Zeroth Law of Turbulence and Its Discontents

The zeroth law of turbulence and its discontents

If in an experiment on turbulent flow all control parameters are fixed, except for the viscosity, which is lowered as much as possible, the energy dissipation per unit mass approaches a nonzero limit. Turbulence - the Legacy of A.N. Kolmogorov Uriel Frisch (1996) An example of the zeroth 2 law of turbulence ρ(ut + u·∇u)+∇p = µ u + f and ∇·u =0 ∇ Some equations for LANL 5 “If my views be correct, a fall of 800 feet will d generate one degree of heat, and the temperature 1 u 2 dx = u·f dx ν ω 2 dx Some equations for LANL 5 dt 2| | − | | of the river Niagara will be raised about one fifth !V !V !V of a degreegH by its1 0fall of4 1608 feet.” (Joule 1845) ∆T = = × K = 0.11K cp 4200 Wikimedia ω = ∇ u gH 10 48 × ∆T = = × K = 0.11K cp 4200 2 1 1 lim ν ω dx =0 cp = 4200JK− kg− ν 0 | | ̸ → !V (For seawater cp is close to 4000MKS) 1 1 cp = 4200JK− kg− 2 ∇ ρ ut + u·∇u + ∇p = µ u This dissipative process requires × " ∇ % molecularg H viscosity.= gα∆ ButT the1 coefficient03 6 m s 1 ′ × × ≈ − # $ !of viscosity ν! is very small and Joule 2 takes it as given, or obvious, that3 the 1 ωt + u·∇ω = ω·∇u + ν ω g′ H = gα∆T 10 6 m s− 6 2 1 × × ≈ ⌫water 10 m s ∇ ! exact value! of ν unimportant. 7 ⇡ √gHH R = 1010 νwater ∼ 7

ˆ u = uˆı + vȷωˆ =(vx uy)k ∴ ω·∇u =0 −

−5 2 −1 νair 10 m s ≈

1 That’s a great story about Joule’s honeymoon, but is it true? He did the calculation, but did he measure the temperature rise? That’s a great story about Joule’s honeymoon, but is it true? He did the calculation, but did he measure the temperature rise?

• The temperature increase, 0.1degree, is smaller than the expected variation of air temperature over H=50m. (For example, the average tropospheric lapse rate is 6.6 degrees per kilometer.) • But is the water in thermal equilibrium with atmosphere? • What about air drag on falling drops, and evaporative cooling of spray? • The temperature increase is buried in a lot of noise and the story is probably apocryphal…..

(Craig Bohren, American Journal of Physics) Another example

5 1 ⌫ 10 ms air ⇡ R 106 ⇠

1 UL Drag = C (R)⇥AU 2 with R = 2 D

and lim CD(R) is non-zero R !1 102 1 2 3 W = CD ρ L U 101 Stokes ﬂow 2

CD CD( ) =0 100 1 6 Frisch(1996)

“Turbulence” Uriel “Turbulence” 10-1 10-1 100 101 102 103 104 105 106 107 R = L U/ν

The drag coefﬁcient means the limit at inﬁnite Reynolds number.

There is no drag without viscosity, but ultimately the value of ν is irrelevant. ν→0 is a singular limit.

From Wikipedia From This is the zeroth law of turbulence. D’Alembert’s paradox “The theory (potential ﬂow), developed in all possible rigor, gives, at least in several cases, a strictly vanishing resistance, a singular paradox which I leave to future Geometers” D’Alembert

“Fluid mechanics is divided between hydraulic engineers who observe phenomena which cannot be explained and mathematicians who explain phenomena which cannot be observed.” Cyril Hinselwood

The zeroth law of turbulence If in an experiment on turbulent flow all control parameters are fixed, except for the viscosity, which is lowered as much as possible, the energy dissipation per unit mass approaches a nonzero limit. “Turbulence - the Legacy of A.N. Kolmogorov” Uriel Frisch (1996) An enclosed incompressible fluid ˆ ∇ u = k ψ 2 × ρ(ut + u·∇u)+∇p = µ∇ u + f and ∇·u =0 2 ρ(ut + u·∇u)+∇p = µ∇ u + f and and ∇·u =0 2 ρ(ut + u·∇u)+∇p = µ∇ u + f and ∇·u =0 u 2 2 =0 ρ(ut + u·∇ρ(uudt)++ u∇1·p∇=u2 )+µ∇∇up+=fµ∇ u + fand 2 and∇·u =0∇·u =0 2|u| dx = u·f dx − ν |ω| dx d dt V V V 1|u|2 dx = u·f dx −Theν power|ω|2 d xintegral! is: ! ! 2 d 2 2 dt V V V 1|u| dx = u·f dx − ν |ω| dx d ! 1 2 ! !d 2 1 2d 2 2 ∇ψ x y ν ζ x y dt 1V 2 ·ω = ∇ ×V·u V 2 2 d d = d2|ud| dx =!2|u| udxf =dx −!uν f d|xω|−dνx !|ω| dx dt !! | | − !!dt V dt V V V V V ω = ∇ × u ! ! ! ! ! ! with the vorticitylim ν |ωω|2 d=x ̸=0∇ × u ν→0 V d 1 2 2 ω = ∇ ×!ωu = ∇ × u ζ dxlimdyν =|ω|2 dνx ̸=0 ∇ζ dxdy 2 ν→0 V dt !! ! − !! | | lim ν |ω|2 dx ̸=0 ν→0 2 ∇ × ρ ut + 2u·∇u +V ∇p = µ∇ u lim ν |ω| dx ̸=0 ! 2 ν→0 V The 2zeroth law" is:! lim ν |ω| dx ̸=0 % ∇ × ρ ut + u·∇u + ∇p = µ∇ u # ν→0 $ d 2 ′′ !V 2 " ∇ % ∇ × ρ ut + u·∇u + ∇2p = µ∇ u F (ζ)dxdy = ν ζ F (ζ)dx ·∇ ·∇ 2 dt !! # − $!! | | ∇ ρ uωt +u·u∇"uω =∇ωp uµ + νu∇ ω % × Turbulence∇t + is+ a singular·∇= ∇ limit∇ 2 2 " × ρ u#t + u u +$ p%= µ∇ u ωt + u·∇ω = ω·∇u + ν∇ ω # (a “dissipative" $ anomaly”). 2 % ω#t + u·∇ω =$ω·∇u + ν∇ ω 2 ·∇ ·∇ 2 ˆ vx uy = ψ u = uωˆıt++vuȷωˆ ω = ω =(uv+x −ν∇uyω)k 2 ∴ ω·∇u =0 ωt + u·∇ω = ω·∇u + ν∇ ω − ∇ ˆ u = uˆı + vȷωˆ =(v − u )k ∴ ω·∇u =0 x y ˆ u = uˆı + vȷωˆ 10−5 m=(2 s−v1 x − uy)k ∴ ω·∇u =0 ˆ ˆ νair ≈ ˆ u2 = uı + vȷω=(vx − uy)k ˆ∴ ω·∇u =0 −5 2 −1 ˆ ˆ ∴ ·∇ vxνair ≈u10y =m s ψ u = uı + vȷω=(vx − uy)k ω u =0 − ∇ −5 2 −1 νair ≈ 10 m s −5 2 −1 νair ≈ 10 m s −5 2 −1 νair 10 m s d ≈1 1 u 2 dx = u·f dx ν ω 2 d 2 1 dt !V | | !V − !V | | 1 1 1 U 3 ε RMS ∼ L

1 4 ν3 / ℓK = ε ! " u =0

d 1 ∇ψ 2 dxdy = ν ζ2 dxdy dt 2 | | − ## ## −18 1/4 coﬀee 10 −4 ℓK = =10 m 10−2 ! " Injection scale L 3D turbulent

ε 2 energy cascade3 URMS 3 1/4 W L ε ν ∼ 3L dimℓK = ε = = URMS ε 3 ε ! " Big whorls have∼ Llittle1/4 whorls kg T ν3 u =0 Which feedℓK on= their1 velocity,4 νε3 / −183 1/4 And little whorlsℓK = have! "lesser whorls ε oceand 1 102 URMS 2 −3 ! " 2 ∇ψ dxdy = ν ζ dxdy And so on to viscosity. ℓK dt=ε| | − =5.56 10 ε =mν [ 1 (u + u )]2 (1) ## −9 ## 2 —i,j uL.=0 F.j,i Richardson ∼10 −18 L1/4 × = ν u2 + u2 + u2 + + w2 (2) coffee 10 −4 # x y z $ z ℓK != −2 "=10 m ⟨ ··· ⟩ 10 d 1 2 2 Inertialcascade ! " Yet again 2 ∇ψ dxdy = ν ζ dxdy d 1 2 W3 L2 1/4 d2t | | ¯ − ε ∇ ## κ ∆b(0) ## 2 ζ dimdx=dy == 3 ν ζ dxdy ε νkg T −≤18 1/H4 dt − | | coffee 10 −5 ℓK = −18 1/4 ℓAside:K = why turbulence=3 10 in am ##ocean 10 −##3 d 1 21 × 2 ℓK = −9 ε=5.56 10 m coffee ∇cup!ψ isdx “stronger”"dy = ν thanζ dx dy 10 × dt 2 | | − d ! !" " 2 ′′ %%turbulence in the ocean.%% d L F ζ 1 x2 y ν 2 ∇ζ F ζ x −y 1/4 ( )d2 ζ ddxdy ==ν ∇ζ dxdy ( )d d 18 dt − | | coffee 10 −5 dt ## ##− | | ℓK = =62 10 m ## uε =0 ## 0.1 W L × d ∇ 2 ′′ dim! ε =" = 3 1 F (ζ)dxdy = ν ζ F (ζ)dxdy kg T

