2. Formulation of the problem and general solution

We seek exact time-dependent velocity field solutions $\boldsymbol{u}(\boldsymbol{x},t)$ to the nonlinear barotropic vorticity equation for a perfect fluid,

(2.1) \begin{align} \frac {\partial \boldsymbol{\omega }}{\partial t} + \boldsymbol{\nabla }\!\boldsymbol{\omega }\,\boldsymbol{u} - \boldsymbol{\nabla }\!\boldsymbol{u}\,\boldsymbol{\omega } +\boldsymbol{\omega } \boldsymbol{\nabla } \boldsymbol{\cdot } \boldsymbol{u} = \boldsymbol{0} , \end{align}

where $\boldsymbol{\omega }(\boldsymbol{x},t) \equiv \boldsymbol{\nabla }\times {\boldsymbol{u}}$ is the total vorticity field. We use $\boldsymbol{\nabla }\!\boldsymbol{a}$ to denote the (covariant) gradient of vector $\boldsymbol{a}(\boldsymbol{x})$ , i.e. the $(1,1)$ tensor defined by its action on any vector $\boldsymbol{b}$ as $\boldsymbol{\nabla }\!\boldsymbol{a}\boldsymbol{b} = (\boldsymbol{\nabla }\!\boldsymbol{a})\boldsymbol{\cdot }\boldsymbol{b} = \boldsymbol{\nabla }\!_{\boldsymbol{b}}\boldsymbol{a} =({\boldsymbol{b}\boldsymbol{\cdot }\boldsymbol{\nabla }})\boldsymbol{a}$ . In Cartesian coordinates, $\boldsymbol{\nabla }\!\boldsymbol{a} =(\partial a_i/\partial x_{\kern-1pt j}) \hat {\boldsymbol{e}}_i\otimes \hat {\boldsymbol{e}}_j$ . It is convenient to express (2.1) using the Cauchy vorticity formula in the spatial description as

(2.2) \begin{align} \boldsymbol{\omega }(\boldsymbol{x},t) = \frac {1}{J_T(\boldsymbol{x},t)} \boldsymbol{\mathsf{F}}_T(\boldsymbol{x},t) \boldsymbol{\omega }_0(\boldsymbol{R}(\boldsymbol{x},t)) ,\quad \boldsymbol{\omega }_0(\boldsymbol{x}) = \boldsymbol{\omega }(\boldsymbol{x},0) . \end{align}

In (2.2), $\boldsymbol{\mathsf{F}}_T(\boldsymbol{x},t) \equiv \partial (\boldsymbol{r}/\partial \boldsymbol{x}_0)(\boldsymbol{R}(\boldsymbol{x},t))$ is the deformation gradient of the total flow in the spatial description, $\boldsymbol{r}(\boldsymbol{x}_0,t)$ is the flow map of the total flow, $\boldsymbol{R}(\boldsymbol{x},t)$ is the inverse flow map and $J_T \equiv \textrm {det}(\boldsymbol{\mathsf{F}}_T)$ .

The total velocity field $\boldsymbol{u}(\boldsymbol{x},t)$ is decomposed into the sum of a time-dependent Beltrami flow $\boldsymbol{\mathcal{U}}(\boldsymbol{x},t)$ and a time-dependent, non-Beltrami background velocity $\boldsymbol{u}_{b}(\boldsymbol{x},t)$ , that is,

(2.3) \begin{align} \boldsymbol{u}(\boldsymbol{x},t) \equiv \boldsymbol{\mathcal{U}}(\boldsymbol{x},t) + \boldsymbol{u}_{b}(\boldsymbol{x},t) . \end{align}

The time-dependent oscillating flow $\boldsymbol{\mathcal{U}}(\boldsymbol{x},t)$ , which is derived from steady Beltrami solutions $\boldsymbol{\mathcal{U}}_0(\boldsymbol{x})$ in the reference initial configuration in § 4, must remain a Beltrami flow at all times, that is,

(2.4) \begin{align} \boldsymbol{\nabla }\times \boldsymbol{\mathcal{U}}(\boldsymbol{x},t) = - k(t) \boldsymbol{\mathcal{U}}(\boldsymbol{x},t) , \quad \forall t , \end{align}

with a time-dependent eigenvalue $-k(t)$ , where $k(t)$ is a time-dependent wavenumber. In the absence of background flow, $\boldsymbol{u}_b=\boldsymbol{0}$ , the vorticity equation requires $\partial _t\boldsymbol{\mathcal{U}}=\boldsymbol{0}$ and, therefore, the time-dependence of the Beltrami flow $\boldsymbol{\mathcal{U}}(\boldsymbol{x},t)$ is completely prescribed from the background velocity $\boldsymbol{u}_b(\boldsymbol{x},t)$ .

The background velocity $\boldsymbol{u}_b(\boldsymbol{x},t)$ must satisfy the vorticity equation, that is, the vorticity equation for the total flow (2.1) in the case $\boldsymbol{\mathcal{U}}=\boldsymbol{0}$ , which, assuming $k(t) \ne 0$ , reduces to

(2.5) \begin{align} \frac {\partial }{\partial t}\left ( k\boldsymbol{\mathcal{U}}\right ) + k\boldsymbol{\nabla } \boldsymbol{\mathcal{U}}\,\boldsymbol{v}_{b} - k\boldsymbol{\nabla }\!\boldsymbol{v}_{b}\,\boldsymbol{\mathcal{U}} + k \boldsymbol{\mathcal{U}} \, \boldsymbol{\nabla }\boldsymbol{\cdot } \boldsymbol{v}_b = \boldsymbol{0} , \end{align}

where the advecting velocity $\boldsymbol{v}_b(\boldsymbol{x},t)$ is defined from the background velocity $\boldsymbol{u}_b(\boldsymbol{x},t)$ and the Beltrami wavenumber $k(t)$ as

(2.6) \begin{align} \boldsymbol{v}_{b} \equiv \boldsymbol{u}_{b} + \frac {\boldsymbol{\nabla }\times \boldsymbol{u}_{b}}{k(t)}. \end{align}

Equation (2.5) follows by substituting $\boldsymbol{u}=\boldsymbol{u}_b + \boldsymbol{\mathcal{U}}$ (2.3) into the vorticity equation and subtracting the (generally nonlinear) vorticity equation satisfied by the background flow $\boldsymbol{u}_b$ (see § 3), so that no background self-advection term is neglected, while the spatial homogeneity $\boldsymbol{\nabla }\!\boldsymbol{u}_b = \boldsymbol{\mathsf{L}}(t)$ is used only to simplify the remaining coupling terms and yield a closed transport law for $k\boldsymbol{\mathcal{U}}$ .

Since the Beltrami fields are isochoric $\boldsymbol{\nabla }\boldsymbol{\cdot }\boldsymbol{\mathcal{U}}=0$ , the divergence of the total flow $\boldsymbol{u}$ in (2.3) is due to the divergence of the background flow $\boldsymbol{u}_{b}$ , which equals the divergence of the advecting flow $\boldsymbol{v}_{b}$ ,

(2.7) \begin{align} \boldsymbol{\nabla }\boldsymbol{\cdot }\boldsymbol{u} = \boldsymbol{\nabla }\boldsymbol{\cdot }\boldsymbol{u}_{b} = \boldsymbol{\nabla }\boldsymbol{\cdot }\boldsymbol{v}_{b} . \end{align}

Equation (2.5) means that the vorticity $-k(t)\boldsymbol{\mathcal{U}}(\boldsymbol{x},t)$ of the Beltrami flow $\boldsymbol{\mathcal{U}}(\boldsymbol{x},t)$ is frozen in the advecting flow $\boldsymbol{v}_b(\boldsymbol{x},t)$ . Assuming that $\boldsymbol{u}_b(\boldsymbol{x},t)$ is a flow solution, the motion problem reduces therefore to find the fields $\{ \boldsymbol{\mathcal{U}},\boldsymbol{v}_{b} \}$ that satisfy the frozen condition (2.5), while $\boldsymbol{\mathcal{U}}$ remains a Beltrami flow (2.4) at all times.

We obtain the time-dependent Beltrami flow $\boldsymbol{\mathcal{U}}(\boldsymbol{x},t)$ from a Beltrami flow in the initial reference configuration $\boldsymbol{\mathcal{U}}_0(\boldsymbol{x})$ that is frozen in $\boldsymbol{v}_b(\boldsymbol{x},t)$ in the following way. Let the flow map of the advecting velocity $\boldsymbol{v}_b(\boldsymbol{x},t)$ be $\boldsymbol{r}_v(\boldsymbol{x}_0,t)$ and its inverse be $\boldsymbol{R}_v(\boldsymbol{x},t)$ . Then, the velocity in the material and spatial descriptions are respectively

(2.8a,b) \begin{align} \tilde {\boldsymbol{v}}_b (\boldsymbol{x}_0,t) \equiv \frac {\partial \boldsymbol{r}_v}{\partial t} (\boldsymbol{x}_0,t) \, \quad \textrm {and} \quad \boldsymbol{v}_b(\boldsymbol{x},t) =\tilde {\boldsymbol{v}}_b (\boldsymbol{R}_v(\boldsymbol{x},t),t) . \quad \end{align}

Introducing the deformation gradient of the advecting flow in the material and spatial descriptions as

(2.9a,b) \begin{align} \tilde {\boldsymbol{\mathsf{F}}}_v(\boldsymbol{x}_0,t) \equiv \frac {\partial \boldsymbol{r}_v}{\partial \boldsymbol{x}_0} (\boldsymbol{x}_0,t) \, \quad \textrm {and} \quad \boldsymbol{\mathsf{F}}_v(\boldsymbol{x},t) = \tilde {\boldsymbol{\mathsf{F}}}_v(\boldsymbol{R}_v(\boldsymbol{x},t),t) , \quad \end{align}

the solution to (2.5) is the Cauchy formula

(2.10) \begin{align} k(t)\boldsymbol{\mathcal{U}}_v(\boldsymbol{x},t) = \frac {1}{J_v(\boldsymbol{x},t)} \boldsymbol{\mathsf{F}}_v(\boldsymbol{x},t) \left ( k_0 \boldsymbol{\mathcal{U}}_0(\boldsymbol{R}_v(\boldsymbol{x},t)) \right ) , \end{align}

where the subscript $v$ in $\boldsymbol{\mathcal{U}}_v$ is introduced to remind of its dependence with the advecting velocity $\boldsymbol{v}_b(\boldsymbol{x},t)$ . The vector $\boldsymbol{\mathcal{U}}_0(\boldsymbol{R}_v(\boldsymbol{x},t))$ in (2.10) is the value of $\boldsymbol{\mathcal{U}}_0$ of the material particle, or spatial point in the reference configuration, which has been (parallel) advected to the point $(\boldsymbol{x},t)$ by the advecting velocity $\boldsymbol{v}_b(\boldsymbol{x},t)$ . The term $\boldsymbol{\mathsf{F}}_v/J_v$ rotates, stretches and compresses the vector $k_0\boldsymbol{\mathcal{U}}_0(\boldsymbol{R}_v(\boldsymbol{x},t))$ in such a way that the final result $k(t)\boldsymbol{\mathcal{U}}_v(\boldsymbol{x},t)$ in (2.10) represents the (frozen) advection of $k_0\boldsymbol{\mathcal{U}}_0$ by the advecting flow $\boldsymbol{v}_b(\boldsymbol{x},t)$ .

Since $\boldsymbol{\mathcal{U}}_v(\boldsymbol{x},t)$ in (2.10) must remain a Beltrami flow at all times while it is (frozen) advected by $\boldsymbol{v}_b(\boldsymbol{x},t)$ , the motion problem is to find the properties of the advecting velocity $\boldsymbol{v}_b(\boldsymbol{x},t)$ which transform a Beltrami flow into a Beltrami flow, that is, that leaves the Beltrami condition invariant.

To find these conditions, let $\boldsymbol{\mathcal{U}}_0(\boldsymbol{x}_0)$ be a Beltrami flow and consider the kinematic identity applied to (2.10) for the transformation of the curl,

(2.11) \begin{align} \boldsymbol{\nabla }_{\boldsymbol{x}} \times \left ( \boldsymbol{\mathsf{F}}_v(t) \boldsymbol{\mathcal{U}}_0(\boldsymbol{R}_v(\boldsymbol{x},t)) \right ) = \frac {1}{J_v(t)} \boldsymbol{\mathsf{F}}_v(t) \, \left ( \boldsymbol{\nabla }_{\boldsymbol{x}_0}\times \boldsymbol{\mathcal{U}}_0(\boldsymbol{x}_0)) (\boldsymbol{R}_v(\boldsymbol{x},t) \right ) . \end{align}

If the homogeneous time-dependent deformation gradient is an isotropic stretch plus a rotation $\boldsymbol{\mathsf{F}}_v = \lambda (t) \boldsymbol{\mathsf{Q}}(t)$ , $\boldsymbol{\mathsf{Q}}(t)\in SO(3)$ , thus $J(t)=\lambda (t)^3$ , and the requirement that $\boldsymbol{\nabla }\times \boldsymbol{\mathcal{U}}=-k(t)\boldsymbol{\mathcal{U}}$ forces $k(t)=k_0/\lambda (t)$ , and therefore,

(2.12) \begin{align} J_v(t) = \left ( \frac {k_0}{k(t)} \right )^{3} ,\quad \textrm {with} \quad J_v(0) = 1 , \end{align}

and the curl of (2.10), divided by $k(t)$ , becomes

(2.13) \begin{align} \boldsymbol{\nabla } \times \boldsymbol{\mathcal{U}}_v (\boldsymbol{x},t) = -k(t) \boldsymbol{\mathsf{F}}_v(t) \boldsymbol{\mathcal{U}}_{0}(\boldsymbol{R}_v(\boldsymbol{x},t)) = - k(t) \boldsymbol{\mathcal{U}}_v(\boldsymbol{x},t) . \end{align}

In other words, since under a Piola transformation the velocity $\boldsymbol{\mathcal{U}}$ transforms as $\boldsymbol{\mathcal{U}} = \boldsymbol{\mathsf{F}}^{-\top } \boldsymbol{\mathcal{U}}_0$ and the vorticity $\boldsymbol{\mathcal{W}}$ as $\boldsymbol{\mathcal{W}} = J^{-1}\boldsymbol{\mathsf{F}} \boldsymbol{\mathcal{W}}_0$ , the Beltrami condition $\boldsymbol{\mathcal{W}}=-k(t)\boldsymbol{\mathcal{U}}$ is preserved if $\boldsymbol{\mathsf{F}}(t)^{\top }\boldsymbol{\mathsf{F}}(t) = J(t) (k(t)/k_0)\boldsymbol{\mathsf{I}}$ . Using the polar decomposition $\boldsymbol{\mathsf{F}}(t)=\boldsymbol{\mathsf{R}}(t)\boldsymbol{\mathsf{U}}(t)$ of the deformation gradient $\boldsymbol{\mathsf{F}}(t)$ into a rotation $\boldsymbol{\mathsf{R}}(t)$ and a positive-definite symmetric right stretch tensor $\boldsymbol{\mathsf{U}}(t)$ , the right Cauchy–Green deformation tensor is $\boldsymbol{\mathsf{C}}(t) \equiv \boldsymbol{\mathsf{F}}^{\top }(t) \boldsymbol{\mathsf{F}}(t) =\boldsymbol{\mathsf{U}}(t)^2$ . Therefore, the right stretch tensor

(2.14) \begin{align} \boldsymbol{\mathsf{U}}^2(t) = J(t)\frac {k(t)}{k_0} \boldsymbol{\mathsf{I}} , \end{align}

and since $\boldsymbol{\mathsf{U}}$ is symmetric positive-definite, we have $\boldsymbol{\mathsf{U}}(t)=\lambda (t)\boldsymbol{\mathsf{I}}$ for some scalar function $\lambda (t)\gt 0$ . Since $J =\textrm {det}\boldsymbol{\mathsf{F}} =\textrm {det}\boldsymbol{\mathsf{U}} =\lambda ^3$ , the deformation gradient must be $\boldsymbol{\mathsf{F}}(t)=(k_0/k(t))\boldsymbol{\mathsf{R}}(t)$ , that is, the deformation gradient is an isotropic stretch and a rotation.

Therefore, $\boldsymbol{\mathcal{U}}(\boldsymbol{x},t)$ remains a Beltrami flow if is initially a Beltrami flow $\boldsymbol{\mathcal{U}}(\boldsymbol{x},0) =\boldsymbol{\mathcal{U}}_0(\boldsymbol{x})$ , with initial wavenumber $k_0\equiv k(0)$ , and the deformation gradient of the background flow is an isotropic stretch and a rotation. This result is independent of the particular geometry of the Beltrami flow $\boldsymbol{\mathcal{U}}_0(\boldsymbol{x})$ . In the more general case in which the deformation gradient $\boldsymbol{\mathsf{F}}_v$ is required only to be only time-dependent $\boldsymbol{\mathsf{F}}_v = \boldsymbol{\mathsf{F}}_v(t)$ , so it may include a time-dependent deviatoric strain $\boldsymbol{\mathsf{D}}_0(t)$ , the Beltrami property (2.13) does not follow from (2.10) for every steady Beltrami flow $\boldsymbol{\mathcal{U}}_0(\boldsymbol{x})$ . Plane Beltrami flows, however, may satisfy (2.13), as is explained in more detail in § 5. We henceforth assume the more general condition for $\boldsymbol{\mathsf{F}}_v(t)$ and consider an advecting velocity field $\boldsymbol{v}_b(\boldsymbol{x},t)$ having a linear dependence on $\boldsymbol{x}$ such that its deformation gradient is only time-dependent $\boldsymbol{\mathsf{F}}_v = \boldsymbol{\mathsf{F}}_v(t)$ .

To obtain the permissible initial Beltrami flows in many different geometries, say Cartesian plane waves, spherical, cylindrical, ellipsoidal, etc., we follow the steady Chandrasekhar–Kendall approach (Morse & Feshbach Reference Morse and Feshbach1953; Chandrasekhar & Kendall Reference Chandrasekhar and Kendall1957; Chandrasekhar & Woltjer Reference Chandrasekhar and Woltjer1958; Woltjer Reference Woltjer1958a , Reference Woltjerb ) which, from a scalar Laplacian eigenfunction $\eta _0(\boldsymbol{x})$ ,

(2.15) \begin{align} {\nabla} ^2\eta _0 = - k_0^2\eta _0 , \end{align}

and a constant unit vector $\hat {\boldsymbol{n}}_0$ , obtains a Beltrami flow, say $\boldsymbol{\mathcal{U}}_{\eta }(\boldsymbol{x})$ , as

(2.16a,b) \begin{align} \boldsymbol{\mathcal{U}}_{\eta }(\boldsymbol{x}) \equiv \boldsymbol{\nabla } \times \big ( \eta _0 \hat {\boldsymbol{n}}_0 - k_0^{-1}\boldsymbol{\nabla }\times (\eta _0 \hat {\boldsymbol{n}}_0) \big ) ,\quad \textrm {such that} \quad \boldsymbol{\nabla } \times \boldsymbol{\mathcal{U}}_{\eta } = - k_0 \boldsymbol{\mathcal{U}}_{\eta } . \quad \end{align}

Using this Beltrami flow $\boldsymbol{\mathcal{U}}_{\eta }(\boldsymbol{x})$ , associated with the scalar Laplacian eigenfunction $\eta _0(\boldsymbol{x})$ we may write, in the case of background isotropic stretch and rotation, as it is shown in § 4, the Beltrami solution as

(2.17) \begin{align} \boldsymbol{\mathcal{U}}(\boldsymbol{x},t) = \boldsymbol{\mathcal{U}}_{\eta v}(\boldsymbol{x},t) = \boldsymbol{\mathsf{F}}_v(t) \boldsymbol{\mathcal{U}}_{\eta }(\boldsymbol{R}_v(\boldsymbol{x},t)) , \end{align}

where the notation $\boldsymbol{\mathcal{U}}_{\eta v}(\boldsymbol{x},t)$ makes explicit that the particular Beltrami solution $\boldsymbol{\mathcal{U}}(\boldsymbol{x},t)$ depends on the Laplacian eigenfunction $\eta _0(\boldsymbol{x})$ and on the advecting flow $\boldsymbol{v}_b(\boldsymbol{x},t)$ or, alternatively, on the background flow $\boldsymbol{u}_b(\boldsymbol{x},t)$ ). Equation (2.17) states that the Beltrami field is advected isochorically by the non-isochoric flow $\boldsymbol{v}_b(\boldsymbol{x},t)$ .

