a. The numerical model

The model describes the flow in a two-layer channel of length Y = 4000 km and width X = 1500 km on a β plane, and the layers have depths H₁ = 1000 m and H₂ = 4000 m. The balance of potential vorticity (QPV)¹,

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includes friction terms F_i = k · ∇ × τ_i/H_i − A_h∇⁶ψ_i where τ₁ is the wind stress vector, and τ₂ is the bottom stress. Subgrid-scale (SGS) effects are modeled by biharmonic lateral friction with hyperviscosity A_h = 10¹⁰ m⁴ s⁻¹. It reflects the parameterization of subgrid momentum transport and serves as energy and enstrophy dissipation. The elevation of the interface is expressed in terms of the geostrophic streamfunction by η = (f₀/g*)(ψ₂ − ψ₁), with f₀ = − 1.263 × 10⁻⁴ s⁻¹ and reduced gravity g*. Lateral boundary conditions ∇²ψ_i = 0, ∇⁴ψ_i = 0 on both walls (the latter establishes zero momentum flux) and integral auxiliary conditions (McWilliams 1977) are standard. A fourth-order accurate formulation of the Jacobian (Arakawa 1966) turned out to be necessary to compute the second-order eddy balances correctly (Wolff et al. 1993). The wind stress is zonal and zonally constant,

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with amplitude τ₀ = 10⁻⁴ m² s⁻². The frictional stress at the bottom is taken either as a linear or quadratic functional of the bottom velocity, that is,

τ₂ϵH₂u₂τ₂μH₂u₂u₂(4)

where ϵ and μ are the corresponding coefficients of linear or nonlinear bottom friction.

We have performed four suites of numerical experiments to explore the sensitivities of the model to the basic parameters ϵ and μ of friction, differential rotation β, and stratification g*. The experiments (see Table 1) are named by three letters; those starting with E and D have linear bottom friction, those starting with N and H have nonlinear bottom friction. The E cases have ϵ = 10⁻⁷ s⁻¹; for the D cases this value is doubled. The N cases have μ = 10⁻⁶ m⁻¹; for the H cases this value is halved. In each suite we have a standard case (*FB, β = 1.1465 × 10⁻¹¹ m⁻¹ s⁻¹, g* = 0.02 m s⁻²) and for each such case there is an offspring with stronger stratification (*G4, doubled g*) and one with half the β value (*B5). Although the Rossby radius λ = [g*H₁H₂/(H₁ + H₂)]^1/2/f₀ exceeds the resolution Δx = Δy = 20 km only by a factor of about 2, the model is eddy resolving (we have made experiments with higher resolution). We will use the notation λ_i = [g*H_i]^1/2/f₀ for the Rossby radius scales of the individual layers.

The integration time of all experiments was 110 years, including a spinup of approximately 20 years. The last 90 years of data were used to determine mean values and eddy covariances. For all these experiments we have evaluated the balance of mean momentum and potential vorticity, and a package of second-, third-, and fourth-order moments of turbulence. Table 1 summarizes the transports and energies of the experiments.

b. The flow regime

Figure 1 shows instantaneous and eddy streamfunction fields of our standard experiment EFB at the end of the integration. A narrow jet has developed with a scale, which is much smaller than the channelwide scale of the wind forcing but much larger than the Rossby radius. It meanders significantly with stronger excursions appearing in the upper layer. The eddy field is defined here as deviation from the time and zonal mean circulation. The instantaneous eddy streamfunction fields show an irregular chain of vortices to the north and to the south of the jet center (particularly in the lower layer). The eddies have a tendency toward an ellipsoidal form, leaning into the direction of the jet. This pattern causes the Reynolds stress convergence (see next section) and thus the concentration of the jet (Holland and Haidvogel 1980). The eddy field is highly correlated in the vertical (see section 3d) but not barotropic, the eddies travel eastward relative to the mean flow, as found by McWilliams et al. (1978) for the most unstable mode of similar channel jets.

Figure 2 shows the profiles of the time and zonally averaged zonal velocities of EFB, DFB, NFB, and HFB. The jet is flanked by side lobes having a similar width as the jet. It is unstable in all cases, the mean shear U = u₁ − u₂ is above the critical value of the Phillip’s inviscid linear instability criterion, U > βλ²₂ (it is not exactly applicable here because the current profiles are not meridionally uniform as in Phillips’ model). Boundary layers are clearly identified at the walls, the scale of these layers has the size of the internal Rossby radius.

