eigenvectors

Module: fridom.shallowwater.grid.cartesian.eigenvectors

Functions

omega(mset, s, k[, use_discrete])	The eigenvalues of the System matrix.
vec_p(mset, s[, use_discrete])	The projection vectors of the system matrix.
vec_q(mset, s[, use_discrete])	The eigenvectors of the system matrix

Eigenvalues and eigenvectors of the system matrix of the shallow water model.

Continuous Case

The System Matrix

Linearizing the shallow water equations (i.e. taking the limit \(Ro \rightarrow 0\)) and performing a spatial Fourier transform, we obtain the following system:

\[\partial_t \boldsymbol{z} = -i \mathbf{A} \cdot \boldsymbol{z}\]

with

\[\begin{split}\boldsymbol{z} = \begin{pmatrix} u \\ v \\ p \end{pmatrix} \quad, \quad \mathbf{A} = \begin{pmatrix} 0 & if & k_x \\ -if & 0 & k_y \\ c^2 k_x & c^2 k_y & 0 \end{pmatrix} \end{split}\]

Eigenvalues

The system matrix has three eigenvalues. One eigenvalue is zero, corresponding to the geostrophic mode:

\[\omega^0 = 0\]

The other two eigenvalues correspond to the inertial-gravity wave modes:

\[\omega^\pm = \pm \sqrt{f^2 + c^2 (k_x^2 + k_y^2)}\]

Eigenvectors

For \(|\boldsymbol{k}| > 0\) the eigenvectors that correspond to the mode \(s=0,+,-\) are given by:

\[\begin{split}\boldsymbol{q^s} = \begin{pmatrix} \omega^s k_x - i f k_y \\ \omega^s k_y + i f k_x \\ f^2 - (\omega^s)^2 \end{pmatrix}\end{split}\]

For \(|\boldsymbol{k}| = 0\) we obtain the eigenvectors for the inertial modes:

\[\begin{split}\boldsymbol{q^s} = \begin{pmatrix} -is \\ s^2 \\ 1 - s^2 \end{pmatrix}\end{split}\]

Projection Vectors

Projection vectors should satisfy

\[{\boldsymbol{p^s}}^* \cdot \boldsymbol{q^{s'}} = \delta_{s,s'}\]

where the star denotes the hermitian transposed. Solving this equation for \(|\boldsymbol{k}| > 0\) yields:

\[\begin{split}\boldsymbol{p^s} = \begin{pmatrix} q^s_x \\ q^s_y \\ c^{-2} q^s_z \end{pmatrix}\end{split}\]

where \(q^s_i\) denotes the \(i\)-th component of the eigenvector \(\boldsymbol{q^s}\).

For the inertial modes (i.e. \(|\boldsymbol{k}| = 0\)), the projection vectors are equal to the eigenvectors. All projection vectors are normalized such that \({\boldsymbol{p^s}}^* \cdot \boldsymbol{p^s} = 1\) holds.

Discrete Case

Lets define the forward and backward finite difference operators as:

\[\delta_x^+ u = \frac{u(x + \Delta x) - u(x)}{\Delta x} \quad, \quad \delta_x^- u = \frac{u(x) - u(x - \Delta x)}{\Delta x}\]

A fourier transform yields the discrete spectral operators:

\[\delta_x^+ u \rightarrow \frac{e^{ik_x \Delta x} - 1}{\Delta x} = i \hat{k}_x^+ u \quad, \quad \delta_x^- u \rightarrow \frac{1 - e^{-ik_x \Delta x}}{\Delta x} = i \hat{k}_x^- u\]

Similarly, we define the forward and backward linear interpolation operators as:

\[\overline{u}^{x+} = \frac{u(x + \Delta x) + u(x)}{2} \quad, \quad \overline{u}^{x-} = \frac{u(x) + u(x - \Delta x)}{2}\]