In addition to the Beltrami condition for $\boldsymbol{\mathcal{U}}(\boldsymbol{x},t)$ , the background flow $\boldsymbol{u}_b(\boldsymbol{x},t)$ must satisfy the vorticity equation and this requirement imposes a constraint to the time-dependent background velocity gradient tensor $\boldsymbol{\mathsf{L}}(t) \equiv \boldsymbol{\nabla }\!\boldsymbol{u}_b(t)$ , which is considered in the next section. Independently of this constraint, the background flow has uniform vorticity $\boldsymbol{\omega }_b(t) = ( \boldsymbol{\nabla }\times \boldsymbol{u}_b )(t)$ and, therefore, we may introduce a vector drift velocity $\boldsymbol{c}(t)$ given by

(2.18) \begin{align} \boldsymbol{c}(t) \equiv \frac {\boldsymbol{\nabla }\times \boldsymbol{u}_b(\boldsymbol{x},t)}{k(t)} = \frac {\boldsymbol{\omega }_b(t)}{k(t)} = \frac {2\boldsymbol{\nu }(t)}{k(t)} , \end{align}

where $\boldsymbol{\nu }(t) \equiv \boldsymbol{\omega }_{b}(t)/2$ is the angular velocity of the background flow $\boldsymbol{u}_b$ . Thus, the advecting velocity

(2.19) \begin{align} \boldsymbol{v}_{b}(\boldsymbol{x},t) = \boldsymbol{u}_{b}(\boldsymbol{x},t) + \boldsymbol{c}(t) \, \end{align}

differs from the background velocity $\boldsymbol{u}_b(\boldsymbol{x},t)$ by the time-dependent drift $\boldsymbol{c}(t)$ .

The subtle difference between the background velocity $\boldsymbol{u}_b$ and the advecting velocity $\boldsymbol{v}_b$ has far-reaching physical consequences. For example, if the background flow $\boldsymbol{u}_b$ is reduced to a steady solid body rotation with constant angular velocity $\boldsymbol{\nu }_0$ , that is, $\boldsymbol{u}_b(\boldsymbol{x}) = \boldsymbol{\nu }_0\times \boldsymbol{x}$ , then the background deformation gradient $\boldsymbol{\mathsf{F}}_v(t) =\boldsymbol{\mathsf{R}}_{\hat {\boldsymbol{\nu }}_0}(\nu _0 t)$ is a constant-rate rotation around the angle defined by the unit vector $\hat {\boldsymbol{\nu }}_0 \equiv \boldsymbol{\nu }_0/\Vert \boldsymbol{\nu }_0\Vert$ . If this external flow $\boldsymbol{u}_b$ is interpreted as generated by an external measuring device acting on a Beltrami field $\boldsymbol{\mathcal{U}}_0(\boldsymbol{x})$ , the presence of this external flow $\boldsymbol{u}_b$ not only causes the Beltrami field $\boldsymbol{\mathcal{U}}(\boldsymbol{x},t)$ to precess, due to the $\boldsymbol{u}_b$ component on the advecting velocity $\boldsymbol{v}_b$ in (2.19), but also to displace with the drift velocity $2\boldsymbol{\nu }_0/k_0$ present in $\boldsymbol{R}_v(\boldsymbol{x},t)$ in (2.17). This displacement velocity $\boldsymbol{c}_0 =2\boldsymbol{\nu }_0/k_0$ , which depends on the amplitude and direction of the background field $\boldsymbol{\nu }_0$ and on the Beltrami wavenumber $k_0$ , is an interaction term between the Beltrami field and the measuring field, and therefore, can be used as a measure of the Beltrami field wavenumber $k_0$ . In this case, correlations between measures $i$ in an ensemble ( $i=1, {\cdots} ,N$ ) using different apparatus configurations $\boldsymbol{\nu }[i]$ , of two Beltrami fields, say $\boldsymbol{\mathcal{U}}_a(\boldsymbol{x},t;k_a)$ and $\boldsymbol{\mathcal{U}}_b(\boldsymbol{x},t;k_b)$ , must depend on the correlation between $\boldsymbol{c}_{a}[i]\equiv 2 \boldsymbol{\nu }_a[i]/k_a[i]$ and $\boldsymbol{c}_{b}[i] \equiv 2 \boldsymbol{\nu }_b[i]/k_b[i]$ . This property, which is independent of the particular geometry of the Beltrami field, has been used, with spherical Beltrami fields (Viúdez Reference Viúdez2025), to suggest a classical fluid mechanics explanation of the correlation experimentally found in quantum particle entanglement experiments. We see here that this drift velocity is a general property of Beltrami fields independent of their particular geometry.

Once a permissible time-dependent, with homogeneous velocity gradient $\boldsymbol{\mathsf{L}}(t)$ , background flow $\boldsymbol{u}_b(\boldsymbol{x},t)$ , or equivalently $\boldsymbol{v}_b(\boldsymbol{x},t)$ , and a particular Laplacian eigenfunction $\eta _0(\boldsymbol{\boldsymbol{x}})$ which specifies the Beltrami flow, are chosen, the resulting oscillatory motion is given by (2.17). An economical way of visualising the time evolution of the Beltrami motion is from the time-dependent scalar density function $\eta (\boldsymbol{x},t)$ ,

(2.20) \begin{align} \eta (\boldsymbol{x},t) \equiv \left (\frac {k(t)}{k_0}\right )^3 \eta _0(\boldsymbol{R}_v(\boldsymbol{x},t)-\boldsymbol{o}(0)) , \end{align}

where $\boldsymbol{o}(0)$ is the value at the initial time $t=0$ of an arbitrary time-dependent vector $\boldsymbol{o}(t)$ . The scalar function $\eta (\boldsymbol{x},t)$ (2.20) is, by construction, frozen in the advecting flow $\boldsymbol{v}_b(\boldsymbol{x},t)$ ,

(2.21) \begin{align} \frac {\partial \eta }{\partial t} + \boldsymbol{\boldsymbol{v}}_b \boldsymbol{\cdot } \boldsymbol{\nabla } \eta + \eta \boldsymbol{\nabla }\boldsymbol{\cdot }\boldsymbol{v}_b = 0 . \end{align}

The Beltrami field $\boldsymbol{\mathcal{U}}(\boldsymbol{x},t)$ can be obtained at any time $t$ from $\eta (\boldsymbol{x},t)$ given by (2.20).

3. Time-dependent background velocity field $\boldsymbol{u}_b(\boldsymbol{x},t)$

The purpose of this section is to obtain the conditions that must satisfy the background flow $\boldsymbol{u}_b(\boldsymbol{x},t)$ to be a solution to the vorticity equation, as well as to obtain the inverse map $\boldsymbol{R}_v(\boldsymbol{x},t)$ . The inverse map $\boldsymbol{R}_v(\boldsymbol{x},t)$ is required to prescribe the time evolution of the Beltrami field $\boldsymbol{\mathcal{U}}(\boldsymbol{x},t)$ described in the previous section.

The background velocity field $\boldsymbol{u}_b(\boldsymbol{x},t)$ , with vorticity $\boldsymbol{\omega }_{b}\equiv \boldsymbol{\nabla }\times \boldsymbol{u}_{b}$ , must satisfy the vorticity equation, that is,

(3.1) \begin{align} \frac {\partial \boldsymbol{\omega }_{b}}{\partial t} + \boldsymbol{\nabla }\! \boldsymbol{\omega }_{b} \, \boldsymbol{u}_{b} - \boldsymbol{\nabla }\! \boldsymbol{u}_{b} \, \boldsymbol{\omega }_{b} + \boldsymbol{\nabla } \boldsymbol{\cdot } \boldsymbol{u}_{b} \, \boldsymbol{\omega }_{b} = \boldsymbol{0} . \end{align}

As explained in § 2, the background flow $\boldsymbol{u}_b(\boldsymbol{x},t)$ must have a homogeneous velocity gradient $\boldsymbol{\mathsf{L}}(t)$ and, consequently, we set

(3.2) \begin{align} {\boldsymbol{u}}_{b}(\boldsymbol{x},t) = \dot {\boldsymbol{o}}(t) + \boldsymbol{\mathsf{L}}(t) (\boldsymbol{x}-\boldsymbol{o}(t)) , \end{align}

where $\boldsymbol{o}(t)$ is an arbitrary time-dependent vector, interpreted as the origin of the spatial frame, and is included in the term $\boldsymbol{\mathsf{L}}(t)(\boldsymbol{x}-\boldsymbol{o}(t))$ to simplify the mathematical expressions. The origin of the spatial frame $\boldsymbol{o}(t)$ as a free parameter allows us to assign the drift velocity $\boldsymbol{c}(t)$ entirely to the advecting flow $\boldsymbol{v}_b(\boldsymbol{x},t)$ setting $\dot {\boldsymbol{o}}(t)\mapsto \boldsymbol{0}$ , or entirely to the background flow $\boldsymbol{u}_b(\boldsymbol{x},t)$ setting $\dot {\boldsymbol{o}}(t) \mapsto -\boldsymbol{c}(t)$ . In the advecting flow, the origin $\boldsymbol{o}_{v}(t)$ of the frame respect to which fluid particle positions are referenced is given by

(3.3) \begin{align} \boldsymbol{o}_v(t) \equiv \boldsymbol{o}(t) + \boldsymbol{\mathsf{F}}(t) \int _0^t \boldsymbol{\mathsf{F}}^{-1}(\tau ) \, \boldsymbol{c}(\tau ) \, \textrm {d}\tau , \quad \boldsymbol{o}_v(0) = \boldsymbol{o}(0) . \end{align}

Thus, we may write

(3.4) \begin{align} \boldsymbol{R}_v(\boldsymbol{x},t) - \boldsymbol{o}(0) = \boldsymbol{\mathsf{F}}^{-1}(t) \left ( \boldsymbol{x} - \boldsymbol{o}_v(t) \right ) \equiv \boldsymbol{z}_v(\boldsymbol{x},t) , \end{align}

which serves as a definition of the material variable $\boldsymbol{z}_v(\boldsymbol{x},t)$ . Thus, the time-dependence of the Beltrami field $\boldsymbol{\mathcal{U}}_{\eta }(\boldsymbol{x},t)$ is fully described by the scalar field

(3.5) \begin{align} \eta (\boldsymbol{x},t) \equiv \frac {1}{J(t)} \eta _0 (\boldsymbol{R}_v(\boldsymbol{x},t)-\boldsymbol{o}(0)) = \frac {1}{J(t)} \eta _0 (\boldsymbol{z}_v(\boldsymbol{x},t)) , \end{align}

through the advection as a density of the Laplacian eigenfunction $\eta _0(\boldsymbol{x}_0)$ by the advecting velocity $\boldsymbol{v}_b(\boldsymbol{x},t)$ .

Since the drift velocity $\boldsymbol{c}(t)$ only depends on time, the background velocity gradient $\boldsymbol{\mathsf{L}}(t)$ is the same for both background $\boldsymbol{u}_b$ and advecting $\boldsymbol{v}_b$ velocity fields,

(3.6) \begin{align} \boldsymbol{\mathsf{L}}(t) = \boldsymbol{\mathsf{L}}_u(t) \equiv \boldsymbol{\nabla }\!\boldsymbol{u}_{b} = \boldsymbol{\mathsf{L}}_v(t) \equiv \boldsymbol{\nabla }\!\boldsymbol{v}_{b} . \end{align}

To facilitate the analysis, it is useful to separate the background velocity gradient $\boldsymbol{\mathsf{L}}(t)$ into its symmetric strain tensor $\boldsymbol{\mathsf{D}}(t)$ and skew-symmetric spin tensor $\boldsymbol{\mathsf{W}}(t)$ components, and then express the strain $\boldsymbol{\mathsf{D}}(t)$ as the sum of the deviatoric traceless strain tensor $\boldsymbol{\mathsf{D}}_0(t)$ and the volumetric strain $(1/3)\textrm {tr}(\boldsymbol{\mathsf{D}}(t))\boldsymbol{\mathsf{I}}$ . That is, when acting on an arbitrary vector $\boldsymbol{e}$ ,

(3.7) \begin{align} {\boldsymbol{\mathsf{L}}}(t) \boldsymbol{e} = \left ( \boldsymbol{\mathsf{D}}(t) + \boldsymbol{\mathsf{W}}(t) \right ) \boldsymbol{e} = \left ( \boldsymbol{\mathsf{D}}_0(t) +\dfrac {1}{3}\delta (t) \boldsymbol{\mathsf{I}}\right ) \boldsymbol{e} + \boldsymbol{\nu }(t)\times \boldsymbol{e} , \end{align}

where the divergence $\delta (t)$ , deviatoric strain $\boldsymbol{\mathsf{D}}_0(t)$ and angular velocity $\boldsymbol{\nu }(t)$ are

(3.8a,b,c) \begin{align} \delta (t) \equiv \boldsymbol{\nabla }\boldsymbol{\cdot } \boldsymbol{u}_b = \textrm {tr}(\boldsymbol{\mathsf{D}}(t)) , \quad \boldsymbol{\mathsf{D}}_0(t) \equiv \boldsymbol{\mathsf{D}}(t) - \frac {1}{3}\textrm {tr}(\boldsymbol{\mathsf{D}}(t)) \boldsymbol{\mathsf{I}} , \quad \boldsymbol{\nu }(t) \equiv \frac {1}{2}\boldsymbol{\omega }_b(t) , \quad\end{align}

where the angular velocity $\boldsymbol{\nu }(t)$ is the axial vector of the spin tensor $\boldsymbol{\mathsf{W}}(t)$ ,

(3.9) \begin{align} \boldsymbol{\mathsf{W}}(t)\boldsymbol{e} = [\boldsymbol{\nu }(t)\times ]\boldsymbol{e} = \boldsymbol{\nu }(t)\times \boldsymbol{e} . \end{align}

Since the relation between the background velocity $\boldsymbol{u}_b(\boldsymbol{x},t)$ and the advecting velocity $\boldsymbol{v}_b(\boldsymbol{x},t)$ is a time-dependent drift,

(3.10) \begin{align} \boldsymbol{v}_b (\boldsymbol{x},t) \equiv \boldsymbol{u}_b + \frac {\boldsymbol{\nabla }\times \boldsymbol{u}_b}{k(t)} = \boldsymbol{u}_b (\boldsymbol{x},t) + \frac {2\boldsymbol{\nu }(t)}{k(t)} \, \end{align}

if $\boldsymbol{u}_b(\boldsymbol{x},t)$ is a flow solution to the vorticity equation, then $\boldsymbol{v}_b(\boldsymbol{x},t)$ is also a flow solution.

From the vorticity equation for the background flow (3.1), the background velocity (3.2) and the background velocity gradient (3.7), we obtain the compatibility condition for the background flow,

(3.11) \begin{align} \dot {\boldsymbol{\nu }} (t) = \left ( \boldsymbol{\mathsf{D}}_0(t) - \frac {2}{3} \delta (t) \boldsymbol{\mathsf{I}} \right ) \boldsymbol{\nu }(t) , \end{align}

which prescribes a constraint between the time rate of change of the angular velocity $\dot {\boldsymbol{\nu }}(t)$ and the actual values of deviatoric and volumetric strains acting on the angular velocity $\boldsymbol{\nu }(t)$ . Clearly, fluid expansion/contraction described by the background divergence $\delta (t)$ is associated with changes in the magnitude $\nu (t)$ of the angular velocity $\boldsymbol{\nu }(t)=\nu (t)\hat {\boldsymbol{\nu }}(t)$ , and not with changes in its direction $\hat {\boldsymbol{\nu }}(t)$ .

It is necessary to time-integrate (3.11) to find $\boldsymbol{\nu }(t)$ and hence obtain the permissible advecting velocity $\boldsymbol{v}_b(\boldsymbol{x},t)$ from which to derive the inverse map $\boldsymbol{R}_{v}(\boldsymbol{x},t)$ . With this objective, it is useful to factorise the right-hand side of the compatibility condition (3.11) introducing the time integral $\varDelta (t)$ of the divergence,

(3.12) \begin{align} \varDelta (t) \equiv \int _0^t \delta (\tau ) \, \textrm {d}\tau , \end{align}

in such a way that the compressibility effects on the time-rate of change of $\boldsymbol{\nu }(t)$ are solved explicitly,

(3.13) \begin{align} \boldsymbol{\nu }(t) = \textrm {e}^{-\tfrac {2}{3}\varDelta (t)} \boldsymbol{\nu }_D(t) , \end{align}

where

(3.14a,b) \begin{align} \dot {\boldsymbol{\nu }}_D (t) = \boldsymbol{\mathsf{D}}_0(t) \boldsymbol{\nu }_D(t) ,\quad \textrm {with} \quad \boldsymbol{\nu }_D(0) = \boldsymbol{\nu }(0) . \quad \end{align}

The vector $\boldsymbol{\nu }_{D}(t)$ is the background angular velocity that is related to the background deviatoric strain tensor and is not affected by the fluid expansion/compression. It remains to solve the more complicated (3.14), which explains how time changes on both amplitude and direction of the angular velocity $\boldsymbol{\nu }_{D}(t)$ are related to the actual values of the deviatoric strain tensor $\boldsymbol{\mathsf{D}}_0(t)$ . The general solution to (3.14) is

(3.15) \begin{align} \boldsymbol{\nu }_{D}(t) = \textrm {OE}[\boldsymbol{\mathsf{D}}_0](t) \boldsymbol{\nu }_0 = \mathcal{T} \textrm {exp}\left ( \int _0^t \boldsymbol{\mathsf{D}}_0(\tau )\, \textrm {d}\tau \right ) \boldsymbol{\nu }_0 , \end{align}

where $\mathcal{T}$ is the time-ordered exponential $\textrm {OE}[a](t)$ , which is the unique solution of the differential equation of initial value

(3.16a,b) \begin{align} \frac { {d}}{\textrm {d}t}\textrm {OE}[a](t) = a(t)\textrm {OE}[a](t) , \quad \textrm {with} \quad \textrm {OE}[a](0) = 1 . \quad \end{align}

For non-commuting $\boldsymbol{\mathsf{D}}_0(t)$ , the fundamental matrix $\boldsymbol{\mathsf{U}}_0(t,t_0)$ in $\boldsymbol{\nu }_{D}(t) = \boldsymbol{\mathsf{U}}_0(t,t_0) \boldsymbol{\nu }_0$ is given by the time-ordered exponential (equivalently its Dyson-series expansion) in (3.15). The close analogy with the quantum evolution operator in time-dependent Schrödinger/quantum field theory dynamics is described in Appendix B.