The gradients of mean potential vorticity (Fig. 3 for EFB) reveal the well-known property of the differing signs in the upper and lower layer in all cases, thus the necessary condition of baroclinic instability is satisfied. The necessary condition for barotropic instability is not satisfied: there is no sign change of the barotropic QPV gradient (lower-left panel). We have separated the QPV gradients into the contributions due to the relative, stretching and planetary vorticity showing that stretching is the dominant contribution. This agrees with the generally assumed scaling (λ/L)² ≪ 1 of relative to stretching vorticity for a large-scale flow with length scale L. In the lower layer, however, the signs of the relative and the stretching parts differ. In the center of the jet their sizes are merely a factor of three different so they add in a way to make the gradient of the total QPV of the same size but different sign as the gradient of relative vorticity. Relative vorticity may thus not be neglected.

c. The balance of zonal momentum

For QG dynamics the zonally and time averaged equations for the balances of zonal momentum and mass can be written as a balance between the eddy QPV flux and the frictional stresses,

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The overbar denotes time and zonal mean, τ₁ is the zonal wind stress, and τ₂ is the zonal bottom stress. SGS friction is omitted because it is negligible in the balance of the mean flow. The time rate of change term is included here for clarity, it is set in curly brackets to indicate that it vanishes due to time averaging. The QPV flux,

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is the two-dimensional divergence of the QG form of the Eliassen–Palm flux. It consists of the Reynolds stress divergence and the vertical divergence of the interfacial form stress, the latter is responsible for the vertical exchange of momentum between the layers. The vertical integral of the QPV flux equals the divergence of the vertically integrated Reynolds stress,

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and integrating further over the channel width, we find the constraint (Bretherton 1966),

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which is central to the conservation of total zonal momentum as a balance between wind stress and bottom stress,

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The second-order eddy-induced covariances appearing in the mean balance of momentum (and QPV) are summarized for all experiments in Fig. 4 (see also Fig. 5). The Reynolds stress in the upper layer is characterized by a central convergence of eastward momentum—establishing a countergradient transport of momentum—and divergence on the flanks. The deep stress divergence is considerably smaller (about a factor of 20) and generally also shows a convergence of eastward momentum in the center (a notable exception is the bottom layer Reynolds stress in the E cases). Though there are clear variations in the overall size, the shape of the stress divergences is surprisingly similar (there is almost no change of the zeroes). The overall balance of the system requires that the magnitude of the interfacial stress must correspond to the wind stress. The Reynolds stress is, however, in some sense a free mode of the system. While the system chooses a small Reynolds stress divergence in the deep layer, the significance of Reynolds stress effects in the directly forced top layer contradicts the scaling derived from linear baroclinic instability theory where the ratio of the Reynolds stress and the interfacial stress is to be of order O(λ/L). It is the failure of this scaling that discriminates our regime from the conditions investigated in most recent parameterization schemes of broad scale ocean circulation (e.g., Treguier et al. 1997) and homogeneous β-plane turbulence (e.g., Held and Larichev 1996) where Reynolds stresses are neglected or vanish identically, respectively. Due to the balance of the QPV flux and the applied stresses at the top and the bottom the QPV flux has different signs in the layers, which are just contrary to the signs of the mean gradient of QPV. The QPV flux in thus down the gradient of mean QPV.

a. Eddy potential enstrophy

The time and zonally averaged balance of eddy potential enstrophy (EPE), defined as , is written in the form

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The first term on the rhs—the divergence of a triple moment—represents the eddy-induced flux of EPE. The next term is the production or destruction due to the gradient of mean QPV, it describes the exchange with the enstrophy of the mean flow. The last term arises from the friction terms in the QPV balance. In the upper layer it is entirely caused by the parameterized SGS motion, in the lower layer there is also a contribution from the bottom friction of the resolved eddy flow. As shown in the lower-left panel of Fig. 6, the SGS contribution exceeds the resolved part. In some cases, the latter is partly positive but then it is always compensated by the negative SGS part. We refer to as “EPE sink”: it includes, besides the true dissipation of EPE by the enstrophy cascade to small scales, also a divergence of a frictionally induced flux of EPE as well as a vertical flux exchanging EPE between the layers. Due to the sixth-order derivatives of the streamfunction occurring in the biharmonic friction the computational burden, which is necessary to separate the above terms in the numerical code, is very high so we have not performed this step.

