Phase transition of the energy flux in the near-inertial wave--mesoscale eddy coupled turbulence
TThis draft was prepared using the LaTeX style file belonging to the Journal of Fluid Mechanics Phase transition of the energy flux in thenear-inertial wave–mesoscale eddy coupledturbulence
Jin-Han Xie Department of Mechanics and Engineering Science at College of Engineering,State Key Laboratory for Turbulence and Complex Systemsand Beijing Innovation Center for Engineering Science and Advanced Technology,Peking University, Beijing, 100871, PR China(Received xx; revised xx; accepted xx)
Wind forcing injects energy into the mesoscale eddies and near-inertial waves (NIWs) inthe ocean, and the NIW is believed to solve the puzzle of mesoscale energy budget byabsorbing energy from mesoscale eddies followed by a forward cascade of NIW energywhich finally dissipates at the ocean interior. This work studies the turbulent energytransfer in the NIW–quasigeostrophic mean mesoscale eddy coupled system based on apreviously derived two-dimensional model which has a Hamiltonian structure and inheritsconserved quantities in the Boussinesq equations (Xie & Vanneste,
J. Fluid Mech. , vol.774, 2015, pp. 147–169). Based on the conservation of energy, potential enstrophy andwave action, we propose a heuristic argument predicting the existence of phase transitionwith changing the relative strength between NIW and mean flow. By running forced-dissipative numerical simulations with varying parameter R , the ratio of the magnitudeof NIW and mean-flow forcing, we justify the existence of phase transition, which isfound to be second-order, around critical value R c . When 0 < R < R c , energy transfersbidirectionally, wave action transfers downscale, and vorticity form strong cyclones. Whilewhen R > R c , energy transfers downscale, wave action transfers bidirectionally, andvortex filaments are dominant. We find the catalytic wave induction (CWI) mechanismwhere the NIW induces a downscale energy flux of the mean flow. The CWI mechanismdiffers from the stimulated loss of balance by the absence of energy conversion from themesoscale eddy to NIW, and it is found to be effective in the toy-model study, makingit potentially important for ocean energetics.
1. Introduction
The mesoscale eddy, which has horizontal scale from one to hundreds of kilometres,contains a significant part of the ocean energy, however, its energy budget remainsnot well understood: wind forcing and large-scale circulations injection energy into themesoscale eddies, and the mesoscale eddies dissipate at the boundaries and be convertedto other types of motions, but the known amount of energy injection is much larger thanthat of energy dissipation (Wunsch & Ferrari 2004; Ferrari & Wunsch 2009). Severalcandidates are thought to be responsible for the mesoscale eddy energy closure, suchas the simultaneous loss of balance (cf. Vanneste 2013) and the stimulated near-inertialwave (NIW) generation, which is also named stimulated loss of balance (Xie & Vanneste2015).There are a variety of mechanisms potentially explaining that NIWs can extract energyfrom balanced flow, and they have been studied in various setups and parameter regimes.Gertz & Straub (2009) run numerical simulations in a periodic box with O(1) Rossby a r X i v : . [ phy s i c s . f l u - dyn ] J un number and find that the energy transfer between 2D and 3D motions is crucial. Alsoin the parameter regime with O(1) Rossby number, Taylor & Straub (2016) consider achannel flow with external forcing acting on both low and high frequencies, and they findthat the Reynolds stresses of NIW act as a kinetic energy sink for mesoscale motions. InBarkan et al. (2017), numerical simulations in a channel flow with external forcing findthe direct extraction dominates while the mechanism of stimulated NIW generation issubdominant. NIW is also found to absorb mean flow energy in frontogenesis (Thomas2012).Applying the generalised-lagrangian mean theory (Andrews & McIntyre 1978; Soward& Roberts 2010) in a variational framework (Salmon 1988, 2013, 2016), Xie & Vanneste(2015) derive an asymptotic coupled model in the parameter regime of small Rossbyand Burger number to describe the interaction between NIWs and quasigeostrophic(QG) mean flow. This model naturally couples the classic models of QG mean flow(cf. Salmon 1998) and the Young-Ben Jelloul (YBJ) equation for the slow evolution ofNIW magnitude (Young & Ben Jelloul 1997), which have been validated in decades ofstudy. It has been extended to include the second harmonic of inertial oscillations byWagner & Young (2016) using the concept of available potential vorticity (Wagner &Young 2015) and been used to study the details of the stimulated loss of balance (Rocha et al. et al. et al. a ), the mechanism for bidirectionalenergy transfer is less understood. The two primary reasons for bidirectional energytransfer are the impact of inertia–gravity waves and the short aspect ratio (Benavides &Alexakis 2017). The toy model of NIW-QG interaction studied in this paper provides asimple setup to study the wave-induced bidirectional energy transfer.The paper is organised as follows. In § + -QG model, which avoids the “ultraviolet catastrophe” ofthe NIW-QG model, in § § §
5, respectively. Appendix Aprovides a derivation of the YBJ + -QG model in a variational approach and discuss thedirections of conserved quantities in this model.
