Modern 4DVar

Differentiable Physics Models

Can we learn to emulate physical models?

CNRS
MEOM

Criteria

Chaotic. The models need to feature chaotic systems that we see in nature

Coupled. The methods should be able to allow us to train parameterizations. This can manifest itself as a missing term within the PDE itself. It can also manifest itself as a multistate system whereby we only observe one state, e.g., a multilayer PDE.

2D Spatiotemporal Structure.

Scale.

Level I

  • Simple Chaotic ODEs

Lorenz 63

dxdt=σ(y−x)dydt=x(ρ−z)−ydzdt=xy−βz\begin{aligned} \frac{dx}{dt} &= \sigma (y - x) \\ \frac{dy}{dt} &= x (\rho - z) - y \\ \frac{dz}{dt} &= xy - \beta z \end{aligned}
(1)

Good For:

  • Great for prototyping
  • Interpretable
  • Low Engineering Efforts
  • Chaotic Nature

Bad For:

  • No Spatiotemporal Structure
  • Testing High-Dimensional Capabilities
  • Testing Scale
  • Parameterizations

Lorenz 96

dxdt=(xi+1−xi−2)xi−1−xi+F\frac{dx}{dt} = (x_{i+1} - x_{i-2})x_{i-1}-x_i+F
(2)

Good For:

  • Great for prototyping
  • Interpretable
  • Low Engineering Efforts
  • Chaotic Nature
  • 1D Spatiotemporal Structure

Bad For:

  • 2D Spatiotemporal Structure
  • Testing High-Dimensional Capabilities
  • Testing Scale
  • Testing Coupled Parameterizations

Lorenz 96 (2 Level)

dxdt=(xi+1−xi−2)xi−1−xi+F−hcb∑jyjdydt=−bc(yj+2−yj−1)yj+1−cyj−hcbxi\begin{aligned} \frac{dx}{dt} &= (x_{i+1} - x_{i-2})x_{i-1}-x_i + F - \frac{h c}{b} \sum_{j}y_j \\ \frac{dy}{dt} &= -b c (y_{j+2} - y_{j-1})y_{j+1}- c y_j - \frac{h c}{b} x_i \end{aligned}
(3)

Level II

  • Simple Ocean PDEs

Quasi-Geostrophic Equations

∂tqk+(ukqk)x+(vkqk)y=Fk+Dk\partial_t q_k + (u_kq_k)_x + (v_kq_k)_y = F_k + D_k
(4)

SSH is linked to the QG equations via the stream function which we can write this as:

ψ=gf0η\psi = \frac{g}{f_0}\eta
(5)

This adds some additional interpretation how the vorticity term can be interpreted when dealing with the SSH over the globe.

ql=∇Hψl⏟Dynamical+(Aψ)k⏟Thermal+fk⏟Planetaryq_l = \underbrace{\boldsymbol{\nabla}_H \psi_l}_{\text{Dynamical}} + \underbrace{(\mathbf{A}\psi)_k}_{\text{Thermal}} + \underbrace{f_k}_{\text{Planetary}}
(6)

We also have . See [Amraoui et al. (2023)Guillou et al. (2021)] for more information about this term.

Shallow Water Equations

Level III

Simple Stacked Ocean PDEs

These are PDE's that exhibit the spatiotemporal structures that are closer to what we are accustomed to seeing in the real world. These models also allow us to incorporate hidden processes. This is done by having stacked models from level II which try to model processes that we cannot directly observe from satellite observations.

Stacked QG

We are going to be using the formulation that is described in the Q-GCM model. The manual can be found here. We write the multi-layer QG equations in terms of the vorticity term, qq, and the stream function term, ψ\psi. We consider the stream function and the potential vorticity to be NZN_Z stacked isopycnal layers.

∂tqk+(ukqk)x+(vkqk)y=Fk+Dk\partial_t q_k + (u_kq_k)_x + (v_kq_k)_y = F_k + D_k
(7)

where the FkF_k and DkD_k are forcing terms for each layer, kk. The vorticity term is defined as

q=1f0∇H2ψ−f0Aψ+β(y−y0)+D~q = \frac{1}{f_0} \boldsymbol{\nabla}_H^2\psi - f_0\mathbf{A}\psi + \beta(y-y_0)+ \tilde{\mathbf{D}}
(8)

where D~\tilde{D} is the dynamic topography and β\beta is the β\beta-plane approximation. The term that links each of the layers together, A\mathbf{A}, is a tri-diagonal matrix that can be written as

A=[1H1g1′−1H1g2′………−1H2g1′1H1(1g1′+1g2′)−1H2g2′………………………−1Hn−1gn−2′1Hn−1(1gn−2′+1gn−1′)−1Hn−1gn−2′………−1Hngn−1′1Hngn−1′]\mathbf{A} = \begin{bmatrix} \frac{1}{H_1 g_1'} & \frac{-1}{H_1 g_2'} & \ldots & \ldots & \ldots \\ \frac{-1}{H_2 g_1'} & \frac{1}{H_1}\left(\frac{1}{g_1'} + \frac{1}{g_2'} \right) & \frac{-1}{H_2 g_2'} & \ldots & \ldots \\ \ldots & \ldots & \ldots & \ldots & \ldots \\ \ldots & \ldots & \frac{-1}{H_{n-1} g_{n-2}'} & \frac{1}{H_{n-1}}\left(\frac{1}{g_{n-2}'} + \frac{1}{g_{n-1}'} \right) & \frac{-1}{H_{n-1} g_{n-2}'} \\ \ldots & \ldots& \ldots & \frac{-1}{H_n g_{n-1}'} & \frac{1}{H_n g_{n-1}'} \\ \end{bmatrix}
(9)

Stacked SW

References
  1. Amraoui, S., and Didier Auroux, Blum, J., & and, E. C. (2023). Back-and-forth nudging for the quasi-geostrophic ocean dynamics with altimetry: Theoretical convergence study and numerical experiments with the future SWOT observations. Discrete and Continuous Dynamical Systems - S, 16(2), 197–219. 10.3934/dcdss.2022058
  2. Guillou, F. L., Metref, S., Cosme, E., Ubelmann, C., Ballarotta, M., Sommer, J. L., & Verron, J. (2021). Mapping Altimetry in the Forthcoming SWOT Era by Back-and-Forth Nudging a One-Layer Quasigeostrophic Model. Journal of Atmospheric and Oceanic Technology, 38(4), 697–710. 10.1175/jtech-d-20-0104.1
Physical Models
Overview
Physical Models
Ocean GCMs