\section{Introduction}

\subsection{Radical meteorology}

It is said that one who only knows a single language knows not his own. Much research
on exoplanets and on the planets in our own solar system, other than earth, is motivated
   by a similar philosophy.

A cosmology in which the earth is a flat, nonrotating disk of finite size is irreconcilable with
 mainstream physics. Such a disk world would rapidly and irreversibly collapse into a sphere.
   Nevertheless, working out the atmospheric and oceanic dynamics of such a world is useful because
it illuminates the tremendous importance of the earth's rotation to weather on the spherical
planet whose surface we actually inhabit.

Exoplanets and the continuing exploration of the  solar system are challenging atmospheric scientists to think far outside the box. This article is in that spirit. However, there is a long tradition of ``radical meteorology" that far preceded exoplanets.
 My colleague, Bill Kuhn, was fond of asking doctoral students on qualifying exams very open-ended questions:  Describe the atmosphere of Titan under early `Dim Sun' " conditions? What happens to Mars if the
sun dies? These questions tested the students' ability to syncretize a grab bag of dynamics, chemistry,
radiation and instrumentation into a comprehensive view of the subject.

   ``Radical meteorology" has its uses on the research frontier as well, even for earth.  Using a hemispheric
general circulation model, B. G. Hunt showed showed that even    fifty days after the sudden termination of all
 incoming solar radiation, a quarter of the initial kinetic energy  remained in the atmosphere.
 Similar though slightly less drastic reductions in solar output must be examined to study
asteroid extinctions \cite{Reid81}, volcanic eruptions \cite{Hunt77} and ``Nuclear Winter", which is the decrease in insolation due to smoke and
atmospheric soot generated by thermonuclear war \cite{Crutzen84}. The earth's rotation rate is
also fair game for climate studies such as \cite{KuhnWalkerMarshall89,Hunt79,Simmonds85}.
 In this tradition, we analyze flow on a disk world.

The recent article by de Marez and Le Corre \cite{MarezLeCorre20} compares numerical simulations
of the North Atlantic Gyre on both flat and spherical earths. They show that the spherical results
are in good agreement with drift observations, discussed further below. The flat earth results
 are quite different form the observed ocean even though the authors assumed that the
disk was rotating. Rotation brings the flat earth model closer to reality, but a rotating disk is
anathema to flat earthers because an underlying tenet is: ``Trust your own senses rather
than mainstream science". If the disk is spinning, then the human senses have been fooled,
and the notion that the earth must be flat because it appears so to the nakd eye is
  cast in doubt.

\subsection{Impossibilities and ``hard" science fiction}

\begin{quotation}
Writing a science fiction story is fun, not work. ...the fun...lies in treating the whole thing as a game.... [T]he rules must be quite simple. For the reader of a science-fiction story, they consist of finding as many as possible of the author's statements or implications which conflict with the facts as science currently understands them. For the author, the rule is to make as few such slips as he possibly can... Certain exceptions are made [e.g., to allow travel faster than the speed of light], but fair play demands that all such matters be mentioned as early as possible in the story...
\end{quotation}
 \hspace*{0.5 in} ------- Hal Clement (Harry Clement Stubbs (1922-2003) in the nonfiction essay
``Whirligig World" included in most editions of his novel \emph{Mission of Gravity} \cite{Clement54}.

The student must approach conceptual exercises such as weather-on-a-flat-earth in the spirit of
so-called ``hard" science fiction. What distinguishes ``hard" science fiction from fantasy and
soft science fiction is the determinantion, expressed by Clement, of working out the science
as carefully as possible while allowing one or more violations of contemporary science for
the sake of the story. Faster-than-light (FTL) travel is the most common exception; hard sf is consoled
by the thought that FTL may be possible in a few centuries.\footnote{Clement's planet Mesklin has a day
only seventeen minutes long, an oblate spheroid more a disk than a sphere. I myself have explored
a toroidal planet in \cite{BoydMoonbow,BoydEarthflight}, an airship-friendly world in
   \cite{BoydHero,BoydVictory,BoydTempest} and starless planets (``anasters") in \cite{BoydElectricSouls,BoydIcosahedral}.}

No such comfort is available for the impossibilities of flat earth as explained below. Outsider science
is sometimes an intriguing venue for science fiction, ocasionally good art, but almost never good science. We can neverthess proceed to
analyze geophysical  fluid flow on a disc in the spirit of Clement and other hard sf writers: to stretch
our minds.

  \section{Gravity}

A crucial assumption of all flat earth theories is that gravity operates on the disk as a force perpendicular to the
surface of the disk, accelerating all objects downward towards the solid surface of the disk at 9.81
meters/second$^{2}$.

The gravity force vector should of course be directed toward the
center of mass of the planet, not normal to a disk. Except at the center of the disk, gravity   will generate a component
  which is radially inward towards the ``North Pole" of the disk. We must assume an imaginary
force, ``fairy-g", which is everywhere downward on the disk, in order to make useful comparisons
and analogies with the flow on a rotating planet. (See Appendix C for further discussion.)

\section{Flat Earth Geometry \& Cartography}

\subsection{Flat earth maps}

Flat earth theorists have proposed a number of geometries including an infinite plane, Derren Nesbit's
square with boundary conditions ( periodicity in longitude, glide reflection at the poles) that make the surface topologically equivalent to the sphere, a square repeated periodically over the infinite plane, various species of sphere projected to a disk by means of a bipolar map  projection and a ``Supernatural Construct" where the last is a frank admission that no ordinary or even extraordinary flat surface is free
of contradictions. By far the most common flat earth concept is a flat disk of finite diameter. \footnote{This is surmounted by a dome
which supports the stars and planets. A planet-sized half-sphere would collapse unless made of a material
several orders of magnitude stronger than any earthly material.  Fortunately this dome of, let us call it
``unobtainabletransparentarium" or ``fairy-glass", is irrelevant to tropospheric meteorology.}
 This is almost universally depicted by mapping continents to the disk using the aximuthal
equidistant projection, known in flat earth circles as the ``Gleason map" \cite{Gleason93}.

The disk is always assumed to be {motionless}.An important justification for the flat earth concept is that human senses are completely trustworthy; therefore, a planet that is percieved as flat and motionless must be
flat and motionless.

Disk cartography uses polar coordinates $(r, \theta)$ with the radial coordinate $r$ as the distance to the North Pole and $\theta$ as the polar angle. The latter is a cyclic coordinate with $\theta \in [0, 2 \pi]$, identical
with longitude. The radial coordinate $r$ is analogous to colatitude. It is convenient to change to a
scaled radial coordinate
\begin{eqnarray}
r = R \rho, \qquad \rho \in [0, 2]
\end{eqnarray}
where $R$ is the distance from the North Pole to the equator.
We shall define the disk that is the direct product of $\rho \in [0, 1]$ and $\theta \in [0, 2 \pi]$ as the ``[disk] Northern Hemisphere" and the annulus which is all points with $\rho \in [1, 2]$ as the ``[disk] Southern Hemisphere" as illustrated in \ref{FigFlatEarth_Sch_Map}.

\begin{figure}[b]
\centerline{\includegraphics[scale=0.5]{FlatEarth_Sch_Map.pdf}}
\caption{The geometry of the disk of the Flat Earth. The ``disk Northern Hemisphere" is shaded.
The North Pole is at the center of the diagram; in terms of polar coordinates $(r, \theta)$, this point is at
$r=0$. The equator is the thick solid curve, a circle of radius $R$ where $R$ is the circumference of the
earth. The dashed circle at $r=2R$ is the South Pole, which is stretched into a circular ``ice wall".
The thin circles of radius $r=R/2$ and $r=(3/2) R$ represent the latitude circles with latitude = 45 degrees N. and 45 degrees S., respectively. The ``disk Southern Hemisphere" is the unshaded annulus; note that the
area of the ``Southern Hemisphere" is three times larger than that of the ``Northern Hemisphere". The latitude circle at 60 degrees south latitude on the disk has a circumference of $(5/3) E$ where $E$ is the equatorial circumference but the same circle on the sphere has a circumference of $E/2$, a factor of $3.33$ difference.}
\label{FigFlatEarth_Sch_Map}
\end{figure}

\subsection{Discussion of why it is impossible to simultaneously match the pole-to-equator distance $R$ and the equatorial circumference $E$ on the disk to the corresponding values on the globe}

On a sphere, the pole-to-equator distance (along a meridian) is equal to \emph{one-quarter} of
the circumference of the equator, 10,000 km and 40,000 km, respectively. If $R$ is the
distance from the pole to the equator on a disk, then the \emph{disk} equator has a circumference
of $E=2 \pi R$. This make it impossible to simultaneously match the pole-to-equator distance $R$ and the equatorial circumference $E$ on the disk to the corresponding values on the globe
as enumerated in Table~\ref{TabDiskRandE} .

If we choose to match the pole-to-equator distances by choosing $R=10,000$ km on the disk,
the disk equator has a circumference of 62,800 km. Such a disk earth has an equatorial circumference
22,800 km larger than the generally accepted value. Interesting!