2. A heuristic argument for the direction of energy transfer
In this section, we propose a heuristic argument applying to a forced-dissipativeturbulent system with three inviscid preserved quantities to predict the transfer directionof preserved quantities across scales.We focus on the dynamics of the interaction between NIW and QG mean flow usingthe model derived by Xie & Vanneste (2015). Since the NIW is described by theslow modulation of the wave amplitude with a fixed frequency equaling to the Coriolisfrequency, the model also preserves wave action due to the phase invariant. Even thoughthe wave action is only asymptotically conserved in the original system, we believe it isstill crucial for the dynamics of the wave-mean flow coupled system considering that theNIW peak is notable in the ocean observation (cf. Ferrari & Wunsch 2009).In this paper, for simplicity and numerical efficiency, we focus on the 2D version of theYBJ-QG model on an f -plane, where the QG mean flow is barotropic ( z -independent),and the NIW has a single mode in vertical. The governing equations are (cf. Xie &Vanneste 2015) M t + J ( ψ, M ) − i N m f ∇ M + i2 M ∇ ψ = 0 , (2.1 a ) q t + J ( ψ, q ) = 0 , (2.1 b )with q = ∇ ψ + i f J ( M ∗ , M ) + f ∇ | M | , (2.1 c )where M is the complex wave amplitude such that u + i v = − i f M e − i ft +i mz with u and v the two horizontal velocities, J denotes the Jacobian, N is the Brunt-V¨ais¨al¨a (buoyancy)frequency, m is the vertical wavenumber of NIW, f is the Coriolis frequency, ψ is thestream function of QG mean flow.The 2D model (2.1) preserves the energy, potential enstrophy and wave action: E = (cid:90) (cid:26) |∇ ψ | + 14 N m |∇ M | (cid:27) d x , (2.2 a ) P = (cid:90) q d x , (2.2 b ) A = (cid:90) | M | d x . (2.2 c )The total energy E contains two quadratic forms of mean flow and wave, and we namethem E QG and E W , respectively. They are defined as E QG = (cid:90) |∇ ψ | d x and E W = (cid:90) N m |∇ M | d x , (2.3)therefore E = E QG + E W . The potential enstrophy P is the integration of the quadraticof PV therefore it contains quadratic, cubic and quartic terms of both mean flow andwave amplitude. But the wave action A is purely a wave quadratic quantity.To understand the transfer direction of conserved quantities, we consider a turbulentpicture similar to those proposed by Kraichnan (1982) and Eyink (1996) to show thedirection of energy and enstrophy cascades in 2D turbulence, and the one used in waveturbulence (cf. Chapter 1 in Cardy et al. k f and they dissipate at both a small wavenumber k anda large wavenumber k . We denote the energy, potential enstrophy and wave actiondissipations at these two wavenumbers as E i , P i and A i ( i = 1 , E f = E + E , P f = P + P , and A f = A + A , (2.4 a )where the symbols with lower index “ f ” denote the injections at the forcing wavenumber.However, different from the 2D turbulence case, where due to the quadratic energy andenstrophy the transfer direction of energy and enstrophy are determined by solving (2.4),we cannot similarly determine the transfer directions of these three quantities, which isthe main topic of this paper, and we will show that the preservation of three quantitiesbrings about a more complicated picture for the transfer directions.We first consider two limiting cases that hint us the transfer directions in general cases.The first case is the mean-flow-dominant case where the NIW is week compared with themean flow, therefore the wave action as a purely wave effect is not dominant, also, thewave effect is of high order in the energy and potential enstrophy. Then, to the leadingorder, the mean flow dominates the energy and potential enstrophy E = (cid:90) (cid:26) |∇ ψ | (cid:27) d x , (2.5 a ) P = (cid:90) |∇ ψ | d x . (2.5 b )Therefore the system recovers the scenario of 2D turbulence: energy transfers upscale andpotential enstrophy transfers downscale. Also, because the feedback to the mean flow isweak, NIW behaves like a passive scalar and the wave action transfers downscale.The other extreme is the NIW-dominant case, where the mean flow is weak comparedwith the NIW. In this case, the potential