Table of Contents

QG Equation

$$\begin{aligned} \mathcal{L}u &\equiv \left( \nabla^2_h + \frac{\partial^2}{\partial p^2} \right) u = f \quad x \in \Omega \\ u &= g \quad x \in \partial\Omega \end{aligned}$$

where Laplacian in spherical coordinates $(\theta,\phi)$ (polar (colatitudinal), azimuthal (longitudinal)) is defined as:

$$\nabla^2_h = \frac{1}{r^2 \sin^2 \phi} \frac{\partial^2}{\partial \theta^2} + \frac{1}{r^2 \sin \phi} \frac{\partial}{\partial \phi} \!\Bigl( \sin \phi \frac{\partial}{\partial \phi} \Bigr)$$

Note that $\theta = \pi/2 - \Phi$ and $\phi = \lambda$, where $(\Phi,\lambda)$ = (latitude, longitude).

Spherical Harmonic

The operator $\nabla^2_h$ has eigenfunctions (Spherical Harmonic) satisfying:

$$\nabla^2_h Y_l = -\lambda_l Y_l$$

Assuming $Y = \Theta(\theta)\Phi(\phi)$ with $2\pi$ periodicity, we find $\lambda_l = l(l+1)$ for $l=0,1,\cdots$. For each $l$:

$$\begin{aligned} \frac{d^2\Theta}{d\theta^2}\frac{1}{\Theta} &= -m^2, \quad m=-l,\cdots,l \\ \text{and } \Phi(\phi) &\text{ satisfies the } \textit{associated Legendre differential equation} \end{aligned}$$

The normalized solutions are:

$$Y_l^m(\theta, \phi) = \sqrt{\frac{2l + 1}{4\pi} \frac{(l - m)!}{(l + m)!}} \, P_l^m(\cos \theta)\, e^{i m \phi}$$

with orthonormality:

$$\langle Y_l^m,Y_{l'}^{m'} \rangle_w = \delta_{m,m'}\,\delta_{l,l'}$$

where the weighted inner product is:

$$\langle f,g \rangle_w \equiv \int_{[0,2\pi]\times[0,\pi]} f \, \bar{g} \,\sin(\theta)d\theta d\phi$$

Spectral Method

Expanding solutions:

$$\begin{aligned} u &= \sum_{l=0}^\infty \sum_{m=-l}^l A_l^m(p)Y_l^m(\theta,\phi) \\ f &= \sum_{l=0}^\infty \sum_{m=-l}^l F_l^m(p)Y_l^m(\theta,\phi) \\ \text{where } F_l^m(p) &= \frac{\langle f,Y_l^m \rangle_w}{\langle Y_l^m,Y_l^m \rangle_w} \end{aligned}$$

Substituting into QG equation yields ODE system:

$$\frac{d^2}{dp^2} A_l^m(p) - l(l+1)A_l^m(p) = F_l^m(p)$$

with boundary conditions:

$$u = g = \sum_{l=0}^\infty \sum_{m=-l}^l G_l^m(p)Y_l^m(\theta,\phi) \quad x \in \partial \Omega$$

Remark

1. Numerical implementation requires careful handling of inner product integrals
2. Real-world QG equations involve scale factors in $\mathcal{L}$ and discrete spherical harmonic transforms
3. This spectral approach converts PDEs into solvable ODE systems