dt − | | Griﬃths and Gayen, minor terms s ##Viscous dissipation## scale 2 2 −18 1/4 m vx 1/4u2y = ψ ocean 10 −3 ¯ ¯ 6 3 vx uy = ψ ℓ b(0) b( H) ν K = =6 10 m − − ∇ ∇ ( z) buoyancy equation10−9 × wb = κ − − 10 d ℓK = ⟨ − × ! ⟩⇒⟨" ⟩ H 1 2 ε 2 2 ⇡ ! " ∇ vx uy = ψ ψ dxd− y =∇ ν ζ ddxdy1 2 ∇ 2 2 2 (Assuming2 ζ dxdy your= ν wrist outputsζ dxdy A.K.A. the t water dt | | v u − ψ d ¯ − ¯ | | ⌫ = b(0) b( H) d Kolmogorov1 2 x scale y 2 0.01W## to stir 0.1kg## of coffee.) 2 u dx = u·f dx ν ω dx wb = κ − − (3) ## dt | | − − ∇| ##| d ⟨ ⟩ H 2 ′′ #V u =0#V #V F (ζ)dxdy = ν ∇ζ F (ζ)dxdy 1 4 dt − | | (4) −18 2 / lim ν ω dx =0 ## ##2 →0 u coﬀee 10ν V | | ̸ −4 wb = ν 2 (5) d 1 2 # ⟨2 ⟩ ⟨|| ∇ || ⟩ u dx = u·f dx ν ω dvxx uy = ψ ℓKd =12 2 −2 =102 m − ∇ t ∇∇ ·∇ ∇ 2 d V 2 | ψ| ρd10uxtd+yu= u νV+ p =ζµ duxdy − · V | | 2 dt ## | ×|$ − ## ∇ ' u momentum equation 2 ν u = wb # ! #" ⟨ # vx ⟩⇒uy = ψ ⟨|| ∇ || ⟩ ⟨ ⟩ % & − ∇ def = ε 1 2 2 d 1 2 2 d 1 2 lim νW ω L2 dx =0 u dx = u¯·f dx ν & ω'( dx) ∇ 2 wb κbz =0 ζ dxdνy→=0 ν ζ dxdy dt V | | −V − V | | dt ## 2dim ε =− ##V |=|| 3 ̸ # # # # T 2 3 kg lim ν ω∆bLdx =0 ν→0 Ra =| | ̸ #V νκ d 1 4 ∇ 2 ′′ F (ζ)dxd−y 18= ν / ζ F (ζ)dx 1 oceandt ## 10 − ## | 1| −3 1 ℓK = −9 =5.56 10 m 10 2 × ! vx u"y = ψ − ∇ d 1 2 ∇ 2 2 ζ dxdy = 2 ν ζ dxdy dt vx uy = −ψ | | ## − ∇ ## d F (ζ)dxdy = ν ∇ζ 2F ′′(ζ)dxdy dt − | | ## ## v u = 2ψ x − y ∇

v u = 2ψ x − y ∇ d 1 u 2 dx = u·f dx ν ω 2 dx dt 2 | | − | | #V #V #V lim ν ω 2 dx =0 ν→0 | | ̸ #V

2 ∇ ρ ut + u·∇u + ∇p = µ u × $ ∇ ' % & 1 Two examples of ﬂows that can never be turbulent according to the zeroth law.

Horizontal Convection and Two-dimensional “turbulence”

Heat In Heat Out DEEP-SEA RESEARCH PART I z=0 ϑ PERGAMON Deep-Sea Research I 45 (1998) 1977-20101 ϑ2 “One of the striking features of the ϑ3 oceanic circulation is the smallness at n^ the ocean surface of the regions whereAbyssal recipesn^ II: ϑ4 deep and bottom water is formed.” DEEP-SEA RESEARCH JS • n^=0 energetics of tidal and wind mixingPART I — Stommel (1962) Jϑ • n^=0 PERGAMON Deep-Sea Research I 45 (1998) 1977-2010 Walter Munk 3 ·*, Carl Wunschb

• University of California-San Diego, Scripps Inst of Oceanography, 9500 Gilman Drive, La Jolla, AbyssalCA 92093-0225, recipes USA II: b Massachusetts Institute of Technology, USA Receivedenergetics 7 November 1997; receivedof tidal in revised and form 13 wind April 1998; mixing accepted 29 June 1998

3 Abstract Walter Munk ·*, Carl Wunschb

Without• University deep of California-Sanmixing, the oceanDiego, wouldScripps Instturn, of within Oceanography, a few thousand9500 Gilman years, Drive, into La Jolla,a stagnant pool of cold salty water with equilibriumCA 92093-0225, maintained USA locally by near-surface mixing and with very weak convectively drivenb Massachusetts surface-intensified Institute of circulation. Technology, USA(This result follows from Sand- strom'sReceived theorem 7 November for a fluid 1997; heated received and cooledin revised at formthe surface.) 13 April In1998; this accepted context 29we June revisit 1998 the 1966 "Abyssal Recipes", which called for a diapycnal diffusivity of 10-4 m2/s (1 cgs) to maintain the abyssal stratification against global upwelling associated with 25 Sverdrups of deep water Abstractformation. Subsequent microstructureAnd the measurements mysterious gave a pelagic diffusivity (away from topography) of 10- 5 m2 /s - a low value confirmed by dye release experiments. WithoutA new solution deep mixing, (without the restriction oceanSandstrom would to constant turn, withinTheorem coefficients) a few thousand leads to approximately years, into a stagnant the same poolvalues of coldof global salty waterupwelling with equilibriumand diffusivity, maintained but we locallyreinterpret by near-surface the computed mixing diffusivity and with as verya surrogate weak convectively for a small driven number surface-intensified of concentrated circulation.sources of buoyancy (This result flux follows (regions from of intenseSand- strom'smixing) theorem from which for a fluidthe waterheated masses and cooled (but notat the the surface.) turbulence) In this are context exported we revisit into thethe ocean1966 "Abyssalinterior. Recipes",Using the whichLevitus called climatology for a diapycnal we find thatdiffusivity 2.1 TW of (terawatts) 10-4 m2/s are (1 cgs)required to maintain to maintain the abyssalthe global stratification abyssal density against distribution global upwelling against 30associated Sverdrups with of deep25 Sverdrups water formation. of deep water formation.The winds Subsequent and tides microstructureare the only possible measurements source of gavemechanical a pelagic energy diffusivity to drive (away the interior from 5 2 topography)mixing. Tidal of dissipation10- m /s -is aknown low value from confirmed astronomy by to dye equal release 3.7 experiments.TW (2.50 ± 0.05 TW from MA 2 newalone), solution but nearly (without all of restriction this has traditionally to constant coefficients)been allocated leads to todissipation approximately in the theturbulent same valuesbottom of boundary global upwelling layers of marginaland diffusivity, seas. However, but we reinterprettwo recent TOPEX/POSEIDONthe computed diffusivity altime- as a tricsurrogate estimates for combineda small number with dynamical of concentrated models suggestsources thatof buoyancy 0.6-0.9 TW flux may(regions be available of intense for mixing)abyssal frommixing. which A recent the water estimate masses of wind-driving (but not the suggests turbulence) 1 TW are of exportedadditional into mixing the oceanpower. interior.All values Using are thevery Levitus uncertain. climatology we find that 2.1 TW (terawatts) are required to maintain the Aglobal surprising abyssal conclusion density distribution is that the equator-to-pole against 30 Sverdrups heat flux of ofdeep 2000 water TW formation.associated with the meridionalThe winds overturning and tides are circulation the only wouldpossible not source exist withoutof mechanical the comparatively energy to drive minute the mechani-interior mixing.cal mixing Tidal sources. dissipation Coupled is known with the from findings astronomy that mixing to equal occurs 3.7 at TW a few (2.50 dominant ± 0.05 TWsites, fromthere M 2 alone), but nearly all of this has traditionally been allocated to dissipation in the turbulent bottom boundary layers of marginal seas. However, two recent TOPEX/POSEIDON altime- tric estimates combined with dynamical models suggest that 0.6-0.9 TW may be available for abyssal*Corresponding mixing. A author.recent estimateFax: 001 619of wind-driving534 6251; e-mail: suggests [email protected]. 1 TW of additional mixing power. All values are very uncertain. A surprising conclusion is that the equator-to-pole heat flux of 2000 TW associated with the 0967-0637/98/$-see front matter 'D 1998 Elsevier Science Ltd. All rights reserved. meridionalPII: S 0 9 6 overturning7 - 0 6 3 7 ( 9 8) circulation 0 0 0 7 0 - 3 would not exist without the comparatively minute mechanical mixing sources. Coupled with the findings that mixing occurs at a few dominant sites, there

*Corresponding author. Fax: 001 619 534 6251; e-mail: [email protected].

0967-0637/98/$-see front matter 'D 1998 Elsevier Science Ltd. All rights reserved. PII: S 0 9 6 7 - 0 6 3 7 ( 9 8) 0 0 0 7 0 - 3 U 3 ε RMS Energy ∼ L Energy 1 4 ν3 / Forming the combination ℓK = Forming the combination ε α1ψ1 × (1)+α2ψ2 × (2) , ! " (20) α1ψ1 × (1)+α2ψ2 × (2) , (20) ! " u =0 implies Horizontal Convection! " implies dE 2 Convection driven by heating and cooling at a 2single2 horizontald 2 1 surface∇2 2 2 eq43 dE = −α ψ ηx U − α µ2 ∇| ψ ψdx|2dy ,= ν ζ dxdy (21) dt = −α2 ψ2ηx U d−t α2µ| |∇|ψ2| , − (21) eq43 ′ ## ## g dt ! " ! " where the total eddy energy is ! " ! −18 "1/4 coffee 10 −5 where the total eddy energy is ℓK = =3 10 m 1 × def ! " 12 1/5 2 2 2 2 Edefνκ=L α1|∇ψ1|− + α2|∇ψ2| + α1α2κ (ψ1 − ψ2) . (22) eq47 ∴ δ 12 =∇L Ra2 1/5 ∇ 2 (11)2 2 ∼E =g′2 #α1| ψ1| + α2| ψ2| + α1α2κ (ψ1 − ψ2) $. ¯b(0) ¯b( H) (22) eq47 ! #" ( z) buoyancy equation $wb = κ − − ⟨ − × ⟩⇒⟨⟩ H On the other hand U (9)is and (12) On the other hand U (9)isκ3Lg′2 1/5 κ 2/5 · 2 U 2 =2 Ra u momentum equation(13) ν u = wb ∼ ν 1 dU L ⟨ 2 ⟩⇒ ⟨|| ∇ || ⟩ ⟨ ⟩ ! " 2 def 12dU = UF − α2µU2 + α2⟨ψ2ηx⟩U. = ε (23) eq53 2 dt = UF − α2µU + α2⟨ψ2ηx⟩U. (23) eq53 dt $ %& ' wb κ¯bz =0 The sum of the relations above is − The sum of the relations above is W L2 d 1 2 2 dim ε = 2= 3 Paparella & Young, (2002) JFM Young, & Paparella d E +12 U2 = UF − α2µ U2 + ⟨|∇ψkg2|2 T. (24) eq57 dt E + 2 U = UF − α2µ U + ⟨|∇ψ2| . (24) eq57 dt % & % −18 1/4 & % & ocean% 10 & −3 ℓK = −9 =6 10 m −1 10 × ρ = ρ0 1 − g−1 b(x,t)! " (25) 0 x The definition of “buoyancy’’: ρ = ρ 1 − g d b( ,t1 )2 ∇ 2 (25) % 2 ζ d&xdy = ν ζ dxdy dt − | | % ## & ## Conventional notation: b = gαd(T − T0) b = gα (T − TF0(ζ))dxdy = ν ∇ζ 2F ′′(ζ)dxdy dt − | | The “conversion” term is ## ## The “conversion” term is def 2 Note an important difference fromC def= RBCα2⟨:ψ2ηx⟩U. vx uy = ψ (26) the zero-flux constraint.C = α2⟨ψ2ηx⟩U. − ∇ (26) 2 Enstrophy vx uy = ψ Enstrophy − ∇ The top-layer time-mean enstrophy power integral is 1 The top-layer time-mean enstrophy power integral is 2 βJ¯1 = −⟨q1Dq1⟩ = −ν⟨(△q1)2 ⟩ . (27) eq58 βJ¯1 = −⟨q1Dq1⟩ = −ν⟨(△q1) ⟩ . (27) eq58 The lower-layer time-mean enstrophy power integral is The lower-layer time-mean enstrophy power integral is 2 2 βJ¯2 = −µ⟨q2△ψ2⟩−ν⟨(△q2)2 ⟩ + ν⟨q2△2 η⟩ . (28) βJ¯2 = −µ⟨q2△ψ2⟩−ν⟨(△q2) ⟩ + ν⟨q2△ η⟩ . (28) The upper layer is simple — the4 lower layer less so. The upper layer is simple — the lower layer less so. For β =0thetop-layerenstrophybalance(27)impliesthatq1 =0orequivalentlythat: For β =0thetop-layerenstrophybalance(27)impliesthatq1 =0orequivalentlythat: 2 ik·x α2κ2 ψˆ2(k)eik·x 2 ˆ2 k ψ1 = α κ ψ2 ( )e 2 . (29) ψ1 = |k|2 + α2κ2 . (29) 'k |k| + α2κ 'k

