The latitude past which a tide cannot split

The reflection that changes the wavelength followed internal tides onto sloping seabeds and into basins that focus them onto single lines, and it ended on a sum that does not yet close. About a terawatt of the Moon’s and the Sun’s tidal energy is converted into internal waves where the barotropic tide flows over ridges and continental slopes. Almost none of it is lost where it is generated. It propagates, sometimes across entire ocean basins, and eventually it has to reach scales of centimetres, where viscosity can turn it into heat and the stratification can be mixed.

A wave a hundred kilometres long does not become a centimetre-scale eddy by itself. Something has to move its energy to shorter scales, and the something is a family of nonlinear interactions between the tide and the smaller internal waves already present. One of them has a clean mechanism, a clean frequency and — unexpectedly — a clean geography, and it is the subject of this essay.

A wave that pumps the waves riding on it #

A large internal wave passing through the ocean strains the water it passes through, rhythmically, at its own frequency. A much smaller internal wave riding on that strain feels a restoring force that is being modulated: the effective stratification it sees, and the shear it lives in, rise and fall with the large wave’s phase.

An oscillator whose restoring force is modulated rather than pushed is the swing that is pumped, not pushed. A child on a swing stands at the bottom and crouches at the ends, twice per swing, and the swing grows. What matters is that the modulation happens at twice the swing’s frequency. Turned around, the same arithmetic says that a modulation at the tide’s frequency pumps oscillators at half the tide’s frequency, and that is what the internal tide does to the waves around it.

The regions of growth are Mathieu’s tongues, the same ones that decide whether a force that averages to nothing can hold something up. The widest of them sits exactly at half the pumping frequency, and its width at small modulation is a quarter of the modulation strength. Inside it the small wave’s amplitude grows exponentially, at a rate proportional to the strength; outside it the wave only beats. There is a second tongue at the tide’s own frequency, but it opens only as the square of the strength and is far narrower. A pumped oscillator grows fastest at half the frequency of whatever pumps it, and the internal tide is a pump of that kind laid across an entire ocean.

This is the parametric subharmonic instability. In the language of waves rather than oscillators it is a resonant triad: the tide at frequency $\omega_0$ and wavevector $\mathbf k_0$ transfers energy to two smaller waves whose frequencies add to $\omega_0$ and whose wavevectors add to $\mathbf k_0$. Among all such triads, the fastest-growing has the two daughter waves at nearly $\omega_0/2$ each, with wavevectors much larger than the tide’s and nearly opposite to one another. So the instability does two things at once. It halves the frequency, and it throws the energy to vertical wavelengths far shorter than the tide’s — which is to say, into shear, which is where mixing starts.

The frequency an internal wave cannot go below #

The instability needs its daughter waves to exist, and on a rotating planet an internal wave cannot have any frequency it likes.

The wave that picks an angle found the upper limit: no internal wave oscillates faster than the buoyancy frequency, because a fluid parcel displaced vertically cannot return faster than its stratification pulls it. The lower limit comes from rotation. A parcel set moving horizontally on a rotating planet, with nothing else acting, does not travel in a straight line but turns in a circle at the inertial frequency, $f = 2\Omega\sin(\text{latitude})$, as the deflection that closes on itself shows. An internal wave slower than that would require the fluid to oscillate more slowly than rotation alone makes it turn, and there is no such wave. Internal waves exist only between the inertial frequency and the buoyancy frequency, and the inertial frequency depends on where on the planet they are.

The semidiurnal lunar tide, $M_2$, has a period of 12.42 hours. Half its frequency is one cycle every 24.84 hours, and the inertial frequency equals that at a latitude where $\sin(\text{latitude}) = \omega_{M_2}/4\Omega$: 28.8°. Equatorward of 28.8° a wave at half the semidiurnal frequency is allowed, and the tide can pump it. Poleward of 28.8° it is not, and that route out of the tide is closed.

The diurnal tide makes the geography sharper still. The luni-solar diurnal tide $K_1$ has a period of exactly one sidereal day — it is the Earth’s rotation itself — so half its frequency is a quarter of $2\Omega$, and its critical latitude is where the sine of the latitude is one quarter: 14.5°. The diurnal tide’s subharmonic is confined to a band within fourteen and a half degrees of the equator, and the diurnal internal tide itself cannot propagate as a free wave poleward of 30°, where its own frequency falls below the inertial one.

Thirty degrees, pulled towards the equator by the Moon #

The number 28.8 is less arbitrary than it looks, and working out where it comes from shows what it would be on another planet or in another epoch.