If we choose to match the equatorial circumferences instead, then the pole-to-equator distance on the
disk is shortened to 6366 km. However, in the metric system, the meter was defined to be 1/10,000,000 of the pole-to-equator distance. Did all the world's navigators and pilots, by land and by air, somehow
not notice that pole-to-equator distance was shorter, by over 3600 km, than the value
published in the standard navigational and geographic tomes. Interesting!

\begin{table}[h] \caption{\label{TabDiskRandE} The North-Pole-to-equator distance $R$ and the circumference
of the equator $E$ for three different models}
\begin{center} \begin{tabular}{|c|c|c|} \hline
Model & $R$ & $E$ \\ \hline
Sphere & 10,000 km & 40,000 km \\
Small Disk & 6366 km & 40,000 km \\
Large Disk & 10,000 km & 62,800 km \\ \hline
 \end{tabular} \end{center} \vspace{5pt} \end{table}

The reason  that $E=4 R$ on the sphere but $E=2 \pi R$ on the disk is that the  pole-to-equator  curve on the surface of a
sphere  is \emph{curved} whereas the ray from pole to equator on the disk is \emph{flat}.
The equatorial terminus of the curve on the sphere is much closer (by $2/\pi)$)  to the axis of the globe than
the equatorial end of the line on the disk is to the North Pole.
 Fig.~\ref{FigFlat_earth_disk_two_poletoequator_curves} is a visual comparison.

\begin{figure}[h]
\centerline{\includegraphics[scale=0.7]{Flat_earth_disk_two_poletoequator_curves.pdf}}
\caption{The vertical dotted line is the spin axis of the globe. The straight solid horizontal line of length
$R$ connects the North Pole with the equator on the  flat disk.  The dashed curve connects pole-to-equator on the sphere and
is also of length $R$. Denote the radius of the globe earth by $\rho$. Because of the curve, the globe equator is only a distance $\rho=(2/\pi) R$ away from the rotation axis. The circumference of
the equator on the disk is $E^{disk}=2 \pi R$; the equatorial circumference on the globe is$ (2/\pi)$ smaller,
$E^{globe} = 4 R = 2 \pi \rho$.
}
\label{FigFlat_earth_disk_two_poletoequator_curves}
\end{figure}

\subsection{Circumference of a circle of latitude}

The isolines of distance to the North Pole are concentric circles on both the sphere and on the disk.
  We shall extend the label of these isolines from the sphere to the disk, and call these
``circles of latitude" in either case. The circumference of these circles of latitude is quite
different for the sphere and the disk:
\begin{eqnarray}
C(latitude) & = & E^{globe} \, \cos(latitude) = 4 R \cos(latitude) \qquad \text{Sphere} \\
     & = & E^{disk}  \left( 1 -  \dfrac{2}{\pi}  latitude \right)
\end{eqnarray}
where latitude is in radians [=360 degrees/($2 \pi$) ].

\begin{figure}[h]
\centerline{\includegraphics[scale=0.7]{Circumference_Circles_Latitude.pdf}}
\caption{ Solid: circumference of latitude circles as a function
 of latitude in radians. Dashed: same but for the disk.
The figure assumes that the circumferences of the equators
are the same.
}
\label{FigCircumference_Circles_Latitude}
\end{figure}

Fig.~\ref{FigCircumference_Circles_Latitude} shows that the circumference of a latitude circle
increases monotonically as we progress from the North Pole (right of the diagram) to the
South Pole (left). In contrast, the circumference of a latitude circle on the globe is a
 maximum on the equator and diminishes to zero at the South Pole.

The Australian cites of Brisbane (-27.5 degrees S, 153 degrees E, and Perth  ( -32 degrees S, 115.5E) are almost on the same circle of latitude. Splitting the difference by using the circumference of $-30$ degrees, the distance between the two cities is the circumference of that latitude circle multiplied
by (153 - 115.5)/360. One obtains 5480 km on the disk, but only 3560 km on the globe. The
actual distance is 3600 km.

The disk thus gives the Perth-to-Brisbance by a number 49 \% too large.  Presumably it is a matter
of intense interest to airlines and long-haul truckersthat the distance beween these two cities is 1920 km larger than
listed in the standard references. Alternatively, the earth is round.

\subsection{Cartographic absurdities}

There are thus three glaring discrepancies between the finite disk/Gleason map and measurements on
the globe:
\begin{enumerate}
\item The southern disk hemisphere has three times the area of the northern disk hemisphere
[And no one noticed?]

\item The circumference of latitude circles monotonically \emph{increases} south of the equator on
the disk, but monotonically \emph{decreases} on the sphere. [And no one noticed?]

\item The pole-to-equator distance is only 64 \% of its spherical value in the small disk model;
alternatively, the circumference of the equator is 57 \%  larger-than-spherical in the large disk model:
it is impossible to match both $R$ and $E$ on the disk to the generally accepted values on the globe.
 [And no one noticed?]
\end{enumerate}

\clearpage

\section{The Effective Gravity, Centrifugal Force and the Oblate
Spheroid Approximation}

The rotation of the earth about its axis
generates a centrifugal force that acts on the solid earth, oceans and all objects on the planetary surface. The centrifugal force acts  \emph{radially outward} from the rotation axis.

Gravity also acts on all that is subject to centrifugal force. Gravity is a vector pointing at the
center of mass of the planet.

The combined effect of gravity and centrifugal
 force is the vector sum of the gravity and
centrifugal force vectors. Meteorologists
denote this sum as the ``effective gravity"
   $\mathbf{g}_{eff}$ as shown in
Fig.~\ref{FigGravityEffective}.

For graphical clarity, the magnitude of the
centrifugal force relative to gravity has
been greatly exaggerated. In reality, the largest
 value of the ratio of centrifugal force divided
by gravity, which occurs at the equator, is
only 1/300. The diagrams of people flung off the
earth by centrifugal force found in many flat
earth videos are hilarious fantasy except to
the innumerate. One observable consequence
of centrifugal force due to the earth's rotation
is that at the equator, one weighs (299/300) of one's weight at the North or South Pole.

The other observable consequence is that a sphere of homogeneous
fluid is deformed into an oblate spheriod by rotation. The change is such that the \emph{effective
gravity vector} is \emph{perpendicular} to the \emph{surface}
of the spinning spheroid at every point on its
   surface.

Newton himself predicted the oblateness as
confirmed by an intensive surveying expedition
led by Maupertuis in the 1740's. The mathematics
of self-gravitating spinning liquid masses in
  equilibrium was developed further by Maclaurin in
the eighteenth century, Jacobi and Poincar\'{e}
in the nineteenth century and ``Black Hole" Chandrasekhar, N. Lebovitz and R. A. Lyttleton in the twentieth century \cite{Chandrasekhar,Lyttleton}. Jacobi predicted that at a high rotation rate,
  the branch of oblate spheroids would bifurcate, and
the stable shape is then a triaxial ellipsoid. The dwarf planet Haumea is a Jacobi ellipsoid with the three axes being roughly 2300 km, 1700 km and 1000 km.

 The Earth, of course, is not liquid (now!), but
sufficiently large solid masses and solid/liquid
masses will plastically deform in response to the
effective gravity as if liquid.
 The flat earth diskworld is \emph{not} an
equilibrium of a self-gravitating mass!

\begin{figure}[h]
\centerline{\includegraphics[scale=0.8]{GravityEffective2.pdf}}
\caption{Cross section of half of an  oblate
   spheroid. The eccentricity is greatly
exaggerated for graphical clarity. The stick figure
is an observer at a point on the planetary surface. At that point (and everywhere on the surface of the spheroid), gravity is a vector pointing at the
center of mass, here shown as the black disk.
The dashed line is the centrifugal force,which is everywhere radially outward from the rotation axis.
The effective gravity $g_{eff}$ is the vector sum of the gravity vector and the centrifugal force
vector. This vector is everywhere perpendicular to the surface of the oblate spheroid.}
\label{FigGravityEffective}
\end{figure}

\section{Hydrostatic Equilibrium}

\subsection{Containment of the Atmosphere}

The atmosphere is both static and dynamic. The simplest approximation to  the static part, usually called the ``mean flow", is obtained by applying a running time average to each of the meteorological fields:
\begin{eqnarray} U \equiv \frac{1}{P} \int_{t-P/2}^{t+P/2} \, u dt
\end{eqnarray}
where $u$ is any  of the wind components or temperature or density or any other meteorological variable and where $U$ is the corresponding mean.  If the period $P$ is anything from a week to a month, the mean is approximately independent of $P$ and is the static part of the flow.To a first approximation, the dynamic part is the  difference between a field and its mean.

 To a very high degree of approximation, the mean is in hydrostatic equilibrium. This means that the force of gravity on a small volume or ``parcel" of fluid is balanced by an upward pressure gradient The vertical momentum equation simplifies to
\begin{eqnarray}
\dfrac{1}{\rho} \dfrac{\partial \, p}{\partial z} = - g
\end{eqnarray}
where $\rho$ is the density, $p$ is the pressure and $g$ is the gravitational constant.

A popular Flat Earth mantra is, ``Pressure cannot exist without a container; otherwise, the
atmosphere would just whoosh away into outer space."

This is true but a physical barrier is unnecessary. The pressure force is a \emph{force}, so
it can only be contained by \emph{another force}, in this case \emph{gravity}.

Hydrostatic balance, combined with the ideal gas flaw and the observation that the temperature
of the mean flow does not vary much with height, implies that the mean pressure must decrease
  \emph{exponentially}   with increasing height above the ground. The air pressure and density are \emph{halved}
for each increase in height of about 6 km. Thus, vacationers who go from sea level to a hike in the
Rocky Mountains at an elevation of 5 km often suffer altitude sickness until their bodies adjust.
Most climbers of Everest have to carry bottles of oxygen because the pressure at the summit is only about
a third of what it is at the surface.

We can imagine confining a tall column of air in hydrostatic equilibrium inside a box. What happens if we open
the lid of the box? Nothing. Gravity and pressure gradient are balanced. There is no acceleration of air
either up or down. What happens if we remove the sides of the container? Nothing. Equilibria are
very boring.

The components of the dynamic part of the flow, which meteorologists call the ``wave", ``eddy" or ``perturbation" flow, are in hydrostatic equilibrium also. However, motions on the scale of an individual thunderstorm or smaller have  deviations from hydrostatic balance that, while small, are dynamically significant.

At large heights, the unbounded atmosphere does suffer interference from other celestial bodies, the solar
wind and so on. Mercury once had an atmosphere, but long ago lost it to space.
Similarly, Mars lost most of its atmosphere although a low density of carbon dioxide remains. However, Earth, Venus and the other planets of the solar system have stronger gravity and will retain (most) of their atmospheres for billions of years to come.
 Through a combination of Jeans escape, polar wind escape, charge exchange escape, photochemical escape, sputtering escape and hydrodynamic escape, the earth is losing about 3 kilograms of hydrogen, 50 grams of helium and small amounts of heavier elements per second. All should be well for at
least a billion years when a combination of hotter sun, runaway greenhouse effect and
atmospheric escape could steal away the seas, but that is a tale for another time.

The atmosphere does not stop at a fixed height; instead, it merely fades away.

