eigenvectors

Module: fridom.nonhydro.grid.cartesian.eigenvectors

Functions

omega(mset, s, k[, use_discrete])	Return eigenvalues of the System matrix.
vec_p(mset, s[, use_discrete])	Return the projection vectors of the System matrix.
vec_q(mset, s[, use_discrete])	Return eigenvectors of the System matrix and the divergence vector.

Eigenvalues and eigenvectors of the system matrix of the nonhydrostatic model.

Continuous Case

The System Matrix

We start from the scaled linearized nonhydrostatic equations in spectral space:

\[\partial_t u = f v - i k_x p\]

\[\partial_t v = -f u - i k_y p\]

\[\partial_t w = \delta^{-2} b - \delta^{-2} i k_z p\]

\[\partial_t b = -wN^2\]

and the diagnostic pressure equation obtained by taking the divergence of the momentum equations:

\[0 = if (k_x v - k_y u) + \delta^{-2} i k_z b + ( k_x^2 + k_y^2 + \delta^{-2} k_z^2) p\]

Solving the diagnostic pressure equation for the pressure and substituting it back into the momentum equations, we obtain the following system of equations:

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

with

\[\begin{split}\boldsymbol{z} = \begin{pmatrix} u \\ v \\ w \\ b \end{pmatrix} \quad, \quad \mathbf{A} = \frac{-1}{\delta^2 k^2} \begin{pmatrix} -if\delta^2k_x k_y & -if\left( \delta^2k_y^2 + k_z^2 \right) & 0 & i k_x k_z \\ if \left( \delta^2 k_x^2 + k_z^2 \right) & if\delta^2k_x k_y & 0 & i k_y k_z \\ -i f k_y k_z & i f k_x k_z & 0 & -i k_h^2 \\ 0 & 0 & iN^2\delta^2 k^2 & 0 \end{pmatrix}\end{split}\]

with the wavenumbers:

\[k^2 = k_h^2 + \delta^{-2} k_z^2 \quad \text{and} \quad k_h^2 = k_x^2 + k_y^2\]

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:

\[\omega^\pm = \pm \sqrt{\frac{f^2 k_z^2 + N^2 k_h^2}{\delta^2 k^2}}\]

Eigenvectors

For the eigenvectors we have to separately consider the case of purely vertical, e.g. \(k_x = k_y = 0\), and the general case of nonzero horizontal wavenumbers. For the purely vertical case, the eigenvectors are:

\[\begin{split}\boldsymbol{q^0} = \begin{pmatrix} 0 \\ 0 \\ 0 \\ 1 \end{pmatrix} \quad, \quad \boldsymbol{q^\pm} = \begin{pmatrix} \mp i \\ 1 \\ 0 \\ 0 \end{pmatrix}\end{split}\]

For the general case of nonzero horizontal wavenumbers, the eigenvectors are:

\[\begin{split}\boldsymbol{q^0} = \begin{pmatrix} -k_y \\ k_x \\ 0 \\ fk_z \end{pmatrix} \quad, \quad \boldsymbol{q^\pm} = \begin{pmatrix} k_z ( -i \omega^\pm k_x + f k_y) \\ k_z ( -i \omega^\pm k_y - f k_x) \\ i\omega^\pm k_h^2 \\ N^2 k_h^2 \end{pmatrix}\end{split}\]

Projection Vectors

Projection vectors should satisfy

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

where the star denotes the hermitian transposed. Add divergent vector \(\boldsymbol{q^d} = \begin{pmatrix} k_x & k_y & k_z & 0 \end{pmatrix}\) such that \((\boldsymbol{q^0}, \boldsymbol{q^+}, \boldsymbol{q^-}, \boldsymbol{q^d})\) form a basis.

For the purely vertical case, the projection vectors are identical to the eigenvectors: \(\boldsymbol{p^s} = \boldsymbol{q^s}\). For the general case of nonzero horizontal wavenumbers, the projection vectors are:

\[\begin{split}\boldsymbol{p^0} = \begin{pmatrix} -N^2 k_y \\ N^2 k_x \\ 0 \\ f k_z \end{pmatrix} \quad, \quad \boldsymbol{p^\pm} = \begin{pmatrix} k_z (-i \omega^\pm k_x + f \gamma k_y) \\ k_z (-i \omega^\pm k_y - f \gamma k_x) \\ i\omega^\pm k_h^2 \\ \gamma k_h^2 \end{pmatrix}\end{split}\]

with

\[\gamma = \frac{k_h^2 + k_z^2}{\delta^2 k^2}\]

The projection vectors are normalized such that \({\boldsymbol{p^s}}^* \cdot \boldsymbol{p^s} = 1\).

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 of equations in spectral space can be written as:

\[\partial_t u = f \hat{1}_x^+ \hat{1}_y^- v - i \hat{k}_x^+ p\]

\[\partial_t v = -f \hat{1}_x^- \hat{1}_y^+ u - i \hat{k}_y^+ p\]

\[\partial_t w = \delta^{-2} \hat{1}_z^+ b - \delta^{-2} i \hat{k}_z^+ p\]

\[\partial_t b = - \hat{1}_z^- w N^2\]

and the diagnostic pressure equation:

\[0 = i (\hat{1}_x^+ \hat{1}_y^- \hat{k}_x^- v - \hat{1}_x^- \hat{1}_y^+ \hat{k}_y^- u) + \delta^{-2} i \hat{1}_z^+ \hat{k}_z^- b + (\hat{k}_x^2 + \hat{k}_y^2 + \delta^{-2} \hat{k}_z^2) p\]

System Matrix

Following the same procedure as in the continuous case, we obtain the discrete system matrix:

\[\begin{split}\mathbf{A} = \frac{1}{\hat{k}^2} \begin{pmatrix} -i \delta^2 \hat{1}_x^- \hat{1}_y^+ \hat{k}_x^+ \hat{k}_y^- f & -i f \hat{1}_x^+ \hat{1}_y^- \left( \delta^2 \hat{k}_y^2 + \hat{k}_z^2 \right ) & 0 & i \hat{1}_z^+ \hat{k}_x^+ \hat{k}_z^- \\ i f \hat{1}_x^- \hat{1}_y^+ \left( \delta^2 \hat{k}_x^2 + \hat{k}_z^2 \right) & i \delta^2 \hat{1}_x^+ \hat{1}_y^- \hat{k}_x^- \hat{k}_y^+ f & 0 & i \hat{1}_z^+ \hat{k}_y^+ \hat{k}_z^- \\ -i \hat{1}_x^- \hat{1}_y^+ \hat{k}_y^- \hat{k}_z^+ f & i \hat{1}_x^+ \hat{1}_y^- \hat{k}_x^- \hat{k}_z^+ & 0 & -i \hat{1}_z^+ \hat{k}_h^2 \\ 0 & 0 & i N^2 \hat{1}_x^- \hat{k}^2 & 0 \end{pmatrix}\end{split}\]

with

\[\hat{k}^2 = \hat{k}_h^2 + \delta^{-2} \hat{k}_z^2 \quad \text{and} \quad \hat{k}_h^2 = \hat{k}_x^2 + \hat{k}_y^2 \quad \text{and} \quad \hat{k}_x^2 = \hat{k}_x^+ \hat{k}_x^-\]

Eigenvalues

\[\omega^0 = 0 \quad \text{and} \quad \omega^\pm = \sqrt{\frac{\hat{1}_x^2 \hat{1}_y^2 f^2 \hat{k}_z^2 + \hat{1}_z^2 N^2 \hat{k}_h^2}{\hat{k}^2}}\]

with

\[\hat{1}_x^2 = \hat{1}_x^+ \hat{1}_x^-\]

Eigenvectors

For the purely vertical case, the discrete eigenvectors are identical to the continuous eigenvectors. For the general case of nonzero horizontal wavenumbers, the discrete eigenvectors are:

\[\begin{split}\boldsymbol{q^0} = \begin{pmatrix} - \hat{1}_x^+ \hat{1}_y^- \hat{1}_z^+ \hat{k}_y^+ \\ \hat{1}_x^- \hat{1}_y^+ \hat{1}_z^+ \hat{k}_x^+ \\ 0 \\ \hat{1}_x^2 \hat{1}_y^2 f \hat{k}_z^+ \end{pmatrix} \quad \text{and} \quad \boldsymbol{q}^\pm = \begin{pmatrix} \hat{k}_z^- ( - i \omega^\pm \hat{k}_x^+ + \hat{1}_x^+ \hat{1}_y^- f \hat{k}_y^+) \\ \hat{k}_z^- ( - i \omega^\pm \hat{k}_y^+ - \hat{1}_x^- \hat{1}_y^+ f \hat{k}_x^+) \\ i \omega^\pm \hat{k}_h^2 \\ \hat{1}_z^- N^2 \hat{k}_h^2 \end{pmatrix}\end{split}\]

The divergent vector is given by \(\boldsymbol{q^d} = (\hat{k}_x^+, \hat{k}_y^+, \hat{k}_z^+, 0)\). All eigenvectors are orthogonal to the divergent vector. Hence no eigenvector project onto divergent velocity fields.

Projection Vectors

Similar to the continuous case, the projection vectors of the purely vertical case are identical to the eigenvectors. Hence, we only show the projection vectors for the general case of nonzero horizontal wavenumbers:

\[\begin{split}\boldsymbol{p^0} = \begin{pmatrix} - \hat{1}_x^+ \hat{1}_y^- \hat{1}_z^+ N^2 \hat{k}_y^+ \\ \hat{1}_x^- \hat{1}_y^+ \hat{1}_z^+ N^2 \hat{k}_x^+ \\ 0 \\ \hat{1}_x^2 \hat{1}_y^2 f \hat{k}_z^+ \end{pmatrix} ~~, \quad \text{and} \quad \boldsymbol{p^\pm} = \begin{pmatrix} \hat{k}_z^- ( -i \omega^\pm \hat{k}_x^+ + \hat{1}_x^+ \hat{1}_y^- f \hat{\gamma} \hat{k}_y^+) \\ \hat{k}_z^- ( -i \omega^\pm \hat{k}_y^+ - \hat{1}_x^- \hat{1}_y^+ f \hat{\gamma} \hat{k}_x^+) \\ i \omega^\pm \hat{k}_h^2 \\ \hat{1}_z^- \hat{\gamma} \hat{k}_h^2 \end{pmatrix}\end{split}\]

with

\[\hat{\gamma} = \frac{\hat{k}_h^2 + \hat{k}_z^2}{\delta^2 \hat{k}^2}\]

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

Return eigenvalues of the System matrix.

Parameters

msetnh.ModelSettings

The model settings.

sint

The mode of the eigenvector. (0, 1, -1)

ktuple[float] | tuple[ndarray]

The wavenumbers.

use_discretebool (default: False)

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

fridom.nonhydro.grid.cartesian.eigenvectors.vec_q(mset: ModelSettings, s: int | str, use_discrete: bool = True) → None

Return eigenvectors of the System matrix and the divergence vector.

Parameters

msetModelSettings

The model settings.

sint | str

The mode of the eigenvector. 0 => geostrophic, “d” => divergent, 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.

Description

For the continuous case, and \(k_x = k_y = 0\), the eigenvectors are:

\[\begin{split}\boldsymbol{q^0} = \begin{pmatrix} 0 \\ 0 \\ 0 \\ 1 \end{pmatrix} \quad, \quad \boldsymbol{q^\pm} = \begin{pmatrix} \mp i \\ 1 \\ 0 \\ 0 \end{pmatrix} \quad, \quad \boldsymbol{q^d} = \begin{pmatrix} k_x \\ k_y \\ k_z \\ 0 \end{pmatrix}\end{split}\]

For the general case of nonzero horizontal wavenumbers, the eigenvectors are:

\[\begin{split}\boldsymbol{q^0} = \begin{pmatrix} -k_y \\ k_x \\ 0 \\ fk_z \end{pmatrix} \quad, \quad \boldsymbol{q^\pm} = \begin{pmatrix} k_z ( -i \omega^\pm k_x + f k_y) \\ k_z ( -i \omega^\pm k_y - f k_x) \\ i\omega^\pm k_h^2 \\ N^2 k_h^2 \end{pmatrix} \quad, \quad \boldsymbol{q^d} = \begin{pmatrix} k_x \\ k_y \\ k_z \\ 0 \end{pmatrix}\end{split}\]

The discrete projection vector is given in the docstring of the eigenvectors.

fridom.nonhydro.grid.cartesian.eigenvectors.vec_p(mset: ModelSettings, s: int | str, use_discrete: bool = True) → None

Return the projection vectors of the System matrix.

Parameters

msetModelSettings

The model settings.

sint

The mode of the eigenvector. 0 => geostrophic, “d” => divergent, 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.

Description

For the continuous case, and \(k_x = k_y = 0\), the projection vectors are:

\[\begin{split}\boldsymbol{p^0} = \begin{pmatrix} 0 \\ 0 \\ 0 \\ 1 \end{pmatrix} \quad, \quad \boldsymbol{p^\pm} = \begin{pmatrix} \mp i \\ 1 \\ 0 \\ 0 \end{pmatrix} \quad, \quad \boldsymbol{p^d} = \begin{pmatrix} k_x \\ k_y \\ k_z \\ 0 \end{pmatrix}\end{split}\]

For the general case of nonzero horizontal wavenumbers, the projection vectors are:

\[\begin{split}\boldsymbol{p^0} = \begin{pmatrix} -N^2 k_y \\ N^2 k_x \\ 0 \\ f k_z \end{pmatrix} \quad, \quad \boldsymbol{p^\pm} = \begin{pmatrix} k_z ( -i \omega^\pm k_x + f \gamma k_y) \\ k_z ( -i \omega^\pm k_y - f \gamma k_x) \\ i\omega^\pm k_h^2 \\ \gamma k_h^2 \end{pmatrix} \quad, \quad \boldsymbol{p^d} = \begin{pmatrix} k_x \\ k_y \\ k_z \\ 0 \end{pmatrix}\end{split}\]

The discrete projection vector is given in the docstring of the eigenvectors.