Analogy between electro-magnetic waves in cold unmagnetized plasma and shallow water inertio-gravity waves in geophysical systems

The nonlinear interaction of intense ultrashort laser pulses with a plasma, e.g. in laser wakefield acceleration of electrons, is described by electro-magnetic fields acting on a cold, unmagnetized plasma [1]. The plasma is considered cold, in the sense that both the electron quiver velocity in the pulse, and the electron longitudinal velocity in the wake, are larger than the electron thermal velocity. The dispersion relation for transverse electro-magnetic waves in such laser-produced plasmas determines whether the plasma is transparent or opaque to laser light in a given frequency range [2] (a complete description of electro-magnetic waves in cold plasma can be found in [3]).

Inertio-gravity (Poincaré) waves are ubiquitous waves in the atmosphere and the oceans. Their propagation mechanism results from a combination of the horizontal Coriolis and the vertical buoyancy restoring forces, and they are responsible for the continuous adjustment mechanism of the large scale flow into a geostrophic balance between the Coriolis force and the horizontal component of the pressure gradient force. Due to their relatively large aspect ratio they can be modeled, to a first approximation, by the linearized dynamics of the rotating shallow water system (a complete description of shallow water inertio-gravity waves in geophysical systems can be found in [4]).

These two fundamental waves of nature seem, at first sight, to have nothing in common with each other. However, their dispersion relations are formally equivalent, where the plasma frequency may be identified with the Coriolis frequency, and the speed of light identified with the surface gravity wave speed. Here we examine the physical propagation mechanisms of the two waves in two physically remote systems, in order to understand why they possess formally equivalent dispersion relations.

The paper is organized as follows. In section 2 we show the equivalency in the wave dispersion relations of the two systems, and then in section 3 we write the linearized equations of the two systems in an equivalent scalar-vector like form. In section 4 we analyze the physical propagation mechanisms of the two waves and show why they result with the same dispersion relation. In section 5 we address some other related analogies and then summarize our analysis in section 6.

Consider a cold collisionless plasma whose ions are assumed to be at rest. The electron mean dynamics is then governed by the charge continuity and momentum equations [3]:

$$\begin{eqnarray}&&\displaystyle \frac{\partial n}{\partial t}=-{\rm{\nabla }}\cdot (n{\bf{u}}),\end{eqnarray} \tag{ 1 }$$

$$\begin{eqnarray}&&\displaystyle \frac{\partial {\bf{u}}}{\partial t}=-({\bf{u}}\cdot {\rm{\nabla }}){\bf{u}}-\displaystyle \frac{e}{m}({\bf{E}}+{\bf{u}}\times {\bf{B}}),\end{eqnarray} \tag{ 2 }$$

and Maxwell's equations:

$$\begin{eqnarray}&&\displaystyle \frac{\partial {\bf{B}}}{\partial t}=-{\rm{\nabla }}\times {\bf{E}},\end{eqnarray} \tag{ 3 }$$

$$\begin{eqnarray}&&\displaystyle \frac{\partial {\bf{E}}}{\partial t}={c}^{2}\,{\rm{\nabla }}\times {\bf{B}}+\displaystyle \frac{e\,n}{{\epsilon }_{0}}{\bf{u}},\end{eqnarray} \tag{ 4 }$$

$$\begin{eqnarray}&&{\rm{\nabla }}\cdot {\bf{E}}=\displaystyle \frac{\rho }{{\epsilon }_{0}}=\displaystyle \frac{{q}_{i}{n}_{i}-e\,n}{{\epsilon }_{0}},\end{eqnarray} \tag{ 5 }$$

$$\begin{eqnarray}&&{\rm{\nabla }}\cdot {\bf{B}}=0,\end{eqnarray} \tag{ 6 }$$

where t denotes the time, ∇ is the 3D nabla operator, n is the electron number density, u is the mean electron velocity, e and m are the electron charge and mass, E and B are the electric and magnetic fields, c is the speed of light, ₀ the vacuum permittivity, ρ the total electric charge density, and q_i and n_i are the ion charge and number density. For the sake of simplicity we neglect the pressure gradient force in the momentum equation (2) that may result from the flow dynamics. Note also that as J = − en u is the electric current density, equation (4) is the actual Ampère's law for plasma.

