the motion equation (3.1), to then obtain the leading approximate solution of the vortex equilibrium form.

Similarly to $\boldsymbol {X}(\theta,t)$, we expand the self-induced velocity along the vortex filament, $\boldsymbol {U}_{{s}}(\theta,t)=( U_{{s}}, V_{{s}}, W_{{s}})(\theta,t)$, in Fourier series:

(3.9a,b)\begin{equation} U_{{s}} = U_0 + \sum_{m\in\mathbb{Z}} U_{{s},m} \, {\rm e}^{{\rm j} m \varphi}, \quad {\rm j} V_{{s}} - W_{{s}} ={-}a\varOmega_0 \, {\rm e}^{{\rm j} \varphi} + \sum_{m\in\mathbb{Z}} V_{{s},m} \, {\rm e}^{{\rm j} (m+1) \varphi}, \end{equation}

where the terms $U_0$ and $-a\varOmega _0 \, {\rm e}^{{\rm j} \varphi }$ are due to the translational and rotational motions of the original helical vortex, respectively. Here $U_{{s},m}$ and $V_{{s},m}$ are the $m$th mode amplitudes of the induced velocity components associated with the modifications of the vortex configuration from the (free-surface) boundary effect. At the leading-order, they are linearly proportional to the $m$th Fourier mode amplitudes of the deformed equilibrium vortex configuration, $\hat {\chi }_m, \hat {\xi }_m$ and $\overline {\hat {\xi }_{-m}}$. For convenience in description, we express $U_{{s},m}$ and $V_{{s},m}$ in symbolic form

(3.10a–c)\begin{equation} U_{{s},m} = \boldsymbol{A}_u(m) \boldsymbol{\cdot}\hat{\boldsymbol{R}}_{m}, \quad V_{{s},m} = \boldsymbol{A}_v(m) \boldsymbol{\cdot}\hat{\boldsymbol{R}}_{m} , \quad \hat{\boldsymbol{R}}_m = [ \hat{\chi}_{m}, \hat{\xi}_{m}, \overline{\hat{\xi}_{{-}m}} ]^{\rm T}, \end{equation}

where $\boldsymbol {A}_u$ and $\boldsymbol {A}_v$ are the $1 \times 3$ coefficient vectors that depend on the mode number $m$ and vortex geometry parameters ($a$ and $e_0$).

For the image-induced velocity $\boldsymbol {U}_{{i}}=(U_{{i}}, V_{{i}}, W_{{i}} )$ and the wave-induced velocity $\boldsymbol {U}_{{w}}=(U_{{w}}, V_{{w}}, W_{{w}})$, under the assumption of $H \gg 1$, the leading-order contributions are from the original helical vortex whereas the effects of the modifications to the equilibrium vortex configuration are of higher order. As the modified helical configuration (and its image) is periodic in the longitudinal direction, both $\boldsymbol {U}_{{i}}$ and $\boldsymbol {U}_{{w}}$ are periodic in $\varphi$. (The proof of the periodicity of $\boldsymbol {U}_{{i}}$ is outlined in § 4.2). We expand the components of $\boldsymbol {U}_{{i}}$ and $\boldsymbol {U}_{{w}}$ in Fourier series and express them in symbolic form

(3.11a,b)\begin{gather} U_{{i}} = \sum_{m\in\mathbb{Z}} U_{{i},m}(a, H) \, {\rm e}^{{\rm j} m \varphi}, \quad {\rm j} V_{{i}} - W_{{i}} = \sum_{m\in\mathbb{Z}} V_{{i},m}(a, H) \, {\rm e}^{{\rm j} (m+1) \varphi}, \end{gather}

(3.12a,b)\begin{gather}U_{{w}} = \sum_{m\in\mathbb{Z}} U_{{w},m}(a, H, F_r) \, {\rm e}^{{\rm j} m \varphi}, \quad {\rm j} V_{{w}} - W_{{w}} = \sum_{m\in\mathbb{Z}} V_{{w},m}(a, H, F_r) \, {\rm e}^{{\rm j} (m+1) \varphi}, \end{gather}

where $U_{{i},m}, V_{{i},m}, U_{{w},m}, V_{{w},m}$ are the associated $m$th Fourier mode amplitudes.

Upon substituting $\boldsymbol {X}$ in (3.7), $\boldsymbol {U}_{{s}}$ in (3.9a,b), $\boldsymbol {U}_{{i}}$ in (3.11a,b) and $\boldsymbol {U}_{{w}}$ in (3.12a,b) into the motion equation (3.1) and matching the coefficients of each harmonic mode, we obtain

(3.13)\begin{gather} \boldsymbol{A}_u(m) \boldsymbol{\cdot}\hat{\boldsymbol{R}}_{m} - m \varOmega_0 \hat{\chi}_m - \hat{u} \delta_m ={-} U_{{i},m} - U_{{w},m}, \end{gather}

(3.14)\begin{gather}\boldsymbol{A}_v(m) \boldsymbol{\cdot}\hat{\boldsymbol{R}}_{m} + (m+1) \varOmega_0 \hat{\xi}_m - {\rm j} \hat{v} \delta_{m+1} + a \hat{\omega} \delta_m ={-} V_{{i},m} - V_{{w},m}, \end{gather}

for $m\in \mathbb {Z}$, where $\delta _m$ denotes the Kronecker delta function that equals 1 if $m=0$ but 0 otherwise. We point out that in (3.13) and (3.14), the equations for different values of $m$ are decoupled. In addition, we have replaced $\varOmega$ by $\varOmega _0$ because the associated quadratic terms, $\hat {\omega }\hat {\chi }_m$ and $\hat {\omega } \hat {\xi }_m$, are of high order and can be neglected.

From (3.13) and (3.14), subject to the constraints (3.8a–c), we can solve for the unknown quantities associated with the modified equilibrium configuration of the helical vortex under a free surface. The solution (for the independent unknowns) can be expressed in the form:

(3.15a,b)\begin{gather} \hat{u} = U_{{i},0} + U_{{w},0}, \quad \hat{\omega} ={-} a^{{-}1} ( V_{{i},0} + V_{{w},0} ), \end{gather}

(3.16)\begin{gather} [\hat{\chi}_1, \hat{\xi}_1, {\rm j} \hat{v} ]^{\rm T} = [ {\boldsymbol{\mathsf{A}}}_0(1) + {\boldsymbol{\mathsf{E}}}_{33} ]^{{-}1} ( \boldsymbol{C}_{{i},1} + \boldsymbol{C}_{{w},1} ), \end{gather}

(3.17)\begin{gather}\hat{\boldsymbol{R}}_{m} = [{\boldsymbol{\mathsf{A}}}_0(m) ]^{{-}1} ( \boldsymbol{C}_{{i},m} + \boldsymbol{C}_{{w},m} ), \quad m \ge 2, \end{gather}

where the $3 \times 1$ vectors, $\boldsymbol {C}_{{i},m}$ and $\boldsymbol {C}_{{w},m}$, and the $3 \times 3$ coefficient matrix, ${\boldsymbol{\mathsf{A}}}_0(m)$, are defined as

(3.18a,b)\begin{gather} \boldsymbol{C}_{{i},m} = [{-}U_{{i},m}, V_{{i},m}, - \overline{V_{{i},-m}} ]^{\rm T}, \quad \boldsymbol{C}_{{w},m} = [{-}U_{{w},m}, V_{{w},m}, - \overline{V_{{w},-m}} ]^{\rm T}, \end{gather}

(3.19)\begin{gather} {\boldsymbol{\mathsf{A}}}_0(m) = {\boldsymbol{\mathsf{A}}}(m) - m \varOmega_0 {\boldsymbol{\mathsf{I}}}_3 , \end{gather}

for $m\geq 1$, ${\boldsymbol{\mathsf{I}}}_3$ being the $3 \times 3$ identity matrix, and the $3 \times 3$ auxiliary matrix ${\boldsymbol{\mathsf{A}}}(m)$ is given by

(3.20)\begin{equation} {\boldsymbol{\mathsf{A}}}(m) = \begin{bmatrix} \boldsymbol{A}_u(m)\\ -\boldsymbol{A}_v(m)\\ \boldsymbol{A}_v({-}m) {\boldsymbol{\mathsf{Q}}} \end{bmatrix} -\varOmega_{0} \begin{bmatrix} 0 & 0 & 0 \\ 0 & 1 & 0\\ 0 & 0 & -1 \end{bmatrix} . \end{equation}

The auxiliary transform matrices ${\boldsymbol{\mathsf{Q}}}$ and ${\boldsymbol{\mathsf{E}}}_{33}$ take the form

(3.21a,b)\begin{equation} {\boldsymbol{\mathsf{Q}}} = \begin{bmatrix} -1 & 0 & 0\\0 & 0 & 1\\0 & 1 & 0\end{bmatrix} , \quad {\boldsymbol{\mathsf{E}}}_{33} = \begin{bmatrix} 0 & 0 & 0\\0 & 0 & 0\\0 & 0 & 1\end{bmatrix}. \end{equation}

The explicit formulae of all coefficient vectors and matrices related to the free-surface boundary effect will be derived in later sections for the respective cases.

