Nonlinear saturation of thermal instabilities
NNonlinear saturation of thermal instabilities
F. J. Beron-Vera a) Department of Atmospheric Sciences, Rosenstiel School of Marine and Atmospheric Science, University of Miami,Miami, Florida 33149, USA (Dated: 26 January 2021)
Low-frequency simulations of a one-layer model with lateral buoyancy variations (i.e., thermodynamicallyactive) have revealed circulatory motions resembling quite closely submesoscale observations in the surfaceocean rather than indefinitely growing in the absence of a high-wavenumber instability cutoff. In this note itis shown that the existence of a convex pseudoenergy–momentum integral of motion for the inviscid, unforceddynamics provides a mechanism for the nonlinear saturation of such thermal instabilities in the zonallysymmetric case. The result is an application of Arnold and Shepherd methods.PACS numbers: 02.50.Ga; 47.27.De; 92.10.Fj I. INTRODUCTION
A simple recipe, introduced in as early as at leastthe late 1960s , to incorporate thermodynamic processes(e.g., those due to heat and freshwater exchanges throughthe air–sea interface) in a one-layer ocean model con-sisted in allowing buoyancy to vary with horizontal po-sition and time, while keeping velocity as independentof the vertical coordinate. The simplicity of the result-ing inhomogeneous-layer model with fields that do notchange with depth—referred to as IL by Ripa to reflect this—promised fundamental understanding ofocean processes which would be difficult to gain by ana-lyzing direct observations or the output from an oceangeneral circulation model. A number of applicationsof the IL , particularly to equatorial dynamics, ap-peared in the 1980s and 1990s supporting this line ofthought .Regrettably, the increase in computational power inthe current century has led to overemphasize reproduc-ing observations in detriment of gaining basic physi-cal insight of the type that ocean modeling based onthe IL system or variants thereof was expectedto provide. A pleasant surprise, however, has been tolearn that layer ocean modeling with reduced thermo-dynamics is regaining momentum . Moreover,the renewed interest in this type of modeling is exceed-ing a pure oceanographic interest. Indeed, applicationsof the IL have been extended to atmospheric dynam-ics, both terrestrial (beyond the original one ) andplanetary .Of particular interest to the present work are the nu-merical simulations of the IL by Holm, Luesink, andPan , which have shown that it can sustain subinertial(i.e., with frequency smaller than the local Coriolis pa-rameter, twice the local Earth’s rotation rate) circulatorymotions (cf. Fig. 1) that resemble quite well submesoscale(1–10 km) features often observed in satellite ocean color a) Electronic mail: [email protected] images. This note is concerned with identifying a mech-anism that can prevent such thermal instabilities fromgrowing indefinitely in the absence of a high-wavenumbercutoff in the IL model . II. A QUASIGEOSTROPHIC IL Consider a low-frequency approximation to the IL model in a reduced-gravity setting (i.e., with the activelayer floating atop a quiescent, infinitely deep layer ofconstant density), which is most appropriate to studynear-surface (mixed-layer) ocean processes. Let x (resp., y ) point eastward (resp., northward) on a zonal β -plane channel, L -periodic and of width W . The quasi-geostrophic IL takes the form ∂ t ¯ ξ + [ ¯ ψ, ¯ ξ ] = R − [ ¯ ψ, ψ σ ] , ∂ t ψ σ + [ ¯ ψ, ψ σ ] = 0 , (1a)with invertibility principle ∇ ¯ ψ − R − ¯ ψ = ¯ ξ − R − ψ σ − βy, (1b)all subjected to no-flow through the coasts, ∂ x ¯ ψ | y =0 ,W =0 , constancy of Kelvin circulations along them, − (cid:82) L ∂ y ¯ ψ | y =0 ,W d x = γ ,W = const , and L -periodicityin x . In (1), [ , ] is the canonical Poisson bracket (Ja-cobian) in R { x, y } and R := √ g b H r | f | > is the Rossbydeformation radius, where g b > is the reference (i.e., inthe absence of currents) buoyancy of the active (relativeto the passive) layer whose thickness is H r > , and f is the Coriolis parameter at the southern coast (its de-pendence on latitude is represented by f + βy ). Theinstantaneous layer thickness, buoyancy, and velocity, h = H r (cid:0) ¯ ψ − ψ σ f R (cid:1) , ϑ = g b (cid:0) ψ σ f R (cid:1) , and u = ∇ ⊥ ¯ ψ ,respectively. Alternative forms of (1) are presented inHolm, Luesink, and Pan , Ripa , Warnerford and Del-lar The quantity ¯ ξ in (1) is a quasigeostrophic ap-proximation to the vertically averaged Ertel’s potentialvorticity , which is not a Lagrangian constant of themodel. This justifies the overbar notation. Note thatby the thermal-wind balance, which is not explicitly re-solved, u would have a vertical shear proportional to a r X i v : . [ phy s i c s . a o - ph ] J a n < x=R y = R FIG. 1. Snapshot of potential vorticity (top) and buoyancyexcess (bottom) from a numerical solution of system (1) ini-tialized close to an unstable uniform zonal flow in a peri-odic channel of the β -plane using a pseudospectral code on a -resolution grid. Note the Kelvin–Helmholtz-like rollupswith scales much smaller than the deformation radius ( R ) ofthe problem. ∇ ⊥ ψ σ . In order for u to read as above upon a verti-cal average over − h ≤ z ≤ , it should be of the form u = ∇ ⊥ ¯ ψ + σ ∇ ⊥ ψ σ , where σ = 1 + zh , which justifiesthe σ subscript (cf. Ripa for details). System (1) preserves energy, E := (cid:90) |∇ ¯ ψ | + R − ¯ ψ (2)(where (cid:82) is a short-hand notation for integration over thezonal-channel domain and operates on everything on itsright); an infinite family of Casimirs, C := a γ + a W γ W + (cid:90) C ( ψ σ ) + ¯ ξC ( ψ σ ) (3)with a ,W = const and C , C ( ) arbitrary; and zonalmomentum M := (cid:90) y ¯ ξ. (4)Furthermore, in Ripa it is shown that equations(1) possess a generalized Hamiltonian structure on the state variables ( ¯ ξ, ¯ ψ, γ ,W ) with Hamil-tonian given by (2) and Lie–Poisson bracket {F , G} := (cid:82) ¯ ξ (cid:2) δ F δ ¯ ξ , δ G δ ¯ ξ (cid:3) + ψ σ (cid:2) δ F δ ¯ ξ , δ G δψ σ (cid:3) + ψ σ (cid:2) δ F δψ σ , δ G δ ¯ ξ (cid:3) for admissible functionals of state F , G .(The variational derivative of a functional F [ ϕ ] = (cid:82) F ( x, y, t ; ϕ, ∂ x ϕ, ∂ y ϕ, ∂ xy ϕ, . . . ) is the uniqueelement δ F δϕ satisfying dd ε (cid:12)(cid:12) ε =0 F [ ϕ + εδϕ ] = (cid:82) δ F δϕ δϕ .)The admissibility condition for the zonal-channel do-main is ∂ x δ F δ ¯ ξ | y =0 ,W = 0 = ∂ x δ F δψ σ | y =0 ,W . This bracketturns out to be the same as that for “low- β ” reducedmagnetohydrodynamics and incompressible, non-hydrostatic, Boussinesq fluid dynamics on a verticalplane ; so the Casimirs in (3), satisfying {F , C} = 0 forall F , have been known prior to the derivation of thederivation of (1). These integrals play a critical rolein the derivation of a-priori stability criteria, discussedbelow. The Hamiltonian formalism enables the linkageof conservation laws with symmetries via Noether’stheorem (e.g., Shepherd ). From a more practicalfluid dynamics standpoint, it provides a framework forderiving flow-topology-preserving stochastic versions of a generalized Hamiltonian model from its analogousEuler–Poincare variational formulation , which canbe used to build parametrizations of unresolvablesubgrid-scale motions . III. STABILITY/INSTABILITY