The lunar semidiurnal tide has two cycles per lunar day, and a lunar day is longer than a sidereal one because the Moon moves along its orbit in the same direction the Earth spins. So the tide’s frequency is twice the difference between the Earth’s rotation rate and the Moon’s orbital rate, $\omega_{M_2} = 2(\Omega - n)$. The critical latitude of its subharmonic is where $2\Omega\sin(\text{latitude}) = \omega_{M_2}/2 = \Omega - n$, which gives

$$\sin(\text{latitude}) = \tfrac12\left(1 - \frac{T_\text{day}}{T_\text{month}}\right),$$

with both periods sidereal. If the Moon did not move, the critical latitude would be exactly 30°. It moves once every 27.32 days against a sidereal day of 0.997 days, and that ratio of 3.65 per cent pulls the latitude equatorward by 1.2 degrees. The line on the ocean is thirty degrees, corrected by the length of the month.

The same formula places the other tides without further work. The solar semidiurnal tide, $S_2$, is set by the year rather than the month, a correction ten times smaller, and its subharmonic’s critical latitude is 29.9°. So the two largest semidiurnal tides draw two lines a degree apart, and between 28.8° and 29.9° the solar tide’s subharmonic is allowed while the lunar one’s is not. The diurnal $K_1$ tide has a period of exactly one sidereal day — it is the Earth’s rotation itself — so its subharmonic’s latitude is where the sine is exactly one quarter, 14.48°, with no correction at all.

In the deep past the numbers were different, and in a readable way. When the Earth spun faster, around six hundred million years ago, with a day of about twenty-two hours and a correspondingly shorter month, the ratio of day to month was slightly different and the $M_2$ subharmonic’s line sat a fraction of a degree elsewhere; on a planet with no moon, only the solar tide’s line near thirty degrees would exist. The latitude is a small piece of orbital mechanics written onto the ocean.

Where the daughters end up #

The waves the tide hands its energy to are not a random choice of waves. At half the semidiurnal frequency and just equatorward of 28.8°, they sit a little above the local inertial frequency — in the near-inertial band, the part of the internal wave spectrum that already carries most of the ocean’s vertical shear, fed by storms that set the surface mixed layer turning in inertial circles.

That is why the instability matters for mixing more than its energy transfer alone would suggest. Energy placed in near-inertial waves with short vertical wavelengths is energy placed where the shear is already concentrated and where the next step — shear instability, overturning, turbulence — is shortest. The subharmonic route delivers tidal energy directly into the band the wind already uses to mix the upper ocean, and it does so only on one side of a line of latitude.

The cut-off is the same shape as others in this collection. Whether rotation matters at all is decided by a single ratio, and for a wave with a period of a day it always does. A pipe will not carry a note below its cutoff, and a plasma lets nothing in below its plasma frequency; in both, a wave below a frequency set by the medium does not propagate. The inertial frequency is that cutoff for a rotating fluid, and its value is set by the medium’s spin and the position on the sphere rather than by any size.

A switch with a finite width #

Where the tongue is wide, the edge on the map is not a line.

Poleward of 28.8° the nearest wave the latitude allows sits at the local inertial frequency, a little above half the tide’s. That wave is detuned from exact resonance, and a detuned oscillator still grows if the detuning is smaller than the tongue’s half-width. So the growth does not stop at 28.8°: it continues, weakening, until the inertial frequency has risen far enough to put every allowed wave outside the tongue. For a modulation strength of 0.1 that happens at 30.0°. For a strong modulation of 0.3 it reaches 31.5°.

In the ocean the modulation strength is roughly the internal tide’s strain — the fractional change in the local stratification its passage produces — and it is usually small, a few per cent to a few tens of per cent near strong generation sites. For the weak tides that fill most of the ocean, the edge is within a degree or so of 28.8°, and for the strong beams near ridges it is smeared by a degree or two. Either way, a few degrees poleward of the critical latitude the mechanism has switched off.

Waves that stop travelling #

The critical latitude is not only where the daughter waves become impossible. Approaching it, they become slow.

The energy of a wave packet travels at its group velocity, a speed distinct from that of its crests. For an internal wave in a rotating fluid the horizontal group speed at a fixed vertical wavelength is proportional to $\sqrt{1 - f^2/\omega^2}$, and as the wave’s frequency approaches the inertial frequency that factor falls to zero. Physically, a wave at the inertial frequency has become an inertial oscillation — fluid turning in circles in place — and a circle in place carries nothing anywhere.

For the half-frequency daughters of the semidiurnal tide, with a vertical wavelength of 500 metres in a typical thermocline, the speed is 14 kilometres a day at the equator, 10 at 20° and 3 at 28°. Energy handed to such waves close to 28.8° therefore stays close to where it was handed over. It accumulates as vertical shear in near-inertial waves, and near-inertial shear is exactly what breaks into turbulence. The latitude that closes the route is also the latitude where the route, while still open, deposits its energy most locally.