3.2.1. Decomposition of perturbation modes

We consider small displacement perturbations $\boldsymbol {y}(\theta,t)$ to the equilibrium vortex configuration $\boldsymbol {X}(\theta,t)$, where $\boldsymbol {y}(\theta,t)$ are assumed to be periodic in $\theta$ (or $\varphi = \theta + \varOmega t$ in the steady moving frame). Following the standard procedure of linear stability analysis, we consider a general Fourier spectral mode of the perturbations, $\boldsymbol {y}(\theta,t) = \boldsymbol {y}_r(\varphi,t) \, {\rm e}^{\pm {\rm j} r \varphi }$, where $r$ is an arbitrary real number and the modal amplitude $\boldsymbol {y}_r$ is periodic in $\varphi$ with the same period of $2{\rm \pi}$ as the base vortex configuration $\boldsymbol {X}(\theta,t)$. Upon expanding $\boldsymbol {y}_r$ in a Fourier series in $\varphi$, we can express the $r$-mode perturbations, $\boldsymbol {y}(\theta,t)=(x, y, z)(\theta,t)$ in the general form

(3.22)\begin{equation} \left.\begin{array}{c@{}}\displaystyle x ={-}{\rm j} \sum_{k\in\mathbb{Z}} [\chi_{k}^{+}(t) \, {\rm e}^{{\rm j} (k+r) \varphi} + \chi_{k}^{-}(t) \, {\rm e}^{{\rm j} (k-r) \varphi} ], \\ \displaystyle y +{\rm j} z = {\rm e}^{{\rm j} \varphi} \sum_{k\in\mathbb{Z}} [ \xi_{k}^{+}(t) \, {\rm e}^{{\rm j} (k+r) \varphi} + \xi_{k}^{-}(t) \, {\rm e}^{{\rm j} (k-r) \varphi} ], \end{array}\right\} \end{equation}

where $\chi _{k}^{\pm }(t)$ and $\xi _{k}^{\pm }(t)$ are the amplitudes of Fourier modes with wavenumbers $k\pm r$, and their time dependencies determine the stability of the vortex filament. Both $k+r$ and $k-r$ Fourier modes are included to ensure that $\boldsymbol {y}(\theta,t)$ is real. It is clear that the perturbations in (3.22) are generally not periodic in $\varphi$, and they become periodic only when $r$ is a rational number.

We point out that in the special case of unbounded fluid (Levy & Forsdyke Reference Levy and Forsdyke1928; Widnall Reference Widnall1972), the modal amplitude of the perturbations $\boldsymbol {y}_r$ is independent of $\varphi$ because the equilibrium vortex configuration is given by the single fundamental Fourier mode only (see (2.3)). The perturbations of the $r$ mode are given by the simple expression $\boldsymbol {y}_r(t) \, {\rm e}^{\pm {\rm j} r \varphi }$. In the presence of a boundary, the equilibrium vortex configuration itself contains multiple Fourier modes in $\varphi$. The evolutions of different Fourier modes of the perturbations are coupled through their interactions with the base vortex. It is, thus, necessary to include a set of coupled Fourier modes in the expression of the $r$-mode perturbations, as given in (3.22), in the present instability analysis.

Note that replacing $r$ by $r\pm 1$ in (3.22), together with corresponding replacements of $[\chi _{k}^{+}, \xi _{k}^{+}]$ by $[\chi _{k \pm 1}^{+}, \xi _{k \pm 1}^{+} ]$ and $[\chi _{k}^{-}, \xi _{k}^{-}]$ by $[\chi _{k \mp 1}^{-}, \xi _{k \mp 1}^{-} ]$, leads to the same expression for $\boldsymbol {y}(\theta,t)$. Moreover, for an arbitrary $r$, there exists an integer $n$ such that $r_0 = |r-n| \le \frac {1}{2}$. Because of these, the instability solution of the problem with the boundary surface is completely described by considering the value of $r$ only in the range of $r \in [0,1/2]$, with $r=0$ representing the super-harmonic perturbation case and $r\in (0,1/2]$ representing the sub-harmonic perturbation case. Note that this is generally true when $k$ takes both even and odd integers in (3.22), but the requisite range of $r$ may vary if $k$ takes only even or odd numbers. In the unbounded fluid case, in contrast, the whole range of $r\in \mathbb {R}$ needs to be considered for a complete description of the instability solution because different modes of the perturbations are not coupled.

3.2.2. Evolution equations of perturbation modes

In linear stability analysis, we take into account only the linear terms related to the small perturbation amplitude in the induced velocity. The contributions to the self-induced velocity from the perturbation $\boldsymbol {y}_{k}(t)$ are

(3.23)\begin{gather} U_{{s}} = \cdots + \sum_{k\in\mathbb{Z}} \left\{ \boldsymbol{A}_u(k+r) + \sum_{m\in\mathbb{Z}} \, {\rm e}^{{\rm j} m \varphi} \hat{\boldsymbol{R}}_m^{\rm T} {\boldsymbol{\mathsf{B}}}_u(m,k+r) \right\} \boldsymbol{\cdot} \boldsymbol{y}_k \, {\rm e}^{{\rm j} (k+r) \varphi}, \end{gather}

(3.24)\begin{gather}{\rm j} V_{{s}} - W_{{s}} = \cdots + \sum_{k\in\mathbb{Z}} \left\{ \boldsymbol{A}_v(k+r) + \sum_{m\in\mathbb{Z}} \, {\rm e}^{{\rm j} m \varphi} \hat{\boldsymbol{R}}_m^{\rm T} {\boldsymbol{\mathsf{B}}}_v(m,k+r) \right\} \boldsymbol{\cdot} \boldsymbol{y}_k \, {\rm e}^{{\rm j} (k+r+1) \varphi}, \end{gather}

where ‘$\cdots$’ represents the contributions from the base equilibrium vortex filament given in (3.9a,b), $\boldsymbol {y}_{k}(t) = [\chi _{k}^{+}(t), \xi _{k}^{+}(t), \overline {\xi _{-k}^{-}(t)}]^{\rm T}$, and $\boldsymbol {A}_u$, $\boldsymbol {A}_v$ are the coefficient vectors introduced in (3.10a,b). The $3 \times 3$ coefficient matrices ${\boldsymbol{\mathsf{B}}}_u$ and ${\boldsymbol{\mathsf{B}}}_v$ are introduced to account for the interaction between the equilibrium modification mode $\hat {\boldsymbol {R}}_m$ and the perturbation mode $\boldsymbol {y}_k$.

For the image-induced velocity, the contributions from the perturbation $\boldsymbol {y}_{k}(t)$ are

(3.25)\begin{gather} U_{{i}} = \cdots + \sum_{k,m\in\mathbb{Z}} \boldsymbol{D}_u(m,k+r) \boldsymbol{\cdot}\boldsymbol{y}_k \, {\rm e}^{{\rm j} (k+m+r) \varphi}, \end{gather}

(3.26)\begin{gather}{\rm j} V_{{i}} - W_{{i}} = \cdots + \sum_{k,m\in\mathbb{Z}} \boldsymbol{D}_v(m,k+r) \boldsymbol{\cdot}\boldsymbol{y}_k \, {\rm e}^{{\rm j} (k+m+r+1) \varphi} , \end{gather}

where ‘$\cdots$’ represents the contributions from the image of the original vortex filament given in (3.11a,b), and $\boldsymbol {D}_u$ and $\boldsymbol {D}_v$ are $1 \times 3$ coefficient vectors that embody the interaction effect of the original vortex image with the perturbation $\boldsymbol {y}_{k}(t)$. As the image effect is strongly affected by the vortex submergence $h$, we derive the explicit formulae of $\boldsymbol {D}_u$ and $\boldsymbol {D}_v$ to the order that matches the order of equilibrium modification modes. The effect of the interactions between the images of equilibrium modifications and perturbations upon the image-induced velocity is of higher order and is thus neglected in the instability analysis.

The contributions to the wave-induced velocity $\boldsymbol {U}_{{w}}$ by the perturbations $\boldsymbol {y}_{k}(t)$ are negligibly small, and are ignored in the stability analysis. This is confirmed by later calculations which show that the wave effect is of importance only when the wave field is near resonant. Under this latter condition, the wave effect causes significant modifications to the equilibrium configuration of the original vortex filament, which has direct influence on the stability of the vortex.