33 3 URMS ε ∼ L

1 4 ν3 / ℓK = ε ! " u =0

d 1 ∇ψ 2 dxdy = ν ζ2 dxdy dt 2 | | − ## ## −18 1/4 coﬀee 10 −5 ℓK = =3 10 m 1 × ! "

¯b(0) ¯b( H) ( z) buoyancy equation wb = κ − − ⟨ − × ⟩⇒⟨⟩ H u · momentum equation ν u 2 = wb ⟨ ⟩⇒ ⟨|| ∇ || ⟩ ⟨ ⟩ def = ε

wb κ¯b =0 $ %& ' − z ∆bL3 2D numerical HC solutions. Fluid sinks in the Ra = center. The circulation is weaker and the νκ 4JEFXBZT $POWFDUJPO thermocline thinner at higher Rayleigh number.

¯b(0) ¯b Density( H) κ∆b(0) Streamfunction ∴ ε = κ − − , (1) H ≤ H 0 as ν 0 with ν/κ ﬁxed. (2) → → z (m) 2 Vallis (2017) W L dim ε = = DEEP-SEA RESEARCH 3 PART I Ra = 10 6 kg T PERGAMON Deep-Sea Research I 45 (1998) 1977-2010 −18 1/4 ocean 10 −3 ℓK = =6 10 m 10−9 × Abyssal recipes II: ! " energetics of tidal and wind mixing

3 z (m) 1 Walter Munk ·*, Carl Wunschb • University of California-San Diego, Scripps Inst of Oceanography, 9500 Gilman Drive, La Jolla, Ra 9 CA 92093-0225, USA = 10 b Massachusetts Institute of Technology, USA Received 7 November 1997; received in revised form 13 April 1998; accepted 29 June 1998 x (m) x (m) Abstract

'JH AtʾF very EFOTJUZ high BOE Ra TUSFBNGVODUJPO the box ﬁlls JO UXP OVNFSJDBMWithout deep TJNVMBUJPOT mixing, the ocean PG UXPEJNFOTJPOBM would turn, within a few TJEFXBZT thousand years, into a stagnant DPOWFDUJPO with XJUI 3BZMFJHIthe densest OVNCFST available PG UPQ BOE pool CPUUPN of cold salty water ʾF with JNQPTFE equilibrium UFNQFSBUVSF maintained locally BU UIF by near-surface UPQ mixing and with MJOFBSMZ EFDSFBTFT GSPN UIF DFOUSF PVUXBSE UIF TJEF BOEvery CPUUPN weak convectively XBMMT BSF driven JOTVMBUJOH surface-intensified BOE UIF circulation. 1SBOEUM (This OVN result follows from Sand- water. (Unlike RBC.) strom's theorem for a fluid heated and cooled at the surface.) In this context we revisit the 1966 CFS JT ʾF UXP EFOTJUZ QMPUT VTF UIF TBNF DPMPVSNBQ CVU UIF TUSFBNGVODUJPO QMPUT EP OPU ʾFSF4 JT2 "Abyssal Recipes", which called for a diapycnal— Munkdiffusivity & of Wunsch 10- m /s (1(1998) cgs) to maintain the B TJOLJOH QMVNF BU UIF DFOUSF XJUI B XFBLFS DJSDVMBUJPOabyssal BOE stratification B UIJOOFS UIFSNPDMJOFagainst global upwelling BU UIF associated IJHIFS 3BZMFJHIwith 25 Sverdrups of deep water formation. Subsequent microstructure measurements gave a pelagic diffusivity (away from OVNCFS topography) of 10- 5 m2 /s - a low value confirmed by dye release experiments. A new solution (without restriction to constant coefficients) leads to approximately the same values of global upwelling and diffusivity, but we reinterpret the computed diffusivity as a surrogate for a small number of concentrated sources of buoyancy flux (regions of intense JG JU JT MBSHFS UIFSF JT OPUIJOH UP CBMBODF JU 0VS EFNBOEmixing) DBOfrom CFwhich TBUJTĕFE the water POMZmasses JG(but UIF not WFSUJDBM the turbulence) TDBMF are PG exported into the ocean UIF NPUJPO BEKVTUT BQQSPQSJBUFMZ BOE GPS � interior. UIJT TVHHFTUTUsing the Levitus UIF climatology TDBMJOHT we find that 2.1 TW (terawatts) are required to maintain the global abyssal density distribution against 30 Sverdrups of deep water formation. The winds and tides are the only possible source of mechanical energy to drive the interior mixing. Tidal dissipation is known from astronomy to equal 3.7 TW (2.50 ± 0.05 TW from ૥ 3B M3B 2 alone), but nearly all of this has traditionally been allocated B C to dissipation in the turbulent bottom boundary layers of marginal seas. However, two recent TOPEX/POSEIDON altime- ÷ ÷ ૜ ৴ tric estimates ÷ combined with dynamical models suggest that 0.6-0.9 TW may be available for ćF WFSUJDBMৰ TDBMF ৴૥ BSJTFT BTʔ B DPOTFRVFODFʢ ૐ PG UIF BOBMZTJT abyssal mixing.૥ BOE A૟ UIFrecent WFSUJDBM ૜ estimate৴ ઼਄ of TJ[F wind-driving PG UIF EPNBJOsuggests 1 TW QMBZT of additional mixing power. ઼਄ All values are very uncertain. OP EJSFDU SPMF <'PS XF NJHIU FYQFDU UIF OPOMJOFBSA surprising UFSNT conclusion UP is CF that TNBMM the equator-to-pole BOE JG UIF heat CVPZBODZ flux of 2000 TW associated with the meridional overturning circulation would not exist without the comparatively minute mechani- UFSN CBMBODFT UIFৰ WJTDPVT UFSN JO UIF SJHIUIBOEcal mixing sources. TJEFT Coupled PG with the BSFfindings NVMUJQMJFE that mixing occurs CZ at a few dominant sites, there BOE 'PS TFBXBUFS ૥մ VTJOH UIF NPMFDVMBS WBMVFT PG BOE *G TNBMM TDBMF UVSCVMFODF FYJTUT ÷ UIFO UIF FEEZ WJTDPTJUZ XJMM MJLFMZ CF TJNJMBS UP UIF FEEZ EJČVTJWJUZ BOE > /VNFSJDBM૥ FYQFSJNFOUT૥ BO FYBNQMF૥Ă JT TIPXO JO 'JH QSPWJEF*Corresponding TVQQPSU author.૜ GPS Fax: UIF ૟001 619 TDBMJOH 534 6251; PGe-mail: [email protected]. BOE B

GFX TJNQMF BOE SPCVTU QPJOUT UIBU IBWF SFMFWBODF UP0967-0637/98/$-see UIF SFBM PDFBO front matter FNFSHF 'D 1998 Elsevier૥Ă Science Ltd. All rights reserved. PII: S 0 9 6 7 - 0 6 3 7 ( 9 8) 0 0 0 7 0 - 3 .PTU PG UIF CPY ĕMMT VQ XJUI UIF EFOTFTU BWBJMBCMF ĘVJE XJUI B CPVOEBSZ MBZFS JO UFNQFSBUVSF OFBS UIF TVSGBDF SFRVJSFE JO PSEFS UP TBUJTGZ UIF UPQ CPVOEBSZ DPOEJUJPO ćF CPVOEBSZ HFUT ¹ UIJOOFS XJUI EFDSFBTJOH EJČVTJWJUZ DPOTJTUFOU XJUI ćJT JT B EJČVTJWF QSPUPUZQF PG UIF PDFBOJD UIFSNPDMJOF ćF IPSJ[POUBM TDBMF PG UIF PWFSUVSOJOH DJSDVMBUJPO JT MBSHF OFBSMZ UIF TDBMF PG UIF CPY ćF EPXOXFMMJOH SFHJPOT UIF SFHJPOT PG DPOWFDUJPO BSF PG TNBMMFS IPSJ[POUBM TDBMF UIBO UIF ¹ VQXFMMJOH SFHJPOT FTQFDJBMMZ BT UIF 3BZMFJHI OVNCFS JODSFBTFT -FU¹ VT OPX USZ UP FYQMBJO TPNF PG UIF GFBUVSFT JO B TJNQMF BOE IFVSJTUJD XBZ CFHJOOJOH XJUI UIF TDBMF PG UIF NPUJPO ANRV332-FL40-09 ARI 10 November 2007 16:16

HC in the laboratory a Hughes & Grifﬁths, (2008) ARFM (the bottom is non-uniformly heated) H

Destabilizing Stabilizing buoyancy input buoyancy input HC “explains” the asymmetry between sinking and rising b regions in the ocean. (Unlike RBC.)

