• Charney, J. G., and J. G. De Vore, 1979: Multiple flow equilibria in the atmosphere and blocking. J. Atmos. Sci.,36, 1205–1216.

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  • Gent, P. R., and J. C. McWilliams, 1982: Intermediate model solutions to the Lorenz equations: Strange attractors and other phenomena. J. Atmos. Sci.,39, 3–13.

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  • Nicolis, C., 1992: Probabilistic aspects of error growth in atmospheric dynamics. Quart. J. Roy. Meteor. Soc.,118, 553–568.

  • ——, 1999: Entropy production and dynamical complexity in a low-order atmospheric model. Quart. J. Roy. Meteor. Soc.,125, 1859–1878.

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  • Van Mieghem, J., 1973: Atmospheric Energetics. Clarendon, 306 pp.

  • View in gallery

    Stable regimes of model (7) as the forcing F increases. Parameter values ν0 = k0 = 1/48, g0 = 8, a1 = a2 = 1, a3 = 3, c = 3/2, h1 = −1, and h2 = h3 = F2 = F3 = 0

  • View in gallery

    Time evolution of y1, y2, y3 as obtained from model (7) in the periodic regime, (a) with F = 0.13; and in the chaotic regime, with (b) F = 0.06. Parameter values as in Fig. 1

  • View in gallery

    Dependence of the (a) space-averaged horizontal entropy production σh,0, (b) vertical entropy production συ,0, and (c) kinetic energy Ek,0 as the forcing amplitude F is increased. In the presence of time-dependent regime F ≥ 0.05 these quantities are obtained through an averaging over the attractors (averaging time ∼3000 days). Dashed curve represents the properties of the Hadley circulation regime beyond its stability region

  • View in gallery

    Time-dependent behavior of (a) σh,0 (solid line) and συ,0 (dashed line) and (b) Ek,0 (dotted line) in a periodic regime corresponding to F = 0.13

  • View in gallery

    As in Fig. 4 but in a chaotic regime with F = 0.06

  • View in gallery

    One-dimensional mapping resulting from the Lorenz (1980) plot of (a) the successive minima of y1; and the corresponding values of (b) σh,0, (c) συ,0, and (d) Ek,0. Parameter values as in Fig. 5

  • View in gallery

    Transient evolution of σh,0 toward the chaotic attractor with F = 0.06. Initial conditions xi = yi = zi = 0.1 (full line) xi = yi = zi = 10−4 (dashed line)

  • View in gallery

    Patterns of the total entropy production along the horizontal and vertical directions as obtained from model (7) with F = 0.04. For this parameter value the system admits two stable fixed points corresponding to antisymmetric dissipation patterns of (a), (b) σh and (c), (d), συ respectively. The values on the isolines have been magnified by a factor of 1000

  • View in gallery

    Dependence of the averaged horizontal entropy production (a) σh,0 and (b) kinetic energy Ek,0 (dots) as the forcing F increases in the case of the quasigeostrophic approximation, Eq. (22). The dashed curves extrapolate σh,0 and Ek,0 beyond the stability region of the Hadley solution (I) and of the two fixed point solutions (II, III)

  • View in gallery

    As in Fig. 7 but for the model of Eq. (22) and F = 0.2

  • View in gallery

    Spatial patterns of the total entropy production (limited here to the horizontal direction) as obtained from model (22) with F = 0.04. For this parameter value the system admits two stable fixed points [(a) and (b)] for which the dissipation patterns are shifted by about half a wavelength. The values on the isolines have been magnified by a factor of 1000

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Dissipation Trends in a Shallow Water Model

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  • 1 Institut Royal Météorologique de Belgique, Brussels, Belgium
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Abstract

The formalism of irreversible thermodynamics is applied to the shallow water model. Entropy production and entropy flow terms are identified, describing the ways dissipation and exchange processes unfold in space and time. Explicit evaluations are carried out in the case of Lorenz’s nine-mode truncation and in the quasigeostrophic limit of the model. A number of systematic trends are identified by studying the way dissipation and kinetic energy vary as the forcing is increased and the system undergoes qualitative changes of behavior between different regimes, from simple symmetric flow to intermittent chaos. The constraints imposed by thermodynamics on the structure of the model equations and, especially, on the parameterization schemes are brought out.