Substituting the perturbations $\boldsymbol {y}(\theta,t)$ for $\boldsymbol {X}(\theta,t)$ in (3.1) and using the self- and image-induced velocities (3.23), (3.24), (3.25) and (3.26) associated with the perturbations for the total velocity $\boldsymbol {U}$ in (3.1), we obtain the differential equations governing the time evolution of perturbation mode amplitudes by matching the coefficients of Fourier harmonic modes ${\rm e}^{{\rm j} (k\pm r) \varphi }$. The system of the evolution equations can be expressed in the following compact form

(3.27)\begin{equation} {-}{\rm j}\frac{\rm d}{{\rm d}t}\, \boldsymbol{y}_k(t) = \tilde{{\boldsymbol{\mathsf{A}}}}(k+r) \boldsymbol{y}_k(t) + \sum_{m{\neq}0} \tilde{{\boldsymbol{\mathsf{B}}}}(m,k-m+r) \boldsymbol{y}_{k-m}(t), \quad k \in \mathbb{Z}, \end{equation}

where the coefficient matrices $\tilde {{\boldsymbol{\mathsf{A}}}}(k)$ and $\tilde {{\boldsymbol{\mathsf{B}}}}(m,k)$ are defined as

(3.28a,b)\begin{equation} \tilde{{\boldsymbol{\mathsf{A}}}}(k) = {\boldsymbol{\mathsf{A}}}_0(k) + {\boldsymbol{\mathsf{A}}}_1(k) + {\boldsymbol{\mathsf{A}}}_2(k) , \quad \tilde{{\boldsymbol{\mathsf{B}}}}(m,k) = {\boldsymbol{\mathsf{B}}}_1(m,k) + {\boldsymbol{\mathsf{B}}}_2(m,k), \end{equation}

with the matrices ${\boldsymbol{\mathsf{A}}}_1$, ${\boldsymbol{\mathsf{A}}}_2$, ${\boldsymbol{\mathsf{B}}}_1$ and ${\boldsymbol{\mathsf{B}}}_2$ given by

(3.29a,b)\begin{gather} {\boldsymbol{\mathsf{A}}}_1(k) ={-} \hat{\omega} \begin{bmatrix} k & 0 & 0 \\ 0 & k+1 & 0 \\0 & 0 & k-1 \end{bmatrix}, \quad {\boldsymbol{\mathsf{A}}}_2(k) = {\boldsymbol{\mathsf{B}}}_2(0,k), \end{gather}

(3.30a,b)\begin{gather}{\boldsymbol{\mathsf{B}}}_1(m,k) = \begin{bmatrix} \hat{\boldsymbol{R}}_m^{\rm T} {\boldsymbol{\mathsf{B}}}_u(m,k) \\ -\hat{\boldsymbol{R}}_m^{\rm T} {\boldsymbol{\mathsf{B}}}_v(m,k) \\ \overline{\hat{\boldsymbol{R}}_{{-}m}^{\rm T}} {\boldsymbol{\mathsf{B}}}_v({-}m,-k) {\boldsymbol{\mathsf{Q}}} \end{bmatrix}, \quad {\boldsymbol{\mathsf{B}}}_2(m,k) = \begin{bmatrix} \boldsymbol{D}_u(m,k) \\ -\boldsymbol{D}_v(m,k) \\ \boldsymbol{D}_v({-}m,-k) {\boldsymbol{\mathsf{Q}}} \end{bmatrix} . \end{gather}

Here matrix ${\boldsymbol{\mathsf{A}}}_0(k)$, as defined in (3.19), represents the effect of the original helical vortex on the perturbations. Matrices ${\boldsymbol{\mathsf{A}}}_1$ and ${\boldsymbol{\mathsf{B}}}_1$ are related to $\hat {\omega }$ and $\hat {\boldsymbol {R}}_m$, respectively, and represent the influence of the modifications of equilibrium configuration on the perturbations. Matrices ${\boldsymbol{\mathsf{A}}}_2$ and ${\boldsymbol{\mathsf{B}}}_2$ originate from the interaction between the vortex image and the perturbations. We note that ${\boldsymbol{\mathsf{B}}}_u$, ${\boldsymbol{\mathsf{B}}}_v$, $\boldsymbol {D}_u$ and $\boldsymbol {D}_v$ are all real, whose explicit formulae will be derived in later sections.

In the special case of unbounded fluid, the evolution equation (3.27) reduces to the simple form: $\dot {\boldsymbol {y}}_k(t) = {\rm j} {\boldsymbol{\mathsf{A}}}_0(k+r) \boldsymbol {y}_k(t)$, where different perturbation modes are decoupled. This equation leads to the stability results of Levy & Forsdyke (Reference Levy and Forsdyke1928) and Widnall (Reference Widnall1972). In the presence of a free surface (or a rigid wall boundary), the perturbation mode $\boldsymbol {y}_k$ is coupled with the modes of different $k$ values (for a specific $r$ value), as (3.27) indicates, resulting in the substantial complexities of the present analysis.

For numerical evaluation, we truncate the number of Fourier modes at a suitable large value $K$ and include the modes of $|k|\le K$ in the analysis. The series summation in (3.27) is then truncated with $|k-m|\le K$. We obtain from (3.27) the system of evolution equations for all the perturbation-related modes considered

(3.31)\begin{equation} \frac{\rm d}{{\rm d}t}\, \boldsymbol{Y}(t) = {\rm j} {\boldsymbol{\mathsf{M}}}(a,e_0,H,F_r,r,K) \boldsymbol{Y}(t), \end{equation}

where $\boldsymbol {Y}(t)$ denotes the amplitude vector ($(6K+3)\times 1$) of perturbation modes

(3.32)\begin{equation} \boldsymbol{Y}(t) = [ \boldsymbol{y}_{{-}K}^{\rm T}, \boldsymbol{y}_{{-}K+1}^{\rm T}, \ldots, \boldsymbol{y}_{0}^{\rm T}, \ldots \boldsymbol{y}_{K-1}^{\rm T}, \boldsymbol{y}_{K}^{\rm T} ]^{\rm T}, \end{equation}

and ${\boldsymbol{\mathsf{M}}}$ is the truncated ($(6K+3) \times (6K+3)$) coefficient matrix which is composed of matrix blocks $\tilde {{\boldsymbol{\mathsf{A}}}}(k+r)$ and $\tilde {{\boldsymbol{\mathsf{B}}}}(m,k-m+r)$. The stability of the helical vortex depends on the eigenvalues $\sigma _\ell$ of ${\rm j} {\boldsymbol{\mathsf{M}}}$. The vortex filament (under the free surface) is stable to the small perturbations in (3.22) if $\operatorname {Re}(\sigma _\ell ) \leq 0$ for all $\ell =1,\ldots, 6K+3$; and unstable if $\operatorname {Re}(\sigma _\ell ) > 0$ for any $\ell =1,\ldots, 6K+3$. The unstable mode shape is given by the eigenvector corresponding to $\sigma _\ell$ with frequency $\operatorname {Im}(\sigma _\ell )$. For later reference, we use $\sigma _R$ to denote the maximum value of $\operatorname {Re}(\sigma _\ell )$ for $\ell =1,\ldots, 6K+3$, representing the growth rate of the most unstable mode.

In the following sections, we derive the explicit formulae of relevant coefficient vectors and matrices and present the results of the equilibrium configuration and stability of the helical vortex filament for the cases of $F_r \rightarrow 0$ and $F_r > 0$, respectively.

4.1.1. Linear terms in ${U}_{{s}}$

We derive the solution for the helical vortex filament with a single harmonic mode modification to its configuration and use linear superposition to obtain the result for the general case involving multiple harmonic modes. The Cartesian coordinates of the filament $\boldsymbol {X}(\varphi )$ are given in terms of the generalised coordinate $\varphi \in \mathbb {R}$ by

(4.1a,b)\begin{equation} X = \varphi - {\rm j} \chi \, {\rm e}^{{\rm j} m \varphi} , \quad Y + {\rm j} Z = ( a + \xi \, {\rm e}^{{\rm j} m \varphi} ) \, {\rm e}^{{\rm j} \varphi} - {\rm j} h . \end{equation}

Here we ignore the translation motion of the filament as it does not affect the self-induced velocity. By the Biot–Savart law, the self-induced velocity at an arbitrary location $\boldsymbol {X}(\varphi )$ along the vortex filament is

(4.2)\begin{equation} \boldsymbol{U}_{{s}}(\varphi) = \frac{1}{2}\int_{\mathbb{R}} \frac{(\boldsymbol{X}(\psi)-\boldsymbol{X}(\varphi)) \times \text{d}\kern0.7pt \boldsymbol{X}(\psi) }{|\boldsymbol{X}(\psi)-\boldsymbol{X}(\varphi)|^3}, \end{equation}

where $\boldsymbol {X}(\psi )$ represents the source point on the vortex filament.

