2.1. Governing equations

We use the non-dimensionalised Boussinesq equations, identical to those detailed in Thomas & Daniel (Reference Thomas and Daniel2020), given by

(2.1a) \begin{align} \frac {\partial \boldsymbol{v}}{\partial t} + \boldsymbol{\hat {z}} \times \boldsymbol{v} + \boldsymbol{\nabla }p + Ro \left ( \boldsymbol{v} \boldsymbol{\cdot }\boldsymbol{\nabla }\boldsymbol{v} + w \frac {\partial \boldsymbol{v}}{\partial z}\right ) = - \nu \Delta ^8 \boldsymbol{v},\\[-28pt]\nonumber\end{align}

(2.1b) \begin{align} \frac {\partial w}{\partial t} + \alpha ^2\left (\frac {\partial p}{\partial z} - b\right ) + Ro \left (\boldsymbol{v} \boldsymbol{\cdot }\boldsymbol{\nabla }w + w \frac {\partial w}{\partial z}\right ) = - \nu \Delta ^8 {w} ,\\[-28pt]\nonumber\end{align}

(2.1c) \begin{align} \frac {\partial b}{\partial t} + w + Ro \left (\boldsymbol{v} \boldsymbol{\cdot }\boldsymbol{\nabla }b + w \frac {\partial b}{\partial z}\right ) = - \nu \Delta ^8 {b} ,\\[-28pt]\nonumber\end{align}

(2.1d) \begin{align} \boldsymbol{\nabla }\boldsymbol{\cdot }\boldsymbol{v} + \frac {\partial w}{\partial z} = 0 .\\[0pt]\nonumber\end{align}

Here $\boldsymbol{v}= u \boldsymbol{\hat {x}} + v \boldsymbol{\hat {y}}$ is the horizontal velocity vector, $w$ is the vertical velocity, $b$ is the buoyancy and $p$ is the fluid pressure normalised with a constant reference density; $\boldsymbol{\hat {x}}$ , $\boldsymbol{\hat {y}}$ and $\boldsymbol{\hat {z}}$ are unit vectors along the $x$ , $y$ and $z$ directions, respectively; $\boldsymbol{\nabla }= \boldsymbol{\hat {x}} \partial /\partial x + \boldsymbol{\hat {y}} \partial /\partial y$ and $\Delta ={\nabla} ^2 + \partial ^2/\partial z^2$ . To obtain an extended inertial range with minimal dissipation, we used hyperdissipation operators of the form $\nu \Delta ^8$ on the right-hand side of the equations in (2.1), with $\nu$ being the hyperviscosity.

For the non-dimensionalisation, we scaled time by the inertial time scale, $1{\kern-1pt}/{\kern-1pt}f$ , with $f$ being the Coriolis frequency. The vertical coordinate $z$ was scaled by $H$ , which can be thought of as an estimate for the depth of the ocean. We consider the dynamics over the $f$ -plane, i.e. constant inertial frequency. We chose a horizontal scale $L$ such that the aspect ratio $L{\kern-1pt}/{\kern-1.5pt}H=N\!/\!f=\alpha \gg 1$ , where $N$ is the constant buoyancy frequency. Here $L$ was then used to scale the horizontal coordinates $\boldsymbol{x} = (x, y)$ . We chose an arbitrary velocity scale $U$ to scale the horizontal velocity $\boldsymbol{v} = (u,v)$ above and then $\textit{HU}{\kern-1.5pt}/{\kern-1.5pt}L$ , obtained using the continuity equation (2.1d ), was used to scale the vertical velocity, $w$ . We scaled $p$ with $\textit{fUL}$ , this being chosen by requiring that the lateral pressure gradient balanced the Coriolis term in the horizontal momentum equation. Finally, balancing the vertical pressure gradient with buoyancy gave us $\textit{fUL}/{\kern-1.5pt}H$ as the scale for buoyancy, $b$ .

The scalings discussed above gave us (2.1) with non-dimensional numbers $\alpha =N\!/\!f$ and Rossby number, $\textit{Ro}=U\!/\!f\!L$ . Observed buoyancy frequencies in the ocean vary between $10^{-3}\,\text{s}^{-1}$ and $10^{-2}\,\text{s}^{-1}$ while $10^{-4}\,\text{s}^{-1}$ is a representative estimate for the Coriolis frequency away from the equator (Emery, Lee & Magaard Reference Emery, Lee and Magaard1984; Vallis Reference Vallis2017; Lien & Sanford Reference Lien and Sanford2019; Li et al. Reference Li, Cheng, Zhu, Trenberth, Mann and Abraham2020; Sallée et al. Reference Sallée, Pellichero, Akhoudas, Pauthenet, Vignes, Schmidtko, Garabato, Sutherland and Kuusela2021). Therefore, $\alpha$ varies in the interval 10–100 and we chose a value in this interval, $\alpha =20$ , for the numerical integrations.

In the absence of dissipation, (2.1) conserves domain-integrated energy, given by

(2.2) \begin{align} \frac {{\rm d}{E}}{{\rm d}{t}} =0, \quad E=\int _D \left (\frac {1}{2}\boldsymbol{v}^2 + \frac {1}{2}\frac {w^2}{\alpha ^2} + \frac {1}{2}b^2\right ){\rm d}x\,{\rm d}y\,{\rm d}z, \end{align}

where the integration is performed over the whole domain $D$ . Hereafter we use $E$ to denote domain-integrated energy, this being the sum of kinetic and potential energy, with suitable subscripts to denote different components.

We numerically integrated (2.1) with a pseudospectral method using RK4 time stepping in a domain $(x,y,z) \in [0, 2 \pi ] \times [0, 2 \pi ] \times [-\pi , 0]$ . Periodic boundary conditions were applied in the $x$ and $y$ directions and free-slip rigid lid boundary conditions were used at the top ( $z = 0$ ) and bottom ( $z = -\pi$ ) boundaries of the domain. Based on multiple iterative numerical integrations, a resolution of $384^3$ grid points with hyperviscosity $\nu = 10^{-31}$ gave us solutions whose dynamics were converged such that the flow features and flow statistics were not sensitive to a further increase in resolution. We used these parameters to generate all the solutions discussed in this work. Convergence was tested with selected simulations, especially at $O(1)$ Rossby numbers, being run with $512$ grid points in each spatial direction.

We decompose the flow fields into barotropic ( $T$ )–baroclinic ( $C$ ) and wave ( $W$ )-balanced ( $G$ ) components. The barotropic fields are obtained by vertically averaging the flow fields and the baroclinic fields are then computed by subtracting the barotropic fields from the total fields. For example,

(2.3) \begin{align} { {\phi }_{_{T} } } (\boldsymbol{x}, t) =\frac {1}{\pi } \int _{-\pi }^{0} \phi (\boldsymbol{x}, z, t)\, {\rm d}z \quad \text{and} \quad { {\phi }_{_{C} } } (\boldsymbol{x}, z, t) = \phi (\boldsymbol{x}, z, t) - { {\phi }_{_{T} } } (\boldsymbol{x}, t) \end{align}

for an arbitrary field $\phi$ . Following the above definition, the baroclinic component vertically integrates to zero:

(2.4) \begin{align} \int _{-\pi }^{0} { {\phi }_{_{C} } } \, {\rm d}z = \int _{-\pi }^{0} \left [ \phi (\boldsymbol{x}, z, t) - { {\phi }_{_{T} } } (\boldsymbol{x}, t) \right ] {\rm d}z = \int _{-\pi }^{0} \phi \, {\rm d}z - \pi { {\phi }_{_{T} } } = 0 .\end{align}

As a result, the domain integral of the product of a barotropic and a baroclinic term vanishes. This leads to orthogonality of the two fields, i.e. flow energy decomposes into the sum of barotropic and baroclinic components. To see this, observe that

(2.5) \begin{align} u^2 = ( { {u}_{_{T} } } + { {u}_{_{C} } } )^2 = { {u}_{_{T} } } ^{\kern -.45em 2} \, + 2\, { {u}_{_{T} } } \, { {u}_{_{C} } } + { {u}_{_{C} } } ^{\kern -.55em 2}. \end{align}

Integrating over the whole domain involves a vertical integration along with horizontal integrals. The $ { {u}_{_{T} } } { {u}_{_{C} } }$ cross-term in (2.5) vanishes on vertical integration. This leads to

(2.6) \begin{align} \int _{0}^{2\pi }\!\!\int _{0}^{2\pi }\!\!\int _{-\pi }^{0} u^2 {\rm d}x {\rm d}y {\rm d}z= \int _{0}^{2\pi }\!\!\int _{0}^{2\pi }\!\!\int _{-\pi }^{0} { {u}_{_{T} } } ^{\kern -.45em 2} {\rm d}x {\rm d}y {\rm d}z+ \int _{0}^{2\pi }\!\!\int _{0}^{2\pi }\!\!\int _{-\pi }^{0} { {u}_{_{C} } } ^{\kern -.55em 2} {\rm d}x {\rm d}y {\rm d}z. \end{align}

Similar manipulations follow for other terms also, leading to the total energy at any time instant being the sum of barotropic and baroclinic energy, i.e.