These equations admit an equilibrium state with E = B = u = 0 and ρ = 0, with an equilibrium electron number density n₀ = q_i n_i /e (indicated by subscript zero) that is uniform in space and time. Denoting perturbations quantities by primes, the linearized equations about the rest equilibrium reads:

$$\begin{eqnarray}&&\displaystyle \frac{\partial n^{\prime} }{\partial t}=-{n}_{0}{\rm{\nabla }}\cdot {\bf{u}}^{\prime} ,\end{eqnarray} \tag{ 7 }$$

$$\begin{eqnarray}&&\displaystyle \frac{\partial {\bf{u}}^{\prime} }{\partial t}=-\displaystyle \frac{e}{m}{\bf{E}}^{\prime} ,\end{eqnarray} \tag{ 8 }$$

$$\begin{eqnarray}&&\displaystyle \frac{\partial {\bf{B}}^{\prime} }{\partial t}=-{\rm{\nabla }}\times {\bf{E}}^{\prime} ,\end{eqnarray} \tag{ 9 }$$

$$\begin{eqnarray}&&\displaystyle \frac{\partial {\bf{E}}^{\prime} }{\partial t}={c}^{2}\,{\rm{\nabla }}\times {\bf{B}}^{\prime} +\displaystyle \frac{e\,{n}_{0}}{{\epsilon }_{0}}{\bf{u}}^{\prime} ,\end{eqnarray} \tag{ 10 }$$

$$\begin{eqnarray}&&{\rm{\nabla }}\cdot {\bf{E}}^{\prime} =-\displaystyle \frac{e}{{\epsilon }_{0}}n^{\prime} ,\end{eqnarray} \tag{ 11 }$$

$$\begin{eqnarray}&&{\rm{\nabla }}\cdot {\bf{B}}^{\prime} =0.\end{eqnarray} \tag{ 12 }$$

As with the original nonlinear equations (1)–(6), this linear set of equations have some redundancies (the divergence of (10), with the use of (11), implies (7), and (9) is consistent with (12)), and plane wave solutions of the form of $f^{\prime} ({\bf{x}},t)=\hat{f}{e}^{i({\bf{k}}\cdot {\bf{x}}-\omega t)}$ (where $f^{\prime}$ represents any perturbed variable at position x and time t, with wavevector k and angular frequency ω) can be described solely from the equation subset (8)–(10). The linear wave solutions are of two types. One is when the electric field is irrotational (${\rm{\nabla }}\times {\bf{E}}^{\prime} =0$, implying that the wavevector is aligned with the perturbation electric field as ${\bf{k}}\times {\bf{E}}^{\prime} =0$), so the perturbation magnetic field is zero, and the dispersion relation is ω = ± ω_p , where ${\omega }_{p}\equiv \sqrt{{n}_{0}{e}^{2}/m\,{\epsilon }_{0}}$ is the plasma frequency for the present setup. The second is the transverse solution when the electric field is solenoidal (i.e. non-divergent, ${\rm{\nabla }}\cdot {\bf{E}}^{\prime} =0$, implying that the wavevector is perpendicular to the perturbation electric field as ${\bf{k}}\cdot {\bf{E}}^{\prime} =0$), so the perturbation electron number density $n^{\prime}$ vanishes, with dispersion relation ${\omega }^{2}={\omega }_{p}^{2}+{\left({kc}\right)}^{2}$, where k = ∣k∣.

Consider in turn the uniformly rotating shallow water (RSW) system (figure 1) of an incompressible (neutrally charged) fluid [4], described by the continuity and momentum equations:

$$\begin{eqnarray}&&\displaystyle \frac{\partial h}{\partial t}=-{{\rm{\nabla }}}_{H}\cdot (h{\bf{v}}),\end{eqnarray} \tag{ 13 }$$

$$\begin{eqnarray}&&\displaystyle \frac{\partial {\bf{v}}}{\partial t}=-({\bf{v}}\cdot {{\rm{\nabla }}}_{H}){\bf{v}}-g{{\rm{\nabla }}}_{H}h+{\bf{v}}\times {\bf{f}}.\end{eqnarray} \tag{ 14 }$$

Here h(x, y, t) denotes the layer's height, v(x, y, t) = (u, v) is the depth-independent horizontal velocity vector within the fluid layer, ∇_H is the horizontal gradient operator and ${\bf{f}}=f\hat{{\bf{z}}}$ is the Coriolis frequency pointing upwards ($\hat{{\bf{z}}}$ is the vertical unit vector), where we take $f={\rm{constant}}$. The latter is equal to twice the rotation frequency of the system (f is positive for counter-clockwise rotation). The term − g∇_H h represents the horizontal pressure gradient force (as the fluid is assumed to be in hydrostatic balance), thus horizontal pressure differences within the layer result solely from differences in the layer's height. The term v × f represents the Coriolis force, and it acts to the right (left) of the flow's motion for positive (negative) values of f.