The box ﬁlls with the lightest available water. (Unlike RBC.)

N.B. non-uniform buoyancy at the bottom! The horizontal scale of the Figure 1 overturning is set by the box. Horizontal convection in a thermally equilibrated laboratory experiment subject to heating and cooling that depends on position along the base of the box (the rate of supply of specific (Unlike RBC.) buoyancy is 1.6 10 6 m3s 3 per unit width in the spanwise direction, using a uniform × − − imposed heat flux at the left-hand end and an imposed boundary temperature of 16◦C at the right-hand end) at a Rayleigh number Ra 2.2 1014 and Prandtl number Pr 4. Passive B = × ≈ dye tracer is introduced into the bottom boundary layer halfway along the tank to visualize the circulation. Panel a shows the full box, whereas panel b is a close-up (approximately one-fourth the tank length) of the left-hand end, showing an asymmetric clockwise circulation extending The critical Rayleigh number through the depth of the box, with a convective mixed layer embedded in a stably stratified boundary layer on the base, an entraining plume against the vertical end wall, and eddies in the is zero. (Unlike RBC.) horizontal outflow from the plume. Figure 1b taken from Mullarney et al. 2004, courtesy of Cambridge University Press.

by AUSTRALIAN NATIONAL UNIVERSITY on 01/15/08. For personal use only. Paparella & Young 2002, Rossby 1998), in which heating and cooling at the same

Annu. Rev. Fluid Mech. 2008.40:185-208. Downloaded from arjournals.annualreviews.org level clearly support a substantial and sustained circulation in the box (Figure 1). Jeffreys (1925) argued on physical grounds that a horizontal density gradient, maintained by heat diffusion from the coils, must lead to a persistent circulation in the tank, which is therefore inconsistent with Sandstrom’s¨ conclusion. However, Jeffreys was unable to predict the magnitude of the horizontal convection, leaving other authors (Defant 1961; Huang 1999, 2004) to suggest that a circulation maintained by diffusion is very weak. Sandstrom¨ himself argued that his experimental conclusion does not apply to the ocean, in which he expected a convective circulation extending to the depth of the mixing from the surface at low latitudes, but in recent years the conclusion has been extrapolated and widely cited in a debate as discounting any

www.annualreviews.org Horizontal Convection 187 • U 3 ε RMS ∼ L 1 4 ν3 / ℓK = ε ! " u =0

d 1 2 2 2 ∇ψ dxdy = ν ζ dxdy ANRV332-FL40-09 ARI 10 November 2007dt 16:16| | − ## ## −18 1/4 coﬀee 10 −5 ℓK = =3 10 m 1 × ! "

¯b(0) ¯b( H) Why are( z) thebuoyancy sinking equation regions wbin= HCκ −so− small? ⟨ − × ⟩⇒⟨⟩ H

u · momentum equation L ν u 2 = wb ⟨ ⟩⇒ ⟨|| ∇ || ⟩ ⟨ ⟩ def a = ε ¯ $ %& ' wb κbz =0 H − the plume W L2 dim ε = = T 3 Destabilizing kg Stabilizing buoyancy input −18 1/4 buoyancy input ocean 10 −3 ℓK = =6 10 m 10−9 × b ! " The short answer:d 1upwards2 buoyancy flux2 in the plume ζ dxdy = ν ∇ζ dxdy is much moredt efficient2 than− downwards| | diffusive ## ## buoyancyd in the interior. But these fluxes cancel. Thus F (ζ)dxdy = ν ∇ζ 2F ′′(ζ)dxdy the convectivedt plumes must− cover| a small| area leaving most of the## box for inefficient## diffusive downwelling. v u = 2ψ x − y ∇

v u = 2ψ x − y ∇ “One of the striking1 features of the oceanic circulation is the smallness at the ocean surface of the regions where deep and bottom water is formed.” — Stommel (1962) Figure 1 Horizontal convection in a thermally equilibrated laboratory experiment subject to heating and cooling that depends on position along the base of the box (the rate of supply of specific buoyancy is 1.6 10 6 m3s 3 per unit width in the spanwise direction, using a uniform × − − imposed heat flux at the left-hand end and an imposed boundary temperature of 16◦C at the right-hand end) at a Rayleigh number Ra 2.2 1014 and Prandtl number Pr 4. Passive B = × ≈ dye tracer is introduced into the bottom boundary layer halfway along the tank to visualize the circulation. Panel a shows the full box, whereas panel b is a close-up (approximately one-fourth the tank length) of the left-hand end, showing an asymmetric clockwise circulation extending through the depth of the box, with a convective mixed layer embedded in a stably stratified boundary layer on the base, an entraining plume against the vertical end wall, and eddies in the horizontal outflow from the plume. Figure 1b taken from Mullarney et al. 2004, courtesy of Cambridge University Press.

by AUSTRALIAN NATIONAL UNIVERSITY on 01/15/08. For personal use only. Paparella & Young 2002, Rossby 1998), in which heating and cooling at the same

www.annualreviews.org Horizontal Convection 187 • Now show that HC cannot satisfy the zeroth law of turbulence.

Turbulence - the Legacy of A.N. Kolmogorov Uriel Frisch (1996) 3 URMS ε Horizontal∼ 3 convectionL is non-turbulent 207 URMS ε 1 4 2. Formulation ∼ Lν3 / We consider a three-dimensionalℓK = rotating fluid (Coriolis frequency f) in a rectan- 3ε 1/4 gular box. The vertical coordinate!ν is "H (2.1) ! 1 " =× b, > (Boussinesq) fluid motion. ! "u =0Dt r => u =0. ¯ ¯ r · ¯ b(0)¯> b( H) ( z) buoyancyd equation1 ∇ 2 wb2 =b(0)κ >b(−H)− ( z) buoyancy equation2 ψ dxdy = ν wbζ =dxκdy − >− ⟨ The−⟨ − boundary×× conditionsdt | on⟩⇒⟨|⟩⇒⟨ the velocity− u =(⟩ u,⟩ v, w) areH;u Hnˆ = 0, where nˆ is the outward normal, and## some combination## of no slip and no· stress. We specify the −18 1/4 buoyancy on the top;coffee an illustrative10 example used−5 below2 2 is uu· · ℓK = =3 ν10 mu u wb momentummomentum equation1 × ν = = wb Power integrals ⟨ ⟨ ! ⟩⇒⟩⇒" ⟨|| ∇⟨2|| ∇|| ⟩ || ⟩⟨ ⟩⟨ ⟩ b(x, y, 0,t)=bmax sindef(⇡defy/L), (2.2) = ε = ε for HC. 2 ¯ ¯ with L/2 < y < L/2. W L $ %& b'(0) b( H) ( z) buoyancy equationdimwbε = κ¯b ==0 wb =$ κ%& −' − We⟨− use an× overbar to denote⟩⇒⟨− thekg horizontalz T 3 average⟩ overH x and y. If the solution is unsteady we also include a time average in the overbar. In any event, we assume that −18 1/4 2 with suucient· momentum timeocean and equation space10 averaging, all overbarredν−3 u = fieldswb are steady. Thus b¯ is a ⟨ ℓK = ¯ −9 ⟩⇒¯ =6 10 ⟨|| ∇m || ⟩ ⟨ ⟩ function only of z. It followsb(0)10 fromb( (2.1),H) × andκ∆ fromb(0)def the no-flux condition at z = H, Combine the ∴ = ε that ε = κ ! − " − , (1) H ≤2 H power integrals. d 1 2 W L∇ 2$ %& ' 20ζ dxdimdasy ε=ν= ν 0= with3¯ ζν/dκxdfixed.y (2) dt −kgwb T|bz =0| . (2.3) ##→ →## Thus there is no net vertical buoyancy−18 1/4 flux through every level z = constant. d 2 ′′ ocean 10 ∇ 2 −3 To measure theFℓ strength(Kζ)d=xdy of= horizontalν =6ζ convection10F (ζm)dxd wey must define a suitable non- dt 10−−9 W | L×| ∴ HC doesdimensional not satisfy measure. the The zerothdim familiarε = law Nusselt= of number turbulence. of the Rayleigh–Benard´ problem is ## ! "kg## T 3 2 not useful becaused from1 (2.3)v2 theu heat fluxψ is∇ zero.2 Instead we construct an alternative 2 ζ xdxdyy== ν ζ dxdy index of the intensitydt of convection−−18 −1/∇ by4 first| solving| the conduction problem ocean## 10 ## −3 ℓK = =6 10 m d −9 2 2 2 ′′ F (ζ)dvxx10dy =uy =ν ψ∇c =0ζ×F, (ζ)dxdy (2.4) dt ! − −" ∇r| | ## ## using the samed boundary1 2 conditions on c as on b.2 Then we define the non-dimensional ζ dxdy = ν 2 ∇ζ dxdy functional 2 vx uy = ψ dt − − ∇ | | ## 1 ## 2 b b dV vx uy = ψ −[b] ∇Z r · r , (2.5) 1 ⌘ c c dV Z r · r where the integral is over the volume1 of the box. In physical terms is the ratio of entropy production in the convecting flow to the entropy production of the conductive solution. One can show using a standard argument that is always greater than unity. In the numerical simulations of 4 we use as a global measure of the strength of horizontal convection. § A new beginning. Cesar’s movie

