Semi-convection in the ocean and in stars: A multi-scale analysis · 2015-08-05 · 344 F. Kupka...

BMeteorologische Zeitschrift, Vol. 24, No. 3, 343–358 (published online April 4, 2015) Article© 2015 The authors

Semi-convection in the ocean and in stars: A multi-scaleanalysis

Friedrich Kupka1∗, Martin Losch2, Florian Zaussinger3 and Thomas Zweigle2,4

1Fakultät für Mathematik, Universität Wien, Austria2Alfred-Wegener-Institut, Helmholtz Zentrum für Polar- und Meeresforschung, Bremerhaven, Germany3Dept. Aerodynamics and Fluid Mechanics, BTU Cottbus-Senftenberg, Germany4Fraunhofer-Institut für Kurzzeitdynamik, Ernst-Mach-Institut, Freiburg, Germany

(Manuscript received August 5, 2014; in revised form January 20, 2014; accepted February 2, 2015)

AbstractFluid stratified by gravitation can be subject to a number of instabilities which eventually lead to a flow thatcauses enhanced mixing and transport of heat. The special case where a destabilizing temperature gradientcounteracts the action of a stabilizing gradient in molecular weight is of interest to astrophysics (inside starsand giant planets) and geophysics (lakes, oceans) as well as to some engineering applications. The detaileddynamics of such a system depend on the molecular diffusivities of heat, momentum, and solute as well assystem parameters including the ratio of the two gradients to each other. Further important properties arethe formation and merging of well-defined layers in the fluid which cannot be derived from linear stabilityanalysis. Moreover, the physical processes operate on a vast range of length and time scales. This hasmade the case of semi-convection, where a mean temperature gradient destabilizes the stratification whileat the same time the mean molecular gradient tends to stabilize it, a challenge to physical modelling and tonumerical hydrodynamical simulation. During the MetStröm project the simulation codes ANTARES andMITgcm have been extended such that they can be used for the simulations of such flows. We present acomparison of effective diffusivities derived from direct numerical simulations. For both stars and the oceanicregimes, the Nusselt numbers (scaled diffusivities) follow similar relationships. Semi-convection quicklybecomes inefficient, because the formation of layers limits vertical mixing. In contrast to the complementarysaltfingering, these layers tend to damp instabilities so that effective diffusivities of salinity (concentration)are up to two orders of magnitudes smaller than in the former case.

Keywords: hydrodynamics, numerical simulation, turbulence

1 Introduction

The purpose of this work was to calculate the semi-convection – and saltfinger – driven turbulent fluxes oftemperature and salinity following previous saltfingerstudies by Fleury and Lueck (1991); Radko (2003)and Schmitt (1988, 2005). Then, these fluxes were usedto determine the parametrization for the effective diffu-sivities of temperature KT and salinity KS . These param-eters are frequently used in estimations of the verticaltransport of temperature (or, actually, heat) and salinityin large scale models. Such an approach is mainly takenbecause of its convenience, since extra diffusion is sim-ple to implement and stabilizes hydrodynamical models.From a physical point of view, the assumption may nothold that the thermodynamical state of the backgroundstratification, which is driving vertical transport througha mean gradient, is changing only by a small amountalong the typical mean free path in the flow. A fluid par-cel advected through a convection zone may be subjectto quite different conditions at the bottom of a zone in

∗Corresponding author: Friedrich Kupka, Fakultät für Mathematik, Uni-versität Wien, Oskar-Morgenstern-Platz 1, 1090 Wien, Austria, e-mail:[email protected]

comparison with its top which can limit the applicabil-ity of such parametrizations.

Hence, parametrizations in terms of diffusivities re-quire thorough studies and awareness about their possi-ble limitations. Direct Numerical Simulations (DNS) arenecessary to achieve that goal. Before we sketch the lay-out of this paper, we first discuss the basic physics, pre-vious research, and the motivation for studies on doublediffusive convection.

Double diffusion in two-component fluids is a pro-cess where the differential molecular diffusion of thefluid components acts on different time scales. The ther-mal diffusion is generally faster than the diffusion ofconcentration, because the latter involves material con-centration transport by random Brownian motion whilethe former is the transport of kinetic energy throughmolecular collisions. In a statically stable stratification(density increases with depth of the fluid), there are twosituations in which double diffusion processes can desta-bilize the fluid and lead to mixing beyond the molecularscales: salt fingering and semi-convection or diffusiveconvection.

For oceanographic salt fingering (Stern, 1960),warm and saline (high concentration) water needs to be

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344 F. Kupka et al.: Semi-convection in the ocean and in stars Meteorol. Z., 24, 2015

stacked over cold and fresh (low concentration) water.This situation, where the salinity gradient destabilizesthe water column, but is balanced by the temperaturegradient, is observed in the Caribbean (Schmitt, 2005)or the Gulf of Lyon in the Mediterranean Sea (Onkenand Brambilla, 2003; Zodiatis and Gasparini, 1996).When small perturbations at the layer interface move awarm, saline water parcel into the cold and fresh envi-ronment below, it quickly looses heat by fast thermaldiffusion. The salt diffusion is 100 times slower so thatthe salinity in the water parcels remains unchanged. Thenow cold and saline water parcel has lost its buoyancyand is accelerated further downward feeding the insta-bility. In the same way, a cold and fresh water parcelfrom the lower layer gains buoyancy when it is movedupwards and warms through diffusion.

The astrophysical equivalent of salt fingering convec-tion is often also discussed under the more general nameof thermohaline convection. Both terms refer to the sce-nario of a stable temperature gradient counteracted bya destabilizing gradient in mean molecular weight. Inastrophysics, this can occur in late stages of stellar evo-lution (off-centre shell burning, a case first described inThomas, 1967) or due to accretion of heavy materialnear the stellar surface (Stothers and Simon, 1969).

The semi-convection case in stars and in oceansrequires the opposite stratification. The concentrationgradient with high concentration below low concentra-tions stabilizes the temperature gradient when fluid withlower temperatures is stacked above fluid with highertemperatures. In stars (e.g. Zaussinger et al., 2013, fora review), such a stratification appears frequently withinregions of nuclear burning (nuclear fusion), for instance,in the core of massive stars where hydrogen gets con-verted into helium at a high rate through the so-calledCNO cycle. The increase of the amount of helium,which has a higher mean molecular weight than hydro-gen, in the centre can lead to a gradient stabilizing thefluid against the destabilizing temperature gradient. Bytriggering convective mixing the latter usually leads toa core region with homogeneous composition. But nearthe upper boundary of that region a stabilizing (helium)concentration gradient will form and it is the structureand time evolution of this transition region, character-ized by “inefficient” or “semi-” convection that has beendebated since the early work of Schwarzschild andHärm (1958). In the interior of (gaseous) giant plan-ets such as Jupiter or Saturn a similar stratification fa-voring semi-convection is expected to occur (Steven-son, 1985; Leconte and Chabrier, 2012; Leconteand Chabrier, 2013). In this case, heavy elements con-tribute a much larger fraction of mass to the fluid inthe deep interior of the planet than found for its atmo-sphere layers. There, semi-convection is rather causedby the initial formation process of the object and altersits evolution over very long timescales (Leconte andChabrier, 2012; Leconte and Chabrier, 2013).

The expression semi-convection has thus far beenused only in astrophysics. Oceanographers describe the

same scenario as diffusive convection. Since this expres-sion can more easily be confused and wrongly appliedto other scenarios (such as the case were both tempera-ture and concentration gradient are stable, but the fluidmay nevertheless be mixed by non-local entrainment offluid from a convectively unstable zone nearby), we useits astrophysical equivalent here and in the following.

In the Arctic ocean, the so-called cold haloclinerepresents a condition for semi-convection. On smallerscales, melting sea-ice also generates fresh and coldwater over warmer and more saline water. Given suchstratification, fast thermal diffusion erodes the temper-ature gradient at the interface of the layers. Once theinterface is destabilized warm and saline fluid parcelsmove up into a cold environment where they quicklyloose heat and thus buoyancy again through fast ther-mal diffusion and are accelerated downwards. By thesame token, cold and fresh fluid parcels can move downwhere they gain buoyancy again and return upwards.Obviously, the sources of instability are much smallerin semi-convection than in salt fingering and mostlyconfined to boundary layers between regions with dif-ferent chemical composition. As a consequence, vigor-ous semi-convection requires a background stratificationthat is only marginally stable.