The factorisation (3.13) suggests that it is convenient to factorise the background deformation gradient $\boldsymbol{\mathsf{F}}(t)$ as

(3.17) \begin{align} \boldsymbol{\mathsf{F}}(t) = \textrm {e}^{\varDelta (t)/3} \boldsymbol{\mathcal{F}}(t) , \end{align}

where the gradient deformation tensor $\boldsymbol{\mathcal{F}}(t)$ carries the isochoric deformation and rotation. Defining the isochoric velocity gradient $\boldsymbol{\mathcal{L}}(t)$ as

(3.18) \begin{align} \boldsymbol{\mathcal{L}}(t) \equiv \boldsymbol{\mathsf{D}}_0(t) + \boldsymbol{\mathsf{W}}(t) = \boldsymbol{\mathsf{L}}(t) -\frac {1}{3}\delta (t) \boldsymbol{\mathsf{I}} , \end{align}

we see that $\boldsymbol{\mathcal{F}}(t)$ satisfies an equation analogous to the kinematic identity, $\dot {\boldsymbol{\mathsf{F}}} = \boldsymbol{\mathsf{L}}\boldsymbol{\mathsf{F}}$ , that is,

(3.19) \begin{align} \dot {\boldsymbol{\mathcal{F}}}(t) = \boldsymbol{\mathcal{L}}(t) \boldsymbol{\mathcal{F}}(t) . \end{align}

The solution to (3.19) is again the time-ordered exponential

(3.20a,b) \begin{align} \boldsymbol{\mathcal{F}}(t) = \textrm {OE}[\boldsymbol{\mathcal{L}}](t) = \mathcal{T}\textrm {exp}\left ( {\int _0^t\boldsymbol{\mathcal{L}}(\tau ) \, \textrm {d}\tau } \right ) , \quad \textrm {with} \quad \boldsymbol{\mathcal{F}}(0)=\boldsymbol{\mathsf{I}} , \quad \end{align}

which provides the flow map $\boldsymbol{r}_v(\boldsymbol{x}_0,t)$ of the advecting flow, $\boldsymbol{v}_b(\boldsymbol{x},t)$ ,

(3.21) \begin{align} \boldsymbol{r}_{v}(\boldsymbol{x}_0,t) = \boldsymbol{o}_v(t) + \textrm {e}^{\varDelta (t)/3} \boldsymbol{\mathcal{F}}(t) (\boldsymbol{x}_0 - \boldsymbol{o}(0)) , \end{align}

and hence, the inverse map

(3.22) \begin{align} \boldsymbol{R}_{v}(\boldsymbol{x},t) = \boldsymbol{o}(0) + \textrm {e}^{-\varDelta (t)/3} \boldsymbol{\mathcal{F}}^{-1}(t) (\boldsymbol{x} - \boldsymbol{o}_v(t)) . \end{align}

The Jacobian $J(t)$ of the advecting flow map $\boldsymbol{r}_v(\boldsymbol{x},t)$ , which is equal to the Jacobian of the background flow map $\boldsymbol{r}_u(\boldsymbol{x},t)$ , is

(3.23) \begin{align} J(t) = J_v(t) = J_u(t) = \textrm {det}(\boldsymbol{\mathsf{F}}(t)) = \textrm {e}^{\varDelta (t)} , \end{align}

and its rate of change,

(3.24) \begin{align} \frac {\dot {J}(t)}{J(t)} = \dot {\varDelta }(t) = \delta (t) = \boldsymbol{\nabla }\boldsymbol{\cdot }\boldsymbol{u} , \end{align}

expresses the rate of change of volume in the background flow. The advecting velocity $\boldsymbol{v}_b$ in the material description is

(3.25) \begin{eqnarray} && \tilde {\boldsymbol{v}}_b(\boldsymbol{x}_0,t) \equiv \frac {\partial \boldsymbol{r}_v}{\partial t}(\boldsymbol{x}_0,t) = \boldsymbol{v}_b(\boldsymbol{r}_v(\boldsymbol{x}_0,t),t) \nonumber \\ &=& \dot {\boldsymbol{o}}_v(t) + \boldsymbol{\mathsf{L}}(t) \boldsymbol{\mathsf{F}}(t) (\boldsymbol{x}_0-\boldsymbol{o}(0)) = \dot {\boldsymbol{o}}_v(t) + \boldsymbol{\mathsf{L}}(t) (\boldsymbol{r}_v(\boldsymbol{x}_0,t)-\boldsymbol{o}_v(t)) \nonumber \\ &=& \dot {\boldsymbol{o}}_v(t) + \frac {1}{3}\delta (t) (\boldsymbol{r}_v(\boldsymbol{x}_0,t)-\boldsymbol{o}_v(t)) \nonumber \\ && + {\mbox{ }} \boldsymbol{\mathsf{D}}_0(t)(\boldsymbol{r}_v(\boldsymbol{x}_0,t)-\boldsymbol{o}_v(t)) + \boldsymbol{\nu }(t) \times \left ( \boldsymbol{r}_v(\boldsymbol{x}_0,t)-\boldsymbol{o}_v(t) \right ) , \end{eqnarray}

which shows in the material description four contributing terms: homogeneous displacement, isotropic expansion/contraction, isochoric stretching and rotation. The background velocity $\boldsymbol{u}_b$ in the material description becomes identical to $\boldsymbol{v}_b$ once the origin $\boldsymbol{o}_v(t)$ is replaced with $\boldsymbol{o}(t)$ ,

(3.26) \begin{eqnarray} && \tilde {\boldsymbol{u}}_b(\boldsymbol{x}_0,t) \equiv \frac {\partial \boldsymbol{r}_u}{\partial t}(\boldsymbol{x}_0,t) = \boldsymbol{u}_b(\boldsymbol{r}_u(\boldsymbol{x}_0,t),t) \nonumber \\ &=& \boldsymbol{v}_b(\boldsymbol{r}_v(\boldsymbol{x}_0,t),t) - \frac {2\boldsymbol{\nu }(t)}{k_0} \nonumber \\ &=& \dot {\boldsymbol{o}}(t) + \frac {1}{3}\delta (t) (\boldsymbol{r}_u(\boldsymbol{x}_0,t)-\boldsymbol{o}(t)) \nonumber \\ && + {\mbox{ }} \boldsymbol{\mathsf{D}}_0(t)(\boldsymbol{r}_u(\boldsymbol{x}_0,t)-\boldsymbol{o}(t)) + \boldsymbol{\nu }(t) \times \left ( \boldsymbol{r}_u(\boldsymbol{x}_0,t)-\boldsymbol{o}(t) \right ) . \end{eqnarray}

Since the background vorticity is spatially uniform, the background vorticity in the spatial and material descriptions coincide $\tilde {\boldsymbol{\omega }}_b(\boldsymbol{x}_0,t) = \boldsymbol{\omega }_b(t) = 2 \boldsymbol{\nu }(t)$ , and the Cauchy vorticity formula for the background flow $\boldsymbol{u}_b$ is

(3.27) \begin{align} \boldsymbol{\nu }(t) = \frac {1}{J(t)} \boldsymbol{\mathsf{F}}(t) \boldsymbol{\nu }_0 , \end{align}

whose time derivative leads, using $\dot {\boldsymbol{\mathsf{F}}}=\boldsymbol{\mathsf{L}}\boldsymbol{\mathsf{F}}$ , to the compatibility condition for the background flow (3.11).

In summary, since there are six time-dependent degrees of freedom in the way the velocity field, the velocity gradient of the background flow $\boldsymbol{\mathsf{L}}(t)$ is defined, one needs to specify six time-dependent functions to obtain a compatible velocity gradient $\boldsymbol{\mathsf{L}}(t)$ . For example, one may specify five components of $\boldsymbol{\mathsf{D}}_0(t)$ and $\delta (t)$ , and then solving for $\boldsymbol{\nu }_D(t)$ using (3.15). With $\boldsymbol{\mathsf{D}}_0(t)$ , $\delta (t)$ and $\boldsymbol{\nu }_D(t)$ available, one may obtain $\boldsymbol{\mathcal{L}}(t)$ using (3.18) and then solve (3.20a ) to obtain $\boldsymbol{\mathcal{F}}(t)$ , and finally the inverse map $\boldsymbol{R}_v(\boldsymbol{x},t)$ using (3.22).

4. Beltrami field $\boldsymbol{\mathcal{U}}(\boldsymbol{x},\boldsymbol{t})$

So far, we have been concerned with the background flow $\boldsymbol{u}_b(\boldsymbol{x},t)$ with a homogeneous velocity gradient $\boldsymbol{\mathsf{L}}(t)$ . In laboratory and field practice, flow diagnostics such as particle image velocimetry (PVI)/laser light sheets, dye or seeded Lagrangian tracers are typically designed to be weakly intrusive, so that at continuum scales the measured velocity may be regarded as unaffected to leading order; nevertheless, any observation ultimately relies on a physical coupling (illumination/scattering, tracer inertia and diffusion, finite-size effects) and it is useful in some contexts to represent such a coupling explicitly as an ‘environment’ acting on a coherent structure. This viewpoint is particularly natural in a broader programme of modelling spin-like degrees of freedom with coherent spherical vortical excitations, where the ‘state’ is defined relative to a chosen axis/observer and external couplings can induce precession or drift, as explored, for example, in the precession of spherical spin-1 vortices (Viúdez Reference Viúdez2025c ), in the fluid-mechanical spin-up/spin-down construction associated with Clauser–Horne–Shimony–Holt inequality tests (Viúdez Reference Viúdez2025), and in Euler solutions reproducing Dirac-type dynamics using multipolar oscillations in spherical geometry (Viúdez Reference Viúdez2025a ). Accordingly, we use the background-flow superposition framework here as a controlled model of environmental coupling that yields observable drift/deflection of coherent modes, without implying that practical macroscopic measurements generally require, or are well approximated by, a dynamically significant interaction flow.

Thus, some practical applications, in particular, those at quantum scales, require including a second flow entity in such a way that the interaction between both flows becomes relevant. For example, at these very small spatial scales, a steady Beltrami flow $\boldsymbol{\mathcal{U}}_0(\boldsymbol{x})$ in a three-dimensional space cannot be observed in isolation; it becomes measurable only through interaction with another flow. Conversely, a large-scale background flow can be detected through the advection of an extensive structure, such as a Beltrami field. In any case, whether the background flow is used to measure the properties of the vortex, or the vortex is used to measure the properties of the background flow, two flow entities are required and what is measured is the interaction between them.

In the set-up explained in the previous sections, the time evolution of the Beltrami flow $\boldsymbol{\mathcal{U}}(\boldsymbol{x},t)$ is completely described by (2.17) once an initial Beltrami flow $\boldsymbol{\mathcal{U}}_0(\boldsymbol{x})$ is given. This initial Beltrami flow can be prescribed by a Chandrasekhar–Kendall (CK) procedure, which provides a Beltrami vector field from a scalar Laplacian eigenfunction $\eta _0(\boldsymbol{x})$ . The Laplacian eigenvalue of $\eta _0(\boldsymbol{x})$ is $-k_0^2$ . Such a procedure is independent of time, but it can be used to provide a time-dependent Beltrami field $\boldsymbol{\mathcal{U}}(\boldsymbol{x},t)$ from a time-dependent function $\eta _0(\boldsymbol{x},t)$ as long as the procedure is generalised to time-dependent flows. The purpose of this section is to explain that generalisation.

Let us assume that $\eta _0(\boldsymbol{x}_0)$ is a Laplacian eigenfunction with eigenvalue $-k^2_0$ . The frozen-in condition of the density $\eta _0(\boldsymbol{x})$ in the advecting flow $\boldsymbol{v}_b(\boldsymbol{x},t)$ is

(4.1a,b) \begin{align} \eta (\boldsymbol{x},t) = \frac {1}{J(t)} \eta _0(\boldsymbol{z}_v(\boldsymbol{x},t)) , \quad \textrm {where} \quad \boldsymbol{z}_v(\boldsymbol{x},t) \equiv \boldsymbol{R}_v( \boldsymbol{x},t) - \boldsymbol{o}(0). \quad \end{align}

These relations immediately imply, in the spatial description, that

(4.2) \begin{align} \frac {\partial \eta }{\partial t} + \boldsymbol{v}_b \boldsymbol{\cdot } \boldsymbol{\nabla }\eta + \eta \boldsymbol{\nabla }\boldsymbol{\cdot }\boldsymbol{v}_b = 0 . \end{align}

The transformation (4.1a ) does not preserve the Laplacian eigenfunction property, unless $\boldsymbol{z}_v(\boldsymbol{x},t)$ (4.1b ) is a similarity, that is, a rigid body rotation and an isotropic expansion (case $\boldsymbol{\mathsf{D}}_0(t)=\boldsymbol{\mathsf{0}}$ ). A direct application of the CK procedure to time-dependent flows requires $\eta (\boldsymbol{x},t)$ be a Laplacian eigenfunction at every time $t$ and that $\boldsymbol{\mathsf{D}}_0(t)=\boldsymbol{\mathsf{0}}$ for general Beltrami flows. We assume therefore implicitly that the deviatoric stretch tensor $\boldsymbol{\mathsf{D}}_0(t)=\boldsymbol{\mathsf{0}}$ . In this case, we have

(4.3) \begin{align} \frac {1}{J(t)} = \frac {k(t)^3}{k_0^3} , \end{align}

and $k(t)$ decays or grows as the inverse cubic root of the volume change induced by the background flow, as expected for a wavenumber carried by an isotropically expanding/compressing medium. The time-independent CK procedure can be recovered by setting $\boldsymbol{\mathsf{F}}(t)=\boldsymbol{\mathsf{I}}$ , so that $J(t)=1$ and $\boldsymbol{z}_v(\boldsymbol{x},t)=\boldsymbol{x}$ . In any case, this time-dependent procedure can be omitted since $\boldsymbol{\mathcal{U}}(\boldsymbol{x},t)$ can be always obtained more directly from $\eta _0(\boldsymbol{x})$ , in a first step computing $\boldsymbol{\mathcal{U}}_{\eta }(\boldsymbol{x})$ using the standard CK procedure (2.16a ) and in a second step using (2.17) to obtain $\boldsymbol{\mathcal{U}}(\boldsymbol{x},t)$ .

There are several, but equivalent, ways to define the time-dependent fields, analogous to those in the CK procedure, $\boldsymbol{\chi }(\boldsymbol{x},t)$ , $\boldsymbol{t}(\boldsymbol{x},t)$ , $\boldsymbol{p}(\boldsymbol{x},t)$ , streamfunction $\boldsymbol{\psi }(\boldsymbol{x},t)$ and $\boldsymbol{\mathcal{U}}(\boldsymbol{x},t)$ . In the following procedure, we define them from their reference initial counterparts $\boldsymbol{\chi }_0(\boldsymbol{x})$ , $\boldsymbol{t}_0(\boldsymbol{x})$ , $\boldsymbol{p}_0(\boldsymbol{x})$ , $\boldsymbol{\psi }_0(\boldsymbol{x})$ and $\boldsymbol{\mathcal{U}}_0(\boldsymbol{x})$ , and hence show their relations. The starting point in the CK procedure is to assume that $\eta _0(\boldsymbol{x})$ is a scalar Laplacian eigenfunction,

(4.4) \begin{align} {\nabla} ^2\eta _0(\boldsymbol{x}) = - k_0^2 \eta _0(\boldsymbol{x}) . \end{align}

The steady vector function $\boldsymbol{\chi }_0(\boldsymbol{x})$ is defined from the scalar Laplacian eigenfunction $\eta _0(\boldsymbol{x})$ as

(4.5) \begin{align} \boldsymbol{\chi }_0(\boldsymbol{x}) \equiv \eta _0(\boldsymbol{x}) \, \hat {\boldsymbol{n}}_0 , \end{align}

where $\hat {\boldsymbol{n}}_0$ is a constant unit vector that defines the direction of the Beltrami field, and from (4.5), or (4.1a ), we define its generalisation by a Piola push-forward transformation

(4.6) \begin{align} \frac {k(t)}{k_0} \boldsymbol{\chi }(\boldsymbol{x},t) \equiv \frac {1}{J(t)} \boldsymbol{\mathsf{F}}(t) \boldsymbol{\chi }_0(\boldsymbol{z}_v(\boldsymbol{x},t)) = \eta (\boldsymbol{x},t) \boldsymbol{\mathsf{F}}(t) \hat {\boldsymbol{n}}_0 = \eta (\boldsymbol{x},t) \boldsymbol{n}(t) , \end{align}

where the time-dependent vector $\boldsymbol{n}(t)$ satisfies

(4.7a,b) \begin{align} \boldsymbol{n}(t) \equiv \boldsymbol{\mathsf{F}}(t)\hat {\boldsymbol{n}}_0 \, \quad \textrm {and} \quad \dot {\boldsymbol{n}}(t) = \boldsymbol{\mathsf{L}}(t) \boldsymbol{n}(t) . \quad \end{align}

The next step in the Chandrasekhar–Kendall procedure is to define from (4.5) the velocity potential $\boldsymbol{\psi }_0(\boldsymbol{x})$ as

(4.8) \begin{align} \boldsymbol{\psi }_0(\boldsymbol{x}) \equiv \boldsymbol{\chi }_0(\boldsymbol{x}) - \frac {\boldsymbol{\nabla } \times \boldsymbol{\chi }_0(\boldsymbol{x})}{k_0} , \end{align}

or, alternatively, the divergence-free functions $\boldsymbol{t}_0(\boldsymbol{x})$ and $\boldsymbol{p}_0(\boldsymbol{x})$ as

(4.9a,b) \begin{align} \boldsymbol{t}_0(\boldsymbol{x}) \equiv \boldsymbol{\nabla } \times \boldsymbol{\chi }_0(\boldsymbol{x}) \,\quad \textrm {and} \quad \boldsymbol{p}_0(\boldsymbol{x}) \equiv \frac {1}{k_0} \boldsymbol{\nabla } \times \boldsymbol{t}_0(\boldsymbol{x}) , \quad \end{align}

which are commonly referred to as the toroidal $(\boldsymbol{t}_0)$ and poloidal $(\boldsymbol{p}_0)$ components. The relation between the poloidal component and the Gauss potential is described in Appendix C. By construction, the fields $\boldsymbol{t}_0$ and $\boldsymbol{p}_0$ are vector Laplacian eigenfunctions,

(4.10a,b) \begin{align} {\nabla }^2 \boldsymbol{t}_0 = -k_0^2 \boldsymbol{t}_0 \,\quad \textrm {and} \quad {\nabla }^2 \boldsymbol{p}_0 = -k_0^2 \boldsymbol{p}_0 , \quad \end{align}

and satisfy

(4.11a,b) \begin{align} \boldsymbol{\nabla }\times \boldsymbol{t}_0 = k_0 \boldsymbol{p}_0 \, \quad \textrm {and} \quad \boldsymbol{\nabla }\times \boldsymbol{p}_0 = k_0 \boldsymbol{t}_0 . \quad \end{align}

Thus, $\boldsymbol{t}_0(\boldsymbol{x})$ and $\boldsymbol{p}_0(\boldsymbol{x})$ are not Beltrami functions, but their helical combinations $\boldsymbol{\mathcal{U}}_{\pm }(\boldsymbol{x}) \equiv \boldsymbol{t}_0(\boldsymbol{x})\pm \boldsymbol{p}_0(\boldsymbol{x})$ are the positive-helicity and negative-helicity Beltrami modes. We therefore define the Beltrami velocity $\boldsymbol{\mathcal{U}}_0$ as

(4.12) \begin{align} \boldsymbol{\mathcal{U}}_0(\boldsymbol{x}) \equiv \boldsymbol{\mathcal{U}}_{-}(\boldsymbol{x}) \equiv \boldsymbol{\nabla }\times \boldsymbol{\psi }_0(\boldsymbol{x}) = \boldsymbol{t}_0(\boldsymbol{x}) - \boldsymbol{p}_0(\boldsymbol{x}) , \end{align}

where the minus sign option is used to be consistent with the minus sign in the definition of $\boldsymbol{\psi }_0(\boldsymbol{x})$ (4.8). We therefore consider $k_0$ as a signed wavenumber, the Beltrami eigenvalue being $-k_0$ , and the wavenumber magnitude $|k_0|$ . The positive sign solution may be recovered from the negative sign solution through the transformation $k_0 \rightarrow -k_0$ . Otherwise, one may define $k_0\equiv |k_0|\gt 0$ and work with both polarised solutions explicitly $\boldsymbol{\mathcal{U}}_{\pm }(\boldsymbol{x})$ which satisfy $\boldsymbol{\nabla }\times \boldsymbol{\mathcal{U}}_{\pm }=\pm k_0\boldsymbol{\mathcal{U}}_{\pm }$ . The velocity $\boldsymbol{\mathcal{U}}_0$ is therefore a Beltrami flow satisfying

(4.13) \begin{align} \boldsymbol{\nabla } \times \boldsymbol{\mathcal{U}}_0 = -k_0 \boldsymbol{\mathcal{U}}_0 . \end{align}

We recall the curl identity applied to a vector function $\boldsymbol{w}(\boldsymbol{z})$ and a vector mapping $\boldsymbol{z}(\boldsymbol{x},t)$ ,

(4.14) \begin{align} \boldsymbol{\nabla }_{\boldsymbol{x}} \times \big ( \boldsymbol{\mathsf{F}}^{-\top }(t)\boldsymbol{w}(\boldsymbol{z}(\boldsymbol{x},t)) \big ) = \frac {1}{J(t)}\boldsymbol{\mathsf{F}}(t) \left ( \boldsymbol{\nabla }_{\boldsymbol{z}} \times \boldsymbol{w} \right )(\boldsymbol{z}(\boldsymbol{x},t)) . \end{align}

In the time-dependent procedure, we define the vector potential $\boldsymbol{\psi }(\boldsymbol{x},t)$ from $\boldsymbol{\psi }_0$ as a pseudovector

(4.15) \begin{align} \frac {k(t)}{k_0} \boldsymbol{\psi }(\boldsymbol{x},t) \equiv \frac {1}{J(t)} \boldsymbol{\mathsf{F}}(t) \boldsymbol{\psi }_0(\boldsymbol{z}_v(\boldsymbol{x},t)) , \end{align}

and the time-dependent $\boldsymbol{t}(\boldsymbol{x},t)$ and $\boldsymbol{p}(\boldsymbol{x},t)$ counterparts to $\boldsymbol{t}_0(t)$ and $\boldsymbol{p}_0(t)$ as

(4.16a,b) \begin{align} \frac {k(t)}{k_0} \boldsymbol{t}(\boldsymbol{x},t) \equiv \frac {\boldsymbol{\mathsf{F}}(t)}{J(t)} \boldsymbol{t}_0( \boldsymbol{z}_v(\boldsymbol{x},t)) ,\quad \frac {k(t)}{k_0} \boldsymbol{p}(\boldsymbol{x},t) \equiv \frac {\boldsymbol{\mathsf{F}}(t)}{J(t)} \boldsymbol{p}_0( \boldsymbol{z}_{v}(\boldsymbol{x},t)) , \quad \end{align}

which satisfy similar relations to (4.9),

(4.17a,b) \begin{align} \boldsymbol{t}(\boldsymbol{x},t) = \frac {1}{k(t)}\boldsymbol{\nabla }\times \boldsymbol{\chi } (\boldsymbol{x},t) \, \quad \textrm {and} \quad \boldsymbol{p}(\boldsymbol{x},t) = \frac {1}{k(t)} \boldsymbol{\nabla }\times \boldsymbol{t}(\boldsymbol{x},t) . \quad \end{align}

Therefore, $\boldsymbol{t}(\boldsymbol{x},t)$ and $\boldsymbol{p}(\boldsymbol{x},t)$ are time-dependent Laplacian eigenfunctions,

(4.18a,b) \begin{align}{\nabla }^2 \boldsymbol{t} = -k(t)^2 \boldsymbol{t} \,\quad \textrm {and} \quad{\nabla }^2 \boldsymbol{p} = -k(t)^2 \boldsymbol{p} . \quad \end{align}

