2.1 Taylor’s solution

Taylor (Reference Taylor1922) considered the axisymmetric flow around a non-precessing sphere (in the laboratory frame) that is translating steadily with non-dimensional velocity $\hat{\boldsymbol{z}}$ through a fluid rotating steadily about $\hat{\boldsymbol{z}}$ . This flow may be described by defining an axisymmetric streamfunction, $\unicode[STIX]{x1D713}(\unicode[STIX]{x1D70E},\unicode[STIX]{x1D703})$ , as in Batchelor (Reference Batchelor1967), and writing the velocity in the radial, polar and azimuthal directions respectively as

(The second term in brackets in the expression for $w$ is included here as we are working in the non-inertial frame of reference of the rotating fluid.) The incompressibility condition (2.1a ) is automatically satisfied. Taylor (Reference Taylor1922) proceeded by posing that the streamfunction be of the separable form $\unicode[STIX]{x1D713}=f(\unicode[STIX]{x1D70E})\sin ^{2}\unicode[STIX]{x1D703}$ . Substitution into the curl of (2.1b ), thus removing the pressure gradient, leads to either the trivial solution $f=0$ or that $f$ must satisfy

where a dash denotes differentiation with respect to $\unicode[STIX]{x1D70E}$ . In the frame of reference rotating with the fluid, but translating with a sphere of radius $\unicode[STIX]{x1D6FF}$ that is not precessing in the laboratory frame, the no-normal flow and no-precession boundary conditions on the boundary of the sphere, together with the far-field velocity boundary condition, are

Equivalently, in terms of $f$

The pressure field is given by

where the pressure $p\rightarrow (8Ro^{2})^{-1}\unicode[STIX]{x1D70E}^{2}\sin ^{2}\unicode[STIX]{x1D703}$ , the hydrostatic pressure field, as $\unicode[STIX]{x1D70E}\rightarrow \infty$ . The vorticity field, $\unicode[STIX]{x1D74E}\equiv \unicode[STIX]{x1D735}\times \boldsymbol{u}$ , is

The solution to (2.3) that satisfies (2.4d,e ) may be written as

for an arbitrary constant $c$ . The remaining no-slip boundary condition in the polar direction, $v(\unicode[STIX]{x1D6FF},\unicode[STIX]{x1D703})=0$ , is satisfied when $f^{\prime }(\unicode[STIX]{x1D6FF})=0$ , forcing $c=\unicode[STIX]{x1D6FF}^{2}-3Ro^{2}$ , and this together with (2.7) can be shown to be equal to Taylor’s solution. The no-precession condition and the no-slip boundary condition in the polar direction are physically motivated choices to close the system. The conditions are no-slip conditions applied to an inviscid fluid in the expectation that the inviscid solution will closely approximate the full viscous solution where no-slip conditions would be rigorously valid.

Figure 2. (a) Streamlines around a unit sphere translating steadily along the axis of rotation ( $r=0$ ) of a rotating fluid in the reference frame of the sphere. Closed streamlines show that fluid is being transported with the sphere. (b) Streamlines around a sphere of vanishing size (white circle at the origin) translating steadily along this axis of rotation of a rotating fluid in the frame of reference of the main body of fluid. Taylor (Reference Taylor1922) observed that the streamlines near the origin resemble those of a Hill’s spherical vortex (Hill Reference Hill1894) but described the analogy between the flow and a spherical vortex as ‘only superficial’.

The streamlines for $\unicode[STIX]{x1D6FF}=1$ , $Ro=(2\unicode[STIX]{x03C0})^{-1}$ are shown in figure 2(a). The nonlinear wavefield in the fluid can be observed (see figure 2 Taylor Reference Taylor1922) as can the closed streamlines that show that fluid is carried with the sphere. Taylor observed that it is not clear how such a flow may be set up; indeed the radiation condition is not everywhere satisfied throughout the wavefield, which prohibits the wavefield being created by the towing of the sphere alone. This observation is consistent with the later findings of Stewartson (Reference Stewartson1958) and Lighthill (Reference Lighthill1967), who both considered the wavefields built up by motion due to localized forcing along the axis of a rotating fluid into initially quiescent fluid and concluded that only columnar modes could exist ahead of the forcing.