Figure 6 shows the zonally averaged EPE balance for EFB. The balances for the upper and the lower layers are similar in structure. The mean gradient term is positive everywhere, that is, the potential vorticity flux is opposite to the gradient (downgradient) of the mean potential vorticity, as found already by inspection of the individual quantities. It is clear that the production and the sink terms must equate in the integrated balance, the triple term is, however, free from such a constraint of balance. The triple term turns out to be of the same magnitude as the production and sink in the middle of the channel (it even exceeds the sink in some cases) and also gives substantial input to the balance in the jet flanks. This disagrees with Shutts (1983), Treguier et al. (1997), and others who have suggested that the triple moment is negligible in large-scale flow. McWilliams and Chow (1981) overestimated the triple moment due to the failure of the Arakawa Jacobian in the enstrophy balance of their model (see Wolff et al. 1993).

The terms in the bottom layer balance are an order of magnitude smaller than those in the top layer but even in this weak eddy regime the eddy-induced transport is significant. We conclude that, at least for the top layer, there cannot be any significant contribution from the above mentioned frictionally induced exchange term in the above separation of . Another outstanding feature of the balance is the local minimum of the sink term in the jet center, which is especially well-pronounced in the lower layer. It mainly arises from the resolved frictional part of but slight contributions are made by the SGS part as well (see Fig. 6, left lower panel). Because these undulations in the sink terms are in phase in the two layers they cannot represent an exchange between layers, and we must attribute them to a frictionally induced transport divergence, which is part of the sink term. This transport counteracts the transport by the triple moment.

b. Eddy energies

The balances of EPE and eddy available potential energy (EAPE), defined as , are very similar in structure:

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Here p_i is the geostrophic pressure and w the vertical velocity of the interface, which is determined by the divergence of the ageostrophic flow in the bottom layer,

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As in (10) we notice in (11) the divergence of a triple moment and a production or destruction due to the gradient of mean interface height (it describes the exchange with the potential energy of the mean flow). There is, however, no friction or dissipation term; instead we find a covariance between the vertical velocity and the thickness perturbation, describing the exchange of EAPE and eddy kinetic energy by eddy induced lifting of the interface.

The eddy kinetic energy (EKE) is balanced according to

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where X_i = (X_i, Y_i) are the frictional (resolved bottom friction and SGS) terms in the ageostrophic momentum equations. They correspond to the QPV friction term F_i = k · ∇ × X_i, as specified in section 2. The ageostrophic fluxes appearing in (13) and the exchange terms between kinetic and potential energy cannot be evaluated from ordinary quasigeostrophic dynamics.²

The balances (11) and (13) are displayed in Fig. 7 with the ageostrophic and SGS terms combined as residual imbalance. Consistent with the dynamics of a baroclinically unstable state the eddy system is driven by the transfer to EAPE from the mean baroclinic energy [by the gradient term in (11)] and further transfer to the EKE [by the ageostrophic exchange term in (11)]. The counterpart of the latter transfer is contained in the ageostrophic imbalance in the balance of EKE in Fig. 7. The mean gradient term in the EKE balance is a loss of EKE everywhere in both layers: the energy cycle is closed—apart from the loss due to friction—by the transfer from eddy to mean kinetic energy [by the gradient term in (13)]. Notice that this property implies an upgradient transport of momentum everywhere in the channel. The eddy-induced transport by the geostrophic terms leads to a loss of eddy energy in all three compartments in the jet center and affects a transfer to the jet flanks.