A fourier transform yields the discrete spectral operators:

\[\overline{u}^{x+} \rightarrow \frac{e^{ik_x \Delta x} + 1}{2} = \hat{1}_x^+ u \quad, \quad \overline{u}^{x-} \rightarrow \frac{1 + e^{-ik_x \Delta x}}{2} = \hat{1}_x^- u\]

Using these discretization operators, the discrete linear system matrix is:

\[\begin{split}\mathbf{A} = \begin{pmatrix} 0 & i f \hat{1}_x^+ \hat{1}_y^- & \hat{k}_x^+ \\ -i f \hat{1}_x^- \hat{1}_y^+ & 0 & \hat{k}_y^+ \\ c^2 \hat{k}_x^- & c^2 \hat{k}_y^- & 0 \end{pmatrix}\end{split}\]

Eigenvalues

The eigenvalues of the discrete system matrix are:

\[\omega^0 = 0 \quad \text{and} \quad \omega^\pm = \sqrt{\hat{1}_x^2 \hat{1}_y^2 f^2 + c^2 (\hat{k}_x^2 + \hat{k}_y^2)}\]

with

\[\hat{1}_x^2 = \hat{1}_x^+ \hat{1}_x^- \quad \text{and} \quad \hat{k}_x^2 = \hat{k}_x^+ \hat{k}_x^-\]

Eigenvectors

The eigenvectors \(\boldsymbol{q^s}\) for \(|\boldsymbol{k}| > 0\) are:

\[\begin{split}\boldsymbol{q^s} = \begin{pmatrix} \omega^s \hat{k}_x^+ - i \hat{1}_x^+ \hat{1}_y^- f \hat{k}_y^+ \\ \omega^s \hat{k}_y^+ + i \hat{1}_x^- \hat{1}_y^+ f \hat{k}_x^+ \\ \hat{1}_x^2 \hat{1}_y^2 f^2 - (\omega^s)^2 \end{pmatrix}\end{split}\]

For \(|\boldsymbol{k}| = 0\) the eigenvectors are identical to the continuous case. The projection vectors are the same as in the continuous case, but by using the discrete eigenvectors.

fridom.shallowwater.grid.cartesian.eigenvectors.omega(mset: ModelSettings, s: int, k: tuple[float] | tuple[ndarray], use_discrete: bool = False) → ndarray

The eigenvalues of the System matrix.

Computes the continuous or discrete eigenvalues as described in eigenvectors.

Parameters

msetModelSettings

The model settings.

sint

The mode of the eigenvector. 0 => geostrophic, 1 => positive inertial-gravity, -1 => negative inertial-gravity

ktuple[float] | tuple[ndarray]

The wavenumber tuple \((k_x, k_y)\).

use_discretebool (default: True)

If True, the discrete eigenvectors are returned. Otherwise, the continuous eigenvectors are returned.

Returns

ndarray

The eigenvalues of the System matrix.

fridom.shallowwater.grid.cartesian.eigenvectors.vec_q(mset: ModelSettings, s: int, use_discrete=True) → State

The eigenvectors of the system matrix

Computes the continuous or discrete eigenvectors as described in eigenvectors.

Parameters

msetModelSettings

The model settings.

sint

The mode of the eigenvector. 0 => geostrophic, 1 => positive inertial-gravity, -1 => negative inertial-gravity

use_discretebool (default: True)

If True, the discrete eigenvectors are returned. Otherwise, the continuous eigenvectors are returned.

Returns

State

The eigenvectors of the system matrix.

fridom.shallowwater.grid.cartesian.eigenvectors.vec_p(mset: ModelSettings, s: int, use_discrete: bool = True) → State

The projection vectors of the system matrix.

Computes the continuous or discrete projection vectors as described in eigenvectors.

Parameters

msetModelSettings

The model settings.

sint

The mode of the eigenvector. 0 => geostrophic, 1 => positive inertial-gravity, -1 => negative inertial-gravity

use_discretebool (default: True)

If True, the discrete eigenvectors are returned. Otherwise, the continuous eigenvectors are returned.

Returns

State

The projection vectors of the system matrix.