Upon introducing $\lambda = \psi - \varphi$ as the integration variable in (4.2), we obtain explicit expressions for the vectors ${\rm \Delta} \boldsymbol {X} = \boldsymbol {X}(\psi ) - \boldsymbol {X}(\varphi )$ and $\text {d}\kern0.7pt \boldsymbol {X}(\psi )$:

(4.3a,b)\begin{gather} {\rm \Delta} X = \lambda - {\rm j} \chi \, {\rm e}^{{\rm j} m \varphi} p_{m} , \quad {\rm \Delta} Y + {\rm j} {\rm \Delta} Z = a \, {\rm e}^{{\rm j} \varphi} p_{1} + \xi \, {\rm e}^{{\rm j}(m+1) \varphi} p_{m+1} , \end{gather}

(4.4a,b)\begin{gather}\text{d}X = ( 1 + m \chi \, {\rm e}^{{\rm j} m \psi} ) \,\text{d}\lambda , \quad \text{d}Y + {\rm j} \text{d}Z = {\rm j}[ a \, {\rm e}^{{\rm j} \psi} + (m+1) \xi \, {\rm e}^{{\rm j}(m+1) \psi} ] \,\text{d}\lambda , \end{gather}

where $p_{m}(\lambda ) \triangleq \, {\rm e}^{{\rm j} m \lambda } - 1$. Using (4.3a,b), neglecting nonlinear terms in $\chi$ and $\xi$, and employing Einstein summation notation for brevity, we obtain

(4.5)\begin{equation} |{\rm \Delta} \boldsymbol{X}|^{{-}3} = (J_{0} + \hat{J}_{1}^{i} y_{i})^{{-}3/2} \approx J_{0}^{{-}3/2} - \tfrac{3}{2} J_{0}^{{-}5/2} \hat{J}_{1}^{i} y_{i} , \end{equation}

where

(4.6a,b)\begin{equation} \hat{J}_{1}^{i} = [{-}2 {\rm j} \lambda p_{m}, a g_{m+1,-1}, a g_{{-}m-1,1} ] , \quad y_i = [\chi \, {\rm e}^{{\rm j} m \varphi}, \xi \, {\rm e}^{{\rm j} m \varphi},\bar{\xi} \, {\rm e}^{-{\rm j} m \varphi}]^{\rm T},\end{equation}

with $g_{m,n} = p_m p_n$. Substituting (4.5) into (4.2) and using (4.3a,b) and (4.4a,b) for ${\rm \Delta} \boldsymbol {X} \times \text {d}\kern0.7pt \boldsymbol {X}$, we obtain the velocity component in the $x$-direction

(4.7)\begin{equation} U_{{s}} = \frac{1}{2} \int_{\mathbb{R}}[\hat{U}_{0} + \hat{U}_{1}^{i} y_{i}] \left[J_{0}^{{-}3/2} - \frac{3}{2} J_{0}^{{-}5/2} \hat{J}_{1}^{i} y_{i}\right] \text{d}\lambda , \end{equation}

where

(4.8a,b)\begin{equation} \hat{U}_{0} = a^{2}(1-\cos\lambda), \quad \hat{U}_{1}^{i} = \frac{a}{2} [0, f_{m+1,-1}, f_{1,-m-1}] , \end{equation}

with $f_{m,n} = m \, {\rm e}^{{\rm j} m \lambda } p_n - n \, {\rm e}^{{\rm j} n \lambda } p_m$. For brevity in description, we define a functional $\mathscr {J}_n$ for any function $f(\lambda )$ as

(4.9)\begin{equation} \mathscr{J}_{n}[f]\triangleq\frac{1}{2}\int_{\mathbb{R}}f(\lambda)J_0^{{-}3/2-n} \text{d}\lambda , \end{equation}

for $n \in \mathbb {N}$. We then rewrite $U_{{s}}$ in (4.7) in the form

(4.10)\begin{equation} U_{{s}} = U_{0} + \check{U}_{1}^{i} y_{i} , \end{equation}

with

(4.11)\begin{equation} \check{U}_{1}^{i} = \mathscr{J}_{0}[\hat{U}_{1}^{i}] - \tfrac{3}{2} \mathscr{J}_{1}[\hat{U}_{0} \hat{J}_{1}^{i}] . \end{equation}

In the general case of a helical vortex filament modified by multiple harmonic modes ${\rm e}^{{\rm j} m \varphi }$ with associated amplitudes $\chi _m$ and $\xi _m$, linear superposition yields

(4.12)\begin{equation} U_{{s}} -U_{0} = \sum_{m} [ \check{U}_1^1(m) \chi_m \, {\rm e}^{{\rm j} m \varphi} + \check{U}_1^2(m) \xi_m \, {\rm e}^{{\rm j} m \varphi} + \check{U}_1^3(m) \overline{\xi_m} \, {\rm e}^{-{\rm j} m \varphi} ] = \sum_m \boldsymbol{A}_u(m) \boldsymbol{\cdot}\boldsymbol{R}_m \, {\rm e}^{{\rm j} m \varphi}, \end{equation}

where the coefficient vector $\boldsymbol {A}_u(m)$ and amplitude vector $\boldsymbol {R}_m$ are defined as

(4.13a,b)\begin{equation} \boldsymbol{A}_u(m) = [ \check{U}_1^1(m), \check{U}_1^2(m), \check{U}_1^3({-}m) ], \quad \boldsymbol{R}_m = [\chi_m, \xi_m, \overline{\xi_{{-}m}} ]^{\rm T} . \end{equation}

Upon introducing two auxiliary vectors

(4.14a,b)\begin{equation} \boldsymbol{J}_1(m) = [{-}2 {\rm j} \lambda p_{m}, a g_{m+1,-1}, a g_{m-1,1} ], \quad \boldsymbol{U}_1(m) = \frac{a}{2} [0, f_{m+1,-1}, f_{1,m-1}], \end{equation}

we rewrite $\boldsymbol {A}_u(m)$ in (4.13a) in a neater form

(4.15)\begin{equation} \boldsymbol{A}_u(m) = \mathscr{J}_{0}[ \boldsymbol{U}_1 ] - \tfrac{3}{2} \mathscr{J}_{1}[ \hat{U}_{0} \boldsymbol{J}_1 ] . \end{equation}

The self-induced velocity components in the $y$- and $z$-directions are obtained similarly and expressed as

(4.16)\begin{equation} {\rm j} V_{{s}} - W_{{s}} ={-}a \varOmega_0 \, {\rm e}^{{\rm j} \varphi} + \sum_{m} \boldsymbol{A}_{v}(m)\boldsymbol{\cdot}\boldsymbol{R}_{m} \, {\rm e}^{{\rm j} (m+1) \varphi}, \end{equation}

where the coefficient vector $\boldsymbol {A}_v(m)$ takes the form

(4.17)\begin{equation} \boldsymbol{A}_v(m) = \mathscr{J}_{0}[ \boldsymbol{V}_1 ] - \tfrac{3}{2} \mathscr{J}_{1}[ \hat{V}_{0} \boldsymbol{J}_1 ] , \end{equation}

with $\hat {V}_{0} = a q_{1}$ and $\boldsymbol {V}_1 = [ a f_{m,1}, q_{m+1}, 0 ]$, where $q_{m} = p_{m} - {\rm j} m \lambda \, {\rm e}^{{\rm j} m \lambda }$.

4.1.2. Quadratic terms in $\boldsymbol {U}_{{s}}$

To determine the interaction coefficients between the modifications to the original helical configuration and the perturbations in the stability analysis, we need to obtain the quadratic terms in the induced velocity. Letting $\{\chi _{m}^{(1)}, \xi _{m}^{(1)}\}$ and $\{\chi _{n}^{(2)}, \xi _{n}^{(2)}\}$ be two sets of harmonic modes added to the original helical vortex filament, we obtain the quadratic terms in the self-induced velocity as

(4.18)\begin{gather} U_{{s}}= \cdots + \sum_{m, n} \boldsymbol{x}_{m}^{(1)} {\boldsymbol{\mathsf{B}}}_{u}(m, n) \boldsymbol{x}_{n}^{(2)} \, {\rm e}^{{\rm j}(m+n) \varphi}, \end{gather}

(4.19)\begin{gather}{\rm j} V_{{s}} - W_{{s}} = \cdots + \sum_{m, n} \boldsymbol{x}_{m}^{(1)} {\boldsymbol{\mathsf{B}}}_{v}(m, n) \boldsymbol{x}_{n}^{(2)} \, {\rm e}^{{\rm j}(m+n+1) \varphi}, \end{gather}

where $\boldsymbol {x}_{m}^{(1)} = [\chi _{m}^{(1)}, \xi _{m}^{(1)}, \overline {\xi _{-m}^{(1)}}]$, $\boldsymbol {x}_{m}^{(2)} = [\chi _{m}^{(2)}, \xi _{m}^{(2)}, \overline {\xi _{-m}^{(2)}}]^{\rm T}$, with the coefficient matrices ${\boldsymbol{\mathsf{B}}}_{u}$ and ${\boldsymbol{\mathsf{B}}}_{v}$ given by