2 kd(U1 U2) µ − =2 =??? 2πL =25λd β Ukd

1 2 1 2 G1 = β + k (U1 U2) ,G2 = β + k (U2 U1) , (22) 2 d − 2 d − =3β , = β . (23) A new beginning. Cesar’s movie − 2 kd(U1 U2) µ − =2 =??? 2πL =25λd β Ukd ∆T =25K, α =2 10−4K−1 , (24) −2 × −2 −2 g 1=10ms2 , bmax =51 2 10 ms (25) G1 = β + k (U1 U2) ,G⇒ 2 = β + k (×U2 U1) , (22) 2 d − 2 d − =3β , = β . (23) − Sandstrom’s (1908) “theorem” κ b ε max “A closed steady(26) circulation can only be maintained if ∆T =25K, α =2≤ 10h −4K−1 , (24) −7 −2 the heat source is below the closed source.” −2 10× 5 10 −2 −2 g =10ms , = bmax×=5× 10 ms (25) (27) ⇒ 5000× −12 −1 So why don’t oceanographers =10 Wkg (28)DEEP-SEA RESEARCH PART I abandon Sandstrom? κ bmax ε PERGAMON Deep-Sea Research I 45 (1998) 1977-2010(26) ≤ h 10−7 5b 10−ψ2 Paparella’s counterexample = × × (27) 5000 Abyssal recipes II: “HC isn’t a vigorous ﬂow” −12 −1 =10 Wkg energetics of tidal and(28) wind mixing

Walter Munk 3 ·*, Carl Wunschb b ψ “HC produces a thin thermocline • University of California-San Diego, Scripps Inst of Oceanography, 9500 Gilman Drive, La Jolla, — you need winds, tides and CA 92093-0225, USA b Massachusetts Institute of Technology, USA breaking IGWs to explain deep There are recent endorsements of the “theorem”: Munk & Wunsch Received 7 November 1997; received in revised form 13 April 1998; accepted 29 June 1998 (1998), Huang (1999), Emmanuel (2001), Wunsch & Ferrari (2004) etc ocean stratiﬁcation”

Abstract “Strict interpretation of the Without deep mixing, the ocean would turn, within a few thousand years, into a stagnant pool of cold salty water with equilibrium maintained locally by near-surface mixing and with theorem is difﬁcult” very weak convectively driven surface-intensified circulation. (This result follows from Sand- strom's theorem for a fluid heated and cooled at the surface.) In this context we revisit the 1966 — Houghton (1977) "Abyssal Recipes", which called for a diapycnal diffusivity of 10-4 m2/s (1 cgs) to maintain the abyssal stratification against global upwelling associated with 25 Sverdrups of deep water formation.Also Subsequent many counterexamples: microstructure measurements Jeffreys gave(1925), a pelagic Rossby diffusivity (1965), (away from 5 2 topography) of 10-Mullarneym /s - aet low al. value (2004), confirmed Wang by & dye Huang release (2005). experiments. A new solution (without restriction to constant coefficients) leads to approximately the same values of global upwelling and diffusivity, but we reinterpret the computed diffusivity as a surrogate for a small number of concentrated sources of buoyancy flux (regions of intense mixing) from which the water masses (but not the turbulence) are exported into the ocean interior. Using6 the Levitus climatology we find that 2.1 TW (terawatts) are required to maintain the global abyssal density distribution against 30 Sverdrups of deep water formation. The winds and tides are the only possible source of mechanical energy to drive the interior mixing.6 Tidal dissipation is known from astronomy to equal 3.7 TW (2.50 ± 0.05 TW from M 2 alone), but nearly all of this has traditionally been allocated to dissipation in the turbulent bottom boundary layers of marginal seas. However, two recent TOPEX/POSEIDON altime- tric estimates combined with dynamical models suggest that 0.6-0.9 TW may be available for abyssal mixing. A recent estimate of wind-driving suggests 1 TW of additional mixing power. All values are very uncertain. A surprising conclusion is that the equator-to-pole heat flux of 2000 TW associated with the meridional overturning circulation would not exist without the comparatively minute mechanical mixing sources. Coupled with the findings that mixing occurs at a few dominant sites, there

*Corresponding author. Fax: 001 619 534 6251; e-mail: [email protected].

0967-0637/98/$-see front matter 'D 1998 Elsevier Science Ltd. All rights reserved. PII: S 0 9 6 7 - 0 6 3 7 ( 9 8) 0 0 0 7 0 - 3 To turn the “theorem” into a ¯ ¯ theorem, and strictly interpret it, ∇ 2 b(0) b( H) use this formula ν u = κ − − 13 Dec 2003 18:52⟨ AR| AR203-FL36-12.tex| ⟩ AR203-FL36-12.sgmH LaTeX2e(2002/01/18) P1: IBD ε But now the debate 3 focusses on whether � is ! "# $ URMS A new beginning. Cesar’s movie ε important for ocean ∼ L 2 302 WUNSCH ⌅ FERRARI Ak newd(U1 beginning.Ucirculation.2) Cesar’s The movieµ majority − =2 =??? 2πL =25λd β 2 position Ukis atd right. 1/4 kd(U1 U2) µ 3 − =2 =??? 2πL =25λd ν β Ukd ℓK = 1 2 1 2 ε G1 = β + kd(U1 U2) ,G2 = β + kd(U2 U1) , (22) 2 1 2 2 1 2 % & G1 = β + −k (U1 U2) ,G2 = β + k (U−2 U1) , (22) =3β , 2 d − = β . 2 d − (23) =3β , −= β . (23) − 1 2 A ballpark estimate ε = ν [ 2 (ui,j + uj,i)] (1) − − − ∆T =25K, α =2 10 4K 1 ,g−4 −1 =10ms 2 , (24) 2 2 2 2 ∆T =25K, α =2 10 K , (24)= ν ux + uy + uz + + wz (2) −2 × × −2 −−22 −2 ' ( g =10ms , bmax =5bmax10=5ms10 ms (25)(25) ⟨ ··· ⟩ ⇒ ⇒ × × Yet again κ bmax ¯ κ bε (26) κ ∆b(0) ε max≤ h (26) ε ≤ h 10−7 5 10−2 −=7 × ×−2 (27) ≤ H 10 5 500010 = × −12× −1 (27) =105000Wkg (28) −12 −1 d =10 Wkg Figure 5(28)Strawman1 ∇ energy2 budget for the global ocean circulation,2 with uncertainties of at least factors2 of 2 andψ possiblydx asd largey = as 10. Topν row of boxesζ representdxdy possible energy dsources.t Shaded| boxes are| the principal energy− reservoirs in the ocean, with crude energy values)) given [in exajoules (EJ) 1018 J, and yottajoules)) (YJ) 1024 J]. Fluxes to and from the reservoirs are in terrawatts (TWs).−18 Tidal1/ input4 (see Munk & Wunsch 1998) of 3.5 TW is the onlycoﬀ accurateee number10 here. Total wind work is in the middle− of5 the range estimated by LueckℓK & Reid= (1984); net wind work on the=6 general circulation10 ism from Wunsch (1998). Heating/cooling/evaporation/precipitation0.1 values are all× taken from Huang & Wang (2003). Value for surface waves% and turbulence& is for surface waves alone, as estimated by Lefevre & Cotton (2001). The internal wave energy estimate is by Munk (1981); the internal tide Griﬃths and Gayen,energy estimate minor is from Kantha terms & Tierney (1997); the Wunsch (1975) estimate is four times larger. Oort et al. (1994) estimated the energy of the general circulation. Energy of the mesoscale is from the Zang & Wunsch (2001) spectrum (X. Zang, personal communication,¯ ¯ 2002). Ellipse indicates the conceivable importance of a loss of balance in theb(0) geostrophicb( H)

( z) buoyancyAnnu. Rev. Fluid Mech. 2004.36:281-314. Downloaded from arjournals.annualreviews.org mesoscale, equation resulting in internal waves and mixing, but of unknownwb = importance.κ Dashed-dot− − ⟨ − × lines indicate energy returned⟩⇒⟨ to the general circulation by mixing,⟩ and are ﬁrst multipliedH by 0. Open-ocean mixing by internal waves includes the upper ocean. by UNIVERSITY OF WASHINGTON - HEALTH SCIENCES LIBRARIES on 04/21/08. For personal use only.

such kinetic energy exist, the wind stress and tidal flows. The tides can account 6 for approximately 1 TW, at¯b most.(0) The wind¯b( fieldH provides) approximately 1 TW— directly—towb the large-scale= κ circulation− and− probably at least another 0.5 TW by (3) generating⟨ inertial⟩ waves and the internalH wave continuum. Taken together, Sandström’s (1908, 1916) and Paparella & Young’s (2002) 6 theorems, the very small, probably negative, contribution to oceanic potential (4) energy by buoyancy exchanges with the atmosphere, and the ready availability of wb = ν u 2 (5) ⟨ ⟩ ⟨|| ∇ || ⟩

u · momentum equation ν u 2 = wb ⟨ ⟩⇒ ⟨|| ∇ || ⟩ ⟨ ⟩ def = ε wb κ¯b =0 ! "# $ − z 1 ¯ ¯ 2¯b(0) b¯b(0)( H)b( H) ν ∇uν 2∇=uκ = κ− − − =−κ z 2b Exercise (surprisingly⟨| ⟨||⟩ | ⟩ easy)H H ∇ ε ε ¯ ¯ % & 2 b(0) b3(3 H) 2 ∇u UURMSRMS ν ! "#!= $κ"# $ε − − = κ z b ⟨| | ⟩ ∼HLL ∇ ε ¯ ¯ % & b(0) b¯b((0)H) ¯b( H) 2 Show that P ∇ · PU∇ · κU U−κ3 − − =−κ z b ! "#⟨ $⟨ ⟩∝ε ⟩∝ RMSH H ⟨− ∇ ⟩ ∼ L UU== the the exact, exact, non-Boussinesq non-Boussinesq compressible compressible velocity velocity ¯ ¯ b(0) b( H) 2 P ∇ · U κP =thetotalpressure− − = κ z b ⟨ ⟩∝ P the totalH pressure ⟨− ∇ ⟩