Let capital letters denote variables that define a basicstate for (1), i.e., a steady solution or equilibrium to (1)with currents. An example is the uniform zonal flow,defined by ¯Ψ = − ¯ U y, Ψ σ = − U σ y, (5)where ¯ U , U σ are constants, with U σ < f R W so Θ = g b (cid:0) − U σ f R y (cid:1) > . Note that, from the thermal-wind re-lation, a basic state with the latter buoyancy distributionwould be consistent with a zonal flow with uniform verti-cal shear given by U σ H r . Thus the study of perturbationsto (5) represents, implicitly, a baroclinic instability prob-lem of the free-boundary type studied in Beron-Vera andRipa . As opposed to classical baroclinic instability , free boundary baroclinic instability has a soft interface,which has a slope, given by ( U σ − ¯ U ) H r f R , in the basic state.The phase speed of infinitesimal normal-mode perturba-tions to (5) is given by c − ¯ U = − ¯ U + U σ + βR | k | R + 2 ± (cid:112) ( ¯ U + U σ + βR ) − U U σ ( | k | R + 1)2 | k | R + 2 , (6) FIG. 2. Minimum wavenumber for instability of a uniformzonal current for various βR U σ values. which extends the result of Ripa , Young and Chen to the β -plane. A sufficient condition for the absence ofgrowing normal modes is ¯ UU σ < . (7) In Fig. 2 I show, as a function of ¯ UU σ , the minimumwavenumber | k | for instability for various βR U σ values.Note that there is stability for ¯ UU σ < for all | k | , as ex-pected. While β can have a stabilizing effect, the lackof a high-wavenumber cutoff of instability when ¯ UU σ > can be consequential for the nonlinear evolution of sys-tem (1), which however tends to show Kelvin–Helmoltz-like circulations that saturate at subdeformation scalesrather than blowing up indefinitely .Condition (7) was shown in Ripa to be an a-prioricondition for the formal stability of basic state (5),i.e., stability under small-amplitude perturbations of ar-bitrary structure. This result followed from the ap-plication of Arnold method. This consists in con-structing an integral of motion which is quadratic tothe lowest order on the deviation from a basic stateand either is positive-definite (Arnold’s first theorem)or negative-definite (Arnold’s second theorem) (cf. Holm et al. , McIntyre and Shepherd ). In both cases, theintegral represents a norm that constrains the growth ofperturbations. For the zonally symmetric basic state (5),such an integral of motion is given by δ H ¯ U := δ ( E + C − ¯ U M ) = (cid:90) |∇ δ ¯ ψ | + R − (cid:18) δ ¯ ψ − ¯ UU σ δψ σ (cid:19) (8)for C in (3) with C = − ¯ U R U σ ψ σ and C = 0 so δ H ¯ U = 0 .Note that (8) is positive-definite when (7) holds. (Thatthe circulation perturbations δγ ,W do not enter in (8)should not be taken as implying positive-semidefinitnessof (8) and hence the possibility of unarrested growthalong their directions in phase space: once initially spec-ified, δγ ,W remain the same at all times. Also notethat when R → ∞ , i.e., the interface is rigid, system(1) reduces to ∂ t ¯ ξ + [ ¯ ψ, ¯ ξ ] = 0 with ∇ ¯ ψ = ¯ ξ − βy , forwhich ¯Ψ = − ¯ U y clearly is stable consistent with (8) be- ing positive-definite in this limit.) Furthermore, (8) coin-cides with the pseudoenergy–momentum ∆ H ¯ U , which isan exact integral of motion of (1). Indeed, H α is a Hamil-tonian for the motion as viewed from an x -translatingframe at constant speed α . This shows that (7) actu-ally is a condition for the formal stability of (5) underfinite-amplitude perturbations. However, (8) cannot beproved to be convex, i.e., to be bounded from below andabove by multiples of an L -norm on the perturbationfield. This precludes one from declaring (5) stable in aLyapunov sense when (7) holds, i.e., the L -distance ofa perturbation to (5) cannot be bounded at all times bya multiple of the initial distance (cf. Holm et al. , McIn-tyre and Shepherd ). Finally, the possibility of provingstability when (7) is violated by seeking conditions un-der which (8) is negative-definite, namely, conditions un-der which δ E = (cid:82) |∇ δ ¯ ψ | + R − δ ¯ ψ can be boundedby − δ C = R − UU σ (cid:82) δψ σ , is ruled out because δ ¯ ψ isnot a (nonlocal) function of δψ σ exclusively. (On the in-variant subspace of system (1), given by { ψ σ = const } , δ ¯ ψ = ( ∇ − R − ) − δ ¯ ξ . Thus on that subspace, relative toa general sheared zonal flow ¯Ψ = − (cid:82) y ¯ U ( y ) , both posi-tive and negative pseudoenergy–momentum integrals ex-ist provided that for all y ∈ [0 , W ] there are constants α such that ¯ U ( y ) − αβ − ¯ U (cid:48)(cid:48) ( y )+ R − ¯ U (cid:48) ( y ) is negative and biggerthan ( κ + R − ) − , respectively, where κ is the gravesteigenvalue of the Helmholtz equation with zero Dirichletboundary conditions at y = 0 , W ; cf. Ripa .) IV. INSTABILITY SATURATION
The fact that the pseudoenergy–momentum (8) is notconvex appears to conspire against the purpose here tobound the growth of perturbations to an unstable basicstate, for which (7) is necessary violated. Let ϕ denotethe state vector. Let the superscript S (resp., U) indicatestable (resp., unstable). Assume that the following con-vexity estimate holds: a (Φ S ) (cid:107) ϕ − Φ S (cid:107) t = t ≤ (cid:107) ϕ − Φ S (cid:107) ≤ A (Φ S ) (cid:107) ϕ − Φ S (cid:107) t = t for < a (Φ S ) ≤ A (Φ S ) < ∞ and (cid:107) (cid:107) representing a certain L -norm. Then one finds: (cid:107) ϕ − Φ U (cid:107) ≤ (cid:107) ϕ − Φ S (cid:107) + (cid:107) Φ S − Φ U (cid:107) ≤ A (Φ S ) (cid:107) ϕ − Φ S (cid:107) t = t + (cid:107) Φ S − Φ U (cid:107) ≈ (cid:0) ( A (Φ S ) + 1 (cid:1) (cid:107) Φ S − Φ U (cid:107) by the trian-gular inequality, application of the convexity estimate, and assuming that ϕ ≈ Φ U initially, respectively. Thisprovides a bound on the growth of ϕ − Φ S in terms ofthe distance between Φ S and Φ U . Note that if thereexists a convex, sign-definite integral I , namely, − f R (10)provided that F has inverse, so ¯Ψ = − ¯ U F F − (Ψ σ ) .Clearly, [ ¯Ψ , Ψ σ ] = 0 , as required for an equilibrium to (1).The set of equilibria actually exceeds the zonal flow class;however, the possibility of deriving a-priori stability con-ditions using Arnold’s method is restricted to this class .With a Casimir defined by C = − R − ¯ U F (cid:82) ψ σ F − ( ψ σ ) and C = 0 in (3), ∆ H ¯ U := (cid:90) (cid:0) |∇ δ ¯ ψ | + R − δ ¯ ψ (cid:1) − R − (cid:90) δψ σ (cid:0) ¯Ψ(Ψ σ + s ) − ¯Ψ(Ψ σ ) (cid:1) d s (11)represents an exact pseudoenergy–momentum, which ispositive-definite when ¯Ψ (cid:48) (Ψ σ ) < . Furthermore if thereare constants c , c such that < c ≤ − ¯Ψ (cid:48) (Ψ σ ) ≤ c < ∞ , (12) then Taylor’s reminder theorem guarantees that (11) isbounded by multiples of the L -norm (cid:107) ( δ ¯ ξ, δψ σ ) (cid:107) λ := (cid:90) |∇ δ ¯ ψ | + R − (cid:0) δ ¯ ψ + λδψ σ (cid:1) , c ≤ λ ≤ c . (13)This convexity estimate guarantees nonlinear stabilityfor the basic state (10) in a Lyapunov sense, i.e., (cid:107) ( δ ¯ ξ, δψ σ ) (cid:107) λ, t>t ≤ (cid:113) c c (cid:107) ( δ ¯ ξ, δψ σ ) (cid:107) λ, t = t , provided that(12) holds, which excludes the case F ( y ) = − U σ y consid- ered above. This stability theorem enables one to a-prioribound the finite-amplitude growth of perturbations toany unstable basic state of system (1) using Shepherd’smethod, even for the linear F ( y ) class, regardless of thefact that for this specific class of equilibrium Lyapunovstability cannot be proved. An upper bound (9) will begiven by (cid:113) c c + 1 times the L -distance (13) between thebasic state (10), with the condition (12), and the unstablebasic state in question.As an example, consider F ( y ) = f R (cid:115) U F σ yf R − , U F σ f > . (14)This gives ¯Ψ(Ψ σ ) = − f R log 12 (cid:18) σ f R (cid:19) , (15)where Ψ σ is restricted to vary from f R to Ψ max σ = f R (cid:113) U F σ Wf R − in a zonal channel ofwidth W . Its derivative ¯Ψ (cid:48) (Ψ σ ) = − Ψ σ f R Ψ σ f R , (16)which is negative since Ψ σ f R = F ( y ) > by (14), makingthe pseudoenergy–momentum in (11) positive-definite.Furthermore, − ¯Ψ (cid:48) (Ψ σ ) is bounded away from zero by c = − ¯Ψ (cid:48) (Ψ max σ ) and from infinity by c = 4 , implyingLyapunov stability for the family of basic states definedby (14). Setting Φ U using (5) under the assumption thatcondition (7) is violated and Φ S using (10) and (14), thebound (9) on the nonlinear growth of perturbations to Φ U with respect to the L -norm (13) with λ = − ¯Ψ (cid:48) (Ψ max σ ) ,i.e., the smallest admissible choice, takes the form: B (¯ µ, µ σ ; ¯ µ F , µ F σ , ν ) = | f | (cid:113) LR · (cid:32)(cid:115) exp µ F σ ν (cid:112) µ F σ ν − (cid:33) · (cid:16)(cid:0) ¯ µ F − ¯ µ (cid:1) (cid:0) ν + ν (cid:1) + 2 ν (cid:112) µ F σ ν − µ F σ ν (cid:90) (cid:18)(cid:113) µ F σ ν yW − µ σ ν yW (cid:19) d yW (cid:33) , (17)where the unstable basic state parameters ¯ µ := ¯ Uf R and µ σ := U σ f R are such that < ¯ µµ σ =: τ , the stable basicstate parameters ¯ µ F := ¯ U F f R and µ F σ := U F σ f R > , andthe channel’s aspect ratio ν := WR > . The top pan-els of 3 show, estimated numerically, B min (¯ µ, µ σ ; ν ) :=min ¯ µ F ,µ F σ B (¯ µ, µ σ ; ¯ µ F , µ F σ , ν ) for two selected values of ν . The bound, which does not depend on the strength ofthe β effect, decreases with ν . Indeed, it decays to zeroas the width of the channel shrinks to zero, as can beanticipated, or as R tends to infinity, limit in which thebasic flow is stable as noted above. As a function of theinstability parameter τ , the bound is multivalued. Thebottom panels of Fig. 3 show B opt ( τ ; ν ) obtained numer-ically by keeping, for each ν , the least attainable valueper τ value (interval), which provides the tightest boundpossible for each unstable basic state (the solid curve isa polynomial fit to the open dots). As can be expected,the obtained optimal bound decreases toward criticality( τ = 0 ). Yet at τ = 0 the bound is not zero (except when ν → ) as it might be desired. A choice of stable basicstate different than (14) could lead to the desired resultand an overall tighter bound.Nonlinear saturation bounds of the type above havebeen shown to exist even for systems subjected to forc-ing and dissipation . Consider, for instance, the caseof system (1) with the ¯ ξ -equation forced (damped?)by R − [ ¯ ψ, ˆ ψ ] , where ˆ ψ is a prescribed function of lati-tude ( y ). This system is Lie–Poisson, with bracket asin (1) and Hamiltonian given by ˆ H = H + R − (cid:82) ˆ ψψ σ . B o p t / f | (cid:113) L R B m i n / f | (cid:113) L R FIG. 3. (top panels) A-priori bound (17) on the nonlineargrowth of perturbations to unstable states of the class (5),minimized over all possible states states defined by (10) and(14). (bottom panels) Optimal bound obtained as a functionof the basic flow’s stability parameter in the unstable range. (In Holm, Luesink, and Pan , ˆ ψ is given the inter-pretation of a bottom (surface in the present case)topography; however, they do not include the corre-sponding topographic- β term in the potential vortic-ity ¯ ξ .) The system thus have the same Casimirs asthe original (unforced, inviscid) system (1), given in(3), and the x -translational symmetry of ˆ ψ makes thezonal momentum (4) to be conserved as well. Fur-thermore, (5) and (10) are