Corresponding author address: Catherine Nicolis, Institut Royal Météorologique de Belgique, Avenue Circulaire 3, B-1180 Brussels, Belgium.

Email: cnicolis@oma.be

Abstract

The formalism of irreversible thermodynamics is applied to the shallow water model. Entropy production and entropy flow terms are identified, describing the ways dissipation and exchange processes unfold in space and time. Explicit evaluations are carried out in the case of Lorenz’s nine-mode truncation and in the quasigeostrophic limit of the model. A number of systematic trends are identified by studying the way dissipation and kinetic energy vary as the forcing is increased and the system undergoes qualitative changes of behavior between different regimes, from simple symmetric flow to intermittent chaos. The constraints imposed by thermodynamics on the structure of the model equations and, especially, on the parameterization schemes are brought out.

Corresponding author address: Catherine Nicolis, Institut Royal Météorologique de Belgique, Avenue Circulaire 3, B-1180 Brussels, Belgium.

Email: cnicolis@oma.be

1. Introduction

Atmospheric flow is one of the most complex phenomena encountered in nature. It involves processes occurring over a wide spectrum of space scales and timescales and gives rise to a variety of behaviors, ranging from climate variability to wave propagation, thermal convection, weak chaos, or fully developed turbulence.

The need to issue reliable predictions of the future states of the atmosphere and climate has stimulated the development of numerical forecasting models incorporating all processes deemed to be relevant for this purpose. This introduces a large number of variables and heavy parameterizations, often at the expense of a deeper understanding, prompting several researchers to develop more compact models representative of certain typical atmospheric situations: the shallow water equations (Lorenz 1980; Gent and McWilliams 1982), quasigeostrophic approximations (Charney and De Vore 1979), or the Boussinesq equations (Saltzman 1962; Lorenz 1963). As a rule, once such models are set up the unknown fields are expanded in a set of linearly independent basis functions and the equations for the expansion coefficients are truncated to a low order. The resulting evolution laws can then be analyzed by the methods of dynamical systems theory. Such studies have provided interesting and efficient ways to characterize the underlying complexity through, for instance, Lyapunov exponents and fractal dimensions (Ott 1993) or the increase of small initial errors arising from sensitivity to initial conditions (Nicolis 1992).

Characterizing complex systems in an intrinsic manner and in a unifying way that is largely independent of the details of the ongoing processes is also the principal aim of thermodynamics. In thermodynamic equilibrium this objective is fulfilled in a remarkably powerful and universal manner, thanks to the extremal properties of the thermodynamic potentials. In the presence of nonequilibrium, purely dissipative, processes and provided that some additional conditions guaranteeing the constancy and symmetry of phenomenological coefficients linking the fluxes of various quantities to the associated constraints are fulfilled, a weak form of universality is also available through the minimum entropy production theorem (De Groot and Mazur 1962). Such results have prompted several authors to seek for extremal properties and, more generally, for global organizing principles in atmospheric physics–related problems (Dutton 1973; Paltridge 1975, 1981; Nicolis and Nicolis 1980; Shutts 1981; Mobbs 1982; Stephens and O’Brien 1995). The success of these attempts has been rather mitigated. In the present work an alternative approach linking atmospheric dynamics to thermodynamics is explored in a case study involving a truncated form of the shallow water equation. First, the behavior of entropy production in the different dynamical regimes generated as the constraints driving the system are varied is monitored, and some unexpected trends are identified. Second, the time evolution of entropy production during the approach to a particular regime is derived and shown not to obey to a variational property. Finally, it is found that the second law of thermodynamics imposes for free, through the positivity of entropy production, conditions on the ways the dissipative processes are to be identified from the model equations.

In section 2 the principal properties of the shallow water model developed by Lorenz (1980) are summarized. From the standpoint of irreversible thermodynamics, the main specificity of this model turns out to be the decoupling of thermal effects from the dynamics. Still, the thermodynamic approach keeps its full meaning and interest. Uncoupling thermal effects or, more precisely, replacing them by an effective forcing amounts to overlooking only the part of dissipation related to heat transfer. On the other hand, the momentum balance equation governing the dynamics contains a second major source of dissipation: the one associated with internal friction (viscosity) effects. Based on these considerations we therefore proceed, in section 3, to the study of the model from the standpoint of irreversible thermodynamics. Explicit analytic expressions for the entropy production and kinetic energy are obtained in terms of the model variables and parameters. The dependence of these quantities on the dynamical regime generated by the system as the principal forcing parameter is varied is analyzed in section 4, and a number of interesting trends are identified. Section 5 is devoted to the quasigeostrophic limit of the model, and the main conclusions are drawn in section 6.