3 1/4 ν 3 1/4 ∇ℓK =· ν ρt + ℓ =(Uερ)=0 K ' ε ( % & 1 2 ε = ν [ (ui,j + uj,i)] (1) U = the exact,ε non-Boussinesq= ν [ 12(u + u )]2 compressible velocity (1) You’ll never doubt= theν u 2 2Boussinesq+i,ju2 + uj,i2 + + w2 (2) %⟨ 2x y2 z2 &··· z ⟩2 approximation= ν uagain.x + uy + uz + + wz (2) Yet again '⟨ (··· ⟩ Yet again P =thetotalpressureκ ∆¯b(0) ε κ ∆¯b(0) ε ≤ H ≤ H Thus the HC energyd source is given by1 /4 1 ∇ψ 2 dxdνy3= ν ζ2 dxdy Sandstorm’s “pistonddt formula”:2 conversion of 1 |ℓ∇Kψ=|2 dxdy =− ν ζ2 dxdy internal energy to)) mechanical2 energy.ε )) dt | | −18 1/4 − ))coffee 10 ' ( )) −5 ℓ = 1 4 =6 10 m K 100−.118 / × ℓcoffee = ' ( =6 10−5 m Griffiths and Gayen,K minor1 terms0.1 2 × ε = ν [ 2 (u%i,j + u&j,i)] (1) Griffiths and Gayen, minor2 terms2 2 2 ¯b(0) ¯b( H) ( z) buoyancy= ν equationux + uy + uz + wb+ w=zκ − − (2) ⟨ − × %⟨ ⟩⇒⟨&··· ⟩ ⟩ ¯b(0)H ¯b( H) ( z) buoyancy equation wb = κ − − Yet again⟨ − × ⟩⇒⟨⟩ H ¯b(0) ¯b( H) wb = κκ ∆¯b(0)− − (3) ⟨ ε⟩ ¯b(0) H¯b( H) wb ≤= κ H − − (4)(3) ⟨ ⟩ H wb = ν u 2 (5) ⟨ ⟩ ⟨|| ∇ || ⟩ (4) d 1 2 2 ∇ψwb d=xνdy =u 2 ν ζ dxdy (5) dt 2 | ⟨ | ⟩ ⟨|| ∇ ||−⟩ )) 1 )) 10−18 11/4 ℓcoffee = =6 10−5 m K 0.1 × ' ( Griffiths and Gayen, minor terms

¯b(0) ¯b( H) ( z) buoyancy equation wb = κ − − ⟨ − × ⟩⇒⟨⟩ H

1 U 3 ε RMS ∼ L

ANRV332-FL40-09 ARI 22 June 2007 20:39 ν3 1/4 ℓ = K ε ! "

1 L 2 ε = ν [ 2 (ui,j + uj,i)] (1) a = ν u2 + u2 + u2 + + w2 (2) #⟨ x y z $··· z ⟩ H But does this look like a laminar flow? Yet again Destabilizing Stabilizing buoyancy input κ ∆¯b(0) buoyancy input ε “We suggest that the “zeroth” law is too restrictive, b ≤ H since according to this strict definition even canonical Rayleigh-Benard convection would not be turbulent!” d 1 ∇ψ 2 dxdy = ν ζ2 dxdy -Scotti & White (2011) dt 2 | | − %% %% 10−18 1/4 ℓcoffee = =3 10−5 m “Horizontal convection can be interpreted in K 1 × terms of a mechanical energy budget, but a ! " detailed understanding has not emerged”Griffiths and Gayen, minor terms -Griffiths & Hughes (2007) Figure 1 Spiegel suggests that examples such as Horizontal convection in a thermally equilibrated laboratory experiment subject to¯b heating(0) ¯b( H) ( z) buoyancyand cooling thatthese equation depends should on position be along called the base “ ofthermalence the box (thewb rate of=”. supplyκ of specific− − buoyancy is 1.6 10 6 m3s 3 per unit width in the spanwise direction, using a uniform ⟨ − × × − − ⟩⇒⟨⟩ H imposed heat flux at the left-hand end and an imposed boundary temperature of 16◦C at the right-hand end) at a Rayleigh number Ra 2.2 1014 and Prandtl number Pr 4. Passive B = × ≈ “These results explain why the convection dye tracer is introduced into the bottom boundary layer halfway along the tank to visualize the circulation. Panel a shows the full box, whereas panel b is a close-up (approximately one-fourth is much stronger than might be inferred the tank length) of the left-hand end, showing¯ an asymmetric¯ clockwise circulation extending through the depth of the box, with a convectiveb(0) mixedb layer( embeddedH) in a stably stratified from previous emphasis on minor terms boundary layer on thewb base, an= entrainingκ plume− against the− vertical end wall, and eddies in the (3) (the buoyancy flux viscous dissipation horizontal outflow from⟨ the⟩ plume. Figure 1b takenH from Mullarney et al. 2004, courtesy of Cambridge University Press. balance and the potential energy budget)” (4) -Gayen, Griffiths, Hughes and Saenz (2007) Paparella & Young 2002, Rossby 1998), in which2 heating and cooling at the same level clearly supportwb a substantial= ν and sustainedu circulation in the box (Figure 1). (5) Jeffreys (1925)⟨ argued⟩ on physical⟨|| ∇ grounds|| ⟩ that a horizontal density gradient, maintained by heat diffusion from the coils, must lead to a persistent circulation in the tank, which is therefore inconsistent with Sandstrom’s¨ conclusion. However, u · momentumJeffreys was unable equation to predict the magnitude of the horizontalν convection,u 2 = leav-wb ⟨ ing other authors (Defant 1961;⟩⇒ Huang 1999, 2004) to suggest⟨|| ∇ that a|| circulation⟩ ⟨ ⟩ maintained by diffusion is very weak. Sandstrom¨ himself argueddef that his experimental conclusion does not apply to the ocean, in which he expected= ε a convective circulation extending to the depth of the mixing from the surface at low latitudes, but in recent years thewb conclusionκ¯b has=0 been extrapolated& and'( widely) cited in − z

www.annualreviews.org Horizontal Convection 187 ∆bL3 • Ra = νκ 1 The second example of ﬂows that do not satisfy the zeroth law is 2D turbulence.

The key feature of 2D turbulence is the robust conservation of energy, and the transfer of energy to large scales. (The inverse energy cascade, negative viscosity, anti-friction etc.)

What would Niagara Falls look like in Flatland? 2 ρ ut + u·∇u +∇p = µ∇ u + f and ∇·u =0 2 ρ ut + u·∇u +∇p = µ∇ u + f and ∇·u =0 = Du ! Dt " = Du #! $%Dt &" d # $% & 1 2 2 · 2 + ·∇ +∇ = 2 |u| d+x = u f dandx − ν |ω| d∇x· =0 ρ ut u u dpd µ1∇ u2 f 2 u t V |u| dx = V u·f dx − ν V |ω| dx d ' 2 ' ' = Du t V V V ! DThet " singular' '3D limit' results # $% & ω = ∇ × u d 1 2 ω = ∇ × u 2 2 |u| dx = u·f dx − ν |ω| dx dfromt V vorticityV productionV ' ' ' lim ν |ω|2 dx ̸=0 ν→0 V 2 lim ν' |ω| dx ̸=0 ω =ν→∇0 Vu ×' ∇ ·∇ ∇ 2 × ρ ut + u u + p = µ∇ 2u ∇ + 2·∇ + ∇ = ×(limρ uνt |ωu| dxu̸=0 p µ∇ u) ν→0 V ( ! ' " ) ! " ·∇ ·∇ 2 ωt + u ω = ω u + ν∇ 2ω ωt + u·∇ω = ω·∇u + ν∇2 ω ∇ × ρ ut + u·∇u + ∇p = µ∇ u ( ) ! " vortex stretching2 ωt + u·∇ω = ω·∇u + ν∇ ω But in 2D ˆ u = uˆı + vȷωˆ =(vx − uy)k ∴ ω·∇u =0

So there is no turbulence in ﬂatland — only ﬂatulence. 1 1

1 The special structure of 2D ﬂuid mechanics

u + v =0 u = v = x y ) y x

= v u = 2⇥ x y r

The curl of the momentum equation ⇣ + ⇣ ⇣ = ⌫ 2⇣ produces the 2D vorticity equation: t x y y x r = v u = 2⌅ 2D Conservationx y ⌥ laws = v u = 2⌅ x y ⌥ d Energy: 1 ⌅ 2 dxdy = ⇥ 2 dxdy ddt 2 | | ZZ 1 ⌅ 2 dxdy = ⇥ ZZ 2 dxdy dt 2 | | ZZ ZZ 2 vx uy = ⌅ d ⌥ Enstrophy: 1 2 dxdy = ⇥ 2 dxdy dt 2 | | ZZ ZZ v u = 2⌅ v x u y = ⌥2⌅ Take the limit ν→0 and observex that enstrophyy ⌥ is bounded by its initial value. Therefore energy is conserved in the limit ν→0.

According to the zeroth law, there is no turbulence in ﬂatland. d Spiegel suggests that 2D turbulence should instead be called 1 2 ﬂatulence. 2 2 u dx =vx uuy =f d⌅x ⇥ dx dt 2| | · ⌥ | | ZV ZV ZV

2 d 1 2 lim ⇥ dx =0 2 u d⇥ x0= u| |f dx ⇧ ⇥ dx dt 2| | ⌅ ZV · | | ZV ZV ZV 2 ⇤ ut + u u + p = µ u ⇥ " lim ⇥ · 2 dx=0 ⌥ # ⇥ 0 | | ⇧ ⌅ ZV 2 t + u = u + ⇥ ⇤ u ·+u u ·+ p =⌥µ 2u ⇥ t · ⌥ " ⌃gHH # R = 1010 ⇥water ⇤ + u = u + ⇥ 2 t · · ⌥ ˆ u = uˆı + v⇥ˆ =(⌃gHHvx u10y)k ) u =0 R = 10 · ⇥water ⇤ 1 1 nx=1024, dt=0.0025, ν=1e−12, β=0, μ=0.007, k =4, p=4 Text f ⇣ = + vorticity ζ at t=1200 xx yy 25 8

6 20 ☛Given the vorticity at t=0, 4

15 2 calculate the streamfunction y 0 and the velocity: u = kˆ × ∇ψ 10 −2

−4 5 −6 d 1 2 2 −8 ∇ 0 ☛Advect the vorticity2 | forψ| dax timedy = dt.−ν ζ dxdy 0 5 10 15 20 25 dt !! !! x Text Stream function ψ at t=1200 d 1 2 2 25 ☛Calculate the new ζstreamfunction.dxdy = −ν |∇ζ| dxdy 30 dt !! 2 !!