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Assuming no topography, equations (13)–(14) admit a rest state solution of mean constant height h = H. Again, denoting the deviation quantities by primes, the linearized RSW equations reads:

$$\begin{eqnarray}&&\displaystyle \frac{\partial h^{\prime} }{\partial t}=-H{{\rm{\nabla }}}_{H}\cdot {\bf{v}}^{\prime} ,\end{eqnarray} \tag{ 15 }$$

$$\begin{eqnarray}&&\displaystyle \frac{\partial {\bf{v}}^{\prime} }{\partial t}=-g{{\rm{\nabla }}}_{H}h^{\prime} +{\bf{v}}^{\prime} \times {\bf{f}}.\end{eqnarray} \tag{ 16 }$$

The linearized RSW system supports horizontal wave solutions, described by the geostrophic mode with dispersion relation ω = 0, as well as inertio-gravity (Poincaré) waves with dispersion relation ${\omega }^{2}={f}^{2}+{({k}_{H}{c}_{s})}^{2}$, where k_H is the magnitude of the horizontal wavevector, and ${c}_{s}=\sqrt{{gH}}$ is the shallow water surface gravity wave phase speed. Introducing the Rossby radius of deformation L_d ≡ c_s /f, the dispersion relation for the inertio-gravity waves can be rewritten as ${\omega }^{2}={f}^{2}[1+{({k}_{H}{L}_{d})}^{2}]$. Thus, in the short wave limit k_H L_d ≫ 1, we obtain gravity waves with ω → ± k_H c_s , whereas in the long wave limit k_H L_d ≪ 1, we obtain inertial oscillations with ω → ± f, a low-frequency cut-off equivalent to ω_p .

The similarity in the dispersion relations, between electro-magentic waves in a cold plasma ${\omega }^{2}={\omega }_{p}^{2}+{\left({kc}\right)}^{2}$ and inertio-gravity waves in the rotating shallow water system ${\omega }^{2}={f}^{2}+{({k}_{H}{c}_{s})}^{2}$, is intriguing, given that the system possesses different physics, and that the linearized RSW system for the three scalar perturbation variables (h, u, v) (omitting the primes hereafter for the perturbation variables)

$$\begin{eqnarray}&&\displaystyle \frac{\partial h}{\partial t}=-H\left(\displaystyle \frac{\partial u}{\partial x}+\displaystyle \frac{\partial v}{\partial y}\right),\end{eqnarray} \tag{ 17 }$$

$$\begin{eqnarray}&&\displaystyle \frac{\partial u}{\partial t}=-g\displaystyle \frac{\partial h}{\partial x}+{fv},\end{eqnarray} \tag{ 18 }$$

$$\begin{eqnarray}&&\displaystyle \frac{\partial v}{\partial t}=-g\displaystyle \frac{\partial h}{\partial y}-{fu},\end{eqnarray} \tag{ 19 }$$

seem, at first sight, to have little in common with the three linearized vector equations (8)–(10) for the three vector fields (u, B, E). However, one can in fact obtain a formal equivalence between the two linearized systems as follows. Starting with the linearized RSW equations, writing it in terms of the divergence δ, vertical component of the vorticity ζ (normalized by f) and the height perturbation (normalized by H) B, given by

$$\begin{eqnarray}\begin{array}{rcl}\delta & \equiv & {{\rm{\nabla }}}_{H}\cdot {\bf{v}}=\left(\displaystyle \frac{\partial u}{\partial x}+\displaystyle \frac{\partial v}{\partial y}\right),\\ \zeta & \equiv & \displaystyle \frac{1}{f}({{\rm{\nabla }}}_{H}\times {\bf{v}})\cdot \hat{{\bf{z}}}=\displaystyle \frac{1}{f}\left(\displaystyle \frac{\partial v}{\partial x}-\displaystyle \frac{\partial u}{\partial y}\right),\\ B & \equiv & \displaystyle \frac{h}{H},\end{array}\end{eqnarray} \tag{ 20 }$$

the linearized RSW equations (17)–(19) are transformed into

$$\begin{eqnarray}&&\displaystyle \frac{\partial \zeta }{\partial t}=-\delta ,\end{eqnarray} \tag{ 21 }$$

$$\begin{eqnarray}&&\displaystyle \frac{\partial B}{\partial t}=-\delta ,\end{eqnarray} \tag{ 22 }$$

$$\begin{eqnarray}&&\displaystyle \frac{\partial \delta }{\partial t}=-{c}_{s}^{2}\,{{\rm{\nabla }}}_{H}^{2}B+{f}^{2}\,\zeta .\end{eqnarray} \tag{ 23 }$$

