2 Hybrid inertial–Alfvén waves

Consider now those waves which satisfy $\boldsymbol{k}\boldsymbol{\cdot }\unicode[STIX]{x1D734}\approx 0$ , so that $|\unicode[STIX]{x1D71B}_{\unicode[STIX]{x1D6FA}}|\ll |\unicode[STIX]{x1D71B}_{B}|$ despite the fact that $\unicode[STIX]{x1D6FA}l\gtrsim 10|\boldsymbol{B}|$ . To leading order in $|\unicode[STIX]{x1D71B}_{\unicode[STIX]{x1D6FA}}|/|\unicode[STIX]{x1D71B}_{B}|$ , (1.4) and (1.5) yield

(2.1a,b ) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D71B}=(\pm )\unicode[STIX]{x1D71B}_{B}\pm \frac{\unicode[STIX]{x1D71B}_{\unicode[STIX]{x1D6FA}}}{2}\approx (\pm )\unicode[STIX]{x1D71B}_{B},\quad \boldsymbol{c}_{g}=(\pm )\boldsymbol{B}_{0}\pm \frac{\unicode[STIX]{x1D734}}{k}, & & \displaystyle\end{eqnarray}$$

where the upper or lower sign in $\pm$ may be chosen independently to that in $(\pm )$ . These are hybrid waves with the frequency of Alfvén waves, which propagate accordingly along field lines, but also disperse energy along the rotation axis at half the speed of low-frequency inertial waves. As (1.3) still holds, they have maximal kinetic, magnetic and cross-helicity, and equipartition of energy, $\hat{\boldsymbol{u}}=(\mp )\hat{\boldsymbol{b}}$ . In the Earth’s core the axial group velocity of these waves is extremely fast; for example, taking $l\sim 10~\text{km}$ , hybrid inertial–Alfvén waves can traverse the core in months. By way of contrast, pure Alfvén waves would traverse the core in decades, whereas convective motions and magnetostrophic waves have time scales measured in centuries.

Since $\boldsymbol{k}\boldsymbol{\cdot }\unicode[STIX]{x1D734}\approx 0$ , rotation appears at leading order in the expression for $\boldsymbol{c}_{g}$ but not in that for $\unicode[STIX]{x1D71B}$ . From a strictly kinematic point of view, one may conceive of these hybrid waves as a pair of linearly polarised Alfvén waves propagating horizontally along $\boldsymbol{B}_{0}$ and superimposed at right angles with a $\unicode[STIX]{x03C0}/2$ phase shift to yield the circularly polarised wave demanded by (1.3). From (2.1) we see that, like low-frequency inertial waves, they focus radiated energy onto the rotation axis and, when the waves emanate from a localised source of buoyancy, this focusing of energy will produce a particularly high energy density within the axially propagating wave packets. We expect, therefore, that these hybrid waves will assume the role adopted by low-frequency inertial waves in the strictly hydrodynamic case, with localised buoyant blobs spontaneously radiating axially elongated wave packets which eventually evolve into quasi-geostrophic vortices.

The curious fact that the axial group velocity is halved in comparison with the equivalent low-frequency inertial waves can be understood as follows. From (1.1) we have the general energy-density energy-flux relationship

(2.2) $$\begin{eqnarray}\displaystyle \frac{\unicode[STIX]{x2202}e}{\unicode[STIX]{x2202}t}=\unicode[STIX]{x1D735}\boldsymbol{\cdot }\left[\left(\boldsymbol{u}\boldsymbol{\cdot }\boldsymbol{b}\right)\boldsymbol{B}_{0}+(\boldsymbol{u}\boldsymbol{\cdot }\boldsymbol{c})\unicode[STIX]{x1D734}+\boldsymbol{c}\times (\unicode[STIX]{x1D734}\boldsymbol{\cdot }\unicode[STIX]{x1D735}\boldsymbol{c})\right], & & \displaystyle\end{eqnarray}$$

where $e=(\boldsymbol{u}^{2}+\boldsymbol{b}^{2})/2$ is the energy density. For axially elongated structures we expect the final term in the divergence to vanish, so the first two terms represent transverse and axial contributions to the energy flux, with the axial energy flux being $-(\boldsymbol{u}\boldsymbol{\cdot }\boldsymbol{c})\unicode[STIX]{x1D734}$ . For the case of a monochromatic wave, (1.3) requires $\hat{\boldsymbol{u}}=\mp k\hat{\boldsymbol{c}}$ , from which $\unicode[STIX]{x1D735}\boldsymbol{\cdot }[\boldsymbol{c}\times (\unicode[STIX]{x1D734}\boldsymbol{\cdot }\unicode[STIX]{x1D735}\boldsymbol{c})]=0$ and the axial energy flux becomes $\pm \boldsymbol{u}^{2}(\unicode[STIX]{x1D734}/k)$ . In the case of low-frequency inertial waves, where $e=\boldsymbol{u}^{2}/2$ , this yields the expected axial group velocity of $\pm 2\unicode[STIX]{x1D734}/k$ , whereas for our hybrid inertial–Alfvén waves (for which $e=\boldsymbol{u}^{2}$ ) the resulting axial group velocity is $\pm \unicode[STIX]{x1D734}/k$ , as in (2.1b ).

