Ai}^{\prime}(-k^{2/3}\sigma_{l1}^{2})\right]+\exp({\rm
i}k{\bar{\sigma}}_{l2})$ $\displaystyle\times\left[\mbox{{\boldmath${\bar{\cal
P}}$}}_{l}{\rm Ai}\left(-k^{2/3}{\bar{\sigma}}_{l1}^{2}\right)-{\rm
i}k^{-1/3}\mbox{{\boldmath${\bar{\cal Q}}$}}_{l}{\rm
Ai}^{\prime}\left(-k^{2/3}{\bar{\sigma}}_{l1}^{2}\right)\right]\bigg{|}^{2},$

i.e., the $k$ dependence of the square of the modulus of the integrand that
appears in (3.61). The frequency $\nu$ of the radiation is related to $k$
through $\nu=(2\pi k)/\omega$.

At harmonic numbers $k$ for which the arguments of the Airy functions in
(4.10) are smaller or of the order of unity, these functions assume values
that are independent of frequency. When their arguments are large, on the
other hand, they reduce to

${\rm
Ai}(-k^{2/3}\sigma_{l1}^{2})\simeq\pi^{-1/2}\sigma_{l1}^{-1/2}k^{-1/6}\cos\left({\textstyle\frac{2}{3}}k\sigma_{l1}^{3}-{\textstyle\frac{1}{4}}\pi\right),$
(4.11)

and

${\rm
Ai}^{\prime}(-k^{2/3}\sigma_{l1}^{2})\simeq\pi^{-1/2}\sigma_{l1}^{1/2}k^{1/6}\sin\left({\textstyle\frac{2}{3}}k\sigma_{l1}^{3}-{\textstyle\frac{1}{4}}\pi\right).$
(4.12)

Equation (4.10) and these limiting values of the Airy functions jointly yield
the dependence

${\tilde{I}}\propto k^{-\beta}$ (4.13)

of the radiation intensity ${\tilde{I}}$ on frequency and the values that the
spectral index $\beta$ can assume in various regimes.

The variables $\sigma_{l1}$ and $\sigma_{l2}$ and the vector functions ${\bf
P}_{l}$ and ${\bf Q}_{l}$ that appear in (4.10) are independent of $k$. When
$|{\bf P}_{l}|\gg k^{-1/2}|{\bf Q}_{l}|$, the vector $\mbox{{\boldmath${\cal
K}$}}_{l}$ and hence $\mbox{{\boldmath${\cal P}$}}_{l}$ and
$\mbox{{\boldmath${\cal Q}$}}_{l}$ are also independent of $k$ and are by a
factor of the order of $k^{1/2}$ larger than $\mbox{{\boldmath${\bar{\cal
P}}$}}_{l}$ and $\mbox{{\boldmath${\bar{\cal Q}}$}}_{l}$ (see (3.55)–(3.60)).
In this case, the possible values of the spectral index $\beta$ are determined
by the relative magnitudes of $|\mbox{{\boldmath${\cal P}$}}_{l}|$ and
$|\mbox{{\boldmath${\cal Q}$}}_{l}|$ only. If $|\mbox{{\boldmath${\cal
P}$}}_{l}|\gg k^{-1/3}|\mbox{{\boldmath${\cal Q}$}}_{l}|$, then $\beta=2/3$
when the Airy functions in the first square bracket in (4.10) are of the order
of unity and $\beta=1$ when these Airy functions have the limiting values
given by (4.11) and (4.12) and so the first square bracket in (4.10) decays as
$k^{-1/6}$. If $|\mbox{{\boldmath${\cal P}$}}_{l}|\ll
k^{-1/3}|\mbox{{\boldmath${\cal Q}$}}_{l}|$, then $\beta=4/3$ when the
arguments of the Airy functions in question are of the order of unity and
$\beta=1$ when the first square bracket in (4.10) decays as $k^{-1/6}$. When
$\sigma_{l1}$ is small and the second term of the first square bracket in
(4.10) dominates, there is a short frequency interval in which the spectral
intensity ${\tilde{I}}$ increases with increasing $k$.

In the opposite regime $|{\bf P}_{l}|\ll k^{-1/2}|{\bf Q}_{l}|$, the factor
$k^{-1/2}$ multiplying ${\bf Q}_{l}$ in (3.59) and (3.60) reduces the value of
the spectral index $\beta$ by $1$ everywhere (see Table 4.2).

| $k\ll\sigma_{l1}^{-3}$ | $k\gg\sigma_{l1}^{-3}$
---|---|---
$|{\bf P}_{l}|\gg k^{-1/2}|{\bf Q}_{l}|$ and $|\mbox{{\boldmath${\cal P}$}}_{l}|\gg k^{-1/3}|\mbox{{\boldmath${\cal Q}$}}_{l}|$ | $\quad 2/3$ | $\quad\,\,\,1$
$|{\bf P}_{l}|\gg k^{-1/2}|{\bf Q}_{l}|$ and $|\mbox{{\boldmath${\cal P}$}}_{l}|\ll k^{-1/3}|\mbox{{\boldmath${\cal Q}$}}_{l}|$ | $\quad 4/3$ | $\quad\,\,\,1$
$|{\bf P}_{l}|\ll k^{-1/2}|{\bf Q}_{l}|$ and $|\mbox{{\boldmath${\cal P}$}}_{l}|\gg k^{-1/3}|\mbox{{\boldmath${\cal Q}$}}_{l}|$ | $\quad 5/3$ | $\quad\,\,\,2$
$|{\bf P}_{l}|\ll k^{-1/2}|{\bf Q}_{l}|$ and $|\mbox{{\boldmath${\cal P}$}}_{l}|\ll k^{-1/3}|\mbox{{\boldmath${\cal Q}$}}_{l}|$ | $\quad 7/3$ | $\quad\,\,\,2$
Table 4.2: Values, in various regimes, of the spectral index $\beta$ (defined
in (4.13)) for the part of the radiation associated with the sharp small-
amplitude modulations (microstructure) of the pulse profile.

###  4.4 Flux density and its rate of decay with distance

Flux density of a radiation is given, in general, by the magnitude of the
Poynting vector, $c|{\bf E}^{\rm uc}|^{2}/4\pi$, which has the dimensions of
erg/(cm${}^{2}\times$sec) in cgs units. In the present case, however, the
linear extents in the azimuthal direction, $R_{P}\delta\varphi_{P}$, of the
focused radiation beams that embody the high-frequency (optical to gamma-ray)
radiation are invariably smaller than $1$ cm at $R_{P}=1$ kpc (see Table 4.1).
The amount of energy that crosses a unit area per unit time is therefore given
by

$\displaystyle{\cal S}$ $\displaystyle=$ $\displaystyle\frac{c}{4\pi}|{\bf
E}^{\rm uc}|^{2}\delta\varphi_{P}$ (4.14) $\displaystyle=$ $\displaystyle
2.79\times
10^{-3}{\hat{I}}\delta\varphi_{P}\left(\frac{{\hat{B}}_{0}d^{2}}{{\hat{P}}D}\right)^{2}\quad\frac{{\rm
erg}}{{\rm cm}^{2}\times{\rm sec}},$

in which $\delta\varphi_{P}$ is in radians (see (4.1), (4.4), (4.7) and
(4.8)). Note that the linear extents in the latitudinal direction,
$R_{P}\delta\theta_{P}$, of the focused radiation beams that embody the high-
frequency radiation are of the order of the light-cylinder radius, $c/\omega$,
in general: these beams remain fully in focus at all distances ${\hat{R}}_{P}$
over the latitudinal interval
$\delta\theta_{P}\simeq|\theta_{PlS}-\lim_{R_{P}\to\infty}\theta_{PlS}|$ which
turns out to be of the order of ${\hat{R}}_{P}^{-1}$ independently of the
values of the other parameters.

In the case of $\alpha=60^{\circ}$, $\theta_{P}=90^{\circ}$, $k_{\rm
u}=10^{7}$ and $D=1$ kpc depicted in figure 4.9, for example, the flux density
${\cal S}$ has the value $32.1\,{\cal S}_{0}$ erg/(sec$\times$cm2), where

${\cal S}_{0}=\left(\frac{{\hat{B}}_{0}d^{2}}{{\hat{P}}}\right)^{2}.$ (4.15)

At latitudes closer to or further away from the critical angle for this
example ($\theta_{P}=90^{\circ}$), the degree of focusing of the radiation
beam and so the value of the flux density ${\cal S}$ is, respectively, higher
or lower.