On the other hand, if we define in the plasma system

$$\begin{eqnarray}&&{\boldsymbol{\delta }}\equiv {\rm{\nabla }}\times {\bf{E}},\hspace{0.5cm}{\boldsymbol{\zeta }}\equiv \displaystyle \frac{m}{e}{\rm{\nabla }}\times {\bf{u}},\end{eqnarray} \tag{ 24 }$$

then the system (8)–(10) reads

$$\begin{eqnarray}&&\displaystyle \frac{\partial {\boldsymbol{\zeta }}}{\partial t}=-{\boldsymbol{\delta }},\end{eqnarray} \tag{ 25 }$$

$$\begin{eqnarray}&&\displaystyle \frac{\partial {\bf{B}}}{\partial t}=-{\boldsymbol{\delta }},\end{eqnarray} \tag{ 26 }$$

$$\begin{eqnarray}&&\displaystyle \frac{\partial {\boldsymbol{\delta }}}{\partial t}=-{c}^{2}\,{{\rm{\nabla }}}^{2}{\bf{B}}+{\omega }_{p}^{2}\,{\boldsymbol{\zeta }}.\end{eqnarray} \tag{ 27 }$$

Hence, within the linear framework, the RSW system formally resembles the plasma one with no background magnetic field, where c_s ↔ c, f ↔ ω_p , with the limitation that the former is of scalar variables and the latter is of vector ones. Furthermore, the gradient operator in the RSW is in the horizontal, whereas in the plasma system it is fully 3D.

We wish to mechanistically understand why the two systems share a similar mathematical formalism, given there is no a priori expectation for them to do so. A schematic of the physical processes is illustrated in figures 2 and 3 for the RSW and plasma system respectively. Consider first the linearized dynamics of the RSW system where equations (21)–(22), repeated at the top of figure 2(a), govern the response to an initial perturbation consisting in convergence (δ < 0, solid blue arrows). Assuming f is positive, the Coriolis force (green thick arrows) acts to the right of the inward motion, and as (21) tells convergence leads to the generation of positive (counterclockwise) vorticity ζ > 0 (red dashed circle in bottom panel). As the shallow water system is incompressible, horizontal convergence simultaneously lifts up the layer's height (according to (22)), yielding a positive height B anomaly (dashed orange curve in middle panel).

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Consider in turn the linearized dynamics in the plasma system where (25)–(26), repeated at the top of figure 3(a), govern the response to an initial perturbation in the electric field. The latter is assumed to be horizontal and circulating clockwise (solid blue arrows and in and out vector signs). This implies that the curl of the electric field ${\boldsymbol{\delta }}=\delta \hat{{\bf{z}}}$ is pointing downward since δ < 0. The electric field apply an electric force (green thick arrows) acting on the electrons on the opposite direction (8). According to (25), this generates a counterclockwise flow circulation ${\boldsymbol{\zeta }}=\zeta \hat{{\bf{z}}}$ that is pointing upwards since ζ > 0 (red dashed circle in bottom panel). Simultaneously, according to Faraday's law in (9) and (26), the clockwise circulation of the electric field yields an upward pointing magnetic field (${\bf{B}}=B\hat{{\bf{z}}}$, with B > 0, orange line in middle panel).

Figure 2(b) shows that, in the RSW system, the new positive height and vorticity anomalies prompt a feedback. As the positive height field anomaly yields a hydrostatic high pressure anomaly, both the pressure gradient force (PGF, magenta thick arrows) and the Coriolis force, acting to the right of the rotational flow, lead to an outward divergent flow, against the initial convergence, generating divergence (δ > 0).

In turn, figure 3(b) shows that in the plasma system the new positive magnetic and vorticity anomalies similarly create a feedback. They act together to generate counterclockwise circulation of the electric field (${\boldsymbol{\delta }}=\delta \hat{{\bf{z}}}$ with δ > 0), in agreement with (27). The Ampère–Maxwell equation (10) indicates that both the curl of the magnetic field and the electron motion generate an electric field. The radial shear of the vertical magnetic field anomaly (indicated by the solid orange arrows in the vertical cross-section) yields a curl pointing counterclockwise in the azimuthal direction (represented by the magenta in and out vector signs), which in turn generates a counterclockwise electric field (blue in and out vector signs). The counterclockwise electron flow (red solid arrows) yields as well an electric field in the direction of their flow. Hence the vorticity ζ associated with the electron flow generates curl of the electric field δ .

By combining the equations in the top panels of figures 2(a) and (b) for the RSW system, or respectively 3(a) and (b) for the plasma system, the initial perturbations and their feedbacks guarantee the presence of an oscillatory response in both systems. These represent waves propagating around a state in which ζ, B and δ and their vector equivalents vanish. At the end of this section we will interpret this state, as it turns out to be nontrivial.