enstrophy is dominated by the wave quarticterms and is assumed to be subdominant, therefore, similar to the wave turbulence,energy and wave action control the turbulent dynamics. To the leading order energy andwave action are expressed as E = 14 N m (cid:90) |∇ M | d x , (2.6 a ) A = (cid:90) | M | d x . (2.6 b )Therefore, the conservations (2.4) imply that E f = 14 N m k A + 14 N m k A , (2.7 a ) A f = A + A , (2.7 b )where A i = | ˆ A ( k i ) | with ˆ · the Fourier transform.Following the idea of Kraichnan (1982) and Eyink (1996), by considering that E f = 14 N m k f A f , (2.8)we can solve (2.7) to obtain A = k f − k k − k A f and A = k f − k k − k A f . (2.9)Then taking the limit k → k → ∞ with fixed finite k f , we obtain A → A f , A → , E → E → E f , (2.10)which implies that the energy transfers downscale while the wave action transfers upscale.The directions of energy transfer are opposite in the NIW-dominant and mean-flow-dominant cases. This implies that when the strength of NIW is intermediate energy cantransfer both upscale and downscale simultaneously, which is consistent with the energytransfer scenario in the oceanic flows. In § a ) impliesthat the resolved time scale behaves as the inverse of the square of wavenumber, whichtends to infinity as the resolution increases. Therefore we consider running numericalsimulations of the YBJ + -QG coupled model proposed by Asselin & Young (2019 a ),which improves the numerical efficiency by retaining specific high-order terms and usingthe reconstitution technique to bound the frequency as wavenumber increases. Asselin& Young (2019 a ) derive the modified YBJ + equation but the form of wave-mean flowcoupling is proposed. We present a variational derivation that mimics the procedure ofderiving the YBJ-QG model in § A.1. And to ensure the validity of using the YBJ + -QGmodel to study the transfer of conserved quantities, we check the argument of transferdirection in YBJ + -QG coupled model in § A.2.
3. Numerical simulation
To explore the transfer of conserved quantities in steady turbulent states, we addexternal forcing and artificial at both large and small scales to the inviscid YBJ + -QGmodel M t + J ( ψ, M ) + i f ∇ − f N m + ∇ M + i2 M ∇ ψ = 0 , (3.1 a ) q t + J ( ψ, q ) = 0 , (3.1 b )with q = ∇ ψ + i f J ( M ∗ , M ) + f ∇ | M | , (3.1 c )whose three-dimensional version is proposed by Asselin & Young (2019 a ). And wevariaionally derive this coupled model in § A.1.Since (3.1) is derived from a variational approach, adding dissipation is artificial.We assume that the inviscid mechanism is not changed by the dissipations and thedissipations monotonically damp the total energy and wave action, so we choose to adddissipations in the equations of M and mean vorticity ∇ ψ instead of q . We will discussthe choice of dissipation terms in details below. We obtain our model for numericalsimulation: ∂ t M + J ( ψ, M ) + P M + i2 ∇ ψM = α ∇ − M + ν ∇ M + R mN k f F , (3.2 a ) ∂ t ∇ ψ + J (cid:0) ψ, ∇ ψ (cid:1) + N ( ψ, M ) = αψ + ν ∇ ψ + F , (3.2 b )where N ( ψ, M ) = f J ( M ∗ , P M ) − J ( P M ∗ , M )] + i f ∇ ( M P M ∗ − M ∗ P M ) − f ∇ · J (cid:0) ∇ ψ, | M | (cid:1) (3.3)with P = i f ∇ − f N m + ∇ . (3.4)Here, in the forcing term, m/ ( N k f ) is a normalized coefficient, F i (i=1 or 2) are temporalwhite-noise external forcing that is centred around wavenumber | k | = k f , and R is atuning parameter that controls the relative strength of energy injection into the NIWand mean-flow components. And when R = 1, the ratio between the wave energy injectionand QG-mean flow energy injection is 1 /