As pointed out in § 3.4, the length of the interval $|\tau_{l{\rm
max}}-\tau_{l{\rm min}}|$ separating the $\tau$ coordinates of the maximum and
minimum of the phase function $f_{lC}$ decreases as ${\hat{R}}_{P}^{-1/2}$
with increasing ${\hat{R}}_{P}$ in a case where this interval is small, i.e.,
when the colatitude $\theta_{P}$ of the observation point has the critical
value $\theta_{PlS}(L,\alpha)$, with $L>{\hat{R}}_{P}$. In particular, if the
observation point has the colatitude $\lim_{R_{P}\to\infty}\theta_{PlS}$ (or
$\pi-\lim_{R_{P}\to\infty}\theta_{PlS}$), then the maximum and minimum of
$f_{lC}$ coalesce into an inflection point only at ${\hat{R}}_{P}\to\infty$,
rather than at a finite distance $L$. (These statements hold true also when
$f_{lC}$ and $\theta_{PlS}$ are replaced by ${\bar{f}}_{lC}$ and
${\bar{\theta}}_{PlS}$, respectively.) In the case illustrated in figure 4.9,
for example, the colatitude of the observation point equals
$\lim_{R_{P}\to\infty}\theta_{P1S}=90^{\circ}$ so that at the finite distance
${\hat{R}}_{P}=10^{13}$, the $\tau$ coordinates of maximum and minimum of
$f_{1C}$ are separated by the short interval $3.05\times 10^{-5}$ degrees. It
follows from the expression for $\partial f_{1C}/\partial\tau$ in (A.1) that
this separation has the value $3.05\times
10^{-5}({\hat{R}}_{P}/10^{13})^{-1/2}$ for all ${\hat{R}}_{P}$.

The enhanced focusing of the radiation with distance that is caused by this
shortening of the separation between the turning points of the phase functions
results in a slower decay rate of the flux density with distance than that
predicted by the inverse-sqaure law. Along colatitudes close to $\theta_{PlS}$
or ${\bar{\theta}}_{PlS}$, the flux density ${\cal S}$ of the radiation
diminishes with increasing distance from its source as ${\hat{R}}_{P}^{-3/2}$
instead of ${\hat{R}}_{P}^{-2}$. This dependence of ${\cal S}$ on
${\hat{R}}_{P}$, or equivalently $D$, is illustrated in figure 4.10 in the
case where $\alpha=60^{\circ}$, $\theta_{P}=90^{\circ}$, $k_{\rm u}=10^{7}$,
and $D$ ranges from $0.1$ to $10^{5}$ kpc, i.e., from a galactic to a
cosmological distance.

Figure 4.10: $log({\cal S}/{\cal S}_{0})$ versus $log(D)$ for
$\alpha=60^{\circ}$, $\theta_{P}=90^{\circ}$ and $k_{\rm u}=10^{7}$. The blue
line with the slope $-3/2$ is the best fit to the red dots whose coordinates
are determined by evaluating (4.14). The violation of the inverse-square law
illustrated in this figure remains in force all the way to infinity whenever
the colatitude of the observation point coincides with or is close to one of
the eight angles given by $\lim_{R_{P}\to\infty}\theta_{PlS}$,
$\lim_{R_{P}\to\infty}{\bar{\theta}}_{PlS}$,
$\pi-\lim_{R_{P}\to\infty}\theta_{PlS}$ and
$\pi-\lim_{R_{P}\to\infty}{\bar{\theta}}_{PlS}$.

The violation of the inverse-square law encountered here (i.e., the fact that
${\cal S}$ is proportional to $D^{-3/2}$ rather than being proportional to
$D^{-2}$) is not incompatible with the requirements of the conservation of
energy because the radiation process discussed in this paper is intrinsically
transitive. Temporal rate of change of the energy density of the radiation
generated by this process has a time-averaged value that is negative (instead
of being zero as in a conventional radiation) at points where the envelopes of
of the wave fronts emanating from the constituent volume elements of the
source distribution are cusped. The difference in the fluxes of power across
any two spheres centred on the star is in this case balanced by the change
with time of the energy contained inside the shell bounded by those spheres
(see appendix C of [2] for a detailed discussion of this point).

##  Appendix A Derivatives of the phase functions $f_{lC}$ and
${\bar{f}}_{lC}$

The first two derivatives with respect to $\tau$ of the function $f_{lC}$ that
appears in the phase of the first exponential in (3.45) are given by

$\frac{\partial
f_{lC}}{\partial\tau}=\frac{\sin\alpha\sin\tau}{\sin\theta}\frac{\partial
f_{lC}}{\partial\theta},$ (A.1)
$\frac{\partial^{2}f_{lC}}{\partial\tau^{2}}=\frac{\sin\alpha\cos^{2}\alpha\cos\tau}{\sin^{3}\theta}\frac{\partial
f_{lC}}{\partial\theta}+\frac{\sin^{2}\alpha\sin^{2}\tau}{\sin^{2}\theta}\frac{\partial^{2}f_{lC}}{\partial\theta^{2}},$
(A.2)

with

$\displaystyle\frac{\partial f_{lC}}{\partial\theta}$ $\displaystyle=$
$\displaystyle({\hat{r}}_{P}^{2}-1)^{-1/2}({\hat{R}}_{P}^{2}\sin^{2}\theta-1)^{-1/2}\left[\csc\theta({\hat{z}}_{P}-{\hat{r}}_{sC}\cos\theta)({\hat{r}}_{P}^{2}{\hat{r}}_{sC}^{2}\sin^{2}\theta-1)^{1/2}\right.$
(A.3)
$\displaystyle\left.-{\hat{r}}_{sC}{\hat{r}}_{P}\sin\theta({\hat{r}}_{P}{\hat{r}}_{sC}\cos\theta-\cot\theta_{P})\right]$
$\displaystyle-(-1)^{l}\cot\alpha\csc^{2}\theta(1-\cot^{2}\alpha\cot^{2}\theta)^{-1/2},$

and

$\displaystyle\frac{\partial^{2}f_{lC}}{\partial\theta^{2}}$ $\displaystyle=$
$\displaystyle({\hat{r}}_{P}^{2}-1)^{-1/2}({\hat{R}}_{P}^{2}\sin^{2}\theta-1)^{-1/2}\Big{\\{}({\hat{r}}_{P}^{2}-1)^{-1/2}({\hat{R}}_{P}^{2}\sin^{2}\theta-1)^{-1/2}$
$\displaystyle\times({\hat{z}}_{P}-{\hat{r}}_{sC}\cos\theta)\Big{[}{\hat{r}}_{P}^{2}{\hat{r}}_{sC}^{2}({\hat{r}}_{P}^{2}{\hat{r}}_{sC}^{2}\sin^{2}\theta-1)^{-1/2}({\hat{z}}_{P}-{\hat{r}}_{sC}\cos\theta)+{\hat{z}}_{P}{\hat{r}}_{sC}\Big{]}$
$\displaystyle-\csc\theta({\hat{r}}_{P}^{2}{\hat{r}}_{sC}^{2}\sin^{2}\theta-1)^{1/2}\Big{[}\cos\theta\frac{\partial{\hat{r}}_{sC}}{\partial\theta}+\csc\theta({\hat{z}}_{P}\cos\theta-{\hat{r}}_{sC})\Big{]}$
$\displaystyle-{\hat{R}}_{P}^{2}\cos\theta({\hat{R}}_{P}^{2}\sin^{2}\theta-1)^{-1}\Big{[}({\hat{z}}_{P}-{\hat{r}}_{sC}\cos\theta)({\hat{r}}_{P}^{2}{\hat{r}}_{sC}^{2}\sin^{2}\theta-1)^{1/2}$
$\displaystyle-{\hat{r}}_{P}{\hat{r}}_{sC}\sin^{2}\theta({\hat{r}}_{P}{\hat{r}}_{sC}\cos\theta-\cot\theta_{P})\Big{]}-{\hat{r}}_{P}^{2}{\hat{r}}_{sC}\Big{(}\sin
2\theta\frac{\partial{\hat{r}}_{sC}}{\partial\theta}+{\hat{r}}_{sC}\cos
2\theta\Big{)}\Big{\\}}$
$\displaystyle+(-1)^{l}\cos\theta\csc^{5}\theta\cot\alpha(1-\cot^{2}\alpha\cot^{2}\theta)^{-3/2}(1-\csc^{2}\alpha\cos
2\theta),$

where

$\frac{\partial{\hat{r}}_{sC}}{\partial\theta}=-\frac{{\hat{r}}_{P}{\hat{r}}_{sC}\sin\theta({\hat{r}}_{P}{\hat{r}}_{sC}\cos\theta-\cot\theta_{P})}{({\hat{r}}_{P}^{2}-1)^{1/2}({\hat{R}}_{P}^{2}\sin^{2}\theta-1)^{1/2}}$
(A.5)

(see (3.16), (3.21) and (3.23)).