\subsection{The Myth of the Firehose Ocean}

The pressure in the ocean increases by one atmosphere (1 bar $\leftrightarrow$ 14.7 pounds per square inch (psi) $\leftrightarrow$ 101 kPa) for every ten meters of depth. The deepest part of the ocean
is 10,984 meters in the Marianas Trench where the pressure is 1,071 atmospheres or 15,570 pounds per
square inch. With so great a pressure difference between the water in the deep ocean and the
pressure of (on average) one atmosphere at the sea surface, why doesn't the
briny ocean erupt upward like the jet of an immensely powerful firehouse aimed at the heavens?

The answer for the sea is the same as for the sky: \emph{hydrostatic equilibrium}. A column of ocean water or air
an inch on each horizontal side and extending from the bottom of the Marianas Trench upward weighs
15,555 pounds if the columns extends only to the ocean surface, or 15,570 pounds if the column extends
100 kilometers into the sky. The weight of the column (15,555 pounds of salt water, 14.7 pounds of air)
is exactly equal to the pressure at the bottom of the column.

If the bottom of the column is moved upward, the weight becomes less and the pressure
becomes less in exact proportion. The change of the pressure with depth is a vertical pressure
gradient. The upward pressure force exactly balances the downward force of gravity to
produce an equilibrium, devoid of movement up or down.

The equilibrium is not perfect; it is disturbed --- slightly -- by the surface gravity waves that rock vessels on the sea and other dynamical phenomena. Metaphorically, dynamics is a dance and
hydrostatic equilibrium is the dance floor. The floor vibrates as the dancers tap and stamp,
but the floor remains the foundation for all the motion above it.

\section{Solid Body Rotation}

\begin{quotation}
No, the
atmosphere is absolutely not velcroed to
 to the frame of reference with the
earth!
\end{quotation}
\hspace*{0.5in} --- Bob Knodel in minute 37 of \cite{GLobebustersDebunkingProfDave}.

The mean east-west velocity is a complicated function of latitude and height because of the transfer of
momentum both latitudinally and vertically by waves. ``Wave-mean flow"  interaction is a topic
too complicated to discuss here but see \cite{Buhler14,BoydOP1,Andrews85}. Furthermore, the mean flow   also contains the Hadley circulation. This is independent of time and therefore part of the mean, but it is a convective circulation, responding to spatially-varying heat sources, which is dynamic. Without wave-mean
flow interaction and the Hadley cell, the mean is a state of ``solid body rotation". It is useful to look at this because, metaphorically,
solid body rotation is the wet plaster wall on which the wave-mean interactions and Hadley flows are painted and absorbed, or if one prefers a different metaphor,   the dance floor on which the dynamics dances.
 Furthermore, an improved approximation of  the static part of the atmosphere subtracts from the mean  the wave-mean flow
effects, which are dynamic.

Solid-body rotation is a static, hydrostatically-balanced flow such that the velocity of
the rotating atmosphere at a given point is identical with that of a solid body at the same point rotating
 with the same angular velocity.
 The velocity is a function only of radial distance to the center of the sphere.

 Viscosity transmits shearing stress from one layer of fluid to another, in this case,
  from one radius to another. Solid-body rotation is equivalent to zero stress and is therefore
the end of this process of viscous adjustment.

It is easy to demonstrate this in a student lab. Fill a cylindrical dish with liquid and place it on a turntable.
When the electric motor is turned on, the fluid in a
thin layer (``boundary layer") accelerates very rapidly  to match the velocity on the rim of the dish;
the fluid over most of the dish is motionless. As time passes, the boundary layer thickens and thickens,
evolving to a state of solid-body rotation in which the tangential velocity is a linear function of the radial distance from any point in the flow to the center of the dish.

Knodel's assertion is wrong; the solid,  spinning sphere controls the static atmosphere. Its ``velcro"
is a combination of gravity (for hydrostatic balance) and viscosity (for transmission of shearing stresses and for evolution to solid-body
rotation).

\section{Hadley Cell}

The equator is hot; the poles are cold. This drives a convective circulation
 with hot air rising at the equator and cold air sinking elsewhere to complete the flow.
 The circulation is independent of longitude (``axisymmetric") if the continent/ocean asymmetries are ignored, which is a useful
first approximation. The name comes from
George Hadley (1685-1768), an English lawyer who published a four page article
 describing and explaining the flow in 1735. Hadley recognized that on a rotating earth,
  the Coriolis force would apply a westward deflection to the surface  flow converging  towards the equator
in both hemispheres. This westward (``easterly") flow is the ``trade winds".

On the flat disk or on a slowly-rotating spherical planet, the Hadley cell is global with the descending
flow at the poles. However, our planet spins so rapidly that the Hadley flow does not reach the poles. Instead,
 the air descends in the subtropics in bands centered at $\pm 30^{0}$ latitude.

The ocean temperature in equatorial regions is very warm, so the
evaporation rate is high,  forcing
 the surface air to be nearly saturated. As this equatorial air rises in the Hadley updraft, the rapid decrease
of pressure forces the air parcels to expand. The ascent is ``adiabatic" in the sense that parcels exchange
little energy with their surroundings. The adiabatic expansion generates ``adiabatic cooling". The lowered
temperature reduces the capacity of the air to hold water as vapor. Saturation occurs, meaning that
the relative humidity is 100 \% and condensation into liquid droplets follows. The phase change generates latent heat
release which warms the air and makes it buoyant, further driving the moist air upward. The liquid water
forms clouds and rain. The heavy, steady rain makes the equatorial region the
\emph{tropical rain forest}.

When the air descends in the subtropics, it is rather dry since much of the water vapor has condensed in
the upward flow at the equator, and rained onto the equatorial jungles. The descending air parcels are compressed and experience
``adiabatic warming". This warming increases the carrying capacity of the descending air, so the
relative humidity decreases sharply even though the absolute amount of water vapor in the
air does not change.

The world's major \emph{deserts} --- the Mojave in the U. S., the Gobi in Asia, the Sahara and Kalahari in Africa, etc. ---
lie in the \emph{descending} regions of the Hadley flow.

Of course, the subtropics are not uniformly desert, and there are also local deserts  outside the subtropics. Mountain ranges and local wind and ocean circulations modify
the global dynamics. Even so, condtions for deserts are very favorable in the dry, descending Hadley flow.

The Hadley circulation is extremely efficient at redistributing heat. The mean atmospheric temperature
is almost uniform over the whole domain of the Hadley cell. There is little variation in mean surface
temperature over the entire tropical band on earth. Venus is very slowly rotating, so the Hadley circulation extends to the poles; the mean surface temperature on Venus has, compared to earth,  a very small latitudinal gradient.

On a flat earth, the Hadley flow should be Venus-like. The Arctic and Antarctic regions would be only be
slightly cooler than the tropics. Florida and Calgary would differ little in temperature.

Also, on a flat, non-rotating earth, the trade winds disappear! Goodbye, Columbus!

\begin{figure}[h]
\centerline{\includegraphics[scale=1.0]{FlatFlow_GlobeFlow.pdf}}
\caption{Schematic of mean, longitude-independent surface  flow on the globe (left)  and flat earth (right). Note that the solid circle is the equator and the dashed circle is the South Pole.
 The top graphs are height-latitude cross-sections. The bottom plots are views looking down from above the North Pole.}
\label{FigFlatFlow_GlobeFlow}
\end{figure}

\section{Midlatitude Synoptic Vortices and Baroclinic Instability}

The latitudinal  bands on our planet, poleward  of the tropics are, on our fast-spinning planet,
occupied not by the Hadley cell, but a circulation cell of the opposite rotation called the ``Ferrell cell".
Because the axisymmetric flow is equatorward at the surface in the Ferrell cell, Coriolis force
deflects the winds eastward, generating the ``prevailing westerlies". This part of the axisymmetric flow
is strongly varying in both latitude and height. It is unstable to small amplitude waves which grow and
grow at the expense of the Ferrel cell. This spontaneous growth of small waves into big vortices is called ``baroclinic instability". It was first
explained  by Jule Charney in 1947. (He employed a tangent plane approximation, the so-called ``beta-plane".)

The waves eventually roll up into vortices. Low pressure centers alternate around a latitude circle
with high pressure centers. The low pressure centers are ``cyclones" in either hemisphere, but spin,
as viewed from above, counterclockwise in the Northern Hemisphere but clockwise in the Southern Hemisphere.
 The rotation is in the opposite sense for high pressure centers, which are ``anticyclones".
 Collectively these are ``synoptic-scale vortices".

These vortices slowly grow and decay, but to within an error of about 10 \%, the pressure force
directed inwards (for cyclones) or outwards (for high-pressure centers) is balanced by Coriolis force.

Hurricanes and typhoons are vortices that always begin in the tropics and are always
cyclones. Again, the sense of rotation in the southern hemisphere is opposite that of the
northern hemisphere. The dominant dynamics is again quasi-steady state with a
balance of Coriolis and pressure forces. In the eyewall region where the tangential
winds are strongest, however, centrifugal force is as important as Coriolis force.

On a flat earth without spin, baroclinic instability would disappear, and with it, most of what
we in the middle latitudes think of as the weather. Fronts, squall lines, and tornado-spawning
  mesoscale systems would be missing, too.

\subsection{Dishpan Laboratory Experiments}

\begin{quotation}
``Approximation is as important as derivation"
\end{quotation}
\hspace*{0.5in} Thomas B. W. Kirk (Harvard lecture,1970)

\bigskip

Dave Fultz showed that baroclinic instability could be captured in the laboratory by taking a cylinder or annulus
of water in a dish on a rotating turntable. (A spinning cylindrical container is a ``dishpan" in
geophysical jargon.) The flow represents a single hemisphere; a temperature
gradient in latitude (radius in cylindrical coordinates) is created by heating the outer rim (``equator") and
cooling the center of the cylinder (``arctic"). The flow is visualized by sprinkling aluminum powder on the
surface of the water. When the apparatus is started from rest and the cylinder spun up to a constant rotation rate, a Hadley-like flow
independent of the polar angle $\theta$ in polar coordinates emerges first. When the flow is  sufficiently strong,
 baroclinic instability sets in, and wavy perturbations  spontaneously amplify, stealing energy from the
axisymmetric ($\theta$-independent) flow. The waves become vortices qualitatively resembling
 synoptic-scale cyclones and anticyclones in the atmosphere \cite{SpenceFultz77,RiehlFultz58}.
Beautiful illustrations are in Van Dyke \cite{vanDyke82}.

