3.1. Overview on the theory of energy fluxes

We start here by revisiting the classical arguments for the spectral energy fluxes in wave turbulence. These arguments appear in Zakharov et al. (Reference Zakharov, L'vov and Falkovich1992). Here, we revisit these arguments to prepare for additional insights into spectral energy transfers that are obtained by using our formalism. Our first application of (2.1) is to isotropic scale-invariant systems. We assume a power-law dispersion relation allowing for three-wave resonant interactions and scale-invariant matrix elements with homogeneity exponent $m$,

(3.1a,b)\begin{equation} \omega_p = \kappa p^\alpha,\ \alpha>1,\quad \left|V_{12}^0\right|^2=V_0^2 p^{2m}\, f\left(\frac{p_1}{p},\frac{p_2}{p}\right), \end{equation}

allowing us to look for general solutions to (1.1) of the form

(3.2)\begin{equation} n_p=A p^{{-}s}. \end{equation}

Let us start by reviewing some classical results for the energy fluxes in such systems, summarized in Chapter 3 of Zakharov et al. (Reference Zakharov, L'vov and Falkovich1992). In direct analogy with the local energy cascades in isotropic turbulence, it is assumed that the interactions are sufficiently local in Fourier space (Kolmogorov Reference Kolmogorov1941; Kraichnan Reference Kraichnan1959; Rose & Sulem Reference Rose and Sulem1978; Eyink Reference Eyink2005) so that one can assume a differential continuity equation for the 1-D spectral energy density

(3.3)\begin{equation} \frac{\partial e_p}{\partial t} = {\rm \pi}(2p)^{(d-1)}\omega_p I_p ={-}\frac{\partial F}{\partial p}, \end{equation}

where the right-hand side of the WKE is interpreted as minus the divergence of a flux $F$. Here, $I_p$ is the collision integral, multiplied by the area of the $d$-dimensional sphere. Supposing that there are no energy sources or sinks in an inertial range $[\epsilon, M]$, taking $\epsilon \to 0$ and $M\to \infty$, and solving for the flux $F$, one obtains

(3.4)\begin{equation} F(p) ={-}{\rm \pi}\int_0^p{\rm d}p'\, (2p')^{(d-1)} \omega_p' I_{p'} . \end{equation}

Interpreting the collision integral as the divergence of a pointwise flux subtends the intuition that energy transfers happen locally in Fourier space. The underlying reasoning involves the following steps. Assume a partition of Fourier space into small boxes of width $\Delta p$. Assume that the time variation of the energy contained in the box between $p$ and $p+\Delta p$, say $\dot e_{[p,p+\Delta p]}$ is due only to the energy exchanges with its two adjacent boxes. Call $F_p$ the net power exchanged at $p$, and $F_{p+\Delta p}$ the net power exchanged at $p+\Delta p$. Express energy conservation for the box under consideration as $\dot e_{[p,p+\Delta p]} = F_p - F_{p+\Delta p}.$ Now take $\Delta _p\to 0$, and obtain (3.3) by standard transition to a continuum representation. In turbulence, the conditions on how fast the correlations have to decay for the transfers to be sufficiently local are studied in Kraichnan (Reference Kraichnan1959) and Eyink (Reference Eyink1994). In wave turbulence, a transposition of the same arguments leads to the statement that if the collision integral is convergent, then the interactions are sufficiently local for the differential conservation picture (3.3) to hold. For this reason, the convergence conditions for the collision integral in (1.1) are named the ‘locality conditions’ (Zakharov et al. Reference Zakharov, L'vov and Falkovich1992). However, we are also not aware of a rigorous proof of this fact.

When locality holds, the expression for the instantaneous energy flux (3.4) is valid in general, in both stationary and non-stationary conditions. The wave turbulence theory of scale-invariant spectra (Zakharov et al. Reference Zakharov, L'vov and Falkovich1992) focuses on the stationary solutions to (1.1). These can be equilibrium ($F=0$) or non-equilibrium ($F={\rm const}.$) solutions, i.e. the Rayleigh–Jeans (RJ) and the Kolmogorov–Zakharov (KZ) solutions, respectively. The KZ spectrum can be obtained dimensionally or via the Zakharov–Kraichnan conformal transformations (Zakharov & Filonenko Reference Zakharov and Filonenko1967a; Zakharov et al. Reference Zakharov1972), and we have

(3.5)\begin{equation} n_p^{RJ} =A p^{-\alpha},\quad n_p^{KZ}=A p^{{-}s_0},\quad \text{with}\ s_0=m+d. \end{equation}

A paradox (only apparent) has to be solved: a constant flux $F\neq 0$ must result from integrating a vanishing integrand in (3.4)! It is convenient to switch to $\omega$ space using the dispersion relation as the change of variables, by defining

(3.6)\begin{equation} I_\omega={\rm \pi} (2p)^{d-1} \left(\frac{{\rm d}\omega}{{\rm d}p}\right)^{{-}1}I_p,\quad \text{so that }\ I_\omega = \omega^{\sigma-2}(V_0A)^2\,\mathcal{I}(s), \end{equation}

where $\mathcal {I}(s)$ is a non-dimensional integral that vanishes in the stationary states, and $\sigma = 2(m+d-s)/\alpha$. Now (3.4) reads

(3.7)\begin{equation} F(\omega) ={-} \int_0^\omega {\rm d}\omega'\, \omega' I_{\omega'} ={-}\omega^\sigma (V_0A)^2\, \frac{\mathcal{I}(s)}{\sigma}. \end{equation}

At the KZ solution (3.5), we have $\sigma =0$, and therefore an indeterminate form $0/0$. This indeterminate form is then regularized by Taylor-expanding $\mathcal {I}(s)$ to first order, or equivalently by using l'Hôpital's rule. We thus obtain

(3.8)\begin{equation} F ={-} (V_0A)^2 \left.\frac{{\rm d}\mathcal{I}}{{\rm d}s}\right|_{s=s_0}, \end{equation}

where the locality conditions ensure that ${\textrm {d}\mathcal {I}}/{\textrm {d}s}|_{s=s_0}$ is finite, with the property that the flux is positive if $s_0>\alpha$, i.e. the KZ spectrum is steeper than the equilibrium spectrum. The solution does not exist if $s_0<\alpha$. Moreover, note that $F$ is independent of $\omega$, consistently with stationarity and corresponding to a constant downscale energy flux in the wave turbulence inertial range.

