Mountains push up, rivers cut down, hillslopes smooth it over. Every landscape is that argument — and you can do the arithmetic.
Descriptive geomorphology names the landform. Process geomorphology asks how fast — how quickly a range rises, how long a scarp takes to round off, how steep a river must be to keep up with uplift. It runs on three ideas: conservation of mass, transport rules, and magnitude versus frequency. Everything here is a back-of-the-envelope calculation you can actually do.
Pairs with Source to Sink — landforms there, rates here
Geomorphology starts at the largest scale, with three habits of thought that everything else in the course hangs off — and with the observation that Earth’s surface elevations are not distributed the way you would expect if the planet were made of one thing.
Three guiding principles
1 · Conservation, and the control volume
Draw a box. Account for everything that enters and everything that leaves, and you know what is inside — at any time. Written as a word equation before any symbols appear:
time rate of change of mass = mass rate in − mass rate out
The classic illustration is sheep on a hillside pasture: they wander in, wander out, are born, and die. Track those four rates and you always know the flock. The same accounting works for heat, ice, water, sediment, regolith, radionuclides and momentum — and starting from the word equation, rather than reaching for a formula, is what keeps a messy problem organised.
2 · Transport rules
Movement is usually driven by a gradient, and when the flux is proportional to that gradient you have a first-order rate law:
Q = k · (∂ something / ∂x)
Heat down a temperature gradient, soil down a topographic slope, solutes down a concentration gradient — the same form each time, with k a property of the material. But the gradient alone is not enough: you also need the rheology, how the material deforms under stress. In one alpine valley, air moves downslope at ~28 m/s, meltwater at ~1 m/s, the glacier at ~3×10−7 m/s, and the mantle beneath it at ~3×10−10 m/s — the same gradient, eighteen orders of magnitude of difference, all of it rheology.
3 · Event size and frequency
Does it happen daily, annually, once a lifetime, or a handful of times in Earth history? Landscapes are shaped by the product of magnitude and frequency, and the biggest event is rarely the most important one. Systems are also thresholded and non-linear, so the response is not proportional to the forcing.
The caveat that matters: uniformitarianism asserts uniformity of process, not uniformity of rate. The physics has not changed; the magnitudes absolutely have. It is likely that some of the most consequential geomorphic events in Earth’s history have never been witnessed by a record-keeping human.
Two drivers, four spheres
Every landscape is the outcome of a contest between forces from below and forces from above.
Endogenic — from within
Deep geophysical processes driving the plates: uplift, mountain building, volcanism, the opening of ocean basins. Powered by Earth’s internal heat. These build relief.
Exogenic — from without
Atmospheric and surface phenomena attacking from above: weathering, rivers, ice, wind, waves, mass movement. Powered by the Sun. These destroy relief.
The four spheres, interacting
Geosphere (lithospheric plates), hydrosphere (surface water, groundwater, ice), atmosphere (the gas blanket, tightly coupled to the hydrosphere) and biosphere (all life). A worked chain: plates collide and raise a range (endogenic, geosphere) → the range forces moist air upward and wrings out precipitation (exogenic, atmosphere) → that water collects into rivers (hydrosphere) → the rivers incise the range and move the sediment downstream (back to the geosphere) → and feed a lake whose ecosystem does its own geomorphic work (biosphere).
Why planets are round, and why Earth is not quite
Size decides shape
Asteroids and small moons look like potatoes because their material strength is enough to hold an irregular shape against the modest internal pressures involved. Above a certain size, interior pressures exceed material strength, the solid flows, and the body relaxes toward hydrostatic equilibrium — a sphere. Picture a planet with a pimple and a dimple: at equal distance from the centre, pressure under the pimple exceeds pressure under the dimple, so material flows from high pressure to low and smooths both away.