2. Here, we add both hyper- and hypo-viscositiesto dissipate the conserved quantities transferred downscale and upscale, respectively.In the governing equations (3.2) we design the forcing and dissipation in the meanvorticity equation instead of the PV equation to achieve that (i) if there is no externalforcing and dissipation the conserved quantities are preserved, and (ii) the effect of viscousterms damp total energy and wave action, i.e. there is no viscous generation of the totalenergy. However, as to the potential enstrophy, viscous generation is nonzero. In otherwords, we choose to damp two out of three preserved quantities and focus on the fluxes ofthese two quantities. It is debatable that this choice is realistic since we did not derive thedissipation terms from the original hydrostatic Boussinesq equations. A potential way toachieve a realistic-scenario study is deriving the dissipative coupled model from viscousprimitive equations, e.g. following the derivation in Rocha (2018). However, the physicalmeaning of large-scale hypoviscosity is uncertain. So considering the balance betweenthe involute derivation and the aim of studying key mechanisms, we choose the abovesimple design as a heuristic starting point to understand the wave-mean flow coupledturbulence.The numerical simulations use a Fourier pseudospectral method with 2/3 dealiasingin space, a resolution 512 ×
512 in a domain of size 2 π × π and a fourth-order explicitRunge–Kutta scheme in time, in which the nonlinear terms are treated explicitly, andlinear terms implicitly use an integrating factor method. We take the forcing wavenumberto be k f = 32 and control the energy injection rate by the external forcing F i as 10 − ,by choosing α = 0 .
01 and ν = 10 − we obtain small- and large-dissipation wavenumbersas k α ≈ . k ν ≈ f = 1, N = 1 and m = 32, thereforethe forcing Burger number is Bu f = 1 / + -QG system recoverthat for the YBJ-QG system (cf. § A.2).3.1.
Dependence on the parameter R In this section, we focus on the dependence of the fluxes of energy and wave actionin the spectral space on the parameter R . Here, the spectral energy flux F E and waveaction flux F A are defined from the equations ∂ t E ( K ) = − ∂ K F E + forcing and dissipation , (3.5 a ) ∂ t A ( K ) = − ∂ K F A + forcing and dissipation , (3.5 b )where conferring to (2.6) we define (cid:90) ∞ E ( K )d K = E and (cid:90) ∞ A ( K )d K = A . (3.6)Here, K = | k | is the magnitude of the wavenunmber. And F E and F A indicate theenergy and wave action transfer from small to large wavenumbers in the spectral space,respectively. -4-3-2-101234 10 -101234 10 Figure 1.
Dependence of the total energy flux and the wave action flux on parameter R , with R = 0 . , . , . .
4. The negative and positive values of F E represent upscale and downscaleenergy transfers, respectively. The black dashed line is 0 for reference. In figure 1, we show the dependence of total energy flux F E and wave action flux F A on parameter R . We observe the transition of energy transfer from upscale tobidirectional to downscale, which justifies our conjecture based on the argument in § R -dependence, in the left panel of Figure 2, we show the R -dependence of normalized upscale and downscale energy fluxes, which are defined as theupscale and downscale energy fluxes divided by the total energy injection, respectively.We find a critical value of R ( E ) c ≈ .
36. When
R < R c , the normalized upscale energytransfer monotonously decrease from 1, which corresponds to the total upscale energytransfer in 2D turbulence, and when R > R c the normalized upscale energy transferequals to zero. When R < R ( E ) c the dependence of normalized downscale energy transferscales as R , and close the critical value we find that the normalized upscale energytransfer scales as ( R ( E ) c − R ) / , implying a second-order phase transition.As to the normalized upscale action transfer shown in the right panel of Figure 2, wefind a critical value R ( A ) c ≈ .
35. When
R < R ( A ) c , the normalized upscale action transferis almost zero, which may be interpreted as a passive-scalar-like wave magnitude; when R > R ( A ) c , the normalized upscale action transfer monotonously increases. Around thecritical value, the normalized upscale action transfer scales as ( R − R ( A ) c ) / , indicatinga second-order phase transition. Here, due to the finite dissipation, when R < R ( A ) c ,the normalized upscale action transfer is small but nonzero, so when fitting the powerfunction near the critical point, we subtracted the small nonzero value at the criticalpoint. Based on the normalized transfers we find two close critical values R ( E ) c ≈ . R ( A ) c ≈ .