In a recent paper by the present author (Nicolis 1999) the irreversible thermodynamics of Lorenz’s three-mode truncation of the thermal convection (Boussinesq) equations has been developed. The main focus of this study was the behavior of entropy production during the successive stages of the phenomenon in the course of time and its connection with Kolmogorov entropy and other indicators of the dynamical complexity. In the present work emphasis is placed on the dependence of thermodynamic quantities on the parameters, rather than on their local variability across the attractor associated to a given dynamical regime.

2. Lorenz’s shallow water model and its nine-mode truncation

In the following we are concerned with a homogeneous, incompressible, uniformly rotating fluid of infinite horizontal extent moving over a topography h(r). The upper free surface of the fluid is at a height H + z(r), where it is understood that the horizontal average of z is zero. Under the additional assumption of hydrostatic equilibrium the momentum balance of the fluid becomes decoupled from the energy balance. Furthermore, since the magnitude of the horizontal components Vx, Vy of the velocity field is much larger than that of the vertical component w, projection of the full, three-dimensional momentum balance equation along the horizontal plane will produce to the dominant order a closed equation of the form (Lorenz 1980)
i1520-0469-57-21-3559-e1
d/dt being the hydrodynamic derivative. Here it is further understood that V and z depend solely on the horizontal coordinates x and y. Further, k is the unit vector along the vertical, f is the Coriolis parameter, g is the acceleration of gravity, and ν is the kinematic viscosity. Equation (1) is usually referred to as the shallow water equation.
Equation (1) needs to be complemented by information on the evolution of z. This is achieved by the balance equation of potential energy Ep to which z is directly related through Ep = g(zh). The only mechanism at the origin of a variation of potential energy is the work per unit time, −gw, done by the gravitational force. Now, w is related to  · V through the incompressibility condition. Combining these different ingredients one arrives at the relation
i1520-0469-57-21-3559-e2a
Lorenz adds to the right-hand side of this equation a forcing term, F, and a damping term, κ2z, both attributed to thermal effects. As pointed out earlier, thermal effects should in principle be decoupled in the framework of the adopted approximations. Nevertheless we shall follow Lorenz in keeping such terms in the dynamics, although their thermodynamic status is subject to caution. We defer to section 3 a more detailed discussion of this point.
Summarizing, the potential energy balance will be written in the form
i1520-0469-57-21-3559-e2b
It constitutes, together with Eq. (1), the set of equations governing our dynamical system.
As well known, the vector, V, can be decomposed into a part deriving from a potential, χ, and a part deriving from a streamfunction, ψ,
Vχkψ.
A low-order truncated version of Eqs. (1) and (2) has been derived by Lorenz (1980) by expanding χ and ψ in three linearly independent functions
ϕiαirLi
where L is a horizontal length scale and αi three horizontal vectors that sum to zero. In reduced variables the expression reads
i1520-0469-57-21-3559-e5
It is convenient to introduce the parameters
i1520-0469-57-21-3559-e6
where i, j, k are in cyclic order. Introducing (5) into Eqs. (1) and (2b) and neglecting all contributions that do not project onto the original basis functions [Eq. (4)] one obtains the following set of equations for the expansion coefficients xi, yi, and zi of the fields χ, ψ, and z:
i1520-0469-57-21-3559-e7
where the time derivatives are taken with respect to the reduced scale τ = ft, and ν0 = νf−1L−2, κ0 = κf−1L−2, g0 = gHf−2L−2. As in (6), Eqs. (7) are defined for each cyclic permutation of (1, 2, 3).