Spin makes it an oblate spheroid
Rotation adds an outward inertial effect strongest at the equator, so mantle flow leaves the equatorial radius longer than the polar radius — a squashed sphere. The flattening ratio is about 1/300, which works out to roughly 21 km of difference in radius. Set that against the planet’s entire topographic range, Everest to the Mariana Trench, of about 20 km: the equatorial bulge is the single largest topographic feature on Earth, and it is not a mountain range. The bimodal split between continents and ocean basins, below, is the second largest.
The hypsometric curve
Plot how much of Earth’s surface sits at each elevation and you do not get one peak. You get two — one scattered about the mean elevation of the continents, one about the mean depth of the ocean basins. That bimodality is the observation isostasy exists to explain. Drag the line and read the curve.
Surface above this level—%
Area above—M km²
Surface below this level—%
Area below—M km²
Back-of-the-envelope
Geomorphology runs on order-of-magnitude estimates: define the problem, sketch it, label the variables, carry the units, and ask whether the answer makes sense. Work these with a calculator and enter the result — the tolerance is generous, because the point is the method, not the third decimal.
Right 0Tried 0
Crust floats on mantle like an iceberg on water. Build a mountain and you must build a root beneath it — and that root is what makes mountains so stubbornly hard to erode away.
4.0 kmtopography
0.0 kmthickness removed
18.0 kmCrustal root
53.0 kmTotal crust
0.0 kmRebound
4.0 kmElevation now
The elevation histogram
The distribution of elevations on Earth is bimodal — two peaks, not one. That single graph is the strongest evidence that continental and oceanic crust are fundamentally different materials, not just high and low ground.
Two thirds of Earth’s surface is ocean floor, and its shape is startlingly orderly — a ridge down the middle of each basin, and the floor sloping away from it with such regularity that a single equation describes it. That order was the observation plate tectonics had to explain.
What the sonar found
The shape of a basin
From a coast: the continental shelf, gently sloping and submerged, then a steep drop to the abyssal plain. But every basin also carries a conspicuous bathymetric high — a mid-ocean ridge — and the floor descends away from it with remarkable consistency. The pattern emerged from Cold War sonar surveys in the 1950s, run by navies looking for submarines rather than for a theory of the Earth.
Depth goes as the square root of age
Plot seafloor depth against the age of the crust and out to about 80 million years it follows √t. That relationship is far too consistent to be coincidence, and explaining it is the thread that runs through this whole module. Age converts to distance through the spreading rate: t = x / (½ spreading rate), because two plates move apart from one ridge.
What the lithosphere actually is
Two equivalent definitions: everything that moves with the migrating surface, and everything that behaves as a solid on geological timescales. Both make it a rheological boundary, not a compositional one — the compositional boundary is the Moho, and it is somewhere else entirely. Since mantle rheology is governed mostly by temperature, the base of the lithosphere is taken as an isotherm, conventionally 1200 °C: above that, mantle deforms fast enough to count as a fluid.
Heat flow out of the Earth
Conservation again — this time of heat. Write the word equation, and the rate of change of heat in a box is heat flowing in minus heat flowing out. To turn that into something you can calculate, you need a transport rule, and for heat it is Fourier’s Law: another first-order rate law, exactly like the ones from Module 1.
Qz = −kz ∂T/∂z ≈ −kz ΔT/Δz
ΔT across the crust1000 °C
Crustal thickness40 km
Conductivity k3.0
Heat flow Q—mW/m²
Geothermal gradient—°C/km
From Fourier to diffusion
Put Fourier’s Law into the conservation statement, do the algebra, and out comes one of the most reused equations in the subject:
∂T/∂t = κ ∂²T/∂z²
The diffusion equation, with κ the thermal diffusivity. Its solution puts the depth to any given isotherm proportional to √t — so the lithosphere thickens as the square root of its age, and therefore as the square root of distance from the ridge.
You have met this equation already in this game. Hillslopes & Scarps uses the identical form for soil moving downslope. Nature abhors a sharp edge: the same mathematics smooths a fault scarp and cools a plate, because both are gradients relaxing.
