README.md

---
pretty_name: GeoSim Mantle Plume Simulation Dataset
license: cc-by-4.0
language:
  - en
tags:
  - pde
  - geodynamics
  - geophysic
  - mantle-convection
  - scientific-machine-learning
  - simulation
size_categories:
  - 100B<n<1T
---

# GeoSim Mantle Plume Simulation Dataset

## Overview

GeoSim contains two-dimensional thermo-mechanical mantle plume simulations generated with **I2VIS**, a geodynamic solver based on Marker-in-Cell and finite-difference methods. The dataset supports spatiotemporal forecasting, surrogate modeling, operator learning, and related scientific machine-learning tasks.

The simulations are divided into two parameter-sampling groups:

- `plume1`: one model parameter is sampled.
- `plume2`: two model parameters are sampled.

Each group provides train and test splits at two resolutions:

- `401x309_physical`: physical field values on the non-uniform I2VIS grid;
- `128x128_norm`: normalized fields conservatively remapped to a 128 x 128 grid;

## Files

Files follow the naming pattern:

```text
geosim_<case>_<split>_<grid>_<values>.hdf5
```

Each field shape is `(n_sample, n_time, H, W)`.

| File | Field shape | Time unit |
| --- | ---: | ---: |
| `geosim_plume1_train_401x309_physical.hdf5` | `(97, 97, 401, 309)` | yr |
| `geosim_plume1_test_401x309_physical.hdf5` | `(12, 131, 401, 309)` | yr |
| `geosim_plume2_train_401x309_physical.hdf5` | `(103, 131, 401, 309)` | yr |
| `geosim_plume2_test_401x309_physical.hdf5` | `(25, 131, 401, 309)` | yr |
| `geosim_plume1_train_128x128_norm.hdf5` | `(97, 97, 128, 128)` | Myr |
| `geosim_plume1_test_128x128_norm.hdf5` | `(12, 131, 128, 128)` | Myr |
| `geosim_plume2_train_128x128_norm.hdf5` | `(103, 131, 128, 128)` | Myr |
| `geosim_plume2_test_128x128_norm.hdf5` | `(25, 131, 128, 128)` | Myr |

## HDF5 Format

All files contain the same nine datasets:

```text
<file>.hdf5
├── nu                 (n_sample, n_time, H, W)  float32
├── pr                 (n_sample, n_time, H, W)  float32
├── ro                 (n_sample, n_time, H, W)  float32
├── t-coordinate       (n_sample, n_time)        float32
├── tk                 (n_sample, n_time, H, W)  float32
├── vx                 (n_sample, n_time, H, W)  float32
├── vy                 (n_sample, n_time, H, W)  float32
├── x-coordinate       (H,)                      float32
└── y-coordinate       (W,)                      float32
```

Field axes are ordered as follows:

| Axis | Meaning |
| ---: | --- |
| 0 | sample |
| 1 | time |
| 2 | x |
| 3 | y |

Access datasets by key rather than relying on HDF5 key order. For spatial meshes, use:

```python
X, Y = np.meshgrid(x, y, indexing="ij")
```

## Fields and Units

| Key | Quantity | `401x309_physical` | `128x128_norm` |
| --- | --- | --- | --- |
| `ro` | density | kg/m³ | normalized |
| `pr` | mechanical pressure | Pa | normalized |
| `tk` | temperature | K | normalized |
| `nu` | log10 viscosity | log10(Pa·s) | normalized |
| `vx` | horizontal velocity | m/s | normalized |
| `vy` | vertical velocity, positive upward | m/s | normalized |
| `t-coordinate` | physical time | yr | Myr |

`pr` is the mechanical pressure unknown solved jointly with `vx` and `vy` in the Stokes-continuity system.

## Coordinates and Time

The physical grid spans **1150 km × 660 km**. It is non-uniform in both directions: horizontal spacing decreases from approximately 4 km near the lateral boundaries to 1 km in the central plume region, while the upper 220 km uses 1 km vertical spacing and the deeper domain gradually coarsens to approximately 9 km.