admissible basic states,and all of the results above carry over mutatis mutan-dis to the forced case. (The only differences appearin the Casimir choices, with C = − ¯ U R U σ ψ σ − ˆ ψR ψ σ and C = 0 for the pseudoenergy–momentum (8) and C = − ˆ ψR ψ σ − ¯ U F R (cid:82) ψ σ F − ( ψ σ ) and C = 0 for that in(11).)The above provides reason to expect (hope) that thebounds discussed here can play a role in arresting thegrowth of small-scale circulatory motions developing indirect numerical simulations of the IL system (1), evenin the forced–dissipative regime. V. CONCLUDING REMARKS
The result of this work adds support tothermodynamically-active-layer ocean modeling asdescribed by the IL system (1), particularly for in-vestigating with confidence in a geometric mechanicsframework the contribution of unresolved submesoscalemotions to transport at resolvable scales in the upperocean, a topic of active research . Interestingly, thetwo-dimensional simulations described in Holm, Luesink,and Pan suggest a scale separation consistent withsatellite ocean color images and statistical analysis ofin-situ Lagrangian ocean observations , but is chal-lenged by three-dimensional surface-quasigeostrophicsimulations , which suggest a continuous inverse energycascade. The extent to which the small-scale circula-tions and the associated scale separation represent apeculiarity of the IL model needs to be assessed, whichis reserved for the future. An appropriate frameworkfor this is provided by models with more vertical reso-lution, and hence better thermodynamics, than the IL model . SUPPLEMENTARY MATERIAL
This paper does not include supplementary material.
AUTHOR’S CONTRIBUTIONS
This paper is authored by a single individual who en-tirely carried out the work.
ACKNOWLEDGMENTS
The author thanks M. Josefina Olascoaga for the ben-efit of discussions on Shepherd’s method. Corrections tothe manuscript by Daniel Karrasch are appreciated.
AIP PUBLISHING DATA SHARING POLICY
This paper does not involve the use of data.
Appendix A: Free waves
For completeness, recall that the free waves of theIL model, i.e., infinitesimally small, normal-modeperturbations to a reference (i.e., quiescent) state of(1) characterized by ¯Ψ = 0 = Ψ σ , are given by : aRossby wave, with frequency ω = − kβ | k | + R − and forwhich δ ¯ ψ (cid:54) = 0 δψ σ (cid:54) = 0 , and an ω = 0 mode, so-calledforce compensating mode , with δ ¯ ψ = 0 = δψ σ orequivalently g b δh + H r δϑ = 0 . Using the Casimir C = R − (cid:82) ψ σ , relative to this reference state, ∆( E + C ) = (cid:82) |∇ δ ¯ ψ | + R − ( δ ¯ ψ + δψ σ ) =: E f is anexact invariant (which can called a free energy). Beingpositive-definite, it prevents the spontaneous growth ofinfinitesimal perturbations to the state with no currents.This result has received less attention than that pertain-ing to the most general reference state, characterized by ¯Ψ = a = const . Upon choosing a Casimir of the form C = a (cid:82) R − ψ σ + ¯ ξ , the following turns out to be a free en-ergy relative to this reference state : E f = δ ( H + C ) = (cid:82) |∇ δ ¯ ψ | + R − δ ¯ ψ ≡ (cid:82) | δ u | + f R (cid:0) δhH r + δϑ g b (cid:1) .Note that this E f is positive-semidefinite, i.e., it canvanish for nonzero perturbations. More specifically,variations of h and ϑ which leave g b h + H r ϑ (for in-finitesimal perturbations this is the force-compensatingmode mentioned above) unaltered do not change E f .Spontaneous growth of such variations cannot be pre-vented by E f conservation . Yet formula (17) with ¯ µ = 0 = µ σ provides an upper bound on their nonlineargrowth. Appendix B: The axisymmetric basic state case
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