(2.7) \begin{align} E(t) = { { {E}_{_{T} } } }(t) + { { {E}_{_{C} } } }(t) .\end{align}

The second decomposition we use is the wave–balance decomposition, it being identical to that used in Thomas & Daniel (Reference Thomas and Daniel2020). This decomposition is based on the linear equations obtained by setting $\textit{Ro} = 0$ and dropping the viscous terms on the right-hand side of (2.1). The wave and balanced fields, denoted with subscripts ‘W’ and ‘G’ evolve according to

(2.8a) \begin{align} \frac {\partial \boldsymbol{{v}}_{_{W} }}{\partial t} + \hat{\boldsymbol{z}}\times \boldsymbol{{v}}_{_{W} } + \boldsymbol{\nabla }{ {{p}}_{_{W} } } &= 0,\\[-10pt]\nonumber \end{align}

(2.8b) \begin{align} \frac {\partial { { {{w}}_{_{W} } } }}{\partial t} + \alpha ^2\!\left (\frac {\partial { { {{p}}_{_{W} } } }}{\partial z} - { { {{b}}_{_{W} } } }\right ) &= 0,\\[-10pt]\nonumber \end{align}

(2.8c) \begin{align} \frac {\partial { { {{b}}_{_{W} } } }}{\partial t} + { { {{w}}_{_{W} } } } &= 0,\\[-10pt]\nonumber \end{align}

(2.8d) \begin{align} \boldsymbol{\nabla }\boldsymbol{\cdot }\boldsymbol{v}_{ _{W} } + \frac {\partial { { {{w}}_{_{W} } } }}{\partial z} &= 0\\[10pt]\nonumber\end{align}

(2.9a) \begin{align} \boldsymbol{\hat z}\times \boldsymbol{v}_{_{G} } + \boldsymbol{\nabla }{ { {{p}}_{_{G} } } } &= 0,\\[-10pt]\nonumber \end{align}

(2.9b) \begin{align} \frac {\partial { { {{p}}_{_{G} } } }}{\partial z} - { { {{b}}_{_{G} } } } &= 0,\\[-12pt]\nonumber\end{align}

(2.9c) \begin{align} { { {{w}}_{_{G} } } } &= 0,\\[-10pt]\nonumber\end{align}

(2.9d) \begin{align} \boldsymbol{\nabla }\boldsymbol{\cdot }\boldsymbol{v}_{_{G} } &= 0 .\\[10pt]\nonumber\end{align}

Notice that the balanced flow is in geostrophic and hydrostatic balance with no vertical velocity. Similar to the baroclinic–barotropic decomposition, the wave–balance decomposition also satisfies the orthogonality property such that the total flow energy is the sum of wave and balanced energy at any given time, i.e.

(2.10) \begin{align} E(t) = { {{E}}_{_{G} } } (t) + { {{E}}_{_{W} } } (t). \end{align}

Demonstration of the orthogonality relation above and implementation of the wave–balance decomposition on the flow fields is detailed in Appendix A.

2.2. Vortex evolution without initial disturbances

To consider a localised coherent vortex idealising coherent eddies in the ocean, as mentioned in the introduction, we set up a Gaussian vortex

(2.11) \begin{align} \zeta = a{\rm e}^{-(\boldsymbol{x}-\boldsymbol{s})^2}, \end{align}

where $\zeta = \boldsymbol{\hat {z}} \boldsymbol{\cdot }\boldsymbol{\nabla }\times \boldsymbol{v}$ is the vertical vorticity, $a$ = $\pm$ 1 decides whether the vortex is a cyclone (+1) or anticyclone (−1) and $\boldsymbol{s} = (\pi , \pi )$ is the vortex centre. We took Fourier transform of the vorticity above and applied to it an exponential filter $ \exp [-4(k_h/{k_h}_{\textit{max}})^{80}]$ , where $k_h=\sqrt {k_x^2 + k_y^2}$ is the magnitude of the horizontal wavenumber vector $(k_x, k_y)$ and $k_{h_{\textit{max}}}$ is the maximum value of $k_h$ based on the resolved wavenumbers. Note that these wavenumbers are in non-dimensional form with $k_h$ effectively being non-dimensionalised by $1/L$ as per the procedure described below (2.1). The filtering operation using an exponential filter ensured that modes close to the maximum resolved wavenumber decayed rapidly to zero while leaving lower modes nearly unaffected. We then computed a streamfunction $\psi$ from the vorticity field $\zeta$ assuming geostrophic and hydrostatic balance, i.e. ${\nabla} ^2 \psi = \zeta$ . The horizontal velocities were obtained as $ { {{u}}_{_{G} } } = -\partial _y \psi$ , $ { {{v}}_{_{G} } } = \partial _x \psi$ , while vertical velocity and buoyancy were set to zero. These fields formed the balanced initial conditions corresponding to the vorticity field in (2.11). This balanced initial vortex configuration (denoted with subscript ‘ $V$ ’ hereafter), initialised with either sign of $ a$ , yields a domain-integrated energy $ { {{E}}_{_{V} } } = 1.354$ .

Figure 1.

Panels (a,b) show the vorticity, $\zeta$ , at the final time of integration, $t = 200$ , for $\textit{Ro} = 1.0$ on the $h$ -plane (a) and vertical plane (b). Panel (c) shows the vertical velocity on the $h$ -plane and the mid-depth plane as a function of time. Panels (d–f) show the difference in vorticity between the initial time and $t=200$ on the $h$ -plane for $\textit{Ro}=1.0$ (d) and $\textit{Ro}=0.1$ (e). The time series of the RMS of the vorticity difference integrated over the whole domain is shown in panel ( f) for $\textit{Ro}=0.1$ and 1.0.

Figure 1 Long description

Panel A: A heatmap shows vorticity on the h-plane at the final time of integration for Ro equal to 1.0. The x-axis ranges from 0 to 2π, and the y-axis ranges from 0 to 2π. The color scale ranges from -1 to 1. Panel B: A heatmap shows vorticity on the vertical plane at the final time of integration for Ro equal to 1.0. The x-axis ranges from 0 to 2π, and the y-axis ranges from -π to π. The color scale ranges from -1 to 1. Panel C: A line graph shows vertical velocity on the h-plane and the mid-depth plane as a function of time. The x-axis represents time (t) ranging from 0 to 200, and the y-axis represents vertical velocity (w_centre) ranging from -10^-2 to 10^-2. Two lines are plotted: one for the h-plane and one for the mid-plane. Panel D: A heatmap shows the difference in vorticity between the initial time and t=200 on the h-plane for Ro equal to 1.0. The x-axis ranges from 0 to 2π, and the y-axis ranges from 0 to 2π. The color scale ranges from -4 times 10^-3 to 4 times 10^-3. Panel E: A heatmap shows the difference in vorticity between the initial time and t=200 on the h-plane for Ro equal to 0.1. The x-axis ranges from 0 to 2π, and the y-axis ranges from 0 to 2π. The color scale ranges from -4 times 10^-3 to 4 times 10^-3. Panel F: A line graph shows the time series of the root mean square (RMS) of the vorticity difference integrated over the whole domain for Ro equal to 0.1 and 1.0. The x-axis represents time (t) ranging from 0 to 200, and the y-axis represents RMS (ζ_t - ζ_t=0) ranging from 0 to 2.5 times 10^-5. Two lines are plotted: one for Ro equal to 0.1 and one for Ro equal to 1.0.

Before introducing disturbances to perturb the vortex, we first integrated (2.1) initialised with the Gaussian vortex alone, with no unbalanced components for different Rossby numbers. A brief summary of the results is given in figure 1. Panels (a) and (b) show the vorticity on the $z=-h=-\pi /100$ plane and $y$ = $\pi$ plane, hereafter referred to as the $h$ -plane and the vertical plane, for the anticyclonic vortex with $\textit{Ro}=1.0$ at $t=200$ . Notice that the vortex remains intact with no signs of disruption. The vertical velocity of the balanced vortex is zero and it remained almost the same during the time evolution. Panel (c) shows the time evolution of vertical velocity $w$ along the vortex centreline, $x=y=\pi$ , denoted as $w_{\textit{centre}}(z,t)$ , on $z=-h$ and $z=-\pi /2$ planes. Of course, although the vortex retains its features to be almost identical to the initial state, weak disturbances were generated with time. This can be seen in panel (d), showing the difference in vorticity on the $h$ -plane between the initial and final time of integration. Notice that the disturbance excited has a rich structure in physical space, although these disturbances are extremely weak and three orders of magnitude below unity (see the colour bar). In panels (a–d) we show results for the highest Rossby number anticyclonic vortex, although similar results were seen for other Rossby numbers as well. Panel (e), for instance, shows the vorticity difference between the initial and final time on the $h$ -plane for $\textit{Ro}=0.1$ . Additionally, the time series of the root-mean-square (RMS) of the vorticity difference given in panel ( f) reveals that the disturbances excited are really weak, causing no noticeable effects on the coherent vortex.

Based on time integrations performed at different Rossby numbers, we found that cyclonic and anticyclonic barotropic balanced Gaussian vortices retained their features, with negligible unbalanced fields generated with time. In a sense these experiments evolving the balanced vortex without any pre-excited disturbance confirm the accuracy and robustness of the numerical integration scheme we implement, emphasising that the numerical disturbances remain quite insignificant and do not perturb the vortex.