Oceanic semi-convection has received very little at-tention in the past (Veronis, 1965). For example, Lin-den (1974) calculated flux ratios of salinity and tem-perature for semi-convection analytically and comparedthe results to experimental data (Turner, 1965). A fewoceanic measurements in the Adriatic Sea (Carnielet al., 2008) and near melting icebergs (Turner, 2010)demonstrated the relevance of semi-convection to verti-cal mixing and stability. Until recently, there have vir-tually been no numerical simulation studies that addresssemi-convection in the ocean. Saltfingers in lab exper-iments, field observations, and numerical simulationshave been described in numerous contexts (e.g. Hup-pert and Moore, 1976; Schmid et al., 2010; Fleuryand Lueck, 1991; Radko, 2003; Schmitt, 1988, 2005).In a recent study, Traxler et al. (2011) and Mirouhet al. (2012) compare oceanic and astrophysical salt fin-gering and semi-convection. Flanagan et al. (2013)have studied the formation of layers in the Arctic Seacaused by semi-convection.

In the following Section 2 we first introduce the basicequations describing the dynamics of physical systemswith double diffusive convection: the Navier–Stokesequations for a compressible fluid coupled to a concen-tration equation and several variants of the Boussinesqapproximation to them which are suitable either for aperfect gas or for liquids. In addition, we introduce phys-ical quantities of interest used in studying the fluxes ofheat and concentration and hence mixing in such sys-tems. We also give further definitions. In Section 3 wedescribe the simulations codes used in our work and pro-vide a brief overview on their underlying mathematicaland physical concepts. Our key results are collected inSection 4, that is, the computation and parametrization

Meteorol. Z., 24, 2015 F. Kupka et al.: Semi-convection in the ocean and in stars 345

of effective diffusivities for the cases we have studied.Section 5 presents our conclusions.

2 Semi-convection and salt-fingering

We first recall the dynamical equations describing thetime evolution of a two-species fluid and several variantsof the Boussinesq approximation to them which we haveused in practice. The equivalence of these variants, asused in both ANTARES and MITgcm, is discussed. Wethen define physical quantities and the main parametersas well as related notation used to describe physical sys-tems for which double-diffusive convection occurs andbriefly discuss some basic physics of semi-convectionand salt-fingering.

2.1 The basic equations

A two-species flow for which double-diffusive convec-tion occurs is modelled by the Navier–Stokes equationsand related conservation laws. They can be recast as

∂

∂t

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

ρρcρue

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

= −∇ ·

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

ρuρcu

ρu ⊗ u + P − σeu + Pu − u · σ

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

+

+

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

00ρgρg · u

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

+ ∇ ·

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

0ρκc∇c

0K∇T

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠. (2.1)

Here, P is the product of the scalar pressure p with theunit tensor I (which in index notation for a tensor ofrank two is represented by Kronecker’s delta and hence∇ · P = ∇ · pI = ∇p, the gradient of p). The meaning ofthe other symbols is standard: ρ and c denote mass den-sity and concentration, u is the velocity and e the densityof internal energy, g = (0, 0,−g)� is the gravitational ac-celeration (which here only has a vertical component ofmagnitude g and � indicates the transposition of the vec-tor), while t denotes time. Furthermore, σ is the viscousstress tensor, κc is the kinematic diffusivity of concentra-tion while K is the heat (or radiative) conductivity whichboth depend on temperature T as well as on concentra-tion and density. Here and in the following, 0 is used torepresent zero without specifically mentioning whetherit has the rank of a scalar, a vector, or tensor of rank two,as this follows from the specific context.

Since the velocities in flows of double-diffusive con-vection are often small compared to the speed of soundwhile the fluctuations of the dynamical variables aresmall compared to their mean value, the Boussinesqapproximation to (2.1) can be used. This requires thatthe mean thermodynamical variables, in particular thepressure p, change only little along the vertical ex-tent of the simulation domain (i.e. the domain height

is small in terms of the local pressure scale height1

Hp = (−∂ ln p/∂z) ≈ P/(ρg), where the latter holds forhydrostatic equilibrium).

In that case we may instead only consider the fluctu-ations of the thermodynamical variables (T , p, ρ, c or,alternatively, salinity S ) and derive dynamical equationsfor the time evolution from the general fluid dynamicalconservation laws for a fully compressible flow (2.1).To this end the potential temperature is introduced as anadiabatically filtered temperature, Θ = T (p0/p)(γ−1)/γ ,where p0 is a constant reference pressure and γ is theratio of specific heats at constant pressure and volume.Thermodynamical variables f (t, x) such as Θ(t, x) aresplit into a mean background state f (an average overspace and time) and its fluctuation f ′ around that value,f = f + f ′. Pressure fluctuations are expressed byconsidering hydrostatic equilibrium for the mean (back-ground) variables. Taking the small thickness of the do-main in terms of Hp into account and that the diffusivi-ties vary but little in such a flow and finally that the flowvelocities are small compared to local sound speed oneeventually arrives (see Spiegel and Veronis, 1960 oralso, e.g., Zaussinger, 2010 for a detailed discussion)at the Boussinesq approximation of (2.1):

∂

∂t

⎛⎜⎜⎜⎜⎜⎜⎝

S ′

uΘ′

⎞⎟⎟⎟⎟⎟⎟⎠ = −∇ ·

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

u(S + S ′)u ⊗ u + P′

ρ0− ν∇u

u(Θ + Θ′)

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠−

−

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎝

0( Θ′

Θ0− S ′

S 0)g

0

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎠

+ ∇ ·⎛⎜⎜⎜⎜⎜⎜⎝

κS∇S ′

0κT∇Θ′

⎞⎟⎟⎟⎟⎟⎟⎠ . (2.2)

which is completed by the incompressibility constraint

∇ · u = 0. (2.3)

Here, we have replaced the concentration c with salin-ity S and the same holds for their fluctuating variablesc′ and S ′ as well as their diffusivities: κS = κc, whileκT = K/(cpρ) and cp is the specific heat at constantpressure. The diffusivities κS and κT are assumed to beconstant as is the kinematic viscosity ν. Similar to (2.1)P′ is the product of the scalar pressure fluctuation p′ andthe unit tensor I. It is a tensor as is ∇u. The quantitiesρ0, Θ0 and S 0 describe the constant background state.The contributions of kinematic viscosity to the equa-tion of (potential) temperature can be neglected, sincethey are small for low Mach number flows includingthose cases for which the Boussinesq approximation ap-plies (cf. also equations II–4–13 to II–4–25 in Lesieur(1997)).

We would like to emphasize here that the fluctuationsf ′ refer to a mean background state which is constant

1We note that typical values for this quantity range from 100 km near thesurface of the solar convection zone to about 50000 km near its bottom, whileit is found to be about 8 km in the lower part of the atmosphere of the Earthand increases from 10 m near the surface to several km near the bottom of anocean.


with respect to horizontal spatial coordinates and for thecases considered here also constant as a function of time.No separate notation is used here to distinguish thesequantities from turbulent fluctuations around some (pos-sibly differently specified) mean. Rather, we considerthe fluctuations f ′ synonymous with turbulent fluctua-tions for which the mean state is given by the Boussinesqreference state.

For constant temporal and spatial averages of thepotential temperature and salinity, we have ∂S /∂t = 0,∂Θ/∂t = 0, ∇S = 0, and ∇Θ = 0, whence we canrewrite (2.2) into the form used in ANTARES,

∂

∂t

⎛⎜⎜⎜⎜⎜⎜⎝

SuΘ

⎞⎟⎟⎟⎟⎟⎟⎠ = −∇ ·

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎝

uSu ⊗ u + P′

ρ0− ν∇u

uΘ

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎠−

−

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎝

0( Θ′

Θ0− S ′

S 0)g

0

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎠

+ ∇ ·⎛⎜⎜⎜⎜⎜⎜⎝

κS∇S0

κT∇Θ

⎞⎟⎟⎟⎟⎟⎟⎠ (2.4)

which is to be solved together with the incompressibilityconstraint (2.3). We refer to this set of equations for thecase of the Boussinesq approximation if a perfect gasequation of state actually holds for the fluid.