If (11) and (13) are combined to the balance of total vertically integrated energy of the system, the exchange terms between potential and kinetic energy cancel and ageostrophic terms remain only in form of divergences. The balance is displayed in the lower-right panel of Fig. 7. It is seen that the sum of all gradient terms drives the eddy system and bottom friction acts as major sink. There is a small net sink from SGS contained in the imbalance, in EFB it is about 40% of the size of the bottom friction term. The dominant role of the imbalance term is, however, a redistribution by ageostrophic eddy motion which counteracts the geostrophic eddy induced transport.

c. QPV and thickness flux

Finally we consider the balances of the eddy induced QPV flux and thickness flux, which read

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We have combined in the expression

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the ageostrophic covariances. We can identify in (14) the divergence of eddy-induced fluxes (by a triple moment) and the production/destruction term associated with the gradient of mean QPV or interface height, respectively. The latter terms describe the exchange of flux with the corresponding transport of mean quantity by the mean current. The meridional flux is coupled by a Coriolis term to the zonal flux. Sink terms are certainly contained in the frictional terms. Contributions in the F′ and Y′ terms from the (resolved) bottom friction can easily be extracted and evaluated. They are of order ϵ times the flux and found to be very small. The remaining SGS contributions have not been evaluated separately but are expected to be small as well. The role and magnitude of the ageostrophic terms are not a priori evident.

The QPV flux balance is displayed in the upper panels of Fig. 8 (in contrast to the scalar balances we have not integrated the flux balances over the layer depths). The contribution from the eddy triple flux, the gradient term, and the Coriolis term are shown individually and the ageostrophic terms are again combined in the remaining imbalance. The balance is easily described: the gradient term acts as a source of northward flux and the ageostrophic terms act as a sink. In the upper layer there is a significant contribution from the triple divergence working as destruction in the jet center. The Coriolis term is a small sink, mainly acting in the flank region. It is obvious from (14) that the magnitudes of the balance terms in the two layers are roughly in proportion to the velocity variance. The balance of thickness flux, shown in the lower panel of Fig. 8, closely corresponds to the balance pattern of the QPV flux. Coriolis and triple terms are here negligible.

d. Timescales of second-order balances

Though we have a drastic difference in the overall size of the balance rates of EPE in the two layers (see Fig. 6), this is compensated by the size of EPE content (see Fig. 5) so that the timescale of the balance terms in the different layers is of the same magnitude, of order 100 days and thus comparable to timescales of frictional processes: the timescale of the bottom friction is 1/ϵ ≈ 100 days, the time of nonlinear bottom friction and the timescale of the SGS parameterization, when taken for the grid scale, are of the same size. The timescale of the rates of energy conversion is clearly of the order of the frictional processes.

The timescales associated with balance rates of the QPV and thickness fluxes are in strong contrast to those of the scalar quantities. In Fig. 10 we compare the timescales of the terms of the QPV flux and thickness flux with some scaling laws explained below. More specifically, the times shown there are associated with the sink rate due to the ageostrophic terms in the flux balances,

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evaluated at the channel center. The T_i differ by an order of magnitude in the layers and T_η is somewhat smaller than T₂. In the bottom QPV flux and the thickness flux we find relaxation scales of the order of 10 days, and in the top layer T₁ ranges from a few hours to a maximum of 1.5 days.

These times are certainly not associated with friction but must be internally established by the flow dynamics. To gain a view of the temporal behavior of the eddy field we have evaluated the covariance of the meridional eddy velocity at various places in the channel. Figure 9 shows the correlation functions for case EFB. Six years of data at five meridional positions (center, on the flanks of the jet, and 500 km away from, near the boundaries) have been used and a zonal average has also been performed by taking the mean over ten points on the same latitude. In the jet the correlation is seen to decrease fairly rapidly after a few days. The microscale of the correlation is roughly three days, the integral scale ranges from about ten days in the center to 50 to 80 days outside the jet closer to the walls where the flow is less disturbed by eddies. For larger lags we find a quasiperiodic behavior at a very low correlation level, expressing the subsequent passing of similar eddies. In turbulence theory these correlation timescales are associated with advection and “overturning” of the eddies (L/u_i and L/υ^′2_i are, indeed, of the order of days but clearly not less than a day). The eddy flow is highly correlated in the vertical, cross-correlations between υ^′₁ and υ^′₂ look similar to the correlations with a maximum of 0.8 to 0.9 at zero lag.

