$\displaystyle\frac{\mathrm{D}u^{\prime}}{\mathrm{D}t^{\prime}}+\frac{u^{\prime}w^{\prime}-u^{\prime}v^{\prime}\tan\theta}{r^{\prime}}-2\varOmega^{\prime}(v^{\prime}\sin\theta-w^{\prime}\cos\theta)=-\frac{1}{\rho^{\prime}}\frac{1}{r^{\prime}\cos\theta}\frac{\partial
p^{\prime}}{\partial\varphi}\,,$ (7.1b)
$\displaystyle\frac{\mathrm{D}v^{\prime}}{\mathrm{D}t^{\prime}}+\frac{v^{\prime}w^{\prime}+u^{\prime
2}\tan\theta}{r^{\prime}}+2\varOmega^{\prime}u^{\prime}\sin\theta+\varOmega^{\prime
2}r^{\prime}\sin\theta\cos\theta=-\frac{1}{\rho^{\prime}}\frac{1}{r^{\prime}}\frac{\partial
p^{\prime}}{\partial\theta}\,,$ (7.1c)
$\displaystyle\frac{\mathrm{D}w^{\prime}}{\mathrm{D}t^{\prime}}-\frac{u^{\prime
2}+v^{\prime
2}}{r^{\prime}}-2\varOmega^{\prime}u^{\prime}\cos\theta-\varOmega^{\prime
2}r^{\prime}\cos^{2}\theta=-\frac{1}{\rho^{\prime}}\frac{\partial
p^{\prime}}{\partial r^{\prime}}-g^{\prime}\,,$

where the material derivative $\mathrm{D}/\mathrm{D}t^{\prime}$ in spherical
coordinates is given by

$\frac{\mathrm{D}\phantom{|}}{\mathrm{D}t^{\prime}}=\frac{\partial}{\partial
t^{\prime}}+\frac{u^{\prime}}{r^{\prime}\cos\theta}\frac{\partial}{\partial\varphi}+\frac{v^{\prime}}{r^{\prime}}\frac{\partial}{\partial\theta}+w^{\prime}\frac{\partial}{\partial
r^{\prime}}\,.$

Here $p^{\prime}$ and $\rho^{\prime}$ are the pressure and density in the
atmosphere, $\varOmega^{\prime}$ is the constant rate of rotation of the
planet, and $g^{\prime}$ is the acceleration due to gravity, taken to be a
constant. (We use primes to denote physical/dimensional variables; they will
be removed when we nondimensionalize.) The conservation of mass in spherical
coordinates takes the form

(7.2)
$\displaystyle\frac{\mathrm{D}\rho^{\prime}}{\mathrm{D}t^{\prime}}+\rho^{\prime}\left(\frac{1}{r^{\prime}\cos\theta}\,\frac{\partial
u^{\prime}}{\partial\varphi}+\frac{1}{r^{\prime}\cos\theta}\,\frac{\partial}{\partial\theta}(v^{\prime}\cos\theta)+\frac{1}{r^{\prime
2}}\,\frac{\partial}{\partial r^{\prime}}\,(r^{\prime
2}w^{\prime})\right)=0\,,$

while the equation of state for an ideal gas reads

(7.3) $p^{\prime}=\rho^{\prime}{\mathfrak{R}}^{\prime}T^{\prime}\,,$

where $T^{\prime}$ is the (absolute) temperature and ${\mathfrak{R}}^{\prime}$
is the gas constant. The first law of thermodynamics should also hold:

(7.4)
$c_{p}^{\prime}\,\frac{{\mathrm{D}}T^{\prime}}{\text{D}t^{\prime}}-\kappa^{\prime}\nabla^{\prime
2}T^{\prime}-\frac{1}{\rho^{\prime}}\,\frac{{\mathrm{D}}p^{\prime}}{{\mathrm{D}}t^{\prime}}=Q^{\prime}\,,$

where

$\nabla^{\prime 2}\equiv\frac{\partial^{2}}{\partial r^{\prime
2}}+\frac{2}{r^{\prime}}\,\frac{\partial}{\partial
r^{\prime}}+\frac{1}{r^{\prime
2}}\Big{(}\frac{1}{\cos^{2}\theta}\,\frac{\partial^{2}}{\partial\varphi^{2}}+\frac{\partial^{2}}{\partial\theta^{2}}-\tan\theta\frac{\partial}{\partial\theta}\Big{)}\,,$

$c_{p}^{\prime}$ is the specific heat and $\kappa^{\prime}/c_{p}^{\prime}$ is
the thermal diffusivity, while $Q^{\prime}$ is a general heat-source term. We
will mainly work with the pressure and the density, so that in our setting the
role of the ideal gas law (7.3) is to specify the temperature, while the
second law of thermodynamics identifies the associated heat sources.

We now introduce the following dimensional scales:

(7.5) $\displaystyle\begin{aligned} R^{\prime}&:\text{ radius of the planet
(as a distance scale)}\\\ H^{\prime}&:\text{ mean width of the
stratosphere}\\\ U^{\prime}&:\text{ horizontal velocity scale}\\\
W^{\prime}&:\text{ vertical velocity scale}\\\
\overline{\rho}^{\prime}&:\text{ average density of the
stratosphere}\,.\end{aligned}$

The inverse Rossby number is defined as

(7.6) $\displaystyle\omega=\frac{\varOmega^{\prime}R^{\prime}}{U^{\prime}}\,,$

and two further important flow-parameters are the shallowness parameter $\mu$
and the ratio $\delta$ between the vertical and horizontal velocity scales,
given by

(7.7)
$\mu=\frac{H^{\prime}}{R^{\prime}}\quad\text{and}\quad\delta=\frac{W^{\prime}}{U^{\prime}}\,.$