Watching one grow #

The integration is the same oscillator the tongues were computed from, followed in time at three positions on the map. At 20° the daughter wave sits at exactly half the tidal frequency and grows by a factor of $e$ every six and a half tidal periods — a hundredfold in about fifteen days. At 29.5°, just past the critical latitude, the nearest allowed wave is detuned by two per cent and still grows, at 0.145 e-folds a period against the resonant 0.157, a result the Floquet calculation had predicted. At 33° the detuning is thirteen per cent and the wave does nothing but beat.

The growth rates are those of an idealised modulated oscillator and not of the ocean. What survives into the ocean is the ordering: resonant growth equatorward, a narrow band of detuned growth just poleward, and nothing beyond.

What the Pacific said #

The prediction has been tested where the geometry is cleanest. The Hawaiian Ridge generates one of the strongest internal tides on Earth, some twenty gigawatts, and its northward beam crosses 28.8° on its way across the North Pacific. Numerical simulations published in 2005 suggested that the beam should lose a large fraction of its energy to subharmonic waves as it approached that latitude, and a cruise along the beam in 2006 went to look.

It found enhanced near-inertial shear near the critical latitude, the signature of energy being handed to waves that stall there. It did not find a collapse of the internal tide’s energy flux: the beam continued northward past 28.8° carrying most of what it had brought. Later, more focused measurements confirmed the instability operating near 29° and transferring energy at rates that were real but modest. The mechanism exists where it should and is not, on its own, the dominant sink for the Hawaiian tide — a result that is informative rather than disappointing, because it bounds how much of the ocean’s mixing budget this one geographical edge can carry.

Where the pumped oscillator stops being the ocean #

The oscillator is a stand-in for a triad. A real subharmonic instability involves daughter waves with definite wavevectors, and its growth rate depends on their vertical wavelength, on the tide’s shear and on the stratification, not on a single modulation strength. The modulated oscillator captures which frequencies are selected and where they are forbidden; its growth rates are illustrative.

The background is not empty. The ocean is filled with a spectrum of internal waves, and the tide loses energy to that spectrum through several other interactions — scattering off the background waves, induced diffusion in wavenumber — which have no critical latitude. The subharmonic route is one of several running in parallel, and which dominates depends on place and on the strength of the tide.

The inertial frequency is not always f. A mesoscale eddy’s vorticity shifts the effective inertial frequency by half the vorticity, so the critical latitude inside a strong eddy is moved by a degree or more. The map’s edge is a line only in an ocean at rest.

The dispersion relation is hydrostatic. The group-speed figure uses the long-wave form. For the shortest daughter waves, and near the buoyancy frequency, nonhydrostatic terms change the speeds, though not the zero at the critical latitude.

And nothing here dissipates. The integration grows without limit; a real daughter wave grows until it breaks or until its own nonlinearity saturates the transfer. Where the energy finally goes, and how much of it mixes the water rather than radiating away, is outside this model.

What the frequency plots leave out #

Every figure is a function of frequency, latitude or time, and the instability is a process in space. A beam of internal tide a hundred kilometres across, crossing 28.8° over several days, sheds energy into a patch of near-inertial waves whose vertical wavelength is hundreds of metres and whose horizontal extent follows the beam. That patch has a shape, a depth and a lifetime, and measuring it is the whole of the observational problem — none of which a curve against latitude can show.

Nor can the figures show the triad’s geometry: the daughter wavevectors nearly equal and opposite, their sum matching the tide’s, and the energy moving through wavenumber space towards the short scales that matter. The oscillator reduces all of that to one frequency, which is the part the geography depends on and not the part the mixing depends on.

Still open: how much of the tide dies near where it is born #

The internal tide’s terawatt is one of the main sources of the mixing that closes the deep ocean’s overturning circulation, and ocean and climate models have to decide where that energy is dissipated. If most of it is lost close to the ridges that generate it, deep mixing is concentrated over rough topography; if most of it travels thousands of kilometres first, the mixing is spread across basins and happens where the beams finally break. The difference changes the modelled circulation.

The subharmonic instability’s critical latitude was one of the first places where a definite prediction about the fate of that energy could be tested, and the answer — real, measurable and modest — has pushed the question onto the other routes: scattering by topography, interaction with eddies, and breaking at distant continental slopes of the kind the reflection that changes the wavelength describes. How the terawatt divides among them is still measured with uncertainties of a factor of two, and it is the number the next generation of mixing parameterisations most needs.

The habit worth carrying away is to look for the frequency a process needs and ask where that frequency is allowed. A mechanism that halves a frequency is switched off wherever half is below a floor, and when the floor depends on position, the mechanism acquires a map — here drawn by the Moon’s period, the Earth’s spin and a factor of two.