3.2. Application of the main statement (2.1) to isotropic systems

Using integration variables in $\omega$ space, in isotropic conditions (1.1) simplifies to

(3.9)\begin{equation} \left. \begin{aligned} & \frac{\partial n_p}{\partial t} = \frac{v_p}{p^{d-1}}\int_{0}^\infty {\rm d}\omega_1 \left( J^{({\rm I})}(\omega_p;\omega_1,\omega_p-\omega_1)+2 J^{({\rm II})}(\omega_p;\omega_1,\omega_1-\omega_p)\right) ,\\ & \text{where}\quad J^{({\rm I})}(\omega_p;\omega_1,\omega_2) = R^0_{12},\quad J^{({\rm II})}(\omega_p;\omega_1,\omega_2) ={-} R^1_{02},\\ & \text{and}\quad\quad R^0_{12} = 4{\rm \pi} \kappa^{3(1-d)/\alpha} \frac{(\omega\omega_1\omega_2)^{({d-1})/{\alpha}}}{v_p v_1v_2}\,\frac{|V^0_{12}|^2\, f^0_{12}}{\Delta_d}. \end{aligned} \right\} \end{equation}

We have used the notation $v_p = \partial \omega /\partial p$, and $\Delta _d$ is defined by angle integration of the wavenumber delta function given space isotropy, with the dimensions of a wavenumber to the $d$th power. We assume a scale-invariant solution

(3.10)\begin{equation} n_p = A \omega_p^{{-}x}. \end{equation}

Using these variables, saying that the interaction kernel $J(\omega _p;\omega _1,\omega _2)$ has homogeneity exponent $\gamma _0-2x$, the KZ solution has exponent $x=(\gamma _0+3)/2$, and the RJ solution has exponent $x=1$.

By formula (2.1), given $A$ and $B$ as two disjoint closed subsets of Fourier space (spanned by $\omega \in \mathbb {R}^+$), the instantaneous power delivered from $A$ to $B$ amounts to

(3.11)$$\begin{gather} \mathcal{P}_{A\rightarrow B} ={-}2^{d-1}{\rm \pi} \int_A {\rm d}\omega' \,\omega' \int_0^\infty {\rm d}\omega_1 \left(\chi_{B}^{({\rm I})}(\omega_1)\,J^{({\rm I})}(\omega';\omega_1,\omega'-\omega_1)\right.\nonumber\\ \left.{}+2 \chi_{B}^{({\rm II})}(\omega_1)\, J^{({\rm II})}(\omega';\omega_1,\omega_1-\omega')\right), \end{gather}$$

where the dependence on $\omega _2$ in the interaction weight is constrained implicitly by $\omega _2=\omega '-\omega _1$ in the first line and by $\omega _2=\omega _1-\omega '$ in the second line. Let us choose $A=[0,\omega ]$, $B=[\omega,+\infty ]$ to make a concrete calculation in a specific case, noting that in principle this corresponds to the computation in (3.7). As represented in figure 1(b), this choice of sets leads to the major simplification

(3.12)\begin{equation} \chi_{B}(\omega_1) = \varTheta(\omega_1-\omega), \end{equation}

where $\varTheta ({\cdot })$ denotes the Heaviside step function. Moreover, as also shown in figure 1(a), the resonant manifold is such that $J^{(\textrm {I})}(\omega ;\omega _1,\omega _1-\omega ')=0$ for $\omega _1>\omega '$, and $J^{(\textrm {II})}(\omega _p;\omega _1,|\omega '-\omega _1|)=0$ for $\omega _1<\omega '$. Thus from (3.11) we obtain

(3.13)\begin{equation} \mathcal{P}_{[0,\omega]\rightarrow [\omega,+\infty)} ={-}2^{d}{\rm \pi} \int_0^\omega {\rm d}\omega' \,\omega' \int_\omega^{+\infty} {\rm d}\omega_1\,J^{({\rm II})}(\omega';\omega_1,\omega_1-\omega'). \end{equation}

We are going to derive (3.7) analytically from (3.13), showing that the main statement (2.1) in § 2 encompasses the standard theory of energy fluxes as a particular case. This proof relies on the detailed conservation property (1.5), from which we see that

(3.14)\begin{equation} \omega_a\,J(\omega_a;\omega_b,\omega_c) + \omega_b\,J(\omega_b;\omega_a,\omega_c) + \omega_c\, J(\omega_c;\omega_a,\omega_b) =0 \end{equation}

for any triad of wavenumbers $\boldsymbol {p}, \boldsymbol {p}_1, \boldsymbol {p}_2$. An independent proof by construction for the isotropic case is given in Appendix B. We suggest that the reader examines this proof for an intuitive graphical interpretation of detailed conservation that relies on the symmetries of the resonant manifold.

Figure 1.

(a) Representation of the resonant manifold in the $\omega _1\unicode{x2013} \omega _2$ space for triads involving wavenumbers $\omega '$, $\omega _1$, $\omega _2$. Here, $\omega$ demarcates the separation between sets $A$ and $B$ (dashed red lines). The three resonant branches are represented as the three solid red lines labelled I, II, III. The shaded area denotes the region satisfying the wavenumber delta function condition, for a value $\alpha >1$. (If $\alpha <1$, then this area becomes disjoint from the frequency condition lines, and there are no resonances.) (b) The values of the characteristic interaction weight $\chi _{B\rightarrow \omega '}$ in the different regions, as given by the main statement (2.1). Highlighted by a red rectangle are the relevant cases for each region, due to the interaction type present in that region. Exploiting the symmetry with respect to the main diagonal and considering only $\omega _2\le \omega _1$ (below the dashed black line), it is clear that $\chi _{B\rightarrow \omega '}=\varTheta (\omega _1-\omega )$.

3.3. Proof of the standard flux formula (3.7)

Property: vanishing self-interactions. The following property holds:

(3.15)\begin{equation} \int_0^\omega {\rm d}\omega'\int_0^\omega {\rm d}\omega_1 \, J(\omega',\omega_1, |\omega'-\omega_1|)=0, \end{equation}

This follows directly from detailed conservation, as shown in Appendix A.

The meaning of this property is that the integral (3.15) quantifies the flux from $[0,\omega ]$ to $[0,\omega ]$, i.e. self-interactions that amount to no net transfer of energy. This leads to the following important corollary of the main statement (2.1).