While T₂ and T_η may correspond to the micro- or macroscale of the correlations this is certainly not true for T₁. This discrepancy is also found when relations to dynamically defined time scales are sought. From the dynamical point of view the most important internal timescales are associated with the baroclinic instability. Following Eady (1949) and Phillips (1954) the growth rates of baroclinically unstable disturbances are described by the timescales

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written here for the two-layer channel geometry.³ The first ignores the presence of the planetary vorticity gradient and may therefore not be appropriate. Both apply strictly, of course, to the linear phase of baroclinic instability and, moreover, to horizontally constant shear. Values for T_Eady and T_Phillips are given in Table 2, with the shear evaluated at the center of the channel. They are of the order of a few days, the Eady timesscale being generally smaller than the Phillips timescale. McWilliams et al. (1978) have determined the linear stability of the mean jet of their channel experiment and found a period of 9 days, which is in favor of the Phillips timescale.

The ratio of the timescales of the flux balances and the Phillips timescale, shown in Fig. 10 versus shear, exemplifies the fundamental problem of scaling. In general T/T_Phillips scatters about an average value by less than one order of magnitude, but this is roughly the range of the T or the Phillips timescale itself. Closer inspection, moreover, reveals clearly that within each group *FB (shown as *), *G4 (shown as ×) and *B5 (shown as +) of the experiments we see a decrease with shear, suggesting that further scaling by β and λ could possibly achieve a power law behavior of the ratio. Any additional scaling by β would affect only the level of the *B5 experiments and similarly, λ-scaling only affects the *G4 experiments. A reasonable power law scaling T ∼ β^aλ^bU^c can easily be found, none, however, in a universal form with a nondimensional coefficient of proportionality, indicating dependence on scales that have not been considered in our experiments. We conclude that T₂ and T_η are of the order of the baroclinic instability timescales but none of the above scalings (18) is successful.

a. QPV and thickness diffusivities

The values of the transfer coefficients k_i for QPV and κ for layer thickness are displayed in Fig. 11. They have been evaluated using zonal and time averages of the eddy fluxes and the mean gradients. The profiles of the QPV diffusivities are rather complex. A local minimum is observed in the jet center, while highest values are found on the flanks of the jet. The coefficients have different shapes in the layers and have a clear ordering of magnitude, k₂ > k₁. Relative to the maximum the valley in k₁ is deeper than in k₂. The cases show significant differences for different parameters of bottom friction, stratification and β, for example, in the width of the central valley and the height of the peaks above the valley, but clear parametrical dependences cannot be evaluated from such a limited suite of experiments. For the low β cases both k_i, and for the high friction cases, the k₂ diffusivity seem to settle on an almost constant shape in the jet region. The diffusivity for layer thickness has a simpler structure, which becomes more evident in the cases with higher friction: the diffusivity is more or less constant in the jet, starting to decrease in the region of the flanks.

Similar shapes for numerically determined diffusivities in a three-layer QG channel flow are reported by McWilliams and Chow (1981). Eddy-flux closure could not, however, stop with the knowledge of the positivity of coefficients. A physically motivated representation of their complex structure must be given. We discuss a few parameterizations and scaling theories that have been suggested to explain the shape and/or the overall size.

b. Parameterizations

All attempts known to us to parameterize the shape of the diffusivities k_i and κ have been performed in terms of algebraic functions of the local vertical shear U. We would like to emphasize, however, that there is no evidence for any single-valued relation between the diffusivities and the shear U in any of our experiments. As exemplified for EFB in Fig. 12, all cases yield multiple-valued relations. Notice, however, that this occurs predominantly in the flank region and not in the central jet.