Taylor (Reference Taylor1922) observed that his analytical solution still exhibited waves even in the limit that the sphere has a vanishingly small radius, i.e. in the limit $\unicode[STIX]{x1D6FF}\rightarrow 0$ . Figure 2(b) shows this solution, in the laboratory frame of reference. The sphere is instantaneously located at the origin, indicated by a white circle. Taylor (Reference Taylor1922) observed that the streamlines near the origin resemble those of a Hill’s spherical vortex (Hill Reference Hill1894). Here the rotation rate has been chosen such that this apparent spherical vortex has a unit radius (given by the largest $Ro$ that satisfies $Ro\sin (Ro^{-1})=\cos (Ro^{-1})$ , i.e. $Ro\approx 0.223$ ). The boundary of the apparent spherical vortex is shown in bold. Taylor (Reference Taylor1922) describes this analogy with Hill’s spherical vortex as superficial as the vortex can only exist when the flow is rotating and a vanishingly small sphere is translating steadily, without precession, along the axis of rotation.

2.2 Hill’s spherical vortex in a rotating fluid

The classical Hill’s spherical vortex is a spherical region of rotational fluid that propagates through an irrotational ambient fluid. The flow is found by constructing solutions inside and outside the spherical vortex and enforcing pressure and velocity continuity across the boundary of the two regions. As with the no-slip conditions enforced on the solid sphere in Taylor’s solution, continuity of velocity is enforced across the boundary of the two inviscid solutions, even though this is not a strict requirement, in the expectation that the solution will closely approximate the behaviour of a real viscous fluid. In a frame of reference moving with the spherical vortex, Hill’s solution is given by

where the arbitrary pressure constant has been chosen without loss of generality such that $p\rightarrow 0$ as $\unicode[STIX]{x1D70E}\rightarrow \infty$ . The solution is in the frame of reference of the spherical vortex and so the velocity in the far field tends to $-\hat{\boldsymbol{z}}$ . The solution is axisymmetric and swirl-free and the velocity can be represented by a streamfunction, $\unicode[STIX]{x1D713}$ , of the form of (2.2a,b ), where

and $w=0$ . The arbitrary constant that may be added to the streamfunction is chosen such that $\unicode[STIX]{x1D713}=0$ on the boundary of the spherical vortex. The pressure and velocity fields are continuous across the boundary of the spherical vortex, $\unicode[STIX]{x1D70E}=1$ .

We now make the following observation; if $\boldsymbol{u}(\unicode[STIX]{x1D70E},\unicode[STIX]{x1D703})$ together with a corresponding pressure field, $p(\unicode[STIX]{x1D70E},\unicode[STIX]{x1D703})$ , solves the non-rotating Euler equations and $\boldsymbol{u}$ can be represented by a streamfunction $\unicode[STIX]{x1D713}(\unicode[STIX]{x1D70E},\unicode[STIX]{x1D703})$ in the form of (2.2a,b ) with $w=0$ , then

solves the rotating Euler equations with the same pressure field $p$ . This is because the swirling component in (2.10), $w=-(2Ro)^{-1}\unicode[STIX]{x1D70E}\sin \unicode[STIX]{x1D703}$ , exactly cancels the background rotation of the fluid. Thus, we have that (2.10) with $\unicode[STIX]{x1D713}$ given by Hill’s inner non-rotating solution ((2.9) for $\unicode[STIX]{x1D70E}\leqslant 1$ ), and the pressure, $p$ ((2.8b ) for $\unicode[STIX]{x1D70E}\leqslant 1$ ) automatically satisfies the incompressibility condition and the rotating equations of motion (2.1a,b ). The solution is Hill’s non-rotating spherical vortex described in a rotating frame of reference. To distinguish this solution from the classical solution we refer to it as the ‘swirling’ Hill’s spherical vortex, even though the azimuthal component of the velocity field exactly cancels the background rotation of the fluid. The observation that Taylor’s sphere in his experiments did not precess about the axis of rotation gives rise to the possibility that there may exist a form of Taylor’s solution that can be matched onto the swirling Hill’s spherical vortex by enforcing a different choice of polar velocity boundary condition to Taylor’s no-slip condition when setting $c$ in (2.7).

To match a solution of the form (2.2) onto the swirling Hill’s spherical vortex such that the velocity field is continuous across the boundary $\unicode[STIX]{x1D70E}=1$ , we require a solution to (2.3) that satisfies, for $\unicode[STIX]{x1D6FF}=1$ , the conditions (2.4d,e ) and yields $v(1,\unicode[STIX]{x1D703})=(3/2)\sin \unicode[STIX]{x1D703}$ . We therefore require $f^{\prime }(1)=-3/2$ , and hence the solution is given by (2.7) with $\unicode[STIX]{x1D6FF}=1$ , $c=1$ , so that