The case of water is more easily dealt with using aslightly different form of these equations. It can also bedirectly derived from (2.1) without assuming a hydro-static background state and for the case of a realisticequation of state, for instance, as in McDougall et al.(2003). This is the form of the Boussinesq approxima-tion actually used in MITgcm for all the simulation runsdiscussed in this paper. It features an equation for T in-stead of Θ and reads:

∂

∂t

⎛⎜⎜⎜⎜⎜⎜⎝

SuT

⎞⎟⎟⎟⎟⎟⎟⎠ = −∇ ·

⎛⎜⎜⎜⎜⎜⎜⎜⎝

uSu ⊗ u + P

ρ0− ν∇u

uT

⎞⎟⎟⎟⎟⎟⎟⎟⎠

+

+

⎛⎜⎜⎜⎜⎜⎜⎜⎝

0( ρρ0

)g0

⎞⎟⎟⎟⎟⎟⎟⎟⎠

+ ∇ ·⎛⎜⎜⎜⎜⎜⎜⎝

κS∇S0

κT∇T

⎞⎟⎟⎟⎟⎟⎟⎠ . (2.5)

We emphasize that both ANTARES and MITgcmconsider a complete equation state (perfect gas or wa-ter) for either the mean background state when assum-ing (2.4) to hold or for the local thermodynamical staterequired to be known in (2.1) and in (2.5). Linear stabil-ity analyses, however, consider a linearized form of theequation of state, i.e.

ρ = ρ0(1 − α(T − T0) + β(S − S 0)). (2.6)

The thermal expansion coefficient α = (−1/ρ)(∂ρ/∂T )and the solutal contraction coefficient β = (1/ρ)(∂ρ/∂S )are evaluated in (2.6) for a thermodynamical referencestate (T0, S 0, p0). Since we consider low Mach numberflows with small fluctuations of the dynamical variablesrelative to either mean reference states or horizontalaverages, the differences between the complete equationof states and their linearized form should be small, aprerequisite for applicability of the stability analyses.

2.2 Double-diffusive convection

2.2.1 General remarks

The magnitude of the turbulent fluxes for semi-convec-tive and salt-fingering sytems depends on the Lewisnumber (Le = κS/κT, the ratio of the molecular diffu-sivities of concentration and heat), the Prandtl number(Pr = ν/κT, the ratio of kinematic viscosity and heat dif-fusivity), and the stratification, characterized by a stabil-ity parameter: the ratio Rsc

ρ of concentration and temper-ature gradients.

We recall that within the Boussinesq approximationthe thermal Rayleigh number RaT, a corresponding so-lute Rayleigh number RaS and the stability parameterRscρ are defined as RaT = gα(Tbottom − Ttop)D3/(κTν),

RaS = gβ(Sbottom − Stop)D3/(κTν) and Rscρ = RaS/RaT.

D is the length scale to which the Rayleigh numbers re-fer to (the size of the simulation domain, the thicknessof a layer, etc.) whereas temperatures T and salinities Sare taken at the bottom and top of the domain which inturn are just a distance D separated from each other. Theratio Rsc

ρ is particularly convenient for studying double-diffusive processes and is more frequently used thanRaS. The superscript “sc” is used here to distinguish itfrom similar parameters and to stress its usefulness indiscussing the stability properties of semi-convection.

Linear theory predicts semi-convection for the sta-bility range 1 ≤ Rsc

ρ ≤ 1+PrLe+Pr (Baines and Gill, 1969;

Stevenson, 1979; Zaussinger et al., 2013), but deter-mining the actual magnitude of the turbulent fluxes aswell as the effective diffusivities for semi-convectiveflows requires direct numerical simulations (DNS).

The conservation laws (2.1)–(2.5) are frequentlytransformed into a non-dimensional form for furtheranalysis and also for their numerical solution. ThePrandtl number Pr, the Lewis number Le, the thermalRayleigh number RaT, the solute Rayleigh number RaS,and the stability parameter Rsc

ρ are common parame-ters used in such transformations and are also used inthe following discussions. We note that for cases as inSection 2.1, where the diffusivities and viscosity are as-sumed to be constant, Pr and Le are constant as well.

We now discuss in more detail the two most interest-ing cases with the gradients of temperature and salinitycounteracting instead of amplifying each other.

2.2.2 Semiconvection, stability and layering

Astrophysical systems such as stars or giant planets ex-tend over many pressure scale heights Hp. It is hencecommon to express stability criteria by means of di-mensionless, logarithmic gradients with respect to the(gas) pressure p. The standard notation for the gra-dients of temperature and mean molecular weight is∇ := ∂ ln T/∂ ln p and ∇μ := ∂ lnμ/∂ ln p while theadiabatic temperature gradient is abbreviated as ∇ad :=(∂ ln T/∂ ln p)ad. In astrophysics the difference ∇ − ∇ad,


known as the superadiabatic temperature gradient, is theusual quantity to determine whether a stratification isthermally stable. In geophysics the potential tempera-ture Θ is common for the same purpose. For the dimen-sionless gradient of potential temperature ∇Θ and a per-fect gas equation of state it is straightforward to show(for instance, Zaussinger, 2010) that ∇Θ = ∇ − ∇ad.

These quantities describe the dynamical stability of asystem that is governed by the Brunt-Väisälä frequency(see, e.g., chapter 6 in Kippenhahn and Weigert(1991)). In general, N2 := (gδ/Hp)(∇ad − ∇ + (ϕ/δ)∇μ),where δ = −(∂ ln ρ/∂ ln T ) and ϕ = (∂ ln ρ/∂ ln μ). For aperfect gas this simplifies to N2 = gH−1

p (∇μ− (∇−∇ad)).If N2 > 0, a vertical displacement leads to a (usuallydamped) oscillation, while for N2 < 0 a vertical dis-placement leads to an exponentially growing instabilityand convection sets in. In a simplified form (δ = φ = 1for a perfect gas) this is known as the Ledoux criterionof convective stability (Ledoux, 1947): ∇μ > ∇ − ∇adimplies stability while for ∇ − ∇ad > ∇μ convectionis predicted to set in. In the spirit of the Boussinesqapproximation the same conditions can be written bycomparing the solute and thermal Rayleigh numbers.With the above definitions of RaS and RaT (note thatthere are different conventions for choosing the sign),0 < RaS < RaT corresponds to the Ledoux unstablecase and 0 < RaT < RaS to the Ledoux stable case.These stability conditions are the result of a well es-tablished linear analysis (see Section 1). In their mostsimple form as is discussed here and sufficient for ourwork, it is assumed that both RaT and RaS are large andin this sense ν is small. Otherwise, slightly more com-plex relations hold which feature additional explicit de-pendencies on Pr and Le. Evidently, for the Ledoux sta-ble case, Rsc

ρ > 1, while for the unstable one, Rscρ < 1.

Staying within the framework of linear stability analy-sis, a layer where Rsc

ρ > 1 is dynamically stable, but canbe vibrationally unstable (Kippenhahn and Weigert,1991): a (vertically) oscillating fluid element does nothave the same temperature as its environment and ex-changes heat with it, that is, it moves non-adiabatically.In a chemically homogeneous case (∇μ = 0) this resultsin damped oscillations. In the non-homogeneous case(∇μ > 0) the oscillations may either also be dampedor they increase slowly (on the time scale of heat dif-fusion). The vibrationally unstable region is character-ized by 1 ≤ Rsc

ρ ≤ 1+PrLe+Pr and it is this parameter regime

the term semiconvective in astrophysics usually refers to(see, e.g., chapter 30.4.2 in Kippenhahn and Weigert(1991)).