Property: Retrieving the standard flux formula for isotropic systems. The standard flux formula (3.7) that is used to calculate the energy flux in isotropic wave turbulence is a direct consequence of (2.1) (main statement) and (3.15).

Proof. Equation (3.13) is derived directly from (2.1), in the particular case of isotropic systems and adjacent control intervals $A=[0,\omega ]$, $B=[\omega,+\infty ]$. Exercising the freedom to add zero (i.e. (3.15)) to (3.13), we obtain

(3.16)\begin{align} \mathcal{P}_{[0,\omega]\rightarrow [\omega,+\infty)} &={-}2^{d-1}{\rm \pi} \int_0^\omega {\rm d}\omega'\, \omega'\left[\int_\omega^{+\infty} {\rm d}\omega_1\,J(\omega',\omega_1, |\omega'-\omega_1|) \right.\nonumber\\ & \quad \left. {}+ \int_0^\omega {\rm d}\omega_1\,J(\omega',\omega_1, |\omega'-\omega_1|) \right]\nonumber\\ & ={-}2^{d-1}{\rm \pi} \int_0^\omega {\rm d}\omega'\,\omega'\int_0^{+\infty} {\rm d}\omega_1\, J(\omega',\omega_1, |\omega'-\omega_1|)\nonumber\\ & = \int_0^\omega {\rm d}\omega' \, \omega' I_{\omega'} = F(\omega), \end{align}

which concludes the proof of validity of the usual flux formula (3.7) starting from the main statement (2.1).

As highlighted in (3.16), note that the classical flux expression $F(\omega )$ (3.7) contains a self-interaction contribution in the interval $[0,\omega ]$. This contribution is vanishing due to (3.15). Moreover, (3.7) requires regularization at the KZ solution (see (3.8)). Equation (3.13) is free of such limitations. We elaborate on these points in § 4 by considering surface capillary waves.

3.4. Quantifying locality: the transfer integral

In order to explore the full potential of the main statement (2.1), let us introduce a slight generalization of (3.13). Performing the outer integration up to a smaller frequency $\tilde \omega <\omega$ allows us to express the power that from the interval $[0,\tilde \omega ]$ is delivered instantaneously to $[\omega, +\infty )$:

(3.17)\begin{equation} \mathcal{P}_{[0,\tilde\omega]\rightarrow [\omega,+\infty)} ={-}2^{d}{\rm \pi} \int_0^{\tilde\omega} {\rm d}\omega' \,\omega' \int_\omega^{+\infty} {\rm d}\omega_1\, J^{({\rm II})}(\omega';\omega_1,\omega_1-\omega'). \end{equation}

This is the wave turbulence analogue of (6.4) in Kraichnan (Reference Kraichnan1959). Recalling that the collision kernel has homogeneity exponent $\gamma _0-2x$, with a change of variables $\varOmega =\omega '/\omega$, we obtain

(3.18)\begin{equation} \mathcal{P}_{[0,\tilde\omega]\rightarrow [\omega,+\infty)} = (V_0A)^2 \,\omega^{y+1} \int_{0}^{\tilde\omega/\omega} {\rm d}\varOmega \,T(\varOmega), \end{equation}

where $y=\gamma _0+2-2x$, with the following.

Definition: transfer integral.

(3.19)\begin{equation} T(\varOmega):={-}2^{d}{\rm \pi} (V_0A)^{{-}2}\;\varOmega^{y} \int_{\varOmega^{{-}1}}^{+\infty} {\rm d}\xi \, J^{({\rm II})}(1;\xi,\xi-1) \end{equation}

is the transfer integral of the problem.

The transfer integral is a non-dimensional function that captures the inter-scale ‘structure’ of the energy transfers between two disconnected regions of Fourier space. In particular, it quantifies the direct transfer by a given frequency (smaller than $\omega$) to all frequencies larger than $\omega$. The integral of $T(\varOmega )$ up to $\tilde \omega /\omega$ gives the distant-transport power exchanged between the two regions $[0,\tilde \omega ]$ and $[\omega,+\infty )$. Because of the scale invariance of the problem, $T(\varOmega )$ is defined uniquely no matter the chosen values of $\tilde \omega$ and $\omega$. It has to be computed only once, and then the boundaries of the two sets enter the problem as the upper integration boundary and as the scaling factor in (3.18).

Using the power $\mathcal {P}$ between adjacent sets, by using (3.10)–(3.18), we are able to express the Kolmogorov constant of the problem (Zakharov et al. Reference Zakharov, L'vov and Falkovich1992) as a function of the transfer integral itself, for the KZ solution:

(3.20)\begin{equation} n_p^{KZ} = \kappa_K \sqrt{\mathcal{P}} \omega^{{-}x_{KZ}},\quad \kappa_K = \left(V_0 \int_0^1T(\varOmega)\,{\rm d}\varOmega \right)^{{-}1}. \end{equation}

This inter-scale decomposition of the Kolmogorov constant is one of the important implications of the main statement (2.1).

How fast $\mathcal {P}_{[0,\tilde \omega ]\rightarrow [\omega,+\infty )}$ tends to zero as $\tilde \omega /\omega \to 0$ describes how ‘local’ or ‘diffuse’ the energy cascade is (Kraichnan Reference Kraichnan1959). This scaling is going to be dictated by the asymptotics of $T(\varOmega )$, and allows us to improve the binary notion of locality (i.e. local/non-local) towards a more quantitative description. How wide should the separation between forcing and dissipation regions be in order to have an inertial range sufficiently disconnected from direct interaction with the boundaries? The transfer integral $T(\varOmega )$ provides a key perspective to tackle this type of question, as will be illustrated in the following sections.