One could imagine different kinds of perturbations, both balanced and unbalanced, that could affect the above coherent vortex configuration. In this study we focus only on unbalanced perturbations in the form of horizontally homogeneous inertial oscillations. We therefore excited inertial oscillations on top of the Gaussian vortex above, initialised by setting $ { {{u}}_{_{W} } } = a_0 \exp { ( - z^2/2 h^2 )}$ , ${ {{v}}_{_{W} } } = { {{w}}_{_{W} } } = { {{b}}_{_{W} } } =0$ . In the linear equations this initial condition will excite inertial oscillations of the form

(2.12) \begin{align} { {{u}}_{_{W} } } + i { {{v}}_{_{W} } } = A(z) {\rm e}^{-it}, \qquad { {{w}}_{_{W} } } = { {{b}}_{_{W} } } = 0 ,\end{align}

where $A(z)=a_0 \exp { ( - z^2/2 h^2 )}$ with $z$ = − $h$ being the location where inertial shear, ${\rm d}A/{\rm d}z$ , is maximum and as mentioned earlier, $h = \pi /100$ . The constant $a_0$ was adjusted to ensure that the wave energy $ { {{E}}_{_{W} } }$ had a specific initial value for each experiment relative to the vortical or balanced energy $ { {{E}}_{_{V} } }$ , thereby allowing us to set an initial wave-to-vortex energy ratio $ ( { {{E}}_{_{W} } } / { {{E}}_{_{V} } } )_{t=0}$ .

Although we set different energy regimes based on domain-integrated wave and vortex energy, the energy ratio can be quite different locally in space. For instance, initially inertial oscillations’ energy is localised in the upper region, leading to a higher wave–vortex energy ratio in the upper part of the domain than in the lower domain. However, as will be seen below, the wave field will propagate downwards following wave–vortex interactions, changing the spatially local wave–vortex energy ratio. Furthermore, the wave field can dissipate and lose its energy with time. Therefore, domain-integrated initial energy is an estimate we use to distinguish regimes initially, although it must be noted that the wave–vortex energy ratio can change with spatial location and time.

At the linear level, the wave and the vortex fields, (2.11) and (2.12), are orthogonal when using the wave–balance decomposition. This means that the vortex is perfectly balanced with no projection on the wave field and vice versa. On the other hand, the inertial oscillation field in (2.12), while being dominated by baroclinic modes, contains about 3 % of its energy in the barotropic mode. Therefore, even in the absence of nonlinear interactions, the wave field contains a small fraction of energy in the barotropic mode, which will slightly contribute towards the barotropic energy of the total flow.

The above set-up was used to excite inertial oscillations as disturbances on top of the coherent balanced vortex and our goal is to explore how the wave–vortex interaction leads to turbulent flows with small-scale dissipative structures and energy exchanges between the fields. This set-up helps us isolate and focus specifically on the interaction between a coherent balanced field and an unbalanced field from amongst many other dynamics that takes place in the real ocean.

From the figures seen above, notice that the vortex occupies a major part of the domain. To explore the flow transitioning to turbulence and small-scale dissipative structure formation due to wave–vortex interactions, we used a domain-scale vortex, thereby providing us with a reasonable inertial range to capture the forward energy cascade. In this process, we end up not having a wide enough wavenumber window to capture any inverse cascade phenomena associated with balanced dynamics (McWilliams Reference McWilliams1989; Marino, Pouquet & Rosenberg Reference Marino, Pouquet and Rosenberg2015; Herbert et al. Reference Herbert, Marino, Rosenberg and Pouquet2016) that might appear during the turbulent interactions. We also note that the periodic domain we use allows perturbations, especially higher frequency waves generated by the interactions, to stay in the neighbourhood of the vortex, thereby further affecting the vortex. In contrast, an unbounded domain would allow waves to propagate away, leading to weaker interactions. The use of periodic boundary conditions therefore provides us with an upper bound on the turbulent energy exchange processes, this being the most extreme case.

2.3. Flow regimes

To cover a broad parameter range with affordable computational resources, we chose five different Rossby numbers, $\textit{Ro}=$ 0.1, 0.3, 0.5, 0.7 and 1.0, and for each of them, we chose five different initial wave–vortex energy ratios, $ ( { {{E}}_{_{W} } } / { {{E}}_{_{V} } } )_{t=0}$ = 0.1, 0.3, 0.5, 0.7 and 1.0, leading to 25 cases. Additionally, for each $\textit{Ro}$ and $ ( { {{E}}_{_{W} } } / { {{E}}_{_{V} } } )_{t=0}$ , we considered anticyclonic and cyclonic vortices, leading to a total of 50 flow regimes. In this study we examine inertial oscillations interacting with balanced vortices in these 50 different regimes.

We integrated (2.1), discussed above, and found that energetic interactions started and gradually the flow settled down with minimal further changes. Based on inspections we set $t=200$ as the final time for integrations, since this time window was seen to capture the key interactions in different regimes. While we used $\textit{Ro}$ in the equations to change the Rossby number, we calculated the effective Rossby number for each flow as $\textit{Ro}_{\textit{eff}} = Ro \zeta _{rms}$ , where $\zeta _{rms}$ is the RMS vorticity computed over the whole domain. This is essentially the non-dimensional form of the RMS of dimensional vorticity divided by the Coriolis frequency. The effective Rossby number changes with time and table 1 shows the maximum value of $\textit{Ro}_{\textit{eff}}$ for each regime. Each cell in the table has $\textit{Ro}_{\textit{eff}}$ for anticyclone and cyclone in that order, the two separated by a comma. Notice that in any specific row, increasing the energy ratio $ ( { {{E}}_{_{W} } } / { {{E}}_{_{V} } } )_{t = 0}$ increases $\textit{Ro}_{\textit{eff}}$ . As will be seen in the following section, increasing the energy ratio leads to more intense interactions and small-scale structures, raising $\textit{Ro}_{\textit{eff}}$ .

Table 1.

Maximum values of $\textit{Ro}_{\textit{eff}}$ for different flow regimes. Each cell shows $\textit{Ro}_{\textit{eff}}$ for anticyclone and cyclonic cases separated by a comma. The flow regimes marked in a grey colour are discussed in greater detail in following sections.

5.1. Physical structure and spectra of decomposed fields

We applied the three decompositions mentioned above to the flow fields. The fast–slow decomposition was the most expensive one, since this decomposition requires storing the flow data over a long enough window to perform the frequency-based decomposition. We applied this decomposition by dividing the total time domain into multiples of $2\pi$ . Following our detailed examination of different time intervals during the flow evolution, below we present results from the time interval $[0, 4\pi ]$ . This time interval was the significant one for the high Rossby number case, since the dominant energy transfers and dissipation of the flow took place in this time interval. In the low Rossby number regime, we found that different time intervals led to qualitatively similar results, with the interval $[0, 4\pi ]$ being a good representative one. Here we present results of different decompositions in the interval $[0, 4\pi ]$ and note that qualitatively similar results were obtained in other time intervals.

Figure 9 shows the instantaneous physical structure of the vorticity field computed using the three flow decompositions for the high Rossby number case. The top row shows the barotropic vorticity on the $h$ -plane, mid-plane and vertical plane. Since the barotropic vorticity has no vertical variability, the structures on different planes are similar. The middle row shows the balanced vorticity, $ { {{\zeta }}_{_{G} } }$ . Notice how we see a lot more small-scale structures, especially close to the top of the domain. Finally, the last row of figure 9 shows the slow vorticity field, $ { {{\zeta }}_{_{S} } }$ . We see that the slow vorticity field shows less small-scale structures than the balanced field, but more than the barotropic field. This is particularly evident comparing panels (d) and (g): while it is difficult to spot the vortex in the balanced field, the slow field better reveals the vortex structure.

Figure 9.

Vertical vorticity, $\zeta$ , based on three different flow decompositions for $\textit{Ro}$ = 1.0 on the $h$ -plane (left column), mid-plane (middle column) and vertical plane (right column) at $t=4\pi$ . Panels (a–c) show barotropic vorticity, panels (d–f) show balanced vorticity and panels (g–i) show slow vorticity.

Figure 10.

Horizontal divergence, $\boldsymbol{\nabla }\boldsymbol{\cdot }\boldsymbol{v}$ , based on three different flow decompositions for $\textit{Ro}$ = 1.0 on the $h$ -plane (left column), mid-plane (middle column) and vertical plane (right column) at $t=4\pi$ . Panels (a–c) show baroclinic divergence, panels (d–f) show wave divergence and panels (g–i) show fast divergence.

Figure 10 Long description

Panel A: A heat map showing baroclinic divergence on the h-plane. The x-axis ranges from 0 to 2π and the y-axis ranges from 0 to 2π. The color scale indicates the intensity of horizontal divergence, with darker colors representing lower values and lighter colors representing higher values. Panel B: A heat map showing baroclinic divergence in the mid-plane. The x-axis ranges from 0 to 2π and the y-axis ranges from 0 to 2π. The color scale indicates the intensity of horizontal divergence, with darker colors representing lower values and lighter colors representing higher values. Panel C: A heat map showing baroclinic divergence in the vertical plane. The x-axis ranges from -π to 2π and the y-axis ranges from -π to 0. The color scale indicates the intensity of horizontal divergence, with darker colors representing lower values and lighter colors representing higher values. Panel D: A heat map showing wave divergence on the h-plane. The x-axis ranges from 0 to 2π and the y-axis ranges from 0 to 2π. The color scale indicates the intensity of horizontal divergence, with darker colors representing lower values and lighter colors representing higher values. Panel E: A heat map showing wave divergence in the mid-plane. The x-axis ranges from 0 to 2π and the y-axis ranges from 0 to 2π. The color scale indicates the intensity of horizontal divergence, with darker colors representing lower values and lighter colors representing higher values. Panel F: A heat map showing wave divergence in the vertical plane. The x-axis ranges from -π to 2π and the y-axis ranges from -π to 0. The color scale indicates the intensity of horizontal divergence, with darker colors representing lower values and lighter colors representing higher values. Panel G: A heat map showing fast divergence on the h-plane. The x-axis ranges from 0 to 2π and the y-axis ranges from 0 to 2π. The color scale indicates the intensity of horizontal divergence, with darker colors representing lower values and lighter colors representing higher values. Panel H: A heat map showing fast divergence in the mid-plane. The x-axis ranges from 0 to 2π and the y-axis ranges from 0 to 2π. The color scale indicates the intensity of horizontal divergence, with darker colors representing lower values and lighter colors representing higher values. Panel I: A heat map showing fast divergence in the vertical plane. The x-axis ranges from -π to 2π and the y-axis ranges from -π to 0. The color scale indicates the intensity of horizontal divergence, with darker colors representing lower values and lighter colors representing higher values.