Linear stability analysis, however, is a very limitedtool for such conditions. Canuto (1999), who givesa detailed discussion of the various stability criteriafor semi-convection and salt-fingers, notes that the lin-ear stability analysis does not account for turbulenceand the consequences of transport of mass, heat, andconcentration. A more complete analysis would actu-ally introduce the turbulent diffusivities, or in our ter-

minology, effective diffusivities, into the stability crite-ria. The dynamically unstable case is characterized by∇ − ∇ad > (KS /KT )∇μ while the vibrationally unsta-ble one requires (KT /KS )(∇ − ∇ad) > ∇μ > (∇ − ∇ad).Since the latter implies KT > KS for a consistent model,the unstable regime is expected to extend somewhatinto the region where Rsc

ρ > 1. Applicability of theso-called Schwarzschild criterion of convective stability(Schwarzschild, 1906), where ∇ > ∇ad describes anunstable state and ∇ < ∇ad a stable one, is found to haveto remain limited to where ∇μ = 0, since only in this caseonce can expect KS = 0. At the bottom line, the mix-ing efficiency is found to be overestimated when impos-ing only the Schwarzschild criterion ∇ > ∇ad in stellarevolution calculations while it is underestimated whenrequiring the Ledoux criterion Rsc

ρ < 1 for convectivemixing to occur. The complete model (Canuto, 1999,2011) predicts a mixing efficiency that is in-between thenaive application of the two criteria and cannot be re-trieved from linear stability analysis alone. We note thatthe discussion of Canuto (1999, 2011) is made with aspecific model approximation in mind, a Reynolds stressdescription of the entire fluid domain which would usu-ally be applied in low resolution global models of oceansor stars which necessarily average over separate double-diffusive layers (when present). In limiting cases of thefull model diffusivities and gradient expressions for thefluxes appear and provide the framework for his discus-sion. It is quite remarkable that other theoretical analy-ses which have quite different starting points such as thedescription of individual layers to which we turn in thefollowing arrive at similar conclusions. Of course, sinceall these discussions are to some extent model depen-dent, they require confirmation by experiments or directnumerical simulation.

An important phenomenon that is not found from lin-ear stability analysis either is the formation of layers inthe Ledoux stable case. With a non-linear analysis Proc-tor (1981) was able to show that in a region limited by

1 ≤ Rscρ ≤ Le−1/2 (2.7)

stable layering has to occur in the semi-convectiveregime (the upper limit is only approximate). The initiallayering process is based on local breaking of gravitywaves, often associated with the so-called Kato instabil-ity. Fig. 1 shows an advanced stage of layer formationin a simulation of semi-convection with the ANTAREScode. A more detailed description of the simulation ofsuch layers with this code is given below in Section 3.1.

Spruit (2013) derived a model for a more accurateprediction of the region where layer formation occursthat is expected to hold for arbitrary values of RaT andfor Pr < 1. Numerical simulations by Zaussinger andSpruit (2013) have shown that oscillatory instabilitiesare damped if Rsc

ρ ≈ Le−1/2, which results in very diffu-sive flows, while overturning cells are found to developfor values closer to 1. The prediction of Spruit (2013)


Figure 1: Layer formation in a snapshot of a simulation ofsemi-convection with idealized microphysics (constant diffusivities,Boussinesq approximation) in a 2D simulation with ANTARES. Theparameters of the simulation are Pr = 1, Le = 0.01, Rsc

ρ = 3,RaT = 5 · 109. The left panel shows temperature, the right one salin-ity. The much sharper interfaces found for the latter are due to thevery low diffusivity of the solute in comparison with heat, as followsfrom Le = 0.01 1. The blue color represents low values, red onehigh values with intermediate values encoded by bright, less intensecolors.

of a critical maximum value of Rscρ for layer formation,

called Rscρmax, was found to agree with the numerical sim-

ulations of Zaussinger and Spruit (2013) within ex-pected accuracy. For RaT = 106, Pr = 1, and Le = 0.1, avalue of Rsc

ρmax ≈ 1.2 is expected in comparison withRscρmax ≈ 1.4 which is actually found from numerical

simulations. The model of Spruit (2013) also providesphysical arguments to justify the following parametriza-tion of thermal and solute fluxes by means of the Nusseltnumbers for the semi-convective regime (i.e. turbulentfluxes in units of diffusive fluxes, for a detailed discus-sion see Section 4.1):

NuS − 1 =q

√Le Rρ

(NuT − 1). (2.8)

Here, q is a fitting parameter in the quantitative com-parison to numerical results and found to be close to 1.Zaussinger and Spruit (2013) also find their numer-ical simulations of semiconvection to support such aparametrization. The layer thickness d itself dependsmainly on the history of the system. Within the samemodelling framework it can be estimated from the so-lute diffusivity for a time interval t as

d =√

2κS NuS t. (2.9)

It is limited from below by the length scale l0 on whichthe thermal diffusion time scale equals the free falltime over a pressure scale height (see Zaussinger andSpruit, 2013),

l0 = (κ2T Hp/g)1/4 < d < Hp, (2.10)

because on length scales smaller than l0 diffusion canexchange heat faster than convection. Moreover, the es-timate has been derived under the Boussinesq approx-imation, which provides an upper limit for d. As isseen in laboratory experiments the layer thickness in-creases in time until a fully mixed single convectionzone is established. However, to explain the continuousre-establishment of layers found in systems heated fromthe bottom, it may additionally be necessary to considereffects introduced by the assumed boundary conditions.Similar to the Rayleigh-Benard system the thermal Nus-selt number can be estimated by a general power law,NuT = a(RaT Pr)b, but the exact values of a and b andtheir region of applicability are still a goal of ongoingresearch activities. Figs. 1 and 2 show the temporal evo-lution of a top and bottom bounded stratified fluid col-umn with a fixed ΔT and ΔS . A steep initial tempera-ture gradient at the bottom induced the development ofplumes, forming the first layer. However, the solute isinitially stratified linearly in the vertical direction. Theheight of the box is set to H = 5, for this multi-layersimulations, and to H = 1 for single layer simulations.The Kato instability leads to three initial layers, whicheventually merge into one single convective zone. Theobserved merging process happens on a single thermal(heat diffusion) time scale (t ≤ τ).

2.2.3 Salt-fingering

We just briefly summarize the differences and similar-ities of the salt-fingering case in comparison with thesemi-convective one. This process can occur despite thestratification is actually dynamically stable in the sensethat a (vertically) displaced fluid parcel is restored backto its original position by buoyancy because ∇ad > ∇,if there were no exchange of heat (and solute) with its(new) environment. For a real fluid with Le < 1, heatexchange by diffusion is faster than the correspondingexchange of solute. Then the heat exchange can lead toa net buoyancy force in the direction of the initial dis-placement. This is called thermal or secular instability(Kippenhahn and Weigert, 1991). In its initial phasethis instability gives rise to a typical “fingering struc-ture”. It may be triggered particularly easily, when thestable temperature gradient is counteracted by an unsta-ble gradient in mean molecular weight (∇μ < 0).

Again, the ratio of the gradients of mean molecu-lar weight and potential temperature allows a distinc-tion between a Ledoux stable and a Ledoux unsta-ble case. In principle, one could use the definition ofRscρ = RaS/RaT � ∇μ/(∇ − ∇ad) to discuss salt-fingering

(e.g., Canuto, 1999), but here we use an alternativequantity more appropriate for the salt-finger case, Rsf

ρ =

RaT/RaS � (∇ − ∇ad)/∇μ, since it leads to symmetricrelations.2

2The direct computation of the stability parameters Rscρ and Rsf

ρ from temper-ature gradients is more convenient for the fully compressible case, for whichthe definition of the Rayleigh numbers can alternatively be defined in terms


(a) t = 0 .0002 τ (b) t = 0 .0005 τ (c) t = 0 .0015 τ (d) t = 0 .005 τ (e) t = 0 .01 τ

(f) t = 0 .03 τ (g) t = 0 .04 τ (h) t = 0 .05 τ (i) t = 0 .07 τ (j) t = 0 .09 τ

Figure 2: Temporal evolution of a semi-convective stack visualized for the solute. The steep temperature gradient at the bottom induces aninitial layer, which grows until 0.01τ (upper row, panels a–e). However, the Kato oscillation triggers the evolution of the second and a thirdlayer, (lower row, panels f–j). The lower row ranges from 0.03 τ to 0.09 τ, where τ is the thermal diffusion time scale for the entire box.Each layer is convectively unstable. At least temporarily the thermal and solute transport from one layer to the next one is by diffusion only.The simulation parameters are Pr = 1, Le = 0.01, Rsc

ρ = 3, RaT = 5 · 109. The simulation has been computed using the ANTARES code.

In this case, Rsfρ < 1 describes the Ledoux unsta-

ble case (N2 < 0 and ∇ − ∇ad > ∇μ) where the fluidis rapidly mixed and thus pre-existing gradients disap-pear unless they are maintained by the boundaries ofthe layer. Rsf

ρ > 1 relates to the Ledoux stable caseof salt-fingering which is of interest here, because theefficiency of mixing and the temporal development ofthe flow depend on the ratio of the different diffusiv-ities (heat, concentration, momentum), and thus on Prand Le. Linear stability analysis allows a first charac-terization of the Ledoux stable parameter regime and asfor semi-convection the evolution time scale of a salt-fingering layer is the thermal diffusion time scale (atleast if Le < 1 and Pr < 1). The actual stability cri-teria and mixing efficiencies depend on effective (tur-bulent) diffusivities (Canuto, 1999) that can only ob-tained from a (non-linear) model or a numerical simula-tion. As in semi-convective systems, the entire double-diffusive system evolves on time scales even longer thanthose of thermal diffusion, because the slower diffusionof the solute plays a role, too.