4.1. Application of the main statement (2.1): transfer integral and the Kolmogorov constant

Let us consider the problem of surface capillary waves in isotropic conditions (Pushkarev & Zakharov Reference Pushkarev and Zakharov2000), for which $d=2$. This system has a dispersion relation with $\alpha =\tfrac 32$, allowing for three-wave resonances. After writing the equation in frequency variables and averaging over the angles in $\boldsymbol {p}$ space as in (3.9), the interaction kernel has homogeneity exponent $\gamma _0-2x$, with $\gamma _0=\tfrac 83$. Therefore, the stationary states of the system, in the form $n_{\boldsymbol {p}}=A\omega ^{-x}$, are the RJ equilibrium spectrum with $x=1$, and the KZ spectrum with $x=(\gamma _0+3)/2=\tfrac {17}{6}$. The convergence conditions of the collision integral determine the locality interval $[\tfrac 56,5]$, which includes both stationary solutions. For the explicit form of the WKE, we refer the reader to Pushkarev & Zakharov (Reference Pushkarev and Zakharov2000). We perform the analytical calculations of the locality conditions in Appendix C. These calculations are not new per se, as they are implied in Pushkarev & Zakharov (Reference Pushkarev and Zakharov2000). Since we were not able to find these calculations in the literature, we included them here. In figure 2(a), we show a numerical evaluation of the non-dimensional collision integral $\mathcal {I}(x)$, vanishing in the two stationary states. In figure 2(b), we show the numerically calculated energy flux between two adjacent sets. This is done in two ways, according to (3.7) and (3.13), showing perfect agreement between the two as proven analytically in (3.16). With the precision adopted, the numerical value so obtained at the KZ solution via (3.13) is identical to the value from the regularization formula (3.8), up to a relative error of the order of $1/1000$. Via the inversion (3.20), this value relates directly to the Kolmogorov constant of the capillary wave problem (Pushkarev & Zakharov Reference Pushkarev and Zakharov2000). In the most up-to-date estimates, a direct comparison with the measured flux finds an agreement within a factor around 1.5–2 from numerical simulations of the equations of motion (Deike et al. Reference Deike, Fuster, Berhanu and Falcon2014b; Pan & Yue Reference Pan and Yue2014; Pan Reference Pan2017), and within a factor of approximately 3–4 from experiments (Deike, Berhanu & Falcon Reference Deike, Berhanu and Falcon2014a).

Figure 2.

(a) Values of the non-dimensional collision integral $\mathcal {I}(x)$ as a function of the spectral exponent $x$ in the locality interval. The two zeros are the RJ and KZ solutions, indicated by magenta and red vertical lines, respectively. (b) Energy flux normalized by its scaling in $\omega$, as a function of $x$. The plot shows perfect agreement between the numerical evaluation of the standard flux formula and our formula. Moreover, note that the latter does not need to be regularized at the KZ solution because it does not contain an indeterminate form $0/0$, and the result is identical to the regularization by l'Hôpital's rule. (c) We represent the metrics introduced in (4.1a,b) to characterize how local the energy transfers are. All solutions are local for $x\in [5/6,5]$, intended as having an integrable collision operator. However, locality as quantified by (4.1a,b) is stronger for small values of $x$, increasing as $x\to 5$ (where $\varOmega _{5\,\%}\to 0$).

Note that the new formula (3.13) can be applied throughout the locality interval, including at the KZ solution, because it does not contain an indeterminate form ‘$0/0$’, as discussed above. Moreover, the decomposition of the power in terms of the transfer integral (3.19) is now available also for the KZ solution. We point out that the integrand of the regularization formula (3.8), containing a logarithmic function, is not equivalent to the transfer-integral decomposition (3.19). Indeed, only the latter can be used to quantify locality and distant-transport scalings.

4.2. Metrics of locality and distant transport

We next exploit the formalism of § 3.4 to decompose the energy flux based on the relative separation of the frequencies involved in the transport. We define the quantities $\varOmega _{5\,\%}$ and $\varOmega _{50\,\%}$ as

(4.1a,b)\begin{align} \int_{0}^{\varOmega_{5\,\%}} {\rm d}\varOmega \,T(\varOmega) := 0.05 \int_{0}^{1} {\rm d}\varOmega \,T(\varOmega) ,\quad \int_{0}^{\varOmega_{50\,\%}} {\rm d}\varOmega \,T(\varOmega) := 0.5 \int_{0}^{1} {\rm d}\varOmega \,T(\varOmega) . \end{align}

The first quantity measures the length of the tail of the transfer integral that contains 5 percent of the total energy transfer. This quantity therefore indicates how far apart two regions in Fourier space have to be for their mutual interactions to be negligible. We define ‘negligible’ to be five per cent of total flux of energy. The second quantity is the median threshold of the transfer integral: half of the energy flux is exchanged within this threshold range, and the other half is exchanged from further than this threshold. Figure 2(c) shows the dependence of $\varOmega _{5\,\%}$ and $\varOmega _{50\,\%}$ on the spectral exponent $x$. The median $\varOmega _{50\,\%}$ is always quite close to $1$, with a minimum around the KZ solution where $\varOmega _{50\,\%}\simeq 0.7$. However, the tail metric $\varOmega _{5\,\%}$ is decreasing from a value around $0.5$ in the neighbourhood of the RJ solution, and tends to zero as $x\to 5$. In particular, at the KZ solution we have $\varOmega _{5\,\%}\simeq 0.2$. This means that frequencies that are separated by up to more than half a decade are still giving a relevant contribution to direct energy transport in the KZ stationary state!

The details of the transfer integral calculations are shown in figure 3, for three different values of $x$: $4.5$, $\tfrac {17}{6}$ and $0.9$. Figures 3(a,c,e) show the magnitude of the interaction kernel. In the first two cases, the type I contributions are positive and the type II negative, corresponding to a direct cascade. In the final case, the signs are exchanged, corresponding to inverse cascade for $x<1$.

Figure 3.

Interaction kernel and transfer integral for three different values of $x$. All plots are in log-log scale. The three solutions here represented are ‘local’: the interaction kernel is integrable. The limiting scalings of the three locality conditions IR, UV1 and UV2 (cf. § 7) are represented by the black dashed lines.

The singularity in $1$ for large values of $x$ behaves like $|\omega _1/\omega -1|^{-x+3}$. With the two integrations in (3.17), the convergence condition must be $x<5$, retrieving the infrared (IR) locality condition. This implies that the transfer integral is dominated by an integrable singularity $T(\varOmega )\simeq (1-\varOmega )^{-x+4}$ for $\varOmega \to 1$, when $x>4$.

The scaling of the interaction kernel for $\omega _1/\omega \gg 1$ is given by $(\omega _1/\omega )^{-x-1/6}$ for $x\simeq 1$, and by $(\omega _1/\omega )^{-x-1/3}$ for $x\gg 1$. For the transfer integral $T(\varOmega )$ to converge, it must have $-x-1/6<-1$, which gives the familiar ultraviolet (UV) locality condition $x>5/6$. Notice the proximity of the case $x=0.9$ to this limit scaling in figure 3(e). By (3.19), this implies an asymptotic scaling $T(\varOmega )\simeq {\varOmega ^{4-x}}$ for $\varOmega \ll 1$. This will be discussed further in § 7.

