chapter_00_3_budget.md

Spectral energy budget

A spectral energy budget is a statistical analysis of energy exchanges between different forms and different scales, carried out in spectral space. The energy flux $\Pi(\mathbf{k}, t)$ is computed by integrating a transfer function $T(\mathbf{k'}, t)$ over all wavenumbers in the interval $[\mathbf{k}, \mathbf{k}_{max}]$. The numerical equivalent of the integration is a cumulative sum over the wavenumbers in the same range. The range of integration and the direction of energy flux for a positive value of $\Pi$ is depicted in @fig:spectral_flux. In this section, we start by deriving the transfer function for the toy model equations. Then, we show that by combining the normal mode decomposition with this calculation, one can distinguish interactions between different forms of energy.

{#fig:spectral_flux width=70%}

Kinetic energy (KE)

In the case of the SWE, kinetic energy is not quadratic, but cubic, $E_K = h {\bf u} \cdot {\bf u} /2$. As a result, the transfer term does not only contain cubic terms but also terms which are of fourth order in the basic flow variables. In turbulence studies, we conventionally encounter a quadratic expression for KE and a cubic expression for the transfer function. It was observed while performing the study on SWE [@augier_shallow_2019] that the contribution from the fourth-order transfer terms were small but not negligible. This complicates the interpretation of the spectral energy budget in the case of the SWE. In the toy model, the analysis is greatly simplified since the expression for energy is quadratic. To identify the transfer terms, we start from the governing equations for the toy model. The rate of change of kinetic energy can be calculated from [@eq:toy], \begin{align*} &\partial_t E_K(\mathbf{k},t) \ &= \frac{1}{2}\pder{t}(\mathbf{\hat u}.\mathbf{\hat u^}) = \frac{1}{2}\left[ \mathbf{\hat u} \cdot\pder[\mathbf{\hat u^}]{t} + \mathbf{\hat u^}\cdot \pder[\mathbf{\hat u}]{t}\right] \ &= \frac{1}{2}\left[ -\hat{u}_i (\widehat{ u_j^r\partial_j u_i })^ - c \hat{u}_i (ik_i \hat{\theta})^* - \hat{u}i (\epsilon{i3k} f \hat{u}_k)^* - ... \text{hermitian conjugate terms} \right] \ &= -\Re\left[ \hat{u}_i (\widehat{ u^r_j\partial_j u_i })^* + c \hat{u}_i (ik_i \hat{\theta})^* + \hat{u}i (\epsilon{i3k} f \hat{u}_k)^* \right], \end{align*} where $^$ represents the hermitian conjugate, and $u_j^r$ represents the rotational component of the velocity, computed using the Helmholtz decomposition. The last term $\hat{u}i (\epsilon{i3k} f \hat{u}_k)^$ is zero due to the fact that velocity is always perpendicular to the Coriolis acceleration. Thus the rate of change of KE, without any approximations can be written as: \begin{equation} \label{eq:dtKE} \pder{t}E_K(\mathbf{k},t) = T_K + C_K, \end{equation} where $T_K$ and $C_K$ represent the transfer and conversion spectral functions respectively, \begin{align} \label{eq:TK} T_K= & -\Re\left[\hat{u}_i (\widehat{ u^r_j\partial_j u_i })^* \right] = -\Re\left[\hat{u}_i^* ik_j \widehat{u^r_j u_i}\right] = \Im\left[\hat{u}_i^* k_j \widehat{u^r_j u_i}\right] ,\ \label{eq:CK} C_K= & -\Re\left[c \hat{u}_i (ik_i \hat{\theta})^* \right] = \Re\left[c \hat{\theta} (ik_i \hat{u}_i)^* \right] = \Re\left[c \kappa^2 \hat{\theta}\hat{\chi}^*\right]. \end{align} In @eq:TK the property that $\Re(iz) = -\Im(z)$ is used. In @eq:CK, the property that Fourier transforms of real functions are hermitian is used [@bracewell_fourier_2014]. In [@eq:TK;@eq:CK] the property that rotational velocity is divergence-free is also used. The conversion function $C_K$ represents the energy converted from APE into KE.