Fig. 3 shows two snapshots of a saltfinger simulationwith the MITgcm code. The fingers appearing duringearly stages (100 sec, left panel) are much more distinctthan the structures appearing during late stages. The do-main was 24.75 cm in height and 8.25 cm in width. Thesimulation starts with a system of two clearly distin-guishable layers. The temperature difference betweenlower and upper boundary was ΔT = 1 ° and choosen asdescribed in Table 2 in Section 4. Late stages of the sim-

of local gradients rather than through temperature or solute differences mea-sured along a distance D. The definition of Rsc

ρ and Rsfρ through Rayleigh

numbers is universal: it hides the difference about how the bouyancy fre-quency is computed in the Boussinesq case as opposed to the fully com-pressible one, where the Rayleigh number can be “local” or refer to averageconditions in an entire layer.

Figure 3: Snapshot of salinity distribution (colors) with temperaturecontours of ΔT = 0.1 ° within a saltfinger simulation with a stabilityratio of Rsf

ρ = 1.33 after 100 sec (left panel) when local instabilitiesstart to develop and with well developed up- and downwelling (salt-)fingers after 6000 sec (right panel). The color scale for salinity is thesame for both panels. The simulation has been performed with theMITgcm code.

ulation such as that one shown in the right-hand panelof Fig. 3 have already lost most information about theirinitial state and thus the latter would have looked quitesimilar in this sense if it had been started from a condi-tion with constant inital gradients in T in S (cf. Zwei-gle, 2011). A more detailed description of the evolutionof such layers with this code is given in Section 3.2.


3 Numerical methods

In the following we describe the improvements made tothe ANTARES simulation code, initially during the Met-Ström project, and the setup of MITgcm, which wererequired for our studies of double-diffusive convection.Since this work is one of the essential results of theproject and existing partial descriptions require to readselected parts of quite a few publications for a full ac-count, we provide a complete summary and detailed ref-erences to each of the original papers in the followingsubsections.

3.1 ANTARES

The ANTARES code (Muthsam et al., 2010) is a multi-purpose simulation program to solve the hydrodynam-ical conservation laws numerically in one, two, andthree spatial dimensions. An initial state of the systemcan be evolved in time given various types of bound-ary conditions. The spatial discretization of advectionoperators (and in some cases also of pressure gradi-ent operators) is based on the weighted essentially non-oscillatory scheme of 5th order (WENO5) proposedby Jiang and Shu (1996), optionally with Marquinaflux splitting (Donat and Marquina, 1996). Diffusionterms are discretized in a compatible way (Happen-hofer et al., 2013). Time integration is performed us-ing strong stability preserving Runge–Kutta methods(Shu and Osher, 1988, Kraaijevanger, 1991, see alsoKupka et al., 2012). The code is fully parallelized fol-lowing the MPI (message passing interface) paradigm(see Muthsam et al., 2010 for further details). With thisframework implementation of an explicit time integra-tion scheme for the Navier–Stokes equation augmentedby an equation for the time evolution of concentrationand the conservation laws for mass and energy, i.e. thesystem (2.1), was readily possible (Zaussinger, 2010).

As mentioned in Section 2.1 the Boussinesq ap-proximation is useful for basic explorations of double-diffusive convection. This holds in particular for com-parisons with oceanographic cases. In Zaussinger(2010) ANTARES was hence extended to solve the dy-namical equations (2.2)–(2.3) or, in practice, (2.3)–(2.4).This setup differs from the compressible case by re-quiring the solution of a Poisson equation for the pres-sure fluctuations which ensures the incompressibilityconstraint (2.3) to hold during time integration. InANTARES this is usually done with the FISHPACKsolver of Adams et al. (2011), a fast finite differencebased solver which has been parallelized by meansof the Schur complement algorithm (Grimm-Strele,2010) during the MetStröm project (Zaussinger, 2010).A time split integration of (2.4) is performed wherethe velocity equation is first integrated in time with-out the pressure term, followed by the computation ofpressure fluctuations under the constraint (2.3). Finally,the velocity field is corrected for the latter. Hence, theWENO5 discretization is only applied to the advection

terms in (2.4) given by their flux functions uS , u ⊗ u,and uΘ and no transformation of the independent vari-ables into their local characteristic fields is necessary tobuild up the WENO5 stencil. This is possible becausethe direction of the numerical flux along each Cartesiancoordinate is uniquely determined by the sign of eachvelocity component. In conjunction with the excellentparallelization through the Schur complement approachan efficient and accurate method to solve (2.3)–(2.4) hasbeen obtained and also successfully compared with di-rect solutions of (2.1) for a parameter region accessibleto fully explicit time integration methods (Zaussinger,2010; Zaussinger and Spruit, 2013).

For numerical simulations of double diffusive con-vection with ANTARES the vertical boundary condi-tions are usually taken to be impermeable for temper-ature and solute, respectively, as well as stress-free.This ignores the distortions of the interfaces by grav-ity waves. In the horizontal direction periodic conditionsare chosen. The initial stratification depends on the in-vestigated problem, however, linear or step-like initialstratifications are used in most cases. Simulations in theBoussinesq approximation require the specification ofLe, Pr, Ra, and Rρ (i.e. either Rsc

ρ or Rsfρ ). For the com-

pressible case the depth of the domain in units of pres-sure scale heights has to be specified, too. The detailedalgorithm for the setup of the initial stratification is de-scribed in Zaussinger (2010) as well as in Zaussingerand Spruit (2013). It is an extension of the proceduredeveloped by Muthsam et al. (1995, 1999) for directnumerical simulations of compressible convection.

Simulations of double diffusive convection with astrong vertical stratification cannot rely on the Boussi-nesq approximation. In such cases the flow speed maybecome large, the mean stratification may no longerbe constant with time, or (in a more general scenario)the diffusivities may be functions of temperature andchemical composition sufficiently sensitive to the vari-ations induced by stratification and flow such that theycan no longer be considered constant. One way to pro-ceed in such cases is to consider analytical approx-imations to the fully compressible equations whichare more refined than the Boussinesq approximation.Kwatra et al. (2009), however, developed an operatorsplitting method that integrates the pressure terms ∇ · Pand ∇ · Pu in (2.1) in a semi-implicit manner withoutfurther approximations to the analytical equations them-selves. The advection operators ∇·ρu, ∇·cρu, ∇·ρu⊗u,and ∇ · eu are integrated explicitly, while the time inte-gration of the pressure terms requires the solution of alinear, generalized Helmholtz equation. This problem isonly slightly more complex than the solution of a Pois-son equation which appears during the numerical inte-gration of (2.2)–(2.3). Full consistency with the equationof state is ensured here since the “predicted pressure”obtained from this elliptic equation is only used in anintermediate step. The physical pressure is recomputedat the end of each stage or step. In practice, no iterationson this step are required. As is the case for (2.2)–(2.3)


a transformation into local characteristic variables is notneeded any more, because the sound speed no longermatters when determining the direction of the numeri-cal flux at the boundary of a grid cell once the advec-tion terms and the pressure terms are treated separately.Happenhofer et al. (2013) demonstrated how to extendthis method to the case of a two-species fluid under thepresence of buoyancy and diffusion of heat and con-centration, i.e. the full system (2.1). Strong scaling wasdemonstrated when a conjugate-gradient based solverfor the solution of the generalized Helmholtz equation iscombined with the Schur complement for parallelization(Happenhofer et al., 2013). This was found to hold forthree orders of magnitudes with respect to the numberof processors, i.e. for up to more than 1000 CPU cores.