- `x-coordinate`: 0 to 1,150,000 m
- `y-coordinate`: 0 to 660,000 m

The 128 x 128 files contain their remapped coordinate arrays. Read `x-coordinate` and `y-coordinate` from each file instead of assuming a fixed normalized coordinate sequence.

For paired physical and normalized products,

```python
t_myr = t_years / 1e6
```

This conversion applies only to `t-coordinate`. The unit is recorded in both metadata locations:

| Product | Root attribute `time_unit` | `t-coordinate` attribute `units` |
| --- | --- | --- |
| `401x309_physical` | `yr` | `yr` |
| `128x128_norm` | `Myr` | `Myr` |

## Normalization and Remapping

The 128 x 128 products use per-field max normalization:

```python
field_norm = field_physical / max_value
```

Train and test files within each case use the same constants.

| Field | plume1 | plume2 |
| --- | ---: | ---: |
| `ro` | 3828.61 | 3829.551 |
| `pr` | 2.18068e10 | 2.1789805e10 |
| `tk` | 2113.0 | 2260.3352 |
| `nu` | 24.0 | 24.000002 |
| `vx` | 7.59748e-09 | 1.9177052e-07 |
| `vy` | 4.59946e-09 | 9.3668184e-08 |

Normalized values are not guaranteed to lie within `[-1, 1]`. Signed velocity components may exceed this range because the constants are not maximum absolute values in both directions.

Spatial remapping uses separable first-order conservative interpolation:

```text
F_target = w_x @ F_source @ w_y.T
```

The weights are source-target cell-overlap fractions on the non-uniform grid. This method preserves constant fields and area-weighted averages.

## Physical Model

### Numerical framework

The production simulations use a two-dimensional Marker-in-Cell formulation on a `401 x 309` non-uniform Eulerian grid. Lagrangian markers carry material identity and history, with an initial density of approximately `5 x 5` markers per cell. Finite-difference and interpolation weights on the non-uniform grid are generated with the Fornberg algorithm.

Gravity is

$$
g_x=0,\qquad g_y=9.80665\ \mathrm{m\,s^{-2}}.
$$

### Mechanical equations

At each mechanical solve, pressure and velocity are obtained from the coupled Stokes-continuity system:

$$
\frac{\partial \tau_{xx}}{\partial x}
+\frac{\partial \tau_{xy}}{\partial y}
-\frac{\partial P}{\partial x}
=-\rho g_x,
$$

$$
\frac{\partial \tau_{yy}}{\partial y}
+\frac{\partial \tau_{xy}}{\partial x}
-\frac{\partial P}{\partial y}
=-\rho g_y.
$$

The continuity residual includes volumetric and compressibility contributions:

$$
R_{\mathrm{cont}}
=\nabla\cdot\mathbf{v}
-d_v
+\beta_{\mathrm{comp}}\frac{P-P_0}{\Delta t}.
$$

The global sparse system is solved with Intel MKL PARDISO. All velocity boundaries are free slip. Marker-to-grid viscosity uses arithmetic averaging, and the final viscosity is limited to

$$
10^{18}\leq\eta\leq10^{24}\ \mathrm{Pa\,s}.
$$

### Rheology

The configured mantle, plume, and crustal materials use Ranalli-style visco-plastic parameterizations. Ductile deformation depends on pressure, temperature, and strain rate. Its general power-law form is

$$
\eta_{\mathrm{ductile}}
\propto
\left[
\eta_0\exp\left(\frac{E+PV}{RT}\right)
\right]^{1/n}
\dot{\varepsilon}_{II}^{(1-n)/n},
$$

with material-dependent diffusion-dislocation transitions. Dry mantle materials use $n=3.5$, while plume material uses $n=4.0$.