The corresponding derivatives of ${\bar{f}}_{lC}$ follow from (A.3),
(LABEL:A4) and the following derivatives of the function $a$ defined in
(3.22):

$\displaystyle\frac{\partial a}{\partial\theta}$ $\displaystyle=$
$\displaystyle\frac{\sin\theta}{({\hat{r}}_{P}^{2}-1)^{1/2}({\hat{R}}_{P}^{2}\sin^{2}\theta-1)^{1/2}}\Bigg{\\{}\frac{{\hat{r}}_{P}^{2}{\hat{r}}_{sC}^{2}({\hat{z}}_{P}-{\hat{r}}_{sC}\cos\theta)}{({\hat{r}}_{P}^{2}{\hat{r}}_{sC}^{2}\sin^{2}\theta-1)^{1/2}}\left[\frac{2a}{({\hat{r}}_{P}^{2}{\hat{r}}_{sC}^{2}\sin^{2}\theta-1)^{1/2}}+1\right]$
(A.6)
$\displaystyle-\frac{({\hat{r}}_{P}^{2}\cos\theta-{\hat{z}}_{P}/{\hat{r}}_{sC})({\hat{r}}_{P}^{2}{\hat{r}}_{sC}^{2}\sin^{2}\theta-1)^{1/2}}{({\hat{r}}_{P}^{2}-1)^{1/2}({\hat{R}}_{P}^{2}\sin^{2}\theta-1)^{1/2}}+\frac{{\hat{R}}_{P}^{2}\cos\theta({\hat{r}}_{P}^{2}{\hat{r}}_{sC}^{2}\sin^{2}\theta-1)^{1/2}}{{\hat{R}}_{P}^{2}\sin^{2}\theta-1}$
$\displaystyle\times\left[\left(\frac{1}{{\hat{r}}_{sC}}-{\hat{r}}_{sC}\sin^{2}\theta_{P}\right)-({\hat{r}}_{P}^{2}{\hat{r}}_{sC}^{2}\sin^{2}\theta-1)^{1/2}\right]\Bigg{\\}}$

and

$\displaystyle\frac{\partial^{2}a}{\partial\theta^{2}}$ $\displaystyle=$
$\displaystyle-\frac{\cot\theta}{{\hat{R}}_{P}^{2}\sin^{2}\theta-1}\frac{\partial
a}{\partial\theta}+\frac{\sin^{2}\theta}{({\hat{r}}_{P}^{2}-1)({\hat{R}}_{P}^{2}\sin^{2}\theta-1)}\Bigg{[}{\hat{r}}_{P}^{2}{\hat{r}}_{sC}^{2}({\hat{z}}_{P}-{\hat{r}}_{zC}\cos\theta)$
(A.7)
$\displaystyle\times\bigg{\\{}-\frac{{\hat{r}}_{P}^{2}{\hat{r}}_{sC}^{2}({\hat{z}}_{P}-{\hat{r}}_{zC}\cos\theta)}{({\hat{r}}_{P}^{2}{\hat{r}}_{sC}^{2}\sin^{2}\theta-1)^{2}}\left[4a+({\hat{r}}_{P}^{2}{\hat{r}}_{sC}^{2}\sin^{2}\theta-1)^{1/2}\right]$
$\displaystyle-\frac{{\hat{r}}_{P}^{2}\cos\theta-{\hat{z}}_{P}/{\hat{r}}_{sC}}{({\hat{r}}_{P}^{2}-1)^{1/2}({\hat{R}}_{P}^{2}\sin^{2}\theta-1)^{1/2}({\hat{r}}_{P}^{2}{\hat{r}}_{sC}^{2}\sin^{2}\theta-1)^{1/2}}+\frac{{\hat{R}}_{P}^{2}\cos\theta}{{\hat{R}}_{P}^{2}\sin^{2}\theta-1}$
$\displaystyle\times\left[\frac{1-{\hat{r}}_{sC}^{2}\sin^{2}\theta_{P}}{{\hat{r}}_{sC}({\hat{r}}_{P}^{2}{\hat{r}}_{sC}^{2}\sin^{2}\theta-1)^{1/2}}-2\right]\bigg{\\}}-{\hat{r}}_{sC}({\hat{r}}_{P}^{2}{\hat{r}}_{sC}\cos\theta-{\hat{z}}_{P})$
$\displaystyle\times\Bigg{\\{}\frac{{\hat{r}}_{P}^{2}{\hat{r}}_{sC}(2{\hat{z}}_{P}-3{\hat{r}}_{sC}\cos\theta)}{{\hat{r}}_{P}^{2}{\hat{r}}_{sC}^{2}\sin^{2}\theta-1}\left[2a+({\hat{r}}_{P}^{2}{\hat{r}}_{sC}^{2}\sin^{2}\theta-1)^{1/2}\right]-\frac{({\hat{r}}_{P}^{2}{\hat{r}}_{sC}^{2}\sin^{2}\theta-1)^{1/2}}{{\hat{r}}_{sC}^{2}({\hat{R}}_{P}^{2}\sin^{2}\theta-1)}$
$\displaystyle\times\left[\frac{{\hat{z}}_{P}({\hat{R}}_{P}^{2}\sin^{2}\theta-1)^{1/2}}{({\hat{r}}_{P}^{2}-1)^{1/2}}+{\hat{R}}_{P}^{2}\cos\theta(1+{\hat{r}}_{sC}^{2}\sin^{2}\theta_{P})\right]\Bigg{\\}}$
$\displaystyle+\frac{{\hat{r}}_{P}^{2}{\hat{r}}_{sC}^{2}({\hat{r}}_{P}^{2}-1)^{1/2}({\hat{R}}_{P}^{2}\sin^{2}\theta-1)^{1/2}}{{\hat{r}}_{P}^{2}{\hat{r}}_{sC}^{2}\sin^{2}\theta-1}\left[2\csc\theta({\hat{z}}_{P}-{\hat{r}}_{sC}\cos\theta)\frac{\partial
a}{\partial\theta}+2{\hat{r}}_{sC}a\right.$
$\displaystyle\left.+{\hat{r}}_{sC}({\hat{r}}_{P}^{2}{\hat{r}}_{sC}^{2}\sin^{2}\theta-1)^{1/2}\right]+\frac{{\hat{R}}_{P}^{2}({\hat{r}}_{P}^{2}{\hat{r}}_{sC}^{2}\sin^{2}\theta-1)^{1/2}}{{\hat{R}}_{P}^{2}\sin^{2}\theta-1}\left({\hat{r}}_{P}^{2}-\sin^{2}\theta_{P}-\frac{{\hat{z}}_{P}}{{\hat{r}}_{sC}}\cos\theta\right)$
$\displaystyle-\frac{{\hat{R}}_{P}^{2}({\hat{r}}_{P}^{2}-1)^{1/2}({\hat{r}}_{P}^{2}{\hat{r}}_{sC}^{2}\sin^{2}\theta-1)^{1/2}}{({\hat{R}}_{P}^{2}\sin^{2}\theta-1)^{3/2}}[{\hat{R}}_{P}^{2}(1+\cos^{2}\theta)-1]$
$\displaystyle\times\left[\frac{1}{{\hat{r}}_{sC}}-{\hat{r}}_{sC}\sin^{2}\theta_{P}-({\hat{r}}_{P}^{2}{\hat{r}}_{sC}^{2}\sin^{2}\theta-1)^{1/2}\right]\Bigg{]}.$

These derivatives can be converted into $\partial a/\partial\tau$ and
$\partial^{2}a/\partial\tau^{2}$ in the same way as was done for the
derivatives of $f_{lC}$ in (A.1) and (A.2).