(4.20)\begin{align} {\boldsymbol{\mathsf{B}}}_{u}(m, n) &= \mathscr{J}_{0}[ {\boldsymbol{\mathsf{U}}}_{2}(m, n)] - \tfrac{3}{2} \mathscr{J}_{1}[ \boldsymbol{J}_{1}^{\rm T}(m) \boldsymbol{U}_{1}(n) + \boldsymbol{U}_{1}^{\rm T}(m) \boldsymbol{J}_{1}(n) + \hat{U}_{0} {\boldsymbol{\mathsf{J}}}_{2}(m, n) ] \nonumber\\ & + \tfrac{15}{4} \mathscr{J}_{2}[\hat{U}_{0} \boldsymbol{J}_1^{\rm T}(m) \boldsymbol{J}_1(n) ] , \end{align}

(4.21)\begin{align} {\boldsymbol{\mathsf{B}}}_{v}(m, n) &= \mathscr{J}_{0}[ {\boldsymbol{\mathsf{V}}}_{2}(m, n)] - \tfrac{3}{2} \mathscr{J}_{1}[ \boldsymbol{J}_{1}^{\rm T}(m) \boldsymbol{V}_{1}(n) + \boldsymbol{V}_{1}^{\rm T}(m) \boldsymbol{J}_{1}(n) + \hat{V}_{0} {\boldsymbol{\mathsf{J}}}_{2}(m, n) ] \nonumber\\ & + \tfrac{15}{4} \mathscr{J}_{2}[\hat{V}_{0} \boldsymbol{J}_1^{\rm T}(m) \boldsymbol{J}_1(n) ] . \end{align}

The auxiliary matrices ${\boldsymbol{\mathsf{J}}}_2(m,n)$, ${\boldsymbol{\mathsf{U}}}_2(m,n)$, and ${\boldsymbol{\mathsf{V}}}_2(m,n)$ are given by

(4.22)\begin{gather} {\boldsymbol{\mathsf{J}}}_2(m,n) = \begin{bmatrix} -2 g_{m,n} & 0 & 0 \\ 0 & 0 & g_{m+1,n-1} \\ 0 & g_{n+1,m-1} & 0 \end{bmatrix} , \end{gather}

(4.23a,b)\begin{gather}{\boldsymbol{\mathsf{U}}}_2(m,n) = \frac{1}{2} \begin{bmatrix} 0 & 0 & 0 \\ 0 & 0 & f_{m+1,n-1} \\ 0 & f_{n+1,m-1} & 0 \end{bmatrix} , \quad {\boldsymbol{\mathsf{V}}}_2(m,n) = \begin{bmatrix} 0 & f_{m,n+1} & 0 \\ f_{n,m+1} & 0 & 0 \\ 0 & 0 & 0 \\ \end{bmatrix} . \end{gather}

The derivation of these formulae is similar to that for $\boldsymbol {A}_u(m)$ and $\boldsymbol {A}_v(m)$ in the preceding section, and the details are omitted here.

5.2.1. Effect of submergence depth on stability

The stability of a single helical vortex in the infinite fluid has been studied extensively by Widnall (Reference Widnall1972). We focus here on the case with an infinite rigid boundary, and investigate the stability under the effect of the boundary controlled by the submergence depth $h = a H$. For specificity, we fix the vortex geometry parameters $a=1.0$ and $e_0=0.1$ and vary $H$ to study how the presence of a rigid boundary influences the stability of the modified helical vortex.

Figure 6 plots the variation of maximum growth rate $\sigma _R$ with perturbation mode number and submergence. The $H=\infty$ curve is the infinite fluid case, where no boundary effect exists. The helical vortex is destabilised by perturbation modes $r \in (\mathcal {R}_1, \mathcal {R}_2)$, where $\mathcal {R}_1 = 0$ and $\mathcal {R}_2 \simeq 0.67$ are the critical mode numbers for the vortex with $a=1.0$ and $e_0 = 0.1$ in the infinite fluid case. In the case of finite $H$, $\sigma _{R}$ in the neighbourhood of $r=\mathcal {R}_1$ slightly decreases as $H$ decreases, and may even become zero for relatively small $H$. This indicates that the (rigid) boundary suppresses the vortex instability for perturbation modes $r\rightarrow \mathcal {R}_1^{+}$. As $r$ increases further away from $\mathcal {R}_1$, $\sigma _{R}$ is almost independent of $H$, implying that the boundary has a negligible effect on the vortex instability for those perturbation modes. As $r\rightarrow \mathcal {R}_2^{-}$, $\sigma _{R}$ increases as $H$ decreases, showing that the boundary effect intensifies the instability. For perturbation modes $r \in [\mathcal {R}_2, 1]$, the helical vortex in infinite fluid is stable. When the vortex filament is close to a wall boundary, however, $\sigma _{R}$ increases to become positive from zero, starting from the modes near $r=1$ and expanding to $r=\mathcal {R}_2$ for deceasing $H$. Thus, the boundary effect destabilises the helical vortex for perturbation modes $r \in [\mathcal {R}_2, 1]$.

Figure 6.

Maximum growth rate $\sigma _R$ as a function of perturbation mode number $r$ for different submergence depths $H$ for a helical vortex with $a=1$ and $e_0=0.1$.

To further understand how the wall effect destabilises the helical vortex, we take the perturbation mode $r=0.9$ as an example and depict the variation of $\sigma _R$ for a wide range of submergence for the same vortex parameters ($a=1.0$ and $e_0=0.1$), see figure 7(a). Here the submergence is given by the parameter $\varepsilon = (2H)^{-1}$. It is seen that the helical vortex becomes unstable to the perturbation mode $r=0.9$ when the submergence is decreased to beyond the threshold value of $\varepsilon > \varepsilon _{0} \simeq 0.09$. Moreover, beyond the threshold submergence $H_0 = (2\varepsilon _0)^{-1}$, $\sigma _R$ increases rapidly as the submergence becomes smaller.

Figure 7.

(a) The maximum growth rate of perturbation mode $r=0.9$ as a function of submergence. (b) The critical submergence parameter $\varepsilon _0$ as a function of perturbation mode number $r$. (Here $a=1.0$ and $e_0=0.1$.)

Mathematically, at the critical value $\varepsilon = \varepsilon _0$, the associated $\sigma (r,\varepsilon _0)$ is the double eigenvalue of coefficient matrix ${\rm j} {\boldsymbol{\mathsf{M}}}(r,\varepsilon _0)$. From the perturbation theory, the matrix eigenvalue with multiplicity $n \ge 2$ can be expanded in a series of $\nu ^{1/n}$ when $O(\nu )$ perturbations are added to the matrix. Therefore, we have the following relation between the leading-order growth rate $\sigma _R(r,\varepsilon )$ and $\varepsilon$:

(5.5)\begin{equation} \sigma_R(r,\varepsilon) \sim \sqrt{\varepsilon^{2}-\varepsilon_{0}(r)^{2}}, \quad \varepsilon \ge \varepsilon_0(r). \end{equation}

To substantiate the validity of this relation, we plot $\sigma _R^2$ vs $\varepsilon ^2$ in the inset of figure 7(a). This relation is validated by the observation that $\sigma _R^2$ is (almost) linearly proportional to $\varepsilon ^2$ for $\varepsilon > \varepsilon _0$.

In figure 7(b), we show the threshold value of $\varepsilon _0(r)$ for perturbation modes $r > \mathcal {R}_2$. It is shown that $\varepsilon _0$ first increases and then decreases to zero as $r$ varies from $\mathcal {R}_2$ to 1, and obtains a maximum $\varepsilon _c \simeq 0.19$ at $r = r_c \simeq 0.7$. Therefore, the helical vortex is unstable to any perturbation modes $r\in (\mathcal {R}_2,1)$ when the vortex is located near the rigid boundary with $H \le H_c = (2 \varepsilon _c)^{-1}$.

5.2.2. Effect of helix geometry on stability

The above discussions are based on a helical vortex filament with fixed geometry parameters $a=1.0$ and $e_0=0.1$. We now examine the effects of vortex geometry parameters on the vortex instabilities. To study the radius (or helix slope) effect, we display in figure 8 the results of $\sigma _R$ as a function of $r$ for two different values of $a=2.0$ and 0.5 with fixed $e_0=0.1$. Three submergence values are considered including the infinite fluid case.

Figure 8.

Maximum growth rate $\sigma _{R}$ as a function of perturbation mode number $r$ for the helical vortex with fixed $e_0=0.1$ and different radius: (a) $a=2.0$ and (b) $a=0.5$ at three different submergence depths $H=3, 5$ and $\infty$.