The relevant data is suggested by the characteristics of persistent large-
scale flow patterns in the stratosphere (see [11, 22, 39]):

Planet |  $R^{\prime}$ |  $H^{\prime}$ |  $g^{\prime}$ |  $\varOmega^{\prime}$ |  $U^{\prime}$ |  $W^{\prime}$ |  $\omega$ |  $\mu$ |  $\delta$
---|---|---|---|---|---|---|---|---|---
Earth |  6371 km |  40 km |  9.8 m/s2 |  $7.27\times 10^{-5}\,\text{rad/s}$ |  50 m/s |  $10^{-3}$ m/s |  9 |  $6\times 10^{-3}$ |  $2\times 10^{-5}$
Jupiter |  69911 km |  270 km |  24.8 m/s2 |  $1.76\times 10^{-4}\,\text{rad/s}$ |  150 m/s |  $10^{-2}$ m/s |  82 |  $4\times 10^{-3}$ |  $6\times 10^{-5}$
Saturn |  58232 km |  200 km |  10.4 m/s2 |  $1.62\times 10^{-4}\,\text{rad/s}$ |  150 m/s |  $10^{-2}$ m/s |  63 |  $3\times 10^{-3}$ |  $6\times 10^{-5}$
Neptune |  24622 km |  200 km |  11.1 m/s2 |  $1.08\times 10^{-4}\,\text{rad/s}$ |  200 m/s |  $10^{-3}$ m/s |  13 |  $8\times 10^{-3}$ |  $5\times 10^{-6}$
Uranus |  25362 km |  150 km |  8.8 m/s2 |  $1.04\times 10^{-4}\,\text{rad/s}$ |  150 m/s |  $10^{-5}$ m/s |  18 |  $6\times 10^{-3}$ |  $6\times 10^{-8}$

Since typically $\delta\leq 10^{-4}$, the time scale $R^{\prime}/U^{\prime}$
is determined by the horizontal flow (the values for Earth, Jupiter, Saturn,
Neptune and Uranus being about 1.5, 5.5, 4.5, 1.4 and 2 days, respectively).
We now define the dimensionless variables $t$, $z$, $u$, $v$, $w$, $\rho$, and
$p$ by

(7.8) $t^{\prime}=\displaystyle\frac{R^{\prime}}{U^{\prime}}\,t\,,\quad
r^{\prime}=R^{\prime}+H^{\prime}z\,,\quad(u^{\prime},v^{\prime})=U^{\prime}(u,v)\,,\quad
w^{\prime}=W^{\prime}w\,,\quad\rho^{\prime}=\overline{\rho^{\prime}}\rho\,,\quad
p^{\prime}=\overline{\rho^{\prime}}U^{\prime 2}p\,,$

and obtain from (7.1)-(7.2) the components of the nondimensional Euler
equation

(7.9) $\displaystyle\displaystyle\frac{\mathrm{D}u}{\mathrm{D}t}+\frac{\delta
uw-uv\tan\theta}{1+\mu z}-2\omega(v\sin\theta-\delta
w\cos\theta)=-\frac{1}{\rho}\frac{1}{(1+\mu z)\cos\theta}\,\frac{\partial
p}{\partial\varphi}$ (7.10)
$\displaystyle\displaystyle\frac{\mathrm{D}v}{\mathrm{D}t}+\frac{\delta
vw+u^{2}\tan\theta}{1+\mu z}+2\omega u\sin\theta+\omega^{2}(1+\mu
z)\sin\theta\cos\theta=-\frac{1}{\rho}\frac{1}{1+\mu z}\,\frac{\partial
p}{\partial\theta}$ (7.11)
$\displaystyle\displaystyle\mu\delta\,\frac{\mathrm{D}w}{\mathrm{D}t}-\mu\,\frac{u^{2}+v^{2}}{1+\mu
z}-2\mu\omega u\cos\theta-\mu\omega^{2}(1+\mu
z)^{2}\cos^{2}\theta=-\frac{1}{\rho}\frac{\partial p}{\partial z}-g\,,$

and the nondimensional equation of mass conservation

(7.12) $\frac{\mathrm{D}\rho}{\mathrm{D}t}+\rho\Big{\\{}\frac{1}{(1+\mu
z)\cos\theta}\,\frac{\partial u}{\partial\varphi}+\frac{1}{(1+\mu
z)\cos\theta}\,\frac{\partial}{\partial\theta}(v\cos\theta)+\frac{\delta}{\mu}\frac{1}{(1+\mu
z)^{2}}\,\frac{\partial}{\partial z}\,\Big{(}(1+\mu
z)^{2}w\Big{)}\Big{\\}}=0\,,$

where

$\frac{\mathrm{D}\phantom{|}}{\mathrm{D}t}=\frac{\partial}{\partial
t}+\frac{u}{(1+\mu
z)\cos\theta}\frac{\partial}{\partial\varphi}+\frac{v}{1+\mu
z}\frac{\partial}{\partial\theta}+\frac{\delta}{\mu}\,w\frac{\partial}{\partial
z}\quad\text{and}\quad g=\frac{g^{\prime}H^{\prime}}{U^{\prime 2}}\,,$

with $g\approx 157$ for Earth, $g\approx 297$ for Jupiter, $g\approx 92$ for
Saturn, $g\approx 58$ for Uranus, and $g\approx 55$ for Neptune.