## References

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# Zero absolute vorticity plane Couette flow as an hydrodynamic representation
of quantum energy states under perpendicular magnetic field

E. Heifetz  Porter school of the Environment and Earth Sciences, Tel Aviv
University, 69978, Israel<EMAIL_ADDRESS>L.R.M. Maas  Institute for
Marine and Atmospheric research Utrecht, University of Utrecht, 3584 CC
Utrecht, NL.  J. Mak  Dept. of Ocean Science and Center for Ocean Research in
Hong Kong and Macau, Hong Kong University of Science and Technology,
Clearwater Bay, Hong Kong SAR.

###### Abstract

Here we extend the Madelung transformation of the Schrödinger equation into a
fluid-like form to include the influence of an external electromagnetic field
on a charged particle. The vorticity of the Madelung fluid is then in the
opposite direction to the imposed magnetic field and equal in magnitude to the
cyclotron angular frequency. When the particle motion is confined to a plane,
perpendicular to an imposed magnetic field, the equivalent flow dynamics is
that of zero absolute vorticity obtained in a quasi 2D rotating frame, where
the cyclotron frequency plays a role equivalent to that of the Coriolis
frequency in a rotating frame. We show how the Landau levels and the extended
modes in the integer quantum Hall effect are all mapped into such zero
absolute vorticity-like plane Couette flows, where the latter exhibit a
geostrophic-like balance between the magnetic force and the gradients of the
quantum (Bohm) potential and the electric force.

††preprint: AIP/123-QED

##  I Introduction

The relation between quantum and fluid mechanics has been established right at
the birth of modern quantum mechanics. Less than a year after Erwin
Schrödinger published his celebrated equation, Erwin Madelung showed (in 1927)
that it can be written in a hydrodynamic-like formMadelung (1927). This
intriguing observation suggested a hydrodynamic approach to quantum mechanics.
The latter however gained modest attention in comparison with the dominant
Copenhagen interpretation. The Madelung equations (ME) remained also
relatively unfamiliar in the fluid dynamics community where the Schrödinger
equation (SE) is mainly implied as a mathematical formulation to find
solutions to the evolution of hydrodynamic waves in different setupsPeregrine
(1983); Maas (1997); Paldor, Rubin, and Mariano (2007).

Recently however, an attempt has been made to re-examine and reinterpret
fundamental quantum phenomena by analyzing their correspondent hydrodynamic
representation. The advantage of this approach is that by mapping quantum
phenomena into classical hydrodynamic ones, we recover physical intuition and
identify familiar flow patterns that shed a new light on the somewhat counter-
intuitive behavior of these phenomena. For instance, from this perspective,
quantum tunneling is enabled due to a local balance between the external
potential barrier and a pressure gradient force exerted by the Madelung fluid,
in a way that the total kinetic (hydrodynamic and internal) energy of the
Madelung fluid remains continuous across the potential barrier Heifetz and
Plochotnikov (2020a). Another example is the problem of a free falling quantum
particle in a gravitational field, that is mapped by the Madelung transform
(JM: it is the transform rather than the equations themselves that do the
mapping) into the dynamics of a 1D stably stratified compressible fluid. The
quantum energy states are mapped into hydrostatic equilibrium states, where
the evaluation of their stability is obtained via the pseudoenergy integral
which in that case is the sum of the kinetic and available potential energies
Heifetz and Plochotnikov (2020b).

In a recent letter in this journal Heifetz and Maas (2021) we discussed a
mapping between the quantum harmonic oscillator ground state and the zero
absolute vorticity plane Couette flow in a rotating frame. Here we show that
this flow pattern is the generic hydrodynamic mapping of quantum eigen-states
with planar channel geometry under an external perpendicular magnetic field.
These eigen-states include the Landau levels, as well as the bulk modes of the
integer quantum Hall effect. In all these cases the Couette flow results from
a geostrophic-like balance between the magnetic force and the gradient of the
superposition of the quantum and the electric potentials.

The paper is organized as follows. In Sec. II we present a simple hydrodynamic
system, in a rotating frame, admitting a steady solution of zero absolute
vorticity plane Couette flow. In Sec. III we derive ME in the absence and in
the presence of an external magnetic field and draw the analogy with zero
absolute vorticity dynamics. Then, in Sec. IV we show how the Landau levels
and the integer quantum Hall effect extended eigen-states are mapped into the
plane Couette flow described in Sec. II. We close in Sec. V by discussing the
results.

##  II Zero absolute vorticity plane Couette flow in a rotating barotropic
compressible system

Consider a barotropic, compressible flow in a counterclockwise rotating
system, with an angular frequency $\Omega=f/2$, where $f$ is the Coriolis
frequency. Viewed from the rotating frame of reference, the flow momentum and
continuity equations read:

${D{\bf u}\over Dt}=-\nabla\left[Q(\rho)+V\right]-{\boldsymbol{f}}\times{\bf
u}\,,$ (1) ${\partial\rho\over\partial t}=-\nabla\cdot(\rho{\bf u})\,.$ (2)

Here $t$ denotes the time and the nabla operator is defined in the Cartesian
coordinates $(x,y,z)$, where $z$ is the vertical coordinate. The velocity
field is given by ${\bf u}=(u,v,w)$, the materiald erivative is
$\mathrm{D}/\mathrm{D}t\equiv\left(\partial/\partial t+{\bf
u}\cdot\nabla\right)$, and ${\boldsymbol{f}}\equiv f{\hat{\bf z}}$, where
${\hat{\bf z}}$ is the vertical unit vector and the axis of rotation.
$Q(\rho)$ is the flow enthalpy (so that $-\nabla Q$ is the pressure gradient
force), $\rho$ is the density and $V$ is a time independent external
potential.

By defining ${\boldsymbol{\omega}}_{a}$ as the absolute flow vorticity (the
flow vorticity viewed from a non-rotating frame of rest), which is the sum of
the flow vorticity ${\boldsymbol{\omega}}$ measured in the rotating frame and
the vorticity contributed by the rotation of the system:

${\boldsymbol{\omega}}_{a}\equiv{\boldsymbol{\omega}}+{\boldsymbol{f}}\,,\hskip
14.22636pt{\boldsymbol{\omega}}=\nabla\times{\bf u}\,,$ (3)

the system (1)-(2) then satisfies the material line equation for
$({\boldsymbol{\omega}}_{a}/\rho)$ (Kundu, Cohen, and Dowling, 2016):

${D\over
Dt}\left({{\boldsymbol{\omega}}_{a}\over\rho}\right)=\left[\left({{\boldsymbol{\omega}}_{a}\over\rho}\right)\cdot\nabla\right]{\bf
u}\,.$ (4)

Thus, for a strictly 2D horizontal flow, in a plane perpendicular to the
rotation axis, the RHS vanishes and $({\boldsymbol{\omega}}_{a}/\rho)$ is
materially conserved (which is a direct consequence of the material
conservation of circulation in a frame of rest). (4) is trivially satisfied
for zero absolute vorticity flows. Furthermore, as ${\bf u}\cdot\nabla{\bf
u}={\boldsymbol{\omega}}\times{\bf u}+\nabla\left(|{\bf u}|^{2}/2\right)$, for
zero absolute vorticity (1) can be written as:

${\partial{\bf u}\over\partial t}=-\nabla\left({|{\bf u}|^{2}\over
2}+Q+V\right)\,,$ (5)

thus for a steady flow, the time independent Bernoulli equation is satisfied:

${|{\bf u}|^{2}\over 2}+{Q}+{V}=\mbox{Be}=\textnormal{constant}\,,$ (6)

where Be is the Bernoulli potential.