However, as figures 2(c) and 3(c) show, these systems also support a non-zero equilibrium state (indicated by subscripts g). In the RSW system this is the celebrated geostrophic balance between pressure gradient and Coriolis forces (top panel), ${{\bf{v}}}_{g}=(g/f)\hat{{\bf{z}}}\times {{\rm{\nabla }}}_{H}h$. This is obtained in the absence of any radial velocity, when a high pressure anomaly B > 0 (orange arrows and in and out vector signs) is located at the center of a clockwise circulation anomaly (ζ < 0). Consequently the geostrophic flow is non-divergent (δ_g = 0). For the plasma system, the nontrivial equilibrium occurs in the absence of an electric field, where the Ampère–Maxwell equation (10) satisfies u_g = (₀ c²/en₀) ∇ ×B. Then δ _g = 0 and the two terms in the RHS of (27) balance each other as ${\omega }_{p}^{2}\,{{\boldsymbol{\zeta }}}_{e}={c}^{2}\,{{\rm{\nabla }}}^{2}{\bf{B}}$. This can be achieved when electrons circulate in a horizontal clockwise direction and balance a magnetic field anomaly that points upward.

Consider now the short wave limit solutions of the RSW system with length-scales much shorter than the Rossby deformation radius. This is an exact limit for the case when the shallow water system is non-rotating (f = 0) and the flow itself is irrotational. For this case, system (21)–(23) reduces to the solution of the non-dispersive surface gravity waves:

$$\begin{eqnarray}&&\displaystyle \frac{\partial B}{\partial t}=-\delta ,\hspace{0.25cm}\displaystyle \frac{\partial \delta }{\partial t}=-{c}_{s}^{2}\,{{\rm{\nabla }}}_{H}^{2}B\hspace{0.5cm}\Longrightarrow \hspace{0.5cm}{\omega }^{2}={\left({k}_{H}{c}_{s}\right)}^{2}.\end{eqnarray} \tag{ 28 }$$

For surface gravity waves, the propagation mechanism can be understood, using figures 2 and 4(a), in terms of the interplay between horizontal divergence/convergence and the undulation of surface elevation. Convergence (divergence) generates ridges (crests) which in turn, by applying a pressure gradient force, generates divergence (convergence). The horizontal velocity and the vertical displacement fields are in phase. Consequently, the resulting pressure gradient force translates the velocity field to the right in concert with the height field, that is translated by the horizontal convergence/divergence field. Hence, for rightward propagation, the divergence field lags the height field by a quarter of a wavelength (for leftward propagation the height and the velocity fields are in anti-phase).

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It is interesting that this non-rotating limit of (irrotational) surface gravity waves corresponds to the limit of the plasma system (25)–(27) in a vacuum, (i.e., in the absence of electron flows, thus ζ = 0), admitting the familiar electro-magnetic wave solution:

$$\begin{eqnarray}&&\displaystyle \frac{\partial {\bf{B}}}{\partial t}=-{\boldsymbol{\delta }},\hspace{0.25cm}\displaystyle \frac{\partial {\boldsymbol{\delta }}}{\partial t}=-{c}^{2}\,{{\rm{\nabla }}}^{2}{\bf{B}}\hspace{0.5cm}\Longrightarrow \hspace{0.5cm}{\omega }^{2}={({kc})}^{2}.\end{eqnarray} \tag{ 29 }$$

The electric and magnetic fields are in phase, where the former is pointing in the y direction and the latter in the z direction (see figure 4(b)). The curl of the electric field is also pointing in the z direction and its negative values translates the magnetic field rightward. In turn, the curl of the magnetic field, pointing in the y direction, translates the electric field rightward as well (for leftward propagation the electric and the magnetic fields are anti-phased).

As illustrated in figure 4(c), for the setups shown in figures 4(a) and (b), despite the difference in physics, both mechanisms converge into the same description in which the δ and the B fields are wavy scalar fields in the x-z plane, where the δ lags B by a quarter of a wavelength and their amplitude ratio ∣δ∣/∣B∣ = ∣ω∣. As B generates δ and − δ generates B, the two fields propagate in concert in the positive x direction. Furthermore, as for electro-magnetic waves in vacuum, shallow water surface gravity waves are non-dispersive. Therefore, as the speed of light, the long surface gravity wave speed c_s constitutes the largest possible attainable value of phase and group speeds (for a given mean layer depth H). This equivalence, between non-rotating shallow water surface gravity wave propagation in a continuous medium, and electro-magnetic wave propagation in vacuum, is intriguing.