Figure 11.

Energy spectra ( $\hat {E}$ ) for $\textit{Ro}$ = 0.1 (left) and 1.0 (right) at $t=4\pi$ . Reference lines with slopes $-16$ and $-2$ are added for comparison. The slope of different spectra varies and $-16$ and $-2$ are representative values around which they fluctuate.

The above comparison indicates that the barotropic and slow fields are more apt to be identified as fields that are less perturbed by fast disturbances. While the linear geostrophic balanced field takes this role at low Rossby numbers, the barotropic and slow fields are more rugged and resistant to perturbations and, therefore, form better variables in high Rossby number flows than the linear balanced field. We also find that the different decompositions led to physical structure plots of vorticity that did not differ much in the low Rossby number case (figures omitted). Following the vorticity field, figure 10 shows instantaneous snapshots of the divergence of the baroclinic, wave and fast fields. We see that all three decompositions give more or less similar qualitative features, with small-scale structures being distributed across the whole domain.

We next examine the energy spectra corresponding to the different flow components, which are given in figure 11 corresponding to the same time as the fields shown in figures 9 and 10. The left column shows the spectra for the low Rossby number case, while the right column shows the high Rossby number spectra. Comparing the two, we see steep spectra, with slopes in the neighbourhood of −16 for all six flow components in the low Rossby number regime. This is an indication that most of the energy in the low Rossby number flows reside at large scales. In contrast, the high Rossby number flow components have much shallower spectra, with slopes in the neighbourhood of −2. At much later times we find that the low Rossby number flow spectra become shallower, approaching a slope of −6. However, the high Rossby number spectra remain close to a slope of −2 even at later times.

Figure 11(c,d) shows that the baroclinic spectra (red curves) contains more energy in the inertial range than the barotropic spectra (blue curves) in panels (a) and (b). Likewise, the fast spectra (purple curves) have more energy than the slow spectra (black curves) in the inertial range. In contrast, the wave spectrum (orange curve) and balanced spectrum (green curve) exhibit comparable energy levels in the inertial range in the high Rossby number case. Following our inference from figure 9, the spectra also emphasise the previous point that the barotropic and slow flow components are more rugged with less energy at small scales when compared with the balanced component that has similar energy levels at small scales as the wave field in the high Rossby number regime. The barotropic and slow fields are therefore more useful at high Rossby numbers than the balanced field.

5.2. Energy transfers and dissipation

We now use the three kinds of flow decompositions to understand energy transfer and dissipation associated with the different modes. To begin with, we apply the wave–balance decomposition to the flow fields and derive the associated energy budget equations from (2.1). Splitting each field into wave and balanced components, such as $u = { {{u}}_{_{W} } } + { {{u}}_{_{G} } }$ for example, gives the x-velocity equation from (2.1a ) in the form

(5.3) \begin{align} \frac {\partial {{u}}}{\partial {t}} &= - Ro \left (( { {{u}}_{_{G} } } + { {{u}}_{_{W} } } ) \frac {\partial {( { {{u}}_{_{G} } } + { {{u}}_{_{W} } } )}}{\partial x} + ( { {{v}}_{_{G} } } + { {{v}}_{_{W} } } ) \frac {\partial {( { {{u}}_{_{G} } } + { {{u}}_{_{W} } } )}}{\partial y} + ( { {{w}}_{_{G} } } \right.\nonumber\\&\quad\left.+ { {{w}}_{_{W} } } ) \frac {\partial {( { {{u}}_{_{G} } } + { {{u}}_{_{W} } } )}}{\partial z} \right ) - \nu \Delta ^8 ( { {{u}}_{_{G} } } + { {{u}}_{_{W} } } )\nonumber\\ &\quad = - Ro \bigg ( \underbrace { { {{u}}_{_{G} } } \frac {\partial { { {{u}}_{_{G} } } }}{\partial x} + { {{v}}_{_{G} } } \frac {\partial { { {{u}}_{_{G} } } }}{\partial y} + { {{w}}_{_{G} } } \frac {\partial { { {{u}}_{_{G} } } }}{\partial z}}_{ N\kern -.15emL ({u_{_{GG}}})} + \underbrace { { {{u}}_{_{W} } } \frac {\partial { { {{u}}_{_{W} } } }}{\partial x} + { {{v}}_{_{W} } } \frac {\partial { { {{u}}_{_{W} } } }}{\partial y} + { {{w}}_{_{W} } } \frac {\partial { { {{u}}_{_{W} } } }}{\partial z}}_{ N\kern -.15emL ({u_{_{WW}}})}\nonumber\\ & \quad + \underbrace { { {{u}}_{_{G} } } \frac {\partial { { {{u}}_{_{W} } } }}{\partial x} + { {{v}}_{_{G} } } \frac {\partial { { {{u}}_{_{W} } } }}{\partial y} + { {{w}}_{_{G} } } \frac {\partial { { {{u}}_{_{W} } } }}{\partial z} + { {{u}}_{_{W} } } \frac {\partial { { {{u}}_{_{G} } } }}{\partial x} + { {{v}}_{_{W} } } \frac {\partial { { {{u}}_{_{G} } } }}{\partial y} + { {{w}}_{_{W} } } \frac {\partial { { {{u}}_{_{G} } } }}{\partial z}}_{ N\kern -.15emL ({u_{_{GW}}})} \bigg )\nonumber\\ &\quad - \underbrace {\nu \Delta ^8 { {{u}}_{_{G} } }}_ {{\dot {d}}_{_{uG}}} - \underbrace {\nu \Delta ^8 { {{u}}_{_{W} } } }_{{\dot {d}}_{_{uW}}}. \end{align}

In (5.3) we retained the nonlinear terms that are responsible for energy transfers and dropped the linear terms. Among the nonlinear terms, $N\kern -.15emL(u_{_{GG}})$ denotes balance–balance interactions, $N\kern -.15emL(u_{_{WW}})$ denotes wave–wave interactions and $N\kern -.15emL(u_{_{GW}})$ captures cross-interactions between wave and balance components. The terms ${\dot {d}}_{_{uG}}$ and ${\dot {d}}_{_{uW}}$ denote dissipation. Following the same strategy, equations similar to (5.3) can be derived from the $v$ , $w$ and $b$ equations of (2.1). Now, taking the three-dimensional Fourier transform of (5.3) and projecting the equations on to the wave and balanced fields following the procedure discussed in Appendix A leads to the following wave and balanced component equations from the $u$ equation:

(5.4a) \begin{align} \frac {\partial {\widehat { { {{u}}_{_{W} } } }}}{\partial {t}} =\, &\widehat {\;{\left (N\kern -.15emL ({u_{_{GG}}})\right )}_{W}} + \widehat {\;{\left (N\kern -.15emL ({u_{_{WW}}})\right )}_{W}} + \widehat {\;{\left (N\kern -.15emL ({u_{_{GW}}})\right )}_{W}} - \widehat {{{\dot d}_{_{uW}}}}, \end{align}

(5.4b) \begin{align} \frac {\partial {\widehat { { {{u}}_{_{G} } } }}}{\partial {t}} =\, &\widehat {\;{\left (N\kern -.15emL ({u_{_{GG}}})\right )}_{G}} + \widehat {\;{\left (N\kern -.15emL ({u_{_{WW}}})\right )}_{G}} + \widehat {\;{\left (N\kern -.15emL ({u_{_{GW}}})\right )}_{G}} - \widehat {{{\dot d}_{_{uG}}}}.\\[12pt]\nonumber \end{align}

Here $\widehat {\;{ (N\kern -.15emL ({u_{_{GG}}}) )}_{W}}$ is the projection of $N\kern -.15emL ({u_{_{GG}}})$ on the wave field. Similarly, other terms follow. Multiplying (5.4a ) by $ \widehat { { {{u}}_{_{W} } } }^{*}$ and adding its complex conjugate yields the time evolution of $ |\widehat { { {{u}}_{_{W} } } }|^2$ . The same steps applied to the $ v$ , $ w$ and $ b$ equations give time-evolving equations for $ |\widehat { { {{v}}_{_{W} } } }|^2$ , $ |\widehat { { {{w}}_{_{W} } } }|^2$ and $ |\widehat { { {{b}}_{_{W} } } }|^2$ . Combining these, we obtain the time-evolving spectral energy equation for the wave energy. Following a similar strategy with balanced mode equations, starting with (5.4b ), gives us a time-evolving balanced energy equation. These are