In numerical simulations of double diffusive convec-tion the diffusive processes may set the most restrictivetime step limit Δt for an explicit integration method. Butas long as the flow velocities are small, the solution mayactually change only by a little amount during Δt. Thus,implicit time integration methods may be desirable. Atleast in principle they allow increasing Δt such that thesimulation proceeds with a time step corresponding tothe change rate of the solution rather than that of an ad-ditive (and possibly negligibly small) contribution to it.Additive splitting techniques promise to be particularlyefficient for the numerical integration of such problems:the non-linear advection operators offer very little poten-tial for speed-up through implicit methods, because un-less the problem is stationary, the solution is expected tovary significantly between two grid cells of size Δx overa time scale t = Δx/|u|. At the same time, the solution ofa large, non-linear system of algebraic equations is ex-pensive in terms of computing time. If instead only theterms related to diffusion are integrated by an implicitmethod, the ensuing (quasilinear and scalar) Helmholtzequations can be solved very efficiently. This has mo-tivated the study and further improvement of implicit-explicit Runge-Kutta (IMEX RK) methods which arestrong stability preserving (SSP). Kupka et al. (2012)discussed the benefits of such integration methods fornumerical simulations of double-diffusive convection.Combining them with semi-implicit methods for thetime integration of pressure gradients for the compress-ible case (Happenhofer et al., 2013) or the Boussinesqapproximation removes the most severe time step limi-tations from numerical simulations of semi-convectionand salt-fingers, at least for the case where Pr < 1.Through extension of this approach to a general addi-tive splitting method, which is work in progress, the timeintegration can proceed along an optimum time step Δtwithout the undue costs of fully implicit time integrationmethods applied to (2.1) or (2.2)–(2.3).

ANTARES permits grid refinement as means of reso-lution optimization (Muthsam et al., 2010). Because thesimulations of double-diffusive convection entail layerformation and merging as well as the formation, merg-ing, and destruction of plumes, fine vortex structures,and other non-stationary features close to the limit of

grid resolution, predefined regions of high resolution areof limited use here. The simulations performed in this re-search have hence been done on single, fixed grids witha resolution optimized according to the physical diffu-sivities represented by the model equations.

We note that the results of all simulations discussedin the present paper which have been computed withANTARES are based on the Boussinesq approximationand repeat that a comparison of these results with calcu-lations based on the fully compressible Equations (2.1)was presented in Zaussinger (2010) and Zaussingerand Spruit (2013). The former also discusses an exten-sive grid of models assuming the Boussinesq approxi-mation. That grid provides the basis for the extrapola-tions to the stellar case to which we turn in Section 4.3.

3.2 MITgcm

The Massachusetts Institute of Technology general cir-culation model (MITgcm) is a general purpose grid-point algorithm that solves the Boussinesq form of theNavier-Stokes equations for an incompressible fluid,here fully non-hydrostatic, with a spatial finite-volumediscretization on a curvilinear computational grid (inthe present context on a three-dimensional Cartesiangrid). The model algorithm is described in Marshallet al. (1997); for online documentation and access to themodel code, see MITgcm Group (2012). The code sup-ports multi-threading and MPI for parallelization andalso vector-cpu architectures; it has been shown to scalefor order(103) CPUs (Hill et al., 2007; Losch et al.,2014). Here, we use the MPI parallelization.

The MITgcm was originally built for large scaleoceanographic and atmospheric applications, but the ro-bust numerics and the non-hydrostatic extension of thesolution algorithm (a pressure correction method) allowssimulations of small scales with very high grid resolu-tion (e.g. Losch, 2004; Losch et al., 2006; Zweigle,2011). The hydrostatic and non-hydrostatic pressurecontributions are obtained from solving two- and three-dimensional elliptic problems implicitly with a pre-conditioned conjugate-gradient method. All other termsin (2.2) (or (2.5)) are stepped forward in time explic-itly. For the direct numerical simulations (DNS) of semi-convection and salt fingering we make use of some ofthe specific features of the MITgcm, in particular, a7th order monotonicity preserving advection scheme fortracers with very little numerical diffusion (Daru andTenaud, 2004).

For the simulations of semi-convection with MIT-gcm which we show here, the model domain is a 384 mmby 128 mm water body (aspect ratio 3:1) with a gridspacing of 1 mm in all directions and periodic in the hor-izontal. The domain size is limited by computational re-quirements resources, but our domain already allows theformation of several convection cells. The simulationsare practically 2D to save computer time, but we allow a


Table 1: Thermal and saline (solute) Rayleigh numbers of simulatedsemi-convection in the ocean and stability ratios. The Rayleigh num-bers are calculated from values at the upper and lower boundaries.In all cases, the Lewis number was Le = 0.01.

Rscρ RaT RaS

SE-1 1.00 2.96 × 106 2.96 × 106

SE-2 1.02 2.96 × 106 3.02 × 106

SE-3 1.05 2.96 × 106 3.11 × 106

SE-4 1.10 2.96 × 106 3.26 × 106

SE-5 1.50 2.96 × 106 4.44 × 106

SE-6 2.00 2.96 × 106 5.92 × 106

few (10) grid points in the y-direction. Flanagan et al.(2013) show that, compared to true 3D-simulations, 2D-simulations of semi-convection tend to overestimate theeffective fluxes for very low stability ratios Rsc

ρ . Still,they conclude that 2D DNS provide an “attractive al-ternative” to 3D DNS, at least for the region of appli-cability identified by comparisons with 3D simulations.The latter are important since the transport propertiesof 2D and 3D flow can be quite different, as also theirresults for low values of Rsc

ρ demonstrate. The size ofthe domain allows the formation of several convectioncells. Temperature and salinity at the bottom and the up-permost layer are restored to the initial values in orderto maintain the stratification and simulate an unlimitedreservoir above and below. Explicit molecular diffusivi-ties of κT = 1.5×10−7 m2 s−1 and κS = 1.5×10−9 m2 s−1,and viscosity (diffusivity of momentum) of ν = 9.3 ×10−7 m2 s−1 give realistic Prandtl and Lewis numbers ofσ = ν

κT= 6.2 and Le =

κSκT

= 0.01. The thermal diffusive

time scale τ is D2/κT = (128 mm)2/1.5 × 10−7 m2 s−1 ≈30 h. Several simulations with different boundary con-ditions are summarized in Table 1. All simulations startfrom step-like initial conditions, inspired by the layersthat form in semi-convection (cf. Fig. 2(h) at t = 0.05τ).Without these initial conditions, the integrations wouldhave to be very long and expensive. We note that wehave focussed here on a parameter range with respectto Rsc

ρ which is dynamically the most interesting one.For smaller values of Rsc

ρ the results from linear stabilityanalysis are rapidly recovered while the diffusive statestypically found for larger values are computationally ex-pensive and at the same time the predictions from stabil-ity analysis are the least certain ones for the intermediatevalues we have investigated here.

Fig. 4 shows a snapshot of a semi-convection simula-tion after 2700 sec und 6500 sec. Through the exchangeof energy (heat), the boundary layer in the middle ofthe domain destabilizes and starts oscillating. These os-cillations develop into approximately five convectioncells that characterize the turbulent mixing process. Theboundary layer between these convective (rolling) cellswas preserved during the entire simulation.

Figure 4: Snapshot of salinity distribution with temperature con-tours of ΔT = 0.02 ° within a simulation of semi-convection withstability ratio Rsc

ρ = 1.1 after 2700 sec (upper panel) when local in-stabilities start to develop and with well developed (rolling) convec-tion cells after 6500 sec (lower panel). The color scale of salinity ischosen to emphasize the structure of the lower fresh layer; similarpatterns are found in the upper layer (not visible).

4 Results4.1 Effective diffusivities in the ocean

Vertically stratified fluids can release potential energywhich eventually leads to (often turbulent) verticalfluxes of heat and concentration. Keeping the limitationsof the diffusion approximation in mind the horizontalaverage of such fluxes, w′χ′, can also be described interms of an effective diffusivity, Kχ = (w′χ′)/(∂zχ), orthrough non-dimensionalized Nusselt numbers, Nuχ =(Kχ)/(κχ). Here, χ is supposed to mean either tempera-ture T , adiabatically filtered temperature Θ, or concen-tration (salinity) S .

If we consider the evolution of the numerical sim-ulations summarized in Table 1 as a function of time,the effective diffusivities KT and KS turn out to reacha maximum near the end of the spin-up from the initialstate. Then, especially for larger stability ratios, the sys-tem equilibrates at lower diffusivities (Fig. 5). To inter-pret these results we note that the effective diffusivitieshave been evaluated here as horizontal averages of fluxesacross a horizontal section at the vertical middle of thedomain, where the interface between the two layers offluid is initially located. For this set of simulations theinterface hardly changes its position with respect to itsvertical location, as is also indicated by the stable (andstationary) horizontal averages of T and S , which ex-plains the physical motivation behind this simple proce-dure. On the other hand, the background gradients ∂zTand ∂zS are computed from the vertical mean gradientsof these quantities over the central 50 % of the domain.We recall that T and S are kept fixed at the top and atthe bottom of the simulation domain during these exper-iments.