To provide the proper context, we first summarise briefly the features of instability we obtain in the unbounded fluid case (which are consistent with those of Widnall Reference Widnall1972). The helical vortex with larger radius $a$ (or smaller slope $l = a^{-1}$) is more unstable. The range of unstable modes $[\mathcal {R}_1, \mathcal {R}_2]$ broadens for vortices of larger radius. As shown in figures 6 and 8, $\mathcal {R}_1$ always equals zero, whereas $\mathcal {R}_2$ increases as $a$ increases. In addition, the growth rate is larger for a vortex with larger radius. From the perspective of physics, the adjacent half-turns of helix are relatively closer to each other when radius is larger, causing the vortex to be more unstable due to mutual-induction instability (Widnall Reference Widnall1972).

In the presence of a rigid boundary, the features of instability results can differ characteristically. In the large-radius case $(a=2.0)$, the curves with different submergences almost coincide with the result in the infinite fluid case, indicating that $\sigma _R$ has a weak dependence on the submergence depth and the rigid boundary effect is negligible. In contrast, $\sigma _R$ varies dramatically with $H$ in the small radius case ($a=0.5$). Similar to the case of $a=1.0$, the boundary effect suppresses the instability of the helical vortex for perturbation modes of $r \sim \mathcal {R}_1$ but excites the instability for perturbation modes of $r > \mathcal {R}_2$. Depending on submergence, the instability due to the boundary effect can be more severe than that due to the self-induced instability (in the infinite fluid).

The above results indicate that helix radius can strongly affect the vortex stability, and the instability induced by the rigid boundary can be more significant in the small radius (or large slope) cases. This phenomenon can be qualitatively explained. For a helical vortex with large helix angle, the adjacent half-turns of vortex image (with respect to the vertical symmetry plane) have approximately the same space location but opposite vorticity direction, hence their effect on the primary vortex roughly cancels out each other, causing the total effect of vortex image much weaker in the large radius case. Mathematically, the boundary effect related terms in the evolution equations of perturbation modes, (4.38) and (4.39), are approximately proportional to $a^{-1}$ or $a^{-2}$. This implies that the boundary effect is weaker for larger radius.

In addition to the helix radius, the core-size of vortex filament can also affect the vortex stability. Figure 9 displays the variations of $\sigma _{R}$ with perturbation mode number $r$ for a helical vortex filament with fixed $a = 1.0$ but different values of $e_0 = 0.2$ and $0.3$ at three different submergences. The comparison of the results shows that for larger $e_0$, the range of unstable perturbation modes $[\mathcal {R}_1, \mathcal {R}_2]$ becomes slightly narrower and the magnitude of growth rate also becomes smaller, indicating that the helical vortex with larger core size is more stable in infinite fluid. In addition, because we do not consider the core-size effect when calculating the image-induced velocity $\boldsymbol {U}_{{i}}$, the growth rate of unstable modes due to the rigid wall effect is expected to be nearly independent of $e_0$. This is consistent with the observation that the profiles of $\sigma _R$ in the range of $1.0 > r > 0.7$ for $e_0 = 0.2$ and $0.3$ are almost identical, as shown in figure 9.

Figure 9.

Maximum growth rate $\sigma _{R}$ as a function of perturbation mode number $r$ for the helical vortex with fixed $a=1$ and different core-size: (a) $e_0=0.2$ and (b) $e_0=0.3$ at three different submergence depths $H=3, 5$ and $\infty$.

5.2.3. Critical parameters $H_0$ and $H_c$

To sum up the boundary effect on vortex stability, we display the results of critical parameters $H_0(r)$ and $H_c$, in figure 10, for various vortex filaments. The modified helical vortex is unstable to perturbation mode $r$ if $H \le H_0(r)$. The profile of function $H_0(r)$, shown in figure 10(a), represents the neutral curve and this figure can be regarded as a stability diagram. Moreover, all vortex filaments (with different radius and core size) retain the same trend that $H_0(r)$ first decreases and then increases as $r$ increases. This means that it is more difficult for the vortex to be destabilised for perturbations with moderate mode number $r \in (\mathcal {R}_2,1)$.

Figure 10.

Critical parameters: (a) $H_0$ for different vortex filaments as a function of perturbation mode number $r$; and (b) $H_c$ and $r_c$ as a function of $a$ with fixed $e_0=0.1$.

Figure 10(b) shows the profiles of $r_c$ and $H_c$ for different helix radius $a$. Note that $H_c$ is the minimum value of $H_0(r)$ achieved at $r = r_c$ and the vortex filament is unstable to all perturbation modes $r>\mathcal {R}_2$ when $H\le H_c$. We fix $e_0=0.1$ in figure 10(b) because core-size $e_0$ has little effect on the instability associated with the rigid boundary effect. It is seen that $r_c$ monotonically increases with $a$. When $a$ is large, $r_c$ rapidly tends to 1 due to the fact that the critical mode number $\mathcal {R}_2$ generally increases with $a$ and the self-induced instability in the infinite fluid dominates. When $a\le 1$, $H_c$ slightly oscillates with its magnitude maintaining a value of $O(2.5)$. When $a>1$, $H_c$ increases rapidly and becomes quite large, which means the vortex is easier to be destabilised. Although the boundary effect decreases as $a$ increases, strong destabilisation in this situation is caused by self-induction as in infinite fluid.

6.3.1. Froude number effect on surface signature

We use the analytic solution derived above to highlight the characteristic features of the induced free surface signature and its dependence on the Froude number. As an example, consider a helical vortex filament with fixed geometry parameters $a = 1.0$ and $e_0 = 0.1$ and submergence $H = 3$. For this example, we have the critical Froude number ${\mathcal {F}} (m) = (0.7955, 0.5625, 0.4593, 0.3978, \ldots ) h^{-{1}/{2}}$, for $m = 1, 2, 3, 4, \ldots$. Figures 11 and 12 display the two components of the free-surface profile in the (transverse) $y$-direction for $\eta _{{h}_m}$ and $\eta _{{w}_m}$, respectively, for the first four modes ($m=1, 2, 3, 4$) at a sample value of $F_r = 0.4/\sqrt {3}$ (i.e. $\Upsilon = 0.4$). As shown in (6.10a), the amplitudes of $\eta _{{h}_m}(y)$ are seen to decay exponentially with increasing $y$ and $m$, indicating that the induced waves from $\eta _{{h}_m}(y)$, $m=1, 2, \ldots,$ are all confined in the local area of the vortex filament and are dominated in magnitude by the lower modes. In addition, the parity of ${\rm Re}[\eta _{{h}_m}(y)]$ with respect to $y$ is the same as the parity of integer $m$, whereas ${\rm Im}[\eta _{{h}_m}(y)]$ has the opposite parity.

Figure 11.

Free-surface elevation component of $\eta _{{h}_m}/2{\rm \pi}$ for the modes $m=1$, 2, 3 and 4. The plotted curves are: ——-, real part; and -$\cdot$-$\cdot$-, imaginary part of $\eta _{{h}_m}/2{\rm \pi}$. (Here $a=1.0$, $e_0 = 0.1$, $H=3$ and $F_r = 0.4/\sqrt {3}$.)

Figure 12.

Free-surface elevation component of $\eta _{{w}_m}/2{\rm \pi}$ for the modes $m=1$, 2, 3 and 4. The plotted curves are: ——-, real part; and -$\cdot$-$\cdot$-, imaginary part of $\eta _{{w}_m}/2{\rm \pi}$. (Here $a=1.0$, $e_0 = 0.1$, $H=3$ and $F_r = 0.4/\sqrt {3}$.)

As $F_r < {\mathcal {F}} (m)$ for $m\leq 3$, $\eta _{{w}_m}$ has the same parity as $\eta _{{h}_m}$, and the amplitudes of $\eta _{{w}_m}$ exhibit the same exponentially decaying features in $y$ as $\eta _{{h}_m}$, see figure 12. Because $F_r > {\mathcal {F}} (m)$ for $m\geq 4$, the free-surface elevation $\eta _{{w}_4}$ displays a wave-like profile at $|y|\gg 1$ with the amplitudes not decaying with increasing $|y|$. The parity of function $\eta _{{w}_m}(y)$ with respect to $y$ breaks, and the amplitude of the wave profile is greater on the $y>0$ side. Moreover, the magnitude of $\eta _{{w}_4}$ is seen to be greater than that of $\eta _{{w}_3}$, indicating that the monotonically decaying feature of the amplitude of $\eta _{{w}_m}(y)$ with $m$ may not be obtained once $F_r$ crosses the value of ${\mathcal {F}} (m)$.