Finally, the time-dependent Beltrami field $\boldsymbol{\mathcal{U}}(\boldsymbol{x},t)$ is defined from $\boldsymbol{\mathcal{U}}_0(\boldsymbol{x})$ as

(4.19) \begin{eqnarray} \boldsymbol{\mathcal{U}}(\boldsymbol{x},t) &\equiv & \frac {k_0}{k(t)} \frac {\boldsymbol{\mathsf{F}}(t)}{J(t)} \boldsymbol{\mathcal{U}}_0( \boldsymbol{z}_v(\boldsymbol{x},t)) = \frac {k_0}{k(t)} \frac {\boldsymbol{\mathsf{F}}(t)}{J(t)} \left ( \boldsymbol{\nabla }_{\boldsymbol{z}}\times \boldsymbol{\psi }_0 \right ) (\boldsymbol{z}_v(\boldsymbol{x},t)) \nonumber \\[5pt]&=& \boldsymbol{\nabla }\times \boldsymbol{\psi }(\boldsymbol{x},t) = \boldsymbol{t}(\boldsymbol{x},t) - \boldsymbol{p}(\boldsymbol{x},t) , \end{eqnarray}

from which the Beltrami property follows,

(4.20) \begin{align} \boldsymbol{\nabla } \times \boldsymbol{\mathcal{U}} = -k(t) \boldsymbol{\mathcal{U}} , \end{align}

and therefore,

(4.21) \begin{align} k(t)\boldsymbol{\mathcal{U}}(\boldsymbol{x},t) = \frac {\boldsymbol{\mathsf{F}}(t)}{J(t)} \left ( k_0 \kern1pt \boldsymbol{\mathcal{U}}_0( \boldsymbol{z}_v(\boldsymbol{x},t)) \right ) . \end{align}

Similarly, the vorticity of the Beltrami field,

(4.22) \begin{align} \boldsymbol{\mathcal{W}}(\boldsymbol{x},t) \equiv \boldsymbol{\nabla }\times \boldsymbol{\mathcal{U}} (\boldsymbol{x},t) = -k(t) \boldsymbol{\mathcal{U}} (\boldsymbol{x},t) = \frac {\boldsymbol{\mathsf{F}}(t)}{J(t)} \boldsymbol{\mathcal{W}}_0(\boldsymbol{z}(\boldsymbol{x},t)) , \end{align}

is just the Piola pushforward operation on the material vorticity $\boldsymbol{\mathcal{W}}_0 =-k_0\boldsymbol{\mathcal{U}}_0$ . Therefore, the total velocity

(4.23) \begin{align} \boldsymbol{u}(\boldsymbol{x},t) \equiv \boldsymbol{\mathcal{U}}(\boldsymbol{x},t) + \boldsymbol{u}_b(\boldsymbol{x},t) , \end{align}

with vorticity

(4.24) \begin{align} \boldsymbol{\omega }(\boldsymbol{x},t) \equiv \boldsymbol{\nabla } \times \boldsymbol{u}(\boldsymbol{x},t) = \boldsymbol{\mathcal{W}}(\boldsymbol{x},t) + 2\boldsymbol{\nu }(t) = -k(t) \boldsymbol{\mathcal{U}}(\boldsymbol{x},t) + 2\boldsymbol{\nu }(t) , \end{align}

satisfies the time-dependent vorticity (2.1).

The key result of this section is that from a scalar Laplacian eigenfunction $\eta _0(\boldsymbol{x})$ and a background flow $\boldsymbol{u}_b(\boldsymbol{x},t)$ with time-dependent homogeneous velocity gradient $\boldsymbol{\mathsf{L}}(t)$ , with vanishing deviatoric strain tensor $\boldsymbol{\mathsf{D}}_0(t)=\boldsymbol{\mathsf{0}}$ , it is possible to obtain a time-dependent Beltrami flow $\boldsymbol{\mathcal{U}}(\boldsymbol{x},t)$ that is materially advected by the background flow $\boldsymbol{v}_b(\boldsymbol{x},t)$ and, therefore, becomes a solution to the time-dependent vorticity equation. Since there are 11 coordinate systems in which the Helmholtz equation is separable, we have $\boldsymbol{\mathcal{U}}_0(\boldsymbol{x})$ modes in 11 different geometries. For example, in Cartesian geometry, with coordinates $\{x,y,z\}$ , the plane wave seed function

(4.25a,b) \begin{align} \eta _0(\boldsymbol{x}) \equiv \hat {\eta }_0 \textrm {e}^{ {i}(k_xx+k_yy+k_zz)} ,\quad \textrm {with} \quad k_0^2 = k_x^2 +k_y^2 + k_z^2 , \quad \end{align}

leads to the family of Arnold–Beltrami–Childress (ABC) flows (Arnold Reference Arnold1965; Childress Reference Childress1970; Dombre et al. Reference Dombre, Frisch, Greene, Hénon, Mehr and Soward1986; Galloway & Frisch Reference Galloway and Frisch1986; Soward Reference Soward1987). In cylindrical coordinates $\{\rho ,\varphi ,z\}$ , the seed function

(4.26) \begin{align} \eta _0(\boldsymbol{x}) \equiv \hat {\eta }_0 R_m(\kappa \rho ) \textrm {e}^{ {i}m\varphi } \textrm {e}^{ {i}k_z z} , \quad R_m(\boldsymbol{\cdot }) \in \big\{ J_m(\boldsymbol{\cdot }) , Y_m(\boldsymbol{\cdot }) , H_m^{(1,2)}(\boldsymbol{\cdot })\big\} , \quad k_0^2 = \kappa ^2+k_z^2 , \end{align}

where $\{ J_m,Y_m,H_m^{(1,2)} \}$ are the cylindrical Bessel and Hankel functions, leads to the cylindrical Kelvin modes (Kelvin Reference Kelvin1880; Dritschel Reference Dritschel1991). In spherical geometry, with coordinates $\{ r,\theta ,\varphi \}$ , the function

(4.27) \begin{align} \eta _0(\boldsymbol{x}) \equiv \hat {\eta }_0 \varPsi _{\ell }(k_0 r)Y_{\ell }^m(\theta ,\varphi ) , \quad \textrm {where} \quad \varPsi (\boldsymbol{\cdot }) \in \{ j_{\ell }(\boldsymbol{\cdot }) , y_{\ell }(\boldsymbol{\cdot })\ , h_{\ell }(\boldsymbol{\cdot }) \} , \end{align}

where $\{j_{\ell } , y_{\ell }, h_{\ell }\}$ are the spherical Bessel and Hankel functions, leads to the spherical vortices, including the Hicks–Moffatt vortex (Hicks Reference Hicks1899; Moffatt Reference Moffatt1969, Reference Moffatt2017; Viúdez Reference Viúdez2022). An application of these spherical modes to physical oceanography is described in Appendix A. In parabolic cylindrical geometry, with coordinates $\{ u , v , z \}$ , the seed function is

(4.28) \begin{align} \eta _0(\boldsymbol{x}) = \hat {\eta }_0 D_{\nu }(\sqrt {\kappa }u) D_{\nu }(\sqrt {\kappa }v) \textrm {e}^{ {i}k_zz} ,\quad k_0^2 = \kappa ^2 + k_z^2 , \end{align}

where $D_{\nu }$ are parabolic-cylinder functions. In confocal ellipsoidal geometry, with coordinates $\{ \lambda , \mu , \nu \}$ , the seed function

(4.29) \begin{align} \eta _0(\boldsymbol{x}) \equiv L(\lambda ) M(\mu ) N(\nu ) \, \end{align}

is defined as the product of Lamé wave functions in ellipsoidal coordinates. And so on, in parabolic three-dimensional geometry, elliptic cylindrical geometry, prolate and oblate spheroidal geometries, and conical geometry. Using a different procedure, namely the Bragg–Hawthorne equation, steady-state solutions in the case of axisymmetric flows were given by Bělík et al. (Reference Bělík, Su, Dokken, Scholz and Shvartsman2020) in cylindrical, spherical, paraboloidal, and prolate and oblate spheroidal geometries. The generalised CK procedure described in this section helps to classify all time-dependent Beltrami flows in different geometries within the same family of flow solutions.

5. Plane Beltrami fields

In this section, we consider the particular case in which the Beltrami fields have a plane wave geometry. In the general case considered in the previous section, the background deviatoric tensor $\boldsymbol{\mathsf{D}}_0(t)$ must vanish to allow the advected Beltrami field to preserve its Beltrami property. In the particular case of a plane Beltrami field, it is shown in the following that it is possible to have $\boldsymbol{\mathsf{D}}_0(t)\ne \boldsymbol{\mathsf{0}}$ as long as $\boldsymbol{\mathsf{D}}_0(t)$ is in the direction of the plane wavevector $\boldsymbol{k}(t)$ .

5.1. Plane Beltrami flows

The Laplacian eigenfunction for a plane wave is

(5.1a,b) \begin{align} {\eta }_0(\boldsymbol{x}_0) = \textrm {e}^{ {i}\boldsymbol{k}_0\boldsymbol{\cdot }\boldsymbol{x}_0} ,\quad \textrm {with} \quad {\nabla} ^2{\eta }_0 = - |\boldsymbol{k}_0|^2 {\eta }_0 . \quad \end{align}

For a linear invertible transformation $\boldsymbol{\mathsf{A}}(t)$ , it follows that

(5.2) \begin{align} {\eta }(\boldsymbol{x},t) \equiv \eta _0(\boldsymbol{\mathsf{A}}(t)\boldsymbol{x}) =\textrm {e}^{ {i} \left (\boldsymbol{\mathsf{A}}(t)^{\top }\boldsymbol{k}_0\right )\boldsymbol{\cdot }\boldsymbol{x}} ,\quad {\nabla} ^2\eta (\boldsymbol{x},t) = - |\boldsymbol{\mathsf{A}}(t)^{\top }\boldsymbol{k}_0|^2 \eta (\boldsymbol{x},t) , \end{align}

that is, ${\eta }(\boldsymbol{x},t)$ is a Laplacian eigenfunction with eigenvalue $- |\boldsymbol{\mathsf{A}}(t)^{\top }\boldsymbol{k}_0|^2$ . This property allows us to use the CK procedure at all times. In the CK procedure, the steady material vector plane waves are

(5.3) \begin{align} \{ \boldsymbol{\chi }_0(\boldsymbol{x}_0) ,\boldsymbol{t}_0(\boldsymbol{x}_0) ,\boldsymbol{p}_0(\boldsymbol{x}_0) \} = \left \{ \hat {\boldsymbol{n}}_0 ,\boldsymbol{k}_0\times \hat {\boldsymbol{n}}_0 ,-\frac {1}{k_0}\boldsymbol{k}_0 \times (\boldsymbol{k}_0\times \hat {\boldsymbol{n}}_0) \right \} \textrm {e}^{ {i}\boldsymbol{k}_0\boldsymbol{\cdot }\boldsymbol{x}_0} . \end{align}

The case $\boldsymbol{k}_0\Vert \hat {\boldsymbol{n}}_0$ produces a zero Beltrami field. In the equivalent time-dependent vector plane waves procedure, the initial seed function is a plane wave

(5.4a,b) \begin{align} {\eta }_0(\boldsymbol{x}_0) = \textrm {e}^{ {i}\boldsymbol{k}_0\boldsymbol{\cdot } (\boldsymbol{x}_0-\boldsymbol{o}(0))} ,\quad {\nabla} ^2{\eta }_0 = - |\boldsymbol{k}_0|^2 {\eta }_0 , \quad \end{align}

whose phase evolves, under the transformation (3.4), as

(5.5a,b) \begin{align} \boldsymbol{k}_0 \boldsymbol{\cdot } \big ( \boldsymbol{\mathsf{F}}^{-1}(t) (\boldsymbol{x} - \boldsymbol{o}_v(t)) \big ) = \boldsymbol{k}(t) \boldsymbol{\cdot } (\boldsymbol{x} - \boldsymbol{o}_v(t)) ,\quad \textrm {where} \; \boldsymbol{k}(t)\equiv \boldsymbol{\mathsf{F}}^{-\top }(t)\boldsymbol{k}_0 . \quad \end{align}

The corresponding scalar density function evolves as

(5.6a,b) \begin{align} \eta (\boldsymbol{x},t) = \frac {1}{J(t)} \textrm {e}^{ {i} \boldsymbol{k}(t)\boldsymbol{\cdot }(\boldsymbol{x} - \boldsymbol{o}_v(t)) } = A(t) \textrm {e}^{ {i} \boldsymbol{k}(t)\boldsymbol{\cdot }\boldsymbol{x}} , \quad A(t) \equiv \frac {1}{J(t)} \textrm {e}^{- {i} \boldsymbol{k}(t)\boldsymbol{\cdot }\boldsymbol{o}_v(t)} , \quad \end{align}

where $A(t)$ is a time-dependent complex-valued amplitude. The rate of change of the time-dependent wavevector is

(5.7) \begin{align} \dot {\boldsymbol{k}}(t) = - \boldsymbol{\mathsf{L}}^{\top }(t)\boldsymbol{k}(t) , \end{align}

and we may check that the rate of change of the phase in the material space is

(5.8) \begin{align} \frac {\partial }{\partial t} \left ( \boldsymbol{r}_v(\boldsymbol{x}_0,t)-\boldsymbol{o}_v(t)) \boldsymbol{\cdot } \boldsymbol{k}(t) \right ) =\boldsymbol{\mathsf{L}}(t) \left [ \boldsymbol{\mathsf{F}}(t)(\boldsymbol{x}_0-\boldsymbol{o}(0)- (\boldsymbol{r}_v(\boldsymbol{x}_0,t)-\boldsymbol{o}_v(t))) \right ] \boldsymbol{k}(t) =0, \end{align}

so that the phase in the material space,

(5.9) \begin{align} \boldsymbol{r}_v(\boldsymbol{x}_0,t)-\boldsymbol{o}_v(t)) \boldsymbol{\cdot } \boldsymbol{k}(t) = (\boldsymbol{x}_0-\boldsymbol{o}(0))\boldsymbol{\cdot } \boldsymbol{k}_0 , \end{align}

is stationary. Since $\eta (\boldsymbol{x},t)$ is a Laplacian eigenfunction at all times $t$ ,

(5.10) \begin{align} {\nabla} ^2\eta (\boldsymbol{x},t) = - k(t)^2\eta (\boldsymbol{x},t) , \end{align}

we carry on with the time-dependent CK procedure and define the corresponding fields

(5.11a,b,c) \begin{align} \boldsymbol{\chi }(\boldsymbol{x},t) \equiv \eta (\boldsymbol{x},t) \hat {\boldsymbol{n}}(t) ,\quad \boldsymbol{t}(\boldsymbol{x},t) \equiv \boldsymbol{\nabla }\times \boldsymbol{\chi }(\boldsymbol{x},t) ,\quad \boldsymbol{p}(\boldsymbol{x},t) \equiv \frac {\boldsymbol{\nabla }\times \boldsymbol{\chi }(\boldsymbol{x},t)}{k(t)} , \quad \end{align}

which become

(5.12a,b) \begin{align} \boldsymbol{t} = \textrm {i}\eta \boldsymbol{k}\times \hat {\boldsymbol{n}} ,\quad \textrm {and} \quad \boldsymbol{p} = \eta \left ( k \hat {\boldsymbol{n}} - \frac {\boldsymbol{k}\boldsymbol{\cdot \hat {\boldsymbol{n}}}}{k} \boldsymbol{k} \right ) . \quad \end{align}

Next, we obtain, from (5.12a,b ), the Beltrami velocity field

(5.13) \begin{align} \boldsymbol{\mathcal{U}}(\boldsymbol{x},t) \equiv \boldsymbol{t}(\boldsymbol{x},t) - \boldsymbol{p}(\boldsymbol{x},t) , \end{align}

and hence, the Beltrami vorticity field $\boldsymbol{\mathcal{W}}(\boldsymbol{x},t)$ as

(5.14) \begin{align} \boldsymbol{\mathcal{W}}(\boldsymbol{x},t) = \boldsymbol{W}(t) \textrm {e}^{ {i}\boldsymbol{k}(t)\boldsymbol{\cdot }\boldsymbol{x}} , \end{align}

where the polarisation vector $\boldsymbol{W}(t)$ of the Beltrami vorticity field is a function of $\{ \boldsymbol{k}(t) , \hat {\boldsymbol{n}}(t) , A(t) \}$ ,

(5.15) \begin{align} \boldsymbol{W}(t) = \boldsymbol{W}(\boldsymbol{k}(t),\hat {\boldsymbol{n}}(t),A(t)) , \end{align}

and is given by

(5.16) \begin{align} \boldsymbol{W}(t) = A(t) \big ( k(t)^2 \hat {\boldsymbol{n}}(t) - (\boldsymbol{k}(t) \boldsymbol{\cdot } \hat {\boldsymbol{n}}(t)) \boldsymbol{k}(t) - \textrm {i} k(t) \boldsymbol{k}(t)\times \hat {\boldsymbol{n}}(t) \big ) . \end{align}

Clearly,

(5.17) \begin{align} \boldsymbol{W}(t)\boldsymbol{\cdot }\boldsymbol{k}(t)=0 , \end{align}

so that the polarisation vector $\boldsymbol{W}(t)$ is tangent to the wavefronts and there is no vorticity component along the propagation direction $\boldsymbol{k}(t)$ . This is equivalent to the property $\boldsymbol{\nabla }\boldsymbol{\cdot }\boldsymbol{\mathcal{W}}(\boldsymbol{x},t)=0$ in terms of the polarisation vector $\boldsymbol{W}(t)$ . The vorticity mode $\boldsymbol{\mathcal{W}}(\boldsymbol{x},t)$ , as well as the velocity mode $\boldsymbol{\mathcal{U}}(\boldsymbol{x},t)$ , are transverse waves. This is analogous to the electromagnetic monochromatic plane waves with wavevector $\boldsymbol{k}$ , where the electric $\boldsymbol{E}(\boldsymbol{x},t)$ and magnetic $\boldsymbol{B}(\boldsymbol{x},t)$ fields,

(5.18a,b) \begin{align} \boldsymbol{E}(\boldsymbol{x},t) \equiv \Re \big\{\boldsymbol{E}_0 \textrm {e}^{ {i}(\boldsymbol{k}\boldsymbol{\cdot }\boldsymbol{x}-\omega t)}\big\} ,\quad \boldsymbol{B}(\boldsymbol{x},t) \equiv \Re \big\{\boldsymbol{B}_0 \textrm {e}^{ {i}(\boldsymbol{k}\boldsymbol{\cdot }\boldsymbol{x}-\omega t)}\big\} , \quad \end{align}

satisfy the Maxwell equations for the propagation of electromagnetic waves in vacuum, which imply

(5.19a,b,c,d) \begin{align} \boldsymbol{k}\boldsymbol{\cdot }\boldsymbol{E}_0 = 0 ,\quad \boldsymbol{k}\boldsymbol{\cdot }\boldsymbol{B}_0 = 0 ,\quad \textrm {i}\boldsymbol{k}\times \boldsymbol{E}_0 = \omega \boldsymbol{B}_0 ,\quad \textrm {i}\boldsymbol{k}\times \boldsymbol{B}_0 = -\frac {\omega }{c_0^2} \boldsymbol{E}_0 . \quad \end{align}

So, in vacuum, the time dependence of the electromagnetic plane waves is purely oscillatory via the phase $\textrm {exp}(-\textrm {i}\omega t)$ , and the polarisations $\boldsymbol{E}_0$ and $\boldsymbol{B}_0$ are constant. This analogy is pursued in § 5.2. The fluid mechanics analogues to the electromagnetic relations (5.19) are

(5.20) \begin{align} & \boldsymbol{k}(t)\boldsymbol{\cdot }\boldsymbol{U}(t) = 0 ,\quad \boldsymbol{k}(t)\boldsymbol{\cdot }\frac {\boldsymbol{W}(t)}{k(t)} = 0 ,\quad \textrm {i}\boldsymbol{k}(t)\times \boldsymbol{U}(t) = \boldsymbol{W}(t), \nonumber \\ & \textrm {i}\boldsymbol{k}(t) \times \left ( \frac {\boldsymbol{W}(t)}{k(t)} \right ) = -k(t) \boldsymbol{U}(t) . \end{align}

Given the relation between $\boldsymbol{k}(t)$ and $\boldsymbol{\mathsf{L}}(t)$ (5.7), the problem is to find the relations between the parameters of the advecting flow and the parameters of the Beltrami flow such that the vorticity polarisation is frozen in the advecting flow, that is,

(5.21) \begin{align} \dot {\boldsymbol{W}}(t) = \left ( \boldsymbol{\mathsf{D}}_0(t) -\frac {2}{3}\delta (t) \boldsymbol{\mathsf{I}} +\boldsymbol{\mathsf{W}}(t) \right ) \boldsymbol{W}(t) , \end{align}

or, in an equivalent way using the Piola transformation, that

(5.22) \begin{align} \boldsymbol{W}(t) =\frac {1}{J(t)} \boldsymbol{\mathsf{F}}(t)\boldsymbol{W}(0) . \end{align}

Time differentiation of (5.22) leads to (5.21). Thus, the polarisation vector $\boldsymbol{W}(t)$ obeys the same time evolution equation as the background vorticity $2\boldsymbol{\nu }(t)$ and $\boldsymbol{W}(t)$ is frozen in the flow $\boldsymbol{v}_b(\boldsymbol{x},t)$ .