Allaby \cite{Allaby95} illustrates a recreation of  Fultz' experiments by middle school students.
 The turntable is a cake dish, spun by hand. The heat source is a set of candles underneath the
outer wall of the dish. The cold source is a cylindrical can filled with ice cubes and placed in the center of
the dish. Paprika is sprinkled on the surface of the water to visualize the flow.

\subsection{Microgravity Laboratory Experiments with Spherical Symmetry}

The cylinder can be replaced by a sphere by (i) using a dielectric fluid and mimicing gravity by a
spherically-symmetric
electric force created by charging the sphere and (ii) eliminating the non-spherically-symmetric force of  gravity by
performing the experiment in free fall (``microgravity").
 The 1985 experiments on the shuttle \emph{Challenger} used an apparatus built by John E.
Hart at the University of Colorado \cite{Hartetal86,HartGlatzmaierToomre86}.
 These experiments have continued on the International Space Station (ISS) \cite{Futteretal13,Zaussingeretal19}.
Experiments on fluids in microgravity have also been performed using aircraft flying  a
parabolic trajectory like NASA's famous ``Vomit Comet". Such flights only provide  30 seconds  of freefall per dive, but this is long enough for some flow experiments \cite{Meyeretal19}.

\subsection{Soap Bubble Vortices}

A group at the  ``Ondes et Matière"  CNRS laboratory in Aquitaine (France) performed  laboratory experiments on vortices in a soap bubble with spherical geometry \cite{Meueletal13}.
  The solid earth is replaced by  a sphere of air; the ocean or atmosphere is modeled by the thin liquid film wrapped uniformly
around it as illustrated in Fig.~\ref{FigVortex_on_Half_Bubble_2plot}.

\begin{figure}[h]
\centerline{\includegraphics[scale=1.0]{Vortex_on_Half_Bubble_2plot.pdf}}
\caption{Left: a thin film uniformly covering a sphere (soap bubble). Right: close-up of a soap bubble
vortex.}
\label{FigVortex_on_Half_Bubble_2plot}

\end{figure}

Thus, the claim that meteorology and physical oceanography are strictly \emph{observational sciences}
 is not entirely true. It is possible to do instructive laboratory experiments even in spherical geometry.

To be sure, it is not possible for laboratory experiments to include all the radiative transfer, photochemistry,
ice physics, etc., included in general circulation numerical models. Thus, modern meteorology is even more so a
  \emph{computational} science.

\section{The Beta Effect}

\subsection{Rossby Waves}

One deficiency of Fultz' spinning cylinders is that the Coriolis parameter is a \emph{constant}, equal to twice
the angular frequency $\Omega$ of the rotating apparatus. On a sphere,
the Coriolis parameter $f$ is $2 \Omega \sin(\text{latitude})$. It thus is a maximum at the North Pole,
diminishes to zero at the equator and is negative everywhere in the southern hemisphere.

The constancy of the Coriolis parameter in experiments with a
flat-bottomed cylinder is unfortunate because
 many geophysical phenomenon depend on the
rate  at which the Coriolis parameter is varying. By convention, the latitudinal derivative of the Coriolis parameter is always denoted by the Greek letter ``beta" [$\beta$]. In the early 20th century, planetary scale waves retrogressing westward relative to the meanl flow were discovered observationally. Carl-Gustav Rossby and Jacob Bjerknes independently provided a theoretical explanation in 1937. These waves, always with the phase velocity westward relative to the mean flow, are now called
``Rossby waves" \cite{Platzman68}. Observations  confirm theory in both the ocean and the atmosphere.

Madden \cite{Madden07} is a good review of atmospheric Rossby normal modes.
The most prominent of these is the so-called ``five-day wave", which propagates westward at 90  m/s, completing one full rotation  of the globe in five days. This mode has longitudinal structure proportional to
$\sin(\lambda)$ where $\lambda$  is longitude in radians. High pressure at the Greenwich meridian is mirrored by low pressure at the international dateline.

Similarly, some modes of atmospheric tides are Rossby waves \cite{ChapmanLindzen70,BoydOP239,BoydOP254}.

High-tech equipment is not required; a barometer and a personal computer (to compute
Fourier transforms) are sufficient to detect these kind of periodicities in atmospheric flow.

\subsection{Stabilizing Long Waves in Baroclinic Instability}

The beta effect is not essential to baroclinic instability, or else Fultz' pioneering experiments with a
rotating cylinder would have failed. However, the beta-effect does significantly modify baroclinic instability by \emph{stabilizing} the long waves, that is, the waves of largest spatial scales. The result is that on the globe, only waves of intermediate scale are unstable. The horizontal scale of these unstable waves is roughly equal to the so-called ``Rossby Deformation Radius" which depends on the strength of the Coriolis parameter.
Without the stabilizing effect of nonzero $\beta$, midlatitude storms would be much larger in scale, and therefore more damaging.

\subsection{The Gulf Stream}

The Gulf Stream is a strong narrow current that runs along the east coast of the United States
from Florida to Cape Hatteras, North Carolina, where the jet turns away from the coast, develops
wavy instabilities and finally evolves to intense, circular vortices called ``Gulf Stream rings".

The Gulf Stream has been known to sailing masters for many centuries.  The first scientific investigation was by Ben Franklin. On his trip back to America in 1775 after his long exile in England, his grandson repeatedly dropped a bucket into the ocean, retrieved it and measured the water temperature with a mercury thermometer; the Gulf Stream is much warmer than the cold North Atlantic and thus can be identified and mapped by temperature alone. Franklin did not publish his map of the Gulf Stream, coauthored with his cousin, a merchant captain, until after the American Revolution. (He was rather  busy during the war as the rebels' ambassador to  France, successfully negotiating French intervention.)

The Gulf Stream is but one instance of a ``western boundary current". Others include
the Brazil Current (South Atlantic), the Somali Current (North Indian Ocean), the Aghulas Current
(eastern coast of Africa), and the Kuroshio Current (North Pacific). Why? And why are there no
\emph{eastern} boundary currents?

After two centuries of failure and futility, the true explanation was given by Henry Stommel in  1948. Science has its urban legends; an oceanographic tall tale is that Stommel was so pleased with his Gulf Stream discovery that he danced a jig  in the sedate hallways of the  Wood Hole Oceanographic Institution.
Many years later, Stommel confirmed to my colleague (and his old friend) Stan Jacobs that the story was in fact true. He was later awarded the Crafoord Prize [the geophysics Nobel Prize]\footnote{The Crafoord Prizes were funded by Dr. Crafoord to provide Nobel prizes in areas of science which were not specified in Alfred Nobel's will. The Crafoord  Prizes are administered by the same organization that awards the Nobel prizes. This body previously instituted the economics Nobel prize, funded by the Bank of Sweden, so Crafoord expected no difficulties. However, the Academy took so much criticism for extending the prizes into the social sciences that instead of renaming the economics prize after the Bank of Sweden and instituting new Nobel prizes in science, the Academy continued the ``Nobel" Prize in economics while administering the new science prizes under the funder's name, despite his opposition.},  though not for dancing.\footnote{Among his many achievements, Stommel coauthored a book entirely devoted to the
Coriolis force \protect\cite{StommelMoore89}.}.

Stommel showed that the beta-effect not only makes Rossby waves propagate westward, it also squeezes jets against the western boundary of the ocean \cite{Stommel48}. The Gulf Stream and all its western boundary current brethren would not exist on a disk, not even on a rotating flat plane.

\subsection{Sverdrup Flow}

In a famous paper, Harald Sverdrup (1947) \cite{Sverdrup47} developed a simple formula, now named for
him, for the depth-averaged north-south mean velocity as driven by wind stress.

Let $\beta$ denote the latitudinal derivative of the
Coriolis parameter. Let $\rho$ be density, assumed constant [an excellent oceanic approximation]. Further
identify
$\overline{v}$ as the mean north-south depth-averaged
oceanic current. Let $\tau^{x}$ and $\tau^{y}$ be the
east-west and north-south components of the
surface wind stress. Lastly, denote the unit vector in the vertical by $\hat{k}$.
If we integrate over depth, assume that the stress is zero at the ocean bottom,
replace the divergence of the horizontal velocity by $\partial w/\partial z$
and assume the vertical velocities are negligible at the boundaries\footnote{The vertical
velocity is not negligible because of surface gravity waves, but
 after we take a running time average or space average to filter the small-scale
surface waves, the \emph{averaged} vertical velocity is negligible.},
we obtain
\begin{equation}
\beta \overline{v} = \frac{1}{\rho} \left( \frac{d \tau^{(y)}} {dx}
 \, - \, \frac{d \tau^{(x)}} {dy}
\right) = \frac{1}{\rho}  \left( \hat{k} \times \vec{\mbox{curl}} \vec{\tau} \right)
\qquad \mbox{[Sverdrup Relation]}
\end{equation}

The trade winds and prevailing westerlies create a wind stress with non-zero curl, which results in
a clockwise gyre in the north Atlantic. In the
absence of the beta effect, the Gulf Stream would be
a broad, slow-moving current instead of the narrow jet
  that is actually observed. The return, southward flow \emph{is} broad and slow-moving; its dynamics are described by the Sverdrup relation.

A flow everywhere equatorward would strip the ocean
dry at high latitude and pile up mass at low latitude. Thus there
must be a region where the Sverdrup relation is inaccurate.
  In a narrow boundary
layer along the western side of the Atlantic, a different balance of forces prevails. In this boundary layer,
  the Gulf Stream is the northward return flow
of the gyre.

Note that the Sverdrup flow is inversely proportional
to $\beta$. Without a Coriolis force, and further,
one in which the Coriolis  parameter varies with
latitude, $\beta=0$ and the Sverdrup flow does not exist.

\section{Ekman Boundary Layers}

On the voyage of the Arctic exploration steamship \emph{Fram} from 1893-1896, the Norwegian scientist and explorer F. Nansen noticed something very odd. Near the Arctic icepack, floating bits of ice provided a natural flow visualization. The wind exerts a stress on the ocean and drives a surface current. The oddity
was that the surface current vector made an angle
twenty to thirty degrees to the right of the wind vector.