Let us use this result to estimate the asymptotic scaling of the distant-transport power:

(4.2)\begin{equation} \mathcal{P}_{[0,\tilde\omega]\rightarrow [\omega,+\infty)} = \int_0^{{\tilde\omega}/{\omega}} T(\varOmega)\,{\rm d}\varOmega \sim \left(\frac{\tilde\omega}{\omega}\right)^{5-x},\quad \text{as} \ \frac{\tilde\omega}{\omega} \to 0. \end{equation}

For the KZ solution, $x=17/6$, this yields $\mathcal {P}_{[0,\tilde \omega ]\rightarrow [\omega,+\infty )} \sim ({\tilde \omega }/{\omega })^{13/6}$.

Thus the energy cascade at the KZ stationary solution of capillary waves can be considered quite strongly local; moreover, the energy transport for spectra that are steeper than KZ becomes more and more diffuse, while for whiter spectra, it becomes more and more local (cf. figure 2). Around equilibrium, $x\simeq 1$, the scaling decay is $({\tilde \omega }/{\omega })^{23/6}$. The analysis presented here suggests a viable approach to quantifying how far from the dissipation and forcing regions one should be in order for direct energy transfers with the boundaries to be fairly negligible. Taking $\varOmega _{5\,\%}$ as a reasonable (albeit arbitrary) cutoff for a notion of ‘negligibility’, for KZ we would obtain at least a factor of $5$ of separation from each boundary. A criterion of this sort would exclude approximately $1.4$ orders of magnitude (half on each side) from being a part of an inertial range fairly independent of both the forcing and the dissipation regions. For surface capillary waves, which are constrained on scales from $0.5$ mm to $17$ mm, there are approximately $2.3$ orders of magnitude of available frequencies, which is not much larger than $1.4$. We refer the reader to § 7 for further discussion.

These and similar quantifications of fluxes and associated level of locality are applicable directly to any wave turbulence system. They open the possibility to an analysis of energy transfers that goes beyond the mere stationary states to explore transients, boundary effects and a scale-by-scale decomposition of the energy transfer contributions.

5.1. Overview on the theory of energy fluxes

A direct extension to anisotropic systems of the theory of energy fluxes reviewed in § 3.1 is possible, in scale-invariant and stationary conditions. It consists of the use of generalized Zakharov–Kraichnan–Kuznetsov conformal transformations (Kuznetsov Reference Kuznetsov1972) to find generalized KZ solutions (Zakharov et al. Reference Zakharov, L'vov and Falkovich1992; Nazarenko Reference Nazarenko2011). Each of these solutions corresponds to the stationary cascade solution of one of the positive-definite conserved quantities of the WKE. In principle, each of these positive invariants also corresponds to an independent equilibrium solution. However, at variance from the isotropic case, these types of equilibrium and non-equilibrium stationary solutions are not the only possible stationary solutions, but only particular ones. In the case of 3-D systems with two effective independent dimensions, there are two families of an infinite number of equilibrium and non-equilibrium solutions, respectively represented by the points of two 1-D curves in the two-dimensional (2-D) plane of possible power-law exponents. Physical examples are the Rossby/drift waves, where there are three positive collision invariants and three KZ solutions (Balk, Zakharov & Nazarenko Reference Balk, Zakharov and Nazarenko1990; Nazarenko Reference Nazarenko2011), or the internal gravity waves, where there is one known collision invariant (the energy) and one corresponding KZ solution (Pelinovsky & Raevsky Reference Pelinovsky and Raevsky1977; Lvov & Tabak Reference Lvov and Tabak2001). Another remarkable recent application is found in Galtier (Reference Galtier2006) and in David & Galtier (Reference David and Galtier2022) to the problem of inertial electron magnetohydrodynamics in plasma physics. One subsequent necessary step in the theory is the verification that these stationary solutions correspond to a convergent collision integral, i.e. that they are local. For internal gravity waves, for instance, there is only one local stationary solution that is found numerically (Lvov et al. Reference Lvov, Tabak, Polzin and Yokoyama2010) and is different from the KZ solution.

The calculation of the fluxes for the anisotropic KZ solutions leads to a regularization similar to (3.7)–(3.8). Here, we make use of simple-minded dimensional analysis to illustrate its properties. For concreteness, we will use the example of horizontally isotropic internal gravity waves. In the hydrostatic approximation, the scale-invariant dispersion relation and matrix elements read (Olbers Reference Olbers1976; Lvov & Tabak Reference Lvov and Tabak2004)

(5.1)\begin{equation} \omega_{\boldsymbol{p}} = \gamma\,\frac{k}{m},\quad \left|V_{12}^0\right|^2=V_0^2 k^{2\mu_k}m^{2\mu_m}\, f\left(\frac{k_1}{k},\frac{k_2}{k},\frac{m_1}{m},\frac{m_2}{m}\right), \end{equation}

where $\gamma$ and $V_0$ are dimensional constants, and $k$ and $m$ are the magnitudes of the (2-D) horizontal and (1-D) vertical wavenumbers, respectively. Moreover, we have $\mu _k=3/2$ and $\mu _m=-1/2$. In an inertial range where no external forcing or dissipation is present, we look for general stationary solutions for the action density of the form

(5.2)\begin{equation} n_{\boldsymbol{p}}=A k^{{-}a} m^{{-}b}. \end{equation}

Let us study energy propagation in the positive quadrant for $k\unicode{x2013} m$, by defining the horizontally averaged wave action $n(k,m)= 4{\rm \pi} k n_{\boldsymbol {p}}$ and energy $e(k,m)=\omega _{\boldsymbol {p}} n(k,m)$. The standard use of the differential conservation equation for energy yields

(5.3)\begin{equation} \frac{\partial e(k,m)}{\partial t} = 4{\rm \pi} k \omega_{\boldsymbol{p}} I_{\boldsymbol{p}} ={-}\frac{\partial F_k(k,m)}{\partial k} - \frac{\partial F_m(k,m)}{\partial m}, \end{equation}

where the dependence on time is implicit, $I_{\boldsymbol {p}}$ is the collision integral, and $F_k$ and $F_m$ are the horizontal and vertical components of the energy flux in $k\unicode{x2013} m$ space. Using (5.1)–(5.2), from dimensional analysis we obtain

