# HG changeset patch # User paugier <pierre.augier@univ-grenoble-alpes.fr> # Date 1698139210 -7200 # Tue Oct 24 11:20:10 2023 +0200 # Node ID 3113319e44c138cc5bf53dd5783f2094e0795e18 # Parent 2f41535b33658b55690c6bc6306099870d7caa35 2022strat_polo_proj: improve forcing description. diff --git a/2022strat_polo_proj/input/main.tex b/2022strat_polo_proj/input/main.tex --- a/2022strat_polo_proj/input/main.tex +++ b/2022strat_polo_proj/input/main.tex @@ -597,16 +597,18 @@ use an anisotropic, poloidal velocity forcing $\hatff = \hatf \, \eep$. The flow is forced at large spatial scales $ \left\{\kk ~ | ~ 5 \leq k/\Delta k_h \leq 20 \right\}$ and small angle $\left\{\kk ~ | ~ |\ok /N - \sin \theta_f| \leq 0.05 \right\}$ where -$\sin \theta_f = 0.3$, meaning that relatively slow internal waves are forced. \Remove{The -correlation time of the forcing is equal to the period of the forced waves $T_c = 2\pi -/(N \sin \theta_f)$.} The forcing scheme is described in Appendix~\ref{appendix:forcing}. \Add{It is not harmonic, and not given by a stochastic differential equation either. Instead, the phase and -the amplitude of the forced wave numbers are randomly changed every renewal time $T_c$ equal to the period of the forced waves $2\pi -/(N \sin \theta_f)$.} \Add{Forcing slow waves is -motivated by oceanic applications, where waves are generated, among other processes, by -slow tides \cite{mackinnon_climate_2017, nikurashin_legg_mechanism_2011}. Low frequency -forcing is also used in order to have a scale separation between forced frequencies and -the \bv frequency so that one can potentially reproduce features of the oceanic -temporal spectra close to $N$.} +$\sin \theta_f = 0.3$, meaning that relatively slow internal waves are forced. The +forcing scheme is described in Appendix~\ref{appendix:forcing}. \Add{It is neither +harmonic nor given by a stochastic differential equation. Instead, a time correlated +forcing is computed via generations of pseudo random numbers and time interpolations.} +Its correlation time is equal to the period of the forced waves $T_c = 2\pi /(N \sin +\theta_f)$. \Add{The forcing is normalized such that the kinetic injection rate $P_K$ +is always equal to 1.} \Add{Forcing slow waves is motivated by oceanic applications, +where waves are generated, among other processes, by slow tides +\cite{mackinnon_climate_2017, nikurashin_legg_mechanism_2011}. Low frequency forcing is +also used in order to have a scale separation between forced frequencies and the \bv +frequency so that one can potentially reproduce features of the oceanic temporal +spectra close to $N$.} The time advancement is performed using the $4^{th}$ order Runge-Kutta scheme. All modes with wave-number modulus larger than $\kmax = 0.8 (n_h/2) \Delta k_h$ are @@ -969,11 +971,12 @@ without vortical modes has no energy in the toroidal velocity. \Add{When vortical modes are present, an important part of the energy is contained in one vortical mode with $(k_h,k_z) = (\Delta k_h, 2 \Delta k_z)$, corresponding to large, nearly vertical -stacked shear layers (Figure~\ref{fig:spectra_khkz}$\rm (a)$). Energy then tends to be accumulated at the smallest horizontal wave vectors, close to shear modes.} When vortical modes are -removed, energy is sill concentrated in the same wave-vectors, but in the form of -poloidal an potential energy (Figure~\ref{fig:spectra_khkz}$\rm (d)$-$\rm (f)$). Except -for this qualitative difference, the toroidal, poloidal, and potential energy spectra -show the same trends. +stacked shear layers (Figure~\ref{fig:spectra_khkz}$\rm (a)$). Energy then tends to be +accumulated at the smallest horizontal wave vectors, close to shear modes.} When +vortical modes are removed, energy is sill concentrated in the same wave-vectors, but +in the form of poloidal an potential energy (Figure~\ref{fig:spectra_khkz}$\rm +(d)$-$\rm (f)$). Except for this qualitative difference, the toroidal, poloidal, and +potential energy spectra show the same trends. \begin{figure} \includegraphics[width=1.0\textwidth]{figure8} @@ -1470,32 +1473,31 @@ \section{Forcing scheme} \label{appendix:forcing} -\Add{Our forcing is designed to excite waves. Phase and amplitude of the forced wave numbers are randomly changed in time every $T_c$ such that the input kinetic energy is equal to unity. Namely, we use the following algorithm:} +\Add{Our forcing is designed to excite waves and computed via generation of pseudo +random numbers with uniform distribution and time interpolation. Namely, we use the +following algorithm:} \begin{algorithm}[H] -$t=0$: generate two random numbers $f_{0\kk}, f_{1\kk} \sim \mathcal{P}$ $\forall \kk$ in the forcing region; $t_0 = 0$; \\ +$t = 0$; $t_0 = 0$; \\ +Generate two complex random fields $\hat f_0(\kk)$ and $\hat f_1(\kk)$; \\ +$\hatff = \hat f_0 \, \eep$;\\ +Normalize $\hatff$ to ensure +\Add{$P_K(t) = \sum\limits_{\kk} ~ \Re \left[ \hatff \cdot \hatvv^* + \frac{\Delta t}{2} |\hatff|^2 \right] = 1$};\\ \While{$t \leq T$}{ - $t \rightarrow t + \Delta t$; \\ + $t = t + \Delta t$; \\ \If{$t - t_0 \geq T_c$}{ - $t_0 \rightarrow t$; \\ - $f_{0 \kk} \rightarrow f_{1 \kk}$; \\ - genererate $f_{1\kk} \sim \mathcal{P}$ + $t_0 = t$; \\ + $\hat f_0 = \hat f_1$; \\ + Generate $\hat f_1$; }{} - $\hatff = \left\{f_{0\kk} - \dfrac{(f_{1\kk} - f_{0\kk})}{2} \left[\cos\left(\dfrac{\pi(t-t_0)}{T_c} \right)+ 1 \right]\right\}~ \eep$ \\ - $\hatff$ are normalized to ensure \Add{$P_K(t) = \sum\limits_{\kk} ~ \Re \left[ \hatff \cdot \hatvv^* \right] = 1$} + $\hatff = \left\{ \hat f_0 - \dfrac{(\hat f_1 - \hat f_0)}{2} \left[\cos\left(\dfrac{\pi(t-t_0)}{T_c} \right)+ 1 \right]\right\}\, \eep$; \\ + Normalize $\hatff$; } %\caption{} \end{algorithm} - -where $\mathcal{P}$ is the probability law which has for distribution -\begin{equation} -p_X(x) = \begin{cases} -\frac{1}{4} ~~ \text{if} ~ \Add{\Re(x)} \in[-1:1] ~ \text{and} ~ \Add{\Im(x)} \in[-1:1] \\ -0 ~~ \text{otherwise} -\end{cases} -\end{equation} -\Remove{and} $\Delta t$ is the time increment at each time step \Add{ and $T$ is the final time of the simulation}. - +where $\Delta t$ is the time increment at each time step \Add{and $T$ is the final time +of the simulation}. \Add{The random complex fields are built such that their inverse +Fourier transform is real and they are null for unforced wavenumbers.} \section{List of simulations}