Substitution of this solution at $\unicode[STIX]{x1D70E}=1$ into (2.5) shows that the pressure on the boundary is given by $p(1,\unicode[STIX]{x1D703})=1/2-(9/8)\sin ^{2}\unicode[STIX]{x1D703}$ , exactly matching the pressure on the boundary of the swirling Hill’s spherical vortex (see (2.8b ) at $\unicode[STIX]{x1D70E}=1$ ). The solution has the property that Hill’s classical solution is recovered in the limit $Ro\rightarrow \infty$ . Thus, we have a complete steady solution to the nonlinear rotating Euler equations whose inner solution is Hill’s spherical vortex with an additional swirling component in the azimuthal direction that cancels out the background rotation of the fluid. This inner solution matches onto an outer solution, with continuous velocity and pressure across the boundary of the vortex. In the far field the velocity tends to the free-stream velocity $-\hat{\boldsymbol{z}}$ and pressure tends to the hydrostatic pressure field $p=(8Ro^{2})^{-1}\unicode[STIX]{x1D70E}^{2}\sin ^{2}\unicode[STIX]{x1D703}$ . We observe that $\unicode[STIX]{x1D713}$ is even in the Rossby number and so the waves that form in a meridional plane ( $\unicode[STIX]{x1D719}=\text{const.}$ ) oscillate according only to the magnitude of the rotation of the system, and not the sign of the direction of rotation, as might be expected on physical grounds. Similarly, as a result of (2.2c ) and (2.11), the azimuthal velocity is odd in the Rossby number and so the azimuthal flow field is reversed if the sign of the direction of rotation of the system is reversed. We see from (2.11) that the wavelength of the inertial waves in the outer fluid is $2\unicode[STIX]{x03C0}Ro$ , as in Taylor (Reference Taylor1922).

Figure 3. (a–c) Plots of the streamfunction for Hill’s spherical vortex for $Ro=\infty$ , $1/4$ , $1/10$ , respectively. The meridional stagnation point in the spherical vortex is indicated by a small black circle and is located at $r=2^{-1/2}$ , $z=0$ . In panel $(c)$ the formation of closed streamlines externally to the spherical vortex may be observed, indicating regions of fluid that are transported with the spherical vortex. (d–f) Plots of the streamfunction for swirling spherical vortices for $Ro=1/10$ . The parameters shown are: (d) $\unicode[STIX]{x1D6FC}=0$ , $\unicode[STIX]{x1D706}=15/2$ (Hill’s spherical vortex); (e) $\unicode[STIX]{x1D6FC}=\unicode[STIX]{x03C0}/2$ , $\unicode[STIX]{x1D706}=15/2$ ; (f) $\unicode[STIX]{x1D6FC}=\unicode[STIX]{x03C0}/2$ , $\unicode[STIX]{x1D706}=15$ .

Figure 3(a–c) shows streamlines in the $(r,z)$ meridional plane of Hill’s spherical vortex and the flow outside the spherical vortex for three different values of the Rossby number. The streamlines represent the flow relative to the translating spherical vortex. The meridional velocity components are always zero at $r=2^{-1/2}$ , $z=0$ , indicated by black circles. Panel (a) is of the non-rotating Hill’s spherical vortex that corresponds to the limit $Ro\rightarrow \infty$ . As the rate of rotation is increased, and the Rossby number reduces from $Ro=1/4$ in panel (b) to $Ro=1/10$ in panel ( $c$ ), inertial waves can be observed in the outer fluid. Below a critical Rossby number it can be seen that these waves begin to overturn. We also observe that closed streamlines appear in the outer fluid in panel (c). These closed streamlines show that above a critical rotation rate, regions of fluid are transported with the spherical vortex, externally to the spherical vortex in the form of concentric vortex rings.

2.3 Swirling spherical vortices in a rotating fluid

We now generalize the method of § 2.2 for the family of swirling spherical vortices identified by Moffatt (Reference Moffatt1969). The spherical vortex, for $\unicode[STIX]{x1D70E}\leqslant 1$ , is described in terms of a streamfunction by

As with Hill’s solution, the streamfunction has the same separable form as Taylor’s streamfunction, specifically $\unicode[STIX]{x1D713}=F(\unicode[STIX]{x1D70E})\sin ^{2}\unicode[STIX]{x1D703}$ , where for $(\unicode[STIX]{x1D706},\unicode[STIX]{x1D6FC})\in \mathbb{R}^{2}$ we have the doubly infinite family of spherical vortices given by