Alternatively we may rewrite (1.1) in terms of Elsasser variables $\unicode[STIX]{x1D74A}^{(\mp )}=\boldsymbol{u}(\mp )\boldsymbol{b}$ and their corresponding vector potentials $\unicode[STIX]{x1D73F}^{(\mp )}$ , to give

(2.3) $$\begin{eqnarray}\displaystyle \frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D74A}^{(\mp )}}{\unicode[STIX]{x2202}t}=(\mp )(\boldsymbol{B}_{0}\boldsymbol{\cdot }\unicode[STIX]{x1D735})\unicode[STIX]{x1D74A}^{(\mp )}+(\unicode[STIX]{x1D734}\boldsymbol{\cdot }\unicode[STIX]{x1D735})(\unicode[STIX]{x1D73F}^{+}+\unicode[STIX]{x1D73F}^{-}). & & \displaystyle\end{eqnarray}$$

for a hybrid wave travelling parallel to $\boldsymbol{B}_{0}$ we have $\unicode[STIX]{x1D74A}^{+}=\mathbf{0}$ , whilst one travelling antiparallel has $\unicode[STIX]{x1D74A}^{-}=\mathbf{0}$ ; in either case, our hybrid waves are governed by

(2.4) $$\begin{eqnarray}\displaystyle \frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D74A}^{(\mp )}}{\unicode[STIX]{x2202}t}=(\mp )(\boldsymbol{B}_{0}\boldsymbol{\cdot }\unicode[STIX]{x1D735})\unicode[STIX]{x1D74A}^{(\mp )}+(\unicode[STIX]{x1D734}\boldsymbol{\cdot }\unicode[STIX]{x1D735})\unicode[STIX]{x1D73F}^{(\mp )}. & & \displaystyle\end{eqnarray}$$

Note the absence of the pre-multiplying factor of two seen in (1.1b ). The dispersion relationship for (2.4) is readily shown to be $\unicode[STIX]{x1D71B}=(\pm )\unicode[STIX]{x1D71B}_{B}\pm (1/2)\unicode[STIX]{x1D71B}_{\unicode[STIX]{x1D6FA}}$ , as in (2.1a ).

Figure 1. Axial velocity $u_{z}$ at $\unicode[STIX]{x1D6FA}t=50$ . Rotation is vertical and gravity acts into the page. (a) Is the non-magnetic case ( $Le=0$ ); (b,c) have a horizontal mean field ( $Le=0.1$ ). (a,b) Show contours of $u_{z}$ in the plane $x=0$ with field lines overlaid in black; (c) shows a 3D isosurface of $|u_{z}|$ at 8 % of its maximum value. Red (blue) indicates $u_{z}>0$ ( $u_{z}<0$ ).

3 The spontaneous radiation of waves from a localised source

Consider the somewhat artificial case of waves radiated from a localised buoyant source introduced into the fluid at time $t=0$ . We restrict ourselves to the rapidly rotating regime $\unicode[STIX]{x1D6FA}l\gg |\boldsymbol{B}|$ thought to be typical of the Earth’s core.

Figure 2. The energy density $e$ in the $y$ – $z$ plane at various times, for $Le=0.1$ , coloured by $\unicode[STIX]{x1D706}=(\boldsymbol{u}^{2}-\boldsymbol{b}^{2})/(\boldsymbol{u}^{2}+\boldsymbol{b}^{2})$ in order to distinguish the three classes of wave.

Figure 3. The energy density $e$ at various times, coloured by normalised cross-helicity $\unicode[STIX]{x1D70E}$ . The solid black line marks a contour at $\unicode[STIX]{x1D70E}=0.9$ , with its lowermost tip highlighted by a black circle, and the dashed red line marks a contour at $e=0.3e_{avg}$ , with its uppermost tip highlighted by a red circle. The blue cross marks the ‘centre’ of the wave packet.