Marshall’s (1981) closure is written in the form k_i ∼ U. It is based on Green’s (1970) and Stone’s (1972) form of the transfer coefficients for an infinitesimal wave growing linearly on a baroclinically unstable mean flow where the eddy transport coefficient of heat is proportional to the lateral gradient of the mean temperature or shear. Held (1978) has extended the Green–Stone concept to disturbances that do not fill the entire height of the fluid but are confined to a lower active portion. This yields a dependence k_i ∼ U⁴. Held and Pavan (1996) have recently suggested a parameterization for a wide jet. On the basis of numerical experiments similar to ours but driven by relaxation to a prescribed shear they find for sufficiently broad jets a parameterization k_i ∼ U^3/2(U − U_crit)^3/2 where U_crit ≈ βλ²₂ is the shear magnitude, which is critical for baroclinic instability. Ivchenko (1985b) pointed out that the Green–Stone closure scheme k_i ∼ U is inappropriate under strongly baroclinic conditions. He noticed that the minimum in the eddy diffusivities could be obtained in an expression placing the QPV gradient (with its central maximum) in the denominator. A more or less concentrated jet⁴ can indeed be obtained with a diffusive parameterization of the form k_i ∼ U/(U² + U²₀) with a free shape parameter U₀. This was tested by Wolff (1990), Sinha (1993), and Ivchenko et al. (1997). A peculiar property of this parameterization should be noted: for small U₀ that is, strongly baroclinic conditions, the transport of QPV is inversely proportional to the gradient of QPV, which contrasts the general concept of a diffusive process: though being downgradient, the transport of QPV is large where the gradient is small and vice versa. In an unforced initial value problem, such a flux leads to an unstable situation.

With exception for this last form (which still is physically not very sound) all these parameterizations fail to simulate the double peak structure of the coefficients and do not yield any substantial concentration of momentum in a jet. A detailed discussion of the failure of various similar forms of the Green–Stone parameterization for the thickness flux in a similar two-layer channel flow is found in McWilliams et al. (1978).

c. Scaling theories

The QPV diffusivity of the top layer is constrained in its magnitude by the momentum balance (5). The steady-state condition of this balance and the dominance of the stretching part of the mean QPV gradient leads to k₁ ≈ τλ²₁/U. Thus k₁ ∼ λ²/U, which, however, is not a parameterization. In the lower layer, and particularly for the thickness diffusion, the shape of the diffusivity could possibly be decoupled from its overall magnitude and thus, scaling could be successful. In particular the κ values show a clear ordering of magnitudes in Fig. 11.

From dimensional reasoning the diffusivity (QPV or thickness) can be represented in a mixing length form, k ∼ L²/T ∼ VL ∼ TV², with appropriate length, time, and velocity scales. The Green–Stone (GS) concept uses the Rossby radius λ supplemented by the Eady timescale T_Eady. In our case the Phillips timescale T_Phillips may be more appropriate. The diffusivities of these models take the form

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Because the diffusive process operates over a scale L, which is certainly larger than λ (see, e.g., Stammer 1998) and correlation properties of the eddy field are not explicitly taken into account in the above expressions, there are nondimensional coefficients in these relations that generally are not of order unity (see, e.g., Visbeck et al. 1997). Larichev and Held (1995) and Held and Larichev (1996) have recently suggested an alternative scaling model. Their scaling law (HL) is derived for fully developed homogeneous β-plane turbulence in the presence of a supercritical shear. They found that the eddy field will be made more barotropic and the inverse energy cascade will expand the eddy scales toward the barotropic Rhines scale, L_R = C/β where C is the barotropic velocity (Rhines 1975). Then, with the scaling C ∼ ξU obtained from energy balance considerations, where ξ = U/(βλ²) is the supercriticality,⁵ and assuming a mixing length form with barotropic velocity and length scales, Held and Larichev arrive at the scaling law

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As indicated in the last equality, the Rhines scale and Eady timescale are apparently working together in the HL scaling.

As an example of the diffusivity scaling we show in Fig. 13 the ratio of the diffusivities and the Eady scaling. All fields are evaluated at the channel center (for values see Table 2). As seen in Fig. 13 neither of k₂ or κ satisfies the GS scaling. Instead we find a decrease of the diffusivities with increasing shear, which is not as strong in case of k₁. As in case of the timescales considered above, a power law scaling k ∼ β^aλ^bU^c can be achieved, but there is no universal form without involving additional scales.

It is evident that this scaling is not appropriate, and similar negative results are obtained for the other scaling laws discussed above. We must conclude that the strong requirement of steady-state and zonal-mean condition and the hard constraint on the eddy QPV flux in top-layer momentum appears to make any scaling attempt of k₁ under these conditions meaningless. Because the Reynolds stress is introduced as a “free” mode of dynamics—it is largely free in terms of shape and magnitude—we have some freedom in the thickness flux and the lower-layer QPV flux but this concerns only the shape, not the overall (integrated) magnitude, which is again set by the wind stress.