Nansen's doctorate was in marine neuroscience and he lacked the mathematics to make a theory. He was, however, a very good scientific evangelist, and he described this non-parallelism of wind and
surface current to any geophysicist who would listen. V. Ekman published a simple theory in 1904.

The Ekman flow is a boundary layer current whose
magnitude falls exponentially with depth. There is a three-way balance of forces between the wind
stress, Coriolis force and viscosity in the Ekman layer. It is the Coriolis force that causes the nonzero angle between the wind stres and the surface current. The angle is
forty-five degrees in Ekman's idealized theory;
 more accurate theories give a somewhat smaller angle as observed by Nansen.

A remarkable feature of oceanic  Ekman layers is that the current vector rotates clockwise in the northern hemisphere with increasing depth. A few dozen meters down, the Ekman current is directly \emph{opposite}
in direction to  the wind stress that drives it!

Ekman's solution, with different boundary conditions, applies also to the atmosphere.
In the atmosphere, the Ekman flow is driven by
the geostrophic flow\footnote{``Geostrophic flow" is shorthand for ``geostrophically-balanced flow", that is a flow in which the dominant terms in each of the two
horizontal momentum equations are pressure gradient force and Coriolis force.} at higher altitudes. The Ekman flow a kilometer up is parallel to the geostrophic
flow. The wind vector rotates counterclockwise with decreasing height so that the surface wind is to the left of the direction of the geostrophic flow above the boundary layer by an angle that is 45 degrees in Ekman's theory and 20 to 30 degrees in observations and improved theories.

 All rotation
directions must be reversed in the southern hemisphere. Ekman theory is singular at the equator where additional terms in the Navier-Stokes equation must be retained to accurately model the observed equatorial boundary layer.

Ekman theory does not replace the usual theories
for hurricanes and synoptic-scale cyclones and anticyclone. Rather,
Ekman theory is a description of the boundary layers that form in the lowest kilometer of the
atmosphere. Above the boundary layer and up to the tropopause, hurricanes
and synoptic cyclones are geostrophically balanced
 as noted earlier.

The Ekman layer underneath a hurricane or synoptic-cyclone is an ageostrophic flow that sends
air spiraling in towards the center of the vortex
where it turns the corner and erupts out of the
boundary layer as a strong vertical flow. (This is called ``Ekman  pumping").  For a
hurricane, the boundary layer radial inflow does not reach the center because of
centrifugal force, which grows as the spiraling blobs
of fluid get closer and closer to the center. Instead, the boundary layer inflow erupts in an annulus which is called the ``eyewall". The upward flow triggers violent
 convection which provides the energy to sustain the
entire vortex. Over the tropical ocean where hurricanes grow, the boundary layer fluid fed to the eyewall is very moist and warm, perfect fuel for a monster storm.

Boundary layers without Coriolis force are common; the
flow over an airplane  is a familiar example. However, for
  Ekman boundary layers and Ekman pumping, Coriolis force is as essential as gills are essential to a fish.

\section{Bubbles}

Blowing soap bubbles is a familiar childhood activity. Who knew those semi-transparent spheres,
floating in the air, are models of our  planet?

A favorite trope of   Flat Earth believers is ``Water always seeks
its own level". This is interpreted as a [spurious] physics law that the [mean] ocean surface
is a plane (and therefore the planet itself is a flat-sided disk).
This steady-flow-is-always flat ``law"  is falsified by a variety of
classical steady flows such as  the draining vortex (Fig.~\ref{FigExploratoriumFreeVortex}),
hydraulic jumps in rivers and soap bubbles (\cite{Meueletal13} and  Fig.~\ref{FigVortex_on_Half_Bubble_2plot}).
   In addition, the mean surface of the ocean is not flat as explained in the next section.

A soap bubble is a sphere of air surrounded by a thin film of liquid, held to the surface by
surface tension. The earth is a sphere of rock and itron
surrounded by a thin film of salt water, held in  place by gravity.

The notion that the ocean surface must always be flat is a Failure of Imagination.
 Sometimes the best refutation of a wrong idea is mathematics. Sometimes,
a better refutation is to blow bubbles.

\section{Ocean Surface Topography and Thermocline/Pyncocline Topography}

The ``geoid" is that particular equipotential surface of the
\emph{effective} gravity which conforms most closely to the mean surface of the ocean, also known as  ``mean sea level". The mean  is what remains after surface waves  have been scrubbed
away by a running time average. The ``effective gravity" is the sum of gravity and centrifugal force as defined earlier.

\begin{figure}[h]
\centerline{\includegraphics[scale=0.5]{ExploratoriumFreeVortex.pdf}}
\caption{A steady draining vortex, also called a ``free vortex" or ``line vortex" at the museum known as the San Francisco Exploratorium.}
\label{FigExploratoriumFreeVortex}

\end{figure}

The TOPEX/Poseidon, Jason-1 and Jason-2 satellites, launched in
1992, 2001 and 2008,   have mapped the topography of the ocean
surface through radar altimeters. The deviations
of the mean from the geoid are only a maximum of a meter and a half,
but Jason-2's altimeter error is only about 3 cm.
    This is sufficient to capture an impressive collection of hills, valleys
and slopes as illustrated in Fig.~\ref{FigOceanTopo}.

\begin{figure}[h]
\centerline{\includegraphics[scale=0.5]{Mean_dynamic_topography_of_global_ocean2010.pdf}}
\caption{Ocean surface topography.}
\label{FigOceanTopo}
\end{figure}

The ocean, like the atmosphere, is hydrostatic equilibrium. The pressure
in a water column at a given depth is proportional to the weight of all the fluid above that depth in the column. A hill in mean ocean surface
topography is a center of high pressure while a valley is a pressure low.

What balances these pressure gradients? For small, intense
ocean vortices, the dominant balance is centrifugal force and horizontal pressure gradient force, a ``cyclostrophic" balance.
 For features with a horizontal length scale of a hundred kilometers or
larger, scale analysis shows that Coriolis force balances pressure force.
 Thus, the satellite altimeter maps, Fig.~\ref{FigOceanTopo}, show us that geostrophically-balanced jets and vortices are as ubiquitious in the ocean as
in the atmosphere.

\clearpage

\section{The Rossby Radius of Deformation and the Geostrophic Adjustment Problem}

A favorite game of children of a certain age is to make an expanding
disk of water waves by tossing a rock into a lake. A  few moments
after the impact, radial expansion and viscosity diminish the
waves to nothing, and the water surface is flat. (Bristol Lake in
 New Hampshire  and Clare Lake in Michigan, my favorite childhood targets, have little surface
wave activity on calm days except when a power boat zooms by.)

Carl-Gustav Rossby recognized long ago that meteorological instruments can fail or otherwise give wildly erroneous
measurements. The effect of a bad
observation is  a point perturbation of
the weather forecast or of the ocean surface, mathematically equivalent to throwing a rock
into a lake. Buoyancy waves will radiate away from the point, as on a
lake, but on a rotating planet, the flow does \emph{not} relax
back to a flat surface, but rather to a \emph{geostrophically-balanced}
time-independent flow. This is final state of the ``Rossby geostrophic adjustment
problem". Many of the finest minds in geophysical fluid dynamics have
refined Rossby's idealized, approximate analysis.

Many of the stationary and traveling bumps in the ocean surface topography are the steady currents and traveling waves excited
by adjustment of the ocean to localized wind stresses, such as those
beneath a hurricane.

The characteristic horizontal length scale for  ``geostrophic adjustment" is the ``Rossby radius of deformation" $R_{D}$. This turns out
to be a fundamental length scale for both the ocean and the
atmosphere. For disturbances whose spatial scale is small
 compared to the Rossby radius, Coriolis force is unimportant whereas
for disturbances whose horizontal scale is large  compared to $R_{D}$,
 Coriolis force is dominant. The synoptic-scale vortices that
dominate weather in the middle latitudes as well as ``Rossby adjustment scars" have length scales roughly equal to the Rossby
radius.

Parenthetically, note that the Rossby radius in the ocean is only
about one-tenth its value in the atmosphere. This means that
an ocean forecasting model must have one hundred times the number
of numerical degrees of freedom (``grid points") as an atmospheric
weather prediction model of comparable accuracy.

By coincidence, the Rossby radius for Mars is about the same as
for the terrestial atmosphere, but the Martian planetary circumference is
much smaller than on earth. The dominant baroclinically unstable
waves  on Mars have three crests and troughs around a circle
of latitude versus roughly six maxima and minima around a
latitude circle. Thus, forecasting Martian weather would be easier
than on earth, if sufficient observational data were available.

The greatest difficulty for both ocean prediction and Martian weather
forecasting is obtaining high resolution data to initialize the model.
 In the latter half of the next section, we discuss a new observational tool that has
vastly increased ocean data.

\section{Two-layer Models or Fun with Oil and Water}

Geophysical fluid dynamicists are like artillery officers: the first shots
are hopefully close but usually not quite on target. With successive
corrections (from forward observers or spotting aircraft or in fluids through
higher order in perturbation series or adding additional small terms),
the salvos are walked into the target.