(5.4)\begin{equation} k \omega_{\boldsymbol{p}} I_{\boldsymbol{p}} = (V_0 A)^2 k^{6-2a} m^{{-}2b}\,\mathcal{I}(a,b), \end{equation}

where $\mathcal {I}(a,b)$ is the non-dimensional collision integral that vanishes in the stationary states. Let us plug this into the right-hand side of (5.3). We follow similar reasoning as for (3.7) in the isotropic case. Let us assume that $F_k=F_k(m)$ and $F_m=F_m(k)$ – a posteriori, the KZ solution is found to enjoy this property. Under this assumption, we integrate in $k$ from $0$ to $k$ to obtain the horizontal component of the flux, and in $m$ from $0$ to $m$ to obtain the vertical component. We obtain

(5.5a,b)\begin{align} F_k ={-}4{\rm \pi}(V_0 A)^2\,\frac{k^{7-2a}m^{{-}2b}}{7-2a}\,\mathcal{I}(a,b),\quad F_m ={-}4{\rm \pi} (V_0 A)^2\,\frac{k^{6-2a}m^{1-2b}}{1-2b}\,\mathcal{I}(a,b). \end{align}

Using the generalized Zakharov–Kraichnan–Kuznetsov transformations, one finds that the generalized KZ solution is the particular one for which the above denominators vanish (Lvov & Tabak Reference Lvov and Tabak2001), setting $a_{KZ}=7/2$, $b_{KZ}=1/2$, the Pelinovsky–Raevsky spectrum (Pelinovsky & Raevsky Reference Pelinovsky and Raevsky1977). Because for this solution also the numerators vanish, in analogy with (3.8), one can regularize the $0/0$ indeterminate form by l'Hôpital's rule to obtain (Zakharov et al. Reference Zakharov, L'vov and Falkovich1992)

(5.6a,b)\begin{equation} F_k^{KZ} = 2{\rm \pi} (V_0A)^2{m^{{-}1}}\left.\frac{\partial\mathcal{I}}{\partial a}\right|_{(a_0,b_0)},\quad F_m^{KZ} =2{\rm \pi} (V_0A)^2{k^{{-}1}}\left.\frac{\partial \mathcal{I}}{\partial b}\right|_{(a_0,b_0)}. \end{equation}

Notice that the KZ solution is the particular case for which the $k$-component is independent of $k$, i.e. $F_k=F_k(m)$, and the $m$-component is independent of $m$, i.e. $F_m=F_m(k)$. This spectrum is known to be non-local (Lvov et al. Reference Lvov, Tabak, Polzin and Yokoyama2010). In particular, the point $(\tfrac 72,\tfrac 12)$ in the $a\unicode{x2013} b$ plane exhibits divergences at both high and low wavenumbers. These divergences can also be seen from (5.6a,b): integrating the flux along any boundary in $k\unicode{x2013} m$ space yields a logarithmic divergence both at low wavenumbers ($k\to 0$, $m\to 0$) and at large wavenumbers ($k\to \infty$, $m\to \infty$). Thus (5.6a,b) can be written only in a formal way, but in practice they have no meaning. It was shown in Lvov et al. (Reference Lvov, Tabak, Polzin and Yokoyama2010) that there is only one stationary solution that is local, with spectral exponent values $a=3.69$, $b=0$. However, for this non-KZ stationary solution, (5.3) is merely stating that the divergence of the flux is zero. Therefore, it is possible to determine only the direction of the flux, but the magnitude of the flux of energy remains undetermined. This is shown in Appendix D.

6.1. Definition of the problem

Here, we illustrate an application of (2.1) to the anisotropic problem of internal gravity waves (Olbers Reference Olbers1976; Caillol & Zeitlin Reference Caillol and Zeitlin2000; Lvov & Tabak Reference Lvov and Tabak2004). This is non-trivial in several ways, involving: (i) physically motivated control volumes in Fourier space with a geometry that is more interesting than simple rectangles in $k\unicode{x2013} m$ space; (ii) a relevant stationary spectrum that is not a KZ solution (the solution $a=3.69$, $b=0$); (iii) a renowned spectrum (the Garrett–Munk spectrum) that is not stationary under the wave kinetic equation evolution; (iv) decomposition of the fluxes in terms of transfer integrals allowing us to quantify the level of locality. Each of these points could not be analysed fully by the standard wave turbulence theory of energy fluxes summarized in § 5. These results were obtained intuitively in Dematteis & Lvov (Reference Dematteis and Lvov2021) and Dematteis, Polzin & Lvov (Reference Dematteis, Polzin and Lvov2022). Here, we provide firm mathematical justification for results of such a type, study locality of internal wave interactions, and analyse the celebrated Garrett–Munk spectrum.

The boundaries in spectral space are defined naturally for internal waves. The frequency $\omega$ takes values in the interval $[f,N]$, where $f$ and $N$ are the inertial and buoyancy frequencies, respectively. These two frequencies give the minimal and maximal frequencies ($f\ll N$) of the problem, with the inertial range between them. The vertical wavenumber $m$ takes values in $[m_{min},m_{max}]$, where $m_{min}=2{\rm \pi} /H$, $m_{max}=2{\rm \pi} /h$, with $H$ and $h$ being the ocean depth and the vertical scale past which internal waves become unstable due to shear instability. Let us assume that the box $\mathcal {A}=[f,N]\times [m_{min},m_{max}]$ is the inertial range, and for $m>m_{min}$ and $\omega >N$, strong turbulence acts as an idealized sink. Suitable energy sources will indeed be necessary at the bottom and left boundaries of the ‘inertial box’ $\mathcal {A}$, in order for an energy cascade to be sustained in time. The inertial box $\mathcal {A}$ is shown in figure 4, in both $k\unicode{x2013} m$ space and $\omega \unicode{x2013} m$ space. The change of variables between the two spaces is prescribed by the dispersion relation (5.1). In figure 4, we also show the streamlines of the energy flux obtained by dimensional arguments in Appendix D, namely (D1a,b), for the stationary state with $a=3.69$, $b=0$. These lines give a sense of the need for a source at low frequencies and low wavenumbers for energy to be delivered to the whole inertial box. We use $\mathcal {A}$ as our input control volume. For the output control volume, $\mathcal {B}$, we consider two possibilities: either $\mathcal {B}_{h}=\{(\omega,m): \omega >N\}$ or $\mathcal {B}_{v}=\{(\omega,m): m>m_{max}\}$. In the first case, the power $\mathcal {P}_{\mathcal {A}\rightarrow \mathcal {B}}$ defines the quantity $\mathcal {P}_{h}$, the instantaneous power transferred ‘horizontally’ through the boundary denoted as $BC$ in figure 4. In the second case, $\mathcal {P}_{\mathcal {A}\rightarrow \mathcal {B}}$ defines the quantity $\mathcal {P}_{v}$, the instantaneous power transferred ‘vertically’ through the boundary denoted as $CD$. The powers $P_{h}$ and $P_{v}$ are calculated rigorously in the next subsection using the main statement (2.1).