Consider now the full systems (21)–(23) in a rotating fluid and (25)–(27) in a plasma instead of in vacuum. These admit the inertio-gravity wave and plasma transverse wave solutions respectively, with the equivalent dispersion relations ${\omega }^{2}={f}^{2}+{\left({{kc}}_{s}\right)}^{2}$ and ${\omega }^{2}={\omega }_{p}^{2}+{\left({kc}\right)}^{2}$. We first note that, for non-stationary modal solutions, ζ = B and ζ = B. In figure 5 we illustrate the wave propagation mechanisms in the two systems. In addition to the gravity wave structure described in figure 4(a), a rotational velocity field in the y direction generates a vorticity field ζ that is equal and in phase with the height field B. The Coriolis force applied on the rotational velocity part joins the pressure gradient force in translating the divergent velocity part. This explains why the inertio-gravity wave phase speed is larger than the gravity one (though the group velocity is smaller than the gravity wave one). By contrast, only the Coriolis force acts on the divergent part to translate in concert the rotational velocity field. Since the Coriolis force is proportional to the velocity field it is applied on, this explains why the divergent field must be larger in magnitude than the rotational part (for leftward propagation the height and the divergent part of the velocity fields should be anti-phased, but the vorticity and the height fields remain in phase).

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For transverse electro-magnetic wave propagation in a plasma, the structure described in figure 4(b) is accompanied by a rotational electron flow field in the horizontal plane, a quarter of a wavelength ahead of the electric field. Consequently, the vorticity field is pointing in the z direction and is in phase with and equal in magnitude to the magnetic field. The new velocity field helps the curl of the magnetic field to translate the electric field, which explains why the phase speed of the wave is larger than that of electro-magnetic waves in vacuum (though the group velocity is obviously smaller than the speed of light in vacuum). By contrast, the electron velocity field is translated only by the electric field (which explains why the curl of the electric field must be larger in magnitude than the vorticity induced by the electron flow). Hence, again, despite differences in physics, the propagation mechanisms of the two waves, in terms of the wavy structure of the B, ζ and δ fields, are the same where ∣B∣ = ∣ζ∣ and δ lags B and ζ by a quarter of a wavelength.

It is worthwhile to mention that although (25)–(27) is an equation set for vector variables, still in free infinite space and in a Cartesian representation (in which unit vectors are invariant in space and time) the transverse plane waves only admit solutions where the three vectors ( ζ , B, δ ) are all aligned with the wave phase lines, thus can be practically considered as scalar variables. Furthermore, since the plasma system considered here is isotropic (in contrast with the RSW system where both gravity and the system rotation vectors are aligned vertically), the wave phase lines orientation can be identified with the z axis of a Cartesian coordinate system without loss of generality. In this case, both the 3D Laplacian acting on B in (27) and the horizontal Laplacian acting on B in (23) become a 1D second-order derivative in the horizontal wave direction of propagation (which is the x direction in figures 4, 5, for the choice of the electric and the electron velocity fields to be aligned in the y direction). This explains why systems (21)–(23) and (25)–(27) yield the same plane wave solutions.

It should be also noted that null solutions of (21)–(23) and (25)–(27) (ζ = δ = B = 0 and ζ = δ = B = 0) do not necessarily imply null solutions of equation sets (17)–(19) and (8)–(10) respectively. Such non-trivial solutions are, respectively, the inertial oscillations for the RSW system governed solely by the Coriolis force:

$$\begin{eqnarray}&&\displaystyle \frac{\partial u}{\partial t}={fv},\ \ \ \displaystyle \frac{\partial v}{\partial t}=-{fu},\ \ \ \Longrightarrow \ \ \ \omega =\pm f,\ \ \ u=-{iv},\end{eqnarray} \tag{ 30 }$$

which represents a clockwise circular motion of the velocity vector (u, v) (where both u and v are constant in space), and the longitudinal plasma oscillations, governed solely by the oscillatory interplay between the irrotational electric and the electron flow fields (but without a magnetic field):

$$\begin{eqnarray}&&\displaystyle \frac{\partial {\bf{u}}}{\partial t}=-\displaystyle \frac{e}{m}{\bf{E}},\ \ \ \displaystyle \frac{\partial {\bf{E}}}{\partial t}=\displaystyle \frac{{{en}}_{0}}{{\epsilon }_{0}}{\bf{u}},\ \ \ \Longrightarrow \ \ \ \omega =\pm {\omega }_{p},\ \ \ {\bf{E}}=i{\omega }_{p}\left(\displaystyle \frac{m}{e}{\bf{u}}\right),\end{eqnarray} \tag{ 31 }$$

lacking any spatial variation as the oscillations are in phase everywhere.