The corresponding pressure field is given by (2.5) with $f$ replaced by $F$ and $Ro$ replaced by $\unicode[STIX]{x1D6FC}^{-1}$ . It follows from (2.12) that $\unicode[STIX]{x1D6FC}$ is a measure of the degree of swirl in the spherical vortex. The special case of Hill’s spherical vortex ring is recovered in the limit $\unicode[STIX]{x1D6FC}\rightarrow 0$ , $\unicode[STIX]{x1D706}=15/2$ . No-normal flow and no-precession conditions are satisfied on the boundary of the swirling spherical vortices as $F(1)=0$ , and hence $u(1,\unicode[STIX]{x1D703})=0$ and $w(1,\unicode[STIX]{x1D703})=-(2Ro)^{-1}\sin \unicode[STIX]{x1D703}$ . The conditions of continuity of pressure and polar velocity across the boundary are satisfied if a solution for $f$ , given by (2.7) with $\unicode[STIX]{x1D6FF}=1$ , can be found such that $f^{\prime }(1)=F^{\prime }(1)$ , and this condition is satisfied when

(Note that the result of § 2.2, $c=1$ , is recovered in the limit $\unicode[STIX]{x1D6FC}\rightarrow 0$ , $\unicode[STIX]{x1D706}=15/2$ .) Hence, any swirling spherical vortex given by (2.12) and (2.13) may be matched onto an oncoming stream in a rotating fluid whose far-field velocity tends to $-\hat{\boldsymbol{z}}$ with continuous pressure and velocity across the boundary of the spherical vortex at $\unicode[STIX]{x1D70E}=1$ . We also see that the outer flow is a singly infinite family given by (2.14) and is equal to the outer flow of the swirling Hill’s spherical vortex solution for all solutions with $c=1$ , i.e. for all $(\unicode[STIX]{x1D706},\unicode[STIX]{x1D6FC})\in \mathbb{R}^{2}$ such that the quantity in square brackets in (2.14) is zero.

Figure 3(d–f) shows streamlines in the $(r,z)$ meridional plane of the swirling spherical vortices for $Ro=1/10$ with $\unicode[STIX]{x1D6FC}=0$ , $\unicode[STIX]{x1D706}=15/2$ (Hill’s swirling spherical vortex), $\unicode[STIX]{x1D6FC}=\unicode[STIX]{x03C0}/2$ , $\unicode[STIX]{x1D706}=15/2$ and $\unicode[STIX]{x1D6FC}=\unicode[STIX]{x03C0}/2$ , $\unicode[STIX]{x1D706}=15$ , respectively. It can be seen that the stronger of the two swirling Moffatt solutions (panel f) corresponds to a larger amplitude wavefield in the outer fluid, with three closed streamlines in the image shown.

Figure 4. (a) Zeros of (2.15a ) for $c=1$ ; as the rotation rate increases the Rossby number, $Ro$ , decreases and new branches of the solution are found. The critical rotation rates at which new branches of solution are found are given by zeros of the system (2.15), indicated by circles. The fifth critical Rossby $Ro=Ro_{c}^{(5)}\approx 0.114$ is indicated by the black circle. ( $b$ ) The streamfunction for $\unicode[STIX]{x1D703}=\unicode[STIX]{x03C0}/2$ and $Ro=Ro_{c}^{(5)}$ . This plot corresponds to the $z=0$ transect of figure 5. The black circles that occur at local minima in $\unicode[STIX]{x1D713}$ correspond to the black stagnation points in figure 5. The white circles that occur at local maxima in $\unicode[STIX]{x1D713}$ correspond to the white stagnation points in figure 5. For $\unicode[STIX]{x1D713}$ lying between pairs of black and white turning points, indicated by the grey regions, the function $\unicode[STIX]{x1D70E}(\unicode[STIX]{x1D713})$ is multivalued, and this corresponds to closed streamlines in the flow. At the chosen Rossby number the fifth set of closed streamlines is about to appear at the inflection point indicated by the dashed line, corresponding to the cusp at $r\approx 4.39$ (black–white circle) in figure 5.