From the preceding discussion we expect at least three types of waves to emerge from the localised source. For those waves which satisfy $|\unicode[STIX]{x1D71B}_{\unicode[STIX]{x1D6FA}}|\gg |\unicode[STIX]{x1D71B}_{B}|$ , which will hold for all wave vectors except those aligned with the transverse plane ( $\boldsymbol{k}\boldsymbol{\cdot }\unicode[STIX]{x1D734}\approx 0$ ), we have both fast inertial and slow magnetostrophic waves. Conversely, those which satisfy the reverse criterion $|\unicode[STIX]{x1D71B}_{\unicode[STIX]{x1D6FA}}|\ll |\unicode[STIX]{x1D71B}_{B}|$ , which requires $\boldsymbol{k}\boldsymbol{\cdot }\unicode[STIX]{x1D734}\rightarrow 0$ for $\unicode[STIX]{x1D6FA}l\gg |\boldsymbol{B}|$ , constitute hybrid inertial–Alfvén waves. (We leave aside the intermediate case $|\unicode[STIX]{x1D71B}_{\unicode[STIX]{x1D6FA}}|\sim |\unicode[STIX]{x1D71B}_{B}|$ , which is not readily classified and in any event unimportant, as we shall see.) Of these three classes of wave, the fastest group velocity belongs to those inertial waves for which $\boldsymbol{c}_{g}$ is closely aligned with the rotation axis, though not so close they violate the criterion $|\unicode[STIX]{x1D71B}_{\unicode[STIX]{x1D6FA}}|\gg |\unicode[STIX]{x1D71B}_{B}|$ . These have an axial group velocity approaching $\boldsymbol{c}_{g}\approx \pm 2\unicode[STIX]{x1D734}/k$ . Next fastest are the hybrid waves, with roughly half the axial group velocity of the fastest inertial waves, $\boldsymbol{c}_{g}=\pm \unicode[STIX]{x1D734}/k$ . Unlike inertial waves, these transport both magnetic energy and magnetic helicity. Finally, we have the sluggish magnetostrophic waves, which emerge slowly and can exhibit a significant energy flux along $\boldsymbol{B}_{0}$ .

Of these three wave classes, hybrid inertial–Alfvén waves are the best candidates for establishing quasi-geostrophic structures. This is partly because they are fast, unlike magnetostrophic waves, and partly because they automatically focus energy onto the rotation axis, unlike the off-axis inertial waves, which disperse radially. A simple example is sufficient to demonstrate these trends.

Figure 4. Spatio-temporal evolution of the inertial–Alfvén wave packet identified in figure 3. (a) Shows the centre moving at the Alfvén velocity along field lines, and (b) tracks its axial velocity; markers correspond to those in figure 3.

4 An illustrative example

Consider the case in which $\unicode[STIX]{x1D734}=\unicode[STIX]{x1D6FA}\boldsymbol{e}_{z}$ , $\boldsymbol{B}_{0}=B_{0}\boldsymbol{e}_{y}$ and $\boldsymbol{g}=-g\boldsymbol{e}_{x}$ and the source is a static buoyant density anomaly of Gaussian profile centred on the origin, with a density perturbation $\unicode[STIX]{x1D70C}^{\prime }\propto -\exp \{-2|\boldsymbol{x}|^{2}/\ell ^{2}\}$ ; this has been previously studied in the quasi-static regime (Moffatt & Loper Reference Moffatt and Loper1994; St Pierre Reference St Pierre1996; Chulliat, Loper & Shimizu Reference Chulliat, Loper and Shimizu2004). The scale $\ell$ is the effective radius of the buoyant blob and the dominant wavenumber for such a disturbance is readily shown to be $k\approx \unicode[STIX]{x03C0}/\ell$ . It is convenient to introduce the ratio of Alfvén to inertial frequencies as the dimensionless Lehnert number, $Le=2B_{0}/\unicode[STIX]{x1D6FA}\ell$ , which is approximately $2B_{0}k/\unicode[STIX]{x03C0}\unicode[STIX]{x1D6FA}$ and of the order of 0.1 for small scales in the Earth. The vorticity equation is then

(4.1) $$\begin{eqnarray}\displaystyle \frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D74E}}{\unicode[STIX]{x2202}t}=\left(\boldsymbol{B}_{0}\boldsymbol{\cdot }\unicode[STIX]{x1D735}\right)\boldsymbol{j}+\left(2\unicode[STIX]{x1D734}\boldsymbol{\cdot }\unicode[STIX]{x1D735}\right)\boldsymbol{u}+\unicode[STIX]{x1D735}\unicode[STIX]{x1D70C}^{\prime }\times \boldsymbol{g}. & & \displaystyle\end{eqnarray}$$