35, but we do not know if they are the same so we keep two symbols forthem. 3.2.
Simulation with R = 0 . R < R c and R > R c in this and next subsections,respectively. Here, since R ( E ) c and R ( A ) c are close, so, e.g., we use the notation R c , andby R < R c we mean R is smaller than both R ( E ) c and R ( A ) c . In this subsection, we showthe details of the simulation with R = 0 . < R c . Figure 2.
The dependence of normalized energy and wave action flux on the parameter R . Here, R ( E ) c = 0 .
36 and R ( A ) c = 0 .
35. The indices E and A represent the energy and action, respectively.And the lower indices up and down denotes the upscale and downscale fluxes, respectively. Theindex c represents the critical point. Figure 3.
Snapshots of mean-flow vorticity and the wave amplitude at the turbulentstatistically steady state in the simulation with R = 0 . We show the snapshots of the of mean-flow vorticity and the wave amplitude at aturbulent statistically steady state in Figure 3. There are more strong cyclones observedcompared with the anticyclone, and the waves are advected by the mean flow. NIW’sconcentration at the anticyclone could explain the observed strong cyclones (Danioux et al. F E into the mean-flow component F mE and thewave component F wE , which are calculated from the mean equation (3.2 b ) and thewave equation (3.2 a ), and they correspond to the transfer of QG mean-flow energyand NIW energy (2.3), respectively. Also, in the transfer of mean energy we distinguishthe contributions from mean effect, J (cid:0) ψ, ∇ ψ (cid:1) , and the wave-mean flow interaction, N ( ψ, M ), in the mean equation (3.2 a ), and name them F m,mE and F m,wE , respectively. It isobserved that the wave energy transfers downscale with constant flux. The mean energytransfers bidirectionally where the mean flow dominantly induces the upscale energytransfer while the NIW dominantly induces the downscale energy transfer. Above theforcing scale, these two effects complete and result in a residue of upscale energy transfer.The wave action flux is shown in the right panel of Figure 4. As we discussed above thatthe NIW almost behaves as a passive scalar, the wave action transfers downscale. -3-2-10123 10 -0.500.511.522.5 10 Figure 4.
Energy and wave action flux for simulation with R = 0 . F E is the total energy flux; F wE is the wave energy flux; F m,mE and F m,wE are the mean-flow-induced and NIW-induced meanenergy fluxes, respectively. All the quantities are normalized by the total energy injection (cid:15) . Figure 5.
Snapshots of mean-flow vorticity and the wave amplitude at t = 2000 in thesimulation with R = 0 . Simulation with R = 0 . R = 0 . > R c .The snapshots of mean-flow vorticity and the wave amplitude at a turbulent statis-tically steady states are shown in Figure 5. Comparing with Figure 3, in the presentsimulation the strong cyclone patches are broken and structures of the forcing scale areobserved in both fields.The energy flux is shown in Figure 6. We observe that the upscale fluxes are almostinvisible, even though this is a 2D system with vorticity advection term, and thecompensation between F m,mE and F m,wE above the forcing scale is not obvious. Nowthe wave action shows a clear bidirectional transfer.
4. Discussion
Mechanism of downscale mean energy transfer
Using the model describing the interaction between NIW and QG mean flow (Xie& Vanneste 2015; Wagner & Young 2016), the stimulated loss of balance (SLOB) (orstimulated NIW generation) mechanism, which explains the energy conversion from theQG mean flow to the NIW is proposed. Considering that the NIW transfers energydownscale and finally dissipate at the ocean interior, which is in the opposite energy0 -3-2-101234 10 -101234 10 Figure 6.
Energy and wave action flux for simulation with R = 0 .