(a) KSCT at interface with linear background gradients

(b) KSCS at interface with linear background gradients

Figure 5: Temporal evolution of effective diffusivities smoothedwith a running mean over 150 seconds. The different line colors referto the experiments SE-1 to 6 in Table 1.

To show the temporal evolution of the effective dif-fusivities we have picked for Fig. 5 one horizontal levelat the interface to compute the fluxes and their corre-sponding effective diffusivities to follow them (for thedifferent simulations) as a function of time. We estimatethe effective diffusivities twice per simulation: (1) whenthey reach their maximum values after about 500 sec-onds (Fig. 5) and (2) as the mean over the equilibriumphase when the convection cells have formed and Rsc

ρ,localalong with the stability has increased. Rsc

ρ,local is evalu-ated from the background gradients computed over thecentral 50 % of the domain rather than from taking thegradients over the entire domain, which are kept fixeddue to the boundary conditions. It is hence a measureof local stability (cf. also Fig. 4.33a and Table 4.5 inZweigle (2011)).

In our simulations we obtain effective diffusivitieswithin a range of KT = 0.29 to 4.27 · 10−6 m2/s andKS = 0.054 to 5.09 · 10−6 m2/s.

Based on the estimates of effective diffusivities, sum-marized in Fig. 6, we present a parametrization for the

(a) KSCT vs Rsc

ρ

(b) KSCS vs Rsc

ρ

Figure 6: Effective diffusivities estimated from simulations(1) when they are maximal, (2) as a mean over the equilibrium state.The different symbols refer to the experiments SE-1 to 6 in Table 1.

effective diffusivities of temperature and salinity,

KT =κT

a1 + b1 ln(Rscρ ), (4.1)

KS = KTTturb, (4.2)

with the turbulent Lewis number

Tturb =c2

a2 + b2Rscρ, (4.3)

with a1 ∼ 0.06, b1 ∼ 0.32, a2 ∼ −3.7, b2 ∼ 4.0, andc2 ∼ 0.1. This parametrization is only valid for stabilityratios Rsc

ρ ∈ [1, 2.8]. In contrast with Eq. (2.7)–(2.10),the relations Eq. (4.1)–(4.3) are purely empirical fittingformulae. Note that some of the diffusivity estimates oftype (1) (maximum values) overlap with those of type(2) (equilibrium phase) for less stable initial conditionssuggesting that the parametrization may be more gen-eral.


Figure 7: Thermal and saline (solute) Nusselt numbers NuT (dia-monds) and NuS (squares) from simulation over density ratio Rsc

ρ .The saline Nusselt numbers are scaled down by a factor of 10. TheNusselt numbers derived from the simulations are compared to thesaline Nusselt number computed from Equation (2.8) with q = O(1)(triangles) and computed from Equations (4.1) and (4.3) (line).

Table 2: Thermal and saline (solute) Rayleigh Numbers in a simu-lated saltfinger situation with Le = 0.1, stability ratios and Rayleighnumbers are comparable to the semi-convection case (see Table 1).

Rsfρ RaT RaS

1.33 1.8 × 108 1.35 × 108

1.33 1.8 × 108 1.35 × 108

1.06 1.8 × 108 1.65 × 108

2.16 1.8 × 108 0.8 × 108

2.66 1.8 × 108 0.65 × 108

Fig. 7 shows the simulated thermal and saline (so-lute) Nusselt numbers NuT and NuS obtained from theeffective diffusivities by Nuχ = Kχ/κχ. The Nusseltnumbers satisfy Equation (2.8) with q = O(1) (blacktriangles). The Nusselt numbers NuS computed with re-lationships (4.1) and (4.3) are drawn for reference (solidline in Fig. 7).

4.2 Comparison to salt fingering in the ocean

Several observational campaigns (Marmorino et al.,1987; Laurent and Schmitt, 1999; Polzin et al.,2001; Bianchi et al., 2002), lab experiments (Taylorand Bucens, 1989) and numerical simulation studies(Özgökmen et al., 1998; Yoshida and Nagashima,2003; Zweigle, 2011) shed light on the magnitude offluxes in saltfingering. For semi-convection there arevery few data available for comparison (Turner, 2010;Kelly et al., 2003).

In the case of the ocean the effective diffusivity oftemperature is of the same order of magnitude for bothsemi-convection and saltfingering (Fig. 8(a)) althoughthe semi-convection diffusivities tend to be lower. How-ever, we have found the effective diffusivity of salinity to

(a) Effective temperature diffusivities for semi-convection (red,blue, magenta) from our simulations and for saltfinger convectionfrom our simulations (cyan) and by other authors (black) for com-parison.

(b) Effective salinity diffusivities for semi-convection (red, blue,magenta) from our simulations and for saltfinger convection fromour simulations (cyan) and by other authors (black) for comparison.

Figure 8: Collection of effective diffusivities from a variety of nu-merical experiments and field observations. (a) temperature diffusiv-ities, (b) salinity diffusivities. Each of them is plotted as a function ofthe respective stability parameter Rρ, i.e. either Rsf

ρ or Rscρ . The cyan

symbols represent values from numerical simulations of salt finger-ing (Table 2), the black symbols are estimates for salt fingering byother authors based on numerical simulations (Merryfield, 20002002), or on observations (Fleury and Lueck, 1991; Laurent andSchmitt, 1999). The effective diffusivities calculated from turbulentfluxes in semi-convection simulations (cf. Fig. 6) are represented byred, blue, and magenta symbols.

be smaller by one to two orders of magnitude for semi-convection in comparison with saltfingering in that case(Fig. 8(b)).

The explanation for the larger salt fluxes (and slightlybigger heat fluxes) in saltfingering follows from the na-ture of the instability as sketched in the introduction.In the saltfingering case, the differential diffusion repre-sents a positive feedback and perturbations are acceler-


ated, as warm and saline water parcels from above looseheat and thus buoyancy and vice versa, cold and freshwaters from below gain heat and buoyancy. In contrast,perturbations in the semi-convection case are damped aswarm and saline water parcels from below loose heatand thus buoyancy and vice versa, so that they are de-celerated.

Hence, the differential diffusion in semi-convectionleads to a negative feedback which in turn supportsstable layers (such as those seen in Fig. 2). These layersform yet another impediment to mixing so that, whilemixing may be efficient within layers, the transport isslowed down by diffusive transition zones in-betweeneach convective layer.

The scenario of saltfingers is different from that ofsemi-convection because there are no diffusive inter-faces through which salinity is transported very slowly.As a result salinity is transported much more efficientlyin the saltfinger case. This is in agreement with anotherfinding of Zweigle (2011): for saltfingers the turbulentfluxes found in the numerical simulations with the MIT-gcm depend only weakly on the Lewis number as long asthe mixing processes have not changed the stratificationto such an extent that the diffusion begins to dominatetransports.

4.3 Extrapolation to stars

To estimate the range of parameter values for semicon-vection zones in main sequence stars,3 we have to quan-tify their physical state and compute the according diffu-sivities. If we use standard stellar evolution models, weobtain microscopic diffusivities of κT = 3 · 104 m2 s−1

and κHe = 10−4 m2 s−1 for the region of interest whichitself has a typical height of H ∼ Hp = 2 · 108 m. Thevalues of these quantities are based on a 15 M� stellarevolution simulation kindly provided by A. Weiss. Wecan use Equations (2.7)–(2.10) to estimate the propertiesof the semi-convection zone. The Rayleigh number canbe found as a function of the layer thickness d and thepressure scale height Hp. We obtain a modified Rayleighnumber Ra∗ = Ra · Pr = 1012 for d/Hp = 0.1 and Ra∗ =

108/Pr for d/Hp = 0.01. The effective Helium diffu-sivity is found to be KHe = 10−1 m2 s−1, three ordersof magnitudes above the microscopic diffusivity. Thisresult is not surprising since the saline (solute) Nusseltnumber is found to be in the order of NuS = 103–104.These estimates are supported by numerical simulations(cf. Fig. 9) for a perfect gas equation of state and Prandtlnumbers between 0.01 and 1.0 (see also (Zaussinger,2010; Zaussinger and Spruit, 2013)). The estimateslead to a total mixing time scale of about 1010 yr. Thisis much longer than the time that stars with sufficientmass to feature semiconvection in their interior actu-ally remain on the main sequence during their evolution

3stars for which nuclear fusion of hydrogen in their central region providesthe main source of energy

(a) NuT as function of Ra∗

(b) NuS as function of NuT

Figure 9: Dependence of the thermal Nusselt number on the modi-fied Rayleigh number Ra∗ = Ra · Pr and the saline (solute) Nusseltnumber NuS for varying stability parameters Rρ := Rsc

ρ . The solidlines are numerical results for Le = 10−2, dashed lines are modelpredictions.