Figure 4.

Representation of the 2-D Fourier space for the anisotropic internal-wave problem: (a) in horizontal–vertical wavenumber $k\unicode{x2013} m$ coordinates; (b) in frequency–vertical wavenumber $\omega \unicode{x2013} m$ coordinates. The streamlines represent the energy flux vector field and are drawn from (D1a,b) for the stationary solution $a=3.69$, $b=0$. Equation (D1a,b) is determined up to an arbitrary factor $C$, but this nonetheless allows us to know the flux direction (thus the streamlines). The change of coordinates from the $k\unicode{x2013} m$ to the $\omega \unicode{x2013} m$ representation is given by the dispersion relation (5.1). The quantities $P_{h}$ and $P_{v}$ are computed rigorously in (6.1a,b). The computations in this paper are performed in $k\unicode{x2013} m$ space, although the physical boundaries are defined naturally in $\omega \unicode{x2013} m$ space. This figure illustrates the equivalence between the two representations.

6.2. Application of the main statement (2.1) and transfer integrals

Applying (2.1), we obtain (Dematteis & Lvov Reference Dematteis and Lvov2021)

(6.1a,b)\begin{equation} \mathcal{P}_{h} = \int_{m_{min}}^{m_{max}} {\rm d}m \,\mathcal{F}_{h}(m), \quad \mathcal{P}_{v} = \int_{({f}/{\gamma})m_{max}}^{({N}/{\gamma})m_{max}} {\rm d}k \,\mathcal{F}_{v}(m), \end{equation}

where

(6.2)\begin{gather} \mathcal{F}_{h}(m) =\frac{N^2}{g}\,(V_0A)^2 \left(\frac{N}{\gamma}\,m\right)^{7-2a}m^{{-}2b} \int_{{f}/{N}}^{1}{\rm d} K \,T_{h}(K), \end{gather}

(6.3)\begin{gather} T_{h}(K)={-}\frac{4{\rm \pi} }{ (V_0A)^{2}}\,K^{6-2a} \int {\rm d}\xi_1\,{\rm d}\xi_2 \sum_l \chi_{K}^{(l)}(\xi_1,\xi_2)\,J^{({ l})}(\xi=1, \mu=1;\xi_1,\xi_2) \end{gather}

and

(6.4)\begin{gather} \mathcal{F}_{v}(k) =\frac{N^2}{g}\,(V_0A)^2 k^{6-2a} m_{max}^{1-2b} \int_{{m_{min}}/{m_{max}}}^{1} {\rm d} M \,T_{v}(M), \end{gather}

(6.5)\begin{gather} T_{v}(M)={-}\frac{4{\rm \pi} }{(V_0A)^{2}}\,M^{{-}2b} \int {\rm d}\xi_1\,{\rm d}\xi_2 \sum_l \chi_{M}^{(l)}(\xi_1,\xi_2)\,J^{({ l})}(\xi=1, \mu=1;\xi_2,\xi_2). \end{gather}

Here, $J^{({ l})}(\xi =1, \mu =1;\xi _1,\xi _2)$ denotes the six resonant branches of the interaction kernel corresponding to the triad of non-dimensional horizontal wavenumbers $\xi =1$, $\xi _1=k_1/k$, $\xi _2=k_2/k$, and vertical wavenumber $\mu =1$. The six resonant conditions determine the values of $\mu _1=m_1/m$ and $\mu _2=m_2/m$. The characteristic interaction weights $\chi _K^{({ l})}(\xi _1,\xi _2)$ are defined by the rules in table 1, taking $\mathcal {B}_{h}=\{\xi :\xi >K^{-1}\}$, with $K^{-1}>1$. The weights $\chi _M^{({ l})}(\xi _1,\xi _2)$ are defined likewise by taking $\mathcal {B}_{v}=\{\mu :\mu >M^{-1}\}$, with $M^{-1}>1$, where the condition $\mu >M^{-1}$ is applied to the non-dimensional vertical wavenumbers $\mu _1$ and $\mu _2$ found as solution of the $l$th resonant branch.

Using similar equations, it was shown in Dematteis & Lvov (Reference Dematteis and Lvov2021) and Dematteis et al. (Reference Dematteis, Polzin and Lvov2022) that both the scaling and the prefactor of the total calculated power are in order-of-magnitude agreement with the observational fine-scale parametrization of oceanic turbulent production (Polzin et al. Reference Polzin, Garabato, Huussen, Sloyan and Waterman2014), up to a factor $1.5$ difference. In a loose sense, these calculations are equivalent to evaluating the Kolmogorov constant for the internal wave problem, i.e. expressing the explicit theoretical relationship between the energy flux and the spectral energy density.

6.3. Metrics of locality and distant transport

The methodology developed in this paper allows us not only to compute the fluxes of energy, but also to analyse the locality of interactions. In relation to their isotropic version (3.18)–(3.19), indeed the expressions (6.1a,b)–(6.5) feature one extra integration along the boundaries of the 2-D inertial box. The quantities $\mathcal {F}_{v}(k)$ and $\mathcal {F}_{h}(m)$ are energy fluxes per unit of $k$ and $m$, respectively. Thanks to scale invariance, their dependence on $k$ and $m$ is given by the scaling relations in (6.2) and (6.4). Therefore, to study level of locality of the interactions, it is sufficient to study $\mathcal {F}_{v}(k=1)$ and $\mathcal {F}_{h}(m=1)$, whose structure is expressed in terms of the transfer integrals $T_{h}$ and $T_{v}$ in (6.3) and (6.5). This is represented in figure 5. For both $T_{h}$ and $T_{v}$, the further from the boundary at $K=1$ or $M=1$, the more non-local (i.e. with large-scale separation) the contribution to the energy transfer. The dashed coloured lines at the right of each plot indicate the analytical leading orders (integrable singularities) from the IR region of the resonant manifold. The scalings on the left side are given by the UV leading orders, multiplied by the factor in (6.3) and (6.5). The shaded areas indicate the leftmost and rightmost contributions to the total flux, in a percentage amount indicated in the figure. Two different spectra in the form (5.2) are studied below (Lvov et al. Reference Lvov, Tabak, Polzin and Yokoyama2010).