Using (5.16), we may express the rate of change of $\boldsymbol{W}(t)$ as

(5.23) \begin{align} \dot {\boldsymbol{W}}(t) = \partial _{\boldsymbol{k}} \boldsymbol{W} \dot {\boldsymbol{k}} + \partial _{\hat {\boldsymbol{n}}} \boldsymbol{W} \dot {\hat {\boldsymbol{n}}} + \partial _{A} \boldsymbol{W} \dot {A} . \end{align}

At every time $t$ , relations (5.7) and (5.21) imply six real linear constraints on the nine real components of $\boldsymbol{\mathsf{L}}(t)$ , so that in the generic case, for non-degenerate $\{\boldsymbol{k}(t), \boldsymbol{W}(t)\}$ , there is a three-parameter family of homogeneous flows with velocity gradient $\boldsymbol{\mathsf{L}}(t)$ in which the vorticity $\boldsymbol{\mathcal{W}}(\boldsymbol{x},t)$ is frozen for any given choice of $\boldsymbol{k}(t)$ , $\hat {\boldsymbol{n}}(t)$ and $A(t)$ . The deviatoric strain tensor $\boldsymbol{\mathsf{D}}_0(t)$ causes the anisotropic stretching and tilting of the polarisation, the divergence $\delta (t)$ is responsible for the isotropic amplification/decay of the magnitude of $\boldsymbol{W}(t)$ , while the background spin causes a rigid body rotation, or precession, of $\boldsymbol{W}(t)$ via $\boldsymbol{\mathsf{W}}(t)\boldsymbol{W}(t)=\boldsymbol{\nu }(t)\times \boldsymbol{W}(t)$ . A Beltrami plane wave $\boldsymbol{\mathcal{W}}(\boldsymbol{x},t)$ (5.14)

(5.24) \begin{align} \boldsymbol{\nabla }\times \boldsymbol{\mathcal{W}} = - k \boldsymbol{\mathcal{W}} \quad \iff \boldsymbol{\mathcal{C}}_{\boldsymbol{k}} \boldsymbol{W}=-k\boldsymbol{W} , \quad \textrm {where} \; \boldsymbol{\mathcal{C}}_{\boldsymbol{k}}\equiv \textrm {i} \boldsymbol{k}\times . \end{align}

Thus, the Beltrami condition implies that the polarisation vector $\boldsymbol{W}(t)$ is an eigenvector of the operator $\boldsymbol{\mathcal{C}}_{\boldsymbol{k}}$ with eigenvalue $-k(t)$ . This means that in the complex representation

(5.25) \begin{align} \boldsymbol{W}(t) = \mathfrak{R}\boldsymbol{W}(t) + \textrm {i} \mathfrak{I}\boldsymbol{W}(t) , \end{align}

the real vectors $\mathfrak{R}\boldsymbol{W}(t)$ and $\mathfrak{I}\boldsymbol{W}(t)$ are orthogonal $\mathfrak{R}\boldsymbol{W}\boldsymbol{\cdot }\mathfrak{I}\boldsymbol{W}=0$ , and have the same norm $||\mathfrak{R}\boldsymbol{W}||=||\mathfrak{I}\boldsymbol{W}||$ , which means that $\boldsymbol{W}(t)$ must have circular polarisation in the plane perpendicular to $\boldsymbol{k}(t)$ .

To find the background velocity gradient $\boldsymbol{\mathsf{L}}(t)$ that preserves the Beltrami condition of the plane wave, we assume that the deviatoric tensor $\boldsymbol{\mathsf{D}}_0(t)$ is axisymmetric about the direction of $\boldsymbol{k}(t)$ and that the spin tensor $\boldsymbol{\mathsf{W}}(t)$ generates a rotation about the same axis. We define orthonormal basis vectors $\{ \hat {\boldsymbol{e}}_1, \hat {\boldsymbol{e}}_2, \hat {\boldsymbol{e}}_3 \}$ such that

(5.26) \begin{align} \hat {\boldsymbol{e}}_3 \equiv \hat {\boldsymbol{k}}(t) = \frac {\boldsymbol{k}(t)}{k(t)} . \end{align}

In that orthogonal basis, the deviatoric $\boldsymbol{\mathsf{D}}_0(t)$ and spin $\boldsymbol{\mathsf{W}}(t)$ tensors are

(5.27) \begin{align} \boldsymbol{\mathsf{D}}_0(t) = \left ( \begin{array}{ccc} \sigma (t) & 0 & 0 \\ 0 & \sigma (t) & 0 \\ 0 & 0 & -2\sigma (t) \end{array} \right ) ,\quad \boldsymbol{\mathsf{W}}(t) = \left ( \begin{array}{ccc} 0 & - \nu (t) & 0 \\ \nu (t) & 0 & 0 \\ 0 & 0 & 0 \end{array} \right ) , \end{align}

and therefore, the velocity gradient is

(5.28) \begin{align} \boldsymbol{\mathsf{L}}(t) = \left ( \begin{array}{ccc} \sigma (t)+\delta (t)/3 & -2\nu (t) & 0 \\[3pt] 2\nu (t) & \sigma (t)+\delta (t)/3 & 0 \\[3pt] 0 & 0 & -2\sigma (t)+\delta (t)/3 \end{array} \right ) . \end{align}

Since $\boldsymbol{\mathsf{L}}(t)$ is block-diagonal with respect to the split $\mathbb{R}^3=\textrm {span}\{\hat {\boldsymbol{e}}_1,\hat {\boldsymbol{e}}_2\} \oplus \textrm {span} \{\hat {\boldsymbol{e}}_3\}$ , the solution $\boldsymbol{\mathsf{F}}(t)$ to $\dot {\boldsymbol{\mathsf{F}}}=\boldsymbol{\mathsf{L}}\boldsymbol{\mathsf{F}}$ has the same block-diagonal form

(5.29) \begin{align} \boldsymbol{\mathsf{F}}(t) = \left ( \begin{array}{cc} \boldsymbol{\mathsf{F}}_{\perp }(t) & 0 \\[3pt] 0 & \mathsf{F}_{\parallel }(t) \end{array} \right ) , \end{align}

where $\boldsymbol{\mathsf{F}}_{\perp }(t)$ is a $2 \times 2$ matrix acting on the transverse plane and $\mathsf{F}_{\parallel }(t)$ is a scalar function acting on $\hat {\boldsymbol{e}}_3$ .

We first solve the transverse block with evolution equation $\dot {\boldsymbol{\mathsf{F}}}_{\perp }(t) =\boldsymbol{\mathsf{L}}_{\perp }(t) \boldsymbol{\mathsf{F}}_{\perp }(t)$ , with $\boldsymbol{\mathsf{F}}_{\perp }(0)=\boldsymbol{\mathsf{I}}_{2}$ , in which the transverse velocity gradient is

(5.30) \begin{align} \boldsymbol{\mathsf{L}}_{\perp }(t) = \left ( \begin{array}{cc} \sigma (t)+\delta (t)/3 & -2\nu (t) \\[3pt] 2\nu (t) & \sigma (t)+\delta (t)/3 \end{array} \right ) . \end{align}

The transverse deformation gradient $\boldsymbol{\mathsf{F}}_{\perp }(t)$ can be explicitly integrated and the solution is

(5.31) \begin{align} \boldsymbol{\mathsf{F}}_{\perp }(t) = \varLambda _{\perp }(t) \boldsymbol{\mathsf{R}}(\theta (t)) , \end{align}

where

(5.32) \begin{align} \varLambda _{\perp }(t) \equiv \textrm {exp} \left ( \int _0^t \left ( \sigma (\tau )+\frac {1}{3}\delta (\tau )\right ) \,\textrm {d}\tau \right ) , \end{align}

and the rotation matrix $\boldsymbol{\mathsf{R}}(\theta )$ and rotation angle $\theta (t)$ are

(5.33a,b) \begin{align} \boldsymbol{\mathsf{R}}(\theta ) = \left ( \begin{array}{cc} \cos \theta & -\sin \theta \\ \sin \theta & \cos \theta \end{array} \right ) ,\quad \theta (t)\equiv \int _0^t \nu (\tau )\,\textrm {d}\tau . \quad \end{align}

Thus, in the transverse plane, the deformation is a uniform stretch $\varLambda _{\perp }(t)$ plus a rotation by an angle $\theta (t)$ .

The axial component of the deformation $\mathsf{F}_{\parallel }(t)$ satisfies $\dot {\mathsf{F}}_{\parallel }(t) ={\mathsf{L}}_{\parallel }(t) {\mathsf{F}}_{\parallel }(t)$ , with ${\mathsf{F}}_{\parallel }(0)=1$ , and

(5.34) \begin{align} {\mathsf{L}}_{\parallel }(t) \equiv -2\sigma (t)+\delta (t)/3 . \end{align}

Therefore, the axial component of the deformation gradient $\mathsf{F}_{\parallel }(t)$ is

(5.35) \begin{align} \mathsf{F}_{\parallel }(t) = \varLambda _{\parallel }(t) = \textrm {exp} \left ( \int _0^t \left ( -2\sigma (\tau )+\frac {1}{3}\delta (\tau )\right ) \,\textrm {d}\tau \right ) . \end{align}

The determinant of the transformation is simply

(5.36) \begin{align} J(t) \equiv \textrm {det} \boldsymbol{\mathsf{F}}(t) =\varLambda _{\parallel }^2(t)\varLambda _{\perp }(t) = \textrm {exp} \left ( \int _0^t \delta (\tau )\,\textrm {d}\tau \right ). \end{align}

The time evolution of the wavenumber $k(t)$ is

(5.37) \begin{align} k(t) = \frac {k_0}{\varLambda _{\parallel }(t)} = k_0 \textrm {exp} \left ( \int _0^t \left ( 2\sigma (\tau )-\frac {1}{3}\delta (\tau )\right ) \,\textrm {d}\tau \right ) , \end{align}

unit vector $\hat {\boldsymbol{n}}(t)$ ,

(5.38) \begin{align} \hat {\boldsymbol{n}}(t)=\boldsymbol{\mathsf{R}}(\theta (t)) \hat {\boldsymbol{n}}_0 ,\quad \hat {\boldsymbol{n}}_0\perp \hat {\boldsymbol{e}}_3 , \end{align}

and the time evolution of the amplitude $A(t)$ is

(5.39) \begin{align} A(t) = A_0\frac {\varLambda _{\parallel }(t)}{\varLambda _{\perp }(t)} = A_0 \textrm {exp} \left ( \int _0^t -3\sigma (\tau ) \,\textrm {d}\tau \right ) . \end{align}

We may summarise the results as follows. An initial helical vector $\boldsymbol{W}(0)$ in $\boldsymbol{k}(0)_{\perp }$ must evolve as $\boldsymbol{W}(t)=(1/J(t))\boldsymbol{\mathsf{F}}(t)\boldsymbol{W}(0)$ and must remain a helical eigenvector of $\textrm {i}\boldsymbol{k}(t)\times$ at all times $t$ , that is, it must conserve its circular polarisation. If the transverse part of $\boldsymbol{\mathsf{F}}(t)$ is a similarity, consisting of a uniform stretch plus a rotation, the Piola evolution keeps $\boldsymbol{W}(t)$ inside the helical subspace in the plane perpendicular to $\boldsymbol{k}(t)$ ; therefore, the Beltrami property survives and the CK representation is possible at all times. In this particular solution, the free parameters of the background flow are the main parameter, namely, the principal deviatoric strain rate $\sigma (t)$ in the two transverse directions, the velocity divergence $\delta (t)$ , and the angular velocity $\nu (t)$ . The anisotropic strain rate $\sigma (t)$ stretches/compresses the wavelength along $\boldsymbol{k}(t)$ and exponentially amplifies or damps the plane-wave amplitude $A(t)$ . The isotropic expansion/compression $\delta (t)$ modifies the growth/decay of the wavenumber $k(t)$ via volume change through $J(t)$ , but cancels out $A(t)$ in this CK–frozen solution. The background rotation rate $\nu (t)$ rigidly rotates the polarisation direction $\hat {\boldsymbol{n}}(t)$ in the plane orthogonal to $\boldsymbol{k}(t)$ without changing $|k(t)|$ or $|A(t)|$ .

From (5.36), (5.37) and (5.39), we obtain a time invariant

(5.40) \begin{align} J(t) A(t)^2k(t)^3 = \textrm {const.} \end{align}

For the CK Beltrami plane-wave, the vorticity amplitude, up to a fixed numerical factor coming from the precise CK normalisation that the vorticity amplitude $\Vert \boldsymbol{\mathcal{W}}\Vert \sim A k^2$ , the velocity $\Vert \boldsymbol{\mathcal{U}}\Vert \sim A k$ implies that the helicity density $\mathcal{H} \equiv \boldsymbol{\mathcal{U}} \boldsymbol{\cdot }\boldsymbol{\mathcal{W}} \sim A^2 k^3$ and, therefore, $H_0 =J(t)A(t)^2k(t)^3$ represents the helicity content of the mode per material volume element.

5.2. Electromagnetic wave analogy: Jones vector and matrix representation

For a monochromatic plane wave propagating along the $z$ -axis, we only need the transverse electric field components, say $E_x$ and $E_y$ . Their complex amplitudes are collected into a column vector called the Jones vector (Jones Reference Jones1941; Collett Reference Collett2005),

(5.41) \begin{align} \boldsymbol{E} = \begin{pmatrix} E_x \\ E_y \end{pmatrix} . \end{align}

For example,

(5.42) \begin{align} \boldsymbol{E}_x = \begin{pmatrix} 1 \\[2pt] 0 \end{pmatrix} ,\quad \boldsymbol{E}_y = \begin{pmatrix} 0 \\[2pt] 1 \end{pmatrix} ,\quad \boldsymbol{E}_R = \frac {1}{\sqrt {2}} \begin{pmatrix} 1 \\[2pt] -\textrm {i} \end{pmatrix} ,\quad \boldsymbol{E}_L = \frac {1}{\sqrt {2}} \begin{pmatrix} 1 \\[2pt] \textrm {i} \end{pmatrix} , \end{align}

represent linear polarisation along $x$ and along $y$ , and right and left circular polarizations, respectively. The relative phase between $E_x$ and $E_y$ is encoded in the complex entries of the Jones vector. The polarisation changes caused by a linear optical element can be represented by a $2\times 2$ complex matrix $\boldsymbol{\mathsf{J}}$ , called the the Jones matrix, acting on the Jones vector,

(5.43) \begin{align} \boldsymbol{E}_{ {out}} = \boldsymbol{\mathsf{J}} \,\boldsymbol{E}_{ {in}} . \end{align}

For example,

(5.44) \begin{align} \boldsymbol{\mathsf{J}} = \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix} ,\quad \boldsymbol{\mathsf{J}}_{{pol},x} = \begin{pmatrix} 1 & 0 \\ 0 & 0 \end{pmatrix} ,\quad \boldsymbol{\mathsf{J}}_{ {ret}}(\phi ) = \begin{pmatrix} 1 & 0 \\ 0 & e^{{i}\phi } \end{pmatrix}, \end{align}

represent free space (no change), ideal linear polariser transmitting only the $x$ -component, phase retarder, waveplate that introduces a phase shift $\phi$ in the $y$ -component. Thus, a Jones matrix is simply the linear operator in the two-dimensional polarisation space that maps an input Jones vector to an output Jones vector.

In the fluid mechanics solution, the polarisation vector $\boldsymbol{W}(t)$ , or the velocity polarisation $\boldsymbol{U}(t)$ , in the plane orthogonal to $\boldsymbol{k}(t)$ plays the role of a Jones vector

(5.45) \begin{align} \boldsymbol{W}(t) \in \boldsymbol{k}(t)^\perp \;\cong \; \mathbb{C}^2, \end{align}

that is, the plane $\boldsymbol{k}(t)^\perp$ is isomorphic to $\mathbb{C}^2$ . The restriction of the time-dependent deformation gradient $\boldsymbol{\mathsf{F}}(t)$ of the background flow to the transverse plane,

(5.46) \begin{align} \boldsymbol{\mathsf{F}}_\perp (t) : \boldsymbol{k}(t)^\perp \to \boldsymbol{k}(t)^\perp , \end{align}

acts on $\boldsymbol{W}(t)$ exactly like a Jones matrix $\boldsymbol{\mathsf{J}}$ (5.43) acts on a Jones vector,

(5.47) \begin{align} \boldsymbol{W}(t) \;\propto \; \boldsymbol{\mathsf{F}}_\perp (t)\,\boldsymbol{W}_0. \end{align}

If $\boldsymbol{\mathsf{F}}_\perp (t)$ is of the form

(5.48) \begin{align} \boldsymbol{\mathsf{F}}_\perp (t) = \lambda (t)\boldsymbol{\mathsf{R}}(\theta (t)) , \end{align}

that is, a scalar stretch $\lambda (t)$ times a rotation $\boldsymbol{\mathsf{R}}(\theta (t))$ in the transverse plane, that is, transverse isotropy about $\boldsymbol{k}(t)$ , then it preserves circular polarisation, and a Beltrami (helical) polarisation remains Beltrami, exactly as an isotropic optical medium preserves circular polarisation. If, instead, $\boldsymbol{\mathsf{F}}_\perp (t)$ contains anisotropic shear (is not a pure stretch plus rotation), then it acts like a birefringent Jones matrix: a circular (Beltrami) polarisation is mapped into an elliptical polarisation in $\boldsymbol{k}(t)^\perp$ , and the mode ceases to be Beltrami, even though the vorticity remains frozen-in. We notice that the Jones matrix transformation is an instantaneous transformation, while its fluid mechanics counterpart (5.47) is a continuous deformation. In practice, the scattering of Beltrami fields may be described, under the Beltrami field approximation, by finite-time, or pulses, of a background velocity gradient $\boldsymbol{\mathsf{L}}(t)$ , and consequently, pulses of the background deformation gradient $\boldsymbol{\mathsf{F}}(t)$ acting on the Beltrami field. An example of a scattering process is given in § 7.

6. Two examples using particular forms of the velocity gradient $\boldsymbol{\mathsf{L}}(t)$

The time-dependent Beltrami flow $\boldsymbol{\mathcal{U}}(\boldsymbol{x},t)$ is completely determined by the initial Beltrami flow $\boldsymbol{\mathcal{U}}_0(\boldsymbol{x})$ and the background flow $\boldsymbol{u}_b(\boldsymbol{x},t)$ . The temporal changes in $\boldsymbol{\mathcal{U}}(\boldsymbol{x},t)$ , apart from those derived from the arbitrary rigid translation $\boldsymbol{o}(t)$ , are due to the time evolution of the background flow $\boldsymbol{u}_b(\boldsymbol{x},t)$ , which are themselves prescribed by the time evolution of the background velocity gradient tensor $\boldsymbol{\mathsf{L}}(t)$ . The nine independent components of $\boldsymbol{\mathsf{L}}(t)$ correspond to the expansion/contraction term $\delta (t)$ (one component), vorticity tensor $\boldsymbol{\mathsf{W}}(t)$ (three components) and deviatoric tensor $\boldsymbol{\mathsf{D}}_0(t)$ (five components, namely two normal/extensional components and three shear-rate components). However, the three components of the angular velocity $\boldsymbol{\nu }(t)$ must satisfy the vorticity equation, which implies that there are six degrees of freedom in the way the time evolution of $\boldsymbol{\mathsf{L}}(t)$ can be arbitrarily prescribed. These degrees of freedom allow us to prescribe the time-evolution of the Beltrami field. This freedom can be viewed as the possibilities to prescribe external time-dependent potential, i.e. conservative, forces in the Euler momentum equation. The introduction of forces, however, is not required in this vorticity equation approach, which only assumes that the flow acceleration $\boldsymbol{a}(\boldsymbol{x},t)$ is irrotational, $\boldsymbol{\nabla }\times \boldsymbol{a}=\boldsymbol{0}$ . This section provides two examples of how particular solutions of $\boldsymbol{\mathsf{L}}(t)$ prescribe the time evolution of a Beltrami field.