In the following we choose a1 = a2 = 1, a3 = 3, c = 3/2, resulting in α1 = (0, 1), α2 = (−3/2, 1/2) and α3 = (3/2, −3/2). Gent and McWilliams (1982) show that, for this choice and for parameter values ν0 = κ0 = 1/48, g0 = 8, h1 = −1, h2 = h3 = F2 = F3 = 0, and F1 = F ≠ 0, Eqs. (7) admit a sequence of transitions as the forcing F varies. These transitions lead from a purely zonal and nearly geostrophic stationary (Hadley) circulation to at least two stable coexisting nonzonal stationary circulations and, finally, to time-dependent circulation patterns that may be time periodic or chaotic. Figure 1 summarizes the bifurcation diagram of these solutions. A single fixed point solution prevailing for small amplitude of the forcing F bifurcates to two coexisting fixed points. A transition to weak chaos is subsequently observed, followed by a succession of periodic and chaotic windows, a large interval of periodic solutions, and eventually intermittent chaos. Figure 2 depicts the time series pertaining to the evolution in the periodic (Fig. 2a) and in the chaotic (Fig. 2b) regimes. The above figures have been obtained by integrating Eqs. (7) using a second-order scheme with a time step of 0.002 time units (1 time unit = 3 h). In the next sections the thermodynamic signature of these different behaviors will be studied and some systematic trends will be identified.

3. Energy dissipation and entropy balance in the shallow water model: Formulation

Our objective in this section is to obtain the energy dissipation and the associated entropy production generated by the model of section 2. To this end we first derive a balance equation for the sum of the horizontal kinetic energy density Ek = V2/2 and of the potential energy density Ep = g(zh). This equation will be cast in the form of the divergence of a vector field plus a sink term,
i1520-0469-57-21-3559-e8
from which the horizontal energy flux JE,h and the horizontal energy dissipation ϕE,h will be identified.
Multiplying both sides of (1) by V and adding to the result Eq. (2b) multiplied by g we obtain
i1520-0469-57-21-3559-eq1
Applying the identity a · divb = diva · bb · a, where a and b are, respectively, a vector and tensor of rank 2, one obtains a cancellation of the −gV · z term and a reduction of this equation to
i1520-0469-57-21-3559-e9
where (V)s is the symmetric stress tensor
i1520-0469-57-21-3559-e10
and jF is defined through the relation
jFgF.
Equation (9) displays the general form of Eq. (8). One is thus entitled to identify JE,h and ϕE,h as, respectively, the argument of the divergence operator and the negative definite term of the right-hand side. In particular,
i1520-0469-57-21-3559-e12
Notice that dissipation comes here entirely from viscous effects in kinetic energy balance. This is a consequence of the decoupling of thermal effects. Furthermore, it is seen that the forcing F and the damping term κ2z of the evolution equation of the height variable z contribute only to the energy flux.
One may now establish a connection between Eqs. (9)–(12) and the entropy balance of irreversible thermodynamics. The main point is that under the conditions of validity of the shallow water model entropy s depends on space and time only through the internal energy e(r, t). Since the total energy u = e + Ek + Ep is conserved, the dissipative part of the entropy balance—the entropy production, σ—will thus be given by the last term of Eq. (9), with the opposite sign and an additional reciprocal temperature factor, T−1, originating from the thermodynamic derivative ∂s/∂e = 1/T. In other words,
i1520-0469-57-21-3559-e13
where ϕE,h is given by Eq. (12).
At this stage one may recall that, although absent in Eq. (1), the vertical component w of the velocity field has actually been responsible for the evolution of the free surface height z [Eqs (2a)–(2b)]. One can identify the contribution of w to dissipation by starting from the symmetric stress tensor [Eq. (10)] of the full 3D velocity field v ≡ (Vx, Vy, w) and computing the augmented formν()2s. One finds three types of contributions. The first displays the derivatives of the horizontal velocity field V with respect to the horizontal coordinates x and y. It is identical, as expected, to the horizontal energy dissipation ν(V)2s, Eq. (12), which was already derived directly from the energy balance. The second contains derivatives of Vx, Vy with respect to the vertical coordinate as well as derivatives of w with respect to x and y. In agreement with the assumptions underlying the derivation of the model of section 2, such cross-derivative terms should be neglected. Finally, the third term contains derivatives of w with respect to the vertical coordinate, which are essential for satisfying the incompressibility condition of the three-dimensional velocity field and should therefore be kept in the description. One arrives thus at the following expression for the contribution of the vertical motion to the dissipation:
i1520-0469-57-21-3559-e14
where the subscript υ now stands for “vertical” and the last equality derives from the incompressibility condition as applied to the three-dimensional velocity field. The vertical coordinate was denoted ζ to avoid confusion with the height variable z.
Substituting the decomposition of Eq. (3) into Eqs. (12)–(14) one obtains the more explicit forms
i1520-0469-57-21-3559-e15a
where the subscripts denote derivatives with respect to the relevant variables.
A similar expression can be derived for the kinetic energy density Ek = υ2/2. Neglecting as before the vertical velocity in front of the horizontal one and using once again Eq. (3) one obtains
EkV2xV2yχxψy2χyψx2
Equations (15)–(16) allow us to evaluate explicitly σh, συ, and Ek in the limit of the nine-mode truncation outlined in section 2. Introducing the first two equations (5), keeping only those space-dependent contributions that belong to the basis functions ϕi [Eq. (4)], one may decompose the above quantities into a space averaged part and a spatially dependent part:
i1520-0469-57-21-3559-e17
For simplicity of notation we use from now on adimensional quantities:
i1520-0469-57-21-3559-e18
The space-averaged part σh,0, συ,0, Ek,0 and the spatially dependent parts σh,s, συ,s, Ek,s are given by the explicit expressions [dropping for simplicity primes in Eq. (18)]
i1520-0469-57-21-3559-e19a