1. (i) $a=3.69$, $b=0$ (stationary state of the internal WKE).

   Figure 5.

   Transfer integrals of internal gravity waves for (a) horizontal transport (6.3), and (b) vertical transport (6.5). For the power-law spectrum (5.2), the stationary solution has exponents $a=3.69$, $b=0$, and the scale-invariant limit of the Garrett–Munk spectrum has exponents $a=4$, $b=0$.

   

   For the horizontal transport, despite having the rightmost $50\,\%$ coming from the $[0.7,1]$ interval (i.e. the median is approximately $0.7$), the heavy tail implies that approximately $12\,\%$ of the power $\mathcal {P}_{h}$ is transferred directly from modes that are smaller than the left boundary of the inertial box, i.e. $\omega < f$ – if we take $N/f=40$, then we obtain a realistic oceanic aspect ratio. Since there are no waves at $\omega < f$, this is not possible. Nevertheless, it indicates that the horizontal transfer is highly non-local – even though the spectrum is ‘local’ in terms of convergence of the collision integral. Even the lowest frequencies in the system are connected energetically with the dissipation region at high frequency in a non-negligible way.

   For the vertical transport, the situation is much more local. The median is around $0.8$, and the rightmost $5\,\%$ of the energy transfer comes from the left of approximately $0.2$. Because a realistic range of vertical scales varies by a factor of the order of $200$, a factor $5$ of distance from the boundary at $m_{max}$ is relatively quite small. Thus the vertical transport is highly local.

2. (ii) $a=4$, $b=0$ (scale-invariant limit of the Garrett–Munk spectrum).

   The horizontal transport power is marginally divergent, due to a $K^{-1}$ singularity as $K\to 0$. Therefore, it is not meaningful to indicate percentage metrics of the contribution.

   The vertical transport is highly local, more so than for the stationary spectrum. Approximately $95\,\%$ of the total power $\mathcal {P}_{v}$ comes from the region within a factor $3$ from the dissipation boundary.

Let us exploit the transfer integrals to define the distant-transport fluxes, neglecting the dimensional prefactors

(6.6)\begin{equation} \left. \begin{aligned} & \mathcal{F}_{{h},[0,\tilde\omega]\rightarrow [\omega,+\infty)} \propto \int_{0}^{\tilde\omega/\omega} {\rm d}K \,T_{h}(K),\\ & \mathcal{F}_{{v},[0,\tilde m]\rightarrow [m,+\infty)} \propto \int_{0}^{\tilde m/m} {\rm d} M \,T_{v}(M). \end{aligned} \right\} \end{equation}

For horizontal transport, we know the analytical scaling $T_{h}(K)\propto K^{3-a}$ as $K\to 0$ (cf. figure 5a). This means that we have $\mathcal {F}_{{h},[0,\tilde \omega ]\rightarrow [\omega,+\infty )}\propto ({\tilde \omega }/{\omega })^{4-a}$ as ${\tilde \omega }/{\omega }\to 0$. This shows that the flux for the Garrett–Munk spectrum is marginally divergent (logarithmic divergence), and that the stationary spectrum has a very weak decay with scaling $({\tilde \omega }/{\omega })^{0.31}$. For vertical transport, we use the numerical scalings shown in figure 5(b). These imply for $\mathcal {F}_{{v},[0,\tilde m]\rightarrow [m,+\infty )}$, a scaling $({\tilde m}/{m})^{1.69}$ for the Garrett–Munk spectrum, and a scaling $({\tilde m}/{m})^{1.45}$ for the stationary spectrum.

We end the section by showing that even when a solution is mathematically non-local, a regularization takes place if one considers physical cutoffs. For the high wavenumber limit of the Garrett–Munk spectrum, indeed a fairly plausible oceanic condition (Pollmann Reference Pollmann2020; Le Boyer & Alford Reference Le Boyer and Alford2021; Thakur et al. Reference Thakur, Arbic, Menemenlis, Momeni, Pan, Peltier, Skitka, Alford and Ma2022), the integral defining the horizontal energy flux has a logarithmic divergence (considering an idealized zero minimal frequency). However, real physical systems have boundaries and other constraints that imply natural lower and upper cutoffs in Fourier space. For instance, oceanic internal waves cannot oscillate at frequencies lower than the Coriolis frequency $f$. Imposing this lower cutoff by hand, the horizontal flux in (6.6) is given by

(6.7)\begin{equation} \mathcal{F}_{{h},[f,\tilde\omega]\rightarrow [\omega,+\infty)} \propto \int_{f/\omega}^{\tilde\omega/\omega} {\rm d}K \,K^{3-a}\propto \frac{1}{4-a}\,\frac{\tilde\omega^{4-a} - f^{4-a}}{\omega^{4-a}}. \end{equation}

For a given frequency $\tilde \omega >f$, the flux is finite and continuous in $a$, with $\lim _{a\to 4^{\pm }}\mathcal {F}_{{h},[f,\tilde \omega ]\rightarrow [\omega,+\infty )}=|{\log (f/\omega )}|$. For $a\ll 4$, the contribution to the flux from $[f,\tilde \omega ]$ is concentrated around $\tilde \omega$. As $a\to 4^-$, the contribution becomes less and less concentrated in $\tilde \omega$. For $a>4$, the contribution becomes more concentrated in $f$ than in $\tilde \omega$, and increasingly so as $a$ increases. Since the non-locality and boundary dependence are smooth functions of $a$, and the transition is continuous in $a=4$, it is clear that interpreting this threshold as a sharp definition of physical realizability/non-realizability is quite fictitious. For a physical system with a finite available range of scales, highly non-local spectra will indeed show strong dependence on the boundaries, as (6.7) demonstrates. If the forcing is varying strongly in time, for instance, this will correspond to transient forcing-driven conditions, with the system being influenced highly by the forcing variability at the lower boundary of Fourier space. However, it may still be of fundamental importance to quantify the associated energy fluxes: the energy fluxes associated with highly non-local and transient spectra of internal waves play a crucial role for the oceanic circulation and climate at large (Polzin et al. Reference Polzin, Garabato, Huussen, Sloyan and Waterman2014).