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**Figure 1.** Isoviscous, bottom-heated thermal convection for $Ra = 10^6$ with a resolution of 150x50. The initial condition is a linearly increasing temperature profile with an elliptical anomaly at the top. Thermal boundary conditions are fixed temperature at the top and bottom and zero heat flux at the sides. All velocity boundary conditions are free slip. Heat diffusion is solved using the Crank–Nicolson method, the Stokes equation using the defect correction method, and temperature advection with the semi-Lagrangian method. Models run until a steady state is reached or up to a maximum of 8000 iterations.
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**Figure 1.** Isoviscous, bottom-heated thermal convection for $Ra = 10^6$ with a resolution of 150x50. The initial condition is a linearly increasing temperature profile with an elliptical anomaly on top. Thermal boundary conditions are fixed temperature at the top and bottom and zero heat flux at the sides. All velocity boundary conditions are free slip. Heat diffusion is solved using the Crank–Nicolson method, the Stokes equation using the defect correction method, and temperature advection with the semi-Lagrangian method. Models run until a steady state is reached or up to a maximum of 8000 iterations.
This exercise revisits **2-D thermal convection** in a fully **nondimensional** (scaled) framework. You will define scaling constants, transform the governing equations, and study how the flow depends on the **Rayleigh number**. The setup follows the **Boussinesq approximation** with isoviscous rheology and bottom heating.
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This exercise revisits **2-D thermal convection** in a fully **nondimensional** (scaled) framework. You will define scaling constants and transform the governing equations, and study how the flow depends on the **Rayleigh number**. The setup follows the **Boussinesq approximation** with isoviscous rheology and bottom heating.
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## Objectives
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**Objectives**
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1. Define physically motivated **scaling constants** and apply the **nondimensional transformations** to the PDEs.
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2. Formulate the **dimensionless** energy, momentum, and mass conservation equations (incl. buoyancy term $Ra,T'$).
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5. Run and compare models for **$Ra = 10^4, 10^5, 10^6$**; discuss plume/slab scale and flow vigor.
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6. Compute diagnostics such as **Nusselt number** and **RMS velocity**; assess approach to steady state.
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## Notes for students
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As the Rayleigh number increases:
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- In the scaled form we assume reference parameters $\eta_0 = 1, g = 1,\kappa = 1,c_p = 1$.
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- If $Ra$ is specified, adjust $\eta_0$ accordingly; otherwise compute $Ra$ from the reference set.
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- Higher $Ra$ requires **finer grids** to maintain stability and accuracy; balance resolution vs. runtime.
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- Keep the documentation for the exercise **brief**: show the key equations, the scaling you used, and a **visualization** (animation or snapshot) of the final temperature/velocity fields.
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-**flow velocities** increase,
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-**convection becomes more vigorous**, and
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- structures such as *slabs* and *plumes* become **finer**.
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Therefore, the **grid resolution** must be adjusted accordingly to ensure **numerical stability** and **accuracy**.
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> However, higher resolution significantly increases computational cost!
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The resolution provided here is sufficient for the Rayleigh numbers listed.
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That said, some numerical methods already show **initial inaccuracies**, so in practice a **higher resolution** is often advisable.
**Figure 3.** Variation in the root mean square velocity with numerical iterations. Empirically, a tolerance of $10^{-15}$ was chosen to define steady state. Low-$Ra$ cases typically reach steady state in fewer than 3000 iterations.
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**Figure 3.** Variation in the root mean square velocity with numerical iterations. Empirically, a tolerance of $3.8^{-3}$ was chosen to define steady state. Low-$Ra$ cases typically reach steady state in fewer than 3000 iterations.
This exercise introduces the **Blankenbach benchmark** (Blankenbach et al., 1989), a widely used reference test for validating numerical models of **mantle convection**. The benchmark compares results from different numerical methods by providing reference values for the **steady-state Nusselt number** and **mean velocity** at various Rayleigh numbers.
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The setup consists of a **2-D rectangular box** with an aspect ratio of 1:1 (height = length = 1000 km). All boundaries are **free-slip**, while thermal boundary conditions are **fixed temperatures** at the top and bottom and **zero heat flux** along the sides. Density varies linearly with temperature according to the **Boussinesq approximation**.
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The main objectives are:
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1. Implementing a **steady-state 2-D isoviscous convection model** under the Boussinesq approximation,
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2. Computing thermal convection for three different **Rayleigh numbers** ($Ra = 10^4, 10^5, 10^6$),
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3. Comparing the computed **Nusselt numbers** and **mean velocities** with the published benchmark values,
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4. Performing a **resolution test** to assess the convergence of numerical results, and
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5. Discussing deviations and numerical stability at higher Rayleigh numbers.
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This benchmark demonstrates how increasing the Rayleigh number strengthens convection, leading to thinner thermal boundary layers and more localized upwellings and downwellings. It also highlights the importance of numerical resolution and stability in high–Rayleigh-number simulations.
**Figure 1.** Isoviscous, bottom-heated thermal convection for $Ra = 10^6$ with a resolution of 100×100.
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The initial condition is a linearly increasing temperature profile with an elliptical anomaly at the top.
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The background color shows the non-dimensional temperature, overlaid by temperature isolines (every 0.05) and centroid velocity vectors. Heat diffusion is solved using the **Crank–Nicolson** method, the Stokes equation using the **defect correction** method, and temperature advection with the **semi-Lagrangian** method.
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Models run until a steady state is reached or up to a maximum of 8000 iterations.
**Figure 2.** Time series of the surface Nusselt number and root-mean-square (RMS) velocity. The steady-state benchmark values are shown as red dashed lines. For details on how these diagnostics are calculated, see the [exercise](https://github.com/GeoSci-FFM/GeoModBox.jl/blob/main/exercises/13_Blankenbach_Benchmark_en.ipynb).
**Figure 3.** Vertical temperature profile at the center of the model domain. Benchmark values for the local maximum and minimum temperatures are shown as black squares.
**Figure 4.** Variation in root-mean-square velocity with numerical iterations. Empirically, a tolerance of $3.8^{-3}$ was chosen to define steady state. Low-$Ra$ cases typically reach steady state in fewer than 3000 iterations.
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