6.1. Case $\boldsymbol{\mathsf{D}}_0(t)=\boldsymbol{\mathsf{0}}$ : background rigid-body rotation with isotropic dilatation

In the case where the deviatoric tensor of the background flow vanishes $\boldsymbol{\mathsf{D}}_0(t)=\boldsymbol{\mathsf{0}}$ , the background flow $\boldsymbol{u}_b(\boldsymbol{x},t)$ is a rigid-body rotation with isotropic expansion/contraction and there is only one degree of freedom available. In this case, the compatibility condition (4.2) reduces to

(6.1) \begin{align} \dot {\boldsymbol{\nu }}(t)=-\frac {2}{3}\delta (t)\boldsymbol{\nu }(t) \, \end{align}

and therefore, only changes in magnitude, and not in orientation, of the background angular velocity $\boldsymbol{\nu }(t)$ are possible, and hence, only changes in the speed of the displacement velocity $\boldsymbol{c}(t)=2\boldsymbol{\nu }(t)/k(t)$ of the Beltrami field. These changes are related to the isotropic expansion/contraction caused by the divergence $\delta (t)$ ,

(6.2) \begin{align} \dot {\nu }(t) = - \frac {2}{3}\delta (t){\nu }(t) . \end{align}

In terms of the time-integrated divergence $\varDelta (t)$ , relation (6.2) is

(6.3a,b) \begin{align} \frac {\dot {\nu }(t)}{\nu (t)} = -\frac {2}{3} \dot {\varDelta }(t) , \quad \textrm {where} \; \dot {\varDelta }(t) = \delta (t) , \quad \end{align}

whose time-integration leads to the solution

(6.4a,b,c) \begin{align} \boldsymbol{\nu }(t) = \boldsymbol{\nu }_0 \textrm {e}^{-\tfrac {2}{3} \varDelta (t)} , \quad \nu (t) = \nu _0 \textrm {e}^{-\tfrac {2}{3} \varDelta (t)} , \quad \boldsymbol{\nu }_0 \equiv \boldsymbol{\nu }(0) , \quad \end{align}

which provides the time-evolution of the background angular velocity $\boldsymbol{\nu }(t)$ as a function of the time-integral of the background divergence $\delta (t)$ . Using Euler’s identity

(6.5a,b) \begin{align} \dot {J}(t)=J(t)\delta (t) , \quad \textrm {where} \quad J(t) \equiv \textrm {det}(\boldsymbol{\mathsf{F}}) , \quad \end{align}

the relation between $\dot {\nu }(t)$ and $\dot {J}(t)$ is

(6.6) \begin{align} \frac {\dot {\nu }(t)}{\nu (t)} = - \frac {2}{3} \frac {\dot {J}(t)}{J(t)} , \end{align}

whose time integration yields the relation between the time-dependent volume and angular velocity,

(6.7) \begin{align} \frac {\nu (t)}{\nu (0)} = \left ( \frac {J(t)}{J(0)} \right )^{-2/3} . \end{align}

Relation (6.3a ) implies that the background velocity $\boldsymbol{u}_b(\boldsymbol{x},t)$ may be written in terms of the time-integrated divergence $\varDelta (t)$ , or in terms of the angular velocity $\nu (t)$ , as

(6.8a) \begin{align} \boldsymbol{u}_b(\boldsymbol{x},t) &= \dot {\boldsymbol{o}}(t) + \frac {1}{3}\dot {\varDelta }(t)(\boldsymbol{x}-\boldsymbol{o}(t)) + \textrm {e}^{-\tfrac {2}{3}\varDelta (t)} \nu _0 \hat {\boldsymbol{\nu }}_0 \times (\boldsymbol{x}-\boldsymbol{o}(t)) \\[-10pt] \nonumber \end{align}

(6.8b) \begin{align} &= \dot {\boldsymbol{o}}(t) - \frac {1}{2} \frac {\dot {\nu }(t)}{\nu (t)}(\boldsymbol{x}-\boldsymbol{o}(t)) + \nu (t)\hat {\boldsymbol{\nu }}_0 \times (\boldsymbol{x}-\boldsymbol{o}(t)) . \\[12pt] \nonumber \end{align}

In the case of vanishing background divergence, $\delta (t)=0$ , the background flow reduces to a constant rotation and arbitrary time-dependent translation given by $\boldsymbol{o}(t)$ .

The advecting velocity $\boldsymbol{v}_b(\boldsymbol{x},t)$ is

(6.9) \begin{align} \boldsymbol{v}_b(\boldsymbol{x},t) = \boldsymbol{u}_b(\boldsymbol{x},t) + \frac {2\textrm {e}^{-\tfrac {2}{3}\varDelta (t)}\nu _0\hat {\boldsymbol{\nu }}_0}{k(t)} = \boldsymbol{u}_b(\boldsymbol{x},t) + \frac {2\nu (t)}{k(t)}\hat {\boldsymbol{\nu }}_0 . \end{align}

In this case, the deformation gradient reduces to the isotropic expansion/contraction and rotation,

(6.10) \begin{align} \boldsymbol{\mathsf{F}}(t) = \textrm {e}^{\varDelta (t)/3} \boldsymbol{\mathsf{R}}(t) , \end{align}

where the rotation tensor

(6.11) \begin{align} \boldsymbol{\mathsf{R}}(t) \equiv \boldsymbol{\mathsf{R}}_{\hat {\boldsymbol{\nu }}}(\vartheta (t)) = \textrm {e}^{[\hat {\boldsymbol{\nu }}\times ] \vartheta (t)} , \end{align}

expresses the rotation around the constant axis defined by $\hat {\boldsymbol{\nu }}$ by an angle $\vartheta (t)$ given by

(6.12a,b) \begin{align} \vartheta (t) \equiv \int _0^t \nu (\tau ) \, \textrm {d}\tau , \quad \textrm {with} \quad \dot {\vartheta }(t) = \nu (t) . \quad \end{align}

These relations imply that any spatial dependence in the Beltrami field given by a dimensionless scalar term in the form ${\boldsymbol{k}}_0 \boldsymbol{\cdot } \boldsymbol{z}_v(\boldsymbol{x},t)$ , where ${\boldsymbol{k}}_0$ is a constant wavevector, or co-vector, transform as

(6.13) \begin{eqnarray} {\boldsymbol{k}}_0 \boldsymbol{\cdot } \boldsymbol{z}_v(\boldsymbol{x},t) &=& {\boldsymbol{k}}_0 \boldsymbol{\cdot } \big ( \textrm {e}^{-\varDelta (t)/3} \boldsymbol{\mathsf{R}}(t)^{\top }(\boldsymbol{x}-\boldsymbol{o}(0)) \big ) = \textrm {e}^{-\varDelta (t)/3} \left ( \boldsymbol{\mathsf{R}}(t) {\boldsymbol{k}}_0 \right ) \boldsymbol{\cdot } (\boldsymbol{x}-\boldsymbol{o}(0)) \nonumber \\[5pt]&=& {\boldsymbol{k}}{(t)} \boldsymbol{\cdot } (\boldsymbol{x}-\boldsymbol{o}(0)) . \end{eqnarray}

That is, if $\boldsymbol{x}-\boldsymbol{o}(0)$ transforms contravariantly $\boldsymbol{z}_v(\boldsymbol{x},t) = (\partial \boldsymbol{r}_v/\partial \boldsymbol{x}_0) (\boldsymbol{x}-\boldsymbol{o}(0)) = \boldsymbol{\mathsf{F}}(t) (\boldsymbol{x} - \boldsymbol{o}(0))$ , then the covector $\hat {\boldsymbol{k}}_0$ transforms covariantly into the time-dependent covector $\boldsymbol{k}(t)$ defined as

(6.14a,b) \begin{align} \boldsymbol{k}(t) \equiv \textrm {e}^{-\varDelta (t)/3}\boldsymbol{\mathsf{R}}(t) {\boldsymbol{k}}_0 ,\quad \textrm {with} \quad \boldsymbol{k}_0\equiv \boldsymbol{k}(0) , \quad \end{align}

which implies that the solid-body rotation plus isotropic expansion/contraction background flow causes a rotation and an expansion/contraction of the initial wavevector ${\boldsymbol{k}}_0$ , that is, an expansion/contraction and precession of the Beltrami field with an angular frequency $\nu (t)$ modulated by the background expansion/contraction.

Since $\boldsymbol{\mathsf{R}}(t)^{\top } = \boldsymbol{\mathsf{R}}_{\hat {\boldsymbol{\nu }}}(-\vartheta (t))$ is a rotation, it preserves norms and, hence, the norm of $\boldsymbol{z}_v(\boldsymbol{x},t)$ transforms as

(6.15) \begin{align} \Vert \textrm {e}^{-\varDelta (t)/3} \boldsymbol{\mathsf{R}}(t)^{\top } ( \boldsymbol{x}-\boldsymbol{o}(t) ) \Vert = \textrm {e}^{-\varDelta (t)/3} \Vert ( \boldsymbol{x}-\boldsymbol{o}(t) ) \Vert . \end{align}

The flow map $\boldsymbol{R}_v(\boldsymbol{x},t)$ of the advecting velocity $\boldsymbol{v}(\boldsymbol{x},t)$ is a composition of a translation, a rotation and a uniform scaling by $\textrm {exp}(-\varDelta (t)/3)$ . The Laplacian is invariant under rigid motions and under uniform scaling, it transforms by a constant factor, consequently,

(6.16) \begin{align} {\nabla} ^2 \eta (\boldsymbol{x},t) = - k_0^2 \eta (\boldsymbol{x},t) . \end{align}

We may define a time-dependent radial scalar wavenumber $k(t)$ as

(6.17) \begin{align} k(t) \equiv k_0 \textrm {e}^{-\varDelta (t)/3} \end{align}

in such a way that the time evolution of $k_0 \Vert \boldsymbol{x}\Vert$ is expressed as the transformation

(6.18) \begin{align} k_0 \Vert \boldsymbol{x}\Vert \mapsto k(t) \Vert ( \boldsymbol{x}-\boldsymbol{o}(t) ) \Vert . \end{align}

Relations (6.17) and (6.4b ) imply that

(6.19) \begin{align} -\frac {\dot {k}(t)}{k(t)} = \frac {\dot {\varDelta }(t)}{3} = \frac {\delta (t)}{3} = -\frac {1}{2}\frac {\dot {\nu }(t)}{\nu (t)}, \end{align}

and therefore, the relation between angular frequency $\nu (t)$ and wavenumber $k(t)$ is

(6.20) \begin{align} \frac {\nu (t)}{\nu _0} = \frac {k^2(t)}{k_0^2} . \end{align}

As a consequence, the background velocity $\boldsymbol{u}_b(\boldsymbol{x},t)$ (6.8a ) can be written in terms of the radial wavenumber $k(t)$ , or in terms of the angular velocity $\nu (t)$ , as

(6.21a) \begin{align} \boldsymbol{u}_b(\boldsymbol{x},t) &= \dot {\boldsymbol{o}}(t) - \frac {\dot {k}(t)}{k(t)}(\boldsymbol{x}-\boldsymbol{o}(t)) + \frac {k^2(t)}{k_0^2} \nu _0 \hat {\boldsymbol{\nu }}_0 \times (\boldsymbol{x}-\boldsymbol{o}(t)) \\[-12pt] \nonumber \end{align}

(6.21b) \begin{align} &= \dot {\boldsymbol{o}}(t) - \frac {1}{2}\frac {\dot {\nu }(t)}{\nu (t)}(\boldsymbol{x}-\boldsymbol{o}(t)) + \nu (t) \hat {\boldsymbol{\nu }}_0 \times (\boldsymbol{x}-\boldsymbol{o}(t)) . \\[12pt] \nonumber \end{align}

The radial velocity ${\dot {k}(t)}/{k(t)}(\boldsymbol{x}-\boldsymbol{o}(t))$ in (6.21a , b ) is a Hubble flow. This relation to the Hubble flow becomes explicit upon defining a dimensionless spatial scale factor $a(t)$ as

(6.22) \begin{align} a(t) \equiv \frac {k_0}{k(t)} . \end{align}

Using material coordinates $(R,t)$ , the flow map $\boldsymbol{r}_H(R,t)$ of this radial Hubble flow is

(6.23) \begin{align} \boldsymbol{r}_{H}(R,t) = \frac {k_0}{k(t)} R \hat {\boldsymbol{r}} = R a(t) \hat {\boldsymbol{r}} . \end{align}

The velocity of the fluid particles is therefore

(6.24a,b,c) \begin{align} \tilde {\boldsymbol{u}}_H(R,t) = R \dot {a}(t) \hat {\boldsymbol{r}} = D(t) H(t) \hat {\boldsymbol{r}} , \quad D(t)\equiv R a(t) , \quad H(t) \equiv \frac {\dot {a}(t)}{a(t)} , \quad \end{align}

which is the Hubble flow (Barnes et al. Reference Barnes, Francis, James and Lewis2006) in a universe with dimensionless scale factor $a(t)$ . Term $D(t)$ is called the proper distance between galaxies, and (6.24c ) is the definition of the Hubble parameter $H(t)$ . In this fluid mechanics description, the expansion of the space is interpreted as the expansion of the fluid by the background flow in which the Beltrami fields are frozen. The time dependence of the wavenumber $k(t)$ of the Beltrami field is used to describe this space expansion.

For example, we may consider Beltrami flows with spherical geometry in which the radial part is given by the spherical Bessel functions $j_{\ell }(k_0 r)$ and the angular part is given by the spherical harmonics $Y_{\ell }^{m}(\theta ,\varphi )$ . The corresponding function $\eta _0(\boldsymbol{x})$ , simplifying $\boldsymbol{o}(t)=\boldsymbol{0}$ , is

(6.25) \begin{align} \eta _0(\boldsymbol{x}) = \sum _{\ell \ge 0}\sum _{|m|\le \ell } \hat {\eta }_{\ell m} j_{\ell }\left ( k_0 r \right ) Y_{\ell }^{m}(\theta ,\varphi ) , \end{align}

where $\hat {\eta }_{\ell m}$ are constants. The simplest mode $\ell =0$ is

(6.26) \begin{align} \eta _0(\boldsymbol{x})=\hat {\eta }_0 j_0(k_0r) , \end{align}

which, using the CK procedure, leads to a Hicks–Moffatt spherical vortex spatially oriented along $\hat {\boldsymbol{n}}_0$ . The next mode $\ell =1$ , in Cartesian coordinates, can be written as

(6.27) \begin{align} \eta _0(\boldsymbol{x}) = j_{1}\left ( k_0 \Vert \boldsymbol{x} \Vert \right ) {\boldsymbol{k}}_0 \boldsymbol{\cdot } \frac {\boldsymbol{x}}{\Vert \boldsymbol{x}\Vert }. \end{align}

Here, we have used the unnormalised real spherical harmonics of order $\ell =1$ and degree $m=\{-1,0,1\}$ , which are $x/r$ , $y/r$ and $z/r$ , with $r\equiv \Vert \boldsymbol{x}\Vert$ . The arbitrary constant $\hat {\eta }_1$ can be used to normalise the radial wavevector $|\hat {\boldsymbol{k}}_0|=1$ . In this case, the time-dependent seed function $\eta (\boldsymbol{x},t)$ becomes

(6.28) \begin{eqnarray} \eta (\boldsymbol{x},t) &=& \left ( \frac {k(t)}{k_0} \right )^3 j_1 \left ( k(t) \Vert \boldsymbol{x}-\boldsymbol{c}(t)\Vert \right ) \frac {\boldsymbol{k}(t)\boldsymbol{\cdot }(\boldsymbol{x}-\boldsymbol{c}(t))}{|k(t)| \Vert \boldsymbol{x}-\boldsymbol{c}(t)\Vert } \nonumber \\ &=& \left ( \frac {k(t)}{k_0} \right )^3 j_1\left ( k(t) \Vert \boldsymbol{x}-\boldsymbol{c}(t)\Vert \right ) \hat {\boldsymbol{k}}(t)\boldsymbol{\cdot }\frac {\boldsymbol{x}-\boldsymbol{c}(t)} {\Vert \boldsymbol{x}-\boldsymbol{c}(t)\Vert } . \end{eqnarray}

This represents a field $\eta _0(\boldsymbol{x})$ that is displacing with velocity $\boldsymbol{c}(t)$ , whose radial part is expanding/contracting by the presence of the time-dependent wavenumber $k(t)$ in the spherical Bessel function $j_1(\boldsymbol{\cdot })$ , and whose angular part is rotating by the time-dependent unit vector $\hat {\boldsymbol{k}}(t)$ , which describes the precession of the Beltrami field.

The same interpretation may be applied to the Beltrami fields in different geometries, and therefore flow expansion/contraction and flow precession are general properties of the Beltrami fields in the presence of a background velocity field with homogeneous but time-dependent divergence $\delta (t)$ and vorticity $2\boldsymbol{\nu }(t)$ . In this background flow, the angular velocity $\boldsymbol{\nu }(t)$ cannot change direction $\hat {\boldsymbol{\nu }}$ and, therefore, the displacement velocity of the Beltrami field $\boldsymbol{c}(t) = 2\boldsymbol{\nu }(t)/k_0$ cannot change its direction either. Changes in the direction of the displacement velocity of the Beltrami field $\boldsymbol{c}(t)$ are related to the presence of a background deviatoric deformation gradient tensor $\boldsymbol{\mathsf{D}}_0(t) \ne \boldsymbol{\mathsf{0}}$ . An example of this possibility is considered in the next subsection.

6.2. Deviatoric deformation gradient $\boldsymbol{\mathsf{D}}_0(t)$ is diagonalisable in the same basis

We consider in this section a deviatoric tensor $\boldsymbol{\mathsf{D}}_0(t)\ne \boldsymbol{\mathsf{0}}$ and assume the Beltrami field approximation. In the case where the deviatoric deformation gradient tensor $\boldsymbol{\mathsf{D}}_0(t)$ is diagonalisable in the same basis at all times,

(6.29a,b) \begin{align} \boldsymbol{\mathsf{D}}_0(t) = \sum _{i=1}^3 \lambda _i(t) \hat {\boldsymbol{e}}_i \otimes \hat {\boldsymbol{e}}_i ,\quad \textrm {with} \quad \sum _{i=1}^3 \lambda _i(t) =0 , \quad \end{align}

the deviatoric deformation gradient $\boldsymbol{\mathsf{D}}_0(t)$ commutes with itself at all times,

(6.30) \begin{align} [ \boldsymbol{\mathsf{D}}_0(t_1),\boldsymbol{\mathsf{D}}_0(t_2)] \equiv \boldsymbol{\mathsf{D}}_0(t_1)\boldsymbol{\mathsf{D}}_0(t_2) - \boldsymbol{\mathsf{D}}_0(t_2)\boldsymbol{\mathsf{D}}_0(t_1) = \boldsymbol{\mathsf{0}} \, \quad \textrm {for all } t_1,t_2 . \end{align}

This condition implies that we can avoid the time-ordered exponential integral in (3.15) and obtain that the isochoric dependence of the background angular velocity $\boldsymbol{\nu }_{D}(t)$ is given by

(6.31a,b) \begin{align} \boldsymbol{\nu }_D(t) = \textrm {e}^{\unicode{x1D793}_0(t)} \boldsymbol{\nu }(0) ,\quad \textrm {where} \quad \unicode{x1D793}_0(t) \equiv \int _0^t \boldsymbol{\mathsf{D}}_0(\tau ) \, \textrm {d}\tau . \quad \end{align}

This relation provides the dynamical dependence that $\boldsymbol{\nu }_D(t)$ and $\boldsymbol{\mathsf{D}}_0(t)$ must satisfy for the background flow to be a solution to the vorticity equation.

The next step is to obtain the relation between $\boldsymbol{\nu }_D(t)$ and the principal deviatoric stretches $\lambda _i(t)$ . To do so, we define the tensor

(6.32) \begin{align} \boldsymbol{\mathsf{A}}(t) \equiv \boldsymbol{\mathsf{D}}_0(t) - \frac {2}{3}\delta (t)\boldsymbol{\mathsf{I}} , \end{align}

in terms of which the vorticity equation for the background flow is expressed as

(6.33) \begin{align} \dot {\boldsymbol{\nu }}(t) = \boldsymbol{\mathsf{A}}(t) \boldsymbol{\nu }(t) . \end{align}

Defining the angular velocity components along the direction of the principal stretches

(6.34a,b) \begin{align} \nu _i(t) \equiv \boldsymbol{\nu }(t)\boldsymbol{\cdot }\hat {\boldsymbol{e}}_i ,\quad \textrm {so that } \quad \boldsymbol{\nu }(t) = \sum _{i=1}^3 \nu _i(t) \hat {\boldsymbol{e}}_i , \quad\end{align}

we obtain the decoupled ordinary differential equations for the angular velocity components $\nu _i(t)$ along the deviatoric stretching directions,

(6.35) \begin{align} \dot {\nu }_i(t) = \left ( \lambda _i(t) - \frac {2}{3}\delta (t) \right )\nu _i(t) \,\quad \textrm {for } i=1,2,3 , \end{align}

which have the solutions

(6.36a,b) \begin{align} \nu _i(t) = \textrm {e}^{-\tfrac {2}{3}\varDelta (t)} \textrm {e}^{\varLambda _i(t)} \nu _i(0) , \quad \textrm {where} \quad \varLambda _i(t) \equiv \int _0^t \lambda _i(\tau ) \, \textrm {d}\tau . \quad \end{align}

Thus, the time evolution of the background angular velocity vector is given by

(6.37) \begin{align} \boldsymbol{\nu }(t) =\textrm {e}^{-\tfrac {2}{3}\varDelta (t)} \sum _{i=1}^3 \nu _i(0) \textrm {e}^{\varLambda _i(t)} \hat {\boldsymbol{e}}_i , \end{align}

which provides the background angular velocity in terms of the time-integrated principal deviatoric stretches $\varLambda _i(t)$ and time-integrated divergence $\varDelta (t)$ . The background expansion/contraction $\delta (t)$ affects the amplitude of the angular velocity, while the deviatoric stretches affect the components of $\boldsymbol{\nu }(t)$ along the principal deviatoric directions.

