The hottest month is not the sunniest

The tilt that makes the seasons puts the longest northern day on 21 June. The warmest month is July or August, and over the ocean it can be September. The Sun is highest at local noon and the warmest part of the afternoon is around three o’clock.

Both observations are the same equation, one cycle apart, and the equation is the simplest one that has a surface in it at all.

One equation, and the two things it says #

Write the surface as a slab of heat capacity $C$ per unit area, absorbing a forcing $Q(t)$ and losing heat in proportion to how far it sits above its equilibrium:

$$C\frac{dT}{dt} = Q(t) - \lambda T .$$

That is a first-order linear equation, and for a sinusoidal forcing its solution is one line of algebra. Writing $Q = Q_0\cos\omega t$ and looking for $T = T_0\cos(\omega t - \phi)$,

$$\phi = \arctan\!\left(\frac{\omega C}{\lambda}\right), \qquad T_0 = \frac{Q_0}{\lambda\sqrt{1 + (\omega C/\lambda)^2}} .$$

Two consequences follow immediately and neither depends on any number.

The lag cannot reach a quarter of a cycle. An arctangent is bounded by 90°, so however massive the surface, its temperature can never peak more than a quarter period after the forcing does. For the annual cycle that is three months and for the daily cycle six hours, and both bounds are approached and never crossed.

The lag and the attenuation are the same number. The dimensionless group $\omega C/\lambda$ appears once in the arctangent and once under the square root, so a surface that lags a long way is necessarily one that barely swings. There is no arrangement in which a place has both a long seasonal delay and a large seasonal range.

What the numbers correspond to #

The heat capacities drawn are in metres of water equivalent, which is a convenient unit because water dominates every case.

A continental interior is about two and a half metres. Most of that is not ground at all — it is the atmospheric column above it, some ten tonnes per square metre of air at roughly a quarter of water’s specific heat, which comes to about 2.4 m of water. The soil contributes the top metre or two, since the annual thermal wave penetrates only a few metres into rock.

A shallow shelf sea is ten metres or so, because the whole water column mixes over a season.

A deep ocean mixed layer is fifty metres or more, set by how far wind stirring reaches before stratification stops it, and it varies through the year: deeper in winter when storms are strong and surface cooling drives convection, shallower in summer when the surface warms and stratifies.

The contrast with a body that has no atmosphere and no ocean is the cleanest way to see how much of the Earth’s thermal inertia is not rock: remove the air and the water and two and a half metres becomes a few centimetres.

That seasonal variation in $C$ is why the real ocean’s lag is not quite what a fixed slab predicts — the layer being heated in spring is thinner than the layer that was cooled in winter — and it is the main reason a linear model with a constant coefficient is only approximately right.

Why a sinusoid is the right thing to force it with #

The insolation at a mid-latitude is not a sinusoid. It is the daily integral over the Sun’s time above the horizon, which at 45° is close to sinusoidal and at 70° is not remotely so.

The lag formula applies to a sinusoid, and the way it is applied here is to the forcing’s fundamental: the insolation curve is Fourier-transformed, its first harmonic taken, and the response computed for that. That is legitimate because the equation is linear — each harmonic is lagged and attenuated independently, and the total response is the sum.

What it means is that the drawn temperature curves are smoother than the drawn forcing, and deliberately so. The higher harmonics are attenuated more, because $\omega$ is larger for each of them, so a slab acts as a low-pass filter on the season as well as a delay.

That filtering is visible in reality. The insolation at 70° has a sharp shoulder where the polar night begins and the temperature record has none, because the second and third harmonics carrying that shoulder are damped by factors of two and three more than the fundamental is. A surface with a memory does not merely remember late; it remembers vaguely.

Why the sea is late and the desert is not #

The observed pattern follows directly and is worth checking against the drawing.

Continental interiors peak in July, about four to six weeks after the solstice, which the 2.4 m curve reproduces at 45.6 days. They also have the largest annual temperature ranges on the planet — central Siberia swings by sixty degrees.

Coastal and maritime climates peak in August, and their ranges are far smaller.

The open ocean’s surface peaks in late August or September at mid-latitudes, some two and a half to three months after the solstice, and its annual range is a few degrees.

All three sit on one curve in the same two variables, and the ordering is forced: there is no maritime climate with a large range and no continental one that peaks in September. It is a rare case of a two-parameter physical law whose predictions can be checked against a world atlas rather than against an instrument, and where the check is a statement about what does not exist.

The same equation, one cycle down #

Reducing the period by a factor of 365 and the heat capacity by a factor of several hundred gives the diurnal case, and it is the same picture.