The opening salvo in the approximation of the ocean is to assume a
layer of incompressible fluid of constant temperature and salinity (``homogeneous fluid"). A far
more realistic approximation is to allow the flow to be continuously
stratified and fully three-dimensional. A useful intermediate is the ``two-layer model". This pretends that the ocean is composed of two immiscible
layers of homogeneous fluid of different densities, such as oil and water.

\begin{figure}[h]
\centerline{\includegraphics[scale=0.3]{xxxxx.pdf}}
\caption{Schematic of two-layer model of the equatorial Pacific Ocean. The upper layer (textured) is
light and hot; the water below the pyncocline (dashed blue line) is heavy, cold abyssal water (unshaded).
In the laboratory, the upper layer would be oil and the lower would be water.
 During normal non-El Ni\~{n}o conditions as shown, the mean sea surface is tilted so that it is highest
in the west [left side of diagram]. The tilt is driven by the wind stress of the trade winds, which applies a westward force to the
upper layer. The pyncocline tilts in the opposite direction from the surface but with typically 200 times the slope of the sea surface.
 The thin black vertical lines bound two columns of water. The hydrostatic pressure at a point on the sea bottom
 is the weight
of the column of water above that point. If there are to be no horizontal pressure gradients in the abyss, then the
two  columns must have equal weight. The left column is taller than the right column because the sea surface is tilted.
 The opposite and much greater slope of the thermocline compensates so that the shorter, rightmost column has less water but a much
higher percentage of dense, cold absyssal water. The leftmost column, despite its greater height, weighs no more than the other
column because the leftmost column has a much greater volume of light, hot upper layer water. For graphical clarity, the drawing is
not at all to scale, and the ratio of surface slope to pyncocline slope, typically 1/200, has been greatly exaggerated.}
\label{FigTwo_Layer_Tilted_Thermocline}
\end{figure}

The two-layer model is surprisingly useful  because to a first approximation, especially in the tropics, the ocean \emph{is} a
two-layer system. The upper fifty or a hundred meters of the ocean
is mixed into a uniform density by turbulent convection driven by
solar heating. Under this ``mixed layer", the abyssal flow is cold
and dark; it varies little in density. The boundary between abyssal
flow and the surface mixed layer is a thin layer of  high gradient in
both density and temperature. The terms ``pyncocline" and ``thermocline" are applied as near-synonyms with the word choice
depending on whether one is displaying a graph with density contours or temperature
isolines. The two-layer model sharpens the thermocline to a jump
discontinuity. In the ocean, the density difference $\delta \rho$ between the
two layers is usually half a percent or less, but the dynamical
consequences of the second layer are profound.

I was first exposed to the wondrous fairyland of two-layer models
when I visited Woods Hole Oceanographic Institution in 1973
  and Jack Whitehead gave me a tour of his laboratory where half a dozen experiments were going simultaneously. Waves can propagate
along the oil-water interface. These are ``baroclinic waves" or ``waves in the first baroclinic vertical mode".  If the density difference between layers is small, as typical of the sea,  the speed of these intrafacial waves is very slow compared to waves on the oil-air interface\footnote{Most oils are less
dense than water, so the oil layer floats above the water layer without
mixing or dissolving into it.}. Indeed, as I watched jets plunge from a
shallow shelf into the abyss and waves run around the edges of a flow basin, it seemed that everything was happening in slow motion, but
without the intermediary of a video camera.

  All of the experiments were on turntables, rotated by speed-variable electric motors. All bow before the Lord Coriolis, who rules the
  oceans and the winds! At least phenomena on the Rossby deformation radius or larger scale.

Ron Smith (Yale) once enlivened and amused an American Meteorological
Society conference by demonstrating the generation of mountain waves using materials purchased entirely at a hobby shop except for a gallon jug of yellowish corn oil bought at the local supermarket. The fluid was confined between the glass panels of an ant farm. The recirculating mean flow was generated by an acquarium pump. The ``mountains" were a block of wood, just thick enough to fit in the three centimeter gap between the two large panels of the ant farm. The upper boundary of the block was machined to have curving
hills and depressions
  that forced the flow to oscillate up and down as it flowed past,
generating stationary waves in the process.

The two-layer model is closely connected with the ocean surface
topography because, assuming there are no horizontal
pressure gradients at the bottom of the sea, the shape of the
thermocline must be a \emph{reverse image} of the ocean surface,
 \emph{amplified}  by the huge factor $1/\delta\rho$, the fractional density
difference between the layers. The reason
is that if an isobath (surface of constant pressure) is horizontal at
some depth, then different columns of seawater must have the same
total weight on that isobath. (See Fig.~\ref{FigTwo_Layer_Tilted_Thermocline}.) Assume $\delta\rho = 1/200$, a typical value. If the ocean surface is one meter above the geoid at one
point but even with the geoid at a different location, then the column
under the hill must be lightened by replacing 200 meters of
dense, lower layer water by lighter mixed layer fluid. This implies that, under the hill,
the thermocline must descend  two hundred meters relative to its
depth at the point where the surface and the geoid coincide.

Thus, the accuracy of satellite  altimeters can be validated  by
measuring temperature versus depth at various points in the
global ocean. An illustration well known in oceanographic circles shows a New Jersey high school teacher deploying an Expendable Bathyhermograph (XBT) into a river for the edification of his students;
an amateur could validate a one-dimensional section by dropping XBTs from a cruise ship. A dozen dropsondes cost only about \$400, far less
than the cost of the cruise ticket.

Although oceanography from a cruise ship is a pleasant dream,
 professional oceanographers have a better  way. More than thirty
nations have collaborated to create a network of 4000 drifting profilers, the ``Argo floats".
 All have a  set of required basic capabilities, but a subset of floats
carries additional sensors or descend to greater depths (``deep Argo" floats). Three different entities in the U. S., France and Japan supply
the basic floats.

Each float  is a little robot submarine that spends most of its time
drifting at a depth of a kilometer. Every ten days, a piston squeezes a mass of compressible mineral oil to slightly change the buoyancy. The
float sinks
  to a depth of about 2 kilometers (or as much as 6 kilometers for the
``deep Argo" floats). It then rises  to the surface, measuring measuring electrical conductivity, temperature and pressure; from these, salinity and density are computed. The float downloads its data to a satellite in
about twenty minutes and then descends a  thousand meters to
await ten days for the next profiling cycle to begin.

The ideal  spatial distribution of the Argo floats is non-uniform. A higher
density of floats in polar waters is desirable because the Rossby radius of
deformation (described earlier), a fundamental spatial scale for
oceanic and atmospheric eddies,  \emph{decreases} with increasing \emph{latitude}.

On a flat, non-rotating earth, the optimum distribution of Argo floats
would be spatially uniform. However, the whole population of vortices generated by baroclinic instability would be missing from the
flat-earth ocean.

The Argo floats are but one of a new generation of \emph{in situ} observing systems that check and cross-validate satellite data.

\section{Equatorial and Coastally-Trapped Waves and ENSO}

El   Ni\~{n}o-Southern Oscillation, known by the acronym ENSO, is a global, recurring part of the climate.
  The trade winds exert a wind stress on the oceans that, to balance, requires the sea surface to
  \emph{tilt} from low at the eastern coastline to high at the western coastline. When the trade winds weaken,
the thermocline alters from tilted to horizontal. This adjustment does not happen instantaneously throughout
the whole ocean; rather it propagates eastward as a wave pulse.

This pulse is composed of a mixture of waves, but the mode with the simplest spatial structure,
which is also the one with the largest
energy, is the ``equatorial Kelvin wave" in the ``first baroclinic mode". The Kelvin wave is concentrated within a narrow
waveguide that extends from three degrees south latitude
to three degrees north latitude. This waveguide is generated by refraction, which in turn is created by  the variations of the Coriolis force with latitude. The
Kelvin wave requires about two months to cross the width of the Pacific Ocean.

When the Kelvin wave  collides  with the Americas, ocean temperature off Central America rises sharply.
In normal conditions with large tilts, strong upwelling brings up nutrients from below.
  This upwelling supports a huge population of anchovies. These fish were for centuries
the main export of Peru. When the water turns warm, however, the upwelling stops, the anchovies die
and the Peruvian fishing industry is soon in a very bad way. Peruvian fisherman noticed that these hot water
episodes tend to occur around Christmas time, so the name ``El Ni\~{n}o", ``the
Christ Child" was attached to this anchovy-killing ocean phenomena.

The mischief is not confined to the equatorial waveguide. After the equatorial Kevin wave collides with the coast of Central America,
    most of its  energy
radiates north and south as ``coastal Kelvin waves". These are so-named because Coriolis force
 creates a coastal waveguide similar to the equatorial waveguide. The result of the poleward propagation of coastal Kelvin waves is a strip of hot water extending far south of the equator and as far north as California.

  The evaporation rate is an exponential function of ocean temperature, so huge amounts of water vapor
form above the hot patch created by coastal Kelvin waves. As this moisture-laden air is pushed into
the mountains of California and is driven up the western slopes, the rising air is adiabatically cooled, yielding clouds first and then torrential rains. The mansions of movie stars in Malibu slide into the sea when the supporting soil is converted into mud.

Meanwhile, Australia goes through severe droughts, the ``Great Dry". When the equatorial Kelvin wave
  sneezes, the whole world catches pneumonia.

The ocean-induced changes in the atmosphere initiate the next phase of the atmospheric cycle known
as the Southern Oscillation. The period of ENSO varies irregularly from five years to seven years.

The equatorial Kelvin wave is equatorial only because the Coriolis force varies with latitude. The
coastal Kelvin wave hugs the coast only because Coriolis force balances the pressure force perpendicular
to the coast. El Ni\~{n}o and the Southern Oscillation would not exist without Coriolis force. ENSO
cannot exist on a flat earth.

\section{Tornadoes}

Dust devils are small vortices that are visible as helical trajectories of sand or dust or fallen leaves or snow. Dust devils are cyclostrophically-balanced, that is to say, centrifugal force (always outward) balances a pressure force which is inward towards the minimum of  pressure on the axis of the vortex. There is no obvious reason why dust devils should spin one way in preference to another, and indeed cyclonic and anticyclonic dust devils are found in roughly equal numbers.

Tornadoes are also cyclostrophically-balanced, but are almost always cyclonic. Why?

The answer is that tornadoes derive their vorticity from their parent, which is either a hurricane or a medium scale weather system called a ``mesocyclone". Mesocyclones in turn inherit their vorticity from large scale weather systems, so-called ``synoptic [scale] cyclones".

Vorticity is a measure of the rotation of a flow at a given point. Strictly speaking, the vorticity is a \emph{vector}; however, for many kinds of dynamics only the vertical component of the vorticity is relevant. In an abuse of notation, the same label is applied both to the vorticity vector and to the scalar which is its vertical component. Henceforth vorticity will be used to denote the vertical component of the vorticity only.

The vorticity at the surface of a flowing brook or river can be measured by dropping a twig in the water.
The vorticity is twice the angular frequency of the rotation of the twig, which thus serves as a very low-tech vorticity meter. (The
factor of two is a mathematical convention of no physical significance.)