We additionally note that, in the fully nonlinear RSW system, the Rossby potential vorticity q ≡ f(ζ + 1)/h (which is not a function of z), is materially conserved [4]:

$$\begin{eqnarray}&&\displaystyle \frac{{D}_{H}}{{Dt}}q\equiv \left(\displaystyle \frac{\partial }{\partial t}+{{\bf{v}}}_{H}\cdot {{\rm{\nabla }}}_{H}\right)q=0.\end{eqnarray} \tag{ 32 }$$

Under linearization, q = f[1 + (ζ − B)]/H, thus (32), is reduced to ∂(ζ − B)/∂t = 0, which implies indeed that for time dependent solutions ζ = B. As pointed out, in the plasma system, it is evident from (25)–(26) that the linearized structure of the electro-magnetic waves must satisfy the vector form equality ζ = B. Hence, it is interesting to examine whether the latter can be obtained from an equivalent version of potential vorticity conservation in the fully nonlinear plasma system. Taking a curl of (2), and using (1), (3) and (6), we obtain:

$$\begin{eqnarray}&&\displaystyle \frac{D}{{Dt}}\left[\displaystyle \frac{({\boldsymbol{\zeta }}-{\bf{B}})}{n}\right]=\left[\displaystyle \frac{({\boldsymbol{\zeta }}-{\bf{B}})}{n}\cdot {\rm{\nabla }}\right]{\bf{u}},\end{eqnarray} \tag{ 33 }$$

which is the material line equation for ( ζ − B)/n, where $D/{Dt}\equiv \left(\partial /\partial t+{\bf{u}}\cdot {\rm{\nabla }}\right)$ is the full 3D material derivative. Hence, ( ζ − B)/n is materially conserved only in the restrictive case where the current flow is on a plane perpendicular to the magnetic field. Nonetheless, for the linearized dynamics, the quadratic term in the RHS of (33) vanishes, and we obtain after linearization that ${n}_{0}^{-1}\partial ({\boldsymbol{\zeta }}-{\bf{B}})/\partial t=0$, which can be regarded as a plasma formal analogue to the linearized potential vorticity conservation law.

There is also a formal analogy between rotating barotropic compressible inviscid flow with shallow water flow [4], and given that the form of (33) looks remarkably like the equation for the (Ertel) PV in the compressible system (with n ↔ ρ and − B ↔ f where the minus sign results from the fact that upward magnetic field acts to the left of electrons' motion), one might wonder if the linearized plasma system could also be compared to the compressible system. Indeed, the 3D polytropic compressible system is given by [4]

$$\begin{eqnarray}&&\displaystyle \frac{\partial \rho }{\partial t}=-{\rm{\nabla }}\cdot (\rho {\bf{u}}),\end{eqnarray} \tag{ 34 }$$

$$\begin{eqnarray}&&\displaystyle \frac{\partial {\bf{u}}}{\partial t}=-({\bf{u}}\cdot {\rm{\nabla }}){\bf{u}}-\displaystyle \frac{1}{\rho }{\rm{\nabla }}p+{\bf{u}}\times {\bf{f}}\,\end{eqnarray} \tag{ 35 }$$

with the general polytropic relation p = a ρ^γ for the two arbitrary constants (a, γ), and ${\bf{f}}=f\hat{{\bf{z}}}$, as before. Linearizing with respect to a rest state density ρ₀, we obtain:

$$\begin{eqnarray}&&\displaystyle \frac{\partial }{\partial t}\left(\displaystyle \frac{\rho ^{\prime} }{{\rho }_{0}}\right)=-{\rm{\nabla }}\cdot {\bf{u}}^{\prime} ,\end{eqnarray} \tag{ 36 }$$

$$\begin{eqnarray}&&\displaystyle \frac{\partial {\bf{u}}^{\prime} }{\partial t}=-{c}_{s}^{2}{\rm{\nabla }}\left(\displaystyle \frac{\rho ^{\prime} }{{\rho }_{0}}\right)+{\bf{u}}^{\prime} \times {\bf{f}},\end{eqnarray} \tag{ 37 }$$

where ${c}_{s}^{2}={\rm{d}}p/{\rm{d}}\rho$ is now the square of the speed of sound, considered constant. Denoting the 3D and horizontal divergence by δ₃ and δ_H , respectively, (36)–(37) can be transformed into:

$$\begin{eqnarray}&&\displaystyle \frac{\partial \zeta }{\partial t}=-{\delta }_{H},\end{eqnarray} \tag{ 38 }$$

$$\begin{eqnarray}&&\displaystyle \frac{\partial B}{\partial t}=-{\delta }_{3},\end{eqnarray} \tag{ 39 }$$

$$\begin{eqnarray}&&\displaystyle \frac{\partial {\delta }_{3}}{\partial t}=-{c}_{s}^{2}\,{{\rm{\nabla }}}_{3}^{2}B+{f}^{2}\,\zeta .\end{eqnarray} \tag{ 40 }$$

where now $B=\rho ^{\prime} /{\rho }_{0}$. While a 3D Laplacian appears now in the RHS of (40), only the horizontal part of the divergence appears in (38). Hence, this system suggests an analogue between electro-magnetic waves in vacuum and sound waves, however the rotating shallow water waves arguably compares better with the cold plasma waves. One should keep in mind however, that this analogue does not hold for a warm plasma, as discussed in the Summary section.

