# ⚛  L1 Principle — Boussinesq Approximation — buoyant flow

**ID:** `L1-182` · **Status:** ⊙ Testnet (genesis catalog)

> **🌐 Domain:** Fluid Dynamics — *Boussinesq buoyancy-driven convection*
> **🎯 Problem class:** nonlinear inverse · **🧮 Solution space:** boussinesq flow T
> **📡 Carrier:** mechanical · **🌫 Noise:** gaussian
> **⚖ Difficulty (δ):** 3 · **⛓ Block:** 41553967

---

## 🧠 1. Introduction

**Boussinesq Approximation — buoyant flow** is a **nonlinear inverse problem** whose unknown lives in **boussinesq flow T** space, within the **Boussinesq buoyancy-driven convection** sub-domain of **Fluid Dynamics**.

Measurements consist of mechanical vibrations or strain signals via a **piv thermocouples** sensing mechanism.

The forward operator applies, in order: L · momentum boussinesq operator; mass-balance: ∂_t ρ + ∇·(ρv) = 0; L · thermal advection operator; detector accumulates flux over the exposure window.

Observations are corrupted by additive Gaussian noise. Conditional stability; mismatch parameters dominate at Omega bounds.

## ⚙ 2. Forward Model

Physical chain: **x** → L · momentum boussinesq → Continuity equation → L · thermal advection → Temporal integration → **y** (detector).

```
y = ∫_t dt `L.thermal_advection` ∂_t ρ + ∇·(ρv) `L.momentum_boussinesq` x + n,    n ~ 𝒩(0, σ²)
```

**Measurement DAG:**

| Primitive | What it does |
|---|---|
| `L.momentum_boussinesq` | L · momentum boussinesq operator |
| `L.continuity` | Mass-balance: ∂_t ρ + ∇·(ρv) = 0 |
| `L.thermal_advection` | L · thermal advection operator |
| `int.temporal` | Detector accumulates flux over the exposure window |

## 🔬 3. Physics Fingerprint

| Property | Value |
|---|---|
| Domain | Fluid Dynamics |
| Sub domain | Boussinesq buoyancy-driven convection |
| Carrier | mechanical |
| Problem class | nonlinear_inverse |
| Solution space | boussinesq_flow_T |
| Noise model | gaussian |
| Integration axis | temporal |
| Difficulty delta | 3 |
| L dag | 3.3 |

## 📡 4. Measurement Model

Conditional stability; mismatch parameters dominate at Omega bounds.

| Metric | Value |
|---|---|
| Metric | PSNR_dB |
| Secondary | SSIM |

## 📏 5. Operating Range (Ω)

**Center problem class:** `boussinesq` · **Forward operator:** `boussinesq_forward`

**Center point:**

| Parameter | Unit | Value |
|---|---|---|
| H | px | 256 |
| W | px | 256 |
| Z | — | 128 |
| Ra | — | 1e+06 |
| Snr db | dB | 30 |
| N times | — | 500 |
| Pr error | — | 0 |
| Bc error | — | 0 |
| Beta error | — | 0 |

**Allowed bounds:**

| Parameter | Unit | Range |
|---|---|---|
| H | px | 32 – 2048 |
| W | px | 32 – 2048 |
| Z | — | 16 – 1024 |
| Ra | — | 1000.0 – 1000000000000.0 |
| Snr db | dB | 0 – 40 |
| N times | — | 50 – 100000 |
| Pr error | — | 0.0 – 0.1 |
| Bc error | — | 0.0 – 0.3 |
| Beta error | — | 0.0 – 0.1 |

## 🎯 6. Tolerance (ε)

**Center tolerance:** 21.0

| Metric | Range |
|---|---|
| Psnr db | 10.0 – 45.0 |

## ⚖ 7. Hardness Function

Hardness scales as **`epsilon_fn`** on **PSNR_dB**, with κ = `400` and δ = `3`.

## 💾 8. Reference Dataset

- **primary** · weight 1.0 · IPFS _(not pinned yet)_

## 9. On-chain Registration

- **Chain hash:** `0x1b4cea7b459c70551bcc14b3dbc7fb45d554692cb471f67e1e34f958934e99cc`
- **Chain tx hash:** `0x0d4d004ab515ad2db329d524d653f012f3d9abf66bb6a38af8eb6e4137f8e773`
- **Chain block:** `41553967`

---

## File Mapping

This bundle consists of: `L1-182.md`, `L1-182.json`.

| File | Role | How to regenerate |
|------|------|-------------------|
| `L1-182.md` | Source of truth — edit this | Human or LLM |
| `L1-182.json` | Structured metadata for the registry | LLM regenerates from the sections above |

**Prompt for your LLM after editing this Markdown:**

> Read the attached Markdown. Regenerate the sibling `.json` so every field matches.
> Preserve the schema documented in the rows above.
> Output each file in its own fenced code block tagged with the filename.
> Output only the JSON object.

_This Markdown was auto-synthesized from the catalog row for `L1-182`._
_Edit it, regenerate the JSON, and submit at [/submit](/submit) to claim the artifact._