QuasiGeostrophic Equations

Quasi-Geostrophic Equations

J. Emmanuel Johnson
CNRS
MEOM

We demonstrate a few case studies of QG models under different parameter regimes that have appeared within the literature.

Note:See this page for more information about the formulations.

Examples

We have various examples which showcase different simulations of various types of the QG model under different parameter configurations.

Idealized QG (TODO)

We can define the completely idealized 1 layer QG model given as

∂tω+det⁡J(ψ,ω)=−μω+ν∇2ω−β∂xψ+Fω=∇ψ\begin{aligned} \partial_t \omega &+ \det\boldsymbol{J}(\psi,\omega) = - \mu\omega + \nu\boldsymbol{\nabla}^2\omega - \beta\partial_x\psi + F \\ \omega &= \boldsymbol{\nabla}\psi \end{aligned}
(1)

This is a very idealistic case with a periodic domain. We showcase how we can have simulations similar to [Frezat et al., 2022] using the above formulation. We demonstrated the different flow regimes which depend upon the parameters, i.e. decay flow, forced flow and β\beta-plane flow.

Realistic Idealized QG (TODO)

We can define a simple 1 layer QG model

∂tq+u⃗⋅q=1ρH∇H×τ⃗−κ∇Hψ+a4ΔH2−β∂xψq=∇ψ\begin{aligned} \partial_t q &+ \vec{\boldsymbol{u}}\cdot q = \frac{1}{\rho H}\boldsymbol{\nabla}_H\times\vec{\boldsymbol{\tau}} - \kappa\boldsymbol{\nabla}_H\psi + a_4\boldsymbol{\Delta}_H^2 - \beta\partial_x\psi \\ q &= \boldsymbol{\nabla}\psi \end{aligned}
(2)

which starts to encapsulate real parameter regimes we can see in nature. Inspired by this course, we showcase how we can have "real-like" simulations using the above formulation.

SSH Free-Run QG (TODO)

Jupyter Notebook

The relationship between SSH and the streamfunction is given by the following relationship.

ψ=gf0η\psi = \frac{g}{f_0}\eta
(3)

So we have a way to approximately model the trajectory of SSH using the QG model. Inspired by [Guillou et al., 2021], we demonstrate a simple free-run simulation using the configurations.

Idealized 2-Layer QG (TODO)

This is a simpler realization of the stacked QG model. In [Laloyaux et al., 2020], they were exploring the effectiveness of a data assimilation method (4DVar) when applied to observation data. They used a simple 2-Layer QG model with the stream function ψk\psi_k and the potential vorticity, qkq_k, as shown in equation (5). However, they have a slight different linking term with no extra forcing or dissipation terms.

q1=∇2ψ1−F1(ψ1−ψ2)+βyq2=∇2ψ2−F2(ψ2−ψ1)+βy+Rs\begin{aligned} q_1 &= \nabla^2\psi_1 - F_1(\psi_1 - \psi_2) + \beta y \\ q_2 &= \nabla^2\psi_2 - F_2(\psi_2 - \psi_1) + \beta y + R_s \end{aligned}
(4)

We showcase how we can recreate a free-run simulation from this configuration.

Idealized Stacked QG (TODO)

Jupyter Notebook

The culmination of the above examples is the stacked QG model given by:

∂tq⃗+u⃗⋅∇q⃗=BF+Dq⃗=(∇2−f02M)ψ+βy\begin{aligned} \partial_t \vec{\boldsymbol{q}} &+ \vec{\boldsymbol{u}}\cdot\boldsymbol{\nabla} \vec{\boldsymbol{q}} = \mathbf{BF} + \mathbf{D} \\ \vec{\boldsymbol{q}} &= \left(\boldsymbol{\nabla}^2 - f_0^2\mathbf{M}\right)\psi + \beta y \end{aligned}
(5)

This is an example that implements an example for the stacked QG model. The underlying mathematics is based on the Q-GCM numerical model. However, the configurations are based upon the papers [Thiry et al. (2022)Thiry et al. (2023)].

References
  1. Frezat, H., Sommer, J. L., Fablet, R., Balarac, G., & Lguensat, R. (2022). A posteriori learning for quasi-geostrophic turbulence parametrization. 10.48550/ARXIV.2204.03911
  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
  3. Laloyaux, P., Bonavita, M., Chrust, M., & Gürol, S. (2020). Exploring the potential and limitations of weak-constraint 4D-Var. Quarterly Journal of the Royal Meteorological Society, 146(733), 4067–4082. 10.1002/qj.3891
  4. Thiry, L., Li, L., & Mémin, E. (2022). Modified (hyper-)viscosity for coarse-resolution ocean models. 10.48550/ARXIV.2204.13914
  5. Thiry, L., Li, L., Roullet, G., & Mémin, E. (2023). Finite-volume discretization of the quasi-geostrophic equations with implicit dissipation. 10.22541/essoar.167397445.54992823/v1
12 Steps to Navier-Stokes
2D Poisson's Equation
Quasi-Geostrophic Model
Formulation