The daily thermal wave penetrates only a few centimetres into soil and a few metres into water, because the penetration depth goes as the square root of the period. So the effective heat capacity for the daily cycle is two or three orders of magnitude smaller than for the annual one — and $\omega$ is 365 times larger, so the product $\omega C$ lands in a similar range.

The observed daily lag over land is two to three hours, which the drawn dry-ground case reproduces — and which is why the hour a sundial reads and the hour a thermometer peaks are different quantities that both track the Sun. Over water it is longer and the swing is almost nothing, which is why sea surface temperature barely varies through a day and why coastal air temperature does.

And the two lags are unrelated in size while being identical in origin. A place can have a short daily lag and a long seasonal one, because the penetration depth differs between the two cycles and therefore so does the effective $C$. Dry sand is a poor conductor and heats and cools violently every day while its annual cycle is damped by the atmosphere above it.

The one place both lags are visible at once #

A coastal site in spring shows both effects operating on the same day and pulling in opposite directions, which is the most direct demonstration of the argument available without instruments.

The sea, with its large $C$, is still near its winter minimum in April and May. The land, with its small one, has already warmed. The temperature difference drives the sea breeze, and its strength through spring is a direct readout of the difference between two lags.

By September the situation reverses: the sea is at or near its annual maximum while the land has begun to cool, and the breeze runs the other way. The annual reversal of a coastal wind is the phase difference between two first-order systems, and it is the only place the lag is felt rather than measured.

The same difference explains why maritime and continental climates at the same latitude have their growing seasons offset by weeks, and why the last frost of spring comes later inland than on a coast at the same latitude despite the coast being cooler in the mean.

The cross-over nobody notices #

There is a case in which the lag exceeds a quarter cycle, and it is worth stating because it looks like a violation and is not.

A subsurface temperature at depth $z$ lags the surface by an amount proportional to $z$, without bound, because the thermal wave propagates downward at a finite speed. At a depth of a few metres the annual wave arrives six months late, so the ground at that depth is warmest in midwinter.

That is a diffusion problem rather than a slab problem, and the slab equation does not describe it. The slab has one temperature; the ground has a profile, and the profile supports a travelling wave. The quarter-cycle bound applies to a lumped system and not to a distributed one.

It is the reason a cellar is cool in summer and mild in winter, and the reason a buried water pipe placed below the frost line does not freeze. The same diffusion equation with a different period sets how deep the daily wave reaches, which is centimetres rather than metres, since the penetration depth goes as the square root of the period. The depth at which the annual wave is inverted is about 2.5 m in ordinary soil, and it is a quantity a builder uses without calling it a phase lag.

A planet with almost no lag, and one with all of it #

Two other bodies bracket the Earth and make the parameter visible.

The Moon has essentially no heat capacity that matters. Its regolith is a poor conductor and there is no atmosphere or ocean, so the daily thermal wave penetrates a few centimetres and the surface follows the Sun almost exactly. Lunar noon is the hottest moment of the lunar day to within a small fraction of it, and the surface swings by nearly three hundred kelvin between day and night. Large swing, no lag — the $\omega C/\lambda \to 0$ limit, drawn as far to the left as the equation goes.

Venus is the other end. Its atmosphere is ninety times the Earth’s by mass and its heat capacity per unit area is correspondingly enormous, so the surface temperature is essentially constant — the same at the poles as at the equator, and the same at midnight as at noon, to within a few kelvin. Its rotation period is 243 days and it makes no difference. Almost no swing, and a lag that has nothing left to delay.

The Earth sits between them and not by much of a margin. Its $\omega C/\lambda$ for the annual cycle runs from about one over a continent to about twenty over the ocean, which is the whole range in which both the lag and the swing are appreciable — and it is appreciable only because the planet is half ocean and half land, so both regimes are present on the same world.

That coexistence is not generic. A planet with a global ocean would be everywhere maritime and a dry one everywhere continental, and the seasonal contrast that drives monsoons, ocean–land breezes and most of the Earth’s weather comes from having both.

One coefficient chosen to fit, and the circularity in that #

The insolation is computed from the orbit and the obliquity, with Kepler’s equation solved for the distance.

The heat capacities are not measured for any particular place; they are representative values, quoted in a unit that makes them comparable.

The damping coefficient is the least secure term. It represents everything that removes heat from a surface in proportion to its temperature excess — outgoing longwave radiation, evaporation, and turbulent exchange with the air — and only the first has a clean theoretical value. A blackbody at 255 K radiates $4\sigma T^3 = 3.8$ $\mathrm{W\,m^{-2}\,K^{-1}}$; the effective climate value including water vapour feedback is nearer 1 to 2, and the surface value including evaporation is larger again.