4. All notation are the sameas those in Figure 4. transfer direction of mesoscale eddy itself, the SLOB is believed to be important forthe energy balance puzzle of oceanic mesoscale eddies, even though Asselin & Young(2019 b ) recently find that the SLOB maybe not effective in the ocean using three-dimensional numerical simulations. However, the works mentioned above focus on initial-value problems, it remains to check whether the interaction between NIW and mesoscaleeddies effectively impact the energetics of the latter in a statistically steady turbulentstate, which is the topic of this paper.We compare the states in the parameter regime R < R c with the SLOB mechanism,as in Figure 2 we find that in this regime the downscale energy flux is proportional tothe injection of NIW energy, which is consistent with the SLOB (cf. Xie & Vanneste(2015)). However, the turbulent-state energy transfer mechanism differs from SLOB. Inthe energy transfer of the R = 0 . R c , the ratio of the energy injection between wave and mean flow is 1 /
16, i.e., themean energy that changes the direction of energy transfer due to the NIW is 16 timesthe injected wave energy, implying a much more substantial impact of the NIW on thedirection of mean energy transfer compared with SLOB. As we are focusing on turbulentstatistically states, it may need to wait for a long time to establish this state. However,considering that wind has been blowing the ocean for a considerable long time, CWI ispotentially crucial for oceanic mesoscale eddies.Both SLOB and CWI show a R -dependence of downscale energy flux, where theformer is explained based on the conserved quantities. In CWI, this dependence can beunderstood from the nonlinear term (3.3) where all terms depend on the quadratic ofthe wave magnitude.We also need to note that CWI differs from the previous mechanism of downscaleenergy transfer in statistically steady states of NIW-mean flow interacting turbulence.Barkan et al. (2017) study the direct extraction (DE) mechanism and SLOB in a wind-driven channel flow, where both mechanisms rely on energy conversion from mean flow toNIWs. But we cannot access the CWI’s impact in their system as we need scale-by-scaleenergy transfer information to distinguish CWI from DE and SLOB. In addition, the1NIW’s induction of downscale mean-flow energy flux is implicitly observed in Taylor &Straub (2016). They study the Reynolds stresses exerted by the near-inertial modes asthe energy sink of the mean flow, and they access the strength of this energy sink in thespectral space. After integrating their Fig. 5 over the wavenumber, we can find a NIW-induced downscale mean-energy flux. But without separating and explicitly calculatingthe energy flux of mean flow and NIW, it is hard to access the importance of energyconversion between NIW and mean flow on the direction of mean flow energy transfer.4.2. Argument for the direction of energy transfer
Our heuristic argument for transfer directions based on three conserved quantities(cf. §
2) is also applicable to other systems, such as the 2D magnetohydrodynamical(MHD) turbulence, where phase transition depending on the relative strength betweenthe magnetic- and fluid-field forcing is observed (Seshasayanan et al. et al. b ). E.g., in 2D inertia-gravity wave turbulence, the injected kinetic first cascades upscale then is convertedto the potential energy which finally transfers downscale. But for complete kinetic-potential energy conversion, the domain size should be larger than the Ozmidov scale,the scale energy conversion happens. If not, the incomplete energy conversion results inbidirectional energy transfer. Here, our argument implies that even when the domain sizeis infinite, bidirectional energy transfer can still happen.4.3. Potential enstrophy flux
In the above section, we pay very little attention to the potential enstrophy fluxbecause the chosen artificial hyper- and hypo-viscosities in (3.2) do not correspond tosign-definitive potential enstrophy “dissipation”. This is because that the YBJ-QG modelis initially derived in a variational framework, which is hard to include the dissipationeffects, we have to design which quantities to dissipate. This situation is similar to thatin shallow water model, where the two out of three conserved quantities are chosen to bedissipated (cf. Jacobson et al. R dependence of the potential enstrophy flux in Figure 7. When R Figure 7.
Potential enstrophy flux for simulation with R = 0 . , . , . .
4. The insetzooms in the black-boxed region. is small, potential enstrophy transfers downscale since the weak wave implies the meanflow’s enstrophy dominance of the potential enstrophy. However, as R increases viscousgeneration is not negligible and the potential energy fluxes are not constants.Since the YBJ-QG coupled model can also be derived in a Eulerian framework (cf.Wagner & Young 2016), it is interesting to perform a derivation starting from the viscousBoussinesq equation to find the suitable dissipation terms (cf. Rocha 2018, Chapter 5,where the vertical viscosity is included).
