Abstract
A rotating ocean has several characteristic lengths. Each is called “the length set by Coriolis.”This paper asks whether they are the same thing: the answer is that they are different things. No new mathematical theorem and no new law is claimed.
First, we line up three lengths. At latitude $45^\circ$: Ekman layer $\mathbf{43.75}$ m, baroclinic deformation radius $\mathbf{48.48}$ km, barotropic deformation radius $\mathbf{1920.86}$ km (Section 2).
Second, this is the core of the paper. Smallest to largest spans $\mathbf{4.3905\times10^{4}}$, and though all contain the Coriolis parameter $f$, nothing else in them overlaps (Section 2).
Third, what separates baroclinic from barotropic is not depth. The ratio is $\mathbf{39.6182}$, and what sets it is the stratification $N$ (Section 3).
Fourth, whether rotation acts is settled by one number. The Rossby number $Ro=U/(fL)$: Gulf Stream $0.0970$, typhoon $0.5818$ (Section 4).
Fifth, the bathtub vortex has $Ro=3232$. A force $3232$ times stronger than Coriolis is turning it (Section 5).
Sixth, the separator is $Ro$. To see Coriolis in a bathtub the flow must be below $\mathbf{0.031}$ mm/s (Section 5).
Scope (what is not claimed)
No new mathematical theorem and no new law is claimed: the Ekman layer, the Rossby deformation radius, the Rossby number, and geostrophic balance are all standard. We do not build ocean dynamics: all we use is three length formulas and one dimensionless number. We do not derive the Ekman spiral: we do not enter the boundary-layer equations. We do not take a roll call of forces: the classification of centrifugal, Coriolis, and pressure-gradient forces is not treated. This is a roll call of lengths. We claim no accuracy for the representative values: eddy viscosity $A_z=0.01$ $\mathrm{m^2/s}$, stratification $N=0.005$ /s, and depths $H=1000$ / $4000$ m are values for showing orders and move by an order with place and season. $A_z$ in particular ranges from $10^{-4}$ to $10^{-1}$. We assert no values for the bathtub vortex: $U=0.1$ m/s and $L=0.3$ m are representative of a typical drain vortex. What this paper asserts is $Ro\gg1$, not the value $3232$. We do not explain the drain vortex’s sense of rotation. What actually decides it (residual initial circulation, vessel shape, how the plug is pulled) is not treated. We show only that it is not Coriolis. Relation to earlier papers: Paper 256 showed that the only scale at which dynamical similarity holds exactly is $1$: the $Ro$ here is likewise a dimensionless number that cannot be held when the scale changes, so a bathtub cannot be a model of an ocean. Paper 117 treated scale invariance fixing an exponent in four ways and wrote the condition that there be exactly one $\Pi$ group. This paper is a case failing that condition, with three lengths standing independently. Paper 140 separated symmetry fixing ratios from dynamics fixing the scale: all three lengths here are on the dynamics side. Paper 112 counted “distinct roots sharing one rhyme”: the three lengths share the rhyme $f$ and differ in root. What is added is lining up three lengths in one table and measuring the span $4.3905\times10^{4}$, confirming that the ratio $39.6182$ is set by stratification, computing the Rossby numbers of five phenomena, and deriving the flow speed $0.031$ mm/s needed to see rotation in a bathtub.
1. Notation, and One Parameter
Setting: On the rotating Earth the Coriolis parameter
$$f=2\Omega\sin\varphi$$appears in every expression ($\Omega$ the rotation rate, $\varphi$ the latitude).
Check 1.1: At latitude $45^\circ$.
| Quantity | Value |
|---|---|
| $\Omega$ | $7.2921159\times10^{-5}$ rad/s |
| $f=2\Omega\sin45^\circ$ | $\mathbf{1.0313\times10^{-4}}$ /s |
| Inertial period $2\pi/f$ | $\mathbf{16.9242}$ hours |
Question: Are the lengths containing $f$ all the same thing?
2. The Core: Three Lengths Spanning $4.39\times10^{4}$
Check 2.1: We take three lengths for the same sea at the same latitude.
| Length | Formula | Value |
|---|---|---|
| Ekman layer depth | $\pi\sqrt{2A_z/f}$, $A_z=0.01$ | $\mathbf{43.75}$ m |
| Baroclinic deformation radius | $NH/f$, $N=0.005$, $H=1000$ | $\mathbf{48.48}$ km |
| Barotropic deformation radius | $\sqrt{gH}/f$, $H=4000$ | $\mathbf{1920.86}$ km |
Observation 2.2 This is the core of the paper. Smallest to largest spans $\mathbf{4.3905\times10^{4}}$: from $44$ metres to $1900$ kilometres, all “lengths set by Coriolis” in one and the same sea.