2.4 Sibling vortices

The critical rotation rate at which the onset of overturning is observed may be found by considering the turning points of the streamlines. If a given streamline, $\unicode[STIX]{x1D713}=\text{const.}$ for $\unicode[STIX]{x1D70E}>1$ , is parameterized by $\unicode[STIX]{x1D703}=\unicode[STIX]{x1D703}(\unicode[STIX]{x1D70E})$ then a necessary condition for overturning is $\text{d}\unicode[STIX]{x1D703}(\unicode[STIX]{x1D70E})/\text{d}\unicode[STIX]{x1D70E}=0$ , that is, when

This expression has a different number of branches of solution for a given Rossby number, $Ro$ , as is shown in figure 4(a) for $c=1$ . There is therefore in this case a minimum rate of rotation below which no closed streamlines in the outer fluid are formed and no fluid is carried along with the spherical vortex. The first critical rotation rate occurs when $\text{d}Ro(\unicode[STIX]{x1D70E})/\text{d}\unicode[STIX]{x1D70E}=0$ , where $Ro=Ro(\unicode[STIX]{x1D70E})$ is determined by (2.15a ). This condition is given by

The critical points determined by simultaneous solutions of (2.15a ) and (2.15b ) are shown for $c=1$ as circles in figure 4(a) and we denote the critical Rossby numbers as $Ro=Ro_{c}^{(n)}$ and the corresponding critical radii as $\unicode[STIX]{x1D70E}=\unicode[STIX]{x1D70E}_{c}^{(n)}$ for $n=1,2,3,\ldots \,$ . The first critical rotation rate, $Ro_{c}^{(1)}$ , that represents the minimum rotation rate for which a closed streamline forms in the flow is found numerically, for Hill’s spherical vortex, to be $Ro_{c}^{(1)}\approx 0.239$ . This rotation rate lies between those shown in figure 3(b,c). The corresponding radius, $\unicode[STIX]{x1D70E}_{c}^{(1)}\approx 2.07$ and $\unicode[STIX]{x1D703}=\unicode[STIX]{x03C0}/2$ gives the location at which the overturning first occurs. For a given Rossby number, the associated number of branches of solutions of (2.15a ) corresponds to the number of closed streamlines in the flow outside the spherical vortex, and hence corresponds to the number of regions of fluid that are advected with the spherical vortex.

Figure 5. A meridional slice through the flow field in the frame of reference moving with a swirling Hill’s spherical vortex ring for $Ro=Ro_{c}^{(5)}\approx 0.114$ . This is the fifth critical Rossby number: we can see four sibling vortex rings (bold lines) have been created in the outer fluid and there is a cusp that has formed (black–white circle at $r\approx 4.39,z=0$ ) on a streamline that will form the next sibling vortex ring when $Ro<Ro_{c}^{(5)}$ .

The value of the streamfunction on the closed streamlines and the location of stagnation points in the flow can be found by considering $\unicode[STIX]{x1D713}$ on $\unicode[STIX]{x1D703}=\unicode[STIX]{x03C0}/2$ . Figure 4(b) shows $\unicode[STIX]{x1D713}(\unicode[STIX]{x1D70E},\unicode[STIX]{x03C0}/2)$ for $\unicode[STIX]{x1D70E}>1$ and $Ro=Ro_{c}^{(5)}\approx 0.114$ , the fifth critical rotation rate (indicated by the black circle in figure 4 a). The turning points in $\unicode[STIX]{x1D713}$ can be seen to appear in pairs of local minima (black circles) and local maxima (white circles). For values of $\unicode[STIX]{x1D713}$ between the local minima and maxima, indicated by the grey bands, the function $\unicode[STIX]{x1D70E}(\unicode[STIX]{x1D713})$ is multivalued, and this corresponds to closed streamlines in the flow. As the flow considered is exactly at the fifth critical rotation rate, the fifth pair of local minima and maxima coincide at the inflection in $\unicode[STIX]{x1D713}$ , where $\unicode[STIX]{x1D70E}\approx 4.39$ and $\unicode[STIX]{x1D713}\approx -9.65$ .

Figure 5 shows streamlines of the flow for a swirling Hill’s spherical vortex in a rotating fluid at $Ro=Ro_{c}^{(5)}$ corresponding to the rotation rate in figure 4(b). The stagnation point in the spherical vortex is, as in the classical solution, at $\unicode[STIX]{x1D70E}=2^{-1/2}$ , $\unicode[STIX]{x1D703}=\unicode[STIX]{x03C0}/2$ . In the outer flow there can be seen to be four closed streamlines, indicated in bold. We observe that the direction of advection around these closed streamlines (anti-clockwise) is opposite to that in the spherical vortex (clockwise), though the vorticity in the closed streamlines may change sign (see (2.6)). The black stagnation points in the closed streamlines correspond to the black local minima in figure 4(b). The white stagnation points, on the boundary of the closed streamlines, correspond to the local maxima in 4(b). A fifth ‘closed streamline’ is about to form at the cusp indicated by the black–white circle at $\unicode[STIX]{x1D70E}\approx 4.39$ . This corresponds to the inflection in $\unicode[STIX]{x1D713}$ in figure 4(b).