5. Summary and conclusion
Using the NIW-QG coupled model (3.2), we study the dependence of energy and waveaction transfer directions on parameter R , which is defined as the ratio of the externalforcing magnitude in the NIW and QG components. We propose a heuristic argumentbased on the inviscid preserved energy, potential enstrophy and wave action to predict theexistence of phase transition, which is justified by numerical simulations. We find a criticalvalue R ( E ) c and R ( A ) c , across which the energy and wave action flux show a second-orderphase transition, respectively. Since R ( E ) c and R ( A ) c are close, they may be the same but weare not sure about it. But for simplicity, in the below summary we keep only one R c . (1)When 0 < R < Rc , the total energy transfers bidirectionally, and the normalized upscaleenergy flux monotonously decrease and reaches zero at R = R c . The normalized upscalewave action flux remains a small close-to-zero value, so we believe the action transfersdownscale. The vorticity field consists of vorticity patches with cyclone dominance, butdue to the wave impact more small scale structures are observed compared with the 2Dturbulence. This is consistent with the information that even in 2D inertia-gravity wavescan induce downscale energy transfer (cf. Xie & B¨uhler (2019 b )). These turbulent statesalso show the consistency with the concentration of NIW at the anticyclones (Danioux et al. R > R c , energytransfers downscale; the wave action transfers bidirectionally, and as R increases theupscale energy transfer increases vortex filaments are the dominant structure, which issimilar to the 3D turbulence.In the forced-dissipative turbulence of NIW-QG mean flow interaction, we discover anew mechanism – catalytic wave induction (CWI), which is responsible for the downscaleenergy transfer. Different from the SLOB mechanism, where the QG mean energyconverts to the wave energy and then cascades downscale, in CWI, the waves play a3catalytic role, and the mean energy transfers downscale without recognizable conversionto the wave energy.We close this paper by pointing out some potential future works relating to theenergy transfer in the NIW-mean flow coupled system: (i) Similar to the SLOB, thedownscale energy flux in CWI is also proportional to the wave energy injection rate,and it is shown to be effective (cf. 4.1) based on our toy model study, but we stillcannot calculate the value of the constant linking the downscale energy flux and thewave energy injection rate. We would like to further explore the dependence of theconstant on parameters such as the Burger number. (ii) This paper’s toy-model studyindicates that CWI is potentially important for the ocean energy energetics, andthere are potential external-forcing-induced sudden changes in the direction of energytransfer in the ocean. These implications remain to be checked and justified in morerealistic simulations and observations. (iii) We need to note that the mechanism ofNIW saturation differs from that in the wave turbulence theory (cf. Zakharov et al. R regime. AcknowledgmentDeclaration of Interests
The authors report no conflict of interest.
Appendix A. YBJ + -QG coupled model A.1.
A variational derivation for the YBJ + -QG model In this section, we modify the variational framework in Xie & Vanneste (2015) toobtain the YBJ + -QG coupled model proposed by Asselin & Young (2019 a ).The reconsitutation by Asselin & Young modifies the dispersion relation of NIW as ω = f + 4Bu4 + Bu f f + σ + , (A 1)where the first f is absorbed in the ansatz of NIW and the second part σ + should beobtained from the YBJ + equation describing the modulation of NIW amplitude. Herewe introduce the Burger number Bu = N k f m . The spirit of YBJ + equation motivates us to introduce the following mappings: χ → K A, (A 2 a ) ∂ z → Nf K , (A 2 b )where u + i v = − i f χ e − i ft and K = (cid:18) ∂ x , ∂ y , fN ∂ z (cid:19) , (A 3)4therefore K · K = L + . Here we have assumed that f and N are constants for simplicity.Applying this mapping to the NIW Lagrangian (cf. (3.23) in Xie & Vanneste (2015)and taking β = 0) we obtain (cid:104)L(cid:105) NIW = − (cid:90) (cid:18) i f (cid:0) L + AD T L + A ∗ − L + A ∗ D T L + A (cid:1) + f ψG ( K A ∗ , K A ) + 14 N |∇K A | (cid:19) d x , (A 4)where D T = ∂ T + J ( ψ, · ) and G ( K