The changes of $\nu _i(t)$ related to the changes of $\lambda _i(t)$ are not independent since the deviatoric stretches must satisfy the condition $\lambda _1+\lambda _2+\lambda _3=0$ . To avoid this lack of independence, we may include the isotropic expansion in the effective stretches

(6.38) \begin{align} \bar {\lambda }_i(t) \equiv \lambda _i(t)-\frac {2}{3}\delta (t) , \end{align}

so that

(6.39a,b) \begin{align} \boldsymbol{\nu }(t) = \sum _{i=1}^3 \nu _i(0) \textrm {e}^{\bar {\varLambda }_i(t)} \hat {\boldsymbol{e}}_i ,\quad \textrm {where} \quad \bar {\varLambda }_i(t) \equiv \int _0^t \bar {\lambda }_i(\tau ) \, \textrm {d}\tau , \quad \end{align}

where now the three effective stretches $\bar {\lambda }_i(t)$ are free parameters. Thus, the components of $\boldsymbol{\nu }(t)$ along the principal deviatoric strain directions change exponentially according to ${\exp }(\bar {\varLambda }_i(t))$ . The direction of $\boldsymbol{\nu }(t)$ changes, unless the growth rates $\bar {\lambda }_i(t)$ along each direction are all equal, which would imply an isotropic expansion, or $\boldsymbol{\nu }(t)$ is initially aligned with a single eigenvector $\hat {\boldsymbol{e}}_N$ , so that $\nu _i(0)=0$ , for $i\ne N$ . In any other cases, the background angular velocity vector $\boldsymbol{\nu }(t)$ , and hence the background vorticity vector $\boldsymbol{\omega }_b(t)=2\boldsymbol{\nu }(t)$ , align over time with the fastest-growing eigen-direction $\hat {\boldsymbol{e}}_M$ , that is, along the direction having the largest eigenvalue $\bar {\lambda }_M(t) \equiv \textrm {max}\{ \bar {\lambda }_1(t) , \bar {\lambda }_2(t), \bar {\lambda }_3(t)\}$ . When $\boldsymbol{\nu }(t)$ is initially aligned with a single eigenvector $\hat {\boldsymbol{e}}_N \in \{ \hat {\boldsymbol{e}}_1,\hat {\boldsymbol{e}}_2,\hat {\boldsymbol{e}}_3\}$ , the background angular velocity $\boldsymbol{\nu }(t)=\nu _N(t)\hat {\boldsymbol{e}}_N$ preserves its direction along $\hat {\boldsymbol{e}}_N$ and scales exponentially. If the initial orientation of the background angular velocity does not correspond to the direction of the largest effective stretching $\hat {\boldsymbol{e}}_N \neq \hat {\boldsymbol{e}}_M$ , any small perturbation at a given time $t_1$ $\delta \nu _M(t_1) \ne 0$ involving a change in the background angular velocity $\boldsymbol{\nu }(t_1)$ in the direction of $\hat {\boldsymbol{e}}_M$ equal to $\delta \nu _M(t_1) \hat {\boldsymbol{e}}_M$ , will produce an exponential change of $\delta \nu _M(t_1)$ , and therefore the state $\boldsymbol{\nu }(t) = \boldsymbol{\nu }_N(t)\hat {\boldsymbol{e}}_N$ is unstable.

We consider now a unidirectional velocity of the Beltrami field. In the particular case of isochoric background motion $\delta (t)=0$ , a unidirectional velocity of a Beltrami field described by the drift $\boldsymbol{c}(t)=2\boldsymbol{\nu }(t)/k_0$ would imply

(6.40) \begin{align} \delta (t)=0 ,\quad \lambda _z(t) = -\lambda _x(t) - \lambda _y(t) . \end{align}

Consequently, the time evolution of the components of the angular velocity $\boldsymbol{\nu }(t)$ are

(6.41) \begin{align} \nu _x(t) = \nu _{x}(0) \textrm {e}^{\varLambda _x(t)} ,\quad \nu _x(t) = \nu _{y}(0) \textrm {e}^{\varLambda _y(t)} ,\quad \nu _z(t) = \nu _{z}(0) \textrm {e}^{-\varLambda _x(t)-\varLambda _y(t)} . \end{align}

In this case, unidirectional velocity is possible if the angular velocity vector $\boldsymbol{\nu }(0)$ at $t=0$ is aligned with one of the strain directions $\hat {\boldsymbol{\nu }}=\hat {\boldsymbol{e}}_{\hat {\imath }}$ for some $\hat {\imath }=\{1,2,3\}$ . To avoid the growth of the angular velocity components along the normal strain directions the stretches should be negative, say $\varLambda _x(t)\lt 0$ and say $\varLambda _y(t)\lt 0$ , but this would imply $-\varLambda _x(t)-\varLambda _y(t)\gt 0$ and therefore the exponential growth of $\nu _z(t)$ . However, for compressible background flows, $\delta (t)\ne 0$ , we may set

(6.42) \begin{align} \lambda _x(t)=\lambda _y(t)=\frac {2}{3}\delta (t) ,\quad \lambda _z(t)=-\frac {4}{3}\delta (t) , \end{align}

so that

(6.43) \begin{align} \dot {\nu }_x(t) = \dot {\nu }_y(t) = 0 ,\quad \dot {\nu }_z(t) = - 2 \delta (t)\nu _z(t) , \end{align}

and therefore,

(6.44) \begin{align} \nu _x(t)=\nu _x(0) ,\quad \nu _y(t)=\nu _y(0) ,\quad \nu _z(t) = \nu _z(0) \textrm {e}^{-2\varDelta (t)} . \end{align}

In this case, the isotropic expansion/contraction cancels the normal stretches along the $\hat {\boldsymbol{x}}$ and $\hat {\boldsymbol{y}}$ directions, and increases/decreases the stretch along the $\hat {\boldsymbol{z}}$ direction, and only the $\hat {\boldsymbol{z}}$ component of the angular background velocity $\nu _z(t)$ is time-dependent.

7. Elastic scattering of Beltrami fields in the Beltrami field approximation

Scattering usually refers to the process by which particles or waves are deflected from their straight path due to interactions with other particles or non-uniformities in a medium, involving a change in the direction, and sometimes energy, of the incoming particle or wave. Usually, the medium is a steady inhomogeneous field. However, we may investigate scattering of a Beltrami field $\boldsymbol{\mathcal{U}}(\boldsymbol{x},t)$ by an unsteady homogeneous velocity field $\boldsymbol{u}_b(\boldsymbol{x},t)$ , under the Beltrami field approximation, assuming that, if the amplitude of the Beltrami flow decays to zero at distances far from the Beltrami field centre, as it happens with spherical Beltrami flows, what the Beltrami vortex experiences along its path is a temporal change in the background homogeneous flow. The process of elastic scattering of a Lamb–Chaplygin dipole (Lamb Reference Lamb1895; Chaplygin Reference Chaplygin1903, Reference Chaplygin2007), a particular case of finite size Beltrami flow in two dimensions, by a background potential flow generated by a central vortex, has been numerically investigated in two-dimensional flows (Zoeller & Viúdez Reference Zoeller and Viúdez2023). These results show that the Lamb–Chaplygin dipole experiences small deformations during the elastic scattering interaction.

As a particular example of scattering, we consider the case in which the background angular velocity $\boldsymbol{\nu }(t)$ evolves from an initial value, say $\boldsymbol{\nu }_1 = {\nu }_1\hat {\boldsymbol{e}}_1$ at $t=t_1$ , to a final value $\boldsymbol{\nu }_2 = {\nu }_{2}\hat {\boldsymbol{e}}_2$ at $t = t_2$ . Changes in the background angular velocity require, to be consistent with the vorticity equation, the presence of a deviatoric tensor $\boldsymbol{\mathsf{D}}_0(t)$ and, hence, we take in this section the assumption of the Beltrami field approximation. The objective is to define the background velocity field $\boldsymbol{v}_b(\boldsymbol{x},t)$ that, satisfying the vorticity equation, accelerates the Beltrami field. For simplicity, we assume isochoric background flow $\delta (t)=0$ . Next, we may define a scalar function $s(t)$ such that $s(t_1)=0$ and $s(t_2)=1$ to model a smooth transition between the scattering angular velocities. For example,

(7.1) \begin{align} s_1(t) \equiv \frac {1}{2} \left ( 1 + \tanh (\mu _0 t) \right ) \in (0,1) ,\quad \mu _0 \gt 0 , \end{align}

so that $s(t)\rightarrow 0$ as $t\rightarrow -\infty$ and $s(t)\rightarrow 1$ as $t\rightarrow +\infty$ . Using the functions $s_{\nu }(t)$ and $s_{\theta }(t)$ , the angular velocity component $\nu (t)$ and phase $\theta _{\nu }(t)$ evolve as

(7.2) \begin{align} \nu (t)=\nu _1 + (\nu _2-\nu _1) s_{\nu }(t) ,\quad \theta _{\nu }(t) = \theta _2 + (\theta _2 - \theta _1) s_{\theta }(t) . \end{align}

Next, we define the time-dependent background angular velocity,

(7.3) \begin{align} \boldsymbol{\nu }(t) \equiv \nu (t) \left ( \sin \left ( \theta _{\nu }(t) \right ) \hat {\boldsymbol{x}} \right ) + \left ( \cos \left ( \theta _{\nu }(t) \right ) \hat {\boldsymbol{z}} \right ) . \end{align}

The rates of change of the frequency and direction are defined as

(7.4a,b) \begin{align} \alpha _{\nu }(t) \equiv \frac {\dot {\nu }(t)}{\nu (t)} \,\quad \textrm {and} \quad \beta _{\nu }(t) \equiv \dot {\theta }_{\nu }(t) . \quad \end{align}

We seek solutions $\boldsymbol{\nu }(t)$ to the vorticity equation such that $\alpha _{\nu }(t)\rightarrow 0$ and $\beta _{\nu }(t) \rightarrow 0$ as $t \rightarrow \pm \infty$ . Defining the orthonormal vectors $\{ \hat {\boldsymbol{\nu }}(t), \hat {\boldsymbol{\nu }}_{\bot }(t)\}$ in the $xz$ -plane as

(7.5a,b) \begin{align} \hat {\boldsymbol{\nu }}(t) \equiv \sin (\theta _v(t)) \hat {\boldsymbol{x}} + \cos (\theta _v(t)) \hat {\boldsymbol{z}} ,\quad \hat {\boldsymbol{\nu }}_{\bot }(t) \equiv \cos (\theta _v(t)) \hat {\boldsymbol{x}} - \sin (\theta _v(t)) \hat {\boldsymbol{z}} , \quad \end{align}

the rate of change of $\nu (t)$ and $\hat {\boldsymbol{\nu }}(t)$ are

(7.6a,b) \begin{align} \boldsymbol{\nu }(t) = {\nu }(t) \hat {\boldsymbol{\nu }}(t) \,\quad \textrm {and} \quad \dot {\hat {\boldsymbol{\nu }}}(t) = \dot {\theta }_{\nu }(t) \hat {\boldsymbol{\nu }}_{\bot }(t) , \quad \end{align}

so that the rate of change of $\boldsymbol{\nu }(t)$ is

(7.7) \begin{align} \dot {\boldsymbol{\nu }}(t) = \dot {\nu }(t) \hat {\boldsymbol{\nu }}(t) + {\nu }(t) \dot {\theta }_{\nu }(t) \hat {\boldsymbol{\nu }}_{\bot }(t) . \end{align}

Then, a simple deviatoric strain tensor $\boldsymbol{\mathsf{D}}_0(t)$ satisfying the compatibility condition,

(7.8) \begin{align} \boldsymbol{\mathsf{D}}_0 \boldsymbol{\nu } = \dot {\boldsymbol{\nu }} = \dot {\nu } \hat {\boldsymbol{\nu }} + {\nu } \dot {\theta }_{\nu } \hat {\boldsymbol{\nu }}_{\bot } , \end{align}

may be written, in tensor form, as

(7.9) \begin{align} \boldsymbol{\mathsf{D}}_0 = \alpha _{\nu } \left ( \hat {\boldsymbol{\nu }} \otimes \hat {\boldsymbol{\nu }} - \hat {\boldsymbol{\nu }}_{\bot } \otimes \hat {\boldsymbol{\nu }}_{\bot } \right ) + \beta _{\nu } \left ( \hat {\boldsymbol{\nu }} \otimes \hat {\boldsymbol{\nu }}_{\bot } + \hat {\boldsymbol{\nu }}_{\bot } \otimes \hat {\boldsymbol{\nu }} \right ) . \end{align}

It may be checked out that $\textrm {tr}\boldsymbol{\mathsf{D}}_0=0$ , $\boldsymbol{\mathsf{D}}_0^{\top } = \boldsymbol{\mathsf{D}}_0$ and that $\boldsymbol{\mathsf{D}}_0 \boldsymbol{\nu } = \dot {\boldsymbol{\nu }}$ . Since, for the example of smooth function $s_1(t)$ given by (7.1), its time derivative

(7.10) \begin{align} \dot {s}_1(t)=\frac {\mu _0}{2}\textrm {sech}^2(\mu _0 t) \end{align}

decays at both ends $t \rightarrow \pm \infty$ , we have $\alpha _{\nu }(t)\rightarrow 0$ and $\beta _{\nu }(t)\rightarrow 0$ as $t \rightarrow \pm \infty$ , and therefore, $\boldsymbol{\mathsf{D}}_0(t)\rightarrow \boldsymbol{\mathsf{0}}$ as $t \rightarrow \pm \infty$ . Thus, $\boldsymbol{\mathsf{D}}_0(t)$ is a localised-in-time strain pulse that smoothly re-orients the angular velocity $\boldsymbol{\nu }(t)$ . The velocity gradient $\boldsymbol{\mathsf{L}}$ of the background flow is

(7.11) \begin{align} \boldsymbol{\mathsf{L}} = \alpha _{\nu } \left ( \hat {\boldsymbol{\nu }} \otimes \hat {\boldsymbol{\nu }} - \hat {\boldsymbol{\nu }}_{\bot } \otimes \hat {\boldsymbol{\nu }}_{\bot } \right ) + \beta _{\nu } \left ( \hat {\boldsymbol{\nu }} \otimes \hat {\boldsymbol{\nu }}_{\bot } + \hat {\boldsymbol{\nu }}_{\bot } \otimes \hat {\boldsymbol{\nu }} \right ) + {\nu } \left ( \hat {\boldsymbol{y}} \otimes \hat {\boldsymbol{\nu }}_{\bot } - \hat {\boldsymbol{\nu }}_{\bot } \otimes \hat {\boldsymbol{y}} \right ) . \end{align}

In the $\hat {\boldsymbol{\nu }}$ – $\hat {\boldsymbol{\nu }}_{\bot }$ plane, the deviatoric strain tensor $\boldsymbol{\mathsf{D}}_0(t)$ (7.9) is

(7.12) \begin{align} \boldsymbol{\mathsf{D}}_{0}|_{ {plane}} = \left ( \begin{array}{cc} \alpha _{\nu } & \beta _{\nu } \\ \beta _{\nu } & -\alpha _{\nu } \end{array} \right ) , \end{align}

whose eigenvalues $\lambda _{\pm }(t)$ are

(7.13a,b) \begin{align} \lambda _{\pm }(t) \equiv \pm \sigma _{\nu }(t) ,\quad \textrm {where} \quad \sigma _{\nu } (t) \equiv \sqrt {\alpha _{\nu }^2(t) + \beta _{\nu }^2(t)} . \quad \end{align}

The eigenvector $\hat {\boldsymbol{e}}_M$ for the largest eigenvalue $\lambda _{+}=+\sigma _{\nu }$ is

(7.14) \begin{align} \hat {\boldsymbol{e}}_{\textrm {M}} \propto \beta _{\nu } \hat {\boldsymbol{\nu }} + (\sigma _{\nu }-\alpha _{\nu }) \hat {\boldsymbol{\nu }}_{\bot } . \end{align}

The objective now is to find the inverse map $\boldsymbol{R}_v(\boldsymbol{x},t)$ of the background advecting velocity

(7.15a,b) \begin{align} \boldsymbol{v}_b(\boldsymbol{x},t) = \boldsymbol{\mathsf{D}}_0(t)\boldsymbol{x} + \boldsymbol{\nu }(t)\times \boldsymbol{x} = \boldsymbol{\mathsf{A}}(t)\boldsymbol{x} , \quad \textrm {where} \quad \boldsymbol{\mathsf{A}}(t) \equiv \boldsymbol{\mathsf{D}}_0(t) + [\boldsymbol{\nu }(t)\times ] , \quad \end{align}

or in material variables,

(7.16) \begin{align} \dot {\boldsymbol{r}} = \boldsymbol{\mathsf{A}}(t)\boldsymbol{r} . \end{align}

To find the deformation gradient $\boldsymbol{\mathsf{F}}(t)$ , we work in the time-dependent frame with unit vectors $\{ \hat {\boldsymbol{\nu }}(t),\hat {\boldsymbol{y}},\hat {\boldsymbol{\nu }}_{\bot }(t)\}$ that co-rotates around the $\boldsymbol{y}$ -axis. To do this, we define the rotation tensor $\boldsymbol{\mathsf{Q}}(t)$ about the $\boldsymbol{y}$ -axis, normal to $\hat {\boldsymbol{\nu }}$ and $\hat {\boldsymbol{\nu }}_{\bot }$ , by an angle $\theta _{\nu }(t)$ ,

(7.17) \begin{align} \boldsymbol{\mathsf{Q}}(t) \equiv \boldsymbol{\mathsf{R}}_{\boldsymbol{y}}(\theta _{\nu }(t)) = \left ( \begin{array}{ccc} \cos (\theta _{\nu }(t)) & 0 & \sin (\theta _{\nu }(t)) \\[3pt] 0 & 1 & 0 \\[3pt] -\sin (\theta _{\nu }(t)) & 0 & \cos (\theta _{\nu }(t)) \end{array} \right ) , \end{align}

whose value at $t=t_1$ is $\boldsymbol{\mathsf{Q}}(t_1) = \boldsymbol{\mathsf{R}}_{\boldsymbol{y}}(\theta _1)$ . We introduce the vector $\boldsymbol{\xi }$ in the rotated frame

(7.18a,b) \begin{align} \boldsymbol{\xi } \equiv \boldsymbol{\mathsf{Q}}^{\top } \boldsymbol{r} ,\quad \boldsymbol{r} = \boldsymbol{\mathsf{Q}}\boldsymbol{\xi } \quad , \end{align}

and therefore,

(7.19a,b) \begin{align} \dot {\boldsymbol{\xi }} = \hat {\boldsymbol{\mathsf{A}}} \boldsymbol{\xi } , \quad \hat {\boldsymbol{\mathsf{A}}} \equiv \boldsymbol{\mathsf{Q}}^{\top }\boldsymbol{\mathsf{A}}\boldsymbol{\mathsf{Q}} - \boldsymbol{\mathsf{Q}}^{\top } \dot {\boldsymbol{\mathsf{Q}}} , \quad \end{align}

expresses the rate of change of $\boldsymbol{\xi }$ in terms of the tensor $\hat {\boldsymbol{\mathsf{A}}}(t)$ . In this tilted reference frame, we introduce an orthogonal basis vector $\{ \hat {\boldsymbol{e}}_x, \hat {\boldsymbol{e}}_y,\hat {\boldsymbol{e}}_z\}$ such that the vectors $\hat {\boldsymbol{\nu }}(t) \mapsto \hat {\boldsymbol{e}}_z$ and $\hat {\boldsymbol{\nu }}_{\bot }(t) \mapsto \hat {\boldsymbol{e}}_x$ , and the different terms in (7.19) are

(7.20a) \begin{align} \boldsymbol{\mathsf{Q}}^{\top }\boldsymbol{\mathsf{D}}_0\boldsymbol{\mathsf{Q}} &= \alpha _v(t) \left ( \hat {\boldsymbol{e}}_z\otimes \hat {\boldsymbol{e}}_z -\hat {\boldsymbol{e}}_x\otimes \hat {\boldsymbol{e}}_x \right ) + \beta _v(t) \left ( \hat {\boldsymbol{e}}_z\otimes \hat {\boldsymbol{e}}_x +\hat {\boldsymbol{e}}_x\otimes \hat {\boldsymbol{e}}_z \right ) , \\[-9pt] \nonumber \end{align}