The lag is a function of a ratio, so an error in $\lambda$ is an error in the answer, and the two $\mathrm{W\,m^{-2}\,K^{-1}}$ used here is chosen to reproduce the observed lags rather than derived from anything. That is an honest circularity and it is worth naming: the figure demonstrates the structure of the lag rather than predicting its size, and a version claiming to predict the size would be claiming a knowledge of $\lambda$ that nobody has.

Heat that does not leave vertically #

Heat does not only leave vertically. The slab model has one loss term proportional to local temperature, and a real surface also exchanges heat sideways — the atmosphere moves energy poleward at a rate comparable to what it radiates, and the ocean does the same. That transport couples the slabs drawn as independent, and its effect is to make every place behave partly like its neighbours.

The heat capacity is not a constant. An ocean mixed layer deepens in winter and shallows in summer, which makes the real system nonlinear and asymmetric: the warming half of the year and the cooling half have different effective capacities, so the response is not the symmetric sinusoid drawn.

And nothing here is a temperature in degrees. Every curve is normalised to its own peak, which is exactly the right choice for comparing timing and exactly the wrong one for comparing magnitude. The amplitudes are in the legend as fractions, and converting them to kelvin needs the forcing’s own amplitude, which varies by a factor of several between the latitudes this equation is applied at.

The lag as a way of measuring the reservoir #

The relation can be inverted, and inverting it is how the depth of an ocean’s mixed layer is estimated from nothing but a temperature record.

Measure the seasonal cycle of sea surface temperature at a point: its amplitude and its lag behind the local insolation. Both give $\omega C/\lambda$, so the pair over-determines it — and the consistency between the two estimates is a check that the first-order model applies at all.

Where they disagree, the disagreement is informative rather than fatal. A site whose lag implies a larger $C$ than its amplitude does is a site where heat is being removed sideways as well as vertically, since horizontal transport reduces the amplitude without delaying it. The residual between two estimates of one parameter is a measurement of the term the model left out, which is the usual way a deliberately simple model earns its place.

The same inversion applied to the whole planet is one of the standard estimates of ocean heat uptake, and it disagrees with the direct measurement — from floats profiling the upper two kilometres — by an amount that has been narrowing for two decades.

Why a first-order model is enough #

It is worth defending the model, because its simplicity is what makes the two conclusions above general rather than fitted.

Nothing in the two results depends on the value of $C$, on the value of $\lambda$, or on the shape of the forcing. The quarter-cycle bound follows from the order of the differential equation; the lag–amplitude trade follows from the same. Any system whose response is governed by one storage term and one loss term has both properties, whatever those terms physically are.

That is why the same arithmetic appears everywhere it has no business appearing — in a low-pass filter, in the response of a chemical reservoir, in the way an orbit’s elements respond to a slow perturbation, and in the damping of a tidal bulge that lags the body raising it. A first-order system is a first-order system, and the lag it produces is always an arctangent of the same group.

What a more elaborate climate model buys is the magnitudes and the geography, and not the structure. The structure is here, and it fits the observed lags to within a week.

Still open: nothing in the equation, and a great deal in its coefficients #

The equation is not in dispute and neither are its two consequences. What is in dispute, and is one of the central quantities in climate science, is $\lambda$ — the rate at which the planet sheds heat per degree of warming.

The Planck term is exact. Everything else — water vapour, clouds, ice albedo, lapse rate — is a feedback that modifies it, and their sum is known to perhaps a factor of two, which is the dominant uncertainty in every projection made from the same equation used here.

It is worth naming the shape of that difficulty, because it is the same one an inference through an intermediary always has: the observable is a ratio, and separating its numerator from its denominator requires a measurement from outside.

And the seasonal lag is one of the few direct constraints on it. A planet with a small $\lambda$ has a long lag and a large swing relative to its forcing; the observed lag and the observed seasonal amplitude together constrain the ratio $\omega C/\lambda$, and an independent estimate of $C$ from ocean heat content then gives $\lambda$. The constraint is weak, because $C$ is uncertain too, and it is genuinely independent of the ones derived from the long-term record.

From here: the same equation with a far larger C #

Everything above has treated the obliquity as a fixed number and the orbit as a fixed ellipse. Neither is, on any timescale longer than a few thousand years, and the variation is what the ice record responds to.

The natural continuation is the one quantity carefully not computed above: the seasonal insolation integrated over a melt season at a high northern latitude, as a function of the three orbital elements, and what an ice sheet with a heat capacity of hundreds of metres of water does with it. It is the same first-order equation with $C$ two orders of magnitude larger, and a lag that reaches the bound.