Available potential energy (APE)

Available potential energy is defined as $$ E_A(\mathbf{r}, t) = \frac{1}{2} { \theta^2 }. $$ By @eq:toy_theta, the rate of change of APE is given by, \begin{align*} \pder{t}E_A(\mathbf{k},t) = & \frac{1}{2}\pder{t}({\hat \theta}{\hat \theta^}) = \frac{1}{2}\left[ \hat{\theta} .\pder[\hat{\theta}^]{t}+ \hat \theta^.\pder[\hat \theta]{t}\right] \ = & \frac{1}{2}\left[ - \hat{\theta} (ik_i\hat{ u_i } + ik_i\widehat{ \theta u^r_i })^ - \hat{\theta}^* (ik_i\hat{ u_i } + ik_i\widehat{ \theta u^r_i }) \right] \ = & -\Re\left[\hat{\theta} (ik_i\hat{ u_i } + ik_i\widehat{ \theta u^r_i })^* \right]. \end{align*} Similar to equation [@eq:dtKE] we write, $$ \label{eq:dtPE} \pder{t}E_A(\mathbf{k},t) = T_A + C_A, $$ where $T_A$ and $C_A$ represent the transfer and conversion spectral functions of APE. Thus, \begin{align} \label{eq:TP} T_A= & -\Re\left[\hat{\theta} (ik_i\widehat{ \theta u^r_i })^* \right] = -\Re\left[\hat{\theta}^* ik_j \widehat{u^r_j \theta}\right] = \Im\left[\hat{\theta}^* k_j \widehat{u^r_j \theta}\right] ,\ \label{eq:CP} C_A= & -\Re\left[c\hat{\theta} (ik_i\hat{u_i })^\right] = -\Re\left[c \kappa^2 \hat{\theta}\hat{\chi}^\right]. \end{align} Comparing @eq:CP with @eq:CK, we see that $C_K = -C_A$, from which we can assert that, equivalent conversion occurs between KE and APE via these terms.

Algorithm for computing spectral energy budget

Using the normal modes ($\bf N$ as shown in @eq:nmode) in spectral space as input, the spectral energy budget can be computed as described below:

1. Compute the inversion matrix $Q$ using @eq:qmat.
2. Divide the normal modes vector, $\bf N$, by the magnitude of Fourier modes, $\kappa$, to obtain $\bf B$ (@eq:bvec).
3. Apply matrix multiplication of $Q$ on $\bf B$ (@eq:uqmatb) to obtain the normal mode decomposition of the primitive variables $\bf U$.
4. Take the expressions for the transfer terms, $T_K$ and $T_A$ (@eq:TK and @eq:TP), and expand the primitive variables $\bf U$ using the decomposition calculated in the previous step. The result is a linear combination of $B^{(0)}, B^{(+)}$ and $B^{(-)}$, the normal modes as shown in @eq:decomp_tensor_u and @eq:decomp_tensor_eta.
5. Compute the transfer terms using the expanded expression for primitive variables term-by-term. While few terms can be computed in spectral space ($C_K, C_A$), where derivatives are involved ($T_K, T_A$) a couple of FFT and inverse FFT would have to be used.
6. Classify the expanded transfer terms into four groups based on the kind of normal modes they are product of: $T_{VVV}, T_{VVW}, T_{VWW}$ and $T_{WWW}$, where the subscript $V$ represents a potential vorticity mode $B^{(0)}$, and $W$ represents a wave mode, $B^{(+)}$ or $B^{(-)}$. The classification does not take into account the order in which the modes appear (i.e., combinations are noted, and not permutations).
7. Store transfer terms for every time instant as a one-dimensional array in $\kappa$ by taking sum along circular wavenumbers shells.
8. After the simulation, load the series of transfer terms. Take a cumulative sum along $\kappa$, which would give the instantaneous spectral energy flux $\Pi(\kappa, t)$.
9. Take a time average of $\Pi(\kappa, t)$ over the interval when the simulation had reached a statistically, stationary state to get the desired spectral energy flux, $\Pi(\kappa)$ decomposed in groups.