As mentioned in the introduction, the dispersion relation for transverse electro-magnetic waves in plasma has an important consequence in laser-produced plasmas. Since k must be a real number for electro-magnetic waves to propagate, a critical value of the plasma frequency ω = ω_c determines whether the plasma is transparent (ω_l > ω_c ) or opaque (ω_l < ω_c ) to laser light of frequency ω_l [2]. Similarly, due to the increase of f toward the Earth pole, near-inertial oceanic inertio-gravity waves (which are ubiquitous waves, generated by wind stress of local storms, whose frequency is close to the low frequency cutoff) tend to propagate equatorward. Super-inertial waves may propagate poleward but then reflect back towards the equator at a nearby turning latitude [5]. Hence, in the absence of strong oceanic shear currents and local vortices, the region poleward of that turning latitude remains 'opaque'.

Although the cold plasma system (with no background magnetic field) and the rotating shallow water system seem physically disparate, this work highlights a non-trivial analogy between the two systems in the linearized regime. The two linearized systems admit a formally equivalent dispersion relation for transverse waves. The vector-scalar analogue suggested here, between the magnetic B and the pressure (height) B fields, and the curl of the electric field δ and the velocity divergence field δ, brings the two systems onto common ground. This allows a similar mechanistic interpretation of the propagation of inertio-gravity waves in the rotating shallow water system and the propagation of electro-magnetic waves in a cold plasma.

The non-trivial analogy is perhaps also interesting in that, if anything, one might have expected the formal analogy to be for the case where there is a vertical background magnetic field in a cold plasma system [6], leading to a non-zero Lorentz force that, at first sight, might be formally identified with the Coriolis force in the rotating shallow water system (with the cyclotron frequency formally identified with the inertial frequency). This is indeed what happens in a warm plasma where the pressure gradient cannot be ignored in the momentum equations and, as a consequence, electron-cyclotron waves (of frequency below the cyclotron frequency) exhibit features analogous to inertial waves (waves of frequency below the inertial frequency). However this is not in fact the case for a cold plasma, where we find it is the plasma frequency rather than the cyclotron frequency that is formally identified with the inertial frequency.

As a consequence of the classical dispersion relation of transverse electro-magnetic waves in plasma, laser pulses interacting with a dense, opaque plasma layer (ω_l < ω_c ) initially reflect at its boundary. However, laser pulses above the relativistic intensity threshold (order of 10¹⁸ W cm⁻², [7]), would heat the plasma electrons to nearly the speed of light and thus increase their mass by the Lorentz factor $\gamma =\sqrt{1-{v}^{2}/{c}^{2}}$. Here the dispersion relation becomes ${\omega }^{2}={\omega }_{p}^{2}/\langle \gamma \rangle +{\left({kc}\right)}^{2}$, where 〈γ〉 is averaged over the plasma volume. This effect, known as relativistic transparency [8], allows intense laser pulses to interact volumetrically with classically opaque plasmas, to efficiently accelerate electrons and ions [9]. An equivalent rotating shallow water system to this case would feature a drop in the effective Coriolis frequency ${f}_{\mathrm{eff}}\equiv f(1+\overline{\zeta })$, due to the generation of anticyclonic (negative vorticity) mean shearing current (for instance when $\overline{\zeta }=-\partial \overline{u}/\partial y\lt 0$, where $\overline{u}(y)$ is a mean, local zonal jet, indicated by an overbar, pointing eastward in the x direction and varies meridionally in the y northward direction). In fact, as suggested by [10], this may allow poleward propagation of oceanic near-inertial waves beyond turning latitudes defined only by the value of f. While highly energetic near-inertial waves may alter the mean flow current [11], it would be interesting to find a real-world example by which oceanic near-inertial waves alter themselves the mean flow and by that allow further poleward propagation into the otherwise 'opaque region'.

While in this paper we investigated the analogy between linearly polarized electro-magnetic waves in plasma and inertio-gravity waves, we note that in the case of circularly polarized electro-magnetic wave the same analogy would introduce variation of the Coriolis force with latitude (the β effect). The latter supports the dynamics of Rossby waves side-by-side to the inertio-gravity waves. This maybe a topic for a follow-up paper.

We wish to thank the two anonymous referees for helping to improve the amuscript. EH is grateful to Roy Barkan for fruitful discussion. This research is supported in part by NSF-BSF grant no. 1 025 495. The order of authorship is alphabetical. JM acknowledges financial support from the RGC Early Career Scheme 2 630 020 and the Center for Ocean Research in Hong Kong and Macau, a joint research center between the Qingdao National Laboratory for Marine Science and Technology and Hong Kong University of Science and Technology.

No new data were created or analysed in this study.