In the absence of torques, angular momentum is conserved, but not the vorticity itself. A good example of vorticity change is a spin in figure skating. The skater's first task is to apply torque to make   her body   rotate with the arms flung wide and the skater so low as to be almost sitting on the ice. The angular momentum of a small bit of matter with respect to a given axis of rotation is given by the product of the mass of the small bit of matter multiplied by its rotation rate and the distance from the rotation axis. To spin very fast, the skater stands up and pulls her legs in tight, as close to the spin axis as possible. Simultaneously, the arms are pulled in and thrust over the head to move the mass of the arms as close to the rotation axis as possible.  Because the distance of much of her body has been moved closer to the spin axis, conservation of angular momentum requires that the rotation rate increase. Without applying any additional torque, the skater  spins much faster. Her vorticity is greatly increased when she stretched her body vertically along the rotation axis.

A similar process called "vortex stretching" allows flows to tremendously concentrate and intensify vorticity. The most extreme example is a tornado.

In addition, the merger of vortices of the same sign combines two small vortices into a single larger vortex,
 a process that also contributes to tornado ferocity.

This transfer of vorticity from large spatial scales to small
is a general tendency in fluid mechanics. This ``cascade" is stopped
only when the spatial scale is so small that viscosity becomes important, as in boundary layers.

Lewis Richardson, the Father of Numerical Prediction, wittily
summarized the vorticity cascade with a paraphrase of
 Jonathan Swift's two-line poem, ``Big fleas have little fleas that feed and scratch and bite'em, and little fleas have lesser fleas and so \emph{ad infinitum}." Richardson wrote,
\begin{quotation}
``Big whirls have little whirls that feed on  their vorticity, and little whirls have lesser whirls and so on to viscosity."
\end{quotation}

Thus, although Coriolis force is not  significant  in the balance  of a mature tornado, the vortex  ultimately derives its vorticity from the rotation of the earth through
 the vorticity cascade from ``big whirls" to ``little whirls". On a \emph{non-rotating} planet, \emph{tornadoes would not exist}.

\section{Meddies}

As noted earlier,  ``Water Always Seeks Its Own Level" is recited endlessly by flat earth proponents,
never with any proof except personal observation. This false ``law"  is interpreted by Flat Earth proponents to mean that the ocean surface always relaxes to a state with a horizontal, uncurved surface,

This is not an accurate description of  the Straits of Gibraltar. The Mediterranean Sea is very shallow and very sunny. There is so much evaporation that the  mean sea level of the Mediterranean Sea is a few centimeters  \emph{lower} than that of the Atlantic Ocean. Cold water, of relatively low salinity, pours steadily from the Atlantic Ocean into the Mediterranean Sea as a surface current eastward through the Straits of Gibraltar. Underneath this eastward surface current there is an undercurrent in the opposite direction. Although warm, the Mediterranean bottom water is very salty and therefore very dense. As it falls into the Atlantic, the salty Mediterranean water sinks to about a kilometer below the surface of the Atlantic. The flow breaks up into lenticular\footnote{``Lenticular" means `lens-shaped".} vortices which are called ``Meddies" by oceanographers \cite{Cartonetal10}.

Meddies are always anticyclonic, that is, the spin is clockwise as viewed from above. These vortices are only a thousand meters thick in the vertical direction, but have diameters of 20   kilometers to  100 kilometers!
  Lifetimes range from five to twenty-five years.

Meddies are easily detected by sounding the ocean with
an expendable bathythermograph, an inexpensive device [as little as \$35 on eBay!] that simultaneously measures
pressure and temperature as it falls into the sea, tugging a communications wire behind. Meddies are
huge lenticular patches of very salty but also very warm water embedded in the cold and less salty
waters of the Atlantic Ocean, and thus are easily identified with the simplest of equipment.

Why do these vortices form? The technical answer is ``conservation of potential vorticity". Although the details are too complicated to be presented here, the important point is that the source of the spin  is the Earth's rotation. This class of vortices would not exist were  it not for the Earth's rotation.

\section{Hemispheric Differences}

Weather and climate are somewhat different between the two hemispheres, but the usual explanation is that these differences are due to the asymmetric distribution of the continents:
there is much more land and much less water in the northern hemisphere than the southern hemisphere.
In addition, the Arctic is a thin sheet of constantly moving and deforming ice above ocean whereas the Antarctic surface is permanent, long-lived ice heaped kilometers thick above solid ground.
When general circulation models are run on supercomputers with land omitted, so-called ``acquaplanet" simulations, the hemispheric differences \emph{vanish} on the \emph{globe}.

On the aquaplanet flat earth \emph{disk}, there are considerable hemispheric differences. It will seem odd and inaccurate to apply ``hemisphere", which literally means ``half-spherical", to a non-spherical geometry like a disk, but no better terminology is available. The ``disk northern hemisphere" is a disk of radius $R$ (shaded in Fig.~\ref{FigFlatEarth_Sch_Map}); by ``disk southern hemisphere" we mean the  annulus of outer radius $2 R$ and inner radius $R$ (unshaded in Fig.~\ref{FigFlatEarth_Sch_Map}). The area of the northern hemisphere disk is $\pi R^{2}$; the area of the entire flat earth, defined by $r \in [0, 2R]$, is $4 \pi R^{2}$. Subtracting the area of the northern hemisphere from the area of the whole gives
\begin{eqnarray}
\text{Area(disk southern hemisphere)} = 3 \, \pi R^{2}
\end{eqnarray}
Thus, the area of the disk southern hemisphere   is   \emph{three times} the area
of the disk northern hemisphere.

This will create huge differences in weather and climate between the hemispheres due solely to geometry. In reality, the observed hemispheric differences
are accurately explained by the  difference    in land/sea fraction and the ice-over-water, ice-over-land difference   between the Arctic and Antarctic.

\section{Summary}

The weather on a non-rotating flat earth would be profoundly different
from that actually observed. The following elements of our atmosphere  and ocean would
be missing on a flat earth:
\begin{enumerate}
\item hurricanes and typhoons
\item synoptic cyclones and anticyclones; baroclinic instability
\item Hadley cell descending in the subtropics;
\item trade winds
\item prevailing westerlies
\item equatorial Kelvin waves
\item coastal Kelvin waves
\item El Ni\~{n}o/Southern Oscillation (ENSO) coupled air-ocean cycle
\item tornadoes
\item Rossby waves
\item The Gulf Stream
\item Gulf Stream rings
\item Meddies
\item Sverdrup flow
\item Geostrophically-balanced jets
\item Ekman  layers.
\item Large temperature differences between Costa Rica and Calgary.
\end{enumerate}
 This list is not exhaustive; a good exercise for students is to extend this catalog.

It must be emphasized that the qualitative arguments given here can only be turned
into a comprehensive, quantitative understanding by applying numerical models and sometimes
 singular perturbation theory. Both the qualitative thinking (here) and supercomputer number-crunching (a topic for future work) are
essential parts of earthly climate forecasting and exoplanet understanding.

In the appendices, we provide further development of key ideas used above, and also discuss the misconceptions
of flat earth videos. Because the video creators have [if flat earth evangelists!] no background or training, the mistakes
  are those of introductory physics. The difference is that freshman physics errors are unlearned, corrected and
forgotten whereas video mistakes are illustrated with gaudy cartoons and quotes that make it easier to understand
how elementary ideas are misconceived and misinterpreted.


\section{Analysis of the Snake Inductance}\label{apx:snake}

Here, we outline the derivation of the snake SQUID array potential, current-phase relation, and inductance. A schematic of the snake array is shown in Fig.~\ref{fig:snake_schem}. We will assume that the array is sufficiently long, so that we can neglect edge effects, and treat the array as translationally invariant\cite{bell2012quantum}.

We will refer to the inductive loops containing a junction and inductors $L_1$ and $L_2$, essentially an rf-SQUID, as the ``small loop" (see Fig.~\ref{fig:snake_schem}). In the small loop, the phase drop across $L_1$ is $\alpha$ and the phase drop across $L_2$ is $\beta$. The junction phase is $\delta$ as usual. The sign convention for the phases is such that $\beta$ and $\delta$ grow from left to right, whereas the direction of $\alpha$ alternates downwards and upwards as the inductive spine of the snake meanders. We will refer to the loop that encloses the full snake array and any inductance $L_b$ external to it (for example, the self-inductance of a bias transformer) as the ``big loop", as shown in Fig.~\ref{fig:snake_squid}. The snake we will consider has $2N$ unit cells, so when implementing a ``dc-SQUID-like" structure (Fig.~\ref{fig:snake_squid}) we will have two $N$-unit-cell arrays in parallel.

From flux quantization of any of the small loops of the snake we have the relation between the phases:
\begin{equation}\label{small_loop}
2\alpha+\beta=\delta,
\end{equation}
and with a total phase drop of $\phi$ across the full array with 2$N$ sections array we have:
\begin{equation}\label{total_phase}
2N\left(\delta-\alpha\right)=\phi.
\end{equation}

We can write the total potential energy of the array and use Eqs.~(\ref{small_loop}) and (\ref{total_phase}) to eliminate $\alpha$ and $\beta$:
\begin{align}\label{potential}
U_s &= 2N\left(\frac{\Phi_0}{2\pi}\right)^2\times\left(\frac{\alpha^2}{2L_1}+\frac{\beta^2}{2L_2}-\frac{1}{L_J}\cos\delta\right)\\
&=2N\left(\frac{\Phi_0}{2\pi}\right)^2\times\left[\frac{\left(\delta-\frac{\phi}{2N}\right)^2}{2L_1}+\frac{\left(\frac{\phi}{N}-\delta\right)^2}{2L_2}-\frac{1}{L_J}\cos\delta\right]
\end{align}

\begin{figure}
\includegraphics[width=\columnwidth]{snake_schem.pdf}
\caption{\label{fig:snake_schem}Schematic of the snake array, with the phases $\alpha$, $\beta$, and $\delta$ indicated. The unit cell is highlighted with a dashed box.}
\end{figure}