(5.5a) \begin{align} \frac {\rm d{}}{{\rm d}{t}} { {{\widehat {E}}}_{_{W} } } (k, t) = \widehat {T}_{_{WWW}}(k, t) + \widehat {T}_{_{WGW}}(k, t) + \widehat {T}_{_{WGG}}(k, t) - { {{\widehat {\dot {D}}}}_{_{W} } } (k, t),\\[-28pt]\nonumber \end{align}

(5.5b) \begin{align} \frac {\rm d{}}{{\rm d}{t}} { {{\widehat {E}}}_{_{G} } } (k, t) = \widehat {T}_{_{GGG}}(k, t) + \widehat {T}_{_{GGW}}(k, t) + \widehat {T}_{_{GWW}}(k, t) - { {{\widehat {\dot {D}}}}_{_{G} } } (k, t),\\[0pt]\nonumber \end{align}

where the left-hand sides denote the rate of change of the wave and balanced energy contained in a particular wavenumber and the right-hand sides contain the different kinds of triadic and dissipation terms that lead to the change. For example, $T_{_{WWW}}$ is a triadic interaction term involving three wave modes while $T_{_{GWW}}$ involves one balanced mode and two wave modes. The last term on the right-hand side of each equation represents dissipation at that wavenumber. Summing (5.5a ) and (5.5b ) over all wavenumbers gives us the integrated energy budget equations

(5.6a) \begin{align} \frac {{\rm d}{ { {{E}}_{_{W} } } }}{{\rm d}{t}} = { {{\dot T}}_{_{WG} } } (t) - { {{\dot D}}_{_{W} } } (t) ,\end{align}

(5.6b) \begin{align} \frac {{\rm d}{ { {{E}}_{_{G} } } }}{{\rm d}{t}} = { {{\dot T}}_{_{GW} } } (t)- { {{\dot D}}_{_{G} } } (t),\\[12pt]\nonumber \end{align}

where $ { {{\dot T}}_{_{WG} } }$ is the balance-to-wave energy transfer rate and $ { {{\dot T}}_{_{GW} } }$ is the wave-to-balance energy transfer rate. Since the wave–balance decomposition is orthogonal, the total conserved energy is the sum of wave and balanced energy. As a result, we have $ { {{\dot T}}_{_{WG} } }$ + $ { {{\dot T}}_{_{GW} } }$ = 0, which implies that the energy lost by the balance mode due to nonlinear interactions should equal energy gained by the wave modes. Here $ { {{\dot D}}_{_{W} } }$ and $ { {{\dot D}}_{_{G} } }$ represent the net dissipation of the wave and balanced components and their sum gives the total flow dissipation. Integrating (5.6a ) and (5.6b ) from $t=0$ to an arbitrary time $t$ gives us

(5.7a) \begin{align} \delta { {{E}}_{_{W} } } (t) = { {{E}}_{_{W} } } (t) - { {{E}}_{_{W} } } (0)= { {{T}}_{_{WG} } } (t) - { {{D}}_{_{W} } } (t) ,\\[-10pt]\nonumber\end{align}

(5.7b) \begin{align} \delta { {{E}}_{_{G} } } (t) = {{{E}}_{_{G} } } (t) - { {{E}}_{_{G} } } (0) = { {{T}}_{_{GW} } } (t) - { {{D}}_{_{G} } } (t) .\\[0pt]\nonumber\end{align}

In (5.7a ), $\delta { {{E}}_{_{W} } } (t)$ is the change in wave energy, while $ { {{T}}_{_{WG} } }$ and $ { {{D}}_{_{W} } }$ are the time-integrated transfer and dissipation, respectively. Likewise, (5.7b ) describes the balanced energy change due to energy transfer and dissipation.

Following a similar procedure as above, we can derive baroclinic–barotropic energy equations. For this, we decompose each field into barotropic and baroclinic components; for example, $u = { {u}_{_{T} } } + { {u}_{_{C} } }$ . Substituting these into Boussinesq equations gives us equations similar to (5.3). Vertical averaging these equations gives us the barotropic mode equations while subtracting the full equations from the vertical averaged equations gives us baroclinic mode equations. Taking the Fourier transform of these equations leads us to

(5.8a) \begin{align} \frac {\partial {\widehat { { {u}_{_{C} } } }}}{\partial {t}} =\, &\widehat {\;{\left (N\kern -.15emL ({u_{_{TT}}})\right )}_{C}} + \widehat {\;{\left (N\kern -.15emL ({u_{_{CC}}})\right )}_{C}} + \widehat {\;{\left (N\kern -.15emL ({u_{_{TC}}})\right )}_{C}} - \widehat {{{\dot d}_{_{uC}}}}, \end{align}

(5.8b) \begin{align} \frac {\partial {\widehat {u_{_{T}}}}}{\partial {t}} =\, &\widehat {\;{\left (N\kern -.15emL ({u_{_{TT}}})\right )}_{T}} + \widehat {\;{\left (N\kern -.15emL ({u_{_{CC}}})\right )}_{T}} + \widehat {\;{\left (N\kern -.15emL ({u_{_{TC}}})\right )}_{T}} - \widehat {{\dot {d}_{_{uT}}}} .\\[10pt]\nonumber\end{align}

We multiply (5.8a ) by $ \widehat { { {u}_{_{C} } } }^{*}$ and add its complex conjugate to get the time-evolution equation for $\vert \widehat { { {u}_{_{C} } } }\vert ^2$ . Following the same procedure for the other fields and combining them gives us an evolution equation for baroclinic energy. Similarly, the barotropic energy equation is derived. These can be written as

(5.9a) \begin{align} \frac {\rm d{}}{{\rm d}{t}} { {\widehat {E}}_{_{C} } } (k, t) = \widehat {T}_{_{CTC}}(k, t) + \widehat {T}_{_{CTT}}(k, t) + \widehat {T}_{_{CCC}}(k, t) - \widehat {\dot {D}}_{_{C}}(k, t) , \\[-28pt] \nonumber \end{align}

(5.9b) \begin{align} \frac {\rm d{}}{{\rm d}{t}} { {\widehat {E}}_{_{T} } } (k, t) = \widehat {T}_{_{TTT}}(k, t) + \widehat {T}_{_{TTC}}(k, t) + \widehat {T}_{_{TCC}}(k, t) - \widehat {\dot {D}}_{_{T}}(k, t). \end{align}

Similar to the wave–balance energy equations in (5.5), (5.9 a,b) contain different baroclinic–barotropic triadic interaction terms and dissipation of the modes on the right-hand side, these being the cause for the energy change in the two modes. Summing (5.9) over all wavenumbers, we obtain the net energy equations:

(5.10a) \begin{align} \frac {{\rm d}{ { {E}_{_{C} } } }}{{\rm d}{t}} = { {\dot T}_{_{CT} } } (t) - { {\dot D}_{_{C} } } (t), \end{align}

(5.10b) \begin{align} \frac {{\rm d}{ { {E}_{_{T} } } }}{{\rm d}{t}} = { {\dot T}_{_{TC} } } (t) - { {\dot D}_{_{T} } } (t) .\\[10pt]\nonumber\end{align}

As in the wave–balance equations, we have $ { {\dot T}_{_{TC} } } + { {\dot T}_{_{CT} } } = 0$ . Integrating (5.10a ) and (5.10b ) from $t=0$ to an arbitrary time $t$ gives us

(5.11a) \begin{align} \delta { {E}_{_{C} } } (t) = { {E}_{_{C} } } (t) - { {E}_{_{C} } } (0) = { {T}_{_{CT} } } (t) - { {D}_{_{C} } } (t) ,\end{align}

(5.11b) \begin{align} \delta { {E}_{_{T} } } (t) = { {E}_{_{T} } } (t) - { {E}_{_{T} } } (0) = { {T}_{_{TC} } } (t)- { {D}_{_{T} } } (t) .\\[10pt]\nonumber\end{align}

Equations (5.11a ) and (5.11b ) give the change in energy in a certain time window with time-integrated transfer and dissipation terms balancing the energy change.

We now derive energy transfer equations for the fast–slow decomposition. Applying the decomposition (5.1) to (2.1) gives us time-evolving equations for the fast and slow fields. For example, the fast $u$ equation takes the form

(5.12) \begin{align} \frac {\partial { { {{u}}_{_{F} } } }}{\partial {t}} = - Ro \underbrace { \left ( u \frac {\partial {u}}{\partial x} + v \frac {\partial {u}}{\partial y} + w \frac {\partial {u}}{\partial z} \right )_{_F} }_{ N\kern -.15emL ( { {{u}}_{_{F} } } ) } - \underbrace {\left (\nu \Delta ^8 { {{u}}_{_{F} } } \right )}_{{\dot {d}}_{_{uF}}} .\end{align}

The terms $N\kern -.15emL( { {{u}}_{_{F} } } )$ represent nonlinear terms, while $ {\dot {d}_{_{uF}}}$ denotes the dissipation term associated with the fast field of the $ u$ equation. Multiplying (5.12) by $ { {{u}}_{_{F} } }$ gives us an evolution equation for $u_{_{F}}^2$ . Combining similar terms from $v$ , $w$ and $b$ equations of (2.1) gives us the fast fields’ energy equation

(5.13) \begin{align} \frac {{\rm d}{ { {{E}}_{_{F} } } }}{{\rm d}{t}} = { {{\dot T}}_{_{FS} } } (t) - { {{\dot D}}_{_{F} } } (t), \end{align}

where $ { {{\dot T}}_{_{FS} } }$ is the domain-integrated energy transfer rate from slow-to-fast mode and $ { {{\dot D}}_{_{F} } }$ is the domain-integrated dissipation rate associated with the fast mode. Following the same procedure used to obtain (5.12) and (5.13), we get the slow energy equation

(5.14) \begin{align} \frac {{\rm d}{ { {{E}}_{_{S} } } }}{{\rm d}{t}} = { {{\dot T}}_{_{SF} } } (t) - { {{\dot D}}_{_{S} } } (t). \end{align}

Note that, for the wave–balance and baroclinic–barotropic decompositions, we kept track of different triadic interaction terms in the energy equations, thereby leading to (5.5) and (5.9). These triadic interaction terms will be used to construct spectral energy fluxes in the next subsection. The fast–slow decomposition is much more demanding in terms of data storage and, therefore, we do not keep track of different triadic interaction terms. Instead, we track the net energy transfer between fast and slow modes using (5.13) and (5.14).