20 20

10 d 2 ′′ 15 ∇ 0 F (ζ)dxdy = −ν | ζ| F (ζ)dx y ☛Keep ind tmind!! that the vorticity!! 10 −10 cannot mix down to arbitrarily small −20 5 scales: this would violate 2 −30 vx − uy = ∇ ψ

0 conservation of energy. 0 5 10 15 20 25 x

2 vx − uy = ∇ ψ

d 1 2 2 2 |u| dx = u·f dx − ν |ω| d dt !V !V !V

lim ν |ω|2 dx =0̸ →0 ν !V g′ g′

2 1/5 νκL − ∴ δ νκL2 1/5= L Ra 1/5 (11) ∴ δ∼ g′ = L Ra−1/5 (11) Consider∼! an idealg′ " ﬂuid so that energy and enstrophy! areand both" conserved. Then (12) and (12) we can make a very′ plausible1/5 argument κ3Lg 2 κ that energyU isκ transferred3Lg′2 1/5= to κlargeRa2/ 5scales. (13) U ∼ ν2 =L Ra2/5 (13) ∼! ν2 " L ! " ∞ 1 2 2 ∞ 21 u 2+ v 2 = E(k, t)dk (14) ⟨ u + v ⟩ = 0 E(k, t)dk (14) 2 ∞ ⟨ ⟩ # 0 1 2 # ∞2 21 ζ 2 = k 2E(k, t)dk (15) ⟨ ζ ⟩ = 0 k E(k, t)dk (15) 2 ⟨ ⟩ # #0

E(k, t)=theenergyspectrum

4 4 The OBF argument

kE(k, t)dk ☛The mean wavenumber of the k¯(t)= energy spectrum is: E(k, t)dk R R (k k¯)2E(k, t)dk k2E(k, t)dk ☛The spectral width is: = k¯2 E(k, t)dk E(k, t)dk R R R R constant t =0

t>0 ☛If nonlinear interactions broaden ) an initially narrow spectrum, then the k, t (

mean wavenumber must decrease. E

k “The net tendency for the bulk of the energy to concentrate in the small wavenumbers means that ﬂuid elements with similarly signed vorticity must tend to group together; in no other way is it possible for the scale of the velocity distribution to increase. We expect therefore that from the original motion there will gradually emerge a few strong isolated vortices and that vortices of the same sign will continue to tend to group together....

Onsager (1949) has arrived at a similar conclusion about the tendency for a small number of strong isolated vortices to form.”

Batchelor, The Theory of Homogeneous Turbulence (1953) The Cray supercomputer came online in 1977 and ﬂatland was settled by Benzi, Fornberg, McWilliams, Santangelo..... ☛A random IC. Ck E(k, 0) = 1+(k/6)4 ☛Emergence of a “vortex gas”. ☛Merger, straining & stripping.

☛Energy is conserved throughout.

☛Nonuniform color scale to show ﬁlaments.

☛At intermediate times there is a scaling law 0.72

Benzi, Patarnello & Santangelo (1987) Santangelo & Patarnello Benzi, %(t) t ⇠ ☛The ﬁnal state is a dipole.

Briscolini & Santangelo (1991), McWilliams (JFM 1984 &1990) 1984 McWilliams(JFM (1991), BriscoliniSantangelo & (x, y, t) What did we just see?

8 4 + ⇥ ⇥ = 3.125 10 t x y y x ⇥ r ⇣ = xx + yy Ck E(k, 0) = with u2 + v2 =1 1+(k/6)4 h i

☛The ﬂow organizes into a dilute vortex gas. ☛Like signed vortices merge into fewer and larger vortices. ☛Mergers jettison ﬁlaments into the chaotic sea of small-scale low-level vorticity. ☛Energy is conserved throughout the evolution. The vortices of two-dimensional turbulence 365

In isolation, the vortices are t =5 impressively axisymmetric.

They can be identiﬁed and counted by a vortex-census algorithm.

30 J. C. Mc Williams

-5 )

y -10 ( s

-15 ) and x ( -20 McWilliams (JFM 1984 &1990) 1984 McWilliams(JFM

-25 -h 0 an FIGURE8. z and y cross-sections of 5 (solidx andand dashedy lines respectively) through a vorticity minimum at t = 40. The abscissa is distance from the centre, where the centre is chosen to give t = 20 best bulk agreement between the two profiles.

2n FIGURE1. &,y) at (a) t = 5 and (b)20. The contour interval is 10. Positive contours are solid, and negative contours are dashed. The zero contour is omitted.

C X FIQURE9. Log,,(gl at t = 16.5 with contours every 0.25 between -0.5 and 1.5. The initial emergence of vortices is not very well understood. But we can say something about the statistics of the emergent vortex gas.

The vortex census indicated that

⇠ (t) = vortices per area t with ⇠ 0.71 0.75 ⇠ ⇠ 1Scaling

Energy is important 1 = 1 ψ 2 dx Evolution of the vortexE (2populationπL)2 ! 2 |∇ | 1Scalingand enstrophy ⇠ 1 1 2 x (t) = vortices per area t with ⇠= 0.712 20ζ .75d 1ScalingEnergy is important⇠ Z ⇠(2πL) ! Batchelor’s predictions 1 1 2 x Energy is important = 2 2 ψ d Follow Batchelor, and assume that energy E (21 πL) ! |∇ | 1 ☛ ϱ 1 and 1 2 x is the only robustly conserved quantity, and =∼ t2 2 2 Zψ ∼dt2 and enstrophy E E(2πL) ! |∇ | use dimensional analysis. 1 1 2 2 x and enstrophy = 2 L 2 ζ d Z (2dim1πL)= ! 2 = E T 1 ζ2 dx Batchelor’s predictionsZ (2πL)2 ! 2 Batchelor’s predictions1 1 2 x The only dimensionally = 2 1 2 ψ d 1 (1) E (2ϱπL) !2 |∇ and| 2 consistent expressions are: ∼1 1t Z ∼1t = ϱ E 1 ψζanddx enstrophy (2) −(2∼πL)t22 ! 2 Z ∼ t2 1E = dx dx′ 1 ζ(x′)G(x′, x)ζ(x)(3) −1(2πL)2 1 2 2x = !2 !ψ d (1) E (2πL)2 ! 4|∇ | ☛ Or try an analogy with colloidal aggregation: ϱ1(t)ζexta(1t) 2 x (4) = ∼ 2 2 ψ d (1) 2E (2πL)1 ! |∇1 | x 1 ⇥˙(t)=Vorticity⇥ extrema= 2 ⇥2(ψζt) d (2) −(21πL) ! 1 ) ζext and⇠x t = 2 2 ψζ d (2) (2πL1) ′ 1 E ′ ′ =− ! dx dx ζ(x )G(x , x)ζ(x)(3) πL 2 2 −(21 def) !√x ! x′ 1 defx′ 1 x′ x x = 2ℓ 2= d4E dand 2 ζτ(= )G( , )ζ( )(3) −ϱ((2t)πζLext)a(!tζ)ext ! ζext (4) ∼ 2 4 ϱ(t)ζexta(t) 1 t (4) Vorticity extrema∼ ϱ(t)= F ℓ2 "τ # Vorticity extrema ζext and E a(t) =ζ typicalext and vortex radius E def √ def 1 2 ξ/2 ℓ =Γ(t) E ζextanda(t) τ t= def √ζext def ζ1ext ℓ = E∼ and τ∼= ζext ζext 1 t ϱ(t)= 1 F 1ℓ2 tτ ϱ(t)= F " # ℓ2 "τ # a(t) = typical vortex radius a(t) = typical vortex radius

2 ξ/2 Γ(t) ζexta(t) t 2 ξ/2 Γ(t) ∼ζexta(t) ∼t ∼ ∼ ∴ ϱ(t) t−ξ a(t) tξ/4 ∴ ϱ(t) ∼t−ξ ⇒ a(t) ∼tξ/4 ∼ ⇒ ∼ 1 1 1Scaling

Energy is important 1 = 1 ψ 2 dx The miserableE (2πL )failure2 ! 2 |of∇ “energy| and enstrophy scaling” means there must be other robustly conserved1 quantities. = 1 ζ2 dx Z (2πL)2 ! 2 Batchelor’s predictions ⇠ (t) = vortices per area t with ⇠ 0.71 0.75 1 ⇠ ⇠ 1 ϱ and ∼ t2 versus Z ∼ t2 E 1 ϱ ∼ t2 E L2 dim = E T 2

1 n n = ζ dx (1) Z (2πL)2 ! n 2 ϱζext a (2) ∼ − t ξ/2 (3) ∼

1 = 1 ψ 2 dx (4) E (2πL)2 ! 2 |∇ | 1 = 1 ψζ dx (5) −(2πL)2 ! 2 1 = dx dx′ 1 ζ(x′)G(x′, x)ζ(x)(6) −(2πL)2 ! ! 2 2 4 ϱ(t)ζexta(t) (7) ∼ Vorticity extrema ζext and E

1 1Scaling

Energy is important 1 E = 1 |∇ψ|2 dx 1Scaling (2πL)2 ! 2 and enstrophy Energy1Scaling is important 1 1 1 2 x Z = 2 1 2 ζ 2dx = !2 ψ d Energy is important π(2Lπ2L) E (21 ) ! |∇ | = 1 ψ 2 dx and enstrophy E (2πL)2 ! 2 |∇ | 1 1 2 and enstrophy = 1 ψ 2 x1 ζ dx E = 2 21|∇ 2| d 2 (1) (2πLZ) =! (2πL) !1 ζ2 dx Z (2πL)2 ! 2 1 1 x = − 2 2 ψζ d (2) 1 !1 2 (21πL) x = 2 1 2 ψ2 xd (1) E =(2πL)2 2 |∇ψ d| (1) E (2π1L) !! |x∇ | x′ 1 x′ x′ x x = d d 2 ζ( )G( , )ζ( )(3) 112 1 ==(2πL) ! 1 ψζ!ψζdxdx (2) (2) 22! 2 2 Conservation ϱ−of−t(2(2 ζπextrema2LL))a t! 4 ∼ ( )1 ext ( ) (4) 1 x x′ 1 ′ 1x′ ′x′ x ′ x = 2 d xd x2 ζ( )Gx( , x)ζ( x)(3)x in vortex= (2coresπL) 2! d ! d 2 ζ( )G( , )ζ( )(3) Vorticity extrema (2πL2) ! 4 ! ϱ(t)ζext2 a(t) 4 (4) There are two conserved quantities:∼ϱ(t)ζexta(tζ)ext and E (4) Vorticity extrema∼ Vorticity extrema ζext and E a(t) = typicalζext and vortex radius E Consequently there is a def √ def 1 Define some stuff ℓ = E and τ = length and a time: ζext ζext def √ def 1 ℓ = E and τ = a(t) = typicala(t) =ζ typical vortexext vortex radius radiusζext (5) Define somer( stut)ff = average separation1 t between vortices (6) Dimensional analysis only tells us that: ϱ(t)= 2 F Γ(ta)(t =) = typical typical vortex circulation radiusℓ "τ of# a vortex (5) (7) f(tr)(t =) = fraction average separation of plane between covered vortices by vortices (6) (8)