As ${\boldsymbol{\omega}}=-{\boldsymbol{f}}$ implies $\left(\partial
v/\partial x-\partial u/\partial y\right)=-f$, the plane Couette flow:

$u(y)=u_{0}+f\,y\,,\qquad u_{0}=u(y=0)\,,$ (7)

is a simple example of such a zero absolute vorticity flow. When both $\rho$
(hence $Q$) and $V$ are only functions of $y$, (7) can be a stationary
solution of (1)-(2), provided that an extended geostrophic balance is
maintained between the Coriolis force and the sum of the pressure gradient
force and the gradient of the external potential $V$:

$fu=-{\partial\over\partial y}(Q+V)\,.$ (8)

In what follows, we show that this simple stationary plane Couette flow is the
generic hydrodynamic representation of fundamental quantum eigen-state
solutions in a rectangular geometry, on a plane perpendicular to an imposed
magnetic field. There, the cyclotron frequency $\omega_{c}$ plays the role of
$f$. To show that, we next derive the hydrodynamic Madelung transformation of
the Schrödinger equation, first (for completeness) for a neutrally charged
quantum particle, and then for a charged particle in the presence of an
electromagnetic field.

##  III Madelung transform of the Schrödinger equation

###  III.1 ME for a neutrally charged particle

The Schrödinger equation (SE), for a non relativistic, neutrally charged,
spinless quantum particle of mass $m$, in the presence of an external scalar
potential $V$ readsMerzbacher (1998):

$i\hbar{\partial\Psi\over\partial t}={\hat{H}}\Psi=\left({{\hat{\bf
p}}^{2}\over 2m}+mV\right)\Psi\,,\qquad{\hat{\bf p}}=-i\hbar\nabla\,.$ (9)

Here ${\hat{H}}$ and ${\hat{\bf p}}$ denote respectively the energy
(Hamiltonian) and momentum operators, acting on the particle wavefunction
$\Psi({\bf r},t)$, where ${\bf r}$ and $t$ denote respectively the position
vector and time. Writing the wavefunction in its polar form:

$\Psi({\bf r},t)=\sqrt{\rho}({\bf r},t)e^{iS({\bf r},t)/\hbar}\,,$ (10)

then $\rho({\bf r},t)$ is the probability density function (PDF) to find the
particle in position ${\bf r}$ at time $t$, and $S$ is the wavefunction phase,
scaled by the reduced Planck constant $\hbar$.

As SE is a complex equation, Madelung Madelung (1927) decomposed it into its
amplitude and phase to obtain two equations. The evolution for the amplitude
is:

${\partial\rho\over\partial t}=-\nabla\cdot\left[\rho\nabla\left({S\over
m}\right)\right]\,.$ (11)

Defining then the velocity according to the de Broglie guiding equation
(Merzbacher, 1998):

${\bf u}=\nabla{\tilde{S}}\,,$ (12)

plays the role of the velocity potential, where tilde denotes hereafter
division by the mass particle $m$, ${\tilde{S}}\equiv S/m$. (11) takes the
form of the familiar continuity equation:

${\partial\rho\over\partial t}=-\nabla\cdot(\rho{\bf u})\,.$ (13)

For the evolution of the phase, Madelung obtained:

${\partial\tilde{S}\over\partial t}=-\left({{\bf u}^{2}\over
2}+{Q}+{V}\right)\,,$ (14)

where

${Q}=-{{\tilde{\hbar}}^{2}\over 2}{\nabla^{2}\sqrt{\rho}\over\sqrt{\rho}}\,,$
(15)

is now the quantum potential (denoted also as the Bohm potential Heifetz and
Cohen (2015)). (14) can be regarded as the quantum Hamilton-Jacobi equation
where ${\tilde{S}}$ plays the role of the action and ${Q}$ is the quantum
correction (for instance, for the electron mass,
${\tilde{\hbar}}^{2}\sim\mathcal{O}(10^{-8}\,\mathrm{m}^{4}\,\mathrm{s}^{-2})$).
Equivalently, (14) can be regarded as the time dependent Bernoulli equation of
the barotropic, inviscid, compressible Madelung fluid where ${Q}$ plays the
formal role of its enthalpy Heifetz and Cohen (2015). As opposed to classical
fluids, this “enthalpy” contains spatial derivatives of the density, and is a
peculiarity of the Madelung fluid.

In the absence of quantized vortices resulting from topological defects, the
Madelung flow is irrotational, i.e., ${\bm{\omega}}=\nabla\times{\bf
u}=\nabla\times(\nabla{\tilde{S}})=0$. Taking the gradient of the two sides of
(14) (and recalling again that $({\bf u}\cdot\nabla){\bf
u}={\bm{\omega}}\times{\bf u}+\nabla({|{\bf u}|^{2}/2})$), Madelung obtained
the Euler-like momentum equation:

${D{\bf u}\over Dt}=-\nabla\left({Q}+{V}\right)\,.$ (16)

Hence the Madelung equations (13) and (16), arising as a transform of the
Schrödinger equation, is formally identified with (1)-(2) in a non-rotating
system (${\boldsymbol{f}}=0$), where the Madelung fluid density $\rho$ is the
PDF of the position of the quantum particle.

For cases where both $\rho$ and $V$ are time independent, the eigenstates of
the time-independent SE ${\hat{H}}{\Psi}=E{\Psi}$ are of the form $S({\bf
r},t)=-E\,t+\eta({\bf r})$ (so that ${\bf u}({\bf r})=\nabla{\tilde{\eta}}$).
These are mapped into stationary anelastic solutions of the Madelung fluid:

$\nabla\cdot(\rho{\bf u})=0\,,$ (17) ${|{\bf u}|^{2}\over
2}+{Q}+{V}={\tilde{E}}=\mbox{Be}\,.$ (18)

Thus, the energy (divided by the particle’s mass) of a quantum eigen-state is
the Bernoulli potential of the corresponding Madelung fluid.

###  III.2 ME for a charged particle in the presence of an external
electromagnetic field

Consider now SE for a charged particle $q$, in the presence of external
magnetic and electric fields, ${\bf B}$ and ${\bf E}$ respectively, where
${\bf B}=\nabla\times{\bf A}$ (${\bf A}$ is the magnetic vector potential),
and ${\bf E}=-\nabla\phi-\partial{\bf A}/\partial t$ ($\phi$ is the electric
scalar potential) fields. The Hamiltonian operator now readsMerzbacher (1998):

${\hat{H}}={1\over 2m}({\hat{\bf p}}-q{\bf A})^{2}+mV\,,\qquad
V={\tilde{q}}\phi\,,$ (19)

where $q$ is the charge of the particle. For particles with non-zero charge,
in order to satisfy the generalized definition of the canonical momentum under
the presence of a magnetic field, ${\bf u}$ (the mechanical momentum per mass
particle) is redefined as (cf. (12)):

${\bf u}=\nabla{\tilde{S}}-{\tilde{q}}{\bf A}\,.$ (20)

With the Coulomb gauge condition $\nabla\cdot{\bf A}=0$, equations (13) and
(14) remain unchanged, and a Helmholtz-like decomposition of the Madelung
velocity field yields:

$\nabla\cdot{\bf
u}=\nabla^{2}{\tilde{S}}\,,\qquad{\bm{\omega}}=\nabla\times{\bf
u}=-{\bm{\omega}}_{c}\,,$ (21)

where $B=|{\bf B}|$, ${\bm{\omega}}_{c}\equiv{\tilde{q}}\,{\bf B}=({\bf
B}/B)\omega_{c}\,$, and $\omega_{c}\equiv{\tilde{q}}B$ is the cyclotron
angular frequency.