So far this is what has been done in Ref.~\onlinecite{naaman2017josephson}, and follows the reasoning in Ref.~\onlinecite{bell2012quantum}. Here we want to also include the effects of an inductor $L_b$ closing the loop, through which we flux-bias the big loop with flux $\left(\frac{\Phi_0}{2\pi}\right)\phi_e$ as in Fig.~\ref{fig:snake_squid}. When doing so we need to account also for the potential energy stored in the inductor $L_b$, and we use flux quantization around the big loop to get:
\begin{equation}
U_b = \left(\frac{\Phi_0}{2\pi}\right)^2\frac{\left(\phi_e-\phi\right)^2}{2L_b},
\end{equation}
and $U_{tot}=U_s+U_b$.

\begin{figure}
\includegraphics[width=1.0in]{squid.pdf}
\caption{\label{fig:snake_squid}Schematic of the snake SQUID, flux-biased through a transformer with self-inductance of $L_b$ and mutual $M$ such that $\phi_e=(2\pi/\Phi_0)\times MI$}
\end{figure}

From here we want to find the equilibrium phase $\delta_0$ that minimizes the potential. We set $\frac{dU}{d\delta}=0$ and $\frac{dU}{d\phi}=0$, and eliminate $\phi$ from the two resulting equations to find $\delta_0$ in terms of the external flux $\phi_e$.
The derivative with respect to $\delta_0$ gives the expression:
\begin{equation}\label{deriv_delta}
\delta_0\left(\frac{1}{L_1}+\frac{1}{L_2}\right)+\frac{1}{L_J}\sin\delta_0=\frac{\phi}{2N}\left(\frac{1}{L_1}+\frac{2}{L_2}\right),
\end{equation}

and the derivative with respect to $\phi$ gives:
\begin{equation}\label{deriv_phi}
\frac{\phi}{2N}=\frac{\frac{\phi_e}{L_b}+\delta\left(\frac{1}{L1}+\frac{2}{L_2}\right)}{\frac{1}{L_1}+\frac{4}{L_2}+\frac{2N}{L_b}}
\end{equation}

Using Eq.~(\ref{deriv_delta}) and Eq.~(\ref{deriv_phi}), and some algebra we finally get an implicit equation for $\delta_0(\phi_e)$:
\begin{widetext}
\begin{equation}\label{equilib}
\delta_0\left(L_1+L_2-L_b\frac{\left(2L_1+L_2\right)^2}{L_b\left(4L_1+L_2\right)+2NL_1L_2}\right)+\frac{L_1L_2}{L_J}\sin\delta_0=\phi_e\frac{L_1L_2\left(2L_1+L_2\right)}{L_b\left(4L_1+L_2\right)+2NL_1L_2}
\end{equation}
\end{widetext}
Note that when $L_b=0$, this equation reduces to Eq.~(\ref{deriv_delta}) as it should, where $\phi_e=\phi$. Eq.~(\ref{equilib}) can be solved numerically to find the equilibrium phase $\delta_0$ as a function of applied flux $\phi_e$.

Another way to get the same result is to consider the current-phase relation $I(\phi)$ of the snake itself, which can be found from the derivative of the potential with respect to the phase drop $\phi$:

\begin{align}\label{current-phase}
I&=\left. \left(\frac{2\pi}{\Phi_0}\right)\frac{dU}{d\phi}\right\rvert_{\delta_0}\\
&=\left(\frac{\Phi_0}{2\pi}\right)\times\left[-\delta_0\left(\frac{1}{L_1}+\frac{2}{L_2}\right)+\frac{\phi}{2N}\left(\frac{1}{L_1}+\frac{4}{L_2}\right)\right]
\end{align}

The external flux that needs to be applied to the ``big loop" to affect a phase drop $\phi$ across the array is $\frac{\Phi_0}{2\pi}\phi_e$, where
\begin{equation}
\phi_e=\phi+\frac{2\pi}{\Phi_0}L_bI(\phi),
\end{equation}
and using Eq.~(\ref{current-phase}):
\begin{equation}
\phi_e=\phi-L_b\left[\delta_0\left(\frac{1}{L_1}+\frac{2}{L_2}\right)-\frac{\phi}{2N}\left(\frac{1}{L_1}+\frac{4}{L_2}\right)\right],
\end{equation}
Which is the same as Eq.~(\ref{deriv_phi}).

The inductance of the snake SQUID, with the array tapped in the middle to form two $N$-section segments in parallel, and closed in the ``big loop" through $L_b$ can be found by taking the second derivative of the potential of Eq.~(\ref{potential}) with respect to the phase $\phi$, and since $\delta_0$ depends on $\phi$ as well, we have to be careful to capture the full derivative:

\begin{equation}
L_s=\frac{N}{2}\left(\frac{\Phi_0}{2\pi}\right)^2\times\left(\frac{\partial^2U}{\partial\phi^2}+\frac{\partial^2U}{\partial\phi\partial\delta_0}\frac{\partial\delta_0}{\partial\phi}\right)^{-1}.
\end{equation}
Carrying out the derivatives we get:
\begin{equation}
L_s=\frac{N}{2}\frac{\left(L_1+L_2\right)L_J+L_1L_2\cos\delta_0}{L_J+\left(4L_1+L_2\right)\cos\delta_0}.
\end{equation}

This is the contribution of the snake itself, with $\delta_0$ given by Eq.~(\ref{equilib}), and we can add $L_b/2$ to each of the SQUID parallel branches.

\begin{figure}[h]
\includegraphics[width=\columnwidth]{snake_ind.pdf}
\caption{\label{inductance}Inductance of the Snake SQUID, on a logarithmic scale, as a function of applied flux (per junction) for several junction $I_c$}
\end{figure}

Figure \ref{inductance} shows the calculated inductance vs flux curves for a device with $N=20$ junctions per arm, small-loop inductances $L_1=2.6$ pH and $L_2=8.0$ pH, $L_b=124$ pH, and for several values of $I_c$. We see that the inductance here is always positive (unlike that of the simple rf-SQUID), and that the depth of inductance modulation depends on the junction critical current, with lower $I_c$ resulting in shallower modulation.

Note that the applied flux in Fig.~\ref{inductance} is per junction, or in other words, a snake with $N$ junctions will require $N$ times more flux bias than a single rf-SQUID to reach the operating point. This is why our amplifier requires a large bias transformer (Fig.~\ref{fig:circuit}), and in general higher pump currents than the traditional dc-SQUID based JPA.

\section{Readout efficiency measurement}\label{apx:efficiency}
Readout efficiency quantifies the amount of information about the qubit state that can be learned per readout photon\cite{QuantEff, vijay2012stabilizing}. In practice, this translates to the rate at which the readout signal-to-noise ratio (SNR) improves as a function of measurement time, $t$. Ideally, this quantity, which we call the SNR flux, is given by:
\begin{equation}\label{eq:ideal_snr}
    \Phi_\mathrm{SNR}^\mathrm{ideal}\equiv\left.\frac{d\text{SNR}}{dt}\right|_\mathrm{ideal}=8\bar{n}\kappa \frac{(2\chi/\kappa)^2}{1+\left(2\chi/\kappa\right)^2},
\end{equation}
where we have included half a photon of noise associated with the input measurement tone, and $\bar{n}$ is evaluated at the readout frequency, $\omega_\text{ro} = (\omega_{r,|0\rangle}+\omega_{r,|1\rangle})/2$, so that both dressed resonator states have the same value of $\bar{n}$.

In reality, not all photons are detected (due to microwave losses) and our phase-preserving amplification chain necessarily adds at least half a photon of noise, see Eq.~(\ref{eq:efficiency}), so in an experiment the SNR flux is smaller than the ideal value by at least a factor of 2. We determine the readout efficiency from the ratio between the experimental and ideal SNR flux Eq.~(\ref{eq:ideal_snr}), $\eta=\Phi_\mathrm{SNR}/\Phi_\mathrm{SNR}^\mathrm{ideal}\le0.5$.

In the experiment, we record the readout SNR vs measurement time, which is a linear function for sufficiently low readout power, with a slope of $\Phi_\mathrm{SNR}$. Finally, using the measured values of $\chi$, $\kappa$, and $\bar{n}$, we calculate the experimental readout efficiency from the slope of a line fit to the data:
\begin{equation}
    \eta=\frac{\Phi_\mathrm{SNR}}{\Phi_\mathrm{SNR}^\mathrm{ideal}}= \frac{d\text{SNR}}{dt}\times\frac{1}{8\bar{n}\kappa}\left[\frac{1+\left(2\chi/\kappa\right)^2}{\left(2\chi/\kappa\right)^2}\right].
\end{equation}

\section{Loss measurement}\label{aps:loss}

We have used a cryogenic microwave calibration platform to characterize losses in our readout assembly. The measurement system is configured according to Ref.~\onlinecite{wang2021cryogenic}, and expanded to allow 2-port SOLT calibration\cite{ranzani2013two}.

We first characterize all components in our readout assembly separately, including cables, circulators, and circuit board traces. We can add up the insertion loss measured for these components to estimate a lower limit on the total loss in our system between the processor and the SNIMPA, $-2.5$~dB for the experiments described here.

When components are cascaded and built into a complete readout assembly, reflections from circulators and connectors can contribute to the overall insertion loss. In addition, electrical delays associated with the electrical lengths of connectors and circuit board traces can add significant frequency dependence to the overall loss. Figure~\ref{fig:loss} shows an $S_{21}$ measurement of a representative integrated readout assembly, similar to that used in the experiment, deembedded from the processor chip reference plane to the signal port of the SNIMPA. The circulators' 4-8 GHz band-pass response can be seen, with an average insertion loss around $-3$~dB. The inset shows a zoom-in on the frequency range corresponding to our readout band (highlighted with dashed lines in the inset), and we can see that indeed the insertion loss varies with frequency.

The details of the insertion loss will vary from one readout line to the next, therefore in the main text we only use the measured minimum loss number, and obtain an upper bound on the SNIMPA added noise. However, as Fig.~\ref{fig:loss} suggests, the total insertion loss is likely higher, and the SNIMPA noise performance is likely consistent with near quantum limited operation.

\begin{figure}[h]
\includegraphics[width=\columnwidth]{loss_figure.pdf}
\caption{\label{fig:loss} Cryogenically calibrated $S_{21}$ measurement of a representative integrated readout assembly, including all components from the processor chip to the amplifier chip. Inset: zoomed in view of the insertion loss in the readout band, indicated by the dashed vertical lines.}
\end{figure}

\clearpage
\end{document}