Age, depth and spreading rate
The lithosphere thickens as it cools, and cooling makes it denser, so it rides lower in the mantle. The result is the “droop” — and because age maps to distance through the spreading rate, a fast-spreading ridge makes a shallow ocean and a slow one makes a deep ocean.
Half-spreading rate2.5 cm/yr
Read at age50 Ma
Depth at that age—m
Distance from ridge—km
Lithosphere thickness—km
The isostatic balance behind the droop
Cooling contracts the lithosphere and raises its density. For mantle material the coefficient of thermal expansion is about 3×10−5 per °C, so a plate averaging 600 °C cooler than the asthenosphere is roughly 2% denser. Float that denser plate isostatically and the arithmetic gives something neat: the ratio of water depth to lithosphere thickness is constant.
ρ/ρ0 = 1/(1 + αvΔT) · D/L = (ρL − ρa) / (ρa − ρw)
Ridge-crest depth D₀2500 m
Depth to read at3800 m
ΔT of the plate600 °C
Density contrast—%
D / L ratio—
Lithosphere thickness L—km
Why there is no old seafloor
No ocean floor anywhere is older than Jurassic, roughly 180 million years. Everything older has been subducted. Continental crust survives for billions of years because it is too buoyant to sink; oceanic crust is on a conveyor with a fixed round trip.
That also makes the age map a record of spreading history. Wide age bands mean fast spreading, narrow ones slow.
And it reaches the rock record. In the Cretaceous, spreading is thought to have run fast, so the ridges were young, hot and voluminous — shallow basins holding less water. With nowhere else to go the water went onto the continents, flooding the Western Interior Seaway across North America. That is why there are marine fossils in Kansas, and coal at the seaway margins.
What moves the plates
Observed speeds
Seven major plates, thousands of kilometres across, moving at 5–15 cm/yr — about the rate a fingernail grows. GPS geodesy now measures this directly, and every one of those vectors has to be taken up at a plate margin by a convergent, divergent, transpressional or transtensional setting. The margin type sets the geomorphology.
Mantle convection is the engine
Convection begins when buoyancy overcomes viscous resistance. The ratio of those two is the dimensionless Rayleigh number: convection starts around Ra ≈ 2000, and the mantle’s exceeds 106. It is convecting vigorously. Lithosphere is born at a ridge, thickens, cools, moves away, and descends at a subduction zone as a slab.
Predicting the speed from a force balance
Because plates are not accelerating, the forces balance: the negative buoyancy of the cold dense slab pulling down, against viscous drag on the slab–mantle interface. Equate them and solve for velocity. It needs the two results from earlier in this module — that thickness goes as √t, and that the density contrast is set by ΔT — and it lands near 0.11 m/yr, about 11 cm/yr. For a calculation built from crude estimates, landing inside the observed range is a real result.
And it was faster before
Viscosity depends steeply on temperature, and Earth is cooling. So plate motions were probably considerably faster in the deep past — which matters for how quickly early mountain belts were built and destroyed.
On soil-mantled hillslopes, sediment moves downhill at a rate proportional to slope. That single rule — diffusion — makes sharp landforms round off over time in a predictable way, which turns a fault scarp into a clock.
4.0 m2a, the vertical offset
20 kyrage of the scarp
5.0 m²/kyrclimate & material dependent
—
Maximum scarp slope
A bedrock river in steady state erodes exactly as fast as the rock rises. That balance fixes how steep it has to be — so a river’s profile is a readout of the uplift it is fighting and the rock it is cutting.
1.0 mm/yrrock uplift
weak rockhigher K = easier to cut
0.45typical range 0.4–0.6
—Steepness index ks
—Channel relief
How we actually measure any of this — and the questions that separate a number from an understanding.
Work it out
Loading…
Correct 0Streak 0
Measuring rates
Useful numbers
Order-of-magnitude anchors worth carrying in your head.