We are interested in the leading-order dynamics as $\mu\to 0$, the physically
relevant regime for the thin-shell stratosphere being characterised by

(7.13) $\delta\ll\mu\ll 1\,,$

so that the flow dynamics is governed at leading-order by the non-dimensional
equations (7.9)-(7.12) in the limit $\mu\to 0$:

(7.14) $\displaystyle\displaystyle\frac{\partial u_{0}}{\partial
t}+\frac{u_{0}}{\cos\theta}\frac{\partial
u_{0}}{\partial\varphi}+v_{0}\frac{\partial
u_{0}}{\partial\theta}-u_{0}v_{0}\tan\theta-2\omega\,v_{0}\sin\theta=-\frac{1}{\rho_{0}}\frac{1}{\cos\theta}\,\frac{\partial
p_{0}}{\partial\varphi}\,,$ (7.15) $\displaystyle\displaystyle\frac{\partial
v_{0}}{\partial t}+\frac{u_{0}}{\cos\theta}\frac{\partial
v_{0}}{\partial\varphi}+v_{0}\frac{\partial
v_{0}}{\partial\theta}+u_{0}^{2}\tan\theta+2\omega\,u_{0}\sin\theta+\omega^{2}\sin\theta\cos\theta=-\frac{1}{\rho_{0}}\,\frac{\partial
p_{0}}{\partial\theta}\,,$ (7.16) $\displaystyle
0=\displaystyle\frac{1}{\rho_{0}}\frac{\partial p_{0}}{\partial z}+g\,,$
(7.17) $\displaystyle\displaystyle\frac{\partial\rho_{0}}{\partial
t}+\frac{u_{0}}{\cos\theta}\frac{\partial\rho_{0}}{\partial\varphi}+v_{0}\frac{\partial\rho_{0}}{\partial\theta}+\frac{\rho_{0}}{\cos\theta}\Big{(}\frac{\partial
u_{0}}{\partial\varphi}+\frac{\partial}{\partial\theta}(v_{0}\cos\theta)\Big{)}=0\,.$

Throughout the stratosphere the main changes in density are in the vertical
direction, with the density decreasing with height (e.g., from about 100 g/cm3
at the bottom of the Earth’s stratosphere to about 1 g/cm3 at its top), so
that we restrict our attention to the setting

(7.18) $\rho_{0}=\rho_{0}(z)\,.$

The flow dynamics is then governed at leading-order by the system

(7.19a) $\displaystyle\frac{\partial u_{0}}{\partial
t}+\frac{u_{0}}{\cos\theta}\frac{\partial
u_{0}}{\partial\varphi}+v_{0}\frac{\partial
u_{0}}{\partial\theta}-u_{0}v_{0}\tan\theta-2\omega\,v_{0}\sin\theta=-\frac{1}{\rho_{0}\cos\theta}\frac{\partial
p_{0}}{\partial\varphi}\,,$ (7.19b) $\displaystyle\frac{\partial
v_{0}}{\partial t}+\frac{u_{0}}{\cos\theta}\frac{\partial
v_{0}}{\partial\varphi}+v_{0}\frac{\partial
v_{0}}{\partial\theta}+u_{0}^{2}\tan\theta+2\omega\,u_{0}\sin\theta+\omega^{2}\sin\theta\cos\theta=-\frac{1}{\rho_{0}}\frac{\partial
p_{0}}{\partial\theta}\,,$ (7.19c) $\displaystyle
0=\frac{1}{\rho_{0}}\frac{\partial p_{0}}{\partial z}+g\,,$ (7.19d)
$\displaystyle\frac{\partial
u_{0}}{\partial\varphi}+\frac{\partial}{\partial\theta}(v_{0}\cos\theta)=0\,.$

For $\rho_{0}$ constant the system (7.19) particularizes to that describing
inviscid flow on the surface of a rotating sphere. This feature is related to
the fact that, due to an ascending temperature with height, the stratosphere
is stably stratified and vertical motion is suppressed.

Equation (7.19c) yields the existence of a stream function,
$\psi(\varphi,\theta,z,t)$, satisfying

(7.20) $u_{0}=-\frac{\partial\psi}{\partial\theta}\quad\text{and}\quad
v_{0}=\frac{1}{\cos\theta}\frac{\partial\psi}{\partial\varphi}\,,$

while the elimination of the dynamic pressure $P_{0}$ between the equations
(7.19a)-(7.19b) gives the vorticity equation

(7.21) $\frac{\partial}{\partial
t}\,\Delta\psi+\frac{1}{\cos\theta}\,\left[\frac{\partial\psi}{\partial\varphi}\,\frac{\partial}{\partial\theta}-\frac{\partial\psi}{\partial\theta}\,\frac{\partial}{\partial\varphi}\right]\left(\nabla^{2}_{\Sigma}\psi+2\omega\sin\theta\right)=0\,,$

in which
$\Delta=\frac{\partial^{2}}{\partial\theta^{2}}-\tan\theta\frac{\partial}{\partial\theta}+\frac{1}{\cos^{2}\theta}\frac{\partial^{2}}{\partial\varphi^{2}}$
is the Laplace-Beltrami operator on the surface of the unit sphere
${\mathbb{S}}^{2}$ and $\Delta\psi$ is the vorticity of the flow.

######  Lemma 2.

If $\psi_{0}(\varphi,\theta)$ solves

(7.22)
$\Delta\psi_{0}=F(\psi_{0})-\frac{1}{4\pi}\,\iint_{{\mathbb{S}}^{2}}F(\psi_{0})\,{\rm
d}\sigma$

for some $F\in C^{1}({\mathbb{R}},{\mathbb{R}})$, then

(7.23)
$\psi(\varphi,\theta,z,t)=\omega\sin\theta+\psi_{0}(\varphi+\omega\,t,\theta)$

is a solution of the vorticity equation (7.21).

###### Proof.

Since $\Delta\sin\theta=-2\sin\theta$, we have

$\Delta(\psi_{0}+\omega\sin\theta)=F(\psi_{0})-2\omega\sin\theta-\frac{1}{4\pi}\,\iint_{{\mathbb{S}}^{2}}F(\psi_{0})\,{\rm
d}\sigma\,,$

so that

$\Sigma\Delta+2\omega\sin\theta=F(\widehat{\psi})-\frac{1}{4\pi}\,\iint_{{\mathbb{S}}^{2}}F(\widehat{\psi})\,{\rm
d}\sigma\,,$

for

$\Psi(\varphi,\theta,t)=\omega\sin\theta+\widehat{\psi}\,,\qquad\widehat{\psi}(\varphi,\theta,t)=\psi_{0}(\varphi+\omega
t,\theta)\,.$

We now compute

$\frac{1}{\cos\theta}\,\left[\frac{\partial\Psi}{\partial\varphi}\,\frac{\partial}{\partial\theta}-\frac{\partial\Psi}{\partial\theta}\,\frac{\partial}{\partial\varphi}\right]\left(\Delta\Psi+2\omega\sin\theta\right)=\frac{1}{\cos\theta}\,\left[\frac{\partial\widehat{\psi}}{\partial\varphi}\,\frac{\partial}{\partial\theta}-\Big{(}\omega\cos\theta+\frac{\partial\widehat{\psi}}{\partial\theta}\Big{)}\,\frac{\partial}{\partial\varphi}\right]\,F(\widehat{\psi})=-\omega
F^{\prime}(\widehat{\psi})\,\frac{\partial\widehat{\psi}}{\partial\varphi}\,.$

On the other hand,

$\frac{\partial}{\partial t}\,\Delta\Psi=\frac{\partial}{\partial
t}\,\Big{(}\Delta\widehat{\psi}-2\omega\sin\theta\Big{)}=\frac{\partial}{\partial
t}\,F(\widehat{\psi})=F^{\prime}(\widehat{\psi})\,\frac{\partial\widehat{\psi}}{\partial
t}=\omega
F^{\prime}(\widehat{\psi})\,\frac{\partial\widehat{\psi}}{\partial\varphi}\,,$

so that $\psi$ solves (7.21). ∎

######  Theorem 11.

Given the vertical density stratification of the stratosphere $\rho_{0}(z)$,
if $\psi_{0}(\varphi,\theta)$ solves

(7.24) $\Delta\psi_{0}=F(\psi_{0})$

for some $F\in C^{1}({\mathbb{R}},{\mathbb{R}})$, then

(7.25)
$\psi(\varphi,\theta,z,t)=\omega\sin\theta+\frac{1}{\sqrt{\rho_{0}(z)}}\,\psi_{0}(\varphi+\omega\,t,\theta)$

with the associated pressure

(7.26)
$p_{0}(\varphi,\theta,z,t)={\mathcal{F}}(\psi_{0}(\varphi+\omega\,t,\theta))-\frac{1}{2}\,\Big{(}\frac{\partial\psi_{0}}{\partial\theta}\,(\varphi+\omega\,t,\theta)\Big{)}^{2}-\frac{1}{2\cos^{2}\theta}\,\Big{(}\frac{\partial\psi_{0}}{\partial\varphi}\,(\varphi+\omega\,t,\theta)\Big{)}^{2}-g\int_{0}^{z}\rho_{0}(s)\,{\rm
d}s\,,$

where ${\mathcal{F}}$ is a primitive of $F$, is a solution of the system
(7.19)-(7.20), describing height-dependent stratospheric planetary flows that
propagate zonally westwards.

###### Proof.

Since

$\Delta\Big{(}\frac{1}{\sqrt{\rho_{0}(z)}}\,\psi_{0}\Big{)}=\frac{1}{\sqrt{\rho_{0}(z)}}\,\Delta\psi_{0}=\frac{1}{\sqrt{\rho_{0}(z)}}\,F(\psi_{0})=G\Big{(}z,\,\frac{1}{\sqrt{\rho_{0}(z)}}\,\psi_{0}\Big{)}\quad\text{with}\quad
G(z,s)=\frac{1}{\sqrt{\rho_{0}(z)}}\,F\big{(}s\sqrt{\rho_{0}(z)}\,\big{)}\,,$

we infer from Lemma 2 that $\psi$ defined by (7.25) solves (7.21) for every
fixed $z$ since (7.24) ensures

$\iint_{{\mathbb{S}}^{2}}F(\psi_{0})\,{\rm d}\sigma=0\,.$

Using (7.20), the Ansatz (7.25) has the following effect on the equations
(7.19a)-(7.19b): on the left sides, only the quadratic terms in $\psi_{0}$
remain and the factor $\frac{1}{\rho_{0}}$ cancels out:

(7.27)
$\begin{cases}\displaystyle\frac{\partial\psi_{0}}{\partial\theta}\frac{\partial^{2}\psi_{0}}{\partial\theta\partial\varphi}-\frac{\partial\psi_{0}}{\partial\varphi}\frac{\partial^{2}\psi_{0}}{\partial\theta^{2}}+\tan\theta\,\frac{\partial\psi_{0}}{\partial\varphi}\frac{\partial\psi_{0}}{\partial\theta}=-\frac{\partial
p_{0}}{\partial\varphi}\,,\\\\[8.5359pt]
-\displaystyle\frac{1}{\cos^{2}\theta}\,\frac{\partial\psi_{0}}{\partial\theta}\frac{\partial^{2}\psi_{0}}{\partial\varphi^{2}}+\frac{1}{\cos^{2}\theta}\,\frac{\partial\psi_{0}}{\partial\varphi}\frac{\partial^{2}\psi_{0}}{\partial\theta\partial\varphi}+\frac{\sin\theta}{\cos^{3}\theta}\,\Big{(}\frac{\partial\psi_{0}}{\partial\varphi}\Big{)}^{2}+\tan\theta\,\Big{(}\frac{\partial\psi_{0}}{\partial\theta}\Big{)}^{2}=-\frac{\partial
p_{0}}{\partial\theta}\,.\end{cases}$

Taking (7.24) into account we see that the left side of (7.27) is precisely
the gradient of the expression

(7.28)
$\frac{1}{2}\,\Big{(}\frac{\partial\psi_{0}}{\partial\theta}\Big{)}^{2}+\frac{1}{2\cos^{2}\theta}\,\Big{(}\frac{\partial\psi_{0}}{\partial\varphi}\Big{)}^{2}-{\mathcal{F}}(\psi_{0})$

with respect to the $(\varphi,\theta)$-variables, and (7.19c) is easily
integrated to yield (7.26). ∎

It is of interest to investigate the stratospheric temperature distribution
associated to the vortices (7.25). With the temperature normalisation

(7.29) $T^{\prime}=\frac{U^{\prime 2}}{{\mathfrak{R}}^{\prime}}\,T\,,$

the equation of state (7.3) takes the nondimensional form

(7.30) $p=\rho T\,.$

Consequently, for the realistic density distribution $\rho_{0}(z)=a\,{\rm
e}^{-bz}$, where $a>0$ is the (nondimensional) average density of the
tropopause and $b>2$ for the atmospheres of our solar system (see the data in
[11]), from (7.26) we obtain the associated stratospheric temperature at
leading order:

(7.31) $T_{0}(\varphi,\theta,z,t)=\frac{ag}{b}+{\rm
e}^{bz}\,\Big{(}\frac{1}{a}\,\widehat{p_{0}}(\varphi,\theta,t)+\frac{g}{b}\Big{)}\,,$

where $\widehat{p_{0}}(\varphi,\theta,t)$ is the atmospheric pressure at the
tropopause. Note that (7.31) captures the increase of the stratospheric
temperature with height.

Planet |  ${\mathfrak{R}}^{\prime}$ |  rounded normalisation factor $U^{\prime 2}/{\mathfrak{R}}^{\prime}$ |  stratospheric temperature range
---|---|---|---
Earth |  287 m2/(s2K) |  9 K |  220 K to 260 K
Jupiter |  3745 m2/(s2K) |  6 K |  90 K to 150 K
Saturn |  3892 m2/(s2K) |  6 K |  110 K to 170 K
Uranus |  3615 m2/(s2K) |  6 K |  55 K to 115 K
Neptune |  3615 m2/(s2K) |  11 K |  55 K to 125 K

######  Remark 3.

(i) Using the kinematic equations for the material derivative (see [31])

(7.32) $u^{\prime}=\frac{{\rm D}\varphi}{{\rm
D}t^{\prime}}\,r^{\prime}\,\cos\theta\,,\qquad v^{\prime}=\frac{{\rm
D}\theta}{{\rm D}t^{\prime}}\,r^{\prime}\,,\qquad w^{\prime}=\frac{{\rm
D}r^{\prime}}{{\rm D}t^{\prime}}\,,$

the Euler equation (7.1) and the equation of mass conservation (7.2) lead to
the axial angular momentum conservation law (see [55])

(7.33) $\rho^{\prime}\,\frac{{\rm D}\ }{{\rm
D}t^{\prime}}\,\Big{\\{}\big{(}u^{\prime}+\varOmega^{\prime}r^{\prime}\cos\theta\big{)}r^{\prime}\cos\theta\Big{\\}}=-\frac{\partial
p^{\prime}}{\partial\varphi}\,.$

The kinematic equations for the material derivative of a flow on the unit
sphere are (see [17])

(7.34) $u_{0}=\cos\theta\,\frac{D}{Dt}\,\varphi\,,\qquad
v_{0}=\frac{D}{Dt}\,\theta\,,$

where

$\frac{D}{Dt}=\frac{\partial}{\partial
t}+\frac{u_{0}}{\cos\theta}\frac{\partial}{\partial\varphi}+v_{0}\,\frac{\partial}{\partial\theta}\,.$

The equations (7.34) are precisely the non-dimensional version of equation
(7.32) with $r^{\prime}\equiv 1$ (and $w^{\prime}\equiv 0$). One can now see
that (7.19a) is precisely the non-dimensional form of (7.33) for flow on a
sphere.

(ii) The motion of individual particles of the flow associated to (7.25)
occurs on a sphere determined by the initial location, and its evolution is
therefore determined by the spherical coordinates $(\varphi(t),\,\theta(t))$.
Setting

(7.35) $\Phi(t)=\varphi(t)+\omega t\,,$

from (7.25), (7.20) and (7.34) we get

$\displaystyle\frac{{\rm d}\ }{{\rm d}t}\,\psi_{0}(\Phi(t),\theta(t))$
$\displaystyle=\frac{1}{\sqrt{\rho(z)}}\,\Big{\\{}\frac{\partial\psi_{0}}{\partial\varphi}\,\frac{{\rm
d}\Phi}{{\rm d}t}+\frac{\partial\psi_{0}}{\partial\theta}\,\frac{{\rm
d}\theta}{{\rm
d}t}\Big{\\}}=\frac{1}{\sqrt{\rho(z)}}\,\Big{\\{}\frac{\partial\psi_{0}}{\partial\varphi}\,\Big{(}\frac{D_{\Sigma}\varphi}{Dt}+\omega\Big{)}+\frac{\partial\psi_{0}}{\partial\theta}\,\frac{D_{\Sigma}\theta}{Dt}\Big{\\}}$
$\displaystyle=\frac{1}{\sqrt{\rho(z)}}\,\Big{\\{}\frac{\partial\psi_{0}}{\partial\varphi}\,\Big{(}\frac{u_{0}}{\cos\theta}+\omega\Big{)}+\frac{\partial\psi_{0}}{\partial\theta}\,v\Big{\\}}$
$\displaystyle=\frac{1}{\sqrt{\rho(z)}}\,\Big{\\{}\frac{\partial\psi_{0}}{\partial\varphi}\,\Big{(}-\frac{\partial\psi_{0}}{\partial\theta}\,\frac{1}{\sqrt{\rho(z)}\,\cos\theta}\Big{)}+\frac{\partial\psi_{0}}{\partial\theta}\,\Big{(}\frac{\partial\psi_{0}}{\partial\varphi}\,\frac{1}{\sqrt{\rho(z)}\,\cos\theta}\Big{)}\Big{\\}}=0\,.$

Consequently the flow occurs along the level sets of $\psi_{0}$, translated
westward at the speed of rotation of the planet. Westward moving persistent
flow patterns that are nearly stationary in the rotating frame of reference
were observed in Saturn’s stratosphere at 40∘, 55∘ and 70∘ N and S, these
flows being remarkably symmetric about Saturn’s equator (see [27]). Similar
coherent high-latitude bands of westward flows in Jupiter’s stratosphere,
persisting for 70 days, were captured in 2000 during the Cassini mission (see
[32] for data and Figure 2 for a visualisation). Terrestrial patterns of a
similar nature also occur but are rather rare events, e.g., major
stratospheric warmings may disrupt the eastward polar vortex and give rise to
westward winds lasting typically a few days (see [31]); these attain the
planet’s speed of rotation at high latitudes. Thus the flow induced by (7.25)
captures physically realistic patterns in suitable latitude bands –
alternating eastward and westward traveling belts being typical for Jupiter
and Saturn. On the other hand, the stratospheric flow for Uranus and Neptune
is highly zonal, featuring a broad retrograde equatorial jet and high-latitude
prograde jets (see Fig. 2). Geostationary flow patterns lasting for decades
occur at about 20∘ latitude on Uranus, while on Neptune they can be observed
near 50∘ latitude but appear to be rather short-lived (see the data in [24]).

(iii) The effect of replacing (7.24) by

$\Delta\psi_{0}=F(\psi_{0})-\frac{1}{4\pi}\,\iint_{{\mathbb{S}}^{2}}F(\psi_{0})\,{\rm
d}\sigma$

(so that the Gauss contraint (2.6) is satisfied) brings about the additive
correction term

$\frac{\psi_{0}(\varphi+\omega
t)}{4\pi\sqrt{\rho_{0}(z)}}\,\iint_{{\mathbb{S}}^{2}}F(\psi_{0})\,{\rm
d}\sigma$

in (7.28) and the compatibility of the horizontal gradient with (7.19c) is not
granted unless we allow for a forcing term as a perturbation of gravity acting
in the radial direction. This feature is replicated if we start with (1.1)
rather than (7.24). Thus an interesting direction for further investigations
is opened up since such forcing terms appear naturally if one accounts for
oblateness: rapidly rotating planets deviate from a perfect sphere by
flattening at the poles and bulging at the Equator (see [17] for further
details in the terrestrial setting).∎

## Appendix: Spherical harmonics

We collect some properties of spherical harmonics relied upon throughout the
paper.

The eigenvalues of the Laplace-Beltrami operator on the unit sphere $\Sigma$,
acting on functions with vanishing spherical average444Without this
restriction, zero would be an eigenvalue with constant eigenvectors. are
$\\{-j(j+1),\ j\in{\mathbb{N}}\\}$. We will denote by $\mathbb{E}_{j}$ the
$j$-th eigenspace, of dimension $2j+1$, associated to the eigenvalue
$-j(j+1)$, and by $\mathbb{P}_{j}$ the corresponding spectral projector. A
basis of $\mathbb{E}_{j}$ is provided by the $(2j+1)$ spherical
harmonics555While in geophysics it convenient to use the latitude
$\theta\in[-\tfrac{\pi}{2},\tfrac{\pi}{2}]$, in quantum mechanics one
typically uses instead the co-latitude or polar angle
$\Theta=\tfrac{\pi}{2}-\theta\in[0,\pi]$. Passing from one set of coordinates
to the other requires only an interchange of $\sin$ and $\cos$ in all explicit
expressions and to keep track of the range of the corresponding angles.

$Y_{j}^{m}(\varphi,\theta)=(-1)^{m}\sqrt{\tfrac{(2j+1)(j-m)!}{4\pi(j+m)!}}\,P_{j}^{m}(\sin\theta){\rm
e}^{{\rm i}m\varphi}\,,\qquad m=-j,\dots,j\,,$

of degree $j$ and zonal number $m$ ($-j\leq m\leq j$), where

$P_{j}^{m}(x)=\frac{1}{2^{j}j!}\,(1-x^{2})^{m/2}\,\frac{{\rm d}^{j+m}}{{\rm
d}^{j+m}x}\,(x^{2}-1)^{j}\,,\qquad m=-j,\dots,j\,,$

are the associated Legendre polynomials, satisfying (see [43])

(A.1) $Y_{j}^{-m}=(-1)^{m}\,\overline{Y_{j}^{m}}\,,\qquad m=-j,\dots,j\,,$

where the overline means complex conjugation. The only zonal spherical
harmonics of degree $j$ are $Y_{j}^{0}$, $Y_{j}^{\pm j}$ are called sectoral
and change only in the longitudinal direction, while for $1\leq|m|\leq l-1$
the spherical harmonics $Y_{j}^{m}$ are called tesseral and vary in both the
longitudinal and latitudinal directions. A real orthonormal basis of spherical
harmonics $\\{R_{j}^{m}\\}$ can be defined in terms of their complex analogues
by setting

$R_{j}^{m}=\begin{cases}\frac{{\rm
i}[Y_{j}^{m}-(-1)^{m}Y_{j}^{-m}]}{\sqrt{2}}=(-1)^{m}\sqrt{\tfrac{(2j+1)(j-|m|)!}{2\pi(j+|m|)!}}\,P_{j}^{|m|}(\sin\theta)\sin(|m|\varphi)\,,\quad&m<0\,,\\\\[5.69046pt]
Y_{j}^{0}=\sqrt{\tfrac{(2j+1)}{4\pi}}\,P_{j}^{0}(\sin\theta)\,,\quad&m=0\,,\\\\[5.69046pt]
\frac{Y_{j}^{m}+(-1)^{m}Y_{j}^{-m}}{\sqrt{2}}=(-1)^{m}\sqrt{\tfrac{(2j+1)(j-|m|)!}{2\pi(j+m)!}}\,P_{j}^{|m|}(\sin\theta)\cos(m\varphi)\,,\quad&m>0\,.\end{cases}$

The first eigenspaces are

  * •

$\mathbb{E}_{1}$, which admits the orthonormal complex basis (with respect to
the spherical surface element ${\rm d}\sigma=\cos\theta\,{\rm d}\theta{\rm
d}\varphi$)

$Y_{1}^{-1}(\varphi,\theta)=\tfrac{1}{2}\sqrt{\tfrac{3}{2\pi}}\,\cos\theta\,{\rm
e}^{-{\rm i}\varphi}\,,\qquad
Y_{1}^{0}(\theta)=\tfrac{1}{2}\sqrt{\tfrac{3}{\pi}}\,\sin\theta\,,\qquad
Y_{1}^{1}(\varphi,\theta)=-\tfrac{1}{2}\sqrt{\tfrac{3}{2\pi}}\,\cos\theta\,{\rm
e}^{{\rm i}\varphi}\,,$

with the corresponding orthonormal real basis

$R_{1}^{-1}(\varphi,\theta)=\tfrac{1}{2}\sqrt{\tfrac{3}{\pi}}\,\cos\theta\,\sin\varphi\,,\qquad
R_{1}^{0}(\theta)=\tfrac{1}{2}\sqrt{\tfrac{3}{\pi}}\,\sin\theta\,,\qquad
R_{1}^{1}(\varphi,\theta)=-\tfrac{1}{2}\sqrt{\tfrac{3}{\pi}}\,\cos\theta\,\cos\varphi\,;$

  * •

$\mathbb{E}_{2}$, which admits the orthonormal complex basis

$Y_{2}^{\pm
1}(\varphi,\theta)=\mp\tfrac{1}{2}\sqrt{\tfrac{15}{2\pi}}\,\sin\theta\cos\theta\,{\rm
e}^{\pm{\rm i}\varphi}\,,\qquad
Y_{2}^{0}(\theta)=\tfrac{1}{4}\sqrt{\tfrac{5}{\pi}}\,(3\sin^{2}\theta-1)\,,\qquad
Y_{2}^{\pm
2}(\varphi,\theta)=\tfrac{1}{4}\sqrt{\tfrac{15}{2\pi}}\,\cos^{2}\theta\,{\rm
e}^{\pm 2{\rm i}\varphi}\,,$

with the corresponding orthonormal real basis

$\displaystyle
R_{2}^{-2}(\varphi,\theta)=\tfrac{1}{4}\sqrt{\tfrac{15}{\pi}}\,\cos^{2}\theta\,\sin(2\varphi)\,,\qquad$
$\displaystyle
R_{2}^{-1}(\varphi,\theta)=\tfrac{1}{2}\sqrt{\tfrac{15}{\pi}}\,\sin\theta\cos\theta\,\sin\varphi\,,\qquad$
$\displaystyle
R_{2}^{0}(\theta)=\tfrac{1}{4}\sqrt{\tfrac{5}{\pi}}\,(3\sin^{2}\theta-1)\,,$
$\displaystyle
R_{2}^{1}(\varphi,\theta)=-\tfrac{1}{2}\sqrt{\tfrac{15}{\pi}}\,\sin\theta\cos\theta\,\cos\varphi\,,\qquad$
$\displaystyle
R_{2}^{2}(\varphi,\theta)=\tfrac{1}{4}\sqrt{\tfrac{15}{\pi}}\,\cos^{2}\theta\,\cos(2\varphi)\,.\qquad$

The only spherical harmonics with modes $j\geq 3$ that we refer to in this
paper are

$Y_{3}^{0}(\theta)=\tfrac{1}{4}\sqrt{\tfrac{7}{\pi}}\,(5\sin^{3}\theta-3\sin\theta)\quad\text{and}\quad
Y_{5}^{0}(\theta)=\tfrac{1}{16}\sqrt{\tfrac{11}{\pi}}\,(63\sin^{5}\theta-705\sin^{3}\theta+15\sin\theta)\,.$

Generally we have

$Y_{j}^{0}(\theta)=\frac{1}{2^{j+1}j!}\sqrt{\frac{2j+1}{\pi
j!}}\,P_{j}^{0}(\sin\theta)\,,$

with $P_{j}^{0}(-x)=(-1)^{j}P_{j}^{0}(x)$ on $(-1,1)$ and $P_{j}^{0}(1)=1$,
while $P_{j}^{0}(0)=0$ for $j$ odd and
$P_{j}(0)=\frac{(-1)^{j/2}j!}{2^{j}(j/2)!}$ for $j$ even. For $j\geq 1$ the
Legendre polynomial $P_{j}^{0}$, of degree $j$, has $j$ distinct simple roots
and $j-1$ local minima and maxima in the interval $(-1,1)$, while $\pm 1$ are
global extrema in $[-1,1]$ with $(P_{j}^{0})^{\prime}(1)=\frac{j(j+1)}{2}$.
While a general formula for the roots $s_{k}$ of $P_{j}$ in $(-1,1)$ is, to
the best of our knowledge, still elusive, their location is quite accurately
described by

$s_{k}\approx\cos\Big{(}\frac{(4k-1)\pi}{4j+2}\Big{)}\,,\qquad k=1,\dots,j\,.$

The fact that for $j\geq 1$ the zonal spherical harmonics $R_{j}^{0}$ has
exactly $j$ nodal domains (connected components of the complement of the set
of zeros) on the sphere $\Sigma$ is generally not replicated by the real
spherical harmonics $R_{j}^{m}$ with $j\geq 2$ and $m\neq 0$, each of the
three real spherical harmonics of degree one having two nodal domains (see
[37]).

Representation theory highlights the relevance of symmetries in the study of
spherical harmonics. An element ${\mathfrak{g}}$ of the group $SO(3)$ of the
rotations of the sphere $\Sigma$ can be parametrized by the Euler angles as

${\mathfrak{g}}(\alpha,\beta,\gamma)={\mathfrak{R}}_{z}(\gamma){\mathfrak{R}}_{y}(\beta){\mathfrak{R}}_{z}(\alpha)\,,$

where ${\mathfrak{R}}_{z}(\gamma)$ and ${\mathfrak{R}}_{y}(\beta)$ represent a
rotation around the $z$-axis by $\gamma$ radians and a rotation around the
$y$-axis by $\beta$ radians. To any ${\mathfrak{g}}\in SO(3)$ we can associate
a rotation $\Lambda_{\mathfrak{g}}$ on $L^{2}(\Sigma)$, defined by
$(\Lambda_{\mathfrak{g}}f)(\eta)=f({\mathfrak{g}}^{-1}\eta)$. The mapping
${\mathfrak{g}}\mapsto\Lambda_{\mathfrak{g}}$ is a unitary representation of
$SO(3)$, and restricting $\Lambda_{\mathfrak{g}}$ to the finite-dimensional
space of spherical harmonics of degree $j\geq 0$, consisting of the linear
combination of the spherical harmonics of degree $j$, one obtains all the
irreducible representations of $SO(3)$, in the sense that there are no genuine
invariant subspaces (see [49]). A rotated spherical harmonic of degree $j$ can
be written as a linear combination of spherical harmonics of degree $j$ by
means of the formula

$\Lambda_{\mathfrak{g}}Y_{j}^{m}=\sum_{|k|\leq
j}{\mathbb{U}}_{l}^{mk}({\mathfrak{g}})Y_{l}^{k}\,,$

where

${\mathbb{U}}_{l}^{mk}({\mathfrak{g}}(\alpha,\beta,\gamma))={\rm e}^{-{\rm
i}(m\gamma+k\alpha)}\,P_{l}^{mk}(\sin\beta)\,,$

with $P_{j}^{mk}$ being the generalized associated Legendre polynomials, given
for $m,\,k\in\\{-j,\dots,j\\}$ by (see [53])

$P_{j}^{mk}(x)=\frac{(-1)^{j-m}}{2^{j}}\,\sqrt{\frac{(j+m)!}{(j-k)!(j+k)!(j-m)!}}(1+x)^{-(m+k)/2}\,(1-x)^{(k-m)/2}\,\frac{{\rm
d}^{j-m}}{{\rm d}^{j-m}x}\,[(1-x)^{j-k}(1+x)^{j+k}]\,.$

Note that the spherical harmonics of degree one can be obtained one from any
another by a rotation.

Expanding a stream function $\psi\in H^{2}({\mathbb{S}}^{2})$ in spherical
harmonics

$\psi(\varphi,\theta)=\sum_{j\geq
1}\sum_{m=-j}^{j}\alpha_{j}^{m}(t)Y_{j}^{m}(\varphi,\theta)\,,$

the associated velocity $U$ and vorticity $\Omega$ are represented by

$\displaystyle U$ $\displaystyle=\Big{(}-\sum_{j\geq
1}\sum_{m=-j}^{j}\alpha_{j}^{m}(t)\frac{\partial
Y_{j}^{m}}{\partial\theta}\,,\ \frac{1}{\cos\theta}\sum_{l\geq
1}\sum_{k=-l}^{l}\alpha_{l}^{k}(t)\frac{\partial
Y_{l}^{k}}{\partial\varphi}\Big{)}\,,$ $\displaystyle\Omega$
$\displaystyle=-\sum_{j\geq
1}\sum_{m=-j}^{j}j(j+1)\,\alpha_{j}^{m}(t)Y_{j}^{m}(\varphi,\theta)\,.$

Since the surface gradients of spherical harmonics are also orthogonal, we
have (see [34])

$\iint_{{\mathbb{S}}^{2}}|U|^{2}\,d\sigma=\sum_{j\geq
1}\sum_{m=-j}^{j}j(j+1)\,|\alpha_{j}^{m}(t)|^{2}\,,\qquad\iint_{{\mathbb{S}}^{2}}|\Omega|^{2}\,d\sigma=\sum_{j\geq
1}\sum_{m=-j}^{j}j^{2}(j+1)^{2}\,|\alpha_{j}^{m}(t)|^{2}\,.$

As a consequence one infers the validity of the sharp Poincaré inequality

(A.2)
$\iint_{{\mathbb{S}}^{2}}|\Omega|^{2}\,d\sigma\geq(n+1)(n+2)\iint_{{\mathbb{S}}^{2}}|U|^{2}\,d\sigma\,,\qquad\psi\in
H^{2}({\mathbb{S}}^{2})\cap\Big{(}\bigcap\limits_{j=1}^{n}{\mathbb{E}}_{j}^{\perp}\Big{)}$

where ${\mathbb{E}}_{j}^{\perp}$ is the orthogonal complement in
$L^{2}({\mathbb{S}}^{2})$ of the $(2j+1)$-dimensional eigenspace
${\mathbb{E}}_{j}$ of the eigenvalue $-j(j+1)$ of the Laplace-Beltrami
operator.

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