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Temperature across the sheet now (solid) against the exact Jaeger solution with no latent heat (dashed) and with all latent heat at once (dotted)
Peak temperature reached against distance from the contact; coloured lines = hornfels facies boundaries
Percent of the sheet solidified and heat delivered to the country rock against time
Starting.
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How to use
Pick a preset (a thin dike, a 30 m sill, a thick sill, a deep warm crust, no latent heat, a cooler rhyolite dike) and press play. The picture is a cross-section: the white lines are the walls of the sheet, the dashed lines are the outer edges of the baked zones. Colour the section by temperature now, by the peak temperature reached, or by the hornfels facies. The five thermocouples log their temperature in graph 3. Try: (1) set L to 0 and compare the red curve with the dashed erf curve in graph 1; (2) double the half-width and check that the freezing time grows four times; (3) raise the depth to 15 km and see how a warm host narrows the temperature contrast but widens the baked zone; (4) change the speed to watch a dike chill in a minute or a sill over millennia. Hover over the section to read the temperature and the peak at any point. Space pauses, R restarts.
The science
A dike (vertical sheet) or sill (horizontal sheet) of magma that squeezes into cold rock is a textbook heat-conduction problem. The lab solves the one-dimensional heat equation across the sheet, symmetric about its centre, with the enthalpy method so that the latent heat of crystallisation is released over the interval between the liquidus and the solidus:
∂h/∂t = κ ∂2T/∂x2 h(T) = T + (L/c)·f(T) f = 0 below Tsol, linear up to 1 at Tliq
Here κ = k/(ρ c) is the thermal diffusivity (about 0.8×10-6 m2/s for rock), c = 1000 J/(kg K) and L the latent heat (about 400 kJ/kg for basalt). Length is measured in units of the half-width a and time in units of a2/κ, so one run describes every size: a 1 m dike has a clock of days, a 300 m sill a clock of thousands of years.
The exact solution without latent heat (Jaeger, 1957) for a sheet of half-width a at T0 in rock at Tr is
T(x,t) = Tr + (T0 − Tr)/2 · [ erf((a − x)/(2√(κt))) + erf((a + x)/(2√(κt))) ]
It gives three results used by the checks: the contact stays at (T0 + Tr)/2 as long as the diffusion length is small against a; the centre keeps Tr + (T0−Tr) erf(a/(2√(κt))), which has lost half its excess at κt/a2 = 1.099; and the heat spreads a distance of order √(κt), so the cooling time scales as a2. Latent heat adds up to L/c to the effective magma temperature (dotted curve in graph 1 is the upper bound where it is all released at once); the model sits between the dashed and the dotted curve and holds the sheet near the solidus for longer.
Baked zones. Wherever the rock is heated, minerals react. The model records the highest temperature each point ever reached (graph 2 and the “peak temperature” view) and reports the distance at which it passed 300, 500 and 650 C, roughly the boundaries of the albite-epidote, hornblende and pyroxene hornfels facies. In reality the width of the aureole is of the order of the sheet width for hot magma in cold rock, and larger when the host is warm (deep) or when hot fluids circulate.
Honest simplifications. (1) Magma and rock have the same κ, c and density, which makes the contact temperature the simple mean; real magma is a little less conductive and denser. (2) Heat leaves by conduction only: no convection in the magma, no hydrothermal circulation in the host, no flow, no repeated pulses. (3) Latent heat is spread linearly over the melting interval; real crystallisation follows the phase diagram. (4) The geometry is a one-dimensional infinite sheet; the dike and sill differ only in orientation, and the 3D picture draws the same temperature field. (5) Metamorphic reactions are only labelled by peak temperature; pressure, fluids and reaction kinetics are not modelled. (6) Presets use round literature-style numbers, not a particular field example.
Checks run in the page. The numerical profile against the erf solution at two times (RMS error); the contact temperature against the mean at early time; the time for the centre to lose half its excess against the root 1.099 of erf(1/(2√τ)) = 1/2; conservation of total enthalpy with latent heat; and the sign of the latent-heat delay.