Brittle yielding follows a Mohr-Coulomb-type relation:

$$
\tau_y=A(\varepsilon_{\mathrm{acc}})
+\lambda B(\varepsilon_{\mathrm{acc}})P,
$$

$$
\eta_{\mathrm{brittle}}
=\frac{\tau_y}{2\dot{\varepsilon}_{II}}.
$$

Cohesion $A$ and friction coefficient $B$ weaken with accumulated strain. When plastic yielding is active, marker strain history evolves as

$$
\varepsilon_{\mathrm{acc}}^{n+1}
=\varepsilon_{\mathrm{acc}}^n
+\Delta t\,\dot{\varepsilon}_{II}.
$$

The weakest active ductile or brittle mechanism controls the effective viscosity. A conditional Peierls mechanism is also included for eligible materials when

$$
T<1473\ \mathrm{K},\qquad
\tau_{II}>10^7\ \mathrm{Pa},\qquad
\dot{\varepsilon}_{II}>0.
$$

### Thermal model

Temperature is advanced with an implicit thermal solve:

$$
\rho C_p\frac{DT}{Dt}
=\nabla\cdot(k\nabla T)+H_r+H_a+H_s.
$$

The configured source terms are radiogenic heating, viscous or shear heating, and simplified adiabatic heating. The latter uses

$$
\frac{\alpha_T T}{C_p}
\left(g_xv_x+g_yv_y\right).
$$

The top and bottom temperatures are fixed at 273 K and 2113 K, respectively. Both lateral thermal boundaries are symmetric.

### Material properties and conditional processes

Material properties are evaluated from the configured material laws, including enabled material and melt corrections. With `densimod=3`, the final density, heat capacity, and thermal conductivity used for marker-to-grid reconstruction are the material-law values. The thermodynamic database participates in the water, hydration, and material-state calculations but does not provide the final reconstructed density field.

The partial-melting routine is evaluated during marker updates and includes pressure- and water-dependent mantle melting. A conditional hydration and material-water subsystem is also enabled and contains an antigorite-stability criterion. These branches depend on the evolving marker state; their activation frequency and spatial extent are not encoded in the released fields.

### Marker transport and time stepping

Markers are advected through the Eulerian velocity field with a four-stage Runge-Kutta scheme. After advection, density, viscosity, heat capacity, thermal conductivity, and related material properties are reconstructed on the Eulerian grid. Consequently, `ro` and `nu` reflect material identity, marker history, constitutive laws, and marker-to-grid reconstruction rather than independent scalar evolution alone.

The solver timestep is adaptive. It is bounded by the configured maximum timestep and by the allowed marker displacement:

$$
\Delta t\leq
\min\left(
\frac{f_{\mathrm{move}}\Delta x_{\mathrm{ref}}}{|v_x|_{\max}},
\frac{f_{\mathrm{move}}\Delta y_{\mathrm{ref}}}{|v_y|_{\max}}
\right).
$$

The thermal solver can reduce it further using

$$
\Delta t_T=\frac{\Delta T_{\max}}{\max|dT/dt|}.
$$

The internal solver timestep is distinct from both the saved output interval and any time normalization used by machine-learning pipelines.

## Usage

```python
import h5py
import numpy as np

path = "geosim_plume2_train_128x128_norm.hdf5"
fields = ["ro", "pr", "tk", "nu", "vx", "vy"]

with h5py.File(path, "r") as f:
    # Read one simulation while keeping the full time axis.
    sample = np.stack([f[key][0] for key in fields], axis=-1)
    time = f["t-coordinate"][0]
    time_unit = f["t-coordinate"].attrs["units"]
    x = f["x-coordinate"][:]
    y = f["y-coordinate"][:]

X, Y = np.meshgrid(x, y, indexing="ij")

print(sample.shape)  # (n_time, H, W, 6)
print(time.shape)    # (n_time,)
print(time_unit)     # Myr
```