(7.20b) \begin{align} \boldsymbol{\mathsf{Q}}^{\top } [\boldsymbol{\nu }\times ] \boldsymbol{\mathsf{Q}} &= [\nu \hat {\boldsymbol{e}}_z\times ] = \nu \left ( \hat {\boldsymbol{e}}_y\otimes \hat {\boldsymbol{e}}_x -\hat {\boldsymbol{e}}_x\otimes \hat {\boldsymbol{e}}_y \right ) , \\[-9pt] \nonumber \end{align}

(7.20c) \begin{align} \boldsymbol{\mathsf{Q}}^{\top } \dot {\boldsymbol{\mathsf{Q}}} &= [\beta _{\nu } \hat {\boldsymbol{e}}_y\times ] = \beta _v \left ( \hat {\boldsymbol{e}}_x\otimes \hat {\boldsymbol{e}}_z -\hat {\boldsymbol{e}}_z\otimes \hat {\boldsymbol{e}}_x \right ) . \\[12pt] \nonumber \end{align}

Therefore, in the $\{ \hat {\boldsymbol{e}}_x, \hat {\boldsymbol{e}}_y,\hat {\boldsymbol{e}}_z\}$ basis, the tensor $\hat {\boldsymbol{\mathsf{A}}}(t)$ is

(7.21) \begin{align} \hat {\boldsymbol{\mathsf{A}}}(t) = \left [ \begin{array}{ccc} -\alpha _{\nu }(t) & -\nu (t) & 0 \\[3pt] \nu (t) & 0 & 0 \\[3pt] 2\beta _{\nu }(t) & 0 & \alpha _{\nu }(t) \end{array} \right ] . \end{align}

Let

(7.22) \begin{align} \boldsymbol{\xi } = X\hat {\boldsymbol{e}}_x + Y \hat {\boldsymbol{e}}_y + Z\hat {\boldsymbol{e}}_z . \end{align}

Equations (7.19) and (7.21) imply that the equations we need to solve are

(7.23a,b,c) \begin{align} \dot {X}=-\alpha _{\nu } X - \nu Y ,\quad \dot {Y}= \nu X ,\quad \dot {Z}= 2\beta _{\nu } X + \alpha _{\nu }Z . \quad \end{align}

From (7.23a,b ), we obtain a separated relation for $Y(t)$ ,

(7.24) \begin{align} \ddot {Y}(t) + \frac {\dot {\nu }(t)}{\nu (t)}\dot {Y}(t) + \nu (t)^2 Y(t) = 0 . \end{align}

This is the equation for a parametric oscillator, where the time-dependent damping term is $\alpha _{\nu }=\dot {\nu }/\nu$ . After the usual change of variable $\mathcal{Y}(t) \equiv \nu (t)^{1/2} Y(t)$ , (7.24) transforms into

(7.25a,b) \begin{align} \ddot {\mathcal{Y}}(t) + \varOmega ^2(t) \mathcal{Y}(t) = 0 , \quad \textrm {where} \quad \varOmega ^2 \equiv \nu ^2 + \frac {1}{4}\left (\frac {\dot {\nu }}{\nu }\right )^2 - \frac {1}{2} \frac {\ddot {\nu }}{\nu } . \quad \end{align}

The solution to (7.25a ) depends on the particular form of $\nu (t)$ . In the simple case of a linear time dependence,

(7.26a,b) \begin{align} \nu (t) = \nu _1+\mu _0 (t-t_1) ,\quad \mu _0 \equiv \frac {\nu _2 - \nu _1}{t_2-t_1} , \quad \end{align}

and the damping term is

(7.27) \begin{align} \alpha _v(t) = \frac {\dot {\nu }(t)}{\nu (t)} = \frac {\mu _0}{\nu (t)} . \end{align}

In the case $\nu _2\ne \nu _1$ , the oscillator equation (7.24) simplifies to

(7.28) \begin{align} \ddot{Y}(t)+\frac{1}{t-t_*}\dot{Y}(t) +\mu_0^2 (t-t_*)^2 Y(t)=0,\quad \textrm{with} \quad t_* \equiv t_1 - \nu_1/\mu_0 , \end{align}

and

(7.29) \begin{align} \frac{\dot{\nu}(t)}{\nu(t)}=\frac{1}{t-t_*}.\end{align}

Defining $z\equiv (\mu_0/2)(t-t_*)^2$ , (7.28) transforms into

(7.30) \begin{align} z^2 \frac{d^2Y}{dz^2} + z \frac{dY}{dz} + z^2 Y =0, \end{align}

which is the Bessel equation of order zero, and therefore the fundamental solutions are

(7.31) \begin{align} Y(t) = c_1 J_0\left(\frac{\mu_0}{2}(t-t_*)^2\right) + c_2 Y_0 \left(\frac{\mu_0}{2}(t-t_*)^2\right). \end{align}

The simplest case, however, is that in which the amplitude of the background angular velocity $\nu (t)$ is constant, that is, $\nu _2=\nu _1=\nu _0$ , and the damping term vanishes $\alpha _{\nu }(t)=0$ . In this case, the parametric oscillator (7.24) simplifies to the undamped oscillator,

(7.32) \begin{align} \ddot {Y}(t) + \nu _0^2 Y(t) = 0 . \end{align}

For undamped oscillations, the deformation gradient $\boldsymbol{\mathsf{F}}_{R}(t;t_1)$ in the rotated reference frame is

(7.33) \begin{align} \boldsymbol{\mathsf{F}}_{R}(t;t_1) \equiv \left [ \begin{array}{ccc} \cos \varTheta (t) & -\sin \varTheta (t) & 0 \\[3pt] \sin \varTheta (t) & \cos \varTheta (t) & 0 \\[3pt] A(t) & B(t) & 1 \end{array} \right ] ,\quad \varTheta (t)\equiv \nu _0(t-t_1) ,\quad \textrm {det} \boldsymbol{\mathsf{F}}_{R}=1 , \end{align}

where the terms depending on the rate of change of the scattering angle $\beta _{\nu }(t)\equiv \dot {\theta }_{\nu }(t)$ are

(7.34a) \begin{align} A(t) &\equiv 2 \int _{t_1}^t \beta _{\nu }(\tau ) \cos (\nu _0(\tau -t_1))\, \textrm {d}\tau , \\[-10pt] \nonumber \end{align}

(7.34b) \begin{align} B(t) &\equiv -2 \int _{t_1}^t \beta _{\nu }(\tau ) \sin (\nu _0(\tau -t_1))\, \textrm {d}\tau . \\[12pt] \nonumber \end{align}

Then, the solution to (7.19) is

(7.35) \begin{align} \boldsymbol{\xi }(t) = \boldsymbol{X}(t;t_1) = \boldsymbol{\mathsf{F}}_R(t;t_1) \boldsymbol{X}_0 . \end{align}

In the co-rotating reference frame the effect of the deformation gradient $\boldsymbol{\mathsf{F}}_R(t;t_1)$ on the $XY$ -plane is a planar rotation with constant angular rate $\nu _0$ ,

(7.36a) \begin{align} X(t) &= X_0 \cos \varTheta (t) -Y_0\sin \varTheta (t) , \\[-10pt] \nonumber \end{align}

(7.36b) \begin{align} Y(t) &= X_0 \sin \varTheta (t) + Y_0\cos \varTheta (t) , \\[10pt] \nonumber \end{align}

where $\{ X_0,Y_0,Z_0\}=\boldsymbol{\xi }(0)=\boldsymbol{\mathsf{Q}}^{\top }(0)\boldsymbol{x}_0$ . The coordinate $Z(t)$ of the particle along the instantaneous rotation axis $\hat {\boldsymbol{\nu }}(t)$ is

(7.37) \begin{align} Z(t)= Z_0 + 2\int _{t_1}^t \beta _{\nu }(\tau ) \left ( X_0 \cos \varTheta (\tau ) -Y_0\sin \varTheta (\tau ) \right ) \, \textrm {d}\tau . \end{align}

In fact, $Z(t)=\boldsymbol{\xi }(t)\boldsymbol{\cdot }\hat {\boldsymbol{e}}_z =\hat {\boldsymbol{\nu }}(t) \boldsymbol{\cdot } \boldsymbol{r}(t)$ . The temporal change of $Z(t)$ is due to temporal change of angular velocity direction, $\beta _{\nu }(t) \equiv \dot {\theta }_{\nu }(t)$ . Term $\beta _{\nu }(t)$ is associated with a tilt-induced shear. The term $2\beta _{\nu }X$ transforms transverse offset $X$ , the coordinate along $\hat {\boldsymbol{e}}_x \mapsto \hat {\boldsymbol{\nu }}_{\bot }$ , into axial motion. When the axis tilts ( $\beta _{\nu }\neq 0$ ), points off the axis are pumped along $\hat {\boldsymbol{\nu }}$ . Thus, $Z(t)$ , is the projection of $\boldsymbol{r}(t)$ on the instantaneous angular-velocity direction.

We need the inverse deformation gradient $\boldsymbol{\mathsf{F}}_{R}(t;t_1)^{-1}$ in the rotated frame, which, from (7.33), is

(7.38) \begin{align} & \boldsymbol{\mathsf{F}}_{R}(t;t_1)^{-1} \nonumber \\ & \equiv \left [ \begin{array}{ccc} \cos \varTheta (t) & \sin \varTheta (t) & 0 \\[3pt] -\sin \varTheta (t) & \cos \varTheta (t) & 0 \\[3pt] -A(t)\cos \varTheta (t)+B(t)\sin \varTheta (t) & -A(t)\sin \varTheta (t)-B(t)\cos \varTheta (t) & 1 \end{array} \right ] . \end{align}

Finally, the solutions, in the laboratory reference frame, for the deformation gradient and the flow map are

(7.39a,b) \begin{align} \boldsymbol{\mathsf{F}}(t) = \boldsymbol{\mathsf{Q}}(t)\boldsymbol{\mathsf{F}}_R(t;t_1) \boldsymbol{\mathsf{Q}}^{\top }(t_1) ,\quad \boldsymbol{r}_v(\boldsymbol{x}_0,t) = \boldsymbol{\mathsf{F}}(t)\boldsymbol{x}_0 , \quad \end{align}

and the inverse deformation gradient and the inverse flow map are

(7.40a,b) \begin{align} \boldsymbol{\mathsf{F}}^{-1}(t) = \boldsymbol{\mathsf{Q}}(t_1) \boldsymbol{\mathsf{F}}_R(t;t_1)^{-1} \boldsymbol{\mathsf{Q}}^{\top }(t) ,\quad \boldsymbol{R}_v(\boldsymbol{x},t) = \boldsymbol{\mathsf{F}}(t)^{-1}\boldsymbol{x} . \quad \text{(a,b)} \end{align}

The inverse map $\boldsymbol{R}_v(\boldsymbol{x},t)$ in (7.40b ) is

(7.41) \begin{align} \boldsymbol{R}_v(\boldsymbol{x},t) = \boldsymbol{\mathsf{R}}_{\boldsymbol{y}}(\theta _{\nu }(t_1)) \boldsymbol{\mathsf{F}}_R(t;t_1)^{-1} \boldsymbol{\mathsf{R}}_{\boldsymbol{y}}(-\theta _{\nu }(t)) \boldsymbol{x} , \end{align}

which is the desired inverse flow map of the advecting flow $\boldsymbol{v}_b(\boldsymbol{x},t)$ as a function of the initial $\theta _1$ and final $\theta _2$ scattering angles.

We provide next a graphical example of elastic scattering produced by a background velocity pulse with parameters $\theta _1=\pi /2$ and $\theta _2=\pi /4$ . The scattering angle difference $\varDelta \theta \equiv |\theta _2-\theta _1|=\pi /4$ has been taken excessively large to visualise the deformation from the Beltrami initial flow. The initial time is $t_1=0$ and the final time is $t_2=1$ . The angular frequency is $\nu _0\equiv 2\pi$ , so that the pulse lasts an interval time of one cycle. The wavenumber $k_0=1$ . The background angular velocity is

(7.42) \begin{align} \boldsymbol{\nu }(t) = 2\pi \left ( \cos \left (\frac {\pi }{4}t\right ) \hat {\boldsymbol{x}} + \sin \left (\frac {\pi }{4}t\right ) \hat {\boldsymbol{z}} \right ) , \quad t\subset [0,1]. \\[-28pt] \nonumber \end{align}

(7.43a,b,c) \begin{align} \boldsymbol{\nu }(t_1) = \nu _0 \hat {\boldsymbol{x}} \,\quad \textrm {and} \quad \boldsymbol{\nu }(t_2) = \frac {\sqrt {2}}{2}\nu _0 (\hat {\boldsymbol{x}}+\hat {\boldsymbol{y}}) ,\quad |\boldsymbol{\nu }(t_1)| =|\boldsymbol{\nu }(t_2)| =|\nu _0| . \\[10pt] \nonumber \end{align}

The scattering angle $\theta (t)$ and its rate of change $\dot {\theta }(t)$ are

(7.44) \begin{align} \theta (t) \equiv \theta _1 + {\beta }_v(t-t_1) ,\quad \dot {\theta } =\beta _v = \frac {\theta _2-\theta _1}{t_2-t_1} = -\frac {\pi }{4} . \end{align}

The rotation tensors are

(7.45) \begin{align} \boldsymbol{\mathsf{Q}}(t) = \left ( \begin{array}{ccc} \sin (\pi t/4) & 0 & \cos (\pi t/4) \\[4pt] 0 & 1 & 0 \\[4pt] -\cos (\pi t/4) & 0 & \sin (\pi t/4) \end{array} \right ) ,\quad \boldsymbol{\mathsf{Q}}(0) = \left ( \begin{array}{ccc} 0 & 0 & 1 \\ 0 & 1 & 0 \\ -1 & 0 & 0 \end{array} \right ) , \end{align}

and the $Z$ -stretchings in the tilted reference frame are

(7.46a,b) \begin{align} A(t;t_1) = - \frac {1}{4} \sin (2\pi t) \,\quad \textrm {and} \quad B(t;t_1) = \frac {1}{2} \left ( \sin (\pi t) \right )^2 . \end{align}

The deformation gradient $\boldsymbol{\mathsf{F}}_{R}(t)$ in the tilted reference frame is

(7.47) \begin{align} \boldsymbol{\mathsf{F}}_{R}(t) = \left ( \begin{array}{ccc} \cos ( 2 \pi t) & -\sin ( 2 \pi t) & 0 \\[4pt]\sin ( 2 \pi t) & \cos ( 2 \pi t) & 0 \\[4pt]-\frac {1}{4}\sin (2\pi t) & \frac {1}{2}\sin ^2(\pi t)) & 1 \end{array} \right ) , \end{align}

and its inverse $\boldsymbol{\mathsf{F}}_{R}(t)^{-1}$ is

(7.48) \begin{align} \boldsymbol{\mathsf{F}}_{R}(t)^{-1} = \left ( \begin{array}{ccc} \cos ( 2 \pi t) & \sin ( 2 \pi t) & 0 \\[4pt] -\sin ( 2 \pi t) & \cos ( 2 \pi t) & 0 \\[4pt] \frac {1}{4}\sin (2\pi t) & \frac {1}{2}\sin ^2(\pi t)) & 1 \end{array} \right ) , \end{align}

and, finally, the inverse map of the advecting flow $\boldsymbol{v}_b(\boldsymbol{x},t)$ is

(7.49) \begin{align} \boldsymbol{R}_v(\boldsymbol{x},t) = \boldsymbol{\mathsf{Q}}(0){\boldsymbol{\mathsf{F}}_R(t)}^{-1}\boldsymbol{\mathsf{Q}}(t)^{\top } \boldsymbol{x} , \quad t\subset [0,1] . \end{align}

Figure 1.

Time evolution of the distributions of $\eta (0,y,z,t)$ (left column), $\eta (x,0,z,t)$ (central column) and $\eta (x,y,0,t)$ (right column), in the case the spherical spin $\ell =1$ vortex dipole is initially aligned along the $z$ -axis, that is, $\hat {\boldsymbol{k}}_0=\hat {\boldsymbol{z}}$ , at the initial time $t=0$ (top row), $t=1/2$ (middle row) and $t=1$ (bottom row). The initial distribution of $\eta (x,y,0,0)=0$ is not included. We notice the evolution of $\eta (x,0,z,t)$ in the $x$ – $z$ plane (mid column) is just a rotation of the field. The vortex recovers its spherical geometry at $t=1$ (bottom row). These distributions are plotted in the frame in which $\boldsymbol{c}(t)=\boldsymbol{0}$ .

Figure 2.

Same as figure 1, but in the case the spherical spin $\ell =1$ vortex dipole is initially aligned along the $x$ -axis, that is, $\hat {\boldsymbol{k}}_0=\hat {\boldsymbol{x}}$ . The initial distribution of $\eta (0,y,z,0)=0$ is not included.

Figure 3.

Isosurfaces of (a) $\eta (x_0,y,z,t)=\pm 0.2$ for $x_0=0$ in the $(y,z,t)$ space in the case the spherical spin $\ell =1$ vortex dipole is initially aligned along the $z$ -axis ( ${\boldsymbol{k}}_0=\hat {\boldsymbol{z}}$ ), and (b) $\eta (x,y_0,z,t)=\pm 0.2$ for $y_0=0$ in the case the vortex dipole is initially aligned along the $x$ -axis ( ${\boldsymbol{k}}_0=\hat {\boldsymbol{x}}$ ).

In the following figures, we represent contours or isosurfaces of $\eta (\boldsymbol{x},t)\equiv \eta _{s1}(\boldsymbol{R}_v(\boldsymbol{x},t))$ where the seed function,

(7.50) \begin{align} \eta _{S1}(\boldsymbol{x}) = j_1(k_0\Vert \boldsymbol{x}\Vert ) \boldsymbol{k}_0 \boldsymbol{\cdot }\frac {\boldsymbol{x}}{\Vert \boldsymbol{x}\Vert } , \end{align}

is representative of the spherical Beltrami vortex of order $\ell =1$ (Viúdez Reference Viúdez2025c ). The order $\ell =0$ function $\eta _{S0}(\boldsymbol{x}) = j_0(k_0\Vert \boldsymbol{x}\Vert )$ is also representative of the spherical Beltrami vortex of order $\ell =1$ , but the precession of the vortex, that is, the rotation of the wavevector $\boldsymbol{k}(t)$ , and its initial orientation is better described by $\eta _{S1}(\boldsymbol{x})$ .

We visualise two examples of scattering. In the first example (figures 1, 3(a) and 4), the vortex is initially aligned along the $\hat {\boldsymbol{z}}$ -axis, that is, $\boldsymbol{k}_0=\hat {\boldsymbol{z}}$ and therefore $\boldsymbol{k}_0 \perp \boldsymbol{\nu }(0)$ , and the vortex precesses around the axis defined by $\boldsymbol{\nu }(t)$ , which is its instantaneous direction of propagation. In the second example (figures 2, 3(b) and 5), the vortex is initially aligned along the $\hat {\boldsymbol{x}}$ -axis, that is, $\boldsymbol{k}_0=\hat {\boldsymbol{x}}$ , and therefore $\boldsymbol{k}_0 \parallel \boldsymbol{\nu }(0)$ , and the vortex does not precess around its direction of propagation.

The initial ( $t=0$ ), middle ( $t=1/2$ ) and final ( $t=1$ ) distributions of $\eta (\boldsymbol{x},t)$ on different planes (figures 1 and 2) display the distortion of the spherical vortex at $t=1/2$ , which implies its departure from a Beltrami field, and its recovery at the end of the background pulse. The final state is the same spherical Beltrami field, but whose wavevector has rotated $\pi /4$ in the $x$ – $z$ plane. The vortex precession is visualised as a double spiral isosurfaces of $\eta (\boldsymbol{x},t)=\pm \eta _0$ in the space $(y,z,t)$ in figure 3(a), The vortex departure from its initial spherical Beltrami configuration is visualised better in the non-precessing case from isosurfaces of $\eta (\boldsymbol{x},t)=\pm \eta _0$ in the space $(y,z,t)$ in figure 3(b). The vortex scattering, in both examples, can be visualised from the superposition of isosurfaces of $\eta (\boldsymbol{x},t_i)=\pm \eta _0$ at different times $t_i$ in figures 4 and 5. The large deformation of the isosurfaces at $t=1/2$ , relative to those at $t=0$ and $t=1$ , is due to the large differential scattering angle $\varDelta \theta =\pi /2$ prescribed. Smaller values of $\varDelta \theta$ , typical of forward scattering, would provide much smaller deformations.

Figure 4.

(a) Front and (b) top views of the superposition of isosurfaces of $\eta (x,y,z,t)=\pm 0.3$ in the $(x,y,z)$ space at three different times $t\in \{0, 1/2,1\}$ for the case where the spherical spin $\ell =1$ vortex dipole is initially aligned along the $z$ -axis.

Figure 5.

(a) Front and (b) top views as in figure 4, but for the case where the spherical spin $\ell =1$ vortex dipole is initially aligned along the $x$ -axis.