Notice that the Madelung fluid divergence field is a variable, but its
vorticity field is dictated by the imposed magnetic field. The latter is equal
to the cyclotron angular frequency and pointing in a direction opposite to the
imposed ${\bf B}$ (where, for an electron with the negative charge $q=-e$, the
vorticity is aligned with the magnetic field). The gradient of (14) yields,
after rearrangement:

${\partial{\bf u}\over\partial t}+\underbrace{\nabla\left({{\bf u}^{2}\over
2}\right)}_{({\bf u}\cdot\nabla){\bf u}-{\bm{\omega}}\times{\bf
u}}=-\nabla{Q}\underbrace{-\left(\nabla\phi+{\partial{\bf A}\over\partial
t}\right)}_{\bf E}{\tilde{q}}\,.$

Since ${\bm{\omega}}=-{\bm{\omega}}_{c}$, we have${\bm{\omega}}\times{\bf
u}=\tilde{q}\,{\bf u}\times{\bf B}$, so adding these terms respectively to
both sides, we obtain the Madelung equation in the presence of the Lorentz
force (JM: normally textbooks write ${\bf u}\times{\bf B}$ so unless there is
a reason for flipping the sign I suggest we stick with convention):

${D{\bf u}\over Dt}=-\nabla{Q}+{\tilde{q}}\,({\bf E}+{\bf u}\times{\bf B})\,.$
(22)

Furthermore, for the case of a time independent magnetic field ($\partial{\bf
A}/\partial t=0$, so that ${\tilde{q}}\,{\bf E}=-\nabla\,V$), we obtain:

${D{\bf u}\over Dt}=-\nabla[Q(\rho)+V]-{\boldsymbol{\omega}}_{c}\times{\bf
u}\,,$ (23)

which is equivalent to (1) with
${\boldsymbol{\omega}}_{c}\leftrightarrow{\boldsymbol{f}}$, where the
equivalent condition for zero absolute vorticity
(${\boldsymbol{\omega}}+{\boldsymbol{\omega}}_{c}=0$) is satisfied by (21).

##  IV Plane Couette flow representation of quantum eigen-states

###  IV.1 General setup

Consider (stationary) eigen-states where the magnetic field is constant and
pointing in the $z$ direction, ${\bf B}=B{\hat{\bf z}}$, and the electric
field, where it exists, is pointing in the $y$ direction with ${\tilde{q}}{\bf
E}(y)=(-\partial V/\partial y){\hat{\bf y}}$, and $V=V(y)$. Furthermore, if we
take $\rho=\rho(y)$, and consequently $Q=Q(y)$, then for the plane Couette
flow ${\bf u}=u(y){\hat{\bf x}}$:

$u(y)=u_{0}+\omega_{c}\,y\,,\qquad u_{0}=u(y=0)\,,$ (24)

we have the extended geostrophic like balance (from (23)):

$\omega_{c}u=-{\partial\over\partial y}\left(Q+V\right)\,.$ (25)

These eigen-state solutions differ from each other by their PDF distribution
$\rho(y)$ and the energy eigen-value ${\tilde{E}}$ corresponding to the
Bernoulli potential. As quantum energy states are quantized, taking the
subscript $n$ to represent those eigen-states where $n=0,1,2...$, the
Bernoulli equation (18) then reads:

${(u_{0}+\omega_{c}y)^{2}\over 2}+Q[\rho_{n}(y)]+V(y)={\tilde{E}}_{n}\,.$ (26)

Taking the Landau Gauge (Tong, 2016) ${\bf A}=-By{\hat{\bf x}}$, we obtain
that $u_{0}={\partial\tilde{S}\over\partial x}$ from (20). For each Fourier
component in the $x$ direction of the wave function (10) (denoted by subscript
$k$), we have $\Psi_{k,n}=\sqrt{\rho_{k,n}(y)}e^{i(kx-E_{n}t/\hbar)}$ and
$u_{0}=\hbar k/m$. Thus, the sinusoidal variation in $x$ of the wave function
contributes only to a shift in the $y$ direction of the zero velocity line of
the Couette flow. Consequently, the hydrodynamic representation of the Fourier
components of these eigen-states are exactly the Couette velocity profiles:

$u_{k}(y)={\tilde{\hbar}}k+\omega_{c}\,y\,,$ (27)

or, equivalently:

$u(Y_{k})=\omega_{c}\,Y_{k}\,,\quad Y_{k}=y+L_{b}^{2}k\,,\quad
L_{b}\equiv\sqrt{{\tilde{\hbar}}\over\omega_{c}}=\sqrt{{\hbar}\over qB}\,,$
(28)

where $L_{b}$ denotes the magnetic length (Tong, 2016).

###  IV.2 Zero electric field - the Landau levels

In the absence of an electric field $V=\textnormal{constant}$, and the
geostrophic-like balance (25) and the Bernoulli equation (18) reduce
respectively to (Fig. 1):

$\omega_{c}u=-{\partial Q\over\partial y}\,,$ (29)
${(u_{0}+\omega_{c}y)^{2}\over 2}+Q[\rho_{n}(y)]={\tilde{E}}_{n}\,,$ (30)

where the spatially uniform electric potential is formally absorbed in
${\tilde{E}}_{n}$.

As (27) and equivalently (28) describe the same unbounded plane Couette flows
for each $k$, just shifted one from each other in the $y$ direction by $\Delta
y=L_{b}^{2}\Delta k$, it is expected that in the absence of an external
potential, such shifts do not change their energy. Therefore all the Fourier
components are degenerate in the sense that they possess the same series of
possible values of the Bernoulli potential. This can be verified explicitly
when substituting (15) in (30):

${1\over
2}\left[(\omega_{c}\,Y_{k})^{2}-{{\tilde{\hbar}}^{2}\over\sqrt{\rho}_{n}}\frac{\mathrm{d}^{2}\sqrt{\rho}_{n}}{\mathrm{d}{Y_{k}}^{2}}\right]={\tilde{E}}_{n}\,,$
(31)

which is identical to the time independent SE for the quantum harmonic
oscillator where $\omega_{c}$ plays the role of the oscillator frequency. Thus
the harmonic potential $(\omega_{c}\,Y_{k})^{2}/2$ is mapped into the kinetic
energy of the Couette flow Heifetz and Maas (2021).

To guarantee localized solutions that vanishes for $|Y_{k}|\rightarrow\infty$,
the harmonic oscillator allows only quantized energy states solutions with
$\mbox{Be}={\tilde{E}}_{n}={\tilde{\hbar}}\omega_{c}\left(n+{1\over 2}\right)$
for non-zero positive integer values of $n$, with the corresponding density
structure:

$\rho_{k,n}(y)={\rho_{n}({Y_{k}})}={1\over
2^{n}n!}\sqrt{\omega_{c}\over\pi{\tilde{\hbar}}}\,H_{n}^{2}\left(\sqrt{\omega_{c}\over{\tilde{\hbar}}}\,{Y_{k}}\right)e^{-{\omega_{c}{Y_{k}}^{2}/{\tilde{\hbar}}}},$
(32)

where $H_{n}$ are the Hermite polynomials of order $n$, and the multiplicative
factors are chosen to normalize the PDF, so that
$\int_{-\infty}^{\infty}\rho_{n}({Y_{k}})\mathrm{d}{Y_{k}}=1$(Merzbacher,
1998).

The energy levels are the Landau levels. From a hydrodynamic perspective, one
may ask for the reason for quantized solutions: why can the Bernoulli
potential take on only discrete set of values, $\mbox{Be}={\tilde{E}}_{n}$,
for the stationary Couette flow, rather than any general set of continuous
values? The reason comes from the peculiar structure of the enthalpy in the
Madelung fluid. Enthalpy in the form of (15) can satisfy the parabolic
structure
${Q}({Y})={\tilde{E}}_{n}-{u^{2}/2}=\mbox{Be}-{(\omega_{c}\,{Y})^{2}/2}\,$
only with specific density structures that correspond to specific discrete
values of Be. Conversely, in classical hydrodynamic the enthalpy of barotropic
fluids is not constrained to be of the form (15), thus there is no counterpart
quantization of Be in the classical setting.

Figure 1: Plane Couette flow representation of the Landau levels. The Couette
flow (green arrows) $u=\omega_{c}Y$ is in the geostrophic-like balance (29),
between the magnetic force (red arrows) and the gradient of the quantum
potential (blue arrows). The latter satisfies (15) for the Hermite squared
polynomials solutions (32) of the PDFs $\rho_{n}$ (filled magenta curves),
corresponding to the discrete energy levels ${\tilde{E}}_{n}$ which are the
permitted Bernoulli potential values of the Couette flow. Despite the
different structure of $\rho_{n}(Y)$, $\mathrm{d}Q(Y)/\mathrm{d}Y$ is
invariant with $n$, hence satisfying the same geostrophic-like balance for the
same Couette flow for all values of $n$. Furthermore, the Landau levels are
degnenerate in the sense that for each wavenumber $k$, the structure
illustrated in this figure is shifted in the $y$ direction by $L_{b}^{2}k$
(according to (28)) without changing the permitted values of the Bernoulli
potential. The mean expectation value of the velocity vanishes according to
(33), as $\rho(Y)$ is symmetric but $u(Y)$ is anti-symmetric.