As (6.9) shows, the induced free-surface shape remains unchanged in a coordinate system that moves at a constant speed $C_0$ in the $x$-direction. The feature of the free surface elevation $\eta (x,y,t)$ is thus represented by its value at a sample time, say, $\eta (x,y,0)$. For the same vortex geometry parameters and $F_r$ values as in figures 11 and 12, figure 13 displays the 3D surface profile and two-dimensional (2D) contour plot of $\eta (x,y,0)$. In the near field of the vortex filament with $y = O(1)$, the surface pattern is dominated by the one-dimensional (1D) wave $(\eta _{{h}_1} + \eta _{{w}_1})\, {\rm e}^{{\rm j} x}$, which has a period of $2{\rm \pi}$ in the $x$-direction. In the far field with large $|y|$, however, the surface pattern becomes 2D and is dominated by the wave mode $\eta _{{w}_4} \, {\rm e}^{4 {\rm j} (x \pm k_0 y)}$. In this case, we have $k_0\approx 0.15 < O(1)$. The wave crest at large $|y|$ is nearly parallel to the $y$-axis as shown in the contours of $\eta (x,y,0)$ (figure 13b).

Figure 13.

(a) Three-dimensional profile and (b) two-dimensional contour plot of free-surface elevation $\eta (x,y,0)$. (Here $a=1.0$, $e_0 = 0.1$, $H=3$ and $F_r = 0.4/\sqrt {3}$.)

As $F_r$ increases, the threshold mode number $M = \lceil \beta ^{-2} \rceil$ causing 2D waves to appear at large $|y|$ reduces, and the wave feature in the $y$-direction becomes more apparent. For a larger value of $F_r = 0.6/\sqrt {3}$, figure 14 shows the free-surface mode profiles of $\eta _{{w}_m}/2{\rm \pi}$ for $m = 1, 2, 3$ and 4, where the 2D wave patterns at large $|y|$ appear for $m\ge 2$ modes. The wave amplitude is seen to decay rapidly with increasing $m$, as shown in figure 14. The far-field wave is dominated by the critical $M=2$ wave mode.

Figure 14.

Free-surface elevation component of $\eta _{{w}_m}/2{\rm \pi}$ for the modes $m=1$, 2, 3 and 4. The plotted curves are: ——-, real part; and -$\cdot$-$\cdot$-, imaginary part of $\eta _{{w}_m}/2{\rm \pi}$. (Here $a=1.0$, $e_0 = 0.1$, $H=3$ and $F_r = 0.6/\sqrt {3}$.)

Figure 15 displays 2D contours of the total free-surface elevation $\eta (x,y,0)$ for different Froude numbers $F_r = 0.6/\sqrt {3}$, $0.8/\sqrt {3}$ and $1.4/\sqrt {3}$, for fixed $a=1.0$, $e_0=0.1$ and $H=3$. The periods of far-field stripe structure in the three subplots of figure 15 are ${\rm \pi}$, $2{\rm \pi}$ and $2{\rm \pi}$, respectively, due to the fact that ${\mathcal {F}} (2) < 0.6/\sqrt {3} < {\mathcal {F}} (1) < 0.8/\sqrt {3} , 1.4/\sqrt {3}$. In the $F_r = 0.8/\sqrt {3}$ case, because $F_r \approx {\mathcal {F}} (1)$, the wave field is nearly resonant with the surface elevation reaching the value of $O(0.5)$. In the $F_r = 1.4/\sqrt {3}$ case, $F_r$ is not close to any ${\mathcal {F}} (m)$ and the amplitude of the total wave elevation decreases to $O(10^{-2})$.

Figure 15.

Contour plots of free-surface elevation $\eta (x,y,0)$ with $F_r=$ (a) $0.6/\sqrt {3}$, (b) $0.8/\sqrt {3}$ and (c) $1.4/\sqrt {3}$. (Here $a = 1.0$, $e_0 = 0.1$ and $H= 3$.)

6.3.2. Far-field wave properties

The above analysis suggests that the amplitude and propagation direction of waves in the far-field regions of the helical vortex are dominated by the threshold mode $M$. The leading-order term of the surface elevation is

(6.11)\begin{equation} \frac{\eta_{{w}_M}}{2{\rm \pi}} ={\rm Re} \{ \mathcal{A}_{{\pm}} \, {\rm e}^{{\rm j} M (x \pm k_0 y - C_0 t)} \} , \end{equation}

as $y\rightarrow \pm \infty$, where

(6.12)\begin{equation} \mathcal{A}_{{\pm}} = 2h^2 F_r^4 C_0^3 (-{\rm j})^{M+1} M C_M \, {\rm e}^{{-}M^2 \beta^2h} (M \beta^2\pm k_0)^M k_0^{{-}1}, \end{equation}

with $k_0 = (M^2 \beta ^4-1)^{1/2}$. It is clear from (6.12) that $\mathcal {A}_{\pm }$ decay exponentially with submergence $h$. The ratio of $|\mathcal {A}_{+}|$ to $|\mathcal {A}_{-}|$, given by

(6.13)\begin{equation} R_A = \frac{|\mathcal{A}_{+}|}{|\mathcal{A}_{-}|} = \frac{(M \beta^2 + k_0)^M}{(M \beta^2 - k_0)^M} = (M \beta^2 + k_0)^{2M}, \end{equation}

is, however, independent of $h$. In addition, the wave propagating direction is controlled by $k_0$ which is also independent of $h$. Another observation from (6.13) is that $R_A > 1$, which means that the wave amplitude in $+y$ direction is always greater than that in $-y$ direction. This non-symmetric property of the wave field, also seen in several figures in the preceding section, is a direct result of the (right-)handedness of helical vortex we consider.

Inversely, we can deduce certain characters of submerged helical vortex based on the properties of generated waves at the far fields. For example, suppose that the far-field waves at the two sides of the submerged vortex filament have amplitudes $|\mathcal {A}_{1}|$ and $|\mathcal {A}_{2}|$ (with $|\mathcal {A}_{1}| > |\mathcal {A}_{2}|$) as well as outward propagation directions $\boldsymbol {k}_1$ and $\boldsymbol {k}_2$, respectively. We then have that the helix axis ($x$-axis in this study) has to be parallel to $\boldsymbol {k}_1 + \boldsymbol {k}_2$. Moreover, we have $k_0 = \tan \theta _0$ with $\theta _0$ being the angle between $\boldsymbol {k}_1$ and the helix axis. The critical mode number $M$ and parameter $\beta$ can be calculated from the relations: $M \beta ^2 = (k_0^2+1)^{1/2}$ and $M = \ln R_A / [2\ln (M \beta ^2 + k_0)]$ with $R_A = |\mathcal {A}_{1}|/|\mathcal {A}_{2}|> 1$. The length scale (or pitch), $L = b^*$, of the helical vortex is obtained from the relation $2{\rm \pi} L = M {\rm \Delta} d$ in terms of the wavelength in the $x$-direction ${\rm \Delta} d$. Furthermore, the radius and submergence of the helical vortex can also be obtained from Fourier analysis of the wave elevations.

7.2.1. Froude number effect

Due to the wave effect, perturbation-related Fourier mode $k$ interacts with modes $k\pm 1$ in addition to $k\pm 2$ modes. As a result, the odd modes $2k+1+r$ and even modes $2k+r$ are coupled. In the stability analysis, both even and odd Fourier modes in (3.22) need to be considered in the representation of perturbations. The stability result for $r\in [1/2, 1]$ is symmetric in $r$ with respect to $r=1/2$, and we need to consider the stability for $r\in [0,1/2]$ only. The truncated coefficient matrix ${\boldsymbol{\mathsf{M}}}$ is a pentadiagonal block matrix containing all perturbation modes $k+r$ with $|k|\le K$. For the results presented in this section, $K=2N=6$ is chosen.

For numerical evaluation, we consider a vortex filament with fixed geometry parameters $a = 1.0$, $e_0 = 0.1$ and $H = 3$, and calculate the maximum growth rate $\sigma _R(r,H,\beta )$ of unstable modes for different values of $r$ and $\beta$. Figure 18 shows the results for representative values of $\beta =0$, 0.4, 0.6 and 0.8. The curves of $\sigma _R$ for different $\beta$ are seen to coincide with each other when $r > \approx 0.1$, indicating that the Froude number effect is weak for perturbation modes with large $r$. On the other hand, $\sigma _R$ for perturbation modes with $r < 0.1$ increases significantly with Froude number. Such a strong Froude number effect is especially apparent for the super-harmonic perturbations (with $r=0$), which are now unstable in contrast to their stability in the $F_r \rightarrow 0$ case.

Figure 18.

Maximum growth rate $\sigma _R$ as a function of $r \in [0, 1/2]$ for varying Froude numbers but fixed vortex geometry parameters $a=1.0$, $e_0=0.1$ and $H = 3$.