The wave-like equation for $\boldsymbol{u}$ continues to be (1.2), whilst that for $\boldsymbol{b}$ acquires a source term:

(4.2) $$\begin{eqnarray}\displaystyle \left[\frac{\unicode[STIX]{x2202}^{2}}{\unicode[STIX]{x2202}t^{2}}-\left(\boldsymbol{B}_{0}\boldsymbol{\cdot }\unicode[STIX]{x1D735}\right)^{2}\right]^{2}\unicode[STIX]{x1D6FB}^{2}\boldsymbol{b}+\left(2\unicode[STIX]{x1D734}\boldsymbol{\cdot }\unicode[STIX]{x1D735}\right)^{2}\frac{\unicode[STIX]{x2202}^{2}\boldsymbol{b}}{\unicode[STIX]{x2202}t^{2}}=\left(\boldsymbol{B}_{0}\boldsymbol{\cdot }\unicode[STIX]{x1D735}\right)^{3}\left[\left(\boldsymbol{g}\boldsymbol{\cdot }\unicode[STIX]{x1D735}\right)\unicode[STIX]{x1D735}\unicode[STIX]{x1D70C}^{\prime }-\boldsymbol{g}\unicode[STIX]{x1D6FB}^{2}\unicode[STIX]{x1D70C}^{\prime }\right]. & & \displaystyle\end{eqnarray}$$

We solve (1.2) and (4.2) subject to the initial conditions $\boldsymbol{u}=\boldsymbol{b}=\mathbf{0}$ , and the associated initial requirements $\unicode[STIX]{x2202}_{t}\unicode[STIX]{x1D74E}=\unicode[STIX]{x1D735}\unicode[STIX]{x1D70C}^{\prime }\times \boldsymbol{g}$ and $\unicode[STIX]{x2202}_{t}^{2}\unicode[STIX]{x1D6FB}^{2}\boldsymbol{u}=(2\unicode[STIX]{x1D734}\boldsymbol{\cdot }\unicode[STIX]{x1D735})(\boldsymbol{g}\times \unicode[STIX]{x1D735}\unicode[STIX]{x1D70C}^{\prime })$ , both of which follow from (4.1). The solution procedure is straightforward in principle: we take the spatial Fourier transforms of (1.2) and (4.2) and solve the resulting equations in Fourier space, subject to the initial conditions. This yields a transformed velocity of

(4.3) $$\begin{eqnarray}\displaystyle \hat{\boldsymbol{u}}\left(\boldsymbol{k},t\right) & = & \displaystyle \hat{\unicode[STIX]{x1D70C}}^{\prime }(\unicode[STIX]{x1D71B}_{\unicode[STIX]{x1D6FA}}^{2}+4\unicode[STIX]{x1D71B}_{B}^{2})^{-1/2} [(\cos \unicode[STIX]{x1D71B}_{F}t-\cos \unicode[STIX]{x1D71B}_{S}t)\boldsymbol{e}_{k}\times \boldsymbol{g}\nonumber\\ \displaystyle & & \displaystyle +\,(\sin \unicode[STIX]{x1D71B}_{F}t+\sin \unicode[STIX]{x1D71B}_{S}t)\boldsymbol{e}_{k}\times (\boldsymbol{g}\times \boldsymbol{e}_{k})\!],\end{eqnarray}$$

where $\hat{\unicode[STIX]{x1D70C}}^{\prime }\propto -\text{exp}\{-k^{2}\ell ^{2}/8\}$ , $2\unicode[STIX]{x1D71B}_{F(S)}=\sqrt{\unicode[STIX]{x1D71B}_{\unicode[STIX]{x1D6FA}}^{2}+4\unicode[STIX]{x1D71B}_{B}^{2}}+(-)\unicode[STIX]{x1D71B}_{\unicode[STIX]{x1D6FA}}$ . The inverse transform, which takes the form of a dispersion integral, is then evaluated numerically.

Figure 5. Axial velocity distribution at $\unicode[STIX]{x1D6FA}t=50$ , visualised by an isosurface of $|u_{z}|$ at 10 % of its maximum value coloured red (blue) for $u_{z}>0$ ( $u_{z}<0$ ), for $10^{4}$ Gaussian buoyant blobs randomly distributed in the vicinity of the $x$ – $y$ plane. Rotation is vertical, ambient magnetic field in the $y$ -direction, and gravity acts in the negative $x$ -direction. Axes in units of $\ell$ .