For each energy state $n$, $\rho_{n}(Y)u(Y)\mathrm{d}Y$ (hereafter dropping
the subscript $k$) is the probability to find the charged particle in between
$Y-\mathrm{d}Y/2<Y<Y+\mathrm{d}Y/2$, moving with the velocity $u(Y)$. The mean
(expectation) value of the Couette flow ${\overline{u}}$, for the Landau
levels are zero, i.e.:

${\overline{u}}\equiv\int_{-\infty}^{\infty}\rho_{n}(y)u(y)\mathrm{d}y=\int_{-\infty}^{\infty}\rho_{n}(Y)\,\omega_{c}\,Y\mathrm{d}Y=0\,,$
(33)

since all of the $\rho_{n}(Y)$ functions are symmetric but $u(Y)$ is anti-
symmetric. This stands in agreement with the geostrophic balance (29), as the
mean value of the quantum potential gradient vanishes
$\int_{-\infty}^{\infty}\rho_{n}(Y)(\mathrm{d}Q/\mathrm{d}Y)\mathrm{d}Y=0$.
This can be verified when substituting ${Q}(Y)$ from (15) and then integrating
by parts when recalling that both $\rho_{n}(Y)$ and
$\mathrm{d}\rho_{n}/\mathrm{d}Y$ vanish at $Y\rightarrow\pm\infty$. It stems
from the more general vanishing of the mean value of the quantum potential
gradient, $\int\rho\,\nabla Q\,\mathrm{d}\Omega=0$, for PDF satisfying
standard boundary conditions ($\rho$ and $\nabla\rho$ vanishing at the
boundaries of the volume domain $\Omega$). (JM: minor inconsistency with
notation, $V$ is also the potential)

The result ${\overline{u}}=0$ is often interpreted as being in agreement with
the classical limit of zero drift of the centers of the circular motions of
charged particles in the plane perpendicular to a constant imposed magnetic
field (positive (negative) charge particles circle clockwise (anti-clockwise)
with the cyclotron angular frequency) (Tong, 2016). This circular motion is
equivalent to the inertial circular motion performed by a fluid particle on an
“f-plane”, resulting from a balance between the Coriolis and the centrifugal
forces acting on the fluid particle (Holton, 2004).

We suggest that we can relate as well a quantum phenomenon in the classical
limit to the Couette flow of the Madelung equations (JM: I am guessing this is
what the sentence is trying to say). Consider a fluid particle at position
${\bf r}(t)=(x(t),y(t))$, circulating clockwise with the cyclotron frequency,
around a center ${\bf r}_{0}=(x_{0},y_{0})$, so that the radius of the circle
is $|{\bf r}-{\bf r}_{0}|$. Then the Cartesian components of the fluid
particle motion are $u_{p}=\omega_{c}(y-y_{0})$ and
$v_{p}=-\omega_{c}(x-x_{0})$. Now, consider an infinite number of all possible
circles, with all possible radii centered at $y_{0}=-L_{b}^{2}k$ ($Y=0$),
where $x_{0}$ varies continuously from $-L_{x}/2$ to $L_{x}/2$. Then the
velocity field, averaged over $x$, resulting from a superposition of all these
circles, is given by
$u=(1/L_{x})\int_{-L_{x}/2}^{L_{x}/2}u_{p}\,\mathrm{d}x_{0}={\tilde{\hbar}}k+\omega_{c}\,y$,
which is the Couette flow as in (27), and
$v=(1/L_{x})\int_{-L_{x}/2}^{L_{x}/2}v_{p}\,\mathrm{d}x_{0}=-\omega_{c}\,x$.
Thus, at the center of the domain, $x=0$, $v$ vanishes due to the cancellation
of positive and negative motion, and in the $y$ direction, every pair of
circles whose centers are located at equal distances $|x_{0}|$, from the left
and the right sides of the domain’s center, $x=0$. For an infinite domain
($L_{x}\rightarrow\infty$), the center can be taken at any point of $x$, so
that we can always find corresponding pairs of circular motions whose
superposition vanishes $v$ at all $x$. Consequently, the resultant mean
superposed flow attributed to all of these possible inertial circles yields
the net Couette flow of (27); see Fig. 2.

Figure 2: Schematic illustration of how a plane Couette flow can be obtained
from a superposition of an infinite number of inertial circles with varying
radii, whose centers are located at the Couette zero velocity line, $Y=0$ (see
more details in text).

While in classical fluid dynamics the value of $u_{0}$ may vary continuously,
for the Landau levels $u_{0}$ is quantized. This leads to the important
property of the Landau levels that the areal density of the charged particles
is proportional to the magnitude of the perpendicular magnetic field. Each
Couette profile $u_{k}$ of (27) or (28) corresponds to a different charged
particle, as each wave number $k=2\pi/\lambda_{x}$ is related to the
wavelength of the quantum wavefunction of a particle. This number is quantized
if we assume periodic boundary conditions at $x=(0,L_{x})$, so that
$\lambda_{x}^{j}=L_{x}/j$, for $j=1,2,3...$ Consequently the allowed
wavenumbers are $k^{j}=\left(2\pi/L_{x}\right)j$, hence the difference in the
$y$ direction between the zero velocity lines of the allowed Couette flows is
$\Delta y_{0}=2\pi(L_{b}^{2}/L_{x})$. Thus, the number of allowed charged
particles in a slab of width $L_{y}$ at an energy level is given by
$N=L_{y}/\Delta y_{0}=(B/\Phi_{0})A$, where $A=L_{x}L_{y}$ is the area of the
slab and $\Phi_{0}\equiv h/q$ is denoted as the quantum magnetic flux (where
$h=2\pi\hbar$ is the Planck constant). Therefore, the areal charge density in
a Landau level $N/A=B/\Phi_{0}$ (not to be confused with the probability
density function, $\rho$, to find a single particle) is proportional to $B$.
Consequently, when $\nu$ number of Landau energy levels are filled, the charge
areal density $n_{2D}\equiv\nu N/A=B\nu\Phi_{0}=Bq\nu/h$.

###  IV.3 Non zero electric field - the integer quantum Hall effect

When adding a constant transverse electric field ${\bf E}=E_{y}{\hat{\bf y}}$
($E_{y}=-\partial\phi/\partial y$, for $\phi=-yE_{y}$), the Couette flow
solution of (27) remains unchanged, but the gradient of the quantum potential
is now accompanied by the constant electric field to balance the magnetic
force, so that (25) reads:

$\omega_{c}u=-\left({\partial Q\over\partial y}+{\tilde{q}}E_{y}\right)\,.$
(34)

This corresponds to the extended bulk modes of the integer quantum Hall effect
(when $q=-e$). The Bernoulli equation (18) in the shifted $Y$ coordinate then
becomes:

${1\over
2}\left[(\omega_{c}\,Y)^{2}-{{\tilde{\hbar}}^{2}\over\sqrt{\rho}}\frac{\mathrm{d}^{2}\sqrt{\rho}}{\mathrm{d}{Y}^{2}}\right]-{\tilde{q}}E_{y}\,(Y-kL_{b}^{2})={\tilde{E}}=\mbox{Be}\,.$
(35)

By making the additional coordinate shift ${\cal Y}\equiv
Y-{\tilde{q}}E_{y}/\omega_{c}^{2}$, (35) can be rewritten, after completing
the square and using (32), as:

$\displaystyle{1\over 2}\left[(\omega_{c}\,{\cal
Y})^{2}-{{\tilde{\hbar}}^{2}\over\sqrt{\rho}}{\partial^{2}\sqrt{\rho}\over\partial{\cal
Y}^{2}}\right]$ $\displaystyle+\left({E_{y}\over
B}\right){\tilde{\hbar}}k-{1\over
2}\left({{\tilde{q}}E_{y}\over\omega_{c}}\right)^{2}$ (36)
$\displaystyle={\tilde{\hbar}}\omega_{c}\left(n+{1\over
2}\right)+\left({E_{y}\over B}\right)\left[{\tilde{\hbar}}k-{1\over
2}\left({E_{y}\over B}\right)\right]={\tilde{E}}_{k,n}=\mbox{Be}\,.$

Hence, the existence of a constant electric field breaks the degeneracy of the
Landau levels and makes the Bernoulli constant take different values for each
allowed set of $(k,n)$. The reason for the breaking of the degeneracy is
demonstrated in Fig. 3. While the zero velocity line of the Couette flow is
located at $Y_{k}=0$, the electric potential zero line is at
$Y_{k}=L_{b}^{2}k$, hence this mismatch depends on $k$. In addition, the
density structure $\rho_{k,n}$ of the Hermite polynomials (32) are centered
around ${\cal Y}_{k}=0$ so that $Y_{k}={\tilde{q}}E_{y}/\omega_{c}^{2}$, i.e.
around $u=\omega_{c}Y_{k}=E_{y}/B$. Therefore, although the Couette velocity
profile itself is not altered by the presence of the constant electric field,
the shift in the $y$ direction between the density structure and the Couette
velocity profile breaks the anti-symmetric structure of $\rho u$ with respect
to $Y_{k}=0$. Consequently, when multiplying (34) by $\rho$ and integrating
over $y$ we obtain that the velocity expectation value is non-zero and
satisfying a geostrophic-like balance between the magnetic and the electric
forces, where the mean effect of the quantum potential gradient vanishes as
before:

$\omega_{c}{\overline{u}}=-{\partial V\over\partial y}=\tilde{q}E_{y}\hskip
7.11317pt\Longrightarrow\hskip 7.11317ptB{\overline{u}}=E_{y}\,.$ (37)

The expectation velocity value is equal to the velocity at the PDF center
line, regardless of the values of $(k,n)$. It obeys a geostrophic like balance
between the magnetic and the electric forces where $E_{y}/B$ is indeed the
classical drift velocity. Hence, the super-positioning of inertial circles,
illustrated in Fig. 2, is applicable as well in the presence of a constant
transverse electric field, when viewed from a frame moving with the drift
velocity.

The expectation value of the charged particle velocity is in agreement as well
with the classical limit of the classical Hall effect. Define the Hall voltage
difference in the $y$ direction as $V_{H}\equiv L_{y}E_{y}$, then the
transverse Ohm’s law for the classical Hall effect is $V_{H}=I_{x}R_{xy}$,
where the electric current in the $x$ direction is $I_{x}\equiv
q\,n_{2D}\,{\overline{u}}\,L_{y}$ and the transverse resistance
$R_{xy}=B/(q\,n_{2D})$. However, while in the classical Hall effect $R_{xy}$
is proportional to the imposed magnetic field, in the integer quantum Hall
effect $n_{2D}=Bq\nu/h$, thus $R_{xy}=h/(q^{2}\nu)$, is independent of $B$.
This results in a series of “plateaus” of constant values of $R_{xy}$, when
plotted against $B$, in the intervals where the integer number of filled
Landau levels remains constant (Tong, 2016).

Figure 3: Bulk modes of the integer quantum Hall effect. The setup is the same
as in Fig. 1, but with an additional constant electric field $E_{y}$ in the
$Y$ direction. The Couette flow is now in a geostrophic-like balance between
the gradient of the quantum potential and the magnetic and electric forces
(the latter is indicated by the magenta arrows). To maintain the balance, the
center of the PDF structures is shifted in the $Y$ direction by
${{\tilde{q}}E_{y}\over\omega_{c}^{2}}$, in order to vanish the gradient of
the quantum potential at the location where the magnetic and the electric
forces are in balance. This shift breaks the anti-symmetric relations between
$\rho$ and $u$, yielding the non zero expectation value
${\overline{u}}=E_{y}/B$ as in (37).

##  V Discussion

The Madelung momentum equation (16) for a quantum particle is somewhat
elusive. It differs from the classical momentum equation by the presence of
the gradient of the quantum potential, but its expectation value generally
vanishes (when applying $\int\rho\,(...)\,\mathrm{d}V$ on the two sides of the
equation). Furthermore, for a charged quantum particle in the presence of an
electromagnetic force (23), the velocity field is generally both divergent and
rotational, but the latter property is not an independent variable but
dictated by the magnetic field. These two subtle issues seem to obscure the
simple representation of the energy states of the Landau levels, and the
extended modes of the integer quantum Hall effect, as sets of simple plane
Couette flows with a constant shear that is equal to the cyclotron frequency.
The geostrophic-like balance of this flow together with the Bernoulli energy
equation it obeys, provide familiarity and physical intuition from the realm
of geophysical fluid dynamics, thus suggesting a different angle of
understanding.

This intuition is however partial. There are no direct fluid mechanics
counterparts to the quantum effects which result both from the structure of
the “quantum enthalpy” and the stream-wise quantized boundary conditions of
the particle’s wavefunction. Nevertheless, it is interesting to compare the
magnetic length scale $L_{b}$ with the Rossby deformation radius length scale
$L_{d}=\sqrt{gH}/f$ (which, roughly speaking, is the length covered in an
inertial period $1/f$ by a wave propagating at long wave speed $\sqrt{gH}$,
where $f$ is the Coriolis frequency, $g$ is gravity and $H$ is the mean layer
thickness). Despite of the enormous scale difference (while $L_{d}$ varies
from $O(10^{4}-10^{6}\,\mathrm{m})$ in the ocean and the atmosphere,
respectively, $L_{b}\approx 2.5\times 10^{-8}\,\mathrm{m}$ for an electron in
a magnetic field of 1 Tesla), if we denote $f=\omega_{c}\equiv\omega$ and
equate $L_{b}$ with $L_{d}$ we obtain $gH={\tilde{\hbar}}\omega_{c}$. Hence,
although the quantized energy levels ${\tilde{E}}_{n}$ have no direct
classical counter-part, the mean potential energy in the geophysical layer
plays the role of the energy difference between two adjacent Landau levels, as
if jumping from one energy level to the adjacent one requires adding another
layer of thickness $H$. Furthermore, in order for $L_{d}$ and $L_{b}$ to play
an equivalent role, $gH/f$ should correspond to a constant, which means that
the mean layer thickness should be proportional to the Coriolis frequency. We
cannot think of any reason, or physical constraint to justify it. However, as
in shallow water system the fluid density is assumed constant, if such a
scenario exists, the mass of a column per unit area, $M/A$, would be
proportional to $f$, which corresponds to the quantum case where
$n_{2D}\propto B$.

We close the article by noting that while the Madelung equations may sometimes
be regarded as a curiosity leading to what might be perceived as superficial
links between quantum mechanics and hydrodynamics, there is in fact a deep
mathematical connection between the quantum system and hydrodynamicsKhesin,
Misiolek, and Modin (2018). In this case the transform between the Schrödinger
and Madelung equations is in fact a symplectomorphism of the corresponding
phase spaces, and fall under a more general and unified geometric framework
with connections to other popular equations in mathematical physicsKhesin,
Misiolek, and Modin (2018). Given the fundamental links, it would be of
interest to see whether quantum mechanical features can advance our
understanding of fluid dynamics or vice-versa. For the former, we note that
the integer quantum Hall effect has been noted have intimate links with
topological invariants, and it would be of interest to see how such
fundamentally quantum mechanical effects manifest for the analogous fluid
system through the bulk-boundary correspondence, some of which have recent
received interest in the fluids communityParker  _et al._ (2020); Delplace,
Marston, and Venaille (2017). For the latter, we note for example that fluid
instabilities have been previously noted to have some formal links with the
breakdown of the quantum Hall effectEaves (2001). Ultimately the goal would be
for one field to be able to predict something we do not already know in the
other, and to that end further work is required in highlighting links between
the two fields through the Madelung formalism, some of which will be our focus
in the near future.

###### Acknowledgements.

JM acknowledges financial support from the RGC General Research Fund 16304021
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.

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