Figure 19 compares the features of $\sigma _R$ obtained with different values of $a$ and $H$ with two fixed $\beta =0$ and 0.6. The results in figure 19 again indicate that the wave effect on growth rate $\sigma _R$ is noticeable only for perturbations of small mode number $r$ (corresponding to long sub-harmonic perturbations), and most significant for super-harmonic ($r=0$) modes. Moreover, the wave effect is shown to be more significant for smaller submergence, as expected, because the induced wave motion decays exponentially with submergence. For the same reason, $\sigma _R$ is weakly dependent on $\beta$ for large $a$, but increases rapidly with $\beta$ for small $a$, as observed in figures 19(c) and 19(d).

Figure 19.

Maximum growth rate $\sigma _{R}$ as a function of perturbation mode number $r$ for $\beta =0$ and 0.6 with various vortex filament parameters: (a) $a = 1.0$, $e_0 = 0.1$, $H = 2.5$; (b) $a = 1.0$, $e_0 = 0.2$, $H = 2.5$; (c) $a = 2.0$, $e_0 = 0.1$, $H = 3$; and (d) $a = 0.5$, $e_0 = 0.1$, $H = 4$.

7.2.2. Reduced-order model for growth rate of $r=0$ perturbation mode

The above results indicate that the wave effect on the helical vortex stability is most noticeable for the $r=0$ perturbation mode. It is of practical interest in the study of vortex dynamics under free surface to provide a simple approximate formula for the estimate of the growth rate of this significant unstable mode in term of $F_r$ and helix parameters. In this subsection, we develop a reduced-order model to describe the quantitative relation between $\sigma _R(r=0)$ and $\beta$ for the $r=0$ perturbation mode.

In figure 20, we plot $\sigma _R(r=0)$ as a function of $\beta$ for fixed $a=1.0$, $e_0=0.1$ and $H = 3$. It is seen that the vortex filament becomes unstable as $\beta$ increases beyond the critical value $\beta _0$, with the growth rate increasing dramatically to infinite as $\beta$ approaches the resonance condition $\beta = 1$. This feature of the solution is similar to that of $\sigma _R(r,\varepsilon )$ of $r > \mathcal {R}_2$ modes in the $F_r \rightarrow 0$ case. We thus expect that the critical condition $\beta = \beta _0$ corresponds to the eigenvalue with multiplicity 2 of coefficient matrix ${\rm j} {\boldsymbol{\mathsf{M}}}$. One also notices that among all $\beta$-related terms in the evolution equation (3.31), only the alteration of rotational speed $\hat {\omega }$ appears in the main diagonal of truncated coefficient matrix ${\boldsymbol{\mathsf{M}}}$. These imply that $\hat {\omega }$ is the major factor through which the wave influences the stability of modified vortex filament. We plot $\sigma _{R}^2$ as a function of $\hat {\omega }$ in the inset of figure 20, from which $\sigma _{R}^2$ is seen to be linearly proportional to $\hat {\omega }$ when $\hat {\omega }$ exceeds the critical value $\omega _0$. This feature can be represented by the formula

(7.6)\begin{equation} \sigma_R(0,H,\beta) = \zeta_1(H) [ \hat{\omega}(H,\beta) - \omega_0(H) ]^{{1}/{2}}, \quad \beta \ge \beta_0(H) , \end{equation}

where $\omega _0(H) = \hat {\omega }(H,\beta _0(H))$ is the threshold of $\hat {\omega }$ to destabilise the vortex filament, and the coefficient $\zeta _1(H)$ embodies the strength of instability which depends on the vortex geometry parameters such as radius and submergence.

Figure 20.

Maximum growth rate $\sigma _R(r=0)$ as a function of $\beta \in [0,1)$ in the case with vortex parameters $a=1.0$, $e_0=0.1$ and $H = 3$.

To find the dependence of $\zeta _1$ and parameters $\beta _0$ and $\omega _0$ on submergence $H$, we plot in figure 21 $\zeta _1$, $\beta _0$ and $\omega _0$ as functions of $H$ for two sets of vortex geometry parameters $(a, e_0)=(1.0, 0.1)$ and $(0.3, 0.1)$. Interestingly, it is observed that $\zeta _1(H) \approx \zeta _F$ with $\zeta _F$ being dependent on the radius and core size but nearly independent of $H$ in both cases.

Figure 21.

Model coefficients $\zeta _1, \beta _0$ and $\omega _0$ as a function of $H$ for a helical vortex filament with: (a) $a=1.0$, $e_0=0.1$; and (b) $a=0.5$, $e_0 = 0.1$. The prediction of $\zeta _1$ by the simple model $\zeta _F =\sqrt {I_2}$ is shown for comparison.

We find that $\zeta _F$ can be estimated by a simple formula. To derive the formula, we ignore all the terms that come from the wall and wave effects except for the alteration of rotational speed $\hat {\omega }$ in the governing evolution equation for the $r=0$ perturbation mode. With this, we obtain the evolution equation

(7.7)\begin{equation} \dot{\boldsymbol{y}}_0(t) = {\rm j} {\boldsymbol{\mathsf{M}}}(0) \boldsymbol{y}_0(t) , \end{equation}

where the coefficient matrix takes the form

(7.8)\begin{equation} {\boldsymbol{\mathsf{M}}}(0) = {\boldsymbol{\mathsf{A}}}_0(0) + {\boldsymbol{\mathsf{A}}}_1(0) = \frac{1}{2} \begin{bmatrix} 0 & I_1 & I_1 \\ 0 & I_2 & I_2 \\ 0 & - I_2 & - I_2 \end{bmatrix} + \begin{bmatrix} 0 & 0 & 0\\ 0 & -\hat{\omega} & 0 \\ 0 & 0 & \hat{\omega} \end{bmatrix} , \end{equation}

with $I_1$ and $I_2$ given by

(7.9)$$\begin{gather} I_1 = \partial_a U_0 = a \mathscr{J}_{0}[1-\cos\lambda] - 3 a^3 \mathscr{J}_{1} [(1-\cos\lambda)^2], \end{gather}$$

(7.10)$$\begin{gather}I_2 = a \partial_a \varOmega_0 = 3 a^2 \mathscr{J}_{1} [(1-\cos\lambda)(\cos\lambda-1+\lambda\sin\lambda)]. \end{gather}$$

The characteristic polynomial of ${\rm j} {\boldsymbol{\mathsf{M}}}(0)$ is $f(\hat {\omega },\sigma ) = \sigma (\sigma ^2 - I_2 \hat {\omega } + \hat {\omega }^2)$. As shown in figure 2, $\varOmega _0$ monotonically increases with $a$ in the interval $a \in (0,3)$ when $e_0\le 0.3$. We thus have $I_2= a \partial _a \varOmega _0 > 0$. The eigenvalues are found to be $\sigma _0 = 0$ and $\sigma _{\pm } = \pm \sqrt {I_2 \hat {\omega } - \hat {\omega }^2} \approx \pm \sqrt {I_2\hat {\omega }}$ (because $\hat {\omega } \ll 1$). This gives the growth rate of the $r=0$ perturbation mode $\sigma _R = \sqrt {I_2 \hat {\omega }}$ from which we obtain $\zeta _F = \sqrt {I_2}$. The prediction of $\zeta _F$ by this simple formula compares well with the numerical solution in figure 21.

From figure 21 it is also seen that $\omega _0$ decreases rapidly to near zero as the submergence parameter $H=h/a$ increases. This is consistent with the fact that the terms characterising the boundary effect in the evolution equation of perturbations are of order $O(h^{-2})$ (or $O({\rm e}^{-h})$), which vanishes at large $h$. The fact that $\omega _0$ is near zero at deep submergence indicates that a small alteration of rotational speed (e.g. due to $F_r$ effect) can cause a deeply submerged vortex to become unstable. Unlike $\omega _0$, $\beta _0(H)$ does not show a robust monotonic dependence on $H$, as shown in figure 21. For $a=1$, $\beta _0$ increases monotonically with increasing $H$, whereas for $a=0.5$ it first decreases with $H$ (at relatively shallow submergence) and then reverses to increase with $H$ (at deep submergence). This feature of $\beta _0(H)$ can be explained qualitatively. Based on (6.16) and (7.1), we have $\hat {\omega } \sim \tilde {g}(a) \beta ^2 (1-\beta ^2)^{-1} \, {\rm e}^{-2h}$, where $\tilde {g}(a) = I_1'(a) I_0(a)$. This leads to $h^{-2} \sim \omega _0 \sim \tilde {g}(a) \beta _0^2 (1-\beta _0^2)^{-1}\, {\rm e}^{-2h}$, from which we obtain $\beta _0 \sim [1 + \tilde {g}(a) h^2 \, {\rm e}^{-2h}]^{-1/2}$. We thus have $\partial _H \beta _0 \propto a H - 1$, which indicates that for fixed radius $a$, $\beta _0$ decreases with $H$ for $H\in (0, a^{-1})$ but increases with $H$ for $H\in (a^{-1}, \infty )$.