(Carnevale, McWilliams, Pomeau, Weiss & Young 1991) Young & Weiss Pomeau, McWilliams, (Carnevale, What canΓ(t) we = typicalado?(t) = circulation typical vortex of a vortex radius (7) Deﬁne some stuf(ﬀt) = fraction of plane covered by vortices (8)

a(t) = typical vortex radius (5) r(t) = average separation between vortices (6) Γ(t) = typical circulation of a vortex (7) f(t) = fraction of plane covered by vortices (8) 1

1 1 1Scaling

Energy is important 1 = 1 ψ 2 dx E (2πL)2 ! 2 |∇ | and enstrophy 1 = 1 ζ2 dx 1Scaling Z (2πL)2 ! 2

1 Energy is important = 1 ψ 2 dx (1) E (2πL)2 ! 2 |∇ | 1 1 2 x = 1 2 1 2 xψ d E = (2πL)2 2 ψζ|∇d | (2) −(2πL) !! and enstrophy 1 x x′ 1 x′ x′ x x = 2 d d 2 ζ( )G( , )ζ( )(3) (2πL) 1! ! 1 2 x = 2 42 2 ζ d ϱ(t)ζexta(t) (4) ∼Z (2πL) ! Vorticity extrema ζext and 1 1 2 x E = 2 ψ d (1) E1Scaling(2πL)2 ! |∇ | def √ def 1 1Scaling1 ℓ =1 E and τ = Energy= is important ψζζextdx ζext (2) πL 2 2 −(2 ) ! 1 1 2 x Energy is important =1 t2 2 ψ d 1 ϱ(t)=E1 ′ 1 2(2FπL′) ! |′∇ | = dx= dx ℓζ(x1")τGψ#(2xdx, x)ζ(x)(3) and− enstrophy(2πL)2 !E !(2πL)2 ! 2 |∇ | 1 1 2 x and enstrophyIntroduce:2 a(4t) = typical= vortex2 radius2 ζ d Comparison ϱ(t)ζexta(t) Z (2πL) ! (4) ∼ 1 1 2 x Define some stuff = 2 2 ζ d with DNS Z 1(2πL) ! Vorticity extrema = 1 ψ 2 dx (1) Nowa consider:(t) = typicalE (2 vortexπL)2 ! radius2 |∇ | (5) ζext and r(t) = average1 separation1 1 2 x1 betweenx vortices (6) = =2 2 ψ2 dE2 ψζ d (1) (2) ΓE(t) =(2 typicalπL) −! circulation(2π|∇L) | ! of a vortex (7) 1 1 f(t=)def = fraction√ = of1 planeψζ dxddef coveredx dx1′ 1 byζ(x vortices′)G(x′, x)ζ(x)(3)(2) (8) ℓ =−(2πEL)−2 and!(2π2L)2 τ! =! 2 1Scaling ζext 2 4 ζext 1 ϱ(t)ζxexta(xt)′ 1 x′ x′ x x (4) = ∼ 2 d d 2 ζ( )G( , )ζ( )(3) −(2πL) ! ! Energy isVorticity important extrema2 14 t ϱ(ϱt()ζtext)=a(t) 1F 1 2 (4) ∼ = 2 ζext andψ dx E (2ℓ πL)"2 !τ #2 |∇ | E Vorticity extrema(Assume that all of the1 energy is and enstrophy due to the vortices.)def √ def 1 ζextℓ = andE and τ = a(t) = typical vortex1 ext radiusE1 2 ext = ζ ζ dx ζ Energy is conserved:Z (2πL)2 ! 2 √ 1 t def ϱ(t)=def F1 Batchelor’s predictionsℓ−=ξ E and τ =2 ξ/4 ∴ ϱ(t) t ζext a(t)ℓ ζextt"τ # ∼ 1 ⇒ ∼ 1 ϱ a(t) =1and typicalt vortex radius Define some stuff ∼ ϱ(tt2)= F Z ∼ t2 E ℓ2 "τ # The circulation of a 2 ξ/2 a(t) = typical vortex radiusΓ(t) ζLext2 a(t) t (5) typical vortexa(t )is: = typicaldim vortex∼= radius∼ E T 2 r(t) = average separation∴ ϱ(t) betweent−ξ vorticesa(t) tξ/4 All statistical (6) properties of the vortex gas can Vorticity moments∴ −are:ξ ∼ ⇒ ξ/4 ∼ Γ(t)Define = typical some stu circulationϱff t of aa( vortext) t be expressed (7) in terms of the exponent �. ∼ ⇒1 n∼ n = ζ dx (1) Definef(t) some = fraction stuff a of(t)Z plane= typical(2 coveredπL vortex)2 ! radius by vortices (8)These (5) relations agree with DNS. n 2 r(t) = averageϱζext separationa between vortices(2) But (6) there is no way to predict �. a(t) = typical vortex∼ radius (5) −ξ/2 r t Γ(t) = typicalt circulation of a vortex(3) (7) ( ) = average∼ separation between vortices (6) Γ(t) = typicalf(t) = circulationfraction1 of plane of a vortex covered by vortices (7) (8) f(t) = fraction of plane covered by vortices (8) 1 = 1 ψ 2 dx 1 (4) E (2πL)2 ! 2 |∇ | 1 = 1 ψζ1dx (5) −(2πL)2 ! 2 1 = dx dx′ 1 ζ(x′)G(x′, x)ζ(x)(6) −(2πL)2 ! ! 2 2 4 ϱ(t)ζexta(t) (7) ∼ Vorticity extrema ζext and E

def √ def 1 ℓ = E and τ = ζext ζext

1 a As another test, we make a model 1 a based on “vortex patch” dynamics 3

a2 A vortex patch moves like a point vortex, except when they get too close to another like-signed patch.

17 vortex trajectories Trajectories of 17 in 2D turbulence point vortices R. BENZI et al.: ON THE STATISTICAL PROPERTIES OF TWO-DIMENSIONAL DECAYING TURBULENCE 815 )

CL) b) Fig. 3. - a) Trajectories of the centres of the 17 largest vortices (vortex area > 0.01); the plot has been obtained from the 512 x 512 simulation with the aid of a vortex recognition algorithm. The time interval plotted is 30 + 40. Thick (thin) lines are used for positive (negative) vortices. Circle sizes are proportional to vortex radii and segment lengths are proportional to velocities. b) The same as in a) but for the 17 point vortices, each one having the same total vorticity r,of the corresponding vortex of the high-resolution simulation. The plot has been obtained from the solution of eqs. (2) starting with the (Benzi, Paternello & Santangelo 1987 Santangelo & Paternello (Benzi, positions of the centres of the corresponding vortices of fig. 1. Note the striking correspondence of the trajectories of a) and b). This defines a field h(x, y) of the local eigenvalues of the trajectories A2= = (a2+i8x - (i32+/ax2)(a2+/3y2)which can be used to 6alyse the stability (see also [31 for a somehow different physical interpretation). It is found that our empirical definition of a vortex corresponds to a stable oscillatory behaviour, while instability eigenvalues are only distributed outside vortices. In particular, the largest instabilities (with eigenvalues - 30) are concentrated on the outer border of the vortices. The dynamical picture of a vortex is therefore given by a stable domain surrounded by a ring of large instability. In turn, this could be used to define a vortex. The complete dynamical picture of the turbulent system seems thus composed by a small and moderately unstable Hamiltonian system, perturbed by a continuous field, charac- terized by much smaller time scales. We concentrate now on the statistical analysis of the system of vortices. In fig. 4 we plot the integral histogram of the enstrophy of the vortices; the figure shows a clear power-law shape with a cut-off at large enstrophies. The corresponding differential distribution function, below the cut-off, can be approximated by cW - Qn;1.3WV, where 0, is the vortex enstrophy. Similar distribution functions are found also for vortex radii, vortex areas and vortex enstrophies; the power-law indices are: - 1.9 for the radii, - 1.5 for the areas and - 1.3 for the energies. This is consistent with roughly having the same mean vorticity value for all the vortices. The power-law shape of vortex probability functions suggests a link with the energy spectrum of the turbulent flow. Indeed, it is known that the most striking disagreement of the phenomenological theories with the simulations is the slope /3 of the power-law energy spectrum in the enstrophy cascading range; the theories of two-dimensional turbulence predicts /3 = - 3 for the energy spectrum [15,16] while numerical experiments [lo, 3,41 show significantly steeper spectra: p = - (4 + 6). We remark that our interest is in the long time The vortex-patch merger rule

Two circular vortex “patches” scrit =1.65(a1 + a2) merge at the critical separation

The “merged vortex” has: a1

⇣ext1 = ⇣ext2 = ⇣ext3 a3 4 4 4 a3 = a1 + a2 a This rule encodes energy 2 conservation (on average). The center of the new vortex is at the mid-point of the line joining the original vortices.

Merger results in lost vortex area. This is an

(Carnevale, McWilliams, Pomeau, Weiss & Young 1991) Young & Weiss Pomeau, McWilliams, (Carnevale, irreversible process occurring within the framework of the reversible system with ν=0. 1Scaling

Energy is important 1 = 1 ψ 2 dx The vortex-patch modelE (2πL)2 ! 2 |∇ | and enstrophy 1 = 1 ζ2 dx Z (2πL)2 ! 2 Move circular top-hat vortex patches, as Batchelor’sAt t=0 predictions there are 600 though they are point vortices vortex patches with equal radius and 300 of each sign.1 1 concentrated at the patch center. ϱ and ∼ t2 Z ∼ t2 E t3/4 ∼ 1 Close binary encounters result in ϱ ∼ t2 merger into a bigger patch. E L2 dim = E T 2

1 n x n = 2 ζ d (1) The crucial ingredient is that Z (2πL) ! n 2 ϱζext a (2) energy conservation is encoded ∼ t−ξ/2 (3) into the merger rule. ∼

1 = 1 ψ 2 dx (4) E (2πL)2 ! 2 |∇ | 1 = 1 ψζ dx (5) Vortex-patch statistics agree with the scaling−(2π Ltheory,)2 ! 2 and the exponent is close to that of 2D turbulence.1 ′ 1 ′ ′ = dx dx ζ(x )G(x , x)ζ(x)(6) −(2πL)2 ! ! 2 2 4 ϱ(t)ζexta(t) (7) ∼ Vorticity extrema ζext and E 1 Another self-similarTHE cascade END