Check 2.3: We write out what enters besides $f$.
| Length | Ingredients besides $f$ | What layer |
|---|---|---|
| Ekman depth | eddy viscosity $A_z$ | the friction layer |
| Baroclinic radius | stratification $N$, depth $H$ | the density layer |
| Barotropic radius | gravity $g$, depth $H$ | the free surface |
Observation 2.4 The right two columns overlap in nothing. Viscosity, stratification, and gravity are mutually independent: all they share is $f$.
Check 2.5: Even the manner of dependence on $f$ differs.
| Length | Dependence on $f$ | Doubling $f$ gives |
|---|---|---|
| Ekman depth | $\propto f^{-1/2}$ | $\times0.7071$ |
| Baroclinic radius | $\propto f^{-1}$ | $\times0.5000$ |
| Barotropic radius | $\propto f^{-1}$ | $\times0.5000$ |
Observation 2.6 Only the Ekman depth has a different exponent. Change the latitude and the very ratios among the three change: measuring one tells you nothing about the other two.
Observation 2.7 Paper 112 counted “six distinct roots sharing one rhyme.”The three lengths share the rhyme $f$ and have entirely distinct roots: that the same parameter appears does not mean the same phenomenon.
3. What Separates Baroclinic from Barotropic Is Not Depth
Check 3.1: We take the ratio of the two deformation radii.
| Quantity | Value |
|---|---|
| Barotropic $/$ baroclinic | $\mathbf{39.6182}$ |
| As a formula, $\sqrt{gH_2} / (NH_1)$ | $f$ cancels |
Observation 3.2 Inside the ratio, $f$ cancels out. So changing the latitude does not change the ratio. It is not rotation that separates them.
Check 3.3: We move the stratification $N$.
| $N$ [/s] | Baroclinic radius | Barotropic $/$ baroclinic |
|---|---|---|
| $0.005$ | $48.48$ km | $39.6182$ |
| $0.010$ | $96.97$ km | $19.8091$ |
Observation 3.4 Doubling $N$ halves the ratio. What sets it is not depth but stratification: the degree to which the sea is layered makes this factor of $39.6$.
Observation 3.5 Were the sea perfectly mixed ($N=0$), the baroclinic radius would be $0$. Only the barotropic one would remain: one of the three lengths exists only because there is stratification.
4. Whether Rotation Acts Is Settled by One Number
Setting: The Rossby number $Ro=U/(fL)$ is the ratio of inertial to Coriolis force. Rotation governs when $Ro\ll1$ and is irrelevant when $Ro\gg1$.
Check 4.1:
| Phenomenon | $U$ [m/s] | $L$ [m] | $Ro$ | Verdict |
|---|---|---|---|---|
| Gulf Stream | $1.0$ | $1.0\times10^{5}$ | $\mathbf{0.0970}$ | rotation acts |
| Atmospheric anticyclone | $10.0$ | $1.0\times10^{6}$ | $0.0970$ | rotation acts |
| Typhoon | $30.0$ | $5.0\times10^{5}$ | $\mathbf{0.5818}$ | rotation acts |
| Laboratory rotating tank | $0.01$ | $0.5$ | $193.94$ | rotation irrelevant |
| Bathtub vortex | $0.1$ | $0.3$ | $\mathbf{3232.29}$ | rotation irrelevant |
Observation 4.2 The Gulf Stream and the atmospheric anticyclone happen to share $0.0970$. The speed is $10$ times larger and the size is too, so the ratio is unchanged: $Ro$ sees only the ratio of $U$ to $L$.
Observation 4.3 The typhoon at $0.5818$ is close to $1$. Rotation acts but does not govern, so near a typhoon’s centre geostrophic balance fails and gives way to gradient-wind balance, a different balance.
5. The Core Continued: The Bathtub Vortex Has $Ro=3232$
Observation 5.1 A bathtub drain vortex has $Ro=\mathbf{3232.29}$. A force $3232$ times stronger than Coriolis is turning it: residual initial circulation, asymmetry of the vessel, and the act of pulling the plug.
Check 5.2: So what would make it visible? To get $Ro<1$ one needs $U<fL$.
| Vessel | Permitted flow speed $U<fL$ |
|---|---|
| Bathtub (radius $0.3$ m) | $\mathbf{0.0309}$ mm/s |
| Tank of diameter $2$ m | $0.1031$ mm/s |
| Pond of radius $100$ m | $10.3126$ mm/s |
Observation 5.3 To see Coriolis in a bathtub the flow must be below $\mathbf{0.031}$ mm/s. A real drain vortex runs at about $100$ mm/s: $3232$ times too fast.