A ∗ , K A ) = 12 (cid:0) |∇K A z | − K A zz ∇ K A ∗ − K A ∗ zz ∇ K A (cid:1) . (A 5)Taking variation of A ∗ we obtain the wave equation L + (cid:0) D T ( L + A ) (cid:1) + i f L + ∇ A + 2i f L + (cid:18) ∇ · (cid:0) ψ ∇ L + A (cid:1) − K · (cid:0) ψ ∇ K A (cid:1)(cid:19) + 12 ∇ K· (cid:16) ψ K / A (cid:17) = 0 . (A 6)Note that (A 6) has three refraction terms that are comparable with that in the first YBJequation (Young & Ben Jelloul 1997).We can multiply ( L + ) − to (A 6) to obtain a complicated version of the YBJ + equation: D T ( L + A )+ i f ∇ A + 2i f ∇· (cid:0) ψ ∇ L + A (cid:1) − i f K· (cid:0) ψ ∇ K A (cid:1) + 12 ∇ K − · (cid:16) ψ K / A (cid:17) = 0 , (A 7)where operator K − is nonlocal in the last refraction term.But for the modeling purpose, we introduce a modified wave potential energy G → G + = 14 ∇ | L + A | , (A 8)corresponding to the vertical averaging procedure introduced by Wagner & Young (2016),then the Lagrangian of NIW become (cid:104)L(cid:105) NIW + = − (cid:90) (cid:18) i f (cid:0) L + AD T L + A ∗ − L + A ∗ D T L + A (cid:1) + f ψ ∇ | L + A | + 14 N |∇K A | (cid:19) d x , (A 9)then taking the variation of A ∗ we recovers the YBJ + equation: D T ( L + A ) + i f ∇ A + i2 ∇ ψL + A = 0 . (A 10)Thus, the potential vorticity (PV) with barotropic mean flow becomes q = ∇ ψ + f N ψ zz + i f J (cid:0) L + A ∗ , L + A (cid:1) + f G + = ∇ ψ + f N ψ zz + i f J (cid:0) L + A ∗ , L + A (cid:1) + f ∇ | L + A | . (A 11)The relabeling symmetry implies the PV conservation q t + J ( ψ, q ) = 0 , (A 12)that together with the YBJ + equation (A 10) form a closed couple system. Then,assuming that the mean flow is barotropic, i.e. ∂ z = 0, and the wave has only one verticalwave wavenumber m , and renaming the variable L + A as M , we obtain the YBJ + -QGmodel (3.1).5A.2. Argument for the transfer of conserved quantities in YBJ + -QG model Note that instead of the variable A used by Asselin & Young (2019 a ) in the aboveequations we introduce M = i L + A/f , where L + A = − f m /N + ∇ /
4, to directlycompare (3.1) with (2.1). We can see that comparing with (2.1 a ) the YBJ + equation(3.1 a ) modifies the linear dispersion term, which bounds the maximum frequency to beresolved as 2 f instead of ∞ , thus, the numerical simulations are accelerated. Meanwhile,the nonlinear terms are not changed, making the system (3.1) remains a good model tostudy the nonlinear dynamics of NIW-mean flow interaction.Same as the YBJ-QG model (2.1), the YBJ + -QG model (3.1) preserves the totalenergy, potential enstrophy and the wave action E = (cid:90) (cid:18) |∇ ψ | + f |L M | (cid:19) d x , (A 13 a ) P = (cid:90) q d x , (A 13 b ) A = (cid:90) | M | d x , (A 13 c )where L = ∇ f N m − ∇ . In the wave-dominant case, the energy and (potential) enstrophy dominates the dynam-ics, so the energy transfers upscale and the (potential) enstrophy transfers downscales.Since the waves are nearly passive, the wave action should transfer downscale.In the mean-flow-dominant case, energy and wave action control the dynamics, andtheir leading order expressions are E = (cid:90) f |L M | d x , (A 14 a ) A = (cid:90) | M | d x . (A 14 b )Again we consider the heuristic scenario that the energy and wave action are injected atan intermediate wavenumber k f and dissipate at both small and large wavenumbers k and k , therefore the conserved energy and wave action imply E f = f k f N m + k A + f k f N m + k A , (A 15 a ) A f = A + A . (A 15 b )Considering the relation (2.8) we can solve (A 15) to obtain A = (cid:0) f m + N k (cid:1) (cid:16) k f − k (cid:17) (4 f m + N k ) ( k − k ) A f and A = (cid:0) f m + N k (cid:1) (cid:16) k f − k (cid:17) (4 f m + N k ) ( k − k ) A f . (A 16)In the limit k → k → ∞ , the leading order energy and wave action dissipationcan be approximated as A = 44 + Bu f A f , A = Bu f f A f , E = 0 and E = E f , (A 17 a )6where Bu f = N k f / (cid:0) f m (cid:1) is the forcing Burger number. It is interesting to observethat the YBJ + -QG model can transfers energy to both wave action to both large andsmall scales depending on the value of forcing Burger number. If we want the energyand action transfer to recover those obtained in the YBJ-QG model (2.10), the forcingBurger number should be much smaller than 4. REFERENCESAlexakis, A. & Biferale, L.
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