We conclude this discussion of diffusive models with a negative statement: the conventional diffusive parameterizations and the scaling theories do not account for the complex structure of the QPV and thickness diffusivities. They fail to reproduce the double peak structure responsible for the concentration of zonal momentum in the jet. They also fail to adjust their magnitudes to any of the above presented scaling laws as consequence of the strong constraint imposed by the momentum balance of the system. We would like to note that more complexity in the model by adding more layers would not lead to a basically different conclusion. In a multilayer model as shown, for example, in Marshall et al. (1993, and there even with topography, realistic coast lines, and nonzonal wind stress) the momentum is constrained to be transferred essentially unchanged through the interior frictionless layers because Reynolds stress effects are negligible, as here in the bottom layer, and—because in a zonal mean flow with QG dynamics—there are no other transport mechanisms than interfacial stresses. This property implies that interfacial stresses in the interior would have to scale as the top and bottom layer thickness fluxes.

a. Parameterization of the flux triple moment

The local relation between the flux and mean gradient in (23) may be based on a false pretense: the divergence of the triple moment may introduce nonlocal behavior. A more precise treatment should also consider the weak coupling to the zonal flux explicitly. We proceed by searching for an advective closure of the triple moment that turns the QPV flux–gradient relation into a local form.

Breaking down the triple moment in the QPV flux balance, , into lower order moments the tensorial character of the moment must be preserved. It is a member of the tensor , so we try the form

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There are not very many possible candidates for the pseudoscalar ω_i. In Fig. 14 we give examples of a regression fit of (24) using the mean relative vorticity,

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with a dimensionless coefficient γ_i to the “data” of the triple moment from the numerical experiments. The fit with constant coefficient γ_i works quite well. The rms between the triple moment “data” and the fitted functionals indicates a misfit of roughly 10% to 30% and the values for the γ_i have an overall consistent pattern of size and sign within all cases. Other such simple choices for the pseudoscalar factor ω_i can hardly be given (the mean QPV, the planetary or stretching, vorticities do not work). It should be noted that the other components of the tensor can be represented by closures corresponding to (24) and (25).

b. Parameterization of the QPV flux

Using (24) and (25) the QPV flux balance appears in the parameterized form

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with K_i = . It is determined by one free constant parameter γ_i, a diffusivity K_i, and either the timescale T_i of the QPV flux relaxation or the covariance . These latter quantities must still be considered to be a function of position. Below we argue that the term involving the derivative of the covariance is small in our experiments and that K_i can be considered constant in the jet region. A coarse model is then developed with two constant parameters in each layer. A more general model based on (26) in a complete second-order closure approach would, however, require additional equations or recipes to determine γ_i, K_i and either of T or .

Comparison of the two γ_i terms in (26) shows for all experiments that is always smaller than (by a factor of roughly 0.2), indicating that the meridional scale of the jet must be smaller than the scale of the covariance. But also, the vanishing of relative vorticity and the derivative of covariance in the center of the jet helps to establish this result. In the following we abandon the second γ_i term to describe a simple shortcut of parameterization.

The concept to express all deviations from a homogeneous state (where γ_i = 0) by the correction term only makes sense if the resulting diffusivity has a simple, almost constant shape. Ignoring the last term in (26) we use in Fig. 15 this relation in the form

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to compare the numerically determined QPV flux—in the form of the k_i—with the expression on the rhs. The structure resulting from fitting the rhs of (27) with constant γ_i and K_i is displayed by the dashdotted curve and the values of these coefficients are given in Table 3. It is clearly seen in Fig. 15 that the γ_i term compensates for the double peak structure of k_i. Notice that in the top layer the fitted curve and the k₁ lie almost on top of each other; for the deep layer there is good agreement in the entire “valley” region of k₂. The coefficient K_i can thus indeed be considered constant over the entire jet region.

We have attempted a scaling of the diffusivity K_i by the scaling laws discussed in section 4c but obtained similar negative results as for the QPV diffusivities. In the present state of this work the K_i as well as the γ_i must be taken without a universal basis. The values of the γ_i in Table 3 clearly show that the correction to conventional QPV diffusion is most important in the top layer where the contribution of the relative vorticity gradient to the QPV gradient is even overcompensated:the correction affects a change of sign of the contribution from the relative vorticity. In the bottom layer the correction is generally small.

c. A closed coarse model

Neglecting the last term in the parameterization (26) the steady state momentum balance (5) becomes

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Though any parameterization of the complete QPV flux cannot consider lateral transport and vertical exchange of momentum in separation, the above equations suggest a simple interpretation in terms of these processes. The contribution arising from the relative vorticity describes lateral transport and the stretching part contains the vertical exchange of momentum between layers. The β term is a source of eastward momentum due to the mere presence of eddies (i.e., nonzero K_i). It appears similar to the Neptune effect investigated by Holloway (1992);a physical interpretation is, however, not easily found. The considerations outlined below suggest to combine the stretching part and the β term in a particular way to enable an interpretation by eddy-induced interfacial friction and a direct driving of the flow by eddies.

Because the diffusivities K₁ and K₂ differ, the stretching part of the QPV gradient in the two layers cannot entirely be attributed to momentum exchange. This part is split apart by rewriting (28) in the form

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where U = u₁ − u₂ and λ²_i = g*H_i/f²₀ as before, and

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The first part of the stretching term in (29) represents momentum exchange between the layers; it has the form of interfacial friction. Because K₂ − K₁ > 0 in all our experiments and thus U^crit_i > 0 as well, the second part of the stretching term in (29) represents a source of eastward momentum where the shear exceeds the corresponding critical value U^crit_i; at other places the flow is retarded.

In comparison to conventional QPV diffusion the most important feature of the coarse model (29) is the direction of the lateral momentum transport by the diffusive term. It is controlled by the parameter γ_i. If γ_i = 0, we are facing the conventional QPV diffusion parameterization where momentum is transferred downward the lateral velocity gradient (pure QPV diffusion with consideration of the relative vorticity implies diffusion of momentum, i.e., downgradient diffusive transfer). With γ₂ > 1 and γ₁ < 0, as found in our experiments, the correction term in the parameterization (26) causes a countergradient transfer of momentum in the upper layer and a stronger downgradient transport in the lower layer. Both these processes thus cause an increase in shear. Remember that this property results from the triple moment in the QPV flux balance.

The direct eddy driving causes as well an increase of shear if U^crit₂ > U > U^crit₁, as in our experiments. The separation of the QPV gradient thus leads to a momentum balance of the jet where interfacial friction is the only sink of eastward momentum in the top layer and the only source in the bottom layer.

The model has to consider the integral constraint (8). Using the present parameterization of the QPV flux the constraint becomes

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It may be satisfied by reducing the number of coefficients. In the regime of our experiments the last term on the rhs is small compared to the (positive) first. Hence, in case of eastward flow where τ₁ > 0 we have U > 0, and thus K₂ > K₁ is required at least in part of the domain. In a westward forced flow where τ₁ < 0 and U < 0 we find the condition K₂ < K₁, as described by Ivchenko et al. (1997) for an experiment of our channel model with westward forcing.

With constant coefficients (28) can be solved analytically.⁶ The relative vorticity term introduces two lateral scales that are independent of the forcing scale. In case of conventional QPV diffusion (i.e., γ_i = 0) these scales are the Rossby radius Λ₁ = λ and a scale Λ₂ = [(K₂/ϵ)(H₁ + H₂)/H₂]^1/2, which results from the balance of diffusion and friction in the bottom layer. Modifications of the sinusoidal behavior of the forcing only occurs in boundary layers at the walls. As shown in Fig. 16 (upper left panel) the exponential boundary layer penetrates quite far into the interior (in this example we have Λ₂ ≈ 200 km ≫ λ) and mimics a sharpening of the jet. In a broader channel this would not occur but the central jet would still be sinusoidal as the forcing. The countergradient transport arising from the triple correction switches the boundary layer behavior to a radiating behavior [both eigenvalues of the linear operator on the lhs of (28) change sign]. Figure 16 (lower-left panel) shows the effect of the correction term: the central jet is concentrated and intensified flanks occur as in the numerical experiments (compare with Fig. 2). The left panels of the figure consider the resulting profiles for an f plane where the solution with constant diffusivity and homogeneous boundary condition on u_i is purely sinusoidal.