(cf. (Salaris and Cassisi, 2005)). The value of the ini-tial layer thickness l0 = 2 · 103 m is established in aboutsome days, however, which is very short compared to anoverall evolution time of more than 107 years (see also(Zaussinger, 2010; Zaussinger and Spruit, 2013)).The enormous differences in the timescales are mainlydue to the extremely small value of the stellar Lewisnumber, Le ≤ 10−9. The same estimation can be donefor layering processes in volcanic lakes (e.g. Lake Kivu)resulting in merging time scales of some months and aninitial layer thickness of l0 = 5 mm.

Is the extrapolation over so many orders of magni-tudes from direct numerical simulations to the stellarcase reliable? The stellar case differs from the geophys-ical one by a very low kinematic viscosity compared toits much larger radiative diffusivity. This explains thevery low values of Pr: the radiative diffusion inside astar in most ranges of temperature and density predom-


inates over contributions from particle collisions. Ther-mal boundary layers are thus much larger than the vis-cous ones and solute boundary layers are even smaller.Simulations used for extrapolation to the stellar regimeshould thus feature Le Pr 1. In practice, a besteffort of Le ≤ 0.1 · Pr ≤ 0.1 can be achieved thoughquite often further compromises have to be made. At theleast, the correct ordering of physical processes takingplace on larger or smaller length scales when comparedto each other can be ensured. We thus expect the ideal-ized simulations to relate to the physical parameter of in-terest in the same sense as well-resolved large eddy sim-ulations should do. Accepting this level of uncertaintywe may thus extrapolate our results to the stellar param-eter regime. Moreover, the extrapolation can be basedon a physical model (Spruit, 2013) underlying (2.8) aslong as this model compares sufficiently well with thesimulations in the parameter regime that can be inves-tigated directly. Currently, there is no known physicalreason why the model (2.8) should break down just be-cause of lowering Pr and Le.

The modified Rayleigh number Ra∗ = Pr·RaT plays acentral role in comparing both regimes. In this context,the Prandtl- and the thermal Rayleigh number in bothregimes require attention. The Prandtl number in semi-convective zones of stars is of the order of 10−7. TheRayleigh number, which scales with the third power ofthe height, is limited by 104 < RaT < 1020, resultingin 10−3 < Ra∗ < 1013. A comparable value for double-diffusive layers in saltwater is about Ra∗ = 109. Thisleads to a broad overlapping range of Ra∗ and conse-quently the possibility to compare both regimes. ThePrandtl number remains as the only uncertain parame-ter, but we could not observe a strong dependency in thedepicted parameter range of Ra∗ and Pr on the Nusseltnumbers and hence on the convective fluxes. This re-sult coincides with the theoretical estimation in Spruit(2013) and Zaussinger and Spruit (2013).

Fig. 9 shows numerical and theoretical results ofthe relation between the thermal and solute fluxesparametrised in terms of the Nusselt numbers. Thepower law for double-diffusive (semi-) convection as de-rived and tested in Zaussinger (2010) and Zaussingerand Spruit (2013) fits over a broad range with numer-ical simulations, as is demonstrated in Fig. 9(a). How-ever, the stability parameter Rsc

ρ plays a central role,which is not covered by the linear stability analysis.Fig. 9(b) depicts the dependency of the Nusselt num-bers, which are related by the square root of the Lewisnumber (see Equation (2.8)). Even this relation is influ-enced by the stability parameter.

At the bottom line the mixing due to semi-convectionis concluded to be a highly inefficient process overstellar evolution time scales (see Zaussinger, 2010;Zaussinger and Spruit, 2013) and if a higher mixingrate were required from astrophysical constraints in lay-ers where semi-convection occurs, additional sources ofmixing would have to be found.

5 Conclusions

In spite of the largely different geometrical scales,double-diffusive mixing processes in stars and in theocean have several aspects in common. In both sys-tems the molecular diffusion of concentration occursmuch more slowly than the diffusion of heat. As a con-sequence, the formation of specific geometrical struc-tures such as “salt-fingers” or “thermohaline staircases”is not restricted to the oceanographic scenarios andsome laboratory counterparts for which these phenom-ena have originally been observed, but can also occurin other physical systems for which comparable ratiosof diffusivities, temperature to concentration gradients,and buoyancy to diffusive time scales hold. In particu-lar, convection inside stars and especially gaseous giantplanets may have similar properties.

A numerical simulation of double-diffusive mixingprocesses either in oceans or in stars involves an enor-mous range of spatial and temporal scales. Hence, sucha simulation will inevitably have some restrictions to itsphysical realism. State-of-the-art numerical algorithmscan help to reduce such limitations. During the Met-Ström project the ANTARES and MITgcm simulationcodes have been extended to make them applicable todirect numerical simulation of double-diffusive convec-tion (Zaussinger, 2010; Zweigle, 2011). For the caseof ANTARES numerical methods have been extended(Happenhofer et al., 2013) and even the developmentof new time integration methods has been motivated(Kupka et al., 2012; Higueras et al., 2014). For astro-physical problems the focus during the project has beenon semiconvection (or diffusive convection) where a sta-bilizing gradient in mean molecular weight counteractsa destabilizing temperature gradient (see Zaussinger,2010; Zaussinger and Spruit, 2013). For oceano-graphic research the focus in the project has been on thecase of salt-fingers (see Zweigle, 2011).

In this paper we discussed some results of bothprojects on semiconvection in parameter spaces of in-terest to both astrophysics and oceanography. In ei-ther system the mixing efficiency is limited by sharplayer interfaces in the background stratification whichtend to damp instabilities. These layers themselves de-velop from smooth background stratifications that fa-vor semi-convection. The layers show a long life time(cf. Zaussinger and Spruit, 2013 for estimates) al-though ultimately, after initial formation, merging pro-cesses occur (cf. also Fig. 2). The complementary dou-ble diffusive process of salt fingering leads to effectivediffusivities for salinity that are one to two orders ofmagnitude higher than for semi-convection (see Fig. 8).

For both stellar and oceanic regimes, the scaled ef-fective diffusivities (the Nusselt numbers) are found tofollow approximately relationship (2.8). However, forthe oceanographic case a purely empirical parametriza-tion, (4.1)–(4.3), is also able to approximate fairly wellthe data on semiconvection from both measurements andour numerical simulations to within measurement uncer-


tainties in spite of its different dependence on the Lewisnumber Le (Fig. 7 and 8). Given the limited parameterrange of the available simulations as a function of Leand the density ratio Rsc

ρ , more extended studies appearnecessary that will clarify in greater detail the relation-ship of the effective diffusivities KT and KS to the ba-sic physical parameters Le,Rsc

ρ (as well as Pr,RaT) andalso quantify more accurately the influence of boundaryconditions or dimensionality (two vs. three spatial di-mensions) on the results as is claimed in recent work ofFlanagan et al. (2013). ANTARES and MITgcm couldreadily be used for such studies to further explore therange of applicability of models such as (2.8).

Acknowledgements

This research has been supported by the DFG within theprojects “Modelling of diffusive and double-diffusiveconvection”, projects KU 1954/3-1 and LO 1143/3-1in SPP 1276/1 and KU 1954/3-2 and LO 1143/3-2in SPP 1276/2 within the interdisciplinary MetStrömproject. F. Zaussinger has also been supported by theAustrian Science Fund FWF through project P20973(project leader H.J. Muthsam). F. Kupka is gratefulfor support by the FWF through projects P21742 andP25229 (project leader). We thank A. Weiss, MPI forAstrophysics, Garching, for providing a stellar evolutionmodel used in this research.

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