Integrating (5.13) and (5.14) from $t=0$ to an arbitrary time $t$ gives us

(5.15a) \begin{align} \delta { {{E}}_{_{F} } } (t) = { {{E}}_{_{F} } } (t) - { {{E}}_{_{F} } } (0) = { {{T}}_{_{FS} } } (t) - { {{D}}_{_{F} } } (t), \end{align}

(5.15b) \begin{align} \delta { {{E}}_{_{S} } } (t) = { {{E}}_{_{S} } } (t) - { {{E}}_{_{S} } } (0) = { {{T}}_{_{SF} } } (t) - { {{D}}_{_{S} } } (t),\\[10pt]\nonumber\end{align}

where in (5.15a ), $\delta { {{E}}_{_{F} } } (t)$ indicates the change in $ { {{E}}_{_{F} } }$ during the time interval of integration, and $ { {{T}}_{_{FS} } }$ and $ { {{D}}_{_{F} } }$ are the time-integrated energy transfer and dissipation of the fast field. Likewise, (5.15b ) describes the slow energy change using transfer and dissipation terms.

We now look at some features of the fast–slow decomposition. Unlike the wave–balance and baroclinic–barotropic decompositions that satisfy orthogonality at any given time such that the flow energy can be split into the sum of the component’s energy, the fast–slow decomposition does not have a similar orthogonality condition. As detailed in Appendix B, the slow and fast components are orthogonal in the frequency space, since the two modes have non-overlapping frequency domains. However, they are not orthogonal in the time domain. Because of this, in general, we have

(5.16a) \begin{align} E(t) \ne { {{E}}_{_{S} } } (t) + { {{E}}_{_{F} } } (t) ,\end{align}

(5.16b) \begin{align} D(t) \ne { {{D}}_{_{S} } } (t) + { {{D}}_{_{F} } } (t), \end{align}

i.e. the total energy and dissipation need not equal the sum of the fast and slow components’ energy and dissipation at a given time instant. On the other hand, wave–balance and baroclinic–barotropic decompositions will have equality for the above conditions.

While the above relationships indicate that the total energy and dissipation departs from the sum of components, the degree of departure depends on the turbulent flow being examined. Looking through the different regimes, we found that the low Rossby number flows had the least departures while the high Rossby number flows had more departures between the total and sum of components. Two extreme cases are illustrated in figure 12 for a certain time window. In the top row we see different energy time series for $\textit{Ro}=0.1$ (panel a) and $\textit{Ro}=1$ (panel b). The blue curve shows slow energy $ { {{E}}_{_{S} } }$ , the red curve shows fast energy $ { {{E}}_{_{F} } }$ , the dashed black curve shows their sum and the solid black curve shows the directly computed total energy. The slow energy remains almost the same while the fast energy changes a little, these changes being highlighted in the left inset of panel (a). Furthermore, notice that the sum of the slow and fast energies (black dashed curve) departs slightly from the total energy computed separately (solid black curve), this being highlighted in the right inset in panel (a). Compared with the low Rossby number case, we see more departures for the total energy from the sum of the slow and fast energies in the high Rossby number limit shown in panel (b), with the same colours used for the variables. Changes are similar to those discussed for energy are seen for dissipation as well, these being highlighted in the bottom row of figure 12. The total dissipation (black solid curve) deviates from the sum of the fast and slow modes dissipation (dashed black curve).

Figure 12.

Energy (a,b) and dissipation (c,d) terms for the slow and fast fields at $ Ro = 0.1$ and $1.0$ .

Figure 12 Long description

The image contains four line graphs showing energy and dissipation terms for slow and fast fields at different Rossby numbers. Panel A: A line graph displays energy over time for a Rossby number of 0.1. The x-axis represents time (t) and the y-axis represents energy. The graph includes four lines: E (black), E sub S (blue), E sub F (red), and E sub S plus E sub F (dashed black). The energy levels for E and E sub S plus E sub F decrease slightly over time, while E sub S remains constant and E sub F shows a minor decline. Panel B: A line graph shows energy over time for a Rossby number of 1.0. The x-axis represents time (t) and the y-axis represents energy. The graph includes four lines: E (black), E sub S (blue), E sub F (red), and E sub S plus E sub F (dashed black). The energy levels for E and E sub S plus E sub F decrease over time, while E sub S remains constant and E sub F shows a significant decline. Panel C: A line graph displays dissipation over time for a Rossby number of 0.1. The x-axis represents time (t) and the y-axis represents dissipation. The graph includes four lines: D (black), D sub S (blue), D sub F (red), and D sub S plus D sub F (dashed black). The dissipation levels for D and D sub S plus D sub F increase over time, while D sub S remains constant and D sub F shows a slight increase. Panel D: A line graph shows dissipation over time for a Rossby number of 1.0. The x-axis represents time (t) and the y-axis represents dissipation. The graph includes four lines: D (black), D sub S (blue), D sub F (red), and D sub S plus D sub F (dashed black). The dissipation levels for D and D sub S plus D sub F increase significantly over time, while D sub S remains constant and D sub F shows a notable increase.

Figure 13.

Energy transfer terms for $\textit{Ro}$ = 0.1 and 1.0.

Results shown in figure 12 emphasise the point made with (5.16): the sum of the fast and slow energy, dissipation, etc. need not add up to the total energy and dissipation instantaneously. Despite this general result, for our regimes, we found that the departures between the total and sum of components were not much, as seen from figure 12 for low and high Rossby number regimes. In other words, the sum of components were roughly close to the total for the regimes we examined.

The above discussion emphasises that the fast and slow components do not have the same kind of orthogonality property seen in wave–balance and baroclinic–barotropic decompositions at a given time instant. However, we can get orthogonality by computing fast and slow energies averaged over a time interval, as shown in Appendix B. Time-averaged fast and slow energies will equal the time-averaged total energy and this strategy is useful for forced-dissipation turbulent flows. In this work, since we are focusing on unforced freely evolving flows where time variability of energy changes are important, we do not examine time-averaged energy budgets.

Having obtained a grip on the three kinds of flow decompositions, we now use the energy transfer equations derived earlier to see the dynamics in the low and high Rossby number regimes. Figure 13 shows the transfer terms with the left column for low Rossby numbers and the right column for high Rossby numbers. Looking at the left column, we see the striking feature that $ { {\dot T}_{_{CT} } }$ , $ { {{\dot T}}_{_{WG} } }$ and $ { {{\dot T}}_{_{FS} } }$ plotted with red curves are similar. The same holds for $ { {\dot T}_{_{TC} } }$ and $ { {{\dot T}}_{_{GW} } }$ , plotted with black curves, these being equal in magnitude and opposite in sign to $ { {\dot T}_{_{CT} } }$ and $ { {{\dot T}}_{_{WG} } }$ as seen from (5.10) and (5.6). Therefore, wave–balance and baroclinic–barotropic energy transfers are quite similar in the low Rossby number regime. The same can be said about fast–slow transfers analysed using the fast mode’s energy budget equation (5.13). We also see that there is not a unidirectional energy transfer in this limit; instead, there is cyclic two-way energy exchange leading to no significant net energy gain or loss by a mode.

As explained earlier, the slow field evolves on a longer time scale than the fast field, and so does the slow energy. Due to this, from the slow energy equation (5.14), the transfer term $ { {{\dot T}}_{_{SF} } }$ evolves much slower than the transfer term $ { {{\dot T}}_{_{FS} } }$ . Additionally, as seen above, since the fast and slow energies do not add up to the total energy, summing (5.13) and (5.14) does not give us the total energy budget. As a result, $ { {{\dot T}}_{_{SF} } } \neq - { {{\dot T}}_{_{FS} } }$ , this being seen in figure 13(e).

A noteworthy feature seen in the transfer terms in the left column of figure 13, with the exception of $ { {{\dot T}}_{_{SF} } }$ , is that the energy transfer oscillations have a period close to $\pi$ , one-half the non-dimensional inertial period or twice the inertial frequency. This behaviour can be understood by re-examining the frequency distribution of different flow components, seen earlier in figure 8(b). Notice that the wave and baroclinic modes have a peak at frequency 1, corresponding to the inertial peak. The same applies for the fast flow component, which also has an inertial peak (not shown in the figure). The balanced and barotropic modes have higher energy levels at lower frequencies, although they have a dominant peak at the inertial frequency of 1. Due to this, the energy associated with these flow components obtained by squaring the flow variables and the energy transfer terms responsible for the energy exchanges will have a dominant frequency at twice the inertial frequency. This leads to oscillations of one-half the inertial period, as seen in the left column of figure 13. Since the slow flow component evolves on a longer time scale, the slow energy and transfer term evolves much slower, as seen for the black curve in figure 13(e).

Energy transfers for the highest Rossby number and equal energy regime is shown in the right column of figure 13. Observe that no similarity is seen in the transfer rates obtained using different decomposition. Additionally, the transfer rates are two orders or more higher than those seen in the low Rossby limit in the left column, indicating stronger interactions and energy transfer at high Rossby numbers.

Figure 14 shows the time-integrated dissipations from the three decompositions, with left and right columns showing low and high Rossby number cases, respectively. Comparing panels (a) and (c), we see that the dissipation of fast, baroclinic and wave modes are an order of magnitude higher than the slow, barotropic and balanced dissipations. Additionally, the dissipations of the fast, baroclinic and wave modes are close to each other. Looking at the right column of figure 14, we see that wave and balanced dissipation, plotted with red curves, are comparable in the high Rossby number regime. In contrast, the barotropic dissipation is about a factor of eight lower than the baroclinic dissipation. Similarly, slow mode dissipation is an order of magnitude lower than the fast mode’s dissipation. This again is an indication that the baroclinic–barotropic and fast–slow decompositions are better suited for the high Rossby number regime than the wave–balance decomposition.

As follows from the energy equations presented earlier, the transfers between modes and dissipation of modes add up to give the net energy change of a mode. Figure 15 shows the energy changes in the different modes. From the left column showing the low Rossby number case, we see that the energy changes in the barotropic and balanced modes are quite similar, with a slight increase over time. The slow energy on the other hand lacks the high frequency oscillations seen in balanced and barotropic energies, although it shows a similar trend as the other two with oscillations on a longer time scale. In the low Rossby number regime shown in panel (c), the energy changes in the baroclinic, wave and fast modes are similar and all of them decrease by an order of magnitude higher than the changes seen in panel (a). Much higher changes are seen for the high Rossby number case shown in the right column, with the baroclinic, wave and fast modes losing energy with time.

Figure 14.

Dissipation of decomposed fields for $\textit{Ro}$ = 0.1 and 1.0.

Figure 14 Long description

Panel A: A line graph shows the dissipation of decomposed fields for Rossby number 0.1. The x-axis represents time (t) and the y-axis represents dissipation values multiplied by 10^-4. Three lines are plotted: black for D_T, red for D_G, and blue for D_S. The blue line shows a steady increase, the black line shows a gradual increase, and the red line remains relatively flat. Panel B: A line graph shows the dissipation of decomposed fields for Rossby number 1.0. The x-axis represents time (t) and the y-axis represents dissipation values multiplied by 10^-1. Three lines are plotted: black for D_T, red for D_G, and blue for D_S. The red line shows a rapid increase, the black line shows a gradual increase, and the blue line shows a moderate increase. Panel C: A line graph shows the dissipation of decomposed fields for Rossby number 0.1. The x-axis represents time (t) and the y-axis represents dissipation values multiplied by 10^-3. Three lines are plotted: black for D_C, red for D_W, and blue for D_F. All three lines show a steady increase. Panel D: A line graph shows the dissipation of decomposed fields for Rossby number 1.0. The x-axis represents time (t) and the y-axis represents dissipation values multiplied by 10^-1. Three lines are plotted: black for D_C, red for D_W, and blue for D_F. The blue line shows a rapid increase, the black line shows a moderate increase, and the red line shows a gradual increase.

The above comparisons emphasise that the three flow decompositions lead to similar flow fields that exhibit similar energy transfers in the low Rossby number limit, especially for the baroclinic–wave–fast components. This implies that any of the three decompositions could be used for practical purposes. In the high Rossby number regime, on the other hand, the baroclinic–barotropic and fast–slow decompositions are more useful and relevant than the wave–balance decomposition.

With regard to energy exchanges, we now make an important observation. From the blue curves in the right column of figure 14, we see that the fast mode dissipation reaches $O(1)$ values while the slow mode dissipation is $O(0.1)$ . Additionally, we see from figure 13( f) that the energy transfer between fast and slow modes is $O(0.1)$ or less. As a result, the energy loss seen in figure 15(d) for the fast mode is almost entirely due to dissipation while the slow mode energy changes, seen in figure 15(b), is an order of magnitude lower. A similar inference can be made using the baroclinic–barotropic flow decomposition: most of the baroclinic energy is lost due to dissipation with very little exchange between the two modes and low magnitude of barotropic dissipation. While the magnitudes change a lot, qualitatively similar conclusions can be drawn in the low Rossby number limit, comparing the left columns of figures 13, 14 and 15. Energy loss of the fast and baroclinic modes are almost entirely due to dissipation; slow and barotropic mode’s dissipation and fast–slow and baroclinic–barotropic energy transfers are quite small.

Therefore, irrespective of Rossby number, we conclude that the energy loss of the flow is almost entirely due to the fast and baroclinic modes losing energy to dissipation. All other energy transfers are relatively small compared with the fast and baroclinic mode’s dissipation. This is an interesting result, since the wave–vortex interaction seems to result in the wave component losing all of its energy to small-scale dissipation. The slow barotropic vortex, even at high Rossby numbers, remains relatively less perturbed while facilitating small-scale dissipation of the fast baroclinic wave field.

Connecting the above insight with the physical structure of the vorticity from different decompositions given in figure 9, notice that the barotropic and slow vorticity fields have minimal small-scale structures across the domain. In contrast, the full vorticity fields shown in §§ 3 and 4 are rich with fine-scale structures scattered throughout the domain, particularly so in the cases we identified as those where transition to turbulence had occurred; recall figure 7. As a result, much of the small-scale structures that formed and dissipated in those cases were associated with the fast field. The slow vortical field has minimal small-scale features across the domain. In other words, although we initialised inertial oscillations and a coherent vortex in different regimes, the transition to turbulence and small-scale dissipation is primarily due to the fast NIW component, with the slow and barotropic vortical component having minimal energy exchanges and dissipation.

Figure 15.

Energy of decomposed fields for $\textit{Ro}$ = 0.1 and 1.0.

5.3. Spectral energy fluxes

We now examine the spectral energy fluxes of the flow using the wave–balance and baroclinic–barotropic decompositions. We use the fluxes to estimate the kind of triadic interactions that dominate energy transfers. Storing several terabytes of data to compute triadic interactions between fast and slow modes turned out to be challenging, since this decomposition needed frequency filtering and, therefore, data needed to be stored at high resolution in time as well. Due to this, here we will not examine the spectral fluxes using the fast–slow decomposition.

To compute the spectral energy flux for wave–balance interactions, we sum (5.5), after dropping dissipation terms, from the maximum wavenumber $ k_{_{max}}$ to an arbitrary wavenumber $ k$ to get

(5.17a) \begin{align} \frac {{\rm d}{ }}{{\rm d}{t}} \sum _{p = k_{max}}^{k} { {{\widehat E}}_{_{W} } } (p, t ) = \underbrace { \varPi _{_{WWW}} (k, t) + \varPi _{_{WGW}} (k, t) + \varPi _{_{WGG}} (k, t) }_{\varPi _{_{W}}} ,\end{align}

(5.17b) \begin{align} \frac {{\rm d}{ }}{{\rm d}{t}} \sum _{p = k_{max}}^{k} { {{\widehat E}}_{_{G} } } (p, t) = \underbrace { \varPi _{_{GGG}} (k, t) + \varPi _{_{GGW}} (k, t) + \varPi _{_{GWW}} (k, t) }_{\varPi _{_{G}}} .\end{align}

Here $ { {{\varPi }}_{_{W} } }$ is the wave flux and $ { {{\varPi }}_{_{G} } }$ is the balanced flux, and each of these are split into different triadic combinations. Since the different triadic terms conserve energy, we have the following properties:

(5.18a) \begin{align} \varPi _{_{GGG}} (0, t) = \varPi _{_{WWW}} (0, t) = 0 ,\end{align}

(5.18b) \begin{align} \varPi _{_{GGW}} (0, t) + \varPi _{_{WGG}} (0, t) =0 ,\end{align}

(5.18c) \begin{align} \varPi _{_{WGW}} (0, t) + \varPi _{_{GWW}} (0, t) = 0 .\\[10pt]\nonumber\end{align}

Figure 16.

Energy fluxes for $\textit{Ro}$ = 0.1 at $t$ = 5.0.

Figure 16 Long description

Panel A: A line graph titled Barotropic mode and its triadic interactions shows the x-axis labeled k ranging from 10^0 to 10^2 and the y-axis labeled with values ranging from -0.5 to 2.0 times 10^-4. The graph includes four lines: blue for Π_TTT, red for Π_TTC, green for Π_TCC, and black for Π_T. The black line starts high and decreases sharply, while the other lines remain close to zero. Panel B: A line graph titled Balanced mode and its triadic interactions shows the x-axis labeled k ranging from 10^0 to 10^2 and the y-axis labeled with values ranging from -0.5 to 2.0 times 10^-4. The graph includes four lines: blue for Π_GGG, red for Π_GGW, green for Π_GWW, and black for Π_G. The black and green lines start high and decrease sharply, while the other lines remain close to zero. Panel C: A line graph titled Baroclinic mode and its triadic interactions shows the x-axis labeled k ranging from 10^0 to 10^2 and the y-axis labeled with values ranging from -2.0 to 0.5 times 10^-4. The graph includes four lines: green for Π_CTT, red for Π_CTC, blue for Π_CCC, and black for Π_C. The black line starts low and increases gradually, while the other lines show varying trends. Panel D: A line graph titled Wave mode and its triadic interactions shows the x-axis labeled k ranging from 10^0 to 10^2 and the y-axis labeled with values ranging from -2.0 to 0.5 times 10^-4. The graph includes four lines: green for Π_WGG, red for Π_WGW, blue for Π_WWW, and black for Π_W. The black line starts low and increases gradually, while the other lines show varying trends.

Figure 17.

Energy fluxes for $\textit{Ro}$ = 0.1 at $t$ = 7.4.

We apply a similar procedure as above for barotropic-baroclinic interactions. Summing (5.9) from $ k_{_{max}}$ to an arbitrary $ k$ , after dropping dissipation terms, gives us

(5.19a) \begin{align} \frac {{\rm d}{ }}{{\rm d}{t}} \sum _{p = k_{max}}^{k} { {\widehat E}_{_{C} } } (p, t ) = \underbrace { \varPi _{_{CCC}} (k, t) + \varPi _{_{CTC}} (k, t) + \varPi _{_{CTT}} (k, t) }_{\varPi _{_{C}}} ,\end{align}

(5.19b) \begin{align} \frac {{\rm d}{ }}{{\rm d}{t}} \sum _{p = k_{max}}^{k} { {\widehat E}_{_{T} } } (p, t) = \underbrace { \varPi _{_{TTT}} (k, t) + \varPi _{_{TTC}} (k, t) + \varPi _{_{TCC}} (k, t) }_{\varPi _{_{T}}} .\end{align}

Here $ { {\varPi }_{_{C} } }$ is the baroclinic flux and $ { {\varPi }_{_{T} } }$ is the barotropic flux, and each of these are split into different triadic combinations. As in the case of wave–balance fluxes, since the different triadic terms conserve energy, we have the following properties:

(5.20a) \begin{align} \varPi _{_{TTT}} (0, t) = \varPi _{_{CCC}} (0, t) = 0 ,\end{align}

(5.20b) \begin{align} \varPi _{_{TTC}} (0, t) + \varPi _{_{CTT}} (0, t) =0 ,\end{align}

(5.20c) \begin{align} \varPi _{_{CTC}} (0, t) + \varPi _{_{TCC}} (0, t) = 0.\\[10pt]\nonumber\end{align}

Since we are working with freely evolving flows that do not reach an equilibrium, we will not obtain a constant flux inertial range. It also does not make sense to time-average fluxes, as is often done in forced-dissipative experiments. Based on our examination of fluxes at different times, here we will therefore present instantaneous snapshots of the spectral fluxes that are representative of the generic nature of the fluxes at different times we looked at.

Figure 18.

Energy fluxes for $\textit{Ro}$ = 1.0 at $t$ = 5.0.

Figure 18 Long description

Panel A: A line graph titled Barotropic mode and its triadic interactions. The x-axis is labeled k and ranges from 10^0 to 10^2. The y-axis ranges from -0.2 to 1.0 times 10^-2. The graph includes four lines: a blue line representing Π_TTT, a red line representing Π_TTC, a green line representing Π_TCC, and a black line representing Π_T. The black line shows a peak around k = 10 and then decreases. Panel B: A line graph titled Balanced mode and its triadic interactions. The x-axis is labeled k and ranges from 10^0 to 10^2. The y-axis ranges from -0.5 to 2.5 times 10^-1. The graph includes four lines: a blue line representing Π_GGG, a red line representing Π_GGW, a green line representing Π_GWW, and a black line representing Π_G. The green line remains relatively constant, while the black line shows a slight decrease. Panel C: A line graph titled Baroclinic mode and its triadic interactions. The x-axis is labeled k and ranges from 10^0 to 10^2. The y-axis ranges from -0.5 to 2.5 times 10^-1. The graph includes four lines: a green line representing Π_CTT, a red line representing Π_CTC, a blue line representing Π_CCC, and a black line representing Π_C. The black line shows a peak around k = 10^2. Panel D: A line graph titled Wave mode and its triadic interactions. The x-axis is labeled k and ranges from 10^0 to 10^2. The y-axis ranges from -1.5 to 1.5 times 10^-1. The graph includes four lines: a green line representing Π_WGG, a red line representing Π_WGW, a blue line representing Π_WWW, and a black line representing Π_W. The black line shows a peak around k = 10^2.

Figures 16 and 17 show the barotropic and baroclinic flux and its decompositions in the left column. The right column of these figures shows the wave and balance flux and its decompositions. The fluxes are shown at two different times. From the top row of figure 16, we see that $\varPi _{_{TCC}}$ almost equals $ { {\varPi }_{_{T} } }$ and $\varPi _{_{GWW}}$ almost equals $ { {{\varPi }}_{_{G} } }$ , while the remaining components are negligible. From the bottom row of figure 16 we see that $\varPi _{_{CTC}}$ almost equals $ { {\varPi }_{_{C} } }$ and $\varPi _{_{WGW}}$ almost equals $ { {{\varPi }}_{_{W} } }$ , with the other components being relatively insignificant. Baroclinic and wave components are therefore losing energy to the barotropic and balanced flow component at this particular time. Qualitatively similar results are seen in figure 17, while the sign of the major fluxes are reversed. Therefore, the baroclinic mode is gaining energy from the barotropic flow and waves are gaining energy from the balanced flow. These periodic energy exchanges are as inferred from the left column of figure 13 in the low Rossby number regime. Importantly, the different flux components in the left and right columns of figures 16 and 17 once again emphasise that the baroclinic–barotropic and wave–balance decomposition leads to similar flow fields with similar energy transfers: across scales and integrated.

The above comparison may be contrasted with the fluxes shown in figure 18 corresponding to a high Rossby number. From the left column we see that both barotropic and baroclinic modes have a forward cascade, with $ { {\varPi }_{_{T} } }$ being closely approximated by $\varPi _{_{TCC}}$ and $ { {\varPi }_{_{C} } }$ being closely approximated by $\varPi _{_{CCC}}$ . In contrast, panel (b) shows that the balanced flow has a forward cascade while waves have an extended range of an inverse cascade. The comparison between the two columns of the figure shows that the wave–balance and baroclinic–barotropic decompositions do not yield qualitatively similar results in the high Rossby number limit. Fluxes at high Rossby numbers at other times were similar to those shown in figure 18, therefore, we omit further figures here. Importantly, the fluxes from the two decompositions do not agree at high Rossby numbers.

The above figures also highlight the range of spatial scales involved in turbulent transfers at low and high Rossby numbers. Notice from the top and bottom rows of figures 16 and 17 that the barotropic flux is negligible across wavenumbers $k \sim 10$ –100, with the dominant values being at large scales or small $k$ . In contrast, the baroclinic flux is spread across the inertial range $k \sim 10$ –100. This is an indication that in the low Rossby number regime, the baroclinic wave mode, which was initially horizontally homogeneous inertial oscillations, transforms significantly by generating smaller spatial scales with an energy transfer across a broad range scale, while the coherent barotropic vortex retains its large-scale structure and has negligible energy transfers beyond large scales. In contrast, we see from figure 18 that the barotropic and baroclinic fluxes are spread across an extended range of scales, $k \sim 10$ –100, this being a reflection of the complete transformation and breakdown of both wave and vortex modes with time in the high Rossby number regime.

We now conclude this section by briefly discussing the key features of energy transfers at other Rossby numbers. In the flow field decompositions we elaborately looked at results for Rossby numbers $\textit{Ro}=0.1$ and $\textit{Ro}=1.0$ with an equal wave–vortex energy ratio, primarily to keep it convenient for readers to follow. The key results were however seen to be similar at other Rossby numbers and energy ratios as well, which is why we omit extra figures here. For instance, the barotropic and slow fields being much more rugged to perturbation by the baroclinic and fast field with relatively less small-scale structures was a feature that persisted across Rossby numbers and energy ratios. Similarly, the energy budget of the flow in general was dominated by the dissipation of the baroclinic and fast fields. Baroclinic–barotropic and fast–slow energy exchanges and dissipation of the barotropic and slow fields were much lower compared with the dissipation of the baroclinic and fast fields.

Looking across the flow regimes, the similarity between energy transfers obtained using the three decompositions seen in the left column of figure 13 was observed only at low Rossby numbers. As the Rossby number increased, we started seeing noticeable differences between the transfers computed using different decompositions. Particularly, the notable periodicity of energy transfers in the left column of figure 13 were clearly seen for all flows with $\textit{Ro} \lt 0.5$ . For $\textit{Ro} = 0.5$ , flows with $ (E_{_{W}} / E_{_{V}})_{t = 0} =0.1$ , 0.3 and 0.5 were also seen to have periodic energy transfers. Looking back at table 1, we see that these flows have their maximum effective Rossby number during the entire flow evolution close to or below 0.1. These cases therefore correspond to asymptotically low Rossby numbers and were characterised by periodic energy transfers. A further increase in the Rossby number or energy ratio was seen to result in non-periodic energy transfers.