In the interests of brevity we restrict our discussion to the case $Le=0.1$ . Figure 1 shows contours of $u_{z}$ , the inverse transform of (4.3), for $z>0$ at $\unicode[STIX]{x1D6FA}t=50$ , with (c) showing a three-dimensional rendering, (b) showing the velocity in the plane $x=0$ , and (a) showing the non-magnetic case ( $Le=0$ ) for comparison. The dispersion pattern in figure 1(b) resembles that which emerges from a buoyant blob in the absence of a magnetic field, dominated by a pair of columnar vortices above the blob – one cyclonic, one anticyclonic – matched by a cyclone–anticyclone pair beneath it. However, there is also clear evidence of off-axis radiation.

Figure 2 shows the energy density $e=(\boldsymbol{u}^{2}+\boldsymbol{b}^{2})/2$ in the $y$ – $z$ plane at various times, restricted to the quadrant $y>0$ , $z>0$ . The intensity of the colour is a measure of $e$ , whilst the hue is determined by $\unicode[STIX]{x1D706}=(\boldsymbol{u}^{2}-\boldsymbol{b}^{2})/(\boldsymbol{u}^{2}+\boldsymbol{b}^{2})$ , as indicated in the rightmost panel. Pure inertial waves are characterised by $\unicode[STIX]{x1D706}=1$ (yellow), hybrid inertial–Alfvén waves by $\unicode[STIX]{x1D706}=0$ (green-blue) and magnetostrophic waves by $\unicode[STIX]{x1D706}\rightarrow -1$ (dark blue). As expected, the inertial waves emerge first, inertial–Alfvén waves second, and magnetostrophic waves last, though the colour pattern is complicated by the fact that slower (more off-axis) inertial waves propagate at a similar speed to the inertial–Alfvén waves, and so the two overlap in space.

We remark that this structure bears some similarity to that found by Jault (Reference Jault2008), who studied the axisymmetric response of the Earth’s outer core to an impulsive rotation of the inner core. In both cases, the waves elongate axially before migrating radially along magnetic field lines. However, it is not clear whether quasi-geostrophy is established by inertial–Alfvén waves (rather than pure inertial waves) in Jault (Reference Jault2008), but they may be responsible.

Figure 3 also shows the energy density, only this time coloured by the normalised cross-helicity $\unicode[STIX]{x1D70E}=|\boldsymbol{u}\boldsymbol{\cdot }\boldsymbol{b}|/e$ . Both inertial and magnetostrophic waves are characterised by $\unicode[STIX]{x1D70E}\approx 0$ (blue/purple) and hybrid inertial–Alfvén waves by $\unicode[STIX]{x1D70E}=1$ (yellow). The solid black line indicates $\unicode[STIX]{x1D70E}=0.9$ and the dashed red line $e=0.3e_{avg}$ (where $e_{avg}$ is an average over all space and time for the five plots shown); we take the area bounded above by $e=0.3e_{avg}$ and below by $\unicode[STIX]{x1D70E}=0.9$ as the region of space occupied by energetic inertial–Alfvén waves, with its centre half-way between the two extrema. This provides an objective means of tracking (some fraction of) the inertial–Alfvén wave packet as a function of time.

From figure 3 we can track the $y$ – $z$ location of the centre of our inertial–Alfvén wave packet, and this is shown as a function of $\unicode[STIX]{x1D6FA}t$ in figure 4. Figure 4(a) shows that the centre of the wave packet propagates along magnetic field lines at the Alfvén speed $\boldsymbol{B}_{0}$ , consistent with (2.1b ), whilst (b) shows it propagating with a mean axial velocity of $c_{gz}\simeq 0.26\unicode[STIX]{x1D6FA}\ell$ . Adopting the approximate result that the dominant wavenumber for a Gaussian disturbance is $k\approx \unicode[STIX]{x03C0}/\ell$ , this translates to $c_{gz}\simeq 0.81\unicode[STIX]{x1D6FA}/k$ , close to the prediction of (2.1b ).

Finally, figure 5 shows the three-dimensional distribution of axial velocity $u_{z}$ at $\unicode[STIX]{x1D6FA}t=50$ associated with a set of $10^{4}$ Gaussian blobs with centres randomly distributed over the $x$ – $y$ plane and in the range $-2\ell <z<2\ell$ . The dispersion pattern is dominated by cyclone–anticyclone pairs, and looks very much like the images obtained from full numerical simulations of planetary dynamos. Using $\ell \sim 10~\text{km}$ (Davidson Reference Davidson2014), the box height would be 400 km, so this is only valid in a local sense – as the waves propagate to larger $z$ we expect both dissipation and reflection to play a role.