Observation 5.4 That the same $3232$ appears twice is no coincidence. It is the definition of $Ro$ itself, but rewriting it as “$3232$ times too fast” shows what would be required.
Observation 5.5 In fact the experiment has succeeded. In a tank $1.8$ m across, left still for over $18$ hours and drained very slowly, the Coriolis sense appears: note that those $18$ hours are close to the inertial period of $16.92$ hours.
Observation 5.6 Separator. What must be moved for rotation to act? Making $U/L$ smaller than $f$. Enlarge the vessel or slow the flow: changing the latitude moves $f$ by at most $\sqrt2$, so that route cannot close the gap.
Observation 5.7 Paper 256 showed that the only scale at which dynamical similarity holds exactly is $1$. Bathtub and ocean differ in $Ro$ by $3.3\times10^{4}$: a bathtub cannot be a model of an ocean.
6. Roll Call and Closing
- $f=1.0313\times10^{-4}$ /s (Check 1.1): Inertial period $16.9242$ hours.
- Ekman layer $43.75$ m (Check 2.1): The friction layer.
- Baroclinic radius $48.48$ km (Check 2.1): The density layer.
- Barotropic radius $1920.86$ km (Check 2.1): The free surface.
- Span $4.3905\times10^{4}$ (Observation 2.2): Same sea, same latitude.
- Different exponents in $f$ (Check 2.5): Only Ekman is $f^{-1/2}$.
- Barotropic$/$baroclinic $=39.6182$ (Check 3.1): $f$ cancels.
- Doubling $N$ halves the ratio (Check 3.3): Stratification sets it.
- Bathtub vortex $Ro=3232.29$ (Check 4.1): Rotation irrelevant.
- Flow needed: $0.031$ mm/s (Check 5.2): One $3232$nd of the real value.
Closing: One sea at one latitude holds three “lengths set by Coriolis.”Ekman layer $\mathbf{43.75}$ m, baroclinic deformation radius $\mathbf{48.48}$ km, barotropic deformation radius $\mathbf{1920.86}$ km: spanning $\mathbf{4.3905\times10^{4}}$. All contain the Coriolis parameter $f$, yet not one of their other ingredients overlaps: viscosity, stratification, gravity. Even the exponent of the dependence on $f$ differs: only the Ekman depth goes as $f^{-1/2}$, so changing the latitude changes the ratios among the three. What separates baroclinic from barotropic is stratification, not depth: $f$ cancels in the ratio, and doubling $N$ turns $39.6$ into $19.8$. And whether rotation acts is settled by one number, the Rossby number $Ro=U/(fL)$: Gulf Stream $0.0970$, typhoon $0.5818$, and the bathtub vortex $\mathbf{3232.29}$. One thing separates them: whether $U/L$ is smaller than $f$. Seeing Coriolis in a bathtub requires flow below $\mathbf{0.031}$ mm/s, and a real drain vortex is $\mathbf{3232}$ times faster. Changing the latitude moves $f$ by at most $\sqrt2$, so that route cannot close the gap.
Revision Record
Second edition (October 2026): the dashes used as separators in the text were replaced by punctuation; section headings containing mathematics, which had been cut off in display, were repaired; no claim, number or reference was changed.
References
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- Yamagishi, Y. (2026). The Allotted Region and the Freedom Within [Paper 112]. Zenodo. 10.5281/zenodo.21962834
- Yamagishi, Y. (2026). Scale Invariance Fixes an Exponent — in Four Different Ways [Paper 117]. Zenodo. 10.5281/zenodo.22005515
- Yamagishi, Y. (2026). Symmetry Fixes the Ratios, Dynamics Fixes the Scale [Paper 140]. Zenodo. 10.5281/zenodo.22081997
- Yamagishi, Y. (2026). The Only Scale at Which Dynamic Similarity Is Exact Is One [Paper 256]. Zenodo. 10.5281/zenodo.22168919
Reference
Yamagishi, Y. (2026). One Sea Holds Three Lengths, Spanning a Factor of $4.39\times10^{4}$ [Paper 291]. YOI: https://www.yamagishiyuuki.com/yoi/291
DOI: 10.5281/zenodo.22171479
ORCID: 0009-0006-8587-3450
PDF (English) · PDF (Japanese) · Date proof of the English PDF (OpenTimestamps .ots) · SHA256SUMS
The record page for this paper (in Japanese, with the PDFs in each language and the date proof): YOI 291. This paper is a preprint and has not been peer reviewed.