4. Parameter dependence and spatiotemporal behavior of entropy production and kinetic energy

In this section we analyze the behavior of the entropy production σh and συ and kinetic energy Ek as the system traverses the bifurcation diagram of Fig. 1.

We first consider the space-averaged parts σh,0, συ,0, and Ek,0. Our main result is summarized in Figs. 3a–c (dots). The following points are worth noticing.

  1. In the range of stability of the Hadley fixed point (0.01 < F ≲ 0.015) all three quantities increase monotonically with the forcing F. Since in this regime all amplitudes but x1 and y1 are zero, σh, συ, and Ek are space independent: σh = x21 + ½y21; συ = x21; Ek = x21 + y21. In actual fact, therefore, the behavior depicted in Fig. 3 in this range pertains to the total entropy production and kinetic energy as well. Notice that σh,0 is about two orders of magnitude larger than συ,0.
  2. The appearance of two coexisting fixed points (nonzonal stationary circulations) beyond the instability of the Hadley solution is marked by a change of slope of the (still increasing) dependence of σh,0, συ,0, and Ek,0 on F. If continued beyond its instability threshold, the Hadley circulation (dashed curve) would be more vigorous and more “expensive” (dissipative) in the horizontal direction, the opposite being true for the dissipation in the vertical direction. Notice that both asymmetric fixed points yield exactly the same values of entropy production and kinetic energy.
  3. The loss of stability of the asymmetric circulation and the emergence of a chaotic regime (F ∼ 0.05) are marked by a jump in the entropy production and kinetic energy (now averaged over the attractor, i.e., over a long time interval) and a subsequent increase with F with a slope comparable to that of the previous regime. The jump is lesser in συ,0.
  4. For values of F above about 0.1, where the system undergoes a complex succession of periodic and chaotic windows, entropy production still steadily increases, albeit in a rather complex manner. At F ≈ 0.25, where intermittency sets in, a jump of entropy production (now quite pronounced in συ,0 as well) and kinetic energy is again observed, followed by a stage of increase with F.

In summary, there is a clear-cut signature, at the thermodynamic level, of both the appearance of new circulation patterns and of the modulation of a given circulation pattern as the forcing increases.

We turn now to the time-dependent properties of the space-averaged entropy production. Figures 4 and 5 depict the time series of σh,0, συ,0, and Ek,0 in the periodic (Fig. 4) and in the chaotic (Fig. 5) regimes. We observe an in-phase evolution of σh,0 and συ,0, a systematic phase lag between them and Ek,0 and a high variability. In particular, the range of values of σh,0 and Ek,0 span more than one order of magnitude.

In Fig. 6a, the mapping generated by the successive minima of the variable y1 is drawn for F = 0.06. One observes an almost symmetric cusplike map, suggesting that some aspects of the overall dynamics can be captured by a low-dimensional attractor (Lorenz 1980). Figures 6b–d depict the corresponding mappings of σh,0, συ,0, and Ek,0. While the mapping of συ,0 keeps the symmetry of y1, those of σh,0 and Ek,0 develop a tail in the region of high values of the respective variables. This structure is due to the sole presence of x variables in συ,0 and the mixing of x and y variables in σh,0 and Ek,0. Notice that in the range of F values considered the values of y variables are numerically larger than those of the x variables.

We now address the transient behavior of entropy production as the system evolves toward its attractor in the chaotic regime. Figure 7 reports the results of two different numerical simulations in which the system is started in different phase space regions corresponding to high and to low values of σh,0. We see that, depending on the initial state, dissipation may both increase or decrease prior to the stabilization on the attractor. This invalidates the possibility of a variational property, whereby dissipation would attain an extremum in the regime of stable operation. The possibility of such a property has aroused in the past considerable interest in the literature, as it would provide a welcome “organizing principle” underlying the complexity of atmospheric dynamics (cf. section 1). A similar (negative) conclusion has been reached in a recent study of Lorenz’s three-mode truncation of the Boussinesq equations by the present author (Nicolis 1999). Naturally, this does not rule out the possibility of other organizing principles of completely different origin underlying the evolution.

We finally comment briefly on the spatial structure of the entropy production and kinetic energy. In Fig. 8 the isolines of the horizontal (upper panel) and vertical (lower panel) dissipation are drawn in a parameter range for which the system admits two simultaneously stable fixed points. As can be seen there is a phase shift in the x direction of about half a wavelength between the dissipation patterns associated to the two states (left vs right panels). Furthermore, there is considerable spatial variability in both horizontal and vertical dissipation, spanning at least one order of magnitude. Notice that kinetic energy (not shown) follows essentially the same pattern as the horizontal dissipation.

5. The quasigeostrophic limit

In his 1980 paper, Lorenz considered the quasigeostrophic limit, in which the synoptic-scale motions to be described are quasi-horizontal and the wind is replaced by its geostrophic value in the momentum balance equation. At the level of the nine-mode truncation of the primitive model equations [Eqs. (7)] this amounts to the following simplifications (Lorenz 1980).

  • Neglect all nonlinear terms and all terms containing xi’s and their time derivatives in the x equations. This allows one to express the zi’s in terms of the y variables by a diagnostic type relation.
  • Neglect all nonlinear or topographic terms containing the xi’s from the other equations. This allows one to eliminate the surviving x terms.

The evolution laws of the resulting three-variable dynamical system read, in the notation of Eqs. (7),
aig0dyig0cakajyjykaiaig0ν0κ0yichkyjchjykFi
Adopting the same parameter values as for the full model of section 2 one can show that Eqs. (22) admit a stable Hadley type solution (y1 ≠ 0, y2 = y3 = 0) for F1 = F between about 0.01 and 0.016. When this solution loses its stability it generates, for F between about 0.016 and 0.10, two asymmetric stationary circulations. Chaos sets in for F between about 0.10 and 0.30, whereas in the range between about 0.30 and 0.45 one observes a succession of periodic and chaotic windows (Gent and McWilliams 1982).
The entropy production and the kinetic energy can be expressed in terms of the model variables in much the same way as in section 3. To ensure consistency with the assumptions underlying the quasigeostrophic limit we keep in Eq. (3) only the contributions to the velocity field coming from the streamfunction ψ. This eliminates the vertical component w (and hence the vertical entropy production συ) and leads to the following expressions for the horizontal entropy production and kinetic energy replacing Eqs. (15a) and (16):
i1520-0469-57-21-3559-e23
Expanding ψ in the basis ϕi as in the first Eq. (5) and truncating to the first three terms, one can further write these equations in the following more explicit form:
i1520-0469-57-21-3559-e25a
where we have displayed the adimensional rather than the original quantities. We notice that the space-dependent parts vanish identically for the Hadley solution. Furthermore, one can show that the values of the total σh and Ek are equal for the two asymmetric stationary circulations arising beyond the instability of the Hadley solution.

Figure 9a (dots) depicts the dependence of the space-averaged entropy production [Eq. (25a)] in terms of the parameter F. The overall behavior (also shared by the space-averaged kinetic energy; Fig. 9b) is quite similar to Fig. 3a, confirming further the robustness of the trends identified in section 4. The upper and lower dashed curves in the figure represent the dependence on F of the entropy production associated to the Hadley and to the asymmetric stationary circulations, extrapolated beyond their domain of stability. We see that the dissipation in the chaotic and in the time-periodic regimes is sandwiched between what would have been an expensive Hadley circulation and an “inexpensive” asymmetric stationary circulation. This is in agreement with the ideas developed in the introduction and in section 4, that in the nonlinear range of nonequilibrium phenomena optimality properties are unlikely to exist.

Figure 10 confirms further the above statement by showing the transient behavior of entropy production as the system evolves toward the attractor. We see, in much the same way as in Fig. 7, that depending on the initial condition one may have an increase or a decrease of dissipation in the course of time.

In Fig. 11 the isolines of the dissipation are drawn in a parameter range of coexistence of two stable fixed points. As in Fig. 8, there is a phase shift in the x direction of about half a wavelength between the two possible dissipation patterns, as well as a considerable spatial variability. The kinetic energy (not shown) follows essentially the same pattern.

6. Conclusions

In this paper the space–time behavior of the entropy production associated to energy dissipation and of the kinetic energy have been analyzed and evaluated explicitly within the framework of Lorenz’s nine-mode truncation of the shallow water equations. Our strategy has been to derive exact expressions for these quantities [Eqs. (15)–(16)] prior to their explicit evaluation in the approximate setting provided by the truncation of the model equations to a finite set of modes. This procedure bypasses difficulties related to energy sinks arising specifically from truncation (Gent and McWilliams 1982) by applying such an approximation only in the final stage of evaluation of the quantities of interest. It has also allowed us to assess the thermodynamic role of the forcing and damping terms in the z equation. These terms were shown not to give rise to extra contributions in the dissipation arising from new irreversible fluxes, but rather to affect dissipation indirectly through their presence in the evolution equations of the state variables.

A first trend that has been identified from our analysis is the systematic increase of entropy production and kinetic energy as the system enters gradually into more complex regimes beyond the instability of the symmetric (Hadley) circulation, such as multiple stationary circulations, periodic, and chaotic solutions. A second, subtler trend is that the Hadley state, if extrapolated beyond its range of stability, would yield higher horizontal dissipation, higher kinetic energy, and lower vertical dissipation than the other more complex regimes encountered when the control parameter F increases. Since vertical dissipation is lower than horizontal one by about two orders of magnitude, this suggests that the complex circulation patterns observed in nature are less dissipative (and thus in a sense more “energy efficient”) than a hypothetical symmetric pattern. An even sharper statement can be made in the quasigeostrophic limit, where dissipation and kinetic energy associated to the time-dependent circulations are literally sandwiched between those of Hadley circulation (upper bound) and those of the multiple stationary circulations (lower bound). We believe that the above trends are not merely features of the specific model considered but rather reflect generic properties of the atmosphere since they are seen to persist when contracting the initial nine-mode model to the three-mode quasigeostrophic one.

The above behavior suggests the existence of an organizing principle operating in the atmosphere. We have shown that this principle is not associated with an extremum property in the strict sense of the term: depending on the initial configuration the system can see its dissipation decrease or increase in time in the course of its evolution toward the attractor.

Irreversible thermodynamics has been applied extensively in the past in atmospheric dynamics in the context of energy exchanges during phase transformations (Van Mieghem 1973). Its systematic application in the general problem of atmospheric dynamics is still in its infancy, yet it would provide extremely useful insights in the characterization of the different patterns and, perhaps, in their prediction. We have outlined an attempt in this direction based on a simple version of the equations of dynamics. Our work could be extended in many directions. A first possibility would be to tackle more realistic models of the atmosphere. The formalism developed in this paper is well suited for this, since it can account for different parameterization schemes, used routinely in such models. A different direction should aim at short-range phenomena such as the irreversible thermodynamics of weather systems. Here transient behavior associated with the genesis of a particular pattern would be especially significant and could be easily accommodated in our formalism.

Acknowledgments

This research is supported, in part, by the Interuniversity Attraction Poles program of the Belgian Office for Scientific, Technical and Cultural affairs.

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Fig. 1.
Fig. 1.

Stable regimes of model (7) as the forcing F increases. Parameter values ν0 = k0 = 1/48, g0 = 8, a1 = a2 = 1, a3 = 3, c = 3/2, h1 = −1, and h2 = h3 = F2 = F3 = 0

Citation: Journal of the Atmospheric Sciences 57, 21; 10.1175/1520-0469(2000)057<3559:DTIASW>2.0.CO;2

Fig. 2.
Fig. 2.

Time evolution of y1, y2, y3 as obtained from model (7) in the periodic regime, (a) with F = 0.13; and in the chaotic regime, with (b) F = 0.06. Parameter values as in Fig. 1

Citation: Journal of the Atmospheric Sciences 57, 21; 10.1175/1520-0469(2000)057<3559:DTIASW>2.0.CO;2

Fig. 3.
Fig. 3.

Dependence of the (a) space-averaged horizontal entropy production σh,0, (b) vertical entropy production συ,0, and (c) kinetic energy Ek,0 as the forcing amplitude F is increased. In the presence of time-dependent regime F ≥ 0.05 these quantities are obtained through an averaging over the attractors (averaging time ∼3000 days). Dashed curve represents the properties of the Hadley circulation regime beyond its stability region

Citation: Journal of the Atmospheric Sciences 57, 21; 10.1175/1520-0469(2000)057<3559:DTIASW>2.0.CO;2

Fig. 4.
Fig. 4.

Time-dependent behavior of (a) σh,0 (solid line) and συ,0 (dashed line) and (b) Ek,0 (dotted line) in a periodic regime corresponding to F = 0.13

Citation: Journal of the Atmospheric Sciences 57, 21; 10.1175/1520-0469(2000)057<3559:DTIASW>2.0.CO;2

Fig. 5.
Fig. 5.

As in Fig. 4 but in a chaotic regime with F = 0.06

Citation: Journal of the Atmospheric Sciences 57, 21; 10.1175/1520-0469(2000)057<3559:DTIASW>2.0.CO;2

Fig. 6.
Fig. 6.

One-dimensional mapping resulting from the Lorenz (1980) plot of (a) the successive minima of y1; and the corresponding values of (b) σh,0, (c) συ,0, and (d) Ek,0. Parameter values as in Fig. 5

Citation: Journal of the Atmospheric Sciences 57, 21; 10.1175/1520-0469(2000)057<3559:DTIASW>2.0.CO;2

Fig. 7.
Fig. 7.

Transient evolution of σh,0 toward the chaotic attractor with F = 0.06. Initial conditions xi = yi = zi = 0.1 (full line) xi = yi = zi = 10−4 (dashed line)

Citation: Journal of the Atmospheric Sciences 57, 21; 10.1175/1520-0469(2000)057<3559:DTIASW>2.0.CO;2

Fig. 8.
Fig. 8.

Patterns of the total entropy production along the horizontal and vertical directions as obtained from model (7) with F = 0.04. For this parameter value the system admits two stable fixed points corresponding to antisymmetric dissipation patterns of (a), (b) σh and (c), (d), συ respectively. The values on the isolines have been magnified by a factor of 1000

Citation: Journal of the Atmospheric Sciences 57, 21; 10.1175/1520-0469(2000)057<3559:DTIASW>2.0.CO;2

Fig. 9.
Fig. 9.

Dependence of the averaged horizontal entropy production (a) σh,0 and (b) kinetic energy Ek,0 (dots) as the forcing F increases in the case of the quasigeostrophic approximation, Eq. (22). The dashed curves extrapolate σh,0 and Ek,0 beyond the stability region of the Hadley solution (I) and of the two fixed point solutions (II, III)

Citation: Journal of the Atmospheric Sciences 57, 21; 10.1175/1520-0469(2000)057<3559:DTIASW>2.0.CO;2

Fig. 10.
Fig. 10.

As in Fig. 7 but for the model of Eq. (22) and F = 0.2

Citation: Journal of the Atmospheric Sciences 57, 21; 10.1175/1520-0469(2000)057<3559:DTIASW>2.0.CO;2

Fig. 11.
Fig. 11.

Spatial patterns of the total entropy production (limited here to the horizontal direction) as obtained from model (22) with F = 0.04. For this parameter value the system admits two stable fixed points [(a) and (b)] for which the dissipation patterns are shifted by about half a wavelength. The values on the isolines have been magnified by a factor of 1000

Citation: Journal of the Atmospheric Sciences 57, 21; 10.1175/1520-0469(2000)057<3559:DTIASW>2.0.CO;2

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