The research method for this study is the multiscale energetics part of the localized multiscale energy and vorticity analysis, or MS-EVA for short (cf. Liang and Robinson, 2005); also to be used is the MS-EVA-based theory of localized finite-amplitude baroclinic and barotropic instabilities (Liang and Robinson, 2007). This is a systematic line of work, involving ingredients from different disciplines such as mathematics and geophysical fluid dynamics. A comprehensive description is beyond the scope of this paper. In the following we present just a very brief introduction. The reader is referred to (Liang, 2016) for details, or, alternatively, to (Liang and Wang, 2018), section 2, for another short introduction but with more details furnished.
MS-EVA is based on a novel functional analysis tool called multiscale window transform (MWT; Liang and Anderson, 2007). With the MWT, one can split a function space into a direct sum of several mutually orthogonal subspaces, each with an exclusive range of time scales, while having its local properties preserved. Such a subspace is termed a "scale window", or simply a "window". A scale window is bounded above and below by two scale levels. In the M-window case, they are bounded above by M scale levels: j0,j1,…,jM-1. (A level j corresponds to a period 2-jτ for a time duration τ.) For convenience, these windows will be denoted with $\varpi=0,1,\ldots,M-1$, respectively.
MWT can be viewed as a generalization of the classical Reynolds decomposition; originally it was developed to faithfully represent the energies (and any quadratic quantities) on the resulting scale windows. This is the key to multiscale energetics analysis. (Liang and Anderson, 2007) found that, for some specially constructed orthogonal filters, there exists a transfer-reconstruction pair, namely, MWT and its counterpart, multiscale window reconstruction (MWR), just like Fourier transform and inverse Fourier transform. In other words, MWR is just like a filter in the traditional sense. What makes it distinctly different is that, for each MWR, there exists an MWT which gives transform coefficients that can be used to represent the energy of the filtered series. (Normally, with a traditional filter there are no such coefficients and hence energy cannot even be represented; see below).
Now, suppose {φnj(t)}n is an orthonormal translational invariant scaling sequence [built from cubic splines; see (Liang and Anderson, 2007) and Fig. 1 in (Liang, 2016)], with j some scale level, n the time step, and t the time variable. Let S(t) be some square integrable function defined on [0,1] (if not, the domain can always be rescaled to [0,1]). (Liang and Anderson, 2007) showed that, in practice, all such functions can be expanded with {φnj(t)}n as a basis; and the resulting transform, \begin{equation} \hat{S}_{n}^{j}=\int_{0}^{1}S(t)\varphi_{n}^{j}(t)dt , \ \ (1)\end{equation} for any scale level j (corresponding to frequency 2j), is called a scaling transform. Given window bounds j0,j1 for a two-window decomposition, S can then be reconstructed on the windows formed above: \begin{eqnarray} S^{\sim 0}(t)&=&\sum_{n=0}^{2^{j_{0}}-1}\hat{S}_{n}^{j_{0}}\varphi_{n}^{j_{0}}(t) , \ \ (2)\\ S^{\sim 1}(t)&=&\sum_{n=0}^{2^{j_{1}}-1}\hat{S}_{n}^{j_{1}}\varphi_{n}^{j_{1}}(t)-S^{\sim 0}(t) , \ \ (3)\end{eqnarray} where ~0, ~1 indicate the corresponding scale windows. With these MWRs, the MWT of S is defined as \begin{equation} \hat{S}_{n}^{\sim\varpi}=\int_{0}^{1}S^{\sim\varpi}(t)\varphi_{n}^{j_{1}}(t)dt , \ \ (4)\end{equation} for windows $\varpi=0,1$ and n=0,1,
,2j1-1. In terms of $\hat{S}_n^{\sim \varpi}$ the above MWRs can be written in one equation: \begin{equation} S^{\sim \varpi}(t)=\sum_{n=0}^{2^{j_{1}-1}}\hat{S}_{n}^{\sim \varpi} \varphi_{n}^{j_{1}}(t) . \ \ (5)\end{equation}
The two equations for $\hat{S}_n^{\sim \varpi}$ and $S^{\sim \varpi}(t)$ form a transform-reconstruction pair for the MWT. Note that the $S^{\sim \varpi}(t)$ are just like the low/high-pass filtered quantities which are defined in physical space, while the transform coefficients $\hat{S}_n^{\sim \varpi}$ (just like Fourier coefficients) can be used to represent multiscale energy——it has been rigorously proven that the energy on scale $\varpi$ is precisely proportional to the square of the MWT coefficients (Liang and Anderson, 2007). For example, the perturbation energy extracted from S(t) is simply $(\hat{S}_n^{\sim 1})^2$ multiplied by some constant. It is by no means the filtered quantities (S~ 1)2, which, however, has been frequently seen in the literature! Moreover, since $\hat{S}_n^{\sim \varpi}$ is localized (time location labeled by n), time variation can be spoken for the resulting energetics even though the multiscale decomposition is performed with respect to time, in contrast to the traditional Reynolds decomposition, which, if performed with respect to time, only results in time-invariant energetics.
With MWT, the available potential energy (APE) and kinetic energy (KE) densities (for convenience, we simply refer to them as APE and KE, unless confusion may arise) for window $\varpi$ can be defined, following Lorenz (1955), as \begin{eqnarray} A_{n}^{\varpi}&=&\frac{1}{2}\varrho(\widehat{T}_{n}^{\sim \varpi})^{2} , \ \ (6) \\ K_{n}^{\varpi}&=&\frac{1}{2}\hat{{v}}_{{\rm h}, n}^{\sim \varpi}\cdot\hat{{v}}_{{\rm h},n}^{\sim \varpi} . \ \ (7)\end{eqnarray} In the above definitions, v h=(u,v) is the horizontal velocity, T is the temperature anomaly [with the mean vertical profile $\overline{T}(z)$ removed], and $\varrho$ is a proportionality depending on the buoyancy frequency. In the absence of diffusion/dissipation, the multiscale energy equations for a geophysical fluid system can now be symbolically written out (location n in the subscript omitted henceforth for simplicity): \begin{eqnarray} \frac{\partial A^{\varpi}}{\partial t}&=&-{\nabla}\cdot{Q}_{\rm A}^{\varpi}+\varGamma_{\rm A}^{\varpi}+b^{\varpi}+\mathfrak{S}_{\rm A}^{\varpi} , \ \ (8)\\ \frac{\partial K^{\varpi}}{\partial t}&=&-{\nabla}\cdot{Q}_{\rm P}^{\varpi}+\varGamma_{\rm K}^{\varpi}-{\nabla}\cdot{Q}_{\rm K}^{\varpi}-b^{\varpi} , \ \ (9)\end{eqnarray} for $\varpi=0,1$. The mathematical expressions and interpretations of the terms in Eqs. (8) and (9) are tabulated in Table 1; their names are the same as many others (e.g., Orlanski and Katzfey, 1991; Chang, 1993; Yin, 2002)a(aNote that the time tendency in Eqs. (8) and (9) in Charney’s model is meaningless since the basic flow has been assumed to be steady. Nevertheless, it has nothing to do with the other terms in the energy equation, in which we are interested most.). It should be noted that all terms are localized both in space and in time; in other words, they are all four-dimensional field variables, distinguished notably from the classical formalisms in which localization is lost in at least one dimension of space-time to achieve the scale decomposition. Processes intermittent in space and time are thus naturally embedded in Eqs. (8) and (9). Figure 1 schematizes the local Lorenz cycle with a two-window decomposition.
Although the terms in Eqs. (8) and (9) have the traditional names, they are distinctly different from those in the traditional formalism. The most distinct terms are $\varGamma_\rm A^\varpi$ and $\varGamma_\rm K^\varpi$. For a scalar field S within a flow v=(u,v,w), the energy transfer from other scale windows to window $\varpi$ rigorously proves (Liang, 2016) to be (now the subscript n is supplied) \begin{equation} \varGamma_{n}^{\varpi}=-E_{n}^{\varpi}{\nabla}\cdot{v}_{\rm S}^{\varpi}= \frac{1}{2}[\widehat{({v}S)}_{n}^{\sim \varpi}\cdot{\nabla}\hat{S}_{n}^{\sim \varpi}-\hat{S}_{n}^{\sim \varpi}{\nabla}\cdot\widehat{({v}S)}_{n}^{\sim \varpi}] , \ \ (10)\end{equation} where $E_n^\varpi=[\varrho(\hat{S}_n^{\sim \varpi)^2}]/2$, with $\varrho$ some constant, is the energy on window $\varpi$ at step n [e.g., if S is the temperature anomaly, then $E_n^\varpi$ is APE; refer to (Liang, 2016) for a detailed explanation], and \begin{equation} {v}_{S}^{\varpi}=\frac{\widehat{(S{v})}_{n}^{\sim \varpi}}{\hat{S}_{n}^{\sim \varpi}} , \ \ (11)\end{equation} is referred to as the S-coupled velocity. According to Eq. (10), $\varGamma_n^\varpi$ has a Lie bracket form, reminding us of the Poisson bracket in Hamiltonian mechanics [a recent reference linking geophysical fluid dynamics (GFD) to Hamiltonian mechanics is referred to (Badin and Crisciani, 2018)]. It also possesses a very important property, \begin{equation} \sum_{\varpi}\sum_{n}\varGamma_{n}^{\varpi}=0 , \ \ (12)\end{equation} as first proposed in (Liang and Robinson, 2005) and later proved in (Liang, 2016). Physically, this implies that the transfer is a mere redistribution of energy among the scale windows, without generating or destroying energy as a whole. This property, though simple to state, does not hold in previous energetic formalisms (see below for a comparison to the classical formalism). To distinguish it from those that may have been encountered in the literature, it is termed "canonical transfer".
Canonical transfer is fundamentally associated with energy conservation among scale windows during nonlinear interactions; this forms the key of the mode-mode interaction. That is to say, a mode may receive energy or lose energy to another mode, but on the whole energy should be conserved, as stated by Eq. (12). To further illustrate how canonical transfer differs from the classical formalism, in the following we consider a special case, i.e., a case with the traditional Reynolds decomposition. Since by (Liang and Anderson, 2007), MWT is a generalization of the Reynolds decomposition (i.e., the mean-eddy decomposition), we can specialize to consider this most particular case. Now, consider a passive tracer S in an incompressible flow, and neglect diffusion for simplicity: \begin{equation} \frac{\partial S}{\partial t}+{\nabla}\cdot({v}S)=0 . \ \ (13)\end{equation} Perform a Reynolds decomposition $S=\overline{S}+S'$ (with $\overline{S}$ and S' respectively denoting the mean and perturbation), and the evolutions of the mean energy and eddy energy (variance) can be shown to be (e.g., Pope, 2000): \begin{eqnarray} \frac{\partial}{\partial t}\left(\frac{1}{2}\overline{S}^{2}\right)+{\nabla}\cdot\left(\frac{1}{2}\bar{{v}}\overline{S}^{2}\right) &=&-\overline{S}{\nabla}\cdot(\overline{{v}'S'}) , \ \ (14)\\ \frac{\partial}{\partial t}\left(\frac{1}{2}\overline{S^{\prime 2}}\right)+{\nabla}\cdot\left(\frac{1}{2}\overline{{v}S^{\prime 2}}\right) &=&-\overline{{v}^{\prime}S^{\prime}}\cdot{\nabla}\overline{S} . \ \ (15)\end{eqnarray} the terms in divergence form are generally understood as the transports of the mean and eddy energies, and those on the right-hand side as the respective energy transfers. The latter are usually used to explain the dynamical source of the mean flow-eddy interaction. Particularly, when S is a velocity component, the right-hand side of the eddy energy equation, $R=-\overline{v'S'}\cdot\nabla\overline{S}$, has been interpreted as the rate of energy extracted by Reynolds stress, or "Reynolds stress extraction" for short, against the mean field to fuel the eddy growth; in the context of turbulence research, it is also referred to as the "rate of the turbulence production" (Pope, 2000). It has also been extensively utilized in dynamic meteorology to explain phenomena such as cyclogenesis, eddy shedding, etc. However, (Holopainen, 1978) and (Plumb, 1983) found that the transport-transfer separation is not unique and hence the resulting transfer seems to be ambiguous. Moreover, the two energy equations do not, in general, sum to zero on the right-hand side. This is not what one would expect of an energy transfer, which by physical intuition should be a redistribution of energy among scale windows, and should not generate nor destroy energy as a whole.
With the MS-EVA formalism, these are not issues any more. In this special case, the energy equations in the form of Eqs. (8) or (9) become [see (Liang, 2016) for rigorous derivation; for a brief illustration, refer to the second section of (Liang and Wang, 2018)] \begin{eqnarray} \frac{\partial}{\partial t}\left(\frac{1}{2}\overline{S}^{2}\right)+{\nabla}\cdot\left(\frac{1}{2}\bar{{v}}\overline{S}^{2}+\frac{1}{2}\overline{{S{v}^{\prime}S^{\prime}}}\right)&=&-\varGamma , \ \ (16)\\ \frac{\partial}{\partial t}\left(\frac{1}{2}\overline{S^{\prime 2}}\right)+{\nabla}\cdot\left(\frac{1}{2}\overline{{v}S^{\prime 2}}+\frac{1}{2}\overline{S{v}^{\prime} S^{\prime}}\right)&=&\varGamma , \ \ (17)\end{eqnarray} where \begin{equation} \varGamma=\frac{1}{2}[\overline{S}{\nabla}\cdot(\overline{{v}^{\prime}S^{\prime}})-\overline{{v}^{\prime}S^{\prime}}\cdot{\nabla}\overline{S}] . \ \ (18)\end{equation} This is in sharp contrast to the traditional one: now, one can see that the right-hand side is balanced. This \(\varGamma\) is the "canonical transfer" in this special case. Previously, (Liang and Robinson, 2007) illustrated, for a benchmark hydrodynamic instability model (Kuo, 1949) whose instability structure is analytically known, the traditional Reynolds stress extraction R does not give the correct source of instability, while \(\varGamma\) does. We further remark that these equations result from the MWT-based multiscale energetics formalism, which are rigorously derived through reconstructing atom-like building blocks of multiscale transports [see (Liang, 2016)]; and they are unique. More specifically, if S is temperature T, then $\varGamma$ is baroclinic canonical transfer $(\varGamma_\rm A)$, \begin{align*} \varGamma_{\rm A}=\frac{\varrho}{2}[\overline{T}{\nabla}\cdot(\overline{{v}^{\prime} T^{\prime}})-\overline{{v}^{\prime} T^{\prime}}\cdot {\nabla}\overline{T}] , \ \ (19a) \end{align*} with $\varrho$ a multiplier depending on the lapse rate (see Table 1); if S is velocity (say u or v), $\varGamma$ is barotropic canonical transfer $(\varGamma_\rm K)$. With a background field $\overline{v}=(\bar u(y, z),0,0), \varGamma_\rm K$ boils down to, in terms of u and v, \begin{align*} \varGamma_{\rm K}=\frac{1}{2}[\bar{u}{\nabla}\cdot(\overline{{v}^{\prime}u^{\prime}})-\overline{{v}^{\prime}u^{\prime}}\cdot{\nabla}\bar{u}] . \ \ (19b) \end{align*} From this formula, it can clearly be seen that, within a stratified baroclinic flow (as in the Charney model), perturbations can also extract kinetic energy from the vertical shear of the basic flow.
It has been established that the canonical transfer terms ($\varGamma_A^\varpi$ and $\varGamma_\rm K^\varpi$) in Eqs. (8) and (9) are very important. Particularly, the mean-to-eddy parts of them correspond precisely to the two important geophysical fluid flow processes, i.e., baroclinic instability and barotropic instability (Liang and Robinson, 2007); details are referred to a recent publication (Liang, 2016). For notational convenience, they are written as BC and BT, respectively. A set of criteria was then derived in (Liang and Robinson, 2007) for instability identification:
(1) A flow is locally unstable if BC+ BT>0, and vice versa;
(2) For an unstable system, if BT>0 and BC≤ 0, the instability the system undergoes is barotropic;
(3) For an unstable system, if BC>0 and BC≤ 0, then the instability is baroclinic; and
(4) If both BT and BC are positive, the system must be undergoing a mixed instability.
Because of their physical meanings, in the following we refer to BT and BC as barotropic transfer and baroclinic transfer, respectively.
We remark that the concept of barotropic and baroclinic instabilities here is in the classical sense (e.g., Pedlosky, 1987): a flow is baroclinically (barotropically) unstable if potential (kinetic) energy is the only form of energy transferred from the mean flow to perturbation fields. However, the energy transfer terms used to infer instabilities, as denoted by BC and BT, are different from the traditional ones. In Eq. (7b), on spatial integration the first term (in a divergence form) on the right-hand side vanishes, whereas the second term $-\overline{v'u'}\cdot\nabla\bar{u}$ does not. The second term can be further divided into two parts: $-\overline{u' v'}\partial\bar{u}/\partial y$ and $-\overline{u'w'}\partial\bar{u}/\partial z$. In the case of quasi-geostrophic flow, because w vanishes to the first order, the kinetic energy transfer related to the vertical shear, $-\overline{u'w'}\partial\bar{u}/\partial z$, is zero. (In fact, because of this, barotropic instability is conventionally believed to be related only to the horizontal shear of the mean flow). But, it cannot be totally ignored. When localized instability is considered, it may appear significant locally, though globally it is still very small. Moreover, in some limiting cases, it could be significant. These are indeed what we will find soon in the Charney model (see below).