Nonlinear equations The governing equations: \( \frac{\partial}{\partial t} \nabla^2 \psi - \text{Pr} \nabla^2 \psi + \text{Ra} \, \text{Pr} \, \frac{\partial \theta}{\partial x} = \mathcal{J}(\nabla^2 \psi, \psi) \) \( \frac{\partial \theta}{\partial t} - \nabla^2 \theta + \frac{\partial \psi}{\partial x} = \mathcal{J}(\theta, \psi) \)
where the nonlinear terms are expressed as a Jacobian operator \(\mathcal{J}\) given by \( \mathcal{J}(f, g) = \frac{\partial f}{\partial x} \frac{\partial g}{\partial y} - \frac{\partial g}{\partial x} \frac{\partial f}{\partial y} \)
Assume free-slip boundary conditions as before

Truncated Galerkin expansion the general solution can be expressed as an infinite Fourier series. Consider a truncated Galerkin expansion \( \psi(x, y, t) = a(t) \sin(\pi y) \sin(k\pi x) + \cdots \) \( \theta(x, y, t) = b(t) \sin(\pi y) \cos(k\pi x) + c(t) \sin(2\pi y) + \cdots \)

• \(a(t)\) and \(b(t)\) correspond to the convection rolls with wavenumber \(k\) in the \(x\)-direction
• The term \(c(t)\) describes the modification of the mean temperature profile due to convection

Obtain the following low-order model \[ \frac{da}{dt} = \underbrace{-\text{Pr} \, \pi^2 (1 - k^2) a}_{{\text{→ } -\sigma x}} \ \underbrace{- \frac{k \pi}{\pi^2 (1 + k^2)} \text{Pr} \, \text{Ra} \, b}_{{\text{→ } +\sigma y}} \] \[ \frac{db}{dt} = \underbrace{-k \pi a}_{{\text{→ } +x}} \ \underbrace{- \pi^2(1 + k^2) b}_{{\text{→ } -y}} \ \underbrace{- k \pi^2 a c}_{{\text{→ } -xz}} \] \[ \frac{dc}{dt} = \underbrace{\frac{1}{2} k \pi^2 a b}_{{\text{→ } +xy}} \, \underbrace{- 4 \pi^2 c}_{{\text{→ } -\beta z}} \] This low-dimensional model of thermal convection is a rescaled version of the one originally introduced by Lorenz in 1963 \(\sigma = \text{Pr}\) \(\rho = \frac{\text{Ra}}{\text{Ra}_c}\) \(\beta = \frac{2\pi^2}{\pi^2 + k^2}\)