Layered quasigeostrophic model¶
\[\,{q_{i}}_t + \mathsf{J}\left(\psi_i\,, q_i\right) + U_i {q_i}_x + V_i {q_i}_y + {Q_i}_y {\psi_i}_x - {Q_i}_x {\psi_i}_y= \text{ssd} - r_{ek} \delta_{i\textsf{N}} \nabla^2 \psi_i\,, \qquad i = 1,\textsf{N}\,,\]
where
\[ {q_i} = \nabla^2\psi_i + \frac{f_0^2}{H_i} \left(\frac{\psi_{i-1}-\psi_i}{g'_{i-1}} - \frac{\psi_{i}-\psi_{i+1}}{g'_{i}}\right)\,, \qquad i = 2,\textsf{N}-1\,,\]\[and\]
\[{q_1} = \nabla^2\psi_1 + \frac{f_0^2}{H_1} \left(\frac{\psi_{2}-\psi_1}{g'_{1}}\right)\,, \qquad i =1\,,\]
\[ {q_\textsf{N}} = \nabla^2\psi_\textsf{N} + \frac{f_0^2}{H_\textsf{N}} \left(\frac{\psi_{\textsf{N}-1}-\psi_\textsf{N}}{g'_{\textsf{N}}}\right) + \frac{f_0}{H_\textsf{N}}h_b\,, \qquad i =\textsf{N}\,,\]\[where the reduced gravity, or buoyancy jump, is\]
\[g'_i \equiv g \frac{\rho_{i+1}-\rho_i}{\rho_i}\,.\]
The inversion relationship in spectral space is
\[\hat{q}_i = \underbrace{\left(\textsf{S} - \kappa^2 \textsf{I}\right)}_{\equiv\textsf{A}}\hat{\psi}_i\,,\]
where the “stretching matrix” is
\[\begin{split}\textsf{S} \equiv f_0^2
\begin{bmatrix}
-\frac{1}{g'_1 H_1} & \frac{1}{g'_1 H_1} & 0 & \dots& \\
0 & & & & & &\\
\vdots & \ddots& \ddots &\ddots & & & &\\
& \frac{1}{g'_{i-1} H_i} & -\left(\frac{1}{g'_{i-1} H_i} + \frac{1}{g'_{i} H_i}\right)& \frac{1}{g'_{i} H_i} \\
& \ddots& \ddots &\ddots & & & &\\
& & & & & \\
& \dots & 0 & \frac{1}{ g'_{\textsf{N}-1} H_\textsf{N}} & -\frac{1}{g'_{\textsf{N}-1} H_\textsf{N}}
\end{bmatrix}\end{split}\]
The forced-dissipative equations in Fourier space are
\[\,{\hat{q}_{i}}_t + ik\,{\hat{\psi}_i} {Q_y} - il\,{\hat{\psi}_i} {Q_x} + (i k U_i + i l V_i) \hat{q}_i+ \mathsf{\hat{J}}\left(\psi_i\,, q_i + \delta_{i\textsf{N}} \tfrac{f_0}{H_\textsf{N}} h_b \right) = \text{ssd} \,, \qquad i = 1,\textsf{N}\,,\]
where the mean potential vorticy gradients are
\[ \textsf{Q}_x = \textsf{S}\textsf{V}\, \qquad \textsf{Q}_y = \beta\,\textsf{I} - \textsf{S}\textsf{U}\,\,,\]\[where the background velocity is\]
\(\vec{\textsf{V}}(z) = \left(\textsf{U},\textsf{V}\right)\).
Energy balance¶
The equation for the energy spectrum,
\[E(k,l) \equiv \underbrace{\frac{1}{2 H}\sum_{i=1}^{\mathsf{N}} H_i \kappa^2 |\hat{\psi}_i|^2}_{\text{kinetic energy}} \,\,\,\,+ \,\,\,\,\,\, \frac{1}{2 H}\underbrace{\sum_{i=1}^{\mathsf{N-1}} \frac{f_0^2}{g'_i}|\hat{\psi}_{i}- \hat{\psi}_{i+1}|^2}_{\text{potential energy}}\,\,\,\,,\]
is
\[\frac{d}{dt} E(k,l) = \underbrace{\frac{1}{H}\sum_{i=1}^{\mathsf{N}} H_i \text{Re}[\hat{\psi}_i^\star \hat{\mathsf{J}}(\psi_i,\nabla^2\psi_i)]}_{I} +
\underbrace{\frac{1}{H}\sum_{i=1}^{\mathsf{N}} H_i\text{Re}[\hat{\psi}_i^\star \hat{\mathsf{J}}(\psi_i,(\mathsf{S} \psi)_i)]}_{II}
+ \underbrace{\frac{1}{H}\sum_{i=1}^{\mathsf{N}} H_i ( k U_i + l V_i)\, \text{Re}[i \, \hat{\psi}^\star_i (\mathsf{S}\hat{\psi}_i)]}_{III} \,\,\,\,\,\,\,\underbrace{- r_{ek} \frac{H_\mathsf{N}}{H} \kappa^2 |\hat{\psi}_{\mathsf{N}}|^2}_{IV}\, .\]
where \(\kappa^2 = k^2 + l^2\) and the terms above represent
I: Spectral divergence of the kinetic energy flux
II: Spectral divergence of the potential energy flux
III: The rate of potential energy generation
IV: The rate of energy dissipation through bottom friction
Using the notation of the two-layer model, the particular case \(\mathsf{N}=2\) is
\[\begin{split}\frac{d}{dt} E(k,l) = \underbrace{\frac{1}{H}\text{Re}[H_1 \hat{\psi}_1^\star \hat{\mathsf{J}}(\psi_1,\nabla^2\psi_1) + H_2 \hat{\psi}_2^\star \hat{\mathsf{J}}(\psi_2,\nabla^2\psi_2)]}_{I} +
\underbrace{ \frac{H_1 H_2}{H^2}\text{Re}[(\hat{\psi}_1-\hat{\psi}_2)^\star \hat{\mathsf{J}}(\psi_1,\hat{\psi}_2])}_{II}\nonumber \\
+ \underbrace{ \frac{H_1 H_2}{H^2} \left[( U_1 - U_2 )\, \text{Re}[i k\, (\hat{\psi}^\star_1+\hat{\psi}^{\star}_2) (\hat{\psi}_2-\hat{\psi}_1)]\right] + ( V_1 - V_2 )\, \text{Re}[i l\, (\hat{\psi}^\star_1+\hat{\psi}^{\star}_2) (\hat{\psi}_2-\hat{\psi}_1)}_{III} \,\,\,\,\,\,\,\underbrace{- r_{ek} \frac{H_1}{H}\kappa^2 |\hat{\psi}_{\mathsf{N}}|^2}_{IV}\, .\end{split}\]
Vertical modes¶
Standard vertical modes are the eigenvectors, \(\mathsf{\phi}_n (z)\), of the “stretching matrix”
\[\textsf{S} \,\mathsf{\phi}_n = -m_n^2\, \mathsf{\phi}_n\,,\]
where the n’th deformation radius is
\[R_n \equiv m_n^{-1}\,.\]
Linear stability analysis¶
With \(h_b = 0\), the linear eigenproblem is
\[\mathsf{A}\, \mathsf{\Phi} = \omega \, \mathsf{B}\, \mathsf{\Phi}\,,\]
where
\[\mathsf{A} \equiv \mathsf{B}(\mathsf{U}\, k + \mathsf{V}\,l) + \mathsf{I}\left(k\,\mathsf{Q}_y - l\,\mathsf{Q}_x\right) + \mathsf{I}\,\delta_{\mathsf{N}\mathsf{N}}\, i\,r_{ek}\,\kappa^2\,,\]
where \(\delta_{\mathsf{N}\mathsf{N}} = [0,0,\dots,0,1]\,,\) and
\[ \mathsf{B} \equiv \mathsf{S} - \mathsf{I} \kappa^2\,.\]\[The growth rate is Im\ :math:`\{\omega\}`.\]