# Explicit and exact solutions concerning the Antarctic Circumpolar Current
with variable density in spherical coordinates

Calin Iulian Martin∗ Faculty of Mathematics, University of Vienna, Oskar
Morgenstern Platz 1, 1090 Vienna, Austria
E-mail address<EMAIL_ADDRESS>and  Ronald Quirchmayr  Department
of Mathematics, KTH Royal Institute of Technology, Lindstedtsvägen 25, 100 44
Stockholm, Sweden
E-mail address<EMAIL_ADDRESS>
###### Abstract.

We use spherical coordinates to devise a new exact solution to the governing
equations of geophysical fluid dynamics for an inviscid and incompressible
fluid with a general density distribution and subjected to forcing terms. The
latter are of paramount importance for the modeling of realistic flows-that
is, flows that are observed in some averaged sense in the ocean. Owing to the
employment of spherical coordinates we do not need to resort to approximations
(e.g. of $f$\- and $\beta$-plane type) that simplify the geometry in the
governing equations. Our explicit solution represents a steady purely-
azimuthal stratified flow with a free surface, that—thanks to the inclusion of
forcing terms and the consideration of the Earth’s geometry via spherical
coordinates—makes it suitable for describing the Antarctic Circumpolar Current
and enables an in-depth analysis of the structure of this flow. In line with
the latter aspect, we employ functional analytical techniques to prove that
the free surface distortion is defined in a unique and implicit way by means
of the pressure applied at the free surface. We conclude our discussion by
setting out relations between the monotonicity of the surface pressure and the
monotonicity of the surface distortion that concur with the physical
expectations.

Keywords: Azimuthal flows, Antarctic Circumpolar Current, spherical
coordinates, Coriolis force, exact explicit and implicit solutions
Mathematics Subject Classification: Primary: 35Q31, 35Q35. Secondary: 35Q86.

∗Corresponding author

##  1\. Introduction

To a large extent, the analytical studies concerning the fluid flows were
(are) concentrated on well-posedness or regularity type results. Our
contribution here aligns with a recent tendency—initiated by Constantin [6, 7,
8, 9] and Constantin and Johnson—[11, 13, 14, 15] of devising explicit and
exact solutions to the governing equations of geophysical fluid dynamics (GFD)
that describe surface waves and their interactions with the underlying
currents which are ubiquitous in Earth’s ocean basins. A selection of further
works in this direction is [4, 5, 26, 32, 34, 39, 40, 41, 42].

More precisely we are concerned here with providing an analytical solution to
the governing equations for water flows (with their boundary conditions) which
represents a flow that moves completely around the Earth on a circular path.
This solution describes an incompressible inviscid stratified steady flow
moving purely in the azimuthal direction, i.e. the velocity profile and the
pressure is described below and up to the free surface as a function of depth
and the angle of latitude. As such, this solution is suitable for the
depiction of the Antarctic Circumpolar Current (ACC), without doubt the most
significant current in Earth’s oceans. Indeed, ACC is the only current that
fully surrounds the polar axis. Its massiveness is reflected by the great area
it occupies—flowing eastwards through the southern regions of Atlantic, Indian
and Pacific Oceans along $23.000$ km, extending in places over $2000$ km in
width, cf. [22, 35, 46, 49] —and as such, by the huge volumes of water it
transports estimated to be between 165 million and 182 million cubic meters of
water every second, cf. [19], which represents more than 100 times the flow of
all the rivers on Earth. Recent mathematical studies investigating ACC’s
properties can be found in [14, 24, 25, 33, 43, 48].

The new aspect of our investigation—compared to the existing mathematical
literature on ACC—is the presence of density stratification. This accommodates
observed sharp changes in water density known as fronts, cf. [47]. The two
main fronts of the ACC are the Subantarctic Front to the north and the Polar
Front further south, see Figure 1.

Figure 1. Variations in density and the mean eastward current speed. The black
lines in the image represent the fronts of ACC. Image credit: Hellen Phillips
(Senior Research Fellow, Institute for Marine and Antarctic Studies,
University of Tasmania), Benoit Legresy (CSIRO) and Nathan Bindoff (Professor
of Physical Oceanography, Institute for Marine and Antarctic Studies,
University of Tasmania)

We remark that the recent study [37] clearly indicates the role of a thorough
knowledge of the impact of enhanced stratification in the Southern Ocean on
the mechanisms behind the glacial-interglacial ocean carbon cycle variations.

Apart from the two aspects pertaining to density variations in ACC we would
like to remark that stratification is of significant importance in the
understanding of GFD in general. Indeed, large scale oceanic processes exhibit
and experience pronounced density variations, most commonly due to
fluctuations in the fluid temperature or salinity, cf. [6, 11, 15, 21, 36,
45]. Therefore, it is of high significance to achieve a comprehensive
analytical understanding of stratified flows which could potentially serve
more applied endeavors. However, allowing for variable density in water flows
greatly complicates the analysis of an already challenging problem. Indeed, we
note that even in the scenario of two-dimensional water flows the literature
manifests a pronounced scarcity: recent steps ahead being undertaken in [1, 3,
10, 11, 12, 17, 20, 23, 27, 28, 29, 30, 50, 51].

Exact and/or explicit solutions describing geophysical flows constitute a
special and very rare event, reason being the intricacy of the involved
mathematical problem. Once they are available they secure ways to produce more
physically realistic and observed flows, by means of asymptotic [16], or
multiple scale methods [11]. Faithfull to this realization, we derive here
explicit solutions for the pressure and for the velocity field beneath the
free surface: the latter is itself given in an implicit form that relates it
to the pressure at the surface. Building on the approaches by Constantin and
Johnson [13, 14], we use spherical coordinates to devise purely-azimuthal,
depth-dependent varying flows, that verify the GFD equations and their
boundary conditions. Moreover, our solutions exhibit a general stratification
(that varies with depth and latitude) and forcing terms believed to be
responsible for the dynamical balance of the ACC, cf. [14, 31]. Finally we
point to [38, 18] for the relevance of stratification to maintain the
equilibrium of ACC: baroclinic instability (arising from stratification)
generates eddy-induced cells (acting to flatten the isopycnals) that
counterbalance the wind-driven Ekman cell (acting to steepen isopycnals).

The organization of the paper is the following: After introducing in Section 2
the governing equations (in spherical coordinates) and their boundary
conditions for geophysical flows, we derive in Section 3.1 explicit solutions
that render the velocity in the azimuthal direction and the corresponding
pressure function. The implicit (exact) solution that describes the free
surface is achieved in Section 3.2 by means of a functional analytic argument.
Relations between the monotonicity of the free surface and the surface
pressure are presented in the last section of the paper.

##  2\. Physical problem and governing equations

In this section we provide the governing equations for geophysical flows
written in spherical coordinates to accommodate the shape of the Earth,
together with the boundary conditions for the free surface and a rigid bed.

We will work in a system of right handed coordinates $(r,\theta,\varphi)$
where $r$ denotes the distance to the centre of the sphere, $\theta\in[0,\pi]$
is the polar angle (the convention being that $\pi/2-\theta$ is the angle of
latitude) $\varphi\in[0,2\pi]$ is the azimuthal angle (the angle of
longitude). While in this coordinate system the North and South poles are
located at $\theta=0,\pi$, respectively, the Equator sits on $\theta=\pi/2$,
the Antarctic Circumpolar Current is situated at $\theta=3\pi/4$. The unit
vectors in this system are

$\displaystyle{\bf e}_{r}$
$\displaystyle=(\sin\theta\cos\varphi,\sin\theta\sin\varphi,\cos\theta),$
$\displaystyle{\bf e}_{\theta}$
$\displaystyle=(\cos\theta\cos\varphi,\cos\theta\sin\varphi,-\sin\theta),$
$\displaystyle{\bf e}_{\varphi}$ $\displaystyle=(-\sin\varphi,\cos\varphi,0),$

with $\bf{e}_{\varphi}$ pointing from West to East and $\bf{e}_{\theta}$ from
North to South, cf. Figure 2.

Figure 2. The spherical coordinate system: $\theta$ is the polar angle,
$\varphi$ is the azimuthal angle (the angle of longitude) and $r$ represents
the distance to the origin.

Throughout this paper we make the following simplifying assumption on the
location of the Antarctic Circumpolar Current. We assume that the angle of
latitude $\theta$ satisfies

$\frac{3\pi}{4}-\frac{\pi}{18}\leq\theta\leq\frac{3\pi}{4}+\frac{\pi}{18}.$
(2.1)

We are guided in our study by the observations made in Maslowe [44] asserting
that the Reynolds number is, in general, extremly large for this oceanic
flows. Accordingly we will consider incompressible and inviscid flows.
However, our analysis incorporates a general density distribution
$\rho=\rho(r,\theta)$.

Denoting with

${\bf u}=u{\bf e}_{r}+v{\bf e}_{\theta}+w{\bf e}_{\varphi}$

the velocity field we have that, cf. [13], the governing equations in the
$(r,\theta,\varphi)$ coordinate system are the Euler’s equations,

$\begin{array}[]{rcl}u_{t}+uu_{r}+\frac{v}{r}u_{\theta}+\frac{w}{r\sin\theta}u_{\varphi}-\frac{1}{r}(v^{2}+w^{2})&=&-\frac{1}{\rho}p_{r}+F_{r}\\\
&&\\\
v_{t}+uv_{r}+\frac{v}{r}v_{\theta}+\frac{w}{r\sin\theta}v_{\varphi}+\frac{1}{r}(uv-w^{2}\cos\theta)&=&-\frac{1}{\rho}\frac{1}{r}p_{\theta}+F_{\theta}\\\
&&\\\
w_{t}+uw_{r}+\frac{v}{r}w_{\theta}+\frac{w}{r\sin\theta}w_{\varphi}+\frac{1}{r}(uw+vw\cot\theta)&=&-\frac{1}{\rho}\frac{1}{r\sin\theta}p_{\varphi}+F_{\varphi},\end{array}$
(2.2)

(where $p(r,\theta,\varphi)$ is denotes the pressure in the fluid and
$(F_{r},F_{\theta},F_{\varphi})$ is the body-force vector) and the equation of
mass conservation

$\frac{1}{r^{2}}\frac{\partial}{\partial r}(\rho
r^{2}u)+\frac{1}{r\sin\theta}\frac{\partial}{\partial\theta}(\rho
v\sin\theta)+\frac{1}{r\sin\theta}\frac{\partial(\rho w)}{\partial\varphi}=0.$
(2.3)

To capture effects of Earth’s rotation we associate $({\bf e}_{r},{\bf
e}_{\theta},{\bf e}_{\varphi})$ with a fixed point on the sphere which is
rotating about its polar axis. Consequently, we consider on the left of (2.2)
the Coriolis term

$2{\bf\Omega}\times{\bf u},$

where

${\bf\Omega}=\varOmega({\bf e}_{r}\cos\theta-{\bf e}_{\theta}\sin\theta),$

(with $\varOmega\approx 7.29\times 10^{-5}$ rad $s^{-1}$ being the constant
rotation speed of Earth) and the centripetal acceleration

${\bf\Omega}\times({\bf\Omega}\times{\bf r}),$

where ${\bf r}=r{\bf e}_{r}$. The latter two quantities add up to

$\displaystyle 2$ $\displaystyle\varOmega\big{[}-w(\sin\theta){\bf
e}_{r}-w(\cos\theta){\bf e}_{\theta}+(u\sin\theta+v\cos\theta){\bf
e}_{\varphi}\big{]}$ $\displaystyle-r\varOmega^{2}\big{[}(\sin^{2}\theta){\bf
e}_{r}+(\sin\theta\cos\theta){\bf e}_{\theta}\big{]}$

with respect to the $({\bf e}_{r},{\bf e}_{\theta},{\bf e}_{\varphi})$ basis.

Concerning the body-force vector it holds that (cf. [14]) the gravitational
constant acts in $r$-direction, a general body-force $G(r,\theta)$ acts in
$\theta$-direction, no force is acting in $\varphi$-direction. That is, the
body-force vector is

$(-g,G(r,\theta),0).$

The equations of motion are filled up by the boundary conditions as follows.
On the free surface $r=R+h(\theta,\varphi)$ we have the dynamic boundary
condition

$p=P(\theta,\varphi),$ (2.4)

and the kinematic boundary condition

$u=\frac{v}{r}\frac{\partial
h}{\partial\theta}+\frac{w}{r\sin\theta}\frac{\partial h}{\partial\varphi}.$
(2.5)

We ask that the bottom of the ocean, described by $r=d(\theta,\varphi)$, is an
impenetrable boundary by demanding

$u=\frac{v}{r}\frac{\partial
d}{\partial\theta}+\frac{w}{r\sin\theta}\frac{\partial d}{\partial\varphi}$
(2.6)

to hold there.

The solutions that we are seeking for, represent a steady flow that propagates
only in the azimuthal direction with no variations in this direction. Thus,
the velocity field is characterized by $u=v=0$ and $w=w(r,\theta)$. Moreover,
$p=p(r,\theta)$, $h=h(\theta)$, $d=d(\theta)$. Consequently, while the two
kinematic boundary conditions (2.5) and (2.6) and the equation of mass
conservation (2.3) are trivially satisfied, the Euler equations governing such
special flows read

$\left\\{\begin{array}[]{rcl}-\frac{w^{2}}{r}-2\varOmega
w\sin\theta-r\varOmega^{2}\sin^{2}\theta&=&-\frac{1}{\rho}p_{r}-g,\\\ &&\\\
-\frac{w^{2}}{r}\cot\theta-2\varOmega
w\cos\theta-r\varOmega^{2}\sin\theta\cos\theta&=&-\frac{1}{\rho
r}p_{\theta}+G(r,\theta),\\\ &&\\\
0&=&-\frac{1}{\rho}\frac{1}{r\sin\theta}p_{\varphi}.\end{array}\right.$ (2.7)

##  3\. Exact explicit and implicit solutions

###  3.1. Explicit solutions for azimuthal velocity and pressure

The first outcomes of this section concern the velocity $w$ and the pressure
$p$. These two quantities can be determined by means of explicit formulas. To
derive the latter we use the third equation in (2.7) to infer that $p$ is
independent of $\varphi$, i.e. $p=p(r,\theta)$. System (2.7) can now be
simplified as

$\displaystyle\rho\frac{(w+\Omega r\sin\theta)^{2}}{r}$
$\displaystyle=p_{r}+g\rho$ (3.1) $\displaystyle\rho
r\cot\theta\frac{(w+\Omega r\sin\theta)^{2}}{r}$
$\displaystyle=p_{\theta}-\rho rG(r,\theta).$ (3.2)

Let us denote

$Z=Z(r,\theta)\coloneqq\frac{(w+\Omega r\sin\theta)^{2}}{r}$ (3.3)

to rewrite the above system as

$\displaystyle\rho Z$ $\displaystyle=p_{r}+g\rho$ (3.4) $\displaystyle(\rho
r\cot\theta)Z$ $\displaystyle=p_{\theta}-\rho rG(r,\theta).$ (3.5)

The elimination of the pressure $p$ from the system (3.4)–(3.5) leads to

$\left(\rho Z\right)_{\theta}-\left(\rho r\cot\theta
Z\right)_{r}=g\rho_{\theta}+\frac{\partial}{\partial r}\big{(}\rho
rG(r,\theta)\big{)}$

which can be rewritten as

$-r\cos\theta\left(\rho rZ\right)_{r}+\sin\theta\left(\rho
rZ\right)_{\theta}=(r\sin\theta)\left[g\rho_{\theta}(r,\theta)+\frac{\partial}{\partial
r}\big{(}\rho rG(r,\theta)\big{)}\right].$ (3.6)

We further set

$U\coloneqq\rho rZ$ (3.7)

and appeal to the method of characteristics to solve (3.6); i.e., we set out
to solve

$-(r\cos\theta)U_{r}+(\sin\theta)U_{\theta}=(r\sin\theta)\left[g\rho_{\theta}(r,\theta)+\frac{\partial}{\partial
r}\big{(}\rho rG(r,\theta)\big{)}\right].$ (3.8)

Along these lines we seek curves $s\rightarrow(r(s),\theta(s))$ satisfying

$\displaystyle\dot{r}(s)=-r(s)\cos\theta(s),\qquad\dot{\theta}(s)=\sin\theta(s).$
(3.9)

Note that any $(r(s),\theta(s))$ that satisfies the previous system also obeys
the equation

$\frac{\mathrm{d}}{\mathrm{d}s}\Big{(}r(s)\sin\theta(s)\Big{)}=0\quad{\rm
for}\,\,{\rm all}\quad s\in\mathbb{R}$ (3.10)

The choice (3.9) transforms equation (3.8) into

$\displaystyle\begin{aligned}
\frac{\mathrm{d}}{\mathrm{d}s}\Big{(}&U\big{(}r(s),\theta(s)\big{)}\Big{)}=\\\
\big{(}&r(s)\sin\theta(s)\big{)}\left[g\rho_{\theta}\big{(}r(s),\theta(s)\big{)}+\frac{\partial}{\partial
r}\Big{(}\rho\big{(}r(s),\theta(s)\big{)}r(s)G\big{(}r(s),\theta(s)\big{)}\Big{)}\right],\end{aligned}$
(3.11)

whose resolution depends upon finding suitable solutions to the characteristic
equations (3.9). The integration of (3.9) yields the general solution
$(\tilde{r}(s),\tilde{\theta}(s))$ given by

$\displaystyle\tilde{r}(s)=c_{1}\mathrm{e}^{s}+c_{2}\mathrm{e}^{-s},\quad
c_{1},c_{2}\in\mathbb{R}$ (3.12)
$\displaystyle\tilde{\theta}(s)=\arccos\left(\frac{1-c\mathrm{e}^{2s}}{1+c\mathrm{e}^{2s}}\right),\quad
c\geq 0,$ (3.13)

where $c,c_{1},c_{2}$ are constants of integration. To find $U$ as a function
of $r$ and $\theta$ we proceed as follows: for given $(r,\theta)$, we search
for an $s_{0}$ such that

$\tilde{\theta}(s_{0})=\theta,\quad\tilde{r}(s_{0})=r.$ (3.14)

We choose now the constant $c$ such that $\tilde{\theta}(0)=\frac{\pi}{2}$.
This gives $c=1$, that is

$\tilde{\theta}(s)=\arccos\left(\frac{1-\mathrm{e}^{2s}}{1+\mathrm{e}^{2s}}\right),$
(3.15)

From the property (3.10) we see that

$\tilde{r}(0)\sin\big{(}\tilde{\theta}(0)\big{)}=\tilde{r}(s_{0})\sin\big{(}\tilde{\theta}(s_{0})\big{)}=r\sin\theta,$

which, after using $\tilde{\theta}(0)=\frac{\pi}{2}$, gives
$\tilde{r}(0)=r\sin\theta$.

It is easy to see that

$s_{0}=\frac{1}{2}\ln\frac{1-\cos\theta}{1+\cos\theta}\eqqcolon f(\theta)$

is the unique element satisfying the first equation in (3.14). To determine
$c_{1}$ and $c_{2}$ we solve

$\displaystyle\tilde{r}(f(\theta))=r$ (3.16)
$\displaystyle\tilde{r}(0)=r\sin\theta$ (3.17)

that is equivalent to

$\displaystyle c_{1}+c_{2}+(c_{2}-c_{1})\cos\theta=r\sin\theta$ $\displaystyle
c_{1}+c_{2}=r\sin\theta,$

whose unique solution (for $\theta\neq\pi/2$) is

$c_{1}=c_{2}=\frac{r\sin\theta}{2}.$

Let us denote with $(\overline{r}(s),\overline{\theta}(s))$ the special
characteristic solution to (3.9) that satisfies (3.14). That is,

$\overline{r}(s)=\frac{r\sin\theta}{2}(\mathrm{e}^{s}+\mathrm{e}^{-s}),\quad\overline{\theta}(s)=\arccos\left(\frac{1-\mathrm{e}^{2s}}{1+\mathrm{e}^{2s}}\right).$

Inserting $(\overline{r}(s),\overline{\theta}(s))$ in the formula (3.11),
integrating afterwards with respect to $s$ from $0$ to $f(\theta)$, taking
into account (3.16)–(3.17), and setting

$H(r,\theta)\coloneqq r\rho(r,\theta)G(r,\theta),$ (3.18)

we obtain

$U(r,\theta)-U(r\sin\theta,\pi/2)=r\sin\theta\int_{0}^{f(\theta)}\left[g\rho_{\theta}(\overline{r}(s),\overline{\theta}(s))+H_{r}(\overline{r}(s),\overline{\theta}(s))\right]\,\mathrm{d}s,$
(3.19)

where we have used that
$\frac{\mathrm{d}}{\mathrm{d}s}\big{(}\overline{r}(s)\sin\overline{\theta}(s)\big{)}=0$
for all $s$, which implies that

$\overline{r}(s)\sin\left(\overline{\theta}(s)\right)=\overline{r}(0)\sin\left(\overline{\theta}(0)\right)=r\sin\theta\,\,{\rm
for}\,\,{\rm all}\,\,s.$

From the definitions of the functions $Z$ and $U$, cf. (3.3) and (3.7),
respectively, we finally obtain that the azimuthal velocity $w$ is given by
the formula

$\displaystyle\begin{aligned} w(r,\theta)=&-\Omega r\sin\theta\\\
&+\sqrt{\frac{F(r\sin\theta)+r\sin\theta\int_{0}^{f(\theta)}\left[g\rho_{\theta}(\overline{r}(s),\overline{\theta}(s))+H_{r}(\overline{r}(s),\overline{\theta}(s))\right]\,\mathrm{d}s}{\rho(r,\theta)}},\end{aligned}$
(3.20)

where $t\mapsto F(t)$ denotes a given arbitrary smooth function.

To determine the pressure $p$ we first infer from (3.3), (3.4), (3.5), (3.7),
(3.18) and (3.19) that

$\displaystyle p_{r}(r,\theta)$
$\displaystyle=-g\rho(r,\theta)+\frac{1}{r}\bigg{(}F(r\sin\theta)+r\sin\theta\int^{f(\theta)}_{0}[g\rho_{\theta}(\overline{r},\overline{\theta})+H_{r}(\overline{r},\overline{\theta})]\,\mathrm{d}s\bigg{)},$
(3.21) $\displaystyle p_{\theta}(r,\theta)$
$\displaystyle=H(r,\theta)+\cot\theta\,\bigg{[}F(r\sin\theta)+r\sin\theta\int^{f(\theta)}_{0}[g\rho_{\theta}(\overline{r},\overline{\theta})+H_{r}(\overline{r},\overline{\theta})]\,\mathrm{d}s\bigg{]}.$
(3.22)

Integrating both sides in (3.21) with respect to $r$ yields that

$p(r,\theta)=C(\theta)-g\int^{r}_{a}\rho(\tilde{r},\theta)\,\mathrm{d}\tilde{r}+\int^{r\sin\theta}_{a\sin\theta}\bigg{[}\frac{F(y)}{y}+\mathcal{F}(y,\theta)\bigg{]}\,\mathrm{d}y,$
(3.23)

where $a\in{\mathbb{R}}$ is a constant, the map $\theta\mapsto C(\theta)$ will
be established in the sequel, and $\mathcal{F}$ is given by

$\mathcal{F}(y,\theta)\coloneqq\int^{f(\theta)}_{0}\bigg{[}g\rho_{\theta}\bigg{(}y\,\frac{\mathrm{e}^{s}+\mathrm{e}^{-s}}{2},\overline{\theta}(s)\bigg{)}+H_{r}\bigg{(}y\,\frac{\mathrm{e}^{s}+\mathrm{e}^{-s}}{2},\overline{\theta}(s)\bigg{)}\bigg{]}\,\mathrm{d}s.$
(3.24)

To determine $C$, we differentiate (3.23) with respect to $\theta$, which, by
an application of the chain rule, yields that

$\displaystyle\begin{aligned}
p_{\theta}(r,\theta)=C^{\prime}(\theta)+H(r,\theta)-H(a,\theta)+\cot\theta\,\big{[}F(r\sin\theta)-F(a\sin\theta)\big{]}.\end{aligned}$
(3.25)

Here we have exploited that for every fixed $y$,

$\displaystyle\mathcal{F}_{\theta}(y,\theta)$
$\displaystyle=f^{\prime}(\theta)\bigg{[}g\rho_{\theta}\bigg{(}y\,\frac{\mathrm{e}^{s}+\mathrm{e}^{-s}}{2},\overline{\theta}(s)\bigg{)}+H_{r}\bigg{(}y\,\frac{\mathrm{e}^{s}+\mathrm{e}^{-s}}{2},\overline{\theta}(s)\bigg{)}\bigg{]}\Bigg{|}_{s=f(\theta)}$
$\displaystyle=\frac{1}{\sin\theta}\bigg{[}g\rho_{\theta}\bigg{(}\frac{y}{\sin\theta},\theta(s)\bigg{)}+H_{r}\bigg{(}\frac{y}{\sin\theta},\theta(s)\bigg{)}\bigg{]},$

which enabled us to write

$\int^{r\sin\theta}_{a\sin\theta}\mathcal{F}_{\theta}(y,\theta)\,\mathrm{d}y=\int^{r}_{a}\big{[}g\rho_{\theta}(\tilde{r},\theta)+H_{r}(\tilde{r},\theta)\big{]}\,\mathrm{d}\tilde{r}.$

A comparison of (3.25) and (3.22) allows us to deduce that

$C^{\prime}(\theta)=F(a\sin\theta)\cot\theta+\mathcal{F}(a\sin\theta,\theta)\,a\cos\theta+H(a,\theta).$
(3.26)

We conclude from (3.23) and (3.26) that the pressure satisfies

$\displaystyle\begin{aligned}
p(r,\theta)=b&-g\int^{r}_{a}\rho(\tilde{r},\theta)\,\mathrm{d}\tilde{r}+\int^{r\sin\theta}_{a\sin\theta}\bigg{[}\frac{F(y)}{y}+\mathcal{F}(y,\theta)\bigg{]}\,\mathrm{d}y\\\
&+\int^{\theta}_{3\pi/4}\big{[}F(a\sin\tilde{\theta})\cot\tilde{\theta}+\mathcal{F}(a\sin\tilde{\theta},\tilde{\theta})\,a\cos\tilde{\theta}+H(a,\tilde{\theta})\big{]}\,\mathrm{d}\tilde{\theta},\end{aligned}$
(3.27)

where $a,b\in{\mathbb{R}}$ are constants, $F$ is an arbitrary smooth function,
and $\mathcal{F}$ is given according to (3.24).

###  3.2. Implicit exact solution for the free surface

By an application of the implicit function theorem we will show the existence
of a unique function $h=h(\theta)$ representing disturbances of the flat water
surface caused by deviations of the pressure distribution from the pressure
required to maintain a free surface following the curvature of the Earth.

According to the dynamic boundary condition at the surface (2.4) and our
assumption of a purely azimuthal flow that may only vary in $\theta$, the
pressure $p$ given by (3.27) and evaluated at the free surface
$r=R+h(\theta)$, denoted by $P(\theta)$, satisfies

$\displaystyle\begin{aligned}
P(\theta)=b&-g\int^{R+h(\theta)}_{a}\rho(r,\theta)\,\mathrm{d}r+\int^{[R+h(\theta)]\sin\theta}_{a\sin\theta}\bigg{[}\frac{F(y)}{y}+\mathcal{F}(y,\theta)\bigg{]}\,\mathrm{d}y\\\
&+\int^{\theta}_{3\pi/4}\big{[}F(a\sin\tilde{\theta})\cot\tilde{\theta}+\mathcal{F}(a\sin\tilde{\theta},\tilde{\theta})\,a\cos\tilde{\theta}+H(a,\tilde{\theta})\big{]}\,\mathrm{d}\tilde{\theta}.\end{aligned}$
(3.28)

Formula (3.28) relates the pressure $P(\theta)$ acting onto the ocean surface
to deformations of the free surface described by $h(\theta)$. To ensure
comparability of the involved variables, we bring equation (3.28) in
dimensionless form. For this purpose we set $h\equiv 0$ in (3.28). This
corresponds to the undisturbed ocean surface following the Earth’s curvature.
The pressure $P_{0}$ that corresponds to the flat surface shape is given by

$\displaystyle\begin{aligned}
P_{0}(\theta)=b&-g\int^{R}_{a}\rho(r,\theta)\,\mathrm{d}r+\int^{R\sin\theta}_{a\sin\theta}\bigg{[}\frac{F(y)}{y}+\mathcal{F}(y,\theta)\bigg{]}\,\mathrm{d}y\\\
&+\int^{\theta}_{3\pi/4}\big{[}F(a\sin\tilde{\theta})\cot\tilde{\theta}+\mathcal{F}(a\sin\tilde{\theta},\tilde{\theta})\,a\cos\tilde{\theta}+H(a,\tilde{\theta})\big{]}\,\mathrm{d}\tilde{\theta}.\end{aligned}$
(3.29)

Let us denote by $P_{a}$ the atmospheric pressure at the ocean surface, which
we assume to be constant along the entire range of the ACC. This is a
reasonable simplification which, in combination with (3.29), implies that the
constant $P_{a}$ can be written as

$P_{a}\equiv
b-g\int^{R}_{a}\rho(r,3\pi/4)\,\mathrm{d}r+\int^{\frac{R}{\sqrt{2}}}_{\frac{a}{\sqrt{2}}}\bigg{[}\frac{F(y)}{y}+\mathcal{F}(y,3\pi/4)\bigg{]}\,\mathrm{d}y.$
(3.30)

Dividing (3.28) by $P_{a}$ yields that the dimensionless pressure
$\mathcal{P}\coloneqq P/P_{a}$ satisfies

$\displaystyle\begin{aligned}
0=&-\mathcal{P}(\theta)+\frac{b}{P_{a}}-\frac{g}{P_{a}}\int_{a}^{[1+\mathcal{h}(\theta)]R}\rho(r,\theta)\,\mathrm{d}r+\frac{1}{P_{a}}\int_{a\sin\theta}^{[1+\mathcal{h}(\theta)]R\sin\theta}\left[\frac{F(y)}{y}+\mathcal{F}(y,\theta)\right]\,\mathrm{d}y\\\
&+\frac{1}{P_{a}}\int_{3\pi/4}^{\theta}[F(a\sin\tilde{\theta})\cot\tilde{\theta}+\mathcal{F}(a\sin\tilde{\theta},\tilde{\theta})a\cos\tilde{\theta}+H(a,\tilde{\theta})]\,\mathrm{d}\tilde{\theta}\eqqcolon\mathfrak{F}(\mathcal{h},\mathcal{P}),\end{aligned}$
(3.31)

where the dimensionless $\mathcal{h}(\theta)\coloneqq\frac{h(\theta)}{R}$
measures the deviation from the flat ocean surface at $r=R$.

According to our assumption (2.1) we restrict the range of $\theta$ to the
compact interval

$I\coloneqq\bigg{[}\frac{3\pi}{4}-\frac{\pi}{18},\frac{3\pi}{4}+\frac{\pi}{18}\bigg{]}$

and denote by $\mathcal{C}(I)$ the space of continuous functions $f\colon
I\to{\mathbb{R}}$, equipped with the supremum norm

$\|f\|=\sup_{\theta\in I}|f(\theta)|.$ (3.32)

Furthermore we denote by $B$ the open ball of radius $10^{-5}$ centered at the
origin in $\mathcal{C}(I)$; we note that any realistic disturbance
$\mathcal{h}$ is contained in $B$. We write now equation (3.31) in the
operatorial form

$\mathfrak{F}\colon B\times\mathcal{C}(I)\to\mathcal{C}(I),$ (3.33)

where $\mathfrak{F}(\mathcal{h},\mathcal{P})$ is the right-hand side of
(3.31). We aim to implicitly resolve the dimensionless surface disturbance
$\mathcal{h}$ as a function of the dimensionless surface pressure
$\mathcal{P}$ by means of the operator equation

$\mathfrak{F}(\mathcal{h},\mathcal{P})=0.$ (3.34)

We note first that (3.34) is clearly satisfied for $\mathcal{h}\equiv 0$ and
the corresponding dimensionless pressure $\mathcal{P}_{0}$, which is obtained
by dividing both sides in (3.29) by $P_{a}$:

$\displaystyle\begin{aligned}
\mathcal{P}_{0}(\theta)=\frac{b}{P_{a}}&-\frac{g}{P_{a}}\int^{R}_{a}\rho(\tilde{r},\theta)\,\mathrm{d}\tilde{r}+\frac{1}{P_{a}}\int^{R\sin\theta}_{a\sin\theta}\bigg{[}\frac{F(y)}{y}+\mathcal{F}(y,\theta)\bigg{]}\,\mathrm{d}y\\\
&+\frac{1}{P_{a}}\int^{\theta}_{3\pi/4}\big{[}F(a\sin\tilde{\theta})\cot\tilde{\theta}+\mathcal{F}(a\sin\tilde{\theta},\tilde{\theta})\,a\cos\tilde{\theta}+H(a,\tilde{\theta})\big{]}\,\mathrm{d}\tilde{\theta}.\end{aligned}$
(3.35)

That is, we have that

$\mathfrak{F}(0,\mathcal{P}_{0})=0.$ (3.36)

Having obtained a trivial solution of (3.34) we aim now to apply the implicit
function theorem in order to find solution flows with non-flat free surface.
To this end we need to compute the derivative of $\mathfrak{F}$ with respect
to the first variable $\mathcal{h}$ at the point $(0,\mathcal{P}_{0})\in
B\times\mathcal{C}(I)$,

$D_{\mathcal{h}}\mathfrak{F}(0,\mathcal{P}_{0})\mathcal{h}=\lim_{t\to
0}\frac{\mathfrak{F}(t\mathcal{h},\mathcal{P}_{0})-\mathfrak{F}(0,\mathcal{P}_{0})}{t},$
(3.37)

A plain calculation yields that

$\displaystyle\begin{aligned}
D_{\mathcal{h}}\mathfrak{F}(0,\mathcal{P}_{0})\,\mathcal{h}(\theta)&=-\frac{gR}{P_{a}}\rho(R,\theta)\,\mathcal{h}(\theta)+\frac{1}{P_{a}}\Big{[}F(R\sin\theta)+R\sin\theta\,\mathcal{F}(R\sin\theta,\theta)\Big{]}\,\mathcal{h}(\theta)\\\
&=\frac{\rho(R,\theta)}{P_{a}}\Big{[}-gR+\big{(}w(R,\theta)+\Omega
R\sin\theta\big{)}^{2}\Big{]}\,\mathcal{h}(\theta),\end{aligned}$

where we employed (3.20) to obtain the last equality. Realistic assumptions on
the maximal size of $w$ yield that $gR\gg(w+\Omega R\sin\theta)^{2}$ for all
$\theta\in I$, cf. Fig. 1. This allows us to infer that

$D_{\mathcal{h}}\mathfrak{F}(0,\mathcal{P}_{0})\colon\mathcal{C}(I)\to\mathcal{C}(I)$
(3.38)

defines a linear topological homeomorphism. By the implicit function theorem
in Banach spaces (cf. [2]) there exists for every sufficiently small
perturbation $\mathcal{P}$ of $\mathcal{P}_{0}$ a unique function
$\mathcal{h}\in\mathcal{C}(I)$ such that (3.31) holds true; this $\mathcal{h}$
is nontrivial.

##  4\. Monotonicity relations

While in the previous section we have exploited the non-dimensional relation
(3.31) to infer that the free surface in the region of the ACC can
locally—around the undisturbed state $(\mathcal{h}\equiv
0,\mathcal{P}_{0})$—be described as a function of the imposed surface
pressure, we establish in this section relations between the monotonicity of
the dimensionless surface pressure $\mathcal{P}$ and surface elevation
$\mathcal{h}$, see Proposition 4.1 below, which are in concordance with
physical expectations. To this end we remark that a bootstrapping procedure
[2] enables us to transfer smoothness properties of $\mathcal{P}$ to
$\mathcal{h}$. Hence assuming a continuously differentiable pressure
$\mathcal{P}$, which we differentiate now with respect to $\theta$ in (3.31)
yields

$\displaystyle\mathcal{P}^{\prime}(\theta)=$
$\displaystyle-\frac{g}{P_{a}}\int_{a}^{[1+\mathcal{h}]R}\rho_{\theta}(r,\theta)\,\mathrm{d}r-\frac{g}{P_{a}}\rho\big{(}(1+\mathcal{h})R,\theta\big{)}\,\mathcal{h}^{\prime}(\theta)R$
$\displaystyle+\frac{\cot\theta}{P_{a}}F\big{(}(1+\mathcal{h})R\sin\theta\big{)}+\frac{\mathcal{h}^{\prime}(\theta)}{P_{a}}\,\frac{F\big{(}(1+\mathcal{h})R\sin\theta\big{)}}{1+\mathcal{h}}-\frac{\cot\theta}{P_{a}}F(a\sin\theta)$
$\displaystyle+\frac{1}{P_{a}}\,\mathcal{F}\big{(}(1+\mathcal{h})R\sin\theta,\theta\big{)}\Big{(}(1+\mathcal{h})R\cos\theta+\mathcal{h}^{\prime}(\theta)\,R\sin\theta\Big{)}$
$\displaystyle-\frac{1}{P_{a}}\,\mathcal{F}(a\sin\theta,\theta)\,a\cos\theta+\frac{1}{P_{a}}\int_{a}^{[1+\mathcal{h}]R}[g\rho_{\theta}(r,\theta)+H_{r}(r,\theta)]\,\mathrm{d}r$
$\displaystyle+\frac{1}{P_{a}}\,\Big{(}F(a\sin\theta)\cot\theta+\mathcal{F}(a\sin\theta,\theta)\,a\cos\theta+H(a,\theta)\Big{)},$

thus

$\displaystyle\mathcal{P}^{\prime}(\theta)$
$\displaystyle=\frac{\mathcal{h}^{\prime}(\theta)}{P_{a}}\Bigg{[}-gR\rho\big{(}(1+\mathcal{h})R,\theta\big{)}+R\,\sin\theta\,\mathcal{F}\big{(}(1+\mathcal{h})R\sin\theta,\theta\big{)}$
$\displaystyle\qquad\qquad\;\,+\frac{F\big{(}(1+\mathcal{h})R\sin\theta\big{)}}{1+\mathcal{h}}\Bigg{]}+\frac{1}{P_{a}}\,H\big{(}(1+\mathcal{h})R,\theta\big{)}$
$\displaystyle\quad+\frac{\cot\theta}{P_{a}}\left[F\big{(}(1+\mathcal{h})R\sin\theta\big{)}+(1+\mathcal{h})R\,\sin\theta\,\mathcal{F}\big{(}(1+\mathcal{h})R\sin\theta,\theta\big{)}\right];$

for convenience we wrote simply $\mathcal{h}$ to mean the evaluation
$\mathcal{h}(\theta)$ in the above calculation. Making use of the formula for
the azimuthal velocity (3.20) we obtain

$\displaystyle\frac{P_{a}}{\rho\big{(}(1+\mathcal{h})R,\theta\big{)}}\,\mathcal{P}^{\prime}(\theta)$
$\displaystyle=\left[-gR+\frac{\Big{(}w\big{(}(1+\mathcal{h})R,\theta\big{)}+\Omega(1+\mathcal{h})R\sin\theta\Big{)}^{2}}{1+\mathcal{h}}\right]\mathcal{h}^{\prime}(\theta)$
(4.1)
$\displaystyle\qquad+\Big{(}w\big{(}(1+\mathcal{h})R,\theta\big{)}+\Omega(1+\mathcal{h})R\sin\theta\Big{)}^{2}\cot\theta$
$\displaystyle\qquad+(1+\mathcal{h})R\,G\big{(}(1+\mathcal{h})R,\theta\big{)}.$

It is apparent from the above formula that, in order to discuss monotonicity
properties of $\mathcal{P}$ and $\mathcal{h}$, we need additional information
on the size of the forcing term $G$. Along these lines, we note that,
according to the discussion in Constantin and Johnson [14], the nonlinear
advection terms in (2.7) are small perturbations of the linear flow $w_{0}$
which is governed by the system

$\begin{array}[]{rcl}-2\Omega
w_{0}\sin\theta&=&-\frac{1}{\rho}\tilde{p}_{r}\\\ &&\\\ -2\Omega
w_{0}\cos\theta&=&-\frac{1}{\rho}\cdot\frac{1}{r}\tilde{p}_{\theta}+G(r,\theta),\end{array}$
(4.2)

with $\tilde{p}=p+\rho gr-\frac{1}{2}\rho r^{2}\Omega^{2}\sin^{2}\theta$ being
the modified pressure absorbing the centripetal and gravitational
acceleration. Using that $\sin\theta\approx\cos\theta$ in the region of the
ACC (recall that we restrict $\theta$ to the interval
$I=\big{[}\frac{3\pi}{4}-\frac{\pi}{18},\frac{3\pi}{4}+\frac{\pi}{18}\big{]}$,
cf. (2.1)), and that the meridional pressure gradient is relatively small
compared to the radial gradient, we have from (4.2) that, at the leading order
balance,

$G(r,\theta)\approx-2\Omega w_{0}\cos\theta.$ (4.3)

We are now able to formulate the following monotonicity result.

######  Proposition 4.1.

Let $\theta\in I$ and assume that the forcing term $G$ satisfies (4.3). Then
the following assertions hold true.

  1. (i)

If $\mathcal{h}^{\prime}(\theta)\geq 0$ then $\mathcal{P}^{\prime}(\theta)<0$.

  2. (ii)

If $\mathcal{P}^{\prime}(\theta)\geq 0$ then $\mathcal{h}^{\prime}(\theta)<0$.

###### Proof.

Let us first consider the sum

$\Big{(}w((1+\mathcal{h})R,\theta)+\Omega(1+\mathcal{h})R\sin\theta\Big{)}^{2}\cot\theta+(1+\mathcal{h})R\,G\big{(}(1+\mathcal{h})R,\theta\big{)}$

and expand it as

$\displaystyle\Big{(}w((1+\mathcal{h})R,\theta)^{2}+2\Omega
R(1+\mathcal{h})w((1+\mathcal{h})R,\theta)\sin\theta\Big{)}\cot\theta$ (4.4)
$\displaystyle+\Omega R(1+\mathcal{h})\left[\Omega
R(1+\mathcal{h})\sin\theta-2w_{0}((1+\mathcal{h})R,\theta)\right]\cos\theta.$

Given that the maximal speed of the ACC does not exceed $3$ m s-1 (cf. [14]),
clearly $\Omega R(1+\mathcal{h})\sin\theta$ is by far greater than $2w_{0}$.
Therefore, the expression in (4.4) is clearly negative for all
$\theta\in[\frac{3\pi}{4}-\frac{\pi}{18},\frac{3\pi}{4}+\frac{\pi}{18}]$.
Also, appealing to the same arguments pertaining to the sizes of
$w_{0},\Omega$ and $R$ we can conclude that

$-gR+\frac{\Big{(}w((1+\mathcal{h})R,\theta)+\Omega(1+\mathcal{h})R\sin\theta\Big{)}^{2}}{1+\mathcal{h}}<0$

within $I$. Thus, assuming $\mathcal{h}^{\prime}(\theta)\geq 0$, we see from
(4.1) that $\mathcal{P}^{\prime}(\theta)<0$, which shows (i). The proof of
(ii) follows by the same arguments. ∎

Acknowledgments
C. I. Martin would like to acknowledge the support of the Austrian Science
Fund (FWF) under research grant P 30878-N32. R. Quirchmayr